text stringlengths 4 2.78M |
|---|
---
abstract: 'Interactions between the lattice and charge carriers can drive the formation of phases and ordering phenomena that give rise to conventional superconductivity, insulator-to-metal transitions, and charge-density waves. These couplings also play a determining role in properties that include electric and thermal conductivity. Ultrafast electron diffuse scattering (UEDS) has recently become a viable laboratory-scale tool to track energy flow into and within the lattice system across the entire Brillouin zone, and to deconvolve interactions in the time domain. Here, we present a detailed quantitative framework for the interpretation of UEDS signals, ultimately extracting the phonon mode occupancies across the entire Brillouin zone. These transient populations are then used to extract momentum- and mode-dependent electron-phonon and phonon-phonon coupling constants. Results of this analysis are presented for graphite, which provides complete information on the phonon-branch occupations and a determination of the $A_1''$ phonon mode-projected electron-phonon coupling strength $\langle g_{e,A_1''}^2 \rangle = \SI{0.035 \pm 0.001}{\square\electronvolt}$ that is in agreement with other measurement techniques and simulations.'
author:
- 'Laurent P. René de Cotret'
- 'Jan-Hendrik Pöhls'
- 'Mark J. Stern'
- 'Martin R. Otto'
- Mark Sutton
- 'Bradley J. Siwick'
title: |
Time- and momentum-resolved phonon population dynamics\
with ultrafast electron diffuse scattering
---
Introduction
============
Elementary excitations and their mutual couplings form the fundamental basis of our understanding of diverse phenomena in condensed matter systems. The interactions between collective excitations of the lattice system (phonons) and charge carriers, specifically, are known to lead to superconductivity, charge-density waves, multiferroicity, and soft-mode phase transitions [@eliashberg1960interactions; @hur2004electric]. These carrier-phonon interactions are also central to our understanding of electrical transport, heat transport, and energy conversion processes in photovoltaics and thermoelectrics [@Zhao2016]. Phonons can themselves be intimately mixed in to the very nature of more complex elementary excitations, as they are in polarons or polaritons, or intertwined with electronic, spin, or orbital degrees-of-freedom, as it now seems is the case for the emergent phases of many strongly-correlated systems that exhibit complex phase diagrams like high-$T_c$ superconductors [@Miyata2017; @kim2012ultrafast; @lanzara2001evidence].
Our inability to fully characterize the nature of elementary excitations and to quantify the strength of their momentum-dependent interactions has been one of the primary barriers to our understanding of these phenomena, particularly in complex anisotropic materials. Ultrafast pump-probe techniques provide an opportunity to study couplings between elementary excitations rather directly. Photoexcitation can prepare a non-equilibrium distribution of quasiparticles or other selected modes whose subsequent relaxation dynamics and coupling to other degrees of freedom can be followed in the time-domain. Under favourable circumstances, qualitatively distinct channels can be disentangled by their associated spectra (response functions) and time-scales. This field has evolved rapidly over the last decade, both from the perspective of the selectivity of the initial excitation and the ability to probe the subsequent dynamics over a broad range of frequencies. For example, spectroscopic pump-probe techniques in the terahertz range have been used to interrogate the link between electrons/holes and optical phonons in hybrid lead halide perovskites [@Lan2019] and to investigate the time-ordering of phenomena behind charge-density waves in titanium diselenide [@Porer2014].
The low-photon momentum associated with optical frequencies, however, prevents the most commonly applied optical photon-in, optical photon-out techniques from providing a full characterization of the wavevector-dependent interactions between elementary excitation. Time-resolved Raman and Brillouin spectroscopies, for example, are limited to the interrogation of zero-momentum (zone-center) phonons for this reason [@Tsen2009; @Yan2009; @Yang2017].
In recent years, non-optical ultrafast techniques have been developed to probe wavevector-dependent dynamics. The most mature of these approaches is time- and angle-resolved photoemission spectroscopy (trARPES), which has been used to assemble a complete picture of the dynamics of the electronic and spin excitations following the photoexcitation of materials [@Johannsen2013; @Gierz2015; @Stange2015; @Yang2017; @Rohde2018].
The ability to directly interrogate wavevector-dependent dynamics within the phonon system, on the other hand, is an extremely recent development. Ultrafast X-ray diffuse scattering [@Trigo2010; @Zhu2015; @Wall2018] is one technique that has the potential to reveal lattice excitation dynamics across the whole Brillouin zone. This approach leverages the remarkable brightness of the beams available from the current generation of X-ray free-electron laser facilities to measure the time dependence of the diffuse (phonon) scattering from materials following photoexcitation.
At the laboratory scale, there has been similar progress made in furthering ultrafast electron beam brightness which has enabled equivalent diffuse scattering experiments to be performed. Ultrafast electron diffuse scattering (UEDS) has the potential to be transformative in that it can provide both a wavevector-resolved view of the coupling between electron and lattice systems [@Harb2016; @Chase2016; @Waldecker2017; @Stern2018; @Konstantinova2018] and the wavevector dependence of the interactions within the phonon system itself. The large scattering cross-section of electrons, combined with the relative flatness of the Ewald sphere, potentially allows for the simultaneous measurement of both the average lattice structure (via Bragg scattering) and lattice excitation dynamics (via diffuse scattering) in specimens as thin as a single atomic layer.
In this work we provide a description of the signals contained in UEDS measurements and a comprehensive and broadly applicable computational method for UEDS data reduction based on density functional perturbation theory (DFPT). Specifically, we present a procedure to recover phonon population dynamics as a function of the phonon branch and wavevector, and a determination of wavevector-dependent (or mode-projected) electron-phonon coupling constants from those phonon population measurements. This method uses only the measured time-resolved UEDS patterns and DFPT determinations of the phonon polarization vectors as inputs. The application of this approach to the case of photodoped carriers in the Dirac cones of thin graphite is demonstrated. The electron-phonon coupling strength to the strongly-coupled $A_1'$ phonon at the ${\boldsymbol{\mathrm{K}}}$-point of the Brillouin zone, and the nonequilibrium optical and acoustic phonon branch populations as a function of time following excitation across the whole Brillouin zone are all determined from the UEDS measurements.
Experimental and Computational methods
======================================
The change in experimental electron scattering intensity of graphite, photoexcited with pulses of light at a fluence of , are presented in Figure \[FIG:graphite\] for a few representative time-delays. This section provides details on the experimental parameters, data processing steps, and computational techniques that are used in this work.
![Schematic diagram of ultrafast electron diffuse scattering experiments. Samples are photoexcited with ultrashort pulses of light at time $t=t_0$. After some delay $\tau$, an ultrashort electron pulse scatters through the sample. Elastic and inelastic signals are collected in transmission geometry. By scanning time-delays $\tau$, a stroboscopic movie of the dynamics can be assembled. This figure also shows the example of a scattering vector ${\boldsymbol{\mathrm{q}}}$, and its associated reduced wavevector ${\boldsymbol{\mathrm{k}}}$, which are related by the nearest Bragg reflection ${\boldsymbol{\mathrm{H}}}$.[]{data-label="FIG:intro"}](intro_to_ued.pdf){width="3.38in"}
Data acquisition
----------------
UEDS measurements are pump-probe experiments in which an ultrafast laser pulse is used to photoexcite a thin single-crystal specimen at $t=t_0$, followed by probing the specimen with an ultrafast electron pulse at $t=t_0 + \tau$, resulting in the acquisition of a transmission electron scattering pattern. By scanning across time-delays $\tau$, the dynamics in the – ($10^{-15} \textrm{--} 10^{-9}$ ) range can be recorded. UEDS data can be acquired coincidentally during ultrafast electron diffraction (UED) experiments with state-of-the-art detection cameras, although UEDS intensities are empirically $10^4$ to $10^6$ times less intense than those of Bragg diffraction. UEDS measurements are inherently statistical. Pump-probe experiments sample many decay processes, incoherently in time. Hence, all possible decay channels are represented, proportionally to their statistical likelihood.
Scattering measurements presented in this work use bunches of $10^7$ electrons, accelerated to , at a repetition rate of . A radio-frequency cavity is used to compress electron bunches to $\approx \SI{150}{\femto\second}$ at the sample, as measured by a home-built photoactivated streak camera [@kassier2010compact]. More detailed descriptions of this instrument are given elsewhere [@Chatelain2012; @Morrison2013; @Otto2017]. Analysis of static diffraction patterns indicate a momentum resolution of , while the range of visible reflections is consistent with a real-space resolution of $<\SI{1}{\angstrom}$. pump laser pulses of () light are used to photoexcite a single-crystal flake of freestanding single crystal natural graphite, provided by Naturally Graphite. The flakes were mechanically exfoliated to a thickness of .
The interrogated film area covers , with a pump spot of full-width half-max (FWHM) ensuring nearly uniform illumination of the probed volume. The film was pumped at a fluence of , resulting in an absorbed energy density of . The scattering patterns are collected with a Gatan Ultrascan 895 camera; a phosphor screen fiber-coupled to a charge-coupled detector (CCD) placed away from the sample. The experiment herein consists of time-delays in the range of . Per-pixel scattering intensity fluctuations over laboratory time reveals a transient dynamic range of $1:10^8$, allowing the acquisition of diffraction patterns and diffuse scattering patterns simultaneously [^1].
{width="100.00000%"}
Processing and corrections {#SEC:corrections}
--------------------------
Scattering intensity patterns are inherently redundant due to the point-group symmetry of the scattering crystal. When this symmetry is not broken by photoexcitation or the dynamical phenomena itself, it is possible to use this redundancy to enhance the signal to noise ratio of a UEDS data set. No dynamical phenomena breaking point-group symmetry was observable within the raw signal to noise of the current measurements, so the measured patterns have been subject to a six-fold discrete azimuthal average based on the $D_{6h}$ point-group of graphite. This discrete rotational average effectively increases the signal-to-noise ratio of this data set by a factor of $\sqrt{6}$ and is therefore an effective data processing step given the weakness of the diffuse scattering signals. “Scattering intensities" is henceforth implied to mean six-fold averaged scattering intensities. Scattering from a few samples with varying thicknesses () was acquired. There was no quantitative difference in the Bragg scattering dynamics, indicating that scattering from these samples is kinematical to a good approximation. The expected effects of multiple scattering on the diffuse scattering intensity distribution will be discussed further below.
Computational methods
---------------------
Structure relaxation was performed using the plane-wave self-consistent field program `PWscf` from the Quantum ESPRESSO software suite [@Giannozzi2017]. The graphite structure was fully relaxed using a ${\boldsymbol{\mathrm{k}}}$-point mesh centered at ${\boldsymbol{\mathrm{\Gamma}}}$ and force (energy) threshold of (). The dynamical matrices were computed on ${\boldsymbol{\mathrm{q}}}$-point grid using a self-consistency threshold of . The resulting graphite structure has the following lattice vectors: $${\boldsymbol{\mathrm{a}}}_1 = a ~ {\boldsymbol{\mathrm{e}}}_1, ~~~~
{\boldsymbol{\mathrm{a}}}_2 = \frac{a}{2} \left( \sqrt{3} ~ {\boldsymbol{\mathrm{e}}}_2 - {\boldsymbol{\mathrm{e}}}_1 \right), ~~~~
{\boldsymbol{\mathrm{a}}}_3 = c ~ {\boldsymbol{\mathrm{e}}}_3,$$ where $a=\SI{2.462}{\angstrom}$, $c=\SI{6.837}{\angstrom}$, and $\left\{ {\boldsymbol{\mathrm{e}}}_i \right\}$ are the usual Euclidean vectors. Graphite has four atoms in the unit cell, with two groups of two atoms forming lattices rotated with respect to each other. This structure respects the symmetries of the $D_{6h}$ point group [^2].
The phonon frequencies $\left\{ \omega_{j,{\boldsymbol{\mathrm{k}}}} \right\}$ and polarization vectors $\left\{ {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$ were computed using the `PHonon` program in the Quantum ESPRESSO software suite, using the B86b exchange-coupled Perdew-Burke-Ernzerhof (B86bPBE) generalized gradient approximation (GGA) and the projector augmented-wave (PAW) method [@Becke1986; @Perdew1996; @Blochl1994]. The cutoff-energy of the wavefunction was set to , while the cutoff energy for the charge density was set to , and a Fermi-Dirac smearing of was applied. To include the dispersion energy between the carbon layers, the exchange-hole dipole moment (XDM) method was used [@Becke2007].
Theory
======
Similar to X-ray scattering, under the kinematical approximation the measurement of the total scattering intensity at scattering vector ${\boldsymbol{\mathrm{q}}}$ and time $t$, $I({\boldsymbol{\mathrm{q}}}, t)$, of an electron bunch interacting with a thin film of crystalline material, can be separated as follows: $$I({\boldsymbol{\mathrm{q}}}, t) = I_0({\boldsymbol{\mathrm{q}}}, t) + I_1({\boldsymbol{\mathrm{q}}}, t) + ... ~ ,$$ where the intensity $I_n$ represents the scattered intensity of an electron that interacted with $n$ phonons. Specifically, $I_0$ represents diffraction, or Bragg scattering, and $I_1$ represents the first-order *diffuse scattering intensity*. The experimentally-observed ratio $I_0/I_1$ ranges between $10^4$ – $10^6$ [^3]. Higher-order terms have much smaller cross-sections, hence much lower contribution to scattering intensity, and are therefore ignored in this work. The expressions for the intensities $I_0$ and $I_1$ are given below: $$\begin{aligned}
I_0({\boldsymbol{\mathrm{q}}}, t) &= N_c I_e \left| \sum_s f_s({\boldsymbol{\mathrm{q}}})~e^{-W_s({\boldsymbol{\mathrm{q}}}, t)}~e^{-i [{\boldsymbol{\mathrm{q}}} \cdot {\boldsymbol{\mathrm{R}}}_s(t)]} \right|^2 \\
I_1({\boldsymbol{\mathrm{q}}}, t) &= N_c I_e \sum_j \frac{n_{j,{\boldsymbol{\mathrm{k}}}}(t) + 1/2}{\omega_{j,{\boldsymbol{\mathrm{k}}}}(t)} |F_{1j}({\boldsymbol{\mathrm{q}}},t)|^2 \label{EQ:diffuse}\end{aligned}$$ where $N_c$ is the number of diffracting cells, $I_e$ is the intensity of scattering from a single event, ${\boldsymbol{\mathrm{q}}}$ is the wavevector (or scattering vector), ${\boldsymbol{\mathrm{k}}} = {\boldsymbol{\mathrm{q}}} - {\boldsymbol{\mathrm{H}}}$ is the reduced wavevector associated to ${\boldsymbol{\mathrm{q}}}$ with respect to the nearest Bragg reflection ${\boldsymbol{\mathrm{H}}}$ (see Figure \[FIG:intro\]), $s$ are indices associated with atoms in the crystal unit cell, ${\boldsymbol{\mathrm{R}}}_s(t)$ is the real-space atomic position of atom $s$, $W_s({\boldsymbol{\mathrm{q}}}, t)$ is the Debye-Waller factor of atom $s$, $f_s({\boldsymbol{\mathrm{q}}})$ are the atomic form factors, $j \in \{1, 2, ..., N \}$ runs over phonon modes, and $\left\{ n_{j,{\boldsymbol{\mathrm{k}}}}(t) \right\}$ and $\left\{ \omega_{j,{\boldsymbol{\mathrm{k}}}}(t) \right\}$ are the population and frequency associated with phonon mode $j$, respectively [@Wang1995; @Xu2005].
The diffuse scattering intensity contribution of each mode $j$ is weighted by a factor, called the *one-phonon structure factor* $|F_{1j}({\boldsymbol{\mathrm{q}}},t)|^2$: $$|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2 = \left|\sum_s e^{-W_s({\boldsymbol{\mathrm{q}}}, t)} \frac{f_s({\boldsymbol{\mathrm{q}}})}{\sqrt{\mu_s}} ({\boldsymbol{\mathrm{q}}} \cdot {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}})\right|^2
\label{EQ:oneph}$$ where $\mu_s$ is the mass of atom $s$, and $\left\{ {\boldsymbol{\mathrm{e}}}_{j, s,{\boldsymbol{\mathrm{k}}}} \right\}$ are the wavevector-dependent polarization vectors associated with phonon mode $j$ for atom $s$. The one-phonon structure factors represent the contribution of phonon mode $j$ on the overall intensity at a specific scattering vector ${\boldsymbol{\mathrm{q}}}$ and time $t$. $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2$ are a measure of two things: the locations in Brillouin zone where the phonon mode polarization vectors $\left\{ {\boldsymbol{\mathrm{e}}}_{j, s,{\boldsymbol{\mathrm{k}}}} \right\}$ are aligned in such a way that they will contribute to diffuse scattering intensity on the detector, expressed via the terms $\left\{ {\boldsymbol{\mathrm{q}}} \cdot {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$; and the strength of the contribution of a single scattering event, including the effect of the instantaneous disorder in the material, expressed via the quantities $\left\{ e^{-W_s({\boldsymbol{\mathrm{q}}}, t)} f_s({\boldsymbol{\mathrm{q}}})/\sqrt{\mu_s} \right\}$.
Instantaneous disorder is described by the Debye-Waller factors $\left\{ W_s({\boldsymbol{\mathrm{q}}}, t) \right\}$. The general expression of the *anisotropic* Debye-Waller factor for atom $s$, $W_s({\boldsymbol{\mathrm{q}}}, t)$, is given below: $$W_s({\boldsymbol{\mathrm{q}}}, t) = \frac{1}{4\mu_s} \sum_{j, {\boldsymbol{\mathrm{k}}}} |a_{j, {\boldsymbol{\mathrm{k}}}}(t)|^2 ~|{\boldsymbol{\mathrm{q}}} \cdot {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}}|^2
\label{EQ:Debye_Waller}$$ where $a_{j, {\boldsymbol{\mathrm{k}}}}(t)$ is the phonon mode vibration amplitude for mode $j$ at reduced wavevector ${\boldsymbol{\mathrm{k}}}$ [@Xu2005]: $$|a_{j, {\boldsymbol{\mathrm{k}}}}(t)|^2 = \frac{2 \hbar}{N_c \omega_{j, {\boldsymbol{\mathrm{k}}}}(t)} \left(n_{j,{\boldsymbol{\mathrm{k}}}}(t) + \frac{1}{2}\right).$$ Debye-Waller factors describe the reduction of intensity at scattering vector ${\boldsymbol{\mathrm{q}}}$ due to the effective deformation of a single atom’s scattering potential that results from the collective lattice vibrations in *all* phonon modes. Wavevector-specific information is in general impossible to extract from the transient changes to the Debye-Waller factors that result from photoexcitation and the non-equlibrium phonon populations that such excitation produces.
The expression for $I_1({\boldsymbol{\mathrm{q}}},t)$ in Equation (and related quantities) apply rigorously under single electron scattering conditions. The most probable type of multiple scattering event affecting $I_1({\boldsymbol{\mathrm{q}}}, t)$ is diffuse scattering followed by secondary Bragg scattering, not multiple or consecutive diffuse scattering events [@Cowley1979; @Wang1995]. A secondary Bragg scattering event only changes the electron wavevector by a reciprocal lattice vector; thus, this type of multiple scattering results in a redistribution of diffuse intensity from lower-order to higher-order Brillouin zones (further from $|{\boldsymbol{\mathrm{q}}}|=0$). However, the wavevector dependence of experimental diffuse intensities is not strongly influenced, even under experimental conditions where such dynamical effects are important. The strength of the scattering selection rules implied by the $\left\{ {\boldsymbol{\mathrm{q}}} \cdot {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$ terms, and described further below, are reduced as the proportion of multiple scattering increases.
Results and Discussion
======================
In this section a comprehensive procedure for ultrafast electron diffuse scattering data reduction will be presented. This approach recovers the time- and wavevector-dependent phonon population dynamics in each of the phonon branches. In addition, the population dynamics of the $A_1'$ phonon, a strongly-coupled optical phonon in graphite, is used to demonstrate the extraction of wavevector-dependent (mode-projected) electron-phonon coupling constants from ultrafast electron diffuse intensities.
Calculation of phonon polarization vectors and eigenfrequencies
---------------------------------------------------------------
The quantitative connection between the observed diffuse intensity $I_1({\boldsymbol{\mathrm{q}}},t)$ and the phonon populations, $\left\{ n_{j,{\boldsymbol{\mathrm{k}}}} \right\}$, is provided by the one-phonon structure factors, $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2$. A determination of $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2$ requires phonon polarization vectors $\left\{ {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$ and associated frequencies $\left\{ \omega_{j,{\boldsymbol{\mathrm{k}}}} \right\}$. Density functional perturbation theory (DFPT) is a widely used, readily-available method to compute these phonon properties.
The separation of frequencies and polarization vectors into modes is key to the calculation of one-phonon structure factors $|F_{1j}({\boldsymbol{\mathrm{q}}},t)|^2$. The phonon frequencies $\left\{ \omega_{j,{\boldsymbol{\mathrm{k}}}} \right\}$ and polarization vectors $\left\{ {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$ were computed independently at every ${\boldsymbol{\mathrm{k}}}$; however, diagonalization routines have no way of clustering eigenvalues and eigenvectors into physically-relevant groups (i.e. phonon branches). Association between atomic motions (given by polarization vectors) and a particular mode is only well-defined near the ${\boldsymbol{\mathrm{\Gamma}}}$-point [@Paulatto2013]. Clustering of phonon properties into phonon branches is described in detail in Appendix \[AP:mode\_clustering\].
The polarization vectors and frequencies, calculated for irreducible ${\boldsymbol{\mathrm{k}}}$-points [^4], were extended over the entire reciprocal space based on crystal symmetries using the `crystals` software package [@RenedeCotret2018].
Debye-Waller calculation {#SEC:debye_waller}
------------------------
A key component of the computation of one-phonon structure factors $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2$ is the calculation of the Debye-Waller factors $\left\{ W_s({\boldsymbol{\mathrm{q}}}, t) \right\}$, representing the instantaneous disorder of the material. Based on Equation , terms of the form $\left\{ W_s({\boldsymbol{\mathrm{q}}},t) \right\}$ are not sensitive reporters on the wavevector dependence of nonequilibrium phonon distributions because their value at every ${\boldsymbol{\mathrm{q}}}$ involves a sum of all mode, at every reduced wavevector ${\boldsymbol{\mathrm{k}}}$. Phonon population dynamics can only affect the magnitude of the Debye-Waller factors. The potential time dependence of the Debye-Waller factors was investigated, via the time dependence of mode populations $\left\{ n_{j,{\boldsymbol{\mathrm{k}}}} \right\}$. Profoundly non-equilibrium distributions of phonon modes were simulated, with all modes populated equivalently to a temperature of except one mode at high temperature [^5]. These extreme non-equilibrium distributions increased the value the terms $\sum_s W_s({\boldsymbol{\mathrm{q}}},t)$ by at most for optical modes, and for acoustic modes. Since these fractional changes are constant across ${\boldsymbol{\mathrm{q}}}$, wavevector-dependent changes in UEDS signals are not impacted significantly by transient changes to the Debye-Waller factors and any time dependence of the one-phonon structure factors $|F_{1j}({\boldsymbol{\mathrm{q}}},t)|^2$ that result from the Debye-Waller factors themselves can be ignored to a good approximation.
One-phonon structure factors
----------------------------
Using the computed Debye-Waller factors from the previous section, the calculation of $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2$ was carried out, from Equation , for the eight in-plane phonon modes of graphite: the longitudinal modes LA, LO1 – LO3, and the transverse modes TA, TO1 – TO3 [^6]. The resulting one-phonon structure factors of a few in-plane modes, with occupations equivalent to a temperature of , are shown in Figure \[FIG:phonon\_struct\_factor\]. One-phonon structure factors $|F_{1j}({\boldsymbol{\mathrm{q}}},t)|^2$ display striking scattering vector dependence (selection rules) based on the nature of phonon polarization vectors $\left\{ {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$. Specifically, near ${\boldsymbol{\mathrm{\Gamma}}}$, the one-phonon structure factor for longitudinal modes is highest in the radial direction, because the polarization of those modes is parallel to ${\boldsymbol{\mathrm{q}}}$. On the other hand, $|F_{1j}({\boldsymbol{\mathrm{q}}},t)|^2$ for transverse modes is highest (near ${\boldsymbol{\mathrm{\Gamma}}}$) in the azimuthal direction for transverse modes, because the polarization of those modes is perpendicular to ${\boldsymbol{\mathrm{q}}}$.
![Calculated one-phonon structure factors $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2$ of selected in-plane modes of graphite, at ($t=t_0$), for ${\boldsymbol{\mathrm{q}}}$ vectors equivalent to the detector area shown in Figure \[FIG:graphite\]. Bright spots indicate locations in reciprocal space where the associated mode contributes strongly to the diffuse scattering intensity.Brillouin zone outlines are overlaid, and their centers (Bragg peaks) are marked with a white dot. While one would expect a wavevector-dependent behavior $|F_{1j}({\boldsymbol{\mathrm{q}}},t)|^2 \propto |{\boldsymbol{\mathrm{q}}}|^2$ from Equation , the gaussian nature of the Debye-Waller factor terms $\left\{ W_s({\boldsymbol{\mathrm{q}}},t) \right\}$ and of the atomic form factor terms $\left\{ f_s({\boldsymbol{\mathrm{q}}})\right\}$ decrease the amplitude of the one-phonon structure factors at larger ${\boldsymbol{\mathrm{q}}}$.[]{data-label="FIG:phonon_struct_factor"}](oneph.pdf){width="1\columnwidth"}
An alternative view of one-phonon structure factors is presented via weighted dispersion curves, an example of which is shown in Figure \[FIG:weighted\_dispersion\]. This presentation allows easy comparison of the relative weights of the one-phonon structure factors along high-symmetry lines for different phonon branches.
{width="100.00000%"}
A cursory inspection of the weighted dispersion curves in Figure \[FIG:weighted\_dispersion\] suggest that there are regions in the Brillouin zone where diffuse intensity is strongly biased towards a single mode (strong scattering selection rule) based on the relative intensities of one-phonon structure factors. Careful analysis reveals that there are very few wavevectors ${\boldsymbol{\mathrm{q}}}$ for which a particular phonon mode’s one-phonon structure factor is strongly dominant. Figure \[FIG:UEDS\_majority\] presents a comparison of the relative intensities of one-phonon structure factors weighted by phonon frequency, which is indicative of mode population as per Equation . Only $37\%$ of wavevectors visible in measurements shown in Figure \[FIG:graphite\] have a mode that contributes over $50\%$ of the quantity $\sum_j |F_{1j}({\boldsymbol{\mathrm{q}}})|^2/\omega_{j,{\boldsymbol{\mathrm{k}}}}$; only $1\%$ of wavevectors have a phonon mode that contributes more than $75\%$.
![Reciprocal space locations where $|F_{1j}({\boldsymbol{\mathrm{q}}}, t_0)|^2/\omega_{j, {\boldsymbol{\mathrm{k}}}}(t_0)$ is dominated by one mode $j$, for ${\boldsymbol{\mathrm{q}}}$ vectors equivalent to the detector area shown in Figure \[FIG:graphite\]. Modes other than those shown here (e.g. TO2) are not dominant anywhere. **a)** $|F_{1j}({\boldsymbol{\mathrm{q}}}, t_0)|^2/\omega_{j, {\boldsymbol{\mathrm{k}}}}(t_0)$ is dominated ($>50\%$) by one mode $j$. White ($\varnothing$) regions account for $63\%$ of the wavevectors, where no phonon mode is dominant. Bragg peaks have been marked by black dots. **b)**: locations where $|F_{1j}({\boldsymbol{\mathrm{q}}}, t_0)|^2/\omega_{j, {\boldsymbol{\mathrm{k}}}}(t_0)$ is dominated ($>75\%$) by one mode $j$. White ($\varnothing$) regions account for $99\%$ of the wavevectors where no mode reaches the $75\%$ threshold.[]{data-label="FIG:UEDS_majority"}](majority.pdf){width="1\columnwidth"}
The results of Figure \[FIG:UEDS\_majority\] show that quantitative answers regarding phonon dynamics from UEDS measurements cannot generally be obtained by inspection; at almost any wavevector ${\boldsymbol{\mathrm{q}}}$, at least two phonon modes contribute significantly to the transient diffuse scattering intensity. Therefore, a more robust procedure, presented in the next section, must be employed to extract wavevector- anndmode-dependent phonon populations from UEDS intensities.
Population dynamics across the Brillouin zone
---------------------------------------------
Transient electron diffuse intensity has been used elsewhere [@Chase2016; @Waldecker2017; @Stern2018; @Konstantinova2018] as an approximation to the population dynamics of particular modes. However, one can extract the transient wavevector-dependent phonon population dynamics $\left\{ \Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t) \right\}$ by combining the measurements of $\Delta I({\boldsymbol{\mathrm{q}}},t)$ with the calculations of one-phonon structure factors and associated quantities presented above.
For many materials (including graphite), the temperature dependence (and hence time dependence) of the phonon mode vibrational frequencies is negligible, because such dependence is proportional to anharmonic couplings between branches [@Calizo2007; @Judek2015]; hence, $\omega_{j, {\boldsymbol{\mathrm{k}}}}(t) \equiv \omega_{j, {\boldsymbol{\mathrm{k}}}}(t_0)$. Moreover, as is discussed in Section \[SEC:debye\_waller\], the temperature dependence (and hence time dependence) of the Debye-Waller factors — and therefore the one-phonon structure factors — has a much smaller magnitude that the variations due to other terms in Equation . Therefore, we have $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|^2 \equiv |F_{1j}({\boldsymbol{\mathrm{q}}}, t_0)|^2$ for all times. In this case, transient scattering intensity at the detector, $\Delta I({\boldsymbol{\mathrm{q}}},t)$, can be expressed as follows: $$\frac{\Delta I({\boldsymbol{\mathrm{q}}},t)}{N_c I_e} = \sum_j \frac{\Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t)}{\omega_{j,{\boldsymbol{\mathrm{k}}}}(t_0)} |F_{1j}({\boldsymbol{\mathrm{q}}}, t_0)|^2
\label{EQ:decomposition}$$ for ${\boldsymbol{\mathrm{k}}}$ away from ${\boldsymbol{\mathrm{\Gamma}}}$, where there might be interference with elastic scattering signals.
At every reduced wavevector ${\boldsymbol{\mathrm{k}}}$ and time-delay $t$, there are $N$ different values $\left\{ \Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t) \right\}$ that must be determined — one for each phonon mode. Since the one-phonon structure factors $|F_{1j}({\boldsymbol{\mathrm{q}}}, t_0)|^2$ vary over the total wavevector ${\boldsymbol{\mathrm{q}}}$, the transient diffuse intensity for at least $N$ Brillouin zones must be considered so that Equation can be solved numerically. A linear system of equations must be solved at every reduced wavevector ${\boldsymbol{\mathrm{k}}}$.
Let $\{ {\boldsymbol{\mathrm{H}}}_1, ..., {\boldsymbol{\mathrm{H}}}_M \mid M \geq N \}$ be the chosen reflections from which to build the system of equations. Then, the transient phonon population of mode $j$ at every ${\boldsymbol{\mathrm{k}}}$ and time $t$, $\Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t)$, solves the linear system: $${\boldsymbol{\mathrm{I}}}_{{\boldsymbol{\mathrm{k}}}}(t) = {\boldsymbol{\mathrm{F}}}_{{\boldsymbol{\mathrm{k}}}} ~ {\boldsymbol{\mathrm{n}}}_{{\boldsymbol{\mathrm{k}}}}(t)
\label{EQ:decomp}$$ where $$\begin{aligned}
{\boldsymbol{\mathrm{I}}}_{{\boldsymbol{\mathrm{k}}}}(t) &=
\frac{1}{N_c I_e}
\begin{bmatrix}
\Delta I({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{H}}}_1, t) & \dots & \Delta I({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{H}}}_M, t)
\end{bmatrix}^T \\
{\boldsymbol{\mathrm{n}}}_{{\boldsymbol{\mathrm{k}}}}(t) &=
\begin{bmatrix}
\Delta n_{1, {\boldsymbol{\mathrm{k}}}}(t)/\omega_{1,{\boldsymbol{\mathrm{k}}}}(t_0) & \dots & \Delta n_{N, {\boldsymbol{\mathrm{k}}}}(t)/\omega_{N,{\boldsymbol{\mathrm{k}}}}(t_0)
\end{bmatrix}^T \\
{\boldsymbol{\mathrm{F}}}_{{\boldsymbol{\mathrm{k}}}} &= \nonumber \\
&\begin{bmatrix}
|F_{11}({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{H}}}_1, t_0)|^2 & \dots & |F_{1N}({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{H}}}_1,t_0)|^2 \\
\vdots & \ddots & \vdots \\
|F_{11}({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{H}}}_M, t_0)|^2 & \dots & |F_{1N}({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{H}}}_M,t_0)|^2
\end{bmatrix}\end{aligned}$$ for ${\boldsymbol{\mathrm{k}}}$ vectors away from the ${\boldsymbol{\mathrm{\Gamma}}}$ point. These linear systems of equations can be solved numerically, provided enough experimental data ($M \geq N$).
The choice to solve for ${\boldsymbol{\mathrm{n}}}_{{\boldsymbol{\mathrm{k}}}}(t)$, rather than for the change in population, is related to the degree of confidence that should be placed in the calculation of phonon polarization vectors and frequencies. The phonon polarization vectors are mostly affected by the symmetries of the crystal. On the other hand, phonon vibrational frequencies might be influenced by non-equilibrium carrier distributions. Solving for the ratio of populations to frequencies, rather than populations, is more robust against the uncertainty in the modelling, because the one-phonon structure factors only take into account the polarization vectors.
We also note that the procedure presented above can be easily extended to (equilibrium) thermal diffuse scattering measurements, where the phonon populations are known at constant temperature, but the phonon vibrational spectrum is unknown. Therefore, using pre-photoexcitation data of a time-resolved experiment, one could infer the phonon vibrational frequencies, which are then used to determine the change in populations using the measurements after photoexcitation. This scheme only relies on the determination of phonon polarization vectors.
### Wavevector-dependent phonon population dynamics in graphite {#wavevector-dependent-phonon-population-dynamics-in-graphite .unnumbered}
We applied this general formalism for wavevector-dependent phonon population decomposition to the transient diffuse intensity patterns of photoexcited graphite shown in Figure \[FIG:graphite\]. Since the diffraction patterns have been symmetrized, one would expect that using intensity data for reflections related by symmetry would be redundant. However, better results were achieved by using the entire area of the detector. We expect this is due to minute misalignment of the diffraction patterns and uncertainty in detector position which are averaged out when using all available data. The Brillouin zones associated with all in-plane Bragg reflections ${\boldsymbol{\mathrm{H}}}$ such that $|{\boldsymbol{\mathrm{H}}}| \leq \SI{12}{\per\angstrom}$ were used, for a total of fourty-four Brillouin zones ($M=44$), many more than the minimum required for the eight in-plane phonon modes of graphite ($N=8$). The physical constraint that $\Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t) > 0 ~ \forall t$ was applied [^7] via the use of a non-negative approximate matrix inversion approach [@NNLS] to solve Equation at every reduced wavevector ${\boldsymbol{\mathrm{k}}}$ and time-delay $t$. Stable solutions were found for reciprocal space points where $|{\boldsymbol{\mathrm{k}}}| > \SI{0.45}{\per\angstrom}$, where there is no interference between elastic $I_0({\boldsymbol{\mathrm{q}}},t)$ and diffuse $I_1({\boldsymbol{\mathrm{q}}},t)$ signals. Figure \[FIG:populations\] presents the direct decomposition of diffuse intensity into wavevector-dependent, transient phonon population changes $\left\{ \Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t) \right\}$, for a few in-plane modes that are particularly relevant to graphite. The discussion of physical processes that explain the wavevector-dependent transient phonon populations follows.
In graphite, optical excitation creates a nonthermal phonon distribution, increasing population primarily in two strongly-coupled optical phonons (SCOP): $A_1'$, located near the ${\boldsymbol{\mathrm{K}}}$ point, and $E_{2g}$, at ${\boldsymbol{\mathrm{\Gamma}}}$ [@Kampfrath2005]. Dynamics measured at the earliest time scales ($<\SI{5}{\pico\second}$) are discussed qualitatively in a previous publication [@Stern2018]. Using the measured population dynamics of Figure \[FIG:populations\], we can track the transfer of energy across the Brillouin zone quantitatively.\
By conservation of both momentum and energy, the anharmonic decay of the two SCOP transfers population into mid-Brillouin zone acoustic modes. The early time-points presented in Figure \[FIG:populations\] confirm that quickly after photoexcitation ($<\SI{500}{\femto\second}$), the transverse optical mode TO2 is strongly populated at ${\boldsymbol{\mathrm{K}}}$, indicative of the expected strong electron-phonon coupling to the $A_1'$ phonon. The transfer of energy away from the TO2 mode is already well underway at , associated with an increase in acoustic modes along the ${\boldsymbol{\mathrm{\Gamma}}}$ – ${\boldsymbol{\mathrm{M}}}$ line. This is in accordance with the phonon band structure, where the mid-point along the ${\boldsymbol{\mathrm{\Gamma}}}$ – ${\boldsymbol{\mathrm{M}}}$ line favors occupancy of the TA mode (Figure \[FIG:weighted\_dispersion\]). This behaviour intensifies from to .
The initial increase ($<\SI{5}{\pico\second}$) of TA population along the ${\boldsymbol{\mathrm{\Gamma}}}$ – ${\boldsymbol{\mathrm{M}}}$ line is in excellent agreement with predicted anharmonic decay probabilities from the $E_{2g}$ phonon [@Bonini2007]. The (small) increase at in LA population at $\tfrac{1}{2}{\boldsymbol{\mathrm{K}}}$ is also in line with calculated decay probabilities from the $E_{2g}$ phonon by anharmonic coupling [^8].
Over longer time scales ($>\SI{25}{\pico\second}$), the TA population has pooled significantly at $\tfrac{1}{3} {\boldsymbol{\mathrm{M}}}$ and ${\boldsymbol{\mathrm{M}}}$. There are no three phonon anharmonic decay processes that start in a purely transverse mode; the only allowed interband transitions are L $\leftrightarrow$ T + T and L $\leftrightarrow$ L + T, where L (T) represents a longitudinal (transverse) mode [@Lax1981; @Khitun2001]; therefore, a build-up of population in the TA mode is expected. At $\tfrac{1}{2}{\boldsymbol{\mathrm{M}}}$, computed lifetimes predict that both LA and TA phonons will favor decay processes into out-of-plane phonons (ZA) [@Paulatto2013] that are not visible (have zero one-phonon structure factor) in the \[001\] zone-axis geometry in which these UEDS experiments were conducted. Similarily, the ZA phonons predominantly decay back into in-plane phonons, implying that the phonon thermalization in the acoustic branches occurs through a mechanism that exchanges in-plane and out-of-plane modes. Additionally, the computed LA and TA anharmonic lifetimes are predicted to significantly drop at $\tfrac{1}{2}{\boldsymbol{\mathrm{K}}}$ and $\tfrac{1}{2}{\boldsymbol{\mathrm{M}}}$, respectively, due to the activation of Umklapp scattering to the ZA phonons. Our measurements corroborate these predictions, as can be seen by TA and LA population at the mid-Brillouin zone (dashed white hexagon) being relatively lower than average. The confirmation of those predictions fundamentally relies on UEDS’ ability to probe the entire Brillouin zone at once.
{width="100.00000%"}
The robustness of such an analysis must be emphasized. The decomposition of transient diffuse intensity change via Equation admits no free parameter. Given sufficient data, a single optimal solution exists.
Wavevector-dependent electron-phonon coupling {#SEC:a1prime}
---------------------------------------------
The flow of energy between electronic and phononic subsystems is typically crudely modelled using the *two-temperature* model [@Allen1987]. This model assumes that the electronic system and the phononic system can each be associated with temperatures $T_e$ and $T_{ph}$, throughout the dynamics. Effectively this approximation assumes that the internal thermalization dynamics of each system is much more rapid than any processes that couple the two system. It is evident from the earlier description of the UEDS data from graphite that this assumption is (rather generally) quite a poor one; the idea that the phononic subsystem is internally thermalized does not hold on the timescales typically associated with energy flows between the electron and phonon systems following photoexcitation. On these timescales the phonon occupations are generally very far from being thermalized. UEDS allows to move beyond the two-temperature approximation; by leveraging momentum-resolution, mode-dependent electron-phonon and phonon-phonon couplings can be extracted from the transient change in mode populations $\left\{ \Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t) \right\}$. Specifically, wavevector-dependent phonon population dynamics determined in the previous section will now be used to determine the electron-phonon and phonon-phonon coupling strength of the $A_1'$ phonon.
The formalism of the two-temperature model can be extended to the *non-thermal lattice model* (NLM) model [@Waldecker2016], where every phonon branch $j$ has its own molar heat capacity $C_{ph,j}$, and temperature $T_{ph,j}$: $$C_e(T_e) \frac{\partial T_e}{\partial t} = \sum_i G_{ep, i}(T_e - T_{ph,i}) + f(t-t_0)
\label{EQ:mode_dep_ttm_electron}$$ $$\begin{gathered}
\label{EQ:mode_dep_ttm_phonon}
\Bigg\{ C_{ph,j}(T_{ph,j}) \frac{\partial T_{ph,j}}{\partial t} = \\
\sum_{i\neq j} G_{ep, i}(T_e - T_{ph,i})
+ G_{pp,ij} (T_{ph,j} - T_{ph,i}) \Bigg\}_{j=1}^{N}\end{gathered}$$ where $f(t-t_0)$ is the laser pulse profile, and $C_e$ and $T_e$ are the electronic heat capacity and electron temperature, respectively [^9]. This model accounts for discrepancies in coupling between the electronic system and certain phonon modes, which occurs for example in graphite — where some modes are strongly-coupled to the electron system via Kohn anomalies [@Piscanec2004].
Observations of transient changes in mode populations are related to mode temperatures $T_{ph,j}(t)$ via the Bose-Einstein distribution: $$n_{j,{\boldsymbol{\mathrm{k}}}}(t) \propto \left[ \exp\left( \frac{\hbar \omega_{j,{\boldsymbol{\mathrm{k}}}}}{k_B T_{ph,j}(t)}\right) -1 \right]^{-1}.
\label{EQ:pop_laurent}$$ We can decompose the above expression with a Laurent series [@Wunsch2005] to show explicitly that the mode population is proportional to temperature, for appropriately high $T_{ph,j}$: $$n_{j,{\boldsymbol{\mathrm{k}}}}(t) \propto \frac{k_B T_{ph,j}(t)}{\hbar \omega_{j,{\boldsymbol{\mathrm{k}}}}} - \frac{1}{2} + O\left( T_{ph,j}^{-1}(t) \right).
\label{EQ:pop_to_temp}$$ Hence, in the case of measurements presented herein, $\Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t) \propto \Delta T_{ph,j}(t)$, where the initial temperature is known to be .
We now use the NLM to extract the couplings to the $A_1'$ mode from population measurements. The differential $A_1'$ phonon population $\Delta n_{A_1'}(t)$ is obtained by integrating over the region of the wavevector-dependent TO2 phonon population, in a circular arc centered at ${\boldsymbol{\mathrm{K}}}$ ($|{\boldsymbol{\mathrm{k}}} - {\boldsymbol{\mathrm{K}}}| \leq \SI{0.3}{\per\angstrom}$). This location is shown in Figure \[FIG:populations\]. In order to correlate the mode population measurements with the NLM, the heat capacities of the electronic system and every phonon mode must be parametrized.
The electronic heat capacity $C_e$ is extracted from experimental work by @Nihira2003: $$\begin{aligned}
C_e(T_e) &= 13.8 ~ T_e \\
&+ 1.16 \times 10^{-3} ~ T_e^2 \nonumber \\
&+ 2.6 \times 10^{-7} ~ T_e^3. \nonumber\end{aligned}$$ Over this range of time-delays, thermal expansion (or contraction, in the case of graphite) has not yet occurred [@Chatelain2014a]. No changes in Bragg peak positions — indicative of lattice parameter changes — is observed within the experimental range of time-delays $\tau \leq \SI{680}{\pico\second}$. We can therefore calculate the heat capacity of each graphite mode $j$ as the heat capacity at constant volume [@ziman1979principles]: $$\begin{gathered}
C_{ph,j}(T_{ph,j}) = \\
k_B \int_0^{\omega_D} d\omega ~ D(\omega)
\left(
\frac{\hbar \omega}{k_B T_{ph,j}}
\right)^2
\frac{e^{\hbar \omega / k_B T_{ph,j}}}{\left( e^{\hbar \omega / k_B T_{ph,j}} - 1\right)^2}\end{gathered}$$ where $k_B$ is the Boltzmann constant, $\omega_D$ is the Debye frequency, and $D(\omega)$ is the phonon density of states. Momentum resolution of UEDS allows for a simplification, where a single frequency contributes to the heat capacity in the $A_1'$ mode — $D(\omega) = \delta(\omega)$. Moreover, we can reduce the number of coupled equations in Equations and . Simultaneous conservation of momentum and energy during the decay of an $A_1'$ phonon can only be satisfied in a few reciprocal space locations. Using first-principles calculations, it is possible to determine the decay probabilities. One such calculation, reported by @Bonini2007, allows us to define an *effective* heat capacity into which the $A_1'$ population drains, $C_l$ [^10]. Therefore, the energy dynamics at ${\boldsymbol{\mathrm{K}}}$ can be specified in terms of a system of three equations: $$\begin{aligned}
C_e(T_e) \frac{\partial T_e}{\partial t}
&= f(t-t_0) \\
&- G_{e,A_1'} ~ (T_e - T_{A_1'}) \nonumber \\
&- G_{e, l} ~ (T_e - T_l) \nonumber \\
C_{A_1'}(T_{A_1'}) \frac{\partial T_{A_1'}}{\partial t}
&= G_{e,A_1'} ~ (T_e - T_{A_1'}) \\
&- G_{A_1', l} ~ (T_{A_1'} - T_{l}) \nonumber\\
C_{l}(T_l) \frac{\partial T_l}{\partial t}
&= G_{e,l} ~ (T_e - T_l) \\
&+ G_{A_1', l} ~ (T_{A_1'} - T_{l}) \nonumber\end{aligned}$$ where $G_{e,A_1'}$, $G_{A_1', l}$, and $G_{e,l}$ are constants. Solving this system of equations gives the temperature evolution of each of the subsystems. The evolution in the $A_1'$ population can be used as a proxy for the mode temperature $T_{A_1'}(t)$; minimizing the difference between observed population dynamics and modelled temperature changes yields the coupling constants $G_{e,A_1'}$, $G_{A_1', l}$, and $G_{e,l}$. The resulting temperature transients are presented in Figure \[FIG:a1prime\]. The extracted coupling constants have been listed in Table \[TAB:coupling\]. This model correctly identifies the strong electron-phonon coupling of the $A_1'$ mode, $G_{e,A_1'}$, as compared with the rest of the relevant modes, $G_{e,l}$.
[c c]{} & Coupling strength \[\]\
\
$G_{e,A_1'}$ & $(6.8 \pm 0.3) \times 10^{17}$\
$G_{A_1', l}$ & $(8.0 \pm 0.5) \times 10^{17}$\
$G_{e,l}$ & $(0.0 \pm 6.0) \times 10^{15}$\
\[TAB:coupling\]
![Evolution of the $A_1'$ mode population in graphite after ultrafast photoexcitation. Differential population measurement of $A_1'$, shown in black (circle), is obtained from the integration of the TO2 mode population in a circular arc centered at ${\boldsymbol{\mathrm{K}}}$ ($|{\boldsymbol{\mathrm{k}}} - {\boldsymbol{\mathrm{K}}}| \leq \SI{0.3}{\per\angstrom}$), visible in Figure \[FIG:populations\]. Error bars are determined from the standard error in the mean of population before photoexcitation ($t<t_0$). The fit to the population change is shown in pink (solid). The effective temperature of the modes in which the $A_1'$ phonon can decay is shown as an orange (dotted) line. **Inset**: temperature dynamics at early times ($<\SI{1000}{\femto\second}$) show that thermalization between the electronic system (purple, dashed) and the $A_1'$ phonon population (pink, solid) is very fast, indicative of strong electron-phonon coupling. The effective temperature of the modes in which the $A_1'$ phonon can decay is shown as an orange (dotted) line.[]{data-label="FIG:a1prime"}](a1prime_evolution.pdf){width="3.38in"}
From the coupling constant $G_{e,A_1'}$, mode-projected electron-phonon coupling value $\langle g_{e,A_1'}^2 \rangle$ can be determined. In the case of the coupling between the electron system and the $A_1'$ phonon, the heating rate of $G_{e,A_1'} = \SI{6.8 \pm 0.3 e17}{\watt \per \meter \cubed \per \kelvin}$ (Table \[TAB:coupling\]) corresponds to a mode-projected electron-phonon coupling value of $\langle g_{e,A_1'}^2 \rangle = \SI{0.035 \pm 0.001}{\electronvolt \squared}$ (see appendix \[AP:mode\_proj\_g\] for details). These values are in agreement with recent trARPES measurements and simulations [@Johannsen2013; @Stange2015; @Rohde2018; @Na2019].
Conclusion
==========
UEDS provides direct access to wavevector-resolved, non-equilibrium phonon populations and is, in this sense, a lattice-dynamical analog of trARPES. A robust and generally-applicable UEDS data reduction method has been described that provides detailed information on transient changes in phonon populations across the entire Brillouin zone that follow photoexcitation in single-crysalline materials. This method takes only the observed UEDS patterns and computed one-phonon structure factors as inputs and can easily be extended with minimal alterations to ultrafast X-ray diffuse scattering. A procedure for computing the required phonon properties using DFPT, and their potential time dependence via the Debye-Waller factors was described in detail. This method was demonstrated for the case of photodoped carriers in the Dirac cones of thin graphite, where the phonon populations were tracked. Finally, the mode dependence of couplings between electron and phonons have been demonstrated at a specific point in the Brillouin zone, where the strongly-coupled optical phonon $A_1'$ is located. Mode-projected electron-phonon coupling value for the $A_1'$ phonon was extracted, using the non-thermal lattice model, and corroborated with numerous other experiments and simulations.
Direct determination of wavevector-dependent, transient phonon populations holds great promise for the study of phenomena that emerge primarily due to the coupling of electronic and lattice degrees of freedom, and specifically those involving strongly anisotropic interactions. In particular, with sufficient time-resolution, the applicability of the Kramers-Heisenberg-Dirac theory to Raman scattering measurements in graphene/graphite could be explored, via the detection of early-times ($<$) phonon populations in the strongly-coupled optical phonon $E_{2g}$ [@Heller2016]. Another potential extension concerns influence of non-equilibrium carrier distributions on phonon vibrational frequencies. Many systems, charge-density wave materials in particular, exhibit phonon modes that harden or soften at high temperatures and selective electronic excitation which can be used to explore such phenomena in greater depth.
Acknowledgements {#acknowledgements .unnumbered}
================
L. P. R. de C. thanks H. Seiler for illuminating discussions regarding the role of the Debye-Waller effect on diffuse intensity, and M. X. Na for providing insights regarding the relationship between heat rates and mode-projected electron-phonon coupling constants.
This research was enabled in part by support provided by Calcul Quebec (www.calculquebec.ca) and Compute Canada (www.computecanada.ca)
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de Recherche du Québec - Nature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI), and the Canada Research Chairs (CRC) program.
### Author contribution {#author-contribution .unnumbered}
M. S. and B. J. S. conceptualized the work. L. P. R. de C. performed the research. L. P. R. de C. and B.J.S. wrote the manuscript. J.-H. P. performed the DFPT calculations. All authors helped edit the article.
Clustering of phonon eigenvalues and eigenvectors into branches {#AP:mode_clustering}
===============================================================
This section describes the clustering of phonon polarization vectors $\left\{ {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$ and frequencies $\left\{ \omega_{j, {\boldsymbol{\mathrm{k}}}}\right\}$ into physically-relevant categories, i.e. branches. The general idea behind the procedure is that phonon properties are continuous. The variation of a property should not display any discontinuity along any path in the Brillouin zone.
Let ${\boldsymbol{\mathrm{P}}}_{j,{\boldsymbol{\mathrm{k}}}}$ be an abstract vector representing the polarization vectors and frequency of branch $j$ at reduced wavevector ${\boldsymbol{\mathrm{k}}}$. We represent ${\boldsymbol{\mathrm{P}}}_{j,{\boldsymbol{\mathrm{k}}}}$ as the following vector: $${\boldsymbol{\mathrm{P}}}_{j,{\boldsymbol{\mathrm{k}}}} =
\begin{bmatrix}
\omega_{j, {\boldsymbol{\mathrm{k}}}} & {\boldsymbol{\mathrm{e}}}_{j,s=1,{\boldsymbol{\mathrm{k}}}} & \dots & {\boldsymbol{\mathrm{e}}}_{j,s=M,{\boldsymbol{\mathrm{k}}}}
\end{bmatrix}^T$$ where the index $s$ runs for all $M$ atoms in the unit cell ($M=4$ in the case of graphite). We define the metric between two abstract vectors ${\boldsymbol{\mathrm{P}}}_{i,{\boldsymbol{\mathrm{k}}}}$ and ${\boldsymbol{\mathrm{P}}}_{j,{\boldsymbol{\mathrm{k}}}}$ as follows: $$\lVert {\boldsymbol{\mathrm{P}}}_{i,{\boldsymbol{\mathrm{k}}}} - {\boldsymbol{\mathrm{P}}}_{j,{\boldsymbol{\mathrm{k}}}'} \rVert =
| \omega_{i, {\boldsymbol{\mathrm{k}}}} - \omega_{j, {\boldsymbol{\mathrm{k'}}}} |^2 + \sum_s \lVert {\boldsymbol{\mathrm{e}}}_{i,s,{\boldsymbol{\mathrm{k}}}} - {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}'} \rVert.$$ A one-dimensional path $\gamma({\boldsymbol{\mathrm{k}}})$ connecting all ${\boldsymbol{\mathrm{k}}}$-points was defined, starting at ${\boldsymbol{\mathrm{\Gamma}}}$. At ${\boldsymbol{\mathrm{\Gamma}}}$, polarization vectors are associated with a mode based on geometry and oscillation frequency. For example, polarization vectors $\left\{ {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \right\}$ parallel to their wavevector ${\boldsymbol{\mathrm{k}}}$ for all atoms $s$ is a longitudinal mode; if the associated frequency is $\approx \SI{0}{\tera\hertz}$, this mode can be labelled longitudinal acoustic. Then, polarization vectors and frequencies at any point along the path were assigned to modes that optimized continuity. That is, the assignment of phonon branches $j$ at $\gamma({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{{\boldsymbol{\mathrm{\Delta}}}}}})$, ${\boldsymbol{\mathrm{P}}}_{j,\gamma({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{{\boldsymbol{\mathrm{\Delta}}}}}})}$ based on the assignment at $\gamma({\boldsymbol{\mathrm{k}}})$, ${\boldsymbol{\mathrm{P}}}_{i,\gamma({\boldsymbol{\mathrm{k}}})}$, minimized the distance $\lVert {\boldsymbol{\mathrm{P}}}_{i,\gamma({\boldsymbol{\mathrm{k}}})} - {\boldsymbol{\mathrm{P}}}_{j,\gamma({\boldsymbol{\mathrm{k}}} + {\boldsymbol{\mathrm{{\boldsymbol{\mathrm{\Delta}}}}}})} \rVert$.
The procedure described above, adapted for numerical evaluation, is part of the `scikit-ued` software package [@RenedeCotret2018].
Calculation of mode-projected electron-phonon coupling from heating rates {#AP:mode_proj_g}
=========================================================================
Consider the coupled equations of the non-thermal lattice model in Equations and . These coupled first-order ordinary differential equations will admit solutions for $T_e(t)$ and $\left\{ T_{ph,j}(t) \right\}$. After photoexcitation ($f(t-t_0) \to 0$), the appropriate summations of those equations yields the following single equation: $$\begin{gathered}
\frac{\partial T_e}{\partial t} - \sum_j \frac{\partial T_{ph,j}}{\partial t} =
\sum_j \Bigg[ \frac{G_{ep,j}}{C_e} ~ (T_e - T_{ph,j})
\\-\sum_i \bigg( \frac{G_{ep,i}}{C_{ph,j}} ~ (T_e - T_{ph,i})
+ \frac{G_{pp,ij}}{C_{ph,j}} ~ (T_{ph,i} - T_{ph,j}) \bigg) \Bigg]\end{gathered}$$ where the temperature dependence of $C_e$ and $\left\{ C_{ph,j} \right\}$ has been omitted for brevity. In the case of graphite, at early times ($<\SI{5}{\pico\second}$), phonon-phonon coupling $G_{pp,ij}$ is much weaker at the ${\boldsymbol{\mathrm{K}}}$-point (Table \[TAB:coupling\]). Therefore, we may simplify the above equation to a more manageable system: $$\begin{aligned}
\frac{\partial T_e}{\partial t} - \sum_j \frac{\partial T_{ph,j}}{\partial t}
&= \sum_j \frac{G_{ep,j}}{C_e} (T_e - T_{ph,j}) \\
&- \sum_{i,j} \frac{G_{ep,i}}{C_{ph,j}}(T_e - T_{ph,i}). \nonumber\end{aligned}$$ By performing a substitution $\lambda = T_e - \sum_j T_{ph,j}$, the equation above simplifies to a familiar situation: $$\dot{\lambda}(t) - a(t) \lambda(t) = 0
\label{EQ:Ode}$$ where $$a(t) = \sum_j \left( \frac{G_{ep,j}}{C_e} - \sum_i \frac{G_{ep,i}}{C_{ph,j}}\right).$$ The time dependence comes from the time-evolution of the individual temperatures. In the case of phonon temperatures, the phonon population dynamics are directly related to temperature dynamics according to Equation . Equation is a separable equation with solution: $$\lambda(t) = \exp{\int dt \left[ a(t) \right]}.$$ For a slow-varying integrand $a(t) \approx a$, then $a = 1/\tau$, where $\tau$ is a compound variable representing the relaxation of the system. This leads to the following form: $$\frac{1}{\tau} \approx \sum_j \left( \frac{G_{ep,j}}{C_e} - \sum_i \frac{G_{ep,i}}{C_{ph,j}}\right).
\label{EQ:tconst}$$ As a specific example, the above expression reduces nicely in the case of the two-temperature model, where all phonon modes are considered to be thermalized with each other, with isochoric heat capacity $C_{ph}$: $$\frac{1}{\tau} = G_{ep} \left( \frac{1}{C_e} - \frac{1}{C_{ph}}\right)$$ and we see that $\tau$ physically represents the relaxation time of the electronic system into the lattice. Equation can be thought of as a sum of relaxation times between the electronic subsystem and specific modes $\tau_{e,j}$: $$\frac{1}{\tau_{e,j}} = \frac{G_{ep,j}}{C_e} - \sum_i \frac{G_{ep,i}}{C_{ph,j}}.$$ The final state in relating heating rates to their mode-projected coupling values requires knowledge about density of states. Because the measurements herein consider only in-plane interactions, we use an approximate electronic density of states for graphene close to the Dirac point [@Neto2009]: $$D_e(\epsilon) = \frac{2 A}{\pi} \frac{|\epsilon|}{(\hbar~v_F)^2}
\label{EQ:graphene_dos}$$ where $A$ is the unit cell area and $v_F = \SI{9.06e5}{\meter \per \second}$ is the Fermi velocity [^11]. The electronic density of states is related to the mode-projected electron-phonon coupling $\langle g_{ep,j}^2 \rangle$ as follows [@Na2019]: $$\frac{\hbar}{\tau_{e,j}} = 2 \pi ~ \langle g_{ep,j}^2 \rangle ~ D_e(\hbar \omega_{\nu} - \hbar \omega_{j, {\boldsymbol{\mathrm{k}}}})$$ where $\hbar \omega_{\nu}$ corresponds to the optical excitation energy ( or ).
[65]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} [****, ()](https://doi.org/10.1126/science.aad3749) ****, [10.1126/sciadv.1701217](https://doi.org/10.1126/sciadv.1701217) () @noop [****, ()]{} @noop [****, ()]{} @noop [**** ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**** ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} [****, ()](https://doi.org/10.1107/S0567739479000061) @noop [****, ()]{} @noop [****, ()]{} [****, ()](https://doi.org/10.1021/nl071033g) [****, ()](https://doi.org/10.1038/srep12422) @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [ ()]{} [****, ()](https://doi.org/10.1021/acsnano.5b07676) @noop [****, ()]{}
[^1]: The intensity fluctuations of pixel values across scattering patterns acquired before photoexcitation are $10^8$ times smaller than the brightest Bragg reflection
[^2]: The space group of this structure is $P6_3/mmc$ (Hermann-Mauguin symbol) or $D_{6h}^4$ (Schoenflies symbol).
[^3]: Detector counts for the brightest Bragg peak reaches as much as 20 000 counts, while the average diffuse feature shown in Figure \[FIG:graphite\] is 0.2 counts.
[^4]: The coverage of irreducible ${\boldsymbol{\mathrm{k}}}$-points is important. Only computing phonon properties along high-symmetry lines is fraught with peril, given that polarization vectors can vary significantly not only along high-symmetry lines, but over the entire Brillouin zone.
[^5]: A maximum of for optical modes, and for acoustic modes. The discrepancy between maximum temperatures represents the fact that the heat capacity of acoustic modes is much higher.
[^6]: The calculation of $|F_{1j}({\boldsymbol{\mathrm{q}}}, t)|$ is trivial for out-of-plane modes ZA, ZO1 – ZO3 because ${\boldsymbol{\mathrm{q}}} \cdot {\boldsymbol{\mathrm{e}}}_{j,s,{\boldsymbol{\mathrm{k}}}} \equiv 0$ for these modes.
[^7]: The positivity constraint $\Delta n_{j,{\boldsymbol{\mathrm{k}}}}(t) > 0 ~ \forall t$ means that the phonon population of a branch cannot drop below its equilibrium level. While not necessary, it leads to more reliable solutions.
[^8]: On the other hand, a monotonic increase in LA population at ${\boldsymbol{\mathrm{\Gamma}}}$ is expected from the decay of the other SCOP, $A_1'$, which has a high probability to decay into a pair of LA-LO modes. However, we were not able to measure population changes close enough to ${\boldsymbol{\mathrm{\Gamma}}}$.
[^9]: The electronic system thermalizes in approximately [@Stange2015], and hence after we can consider the electronic system to be well-described by a single temperature $T_e$.
[^10]: This effective heat capacity $C_l$ is composed of 9% chance to decay into two TA modes, 36% change to decay into a TA mode and an LA mode, and 55% chance to decay into either an LA and TA, or LO and LA.
[^11]: Note that factors of $\hbar$ are often ignored, including in the accompanying reference.
|
---
abstract: 'We define and study a class of graphs, called 2-stab interval graphs (2SIG), with boxicity 2 which properly contains the class of interval graphs. A 2SIG is an axes-parallel rectangle intersection graph where the rectangles have unit height (that is, length of the side parallel to $Y$-axis) and intersects either of the two fixed lines, parallel to the $X$-axis, distance $1+\epsilon$ ($0 < \epsilon < 1$) apart. Intuitively, 2SIG is a graph obtained by putting some edges between two interval graphs in a particular rule. It turns out that for these kind of graphs, the chromatic number of any of its induced subgraphs is bounded by twice of its (induced subgraph) clique number. This shows that the graph, even though not perfect, is not very far from it. Then we prove similar results for some subclasses of 2SIG and provide efficient algorithm for finding their clique number. We provide a matrix characterization for a subclass of 2SIG graph.'
author:
- |
[Sujoy Kumar Bhore$^{(a)}$]{}, [Dibyayan Chakraborty$^{(b)}$]{}, [Sandip Das$^{(b)}$]{},\
[Sagnik Sen$^{(c)}$]{}\
\
[$(a)$ Ben-Gurion University, Beer-Sheva, Israel]{}\
[$(b)$ Indian Statistical Institute, Kolkata, India]{}\
[$(c)$ Indian Statistical Institute, Bangalore, India]{}
bibliography:
- 'science.bib'
title: On a special class of boxicity 2 graphs
---
**Keywords:** boxicity, chromatic number, clique number, perfect graph, matrix characterization.
Introduction
============
A *geometric intersection graph* [@golumbic] is a graph whose vertices are represented by geometric objects and two vertices are adjacent if their corresponding geometric objects intersect. *Boxicity* [@krato2] of a graph $G$ is the minimim $k$ such that $G$ can be expressed as a geometric intercestion graph of of axes-parallel $k$ dimensional rectangles. The class of *boxicity $k$ graphs* is the class of graphs with boxicity at most $k$. The class of graphs with boxicity $1$ is better known as *interval graphs* [@golumbic] (intersection of real intervals) while the class of graphs with boxicity $2$ is better known as *rectangle intersection graphs* [@krato2] (intersection of axes-parallel rectangles).
It is known that several questions (for example, recognition, determining clique number, determining chromatic number) that are $NP$-hard in general becomes polynomial time solvable when restricted to the class of interval graphs while they remain $NP$-hard for the family of graphs with boxicity $k$ (for $k \geq 2$) [@krato2]. The reason for this dichotomy is probably because interval graphs are *perfect* (defined in Section \[preliminaries\]) while boxicity $k$ (for $k \geq 2$) graphs are not perfect (those questions are polynomial time solvable for perfect graphs as well) [@golumbic].
Naturally we are interested in exploring the objects that lie in between, that is, the proper subclasses of graphs with boxicity 2 that contains all interval graphs. Several such graph classes have been defined and studied [@zhang] [@hell] [@brand]. In this article, we too define such a graph class and study its different aspects. We keep in mind that ‘perfectness’ is probably the key word here. Our class of graphs is not perfect but it contains all interval graphs and is a proper subclass of boxicity 2 graphs. Moreover, our graph class is based on local structures of boxicity 2 graphs in some sense. Thus, the study of this class may help us understand the structure of boxicity 2 graphs in a better way.
As a matter of fact, the definition of our graph class is motivated from the definition of a well-known class of perfect graphs, the split graphs. A *split graph* is obtainted by putting edges between a clique and a set of independent vertices [@golumbic]. Note that a complete graph and an independent set are the two extreme trivial examples of perfect graphs. So when we put edges between these two types of perfect graphs, what we obtain is again perfect.
Motivated by this example, we wondered what would happen if we put edges between other kinds of perfect graphs. We take two interval graphs and put edges in between, following a particular rule. What we obtain is a class of geometric intersection graphs, not perfect, with certain properties which enables us to call them “nearly perfect” [@gyar]. That is, the chromatic number of each induced subgraph is bounded by a function of its clique number; a linear function in our case.
Let $y = 1$ be the *lower stab line* and $y = 2+ \epsilon $ be the *upper stab line* where $ \epsilon \in (0,1)$ is a constant. Now consider axes-parallel rectangles with unit height (length of the side parallel to $Y$-axis) that intersects one of the stab lines. A *2-stab interval graph (2SIG)* is a graph $G$ that can be represented as an intersection graph of such rectangles. Such a representation $R(G)$ of $G$ is called a *2-stab representation* (for example, see Fig. \[bridge\]). A 2SIG may have more than one 2-stab representation.
(0,0) circle (2pt) [node\[left\][$a$]{}]{}; (1,0) circle (2pt) [node\[right\][$b$]{}]{}; (1.5,1) circle (2pt) [node\[right\][$c$]{}]{}; (-.5,1) circle (2pt) [node\[left\][$e$]{}]{}; (.5,2) circle (2pt) [node\[above\][$d$]{}]{};
(0,0) – (1,0); (-.5,1) – (.5,2); (1.5,1) – (.5,2);
(-.5,1) – (0,0); (1,0) – (1.5,1);
(4,.2) – (5,.2); (4,1.2) – (5,1.2); (4,.2) – (4,1.2); (5,.2) – (5,1.2);
at (3.9,.1) [$a$]{};
(4.8,0) – (5.8,0); (4.8,1) – (5.8,1); (4.8,0) – (4.8,1); (5.8,0) – (5.8,1);
at (5.9,-.1) [$b$]{};
(3.5,1) – (4.5,1); (3.5,2) – (4.5,2); (3.5,1) – (3.5,2); (4.5,1) – (4.5,2);
at (3.4,2.1) [$e$]{};
(4.4,1.5) – (5.4,1.5); (4.4,2.5) – (5.4,2.5); (4.4,1.5) – (4.4,2.5); (5.4,1.5) – (5.4,2.5);
at (4.9,2.7) [$d$]{};
(5.2,.7) – (6.2,.7); (5.2,1.7) – (6.2,1.7); (5.2,.7) – (5.2,1.7); (6.2,.7) – (6.2,1.7);
at (6.3,1.8) [$c$]{};
(2.5,1.6) – (7.5,1.6);
at (7.7,1.8) [$y = 2 + \epsilon $]{};
(2.5,.6) – (7.5,.6);
at (7.3,.4) [$y = 1 $]{};
Notice that, given a representation $R(G)$ of $G$, each such rectangle intersects exactly one stab line partitioning the vertex set $V(G)$ in two disjoint parts, the *lower partition* $V_1$ (vertices with corresponding rectangles intersecting the lower stab line) and the *upper partition* $V_2$ (vertices with corresponding rectangles intersecting the upper stab line). Observe that such a vertex partition depends on the representation and is not unique. In the remainder of the article, whenever we speak about a 2SIG with a vertex partition $V(G) = V_1 \sqcup V_2$ we will mean the partitions are lower and upper partition due to a representation.
Also note that the induced subgraphs $G[V_1]$ and $G[V_2]$ are interval graphs with intervals corresponding to the projection of their rectangles on $X$-axis. Hence, indeed, a 2SIG is obtained by putting some edges between two different interval graphs. Also, observe that the definition of 2SIG does not depend on the specific value of the constant $ \epsilon $ as long as it belongs to the interval $(0,1)$. Furthermore, observe that a rectangle interval graph with rectangles with unit height locally looks like a 2-stab interval graph.
The article is organized in the following manner. In Section \[preliminaries\] we present the necessary definitions, notations and some observations. We study the clique number and the chromatic number of 2SIG in Section \[clique number\] and justify our claim that 2SIG and some of its subclasses are “nearly perfect" even though not perfect. We provide a matrix characterization for a subclass of 2SIG graph in Section \[sec matrix\]. Finally, we conclude the article in Section \[conclusion\].
Preliminaries
=============
The *clique number* $\omega(G)$ of a graph $G$ is the *order* (number of vertices) of the biggest complete subgraph of $G$. A *$k$-coloring* of a graph $G$ is an assignment of $k$ colors to the vertices of $G$ such that adjacent vertices receive different colors. The *chromatic number* $\chi(G)$ of a graph $G$ is the minimum $k$ such that $G$ admits a $k$-coloring. A graph $G$ is *perfect* if $\omega(H) = \chi(H)$ for all induced subgraph $H$ of $G$.
A graph $G$ is *$\chi$-bounded* if $\chi(H) \leq f(\omega(H))$ for all induced subgraph $H$ of $G$ where $f$ is a bounded integer-valued function [@gyarfas]. This is what we meant when we used the informal term “nearly perfect".
Recall the definition of 2-stab interval graphs from the previous section. Now by putting more restrictions on our definition of 2SIG we obtain a few other interesting subclasses of 2SIG that we are going to study in this article.
A *2-stab unit interval graph (2SUIG)* is a 2SIG with a representation where each rectangle is a unit square. The corresponding representation is a *2SUIG representation*. A *proper 2-stab interval graph (proper 2SIG)* is a 2SIG with a representation where the projection of a rectangle on $X$-axis does not properly contain the projection of any other rectangle on $X$-asis. A *2-stab independent interval graph (2SIIG)* is a 2SIG with a representation where the upper partition induces an independent set. The corresponding representation is a *2SIIG representation*.
(0,0) circle (2pt) [node\[left\]]{}; (2,0) circle (2pt) [node\[right\]]{}; (4,0) circle (2pt) [node\[left\]]{}; (6,0) circle (2pt) [node\[right\]]{}; (8,0) circle (2pt) [node\[left\]]{}; (10,0) circle (2pt) [node\[right\]]{};
(0,2) circle (2pt) [node\[left\]]{}; (2,2) circle (2pt) [node\[left\]]{}; (6,2) circle (2pt) [node\[left\]]{}; (8,2) circle (2pt) [node\[left\]]{};
(0,0) – (10,0);
(0,0) – (0,2); (2,0) – (2,2); (6,0) – (6,2); (6,0) – (8,2);
(4,0) .. controls (6,-.8) .. (8,0);
Let us fix a representation $R(G)$ of a 2-stab interval graph $G$ with corresponding lower and upper partitions $V_1$ and $V_2$, respectively. Then the set of *bridge edges* $E_B$ is the set of edges (depicted using “dashed" edges in the figures) between the vertices of $V_1$ and $V_2$ while the set *bridge vertices* $V_B$ is the set of vertices incedent to bridge edges (see Fig. \[bridge\]). For some $v \in V(G)$, the set of *bridge neighbors* $N_B(v)$ is the set of all vertices adjacent to $v$ by a bridge edge. A *bridge triangle* is a triangle (induced $K_3$) in which exactly two of its edges are bridge edges (note that, no triangle of $G$ can have exactly one or three edges from $E_B$). A *bridge triangle free 2SUIG* is a graph with at least one 2SUIG representation without any bridge triangle (see Fig. \[bridgetri\]).
An *orientation* $\overrightarrow{G}$ of a graph $G$ is obtained by replacing its edges with *arcs* (ordered pair of vertices). An orientation $\overrightarrow{G}$ of $G$ is a *transitive orientation* if for each pair of arcs $(a,b)$ and $(b,c)$ we have the arc $(a,c)$ in $\overrightarrow{G}$. We know that the complement of an interval graph admits a transitive orientation [@mcconne]. Let $\overrightarrow{I^c}$ be a transitive orientation of the complement graph of an interval graph $I$.
Relation with other graph classes
=================================
The class of 2SIG graphs can be thought of as a generalization of interval graphs. So, we wondered if there is any relation between 2SIG graphs and other generalization of interval graphs, such as, 2-interval graphs. A *2-interval graph* is a geometric intersection graph where each vertex corresponds to two real intervals [@2interval].
\[2-int\] All bridge triangle free 2SUIG graphs are 2-interval graphs.
Let $G=(V_1\sqcup V_2,E)$ be a bridge triangle free 2SUIG. Note that $G[V_1]$ and $G[V_2]$ induces two unit interval graphs. We can assign an intervals to each of the vertices of $G$ such that the intersection graph of those intervals is the graph isomorphic to the disjoint union of $G[V_1]$ and $G[V_2]$. These intervals are the first set of intervals assigned to the vertices of $G$.
Now we want to assign a second set of intervals to the vertices of $G$ such that they intersects to represent the remaining edges of $G$. Note that the only edges that are not represented yet are exactly the set of bridge edges. As $G$ is bridge triangle free 2-stab unit interval graph, the set of bridge edges induces an interval graph isomorphic to disjoint union of paths. Thus, it is possible to assign a second set of intervals, each of them completely disjoint from the intervals belonging to the first set of intervals, to the vertices such that the intersection graph is isomorphic to the graph induced by bridge edges of $G$.
It is well known that proper interval graphs are equivalent to unit interval graphs [@bogart]. Interestingly, an analogous result exists for 2SUIG graphs.
The class of proper 2SIG graphs is equivalent to the class of 2SUIG graphs.
It is easy to observe that a 2SUIG representation of a graph is also a proper 2SIG representation.
Let $G$ admits a proper 2SIG representation $R$. Assume that a vertex $v$ of $G$ corresponds to a rectangle $r_v$. Let the projection of $r_v$ on $X$-axis be the interval $I_x(r_v)$ and let the projection of $r_v$ on $Y$-axis be the interval $I_y(r_v)$. Thus, the rectangle $r_v$ is nothing but the cross product $I_x(r_v) \times I_y(r_v)$ of the two intervals.
Consider the intervals obtained from projecting the rectangles on $X$-axis. The intersection graph of these intervals will give us a proper interval graph $P$ according to the definition of a proper 2SIG. We know that every proper interval graph has a unit interval representation [@golumbic]. Let $U$ be such a representation of $P$. Note that $P$ has all edges of $G$ but may have some additional edges as well. Those additional edges $uv$ are precisely those for which $I_x(r_u) \times I_x(r_v) \neq \emptyset $ and $I_y(r_u) \times I_y(r_v) = \emptyset $. Let $U_v$ be the interval corresponding to a vertex $v$ in $U$. Now consider the rectangle $r'_v = U_v \times I_y(r_v)$ for each vertex $v$ of $G$. Note that these rectangles are unit rectangles and their intersection graph is $G$. Also note that, as we have not changed the $Y$-co-ordinates of the rectangles, the so obtained representation is still a 2SIG representation. Hence our new representation is indeed a 2SUIG representation of $G$.
Clique number and chromatic number {#clique number}
==================================
A 2SUIG graph is obtained by putting some edges between two interval graphs. The perfectness of an interval graph implies that it has chromatic number equal to its clique number. Hence the observation follows.
\[uplow\] Let $G = (V_1 \sqcup V_2, E)$ be a 2SIG graph with a given vertex partition.
- Then $max\{\omega(G[V_1]), \omega(G[V_2])\} \leq \omega(G) \leq \omega(G[V_1]) + \omega(G[V_2])$.
- Then $max\{\chi(G[V_1]), \chi(G[V_2])\} \leq \chi(G) \leq \chi(G[V_1]) + \chi(G[V_2])$.
Let $H$ be a 2SIG with no bridge edges. Then for $H$ both lower bounds of Observation \[uplow\] are tight. Now note that even a complete graph with any vertex partition admits a 2SIG representation. In that case, both the upper bounds of Observation \[uplow\] are tight. As the clique number and the chromatic number of an interval graph can be computed in linear time, given a 2SIG with a vertex partition, the lower and upper bounds of Observation \[uplow\] can be obtained in linear time as well. As any induced subgraph of a 2SIG is again a 2SIG we have the following result as a direct corollary of the above theorem.
Given any 2SIG graph $G$ we have $\chi(H) \leq 2\omega(H)$ for all induced subgraph $H$ of $G$.
Let $G$ be a 2SIG with a vertex partition $V(G) = V_1 \sqcup V_2$. As $G[V_i]$ is an interval graph we have $\omega(G[V_i]) = \chi(G[V_i])$ for all $i \in \{1,2\}$. Then by Observation \[uplow\] we have
$$\begin{aligned}
\nonumber
\chi(G) &\leq \chi(G[V_1]) + \chi(G[V_2]) = \omega(G[V_1]) + \omega(G[V_2]) \nonumber \\
&\leq 2max\{\omega(G[V_1]), \omega(G[V_2])\} \leq 2\omega(G).
\nonumber\end{aligned}$$
This completes the proof.
So in particular 2SIG graphs are $\chi$-bounded which is not surprising as Gyárfás [@gyarfas] showed that all boxicity 2 graphs are $\chi$-bounded by a quadratic function. We showed that 2SIGs are, in fact, $\chi$-bounded by a linear function. It is known that square intersection graphs are $\chi$-bounded by a linear function [@kostochka].
Now we focus on some of the subclasses of 2SIG. First, we show that 2SIIG graphs are $\chi$-bounded by a better function.
\[i2sigcol\] Let $G = (V_1 \sqcup V_2, E)$ be a 2SIIG graph with a given vertex partition. Then $\omega(G) \leq \chi(G) \leq \omega(G)+1$.
Moreover, we can enumerate all the maximal cliques and hence, can compute the clique number $\omega(G) $ of $G$ in $O(|V|+|E|)$ time, where $|V|$ is the number of vertices in $G$.
Note that, the lower partition $V_1$ induces an interval graph and the upper partition $V_2$ induces an independent set. Hence $\omega(G[V_1]) = \chi(G[V_1])$ while $\omega(G[V_2]) = \chi(G[V_2]) = 1$. Also, Observation \[uplow\] implies $\omega(G[V_1]) \leq \omega(G) \leq \omega(G[V_1]) +1$ and $\chi(G[V_1]) \leq \chi(G) \leq \chi(G[V_1]) +1$ which implies this result.
We know that it is possible to enumerate all the maximal cliques and to compute the clique number of $G[V_1]$ in linear time [@golumbic]. For a vertex $v$ in $G[V_2]$, $N_B(v)$ induces an interval graph. The maximal cliques containing $v$ can be enumerated in $O(d)$ time, where $d$ is the degree of the vertex $v$. So, in $O(|V|+|E|)$ time we can compute the clique number of the graph. Hence we are done.
Note that, if the intersection representation of a boxicity 2 graph, $G=(V,E)$, is given then it is possible to compute the clique number of $G$ in $O(|V|log|V|+|V| \cdot K)$ time, where $K$ is the size of the maximum clique [@snandy].
Here we provide a quadratic time solution for the same problem for 2SIIG, a subclass of boxicity 2 graphs, but we do not require the intersection model as our input in this case. It is enough if the vertex partition of the graph is provied. We can prove a similar result for yet another subclass of boxicity 2 graphs, the 2SUIG graphs. The proof is more involved.
\[allmaxclique\] For any 2SUIG graph $G$ we can enumerate all the maximal cliques and hence, can compute the clique number $\omega(G) $ in polynomial time.
To prove the above result we need to prove the following lemmas.
\[orient\] Let $G = (V_1 \sqcup V_2, E)$ be a 2SUIG graph with a given partition. Then there exist transitive orientations $\overrightarrow{G[V_1]^c}$ and $\overrightarrow{G[V_2]^c}$ such that for every pair of bridge edges $u_1v_1$ and $u_2v_2$ with $u_1u_2, v_1v_2 \notin E(G)$ we have the two arcs $(u_1,u_2)$ and $(v_1,v_2)$ in the orientations. Moreover, such orientations can be found in polynomial time.
Take $\overrightarrow{G[V_1]^c}$ and $\overrightarrow{G[V_2]^c}$ such that the statement does not hold. The rectangles corresponding to $u_1$ and $v_1$ along with their intersection divide the region between the axis parallel lines into two disjoint parts. Hence, the intersection between the rectangles corresponding to $u_2$ and $v_2$, cannot be created without any intersection between $u_2$ and $u_1$ or between $v_2$ and $v_1$. This contradicts the premise of the lemma. Since, the given graph has representation with the given partition, there must exist $\overrightarrow{G[V_1]^c}$ and $\overrightarrow{G[V_2]^c}$ such that the lemma holds.
Moreover, it is possible to compute such an orientation of $\overrightarrow{G[V_1]^c}$ and $\overrightarrow{G[V_2]^c}$ in $O(|V_1|+|V_2|+|E|)$ time since the transitive orientation of the complement of a connected unit interval graph is unique up to reversal [@Ibarra20091737].
So, given a 2SUIG $G = (V_1 \sqcup V_2, E)$ with a partition we can fix transitive orientations $\overrightarrow{G[V_1]^c}$ and $\overrightarrow{G[V_2]^c}$ as in Lemma \[orient\]. Now given a bridge vertex $v \in V_i$ a vertex $v' \in V_i$ is its preceeding bridge vertex if each directed path from $v'$ to $v$ does not go through any other bridge vertex. The set of all preceeding bridge vertices of $v$ is denoted by $PBV(v)$.
Now we will assign integer labels to the bridge edges of $G$. Let $uv \in E_B$ and let $\mathscr{B}(uv)$ be the set of all bridge edges with one vertex incident to it lying completely to the left (that is, strictly less with respect to the transitive orientation) of $u$ or $v$. We assign the integer label $[e]$ to each edge $e= uv \in E_B$ inductively as follows:
$$[e] =
\begin{cases}
0 & \text{ if PBV(u) = PBV(v)= $\phi$},\\
i+1 & \text{ otherwise, where $i=max \{ [e']|e' \in \mathscr{B}(e) \}$ }.
\end{cases}$$
Let $E_{B}^{i}$ be the bridge edges with label $i$.
\[biclique\] In a $2SUIG$ graph the maximal cliques induced by the vertices incident to edges of $E_B^i$ can be enumerated in polynomial time.
Let $G=(V_1\sqcup V_2,E)$ be a $2SUIG$ graph with a given partition. Let $G'$ be the subgraph induced by the vertices incedent to the edges of $E_B^i$ for some fixed index $i$.. The vertices of $G'$ belonging to the same stab line create a clique (not necessarily maximal in $G$). Consider the subgraph $G'' \subseteq G'$ containing only edges of $E_B^i$. Note that, this graph is a bipartite graph. Any maximal bipartite clique in $G''$ creates a maximal clique in $G$. All maximal bipartite cliques of $G''$ can be enumerated in polynomial time [@Alexe] (since we can have at most $O(|V(G'')|)$ maximal bipartite cliques). Therefore, the maximal cliques created by the union of the endpoints of $E_B^i$ can be evaluated in polynomial time.
Now we will show that it is not possible to have bridge edges with different labels in the same maximal clique of a 2SUIG $G$.
\[noclique\] Bridge edges with different labels are not part of the same maximal clique in a $2SUIG$.
Let $e,e'$ be two bridge edges and without loss of generality assume $[e] < [e']$. Then by definition at least one vertex incident to $e$ is not adjacent to one of the vertices incident to $e'$. Hence we are done.
Now we are ready to prove our main result.
***Proof of Theorem \[allmaxclique\]:*** Let $G= (V_1\sqcup V_2,E)$ be a $2SUIG$ graph with a given partition. We can enumarate all the maximal cliques of $G$ containing at least one bridge edge using Lemma \[biclique\] and Lemma \[noclique\] in polynomial time. The maximal cliques of $G[V_1]$ and $G[V_2]$ can be enumerated in polynomial time as they are unit interval graphs.
We could not provide a better $\chi$-bound function for 2SUIG graphs than the one in Observation \[i2sigcol\]. However, we can provide a better $\chi$-bound function for bridge triangle-free 2SUIG graphs.
\[th trifreecol\] Let $G$ be a bridge triangle free 2SUIG. Then $\omega(G) \leq \chi(G) \leq \omega(G)+1$.
However, we will need to prove some lemmas before proving this result.
\[3col\] The bridge vertices of a triangle free 2SUIG graph can be coloured using 2 colors.
Let $G=(V_1\sqcup V_2,E)$ be a triangle free 2SUIG graph with a given partition and representation. Then $G[V_1]$ and $G[V_2]$ are disjoint union of paths. Let $u < v$ if the interval corresponding to $u$ lies in the left of the interval corresponding to $v$ (we compare the starting points) for any $u,v \in V_i$ where $i \in \{1,2\}$. Furthermore, we say that $e=uv < u'v'=e'$ if $u<u'$ or $v<v'$ where $e, e' \in E_B$, $u,u' \in V_1$ and $v,v' \in V_2$.
We prove the statement using induction on the number of bridge edges. For $i=1$ the graph is a tree, hence admits a 2-coloring.
Assume that all bridge triangle-free 2SUIG with at most $k$ bridges admits a 3-coloring such that the bridge vertices received only two of the three colors. Let $G$ be a bridge triangle-free 2SUIG with $k+1$ bridges. Let $e'=u'v'$ be an edge of $G$ such that $e< e'$ for all $e \in E_B$. Delete $e'$ from $G$ to obtain the graph $G'$. Note that $G'$ admits a 3-coloring where all the bridge edges received only two of the three colors. Let $e''= uv$ be the edge of $G'$ such that $e< e''$ for all bridge edge $e$ of $G'$. Suppose they received the colors $c_1$ and $c_2$. The subgraph induced by the paths $u$ to $u'$ and $v$ to $v'$ is a cycle. If it is an even cycle then we are done. Otherwise, we can always assign the third color to a non-bridge vertex of the cycle and complete the required coloring.
Note that the proof of our above result has an algorithmic aspect as well and it is not difficult to observe the following result:
\[trifreecol\] The chormatic number of a triangle free 2SUIG graph can be decided in polynomial time.
Now we are ready to prove our main result.
***Proof of Theorem \[th trifreecol\]:*** Let $G= (V_1\sqcup V_2,E)$ be a bridge triangle free 2SUIG with a given partition. Note that $G[V_1]$ and $G[V_2]$ are unit interval graphs. We prove the statement using induction on clique number $\omega(G)$. Note that the theorem is true for graphs $G$ with $\omega(G)=2$ by Lemma \[trifreecol\]. Assume that the theorem is true for all bridge triangle free 2SUIG $G$ with $\omega(G)\leq m$. Let $G=(V_1\sqcup V_2,E)$ be a bridge triangle free 2SUIG with $\omega(G)= m+1$. We delete a maximal independent set from $G$ to obtain the graph $G'$. Note that $\omega(G') \leq m$ and hence admits a $(m+1)$-coloring by our induction hypothesis. Now we extend this coloring by assigning a new color to the vertices of the deleted maximal independent set to obtain a $(m+2)$-coloring of $G$.
Matrix characterization {#sec matrix}
=======================
A graph $G$ is a *2-stab unit independent interval graph (2SUIIG)* if it admits a 2SUIG representation where the upper partition induces an independent set. The corresponding representation is a *2SUIIG representation*. The corresponding vertex partition $V_1 \sqcup V_2$ is a *strict partition* if $G[V_1 \cup \{v\}]$ is not a unit interval graph for any $v \in V_2$ (upper partition). It is easy to see that a graph is a $2SUIIG$ if and only if it has a strict partition. For the rest of the section denote the $x$-coordinate of the bottom-left corner of a unit square representing a vertex $v$ by $s_v$.
Note that the class of unit interval graphs is a subclass of 2SUIIG. Moreover, note that the class of 2SUIIG is not perfect as the 5-cycle admits a 2UIIG representation. Now we characterize the adjacency matrix of a 2SUIIG. We will define some matrix forms for that. The matrices we consider are 0-1 matrices. The element in the $i^{th}$ row and $j^{th}$ column of a matrix $\mathscr{M}$ is denoted by $\mathscr{M}_{ij}$. Also, the $i^{th}$ row and $j^{th}$ column of $\mathscr{M}$ is denoted by $\mathscr{M}_{i*}$ and $\mathscr{M}_{*j}$, respectively. Furthermore, $First(\mathscr{M}_{i*})$ and $Last(\mathscr{M}_{i*})$ denotes the column indices of the first and last non-zero entries of $\mathscr{M}_{i*}$, respectively.
A *stair normal interval representation (SNIR) matrix* $\mathscr{A}$ is a 0-1 matrix with the following properties:
- The 1’s in a row are consecutive.
- For $j < i$ we have $First(\mathscr{A}_{j*}) \leq First(\mathscr{A}_{i*})$ and $Last(\mathscr{A}_{j*}) \leq Last(\mathscr{A}_{i*})$.
Mertzios [@Mer] showed that a graph is a unit interval graph if and only if its adjacency matrix is a SNIR matrix. Let $[u \leadsto v]$ denote a longest directed path (not necessarily unique) in $\overrightarrow{I^c}$ from $u$ to $v$ and its length is denoted by $l_{uv}$.
A *proper stab adjacency (PSA) matrix* $\mathscr{A}$ is a 0-1 matrix with the following properties:
- The 1’s in a row are consecutive and each row has at most two 1’s.
- For $j < i$ and $\displaystyle\sum\limits_{k} \mathscr{A}_{jk} =2$ we have $First(\mathscr{A}_{j*}) \leq First(\mathscr{A}_{i*})$.
- For $j < i$ and $\displaystyle\sum\limits_{k} \mathscr{A}_{jk} =1$ we have $First(\mathscr{A}_{j*}) \leq First(\mathscr{A}_{i*}) + 1$. Equality holds only when $\displaystyle\sum\limits_{k} \mathscr{A}_{ik} =2$.
An *independence stair stab representation (ISSR) matrix*
$$\mathscr{A} _{(m+n) \times (m +n)} = \left[ \begin{array}{c|c}
\mathscr{A'}_{m \times m} & \mathscr{A''}_{m \times n} \\
\hline
\mathscr{A''}_{n \times m}^t & 0_{n \times n} \end{array} \right]$$
is a 0-1 matrix with the following properties:
- The submatrix $\mathscr{A'}$ is a SNIR matrix.
- The submatrix $\mathscr{A''}$ is a PSA matrix.
Note that using the characterization given by Mertzios [@Mer] the SNIR submatrix $\mathscr{A'}$ corresponds to a a unit interval graph $I$ (say). Let $\overrightarrow{I^c}$ be any transitive orientation of its complement.
- Let $j <i$, $m < k \leq m+n$, $\mathscr{A}_{ik} = 1$ and $\mathscr{A}_{jk}=1$. Let the rows $\mathscr{A}_{i*}$ and $\mathscr{A}_{j*}$ correspond to the vertices $u$ and $v$, respectively, of $I$. Then there is no directed path of length at least two from $v$ to $u$ in $\overrightarrow{I^c}$.
- Let $m < k \leq l \leq m+n$, $\mathscr{A}_{ik} = 1$ and $\mathscr{A}_{jl}=1$. Suppose $u_1$ and $u_p$ are the vertices corresponding to the rows $\mathscr{A}_{i*}$ and $\mathscr{A}_{j*}$, respectively. If $[u_1u_2...u_p]$ is a shortest path between $u_1$ and $u_p$ in $I$, then $(l-k) \leq p + 1$.
Now we are ready to state our main result:
\[matrix\] A graph is a 2SUIIG graph if and only if it can be represented in ISSR matrix form.
To prove Theorem \[matrix\] we need to prove some lemmas.
\[ubn\] Let $G= (V_1 \sqcup V_2,E)$ be a 2SUIIG with upper partition $V_2$. Then for a vertex $v \in V_1$ we have $|N_B(v)| \leq 2$.
Moreover, if $N_B(v) = \{x,z\}$ then it is not possible to have a vertex $y \in V_2$ with $s_x < s_y < s_z$.
Let $x,y,z \in V_2$ be such that $s_x < s_y < s_z$. As $x,y,z$ are independent, we must have $s_x +1 < s_y < s_y +1 < s_z$. If a vertex $v \in V_1$ is adjacent to both $x$ and $z$ then we must have $s_v \leq s_x +1$ and $s_z \leq s_v +1$. This implies $s_v \leq s_x +1 < s_y < s_z -1 \leq s_v$, a contradiction.
\[ubn\_clique\] Let $G= (V_1 \sqcup V_2,E)$ be a 2SUIIG with upper partition $V_2$. Then for two vertices $u,v \in V_1$ we have $\left|\displaystyle\bigcup\limits_{k=1}^{p} N_B(u_k)\right| \leq p+1$ where $u_k$ are the vertices of the shortest path $(P)$ between $u,v$ in $G[V_1]$ and $p$ is the length of $P$.
Let $N=\displaystyle\bigcup\limits_{k=1}^{p} N_B(u_k)$ and $|N| > p+1$. Let there is a representation $R$ of $G$. Without loss of generality assume $s_u < s_v$ in $R$. There must be a vertex $y,z\in N$ such that $(z,u),(y,v) \in E$, $s_u-1<s_z<s_u$ and $s_z+p+1+\epsilon$, $0<\epsilon<1$. In any representation of $G$, $|s_u - s_v| $ is at most $p-\epsilon$,$0<\epsilon<1$. Then the edge between $y,v$ cannot be realised leading to a contradiction.
The above lemmas gives an upper bound on the number of bridge neighbours of a vertex in $V_1$.
\[lbc2\] Let $G= (V_1 \sqcup V_2,E)$ be a 2SUIIG with upper partition $V_2$. For vertices $u,v \in V_1$ with $|N_B(u)| = 2$ and $s_u<s_v$ there exists $w\in N_B(u)$ such that for any $x \in N_B(v)$ we have $s_w \leq s_x$.
Let there are vertices $w,x \in V_2$ such that $s_x < s_w$ where $w\in N_B(u)$ and $x \in N_B(v)\setminus N_B(u)$. Then the union of the unit squares corresponding to $u$ and $w$ divides the region enclosed by the stab lines into two disjoint parts. Then the intersection of the unit squares corresponding to $v$ and $x$ cannot be realised.
(0,0) circle (2pt) [node\[below\][$w$]{}]{}; (1,0) circle (2pt) [node\[below\][$x$]{}]{}; (2,0) circle (2pt) [node\[below\][$y$]{}]{}; (3,0) circle (2pt) [node\[below\][$z$]{}]{};
(1,1) circle (2pt) [node\[above\][$a$]{}]{}; (3,1) circle (2pt) [node\[above\][$b$]{}]{};
(0,0) – (3,0);
(0,0) – (1,1); (2,0) – (1,1); (1,0) – (3,1); (2,0) – (3,1);
(5,-.3) – (6,-.3) – (6,.7) – (5,.7) – (5,-.3);
at (4.9,-.4) [$w$]{};
(5.5,-.6) – (6.5,-.6) – (6.5,.4) – (5.5,.4) – (5.5,-.6);
at (5.4,-.7) [$x$]{};
(5.2,.6) – (6.2,.6) – (6.2,1.6) – (5.2,1.6) – (5.2,.6);
at (5.1,1.7) [$a$]{};
(5.8,-.2) – (6.8,-.2) – (6.8,.8) – (5.8,.8) – (5.8,-.2);
at (6.9,-.3) [$y$]{};
(6.3,.3) – (7.3,.3) – (7.3,1.3) – (6.3,1.3) – (6.3,.3);
at (7.4,.4) [$b$]{};
(6.6,-1) – (7.6,-1) – (7.6,0) – (6.6,0) – (6.6,-1);
at (7.7,-1.1) [$z$]{};
(4,-.12) – (8.5,-.12);
(4,1) – (8.5,1);
\[lbc1\] Let $G= (V_1\sqcup V_2,E)$ be a 2SUIIG with upper partition $V_2$. For vertices $x,y \in V_1$ with $|N_B(x)| = 1$ and $s_x<s_y$ we have $s_b - 2 < s_a$ where $b\in N_B(x)$ and $a \in N_B(y)$.
Let we have vertices $x,y \in V_1$ with $|N_B(x)| = 1$ and $s_x<s_y$ we have $s_a \leq s_b - 2$ where $b\in N_B(x)$ and $a \in N_B(y)$. Now $s_b < s_x+1$. If $s_a \leq s_b - 2$ then to realize the intersection between $y,a$ we will have $s_y < s_x$, which is a contradiction. Note that the bound $s_b - 2$ is tight as we can have a situation illustrated in Fig. \[fig adjacency\]. Moreover in such situations $b$ must be adjacent to $y$ also.
Intuitively, Lemma \[lbc2\] and Lemma \[lbc1\] show that in a 2SUIIG representation, the ordering of the intervals corresponding to the vertices in $G[V_1]$ fixes an ordering of the intervals corresponding to the vertices in $G[V_2]$.
Let $G= (V_1\sqcup V_2,E)$ be a 2SUIIG with upper partition $V_2$. Assume that $|V_1| = m$ and $|V_2| = n$ We know that it admits a strict partition. From this strict partition we will construct a matrix which we will prove is a PSA matrix. Note that $G[V_1]$ is a unit interval graph and hence Mertzios [@Mer] provides a (adjacency) SNIR matrix $\mathscr{A}_{m\times m}$ of $G[V_1]$. This matrix is obtained by putting the vertices of $V_1$ in a particular order. We will show that, we can obtain a particular order of $V_2$ such that the biadjacency matrix $\mathscr{A}^{'}_{m\times n}$ of $V_1$ (taken in the same order as above) and $V_2$ is a PSA matrix.
Call it $\mathscr{A}_{m\times m}$. $\mathscr{A}^{'}_{m\times n}$ be a zero one matrix. An entry $\mathscr{A}^{'}_{ij}$ is 1 if there is a bridge between $i,j$. The ordering of the rows of $\mathscr{A}^{'}$ remains same as the ordering of the rows in $\mathscr{A}$. The ordering of the columns of $\mathscr{A}^{'}$ corresponds to an ordering of the vertices of $V_2$.
We order the vertices of $V_2$ using the following prescribed rule. Recall that the adjacency matrix $\mathscr{A}$ of $G[V_1]$ produces an interval intersection representation of (may not be unique) the graph. Fix one such representation $R$. Suppose we have $u,v \in V_1$ and $u',v' \in V_2$ such that $u' \in N_B(u)\setminus N_B(v)$ and $v' \in N_B(v)$. If $u,v$ are not twins (that is, have the same set of neighbors) in $G[V_1]$ and $s_u < s_v$ in $R$ then we want $u'< v'$ in our ordering of $V_2$. In any other case we take an arbitrary ordering between a pair of vertices $u',v' \in V_2$. This is a well defined algorithm for getting the ordering due to the previously proved lemmas.
\[PSA\] The matrix $\mathscr{A}^{'}$ is a PSA matrix.
Suppose $\mathscr{A}^{'}$ is not PSA matrix. Note that, there cannot be any zero column in it as we have assumed a strict partition of the graph. Now we will check all the properties mentioned in the definition of the PSA matrix.
*Property $(i)$:* Due to Lemma \[ubn\] each vertex in $V_1$ have at most two neighbours and they are consecutive.
*Property $(ii)$:* Suppose a vertex in $v \in V_1$ has two bridge neighbours and $w$ be the left most bridge neighbour of $v$. $u$ be another vertex in $V_1$ lying to the right of $v$. From Lemma \[lbc2\], we know that the bridge neighbours of $u$ will lie to the right of $w$. Hence, this property is also satisfied.
*Property $(iii)$:* Assume that this property is not satisfied and $x,y$ be the vertices corresponding to the rows violating the property. Now $s_x < s_y$. Let $b$ be the bridge neighbour of $x$. There is a bridge neighbour $a$ of $y$ such that there is another vertex $c \in V_2$ with $s_a < s_c < s_b$. Hence, $s_b - 2 > s_a$. This contradicts Lemma \[lbc1\]. $\hfill \square$
Now we are ready to prove our main theorem.
***Proof of Theorem \[matrix\]:*** First we will prove the “if" part. Let $G= (V_1\sqcup V_2,E)$ be a 2SUIIG with upper partition $V_2$. Then by the discussions above Lemma \[PSA\] we obtained an ordering of the vertices of $V_1$ and $V_2$. Consider the ordering of $V_1 \sqcup V_2$ by putting the vertices of $V_1$ in the previously obtained order followed by the vertices of $V_2$ (in previously obtained order as well). This ordering will give us a matrix of the following type:
$$Adj(G) = \left[ \begin{array}{c|c}
\mathscr{A} & \mathscr{A'} \\
\hline
\mathscr{A'}^t & 0 \end{array} \right]$$
From the previous lemmas we know that the above matrix satisfies the first two properties for being a ISSR matrix. Now we check the remaining two properties.
*Property $(iii)$:* As a consequence of Lemma \[ubn\] any vertex in $V_2$ can have at most two bridge neighbours which are pair wise independent.
*Property $(iv)$:* As a consequence of Lemma \[ubn\_clique\] for any vertices in $G[V_1]$, the vertices of the shortest path between $u,v$ in $G[V_1]$ can have at most a total of $p+1$ bridge neighbours where $p$ is the length of the shortest path.
For the “only if” part, given a ISSR matrix we can construct a unit interval graph (and generate the intervals corresponding to the vertices) with the SNIR sub-matrix of it. Now we show that the column ordering of the PSA sub-matrix generates the intervals of the rest of the vertices. If the PSA sub-matrix have only one non-zero cell, then there is only one vertex, $x$. Due to property (iii) of ISSR matrix the unit square $s_x$ can be generated and the bridge edge can be realized. Assume that the theorem is true for all ISSR matrix whose PSA sub-matrix have $k$ columns. Consider a ISSR matrix $M$ whose PSA sub-matrix have $m=k+1$ columns. Consider $M'$ as the matrix which is obtained by deleting the last columns of the PSA sub-matrix of $M$. From induction hypothesis, there is a 2SUIIG representation of $M'$, say $R$. Let $x$ be a vertex corresponding to the $(k+1)^{th}$ column. Let $Y$ be the set of vertices corresponding to the rows $i$ such that $M_{i(k+1)} = 1$. $y$ be an element of $Y$. So, in any 2SUIIG representation of $M$, $|s_y - s_x | \leq 1$.
**case 1.** Let there is a vertex $y \in Y$ such that for any $s_x$ with $s_y < s_x < s_y + 1$ in $R$, $G[V_2]$ remains independent but the bridge edge $xy$ cannot be realised. The bridge edge $xy$ cannot be realised implies there are $u,v$ in $V_2,V_1$ respectively such that $s_u < s_x,s_y < s_v$ and $u,v$ are adjacent. If $y,u$ were adjacent, then we could have realised the bridge edge between $x,y$. Again if $v,x$ were not adjacent then $M$ would violate property (iii) of the definition of PSA matrix. The adjacency of $v,x$ ensures that the bridge edge between $x,y$ can be realised.
**case 2.** Let there is a vertex $y \in Y$ such that for any $s_x$ with $s_y < s_x < s_y + 1$ in $R$, $G[V_2]$ does not remain independent. This implies that there exists $u,v$ in $R$ such that if $s_y < s_x < s_y + 1$ and $G[V_2]$ is an independent set then the bridge edge between $u,v$ cannot be realised. Let $Z=~\{a:~a~\in~V_2~\text{ and }~s_u-1~< s_a <~s_y+1~\text{ in R}\}$. Let $P=[u_1u_2\ldots u_k]$ be the shortest path from $u$ to $y$ in $G[V_1]$ where $u_1=u$ and $u_k=y$. Now $ |Z| = k+1$, otherwise the bridge edge between $x,y$ can be realised. But then $M$, violates the definition of ISSR matrix.
Property (iii) of ISSR matrix insures that for all $y,z \in Y$ with $s_y < s_z$ in $R$, if $s_y < s_x < s_y+1$ we can have $|s_z - s_x | \leq 1$. Hence, all the bridge edges between $x$ and the elements of $Y$ can be realised. This completes the proof.
Conclusion
==========
The complexity of recognizing 2SIG is not known and hence is an interesting future problem. Domination number of 2SIG is one of the prospective areas of research as the problem is polynomial time solvable for interval graphs but is $NP$-hard for boxicity 2 graphs. One can also generalize the concept to define $k$-stab interval graphs and study its different aspects.
|
---
abstract: |
Existing traffic engineering (TE) solutions performs well for software defined network (SDN) in average cases. However, during peak hours, bursty traffic spikes are challenging to handle, because it is difficult to react in time and guarantee high performance even after failures with limited flow entries.
Instead of leaving some capacity empty to guarantee no congestion happens due to traffic rerouting after failures or path updating after demand or topology changes, we decide to make full use of the network capacity to satisfy the demands for heavily-loaded peak hours. The TE system also needs to react to failures quickly and utilize the priority queue to guarantee the transmission of loss and delay sensitive traffic. We propose TED, a scalable TE system that can guarantee high throughput in peak hours. TED can quickly compute a group of maximum number of edge-disjoint paths for each ingress-egress switch pair. We design two methods to select paths under the flow entry limit. We then input the selected paths to our TE to minimize the maximum link utilization. In case of large traffic matrix making the maximum link utilization larger than 1, we input the utilization and the traffic matrix to the optimization of maximizing overall throughput under a new constrain. Thus we obtain a realistic traffic matrix, which has the maximum overall throughput and guarantees no traffic starvation for each switch pair. Experiments show that TED has much better performance for heavily-loaded SDN and has $10\%$ higher probability to satisfy all ($> 99.99\%$) the traffic after a single link failure for G-Scale topology than Smore under the same flow entry limit.
author:
- |
Che Zhang\
SUSTech
- |
Shiwei Zhang\
SUSTech
- |
Yi Wang\
SUSTech
- |
Weichao Li\
SUSTech
- |
Bo Jin\
SUSTech
- |
Ricky K. P. Mok\
CADIA/UC San Diego
- |
Qing Li\
SUSTech
- |
Hong Xu\
CityU of Hong Kong
- 'Paper \#XXX, pages'
bibliography:
- 'main.bib'
date:
-
-
title:
- 'Scalable Traffic Engineering for Higher Throughput in Heavily-loaded Software Defined Networks'
- XXX
---
Introduction {#sec:introduction}
============
CAIDA [@ref:inferring] observed that congestion was not widespreaded on the peer/provider interdomain links during their measurement period. However, Akamai said that on Dec. 11, 2018, the volume of data passed 72 Tbps, which equated to delivering more than 10 million DVDs per hour, was largely attributed to streaming of live sports events, etc [@akamai].
Several solutions work well in non-peak hours. However, they are not suitable for peak hours to handle high volume traffic bursts and guarantee high performance even after failures. We can classify these solutions into three categories. The first category is to proactively consider failures when formulating the TE problem. However, these solutions neither may not be scalable to large network due to exponential number of all possible failure scenarios [@ref:joint; @ref:teavar] nor require to reserve significant portion of network capacity in order to guarantee no congestion happens for arbitrary k faults with rescaling [@ref:ffc]. The second way is to periodically update the paths and weights according to current traffic demand matrix, but it also needs to reserve bandwidth to prevent network congestion or network looping caused by path update events [@ref:swan]. The third type is to pre-calculate and deploy a large number of static paths to avoid a series of problems caused by path updates, e.g., Smore’s semi-oblivious routing [@ref:smore]. They often use minimizing the maximum link utilization (MLU) as the optimization target. Although minimizing the maximum link utilization can make network load balanced, it will limit the whole throughput and lead to large amount of packet loss when the load is heavy as it requires a constrain to satisfy all the demands. Maximizing the whole throughput is a natural objective for heavy load, but optimize the whole throughput only may lead to traffic starvation for some ingress-egress switch pairs.
Moreover, Google found that off-the-shelf switch chips impose hard limits on the size of each flow table [@ref:b4andafter], which also limit the number of deployable static paths especially for large scale networks as traffic engineering systems usually implement flow group matching using ACL tables to leverage their generic wildcard matching capability. Motivated by [@ref:b4andafter] and the design of the third type of TE method [@ref:smore], we design TED to solve the above problems. TED utilizes the advantages of both optimizations to maximizing the throughput with no starvation for each switch pair guaranteed.
TED is a TE system that is fast, scalable, and simple to be used. TED typically includes three phases: I) path set computation, II) path selection, and III) weight computation & bandwidth allocation. By extending Dinic’s algorithm [@ref:dinic], TED can compute a group of maximum number of edge-disjoint paths between each ingress-egress switch pair with time complexity $O(n^2m)$ (The algorithm is called “Custom" to remind users that they can also use other path computation methods in our TED system). Using the limitation of the number of flow entries for each switch (router) in the WAN network topology, TED can search the maximum path budget with time complexity $O(n^2)$ (“hardnop"). Furthermore, we design a two-step path selection method (“program") to select paths using 0-1 integer linear programming (Fig. 10c).
For TE optimization, the goal of TED is to minimize the maximum link utilization. We do not directly apply the weights when its result $Z>1$ ($Z$ stands for the maximum link utilization) to avoid network congestion. Instead, Phase IV of TED is triggered. Input $TM$ and $Z$ to a new TE optimization with objective of maximizing the overall throughput under link capacity constraint, weights and bandwidth allocation are re-computed under new constraint $TM/Z \leq TM' \leq TM$. Phase IV not only guarantees no congestion even when $Z>1$, but also can fully utilize the network capacity under the guarantee that each switch pair can at least meet $1/Z$ of its demand. Another feature of TED is customizable. The operator or network slicing user can apply other path set computation algorithms, limit the use of flow entries, or reserve a portion of link capacity for robustness.
Evaluation shows TED has much better performance for heavily-loaded networks (Fig. 8a) and has 10% more possibility to satisfy all the traffic ($P(T>99.99\%)$) after single link failures than Smore (Fig. 11a, under the same flow entry limit and topology). TED has only less than 10% $s-t$ pairs assigned an average path length, which is longer than the maximum average path length computed by Smore (Fig.11b), but Smore uses 33% or more flow entries in some switches (Fig. 7a, 7b).
Background {#sec:background}
==========
The most commonly used way to solve the multi-commodity flow problem [@ref:networkflow], liner programming takes the input of $(s_i,t_i,d_i)$ pairs and the paths between $(s_i,t_i)$, where $s_i$ and $t_i$ are the $i^{th}$ pair of ingress ($s$) and egress ($t$) routers with demand $d_i$ (refer to Fig. \[fig:sys\] Phase II).
We use a simple example to illustrate the influence of selection of paths to bandwidth allocation result of TE and reliability of the network. Fig. \[fig:moti\] (a) shows that if we select two shortest paths, indicated with red dotted line ($1 \rightarrow 2 \rightarrow 3 \rightarrow 5 \rightarrow 6$) and blue solid line ($1 \rightarrow 4 \rightarrow 5 \rightarrow 6$ ) for $s-t$ pair $(1,6)$. Because the two paths share the link $5 \rightarrow 6$, each of them can only obtain half of link capacity. SWAN [@ref:swan] finds that each ingress-egress switch pair needs to have at least 15 shortest paths to fully utilize the overall network throughput. In contrast, by selecting two edge-disjoint paths as shown in Fig \[fig:moti\] (b), the available bandwidth of both paths is doubled. Even though the two disjoint paths are not the shortest ones, they are more robust to single link failures.
![Comparison of bandwidth allocation between selecting two shortest paths (a) and two edge disjoint paths (b).[]{data-label="fig:moti"}](motiexample.pdf){width="0.6\linewidth"}
We further analyzed Google G-Scale Topology [@ref:b4] to verify our previous conclusion. We selected $k$-shortest paths, where $k$ is the maximum number of edge-disjoint paths, for any $s-t$ pair. We found that, on average, around five $s-t$ pairs had at least two paths that are vulnerable to any single link failure. In the worst case, 13.6% pairs had at least two paths with shared bottleneck, which is undesirable for WANs with high link utilization. Although 26.3% of single link failures did not impact on more than one $k$-shortest path for any node pair, the remaining 73.7% single link failure can cause significant impact to the network because of the shared link among selected paths. A suitable path selection is the key of ensuring high reliability and availability in WANs. Specifically, selecting edge-disjoint paths can make network more fault-tolerance and even improve the overall network throughput with a chance of a slight sacrifice of latency.
![Using edge-disjoint paths computation method in previous TE system architecture.[]{data-label="fig:sys"}](path-selection-system-enold.pdf){width="1.1\linewidth"}
![When $TM$ is large, using the weights computed by minimize the maximum link utilization may lead to congestion. Naive way to limit the bandwidth of each path as $TM/Z$ may lead to inefficiency using of network capacity.[]{data-label="fig:sysnew1"}](path-selection-system-en5new.pdf){width="1.0\linewidth"}
Next, we will show why we cannot directly apply traditional TE system. Fig. \[fig:sys\] illustrate our attempt to combine edge disjoint paths computation with traditional two-phase TE system model. In phase I, we select the maximum number of edge disjoint paths. After that, we employ the paths in phase I and using minimizing maximum link utilization optimization to compute the bandwidth allocation and corresponding weights in phase II. It seems to work well, but first, WANs need to guarantee all-to-all connections and when the number of flow entries are limited, we cannot simply select commonly used 3 or 4 paths [@ref:joint; @ref:b4; @ref:smore] between each ingress-egress switch pair. It is because the result can be either limited flow entries cannot support those paths especially in large WANs or path diversity decreases due to the “3 or 4" limitation.
With development of network measurement like [@ref:bwe] and SDN-based system used in Internet like Google’s Espresso [@espresso] and Facebook’s Edge fabric [@edgefabric], we believe that the estimated or measured $TM$ can be more accurate. TED intends to use minimizing the *maximum link utilization* ($Z$) optimization to compute bandwidth allocation and corresponding weights, as it has a constraint to meet all demands in $TM$ and balances the traffic. Furthermore, the implementation of such optimization is easier than minimizing the overall network congestion which is a convex optimization and hard to select a proper penalty function to use [@ref:joint].
The $TM$ increases as the use and expectation of high quality (High definition, 4K, or even 8K) video streaming, IPTV, and video conferencing [@edgefabric]. Then $Z$ may be larger than 1 as the optimization of minimizing the maximum link utilization has no link capacity constraint (in this case, the large $TM$ may not be satisfied and the optimization will have no result that satisfies all constraints). Although when $Z>1$ we can simply allocate bandwidth as $TM/Z$ without changing the weights to guarantee no congestion happens, the network may not be fully used as shown in the example of Fig. \[fig:sysnew1\].
Design Overview {#sec:design}
===============
{width="100.00000%"}
In this section, we present the design of TED. Fig. \[fig:sysnew4\] depicts the architecture of TED in a four-phase TE system model.
We compute the maximum number of edge disjoint paths between each ingress-egress router pair as the path set in Phase I. We add a path selection phase II to deal with flow entries’ limitation and weight re-computation & bandwidth re-allocation phase IV to deal with large $TM$ that makes $Z>1$.
Our prioritised objective is to 1. meet the $TM$, 2. minimize failure impact, 3. guarantee at least $TM/Z$ is satisfied, when $TM$ can not be satisfied without causing congestion and 4. maximize overall network throughput. In other words, minimizing network congestion so that the impacted traffic are as little as possible after failures. And such traffic can still be satisfied almost maximally after rerouting or using the failover mechanism. Even for network failures in peak hours, we can still guarantee the transmission of loss and delay sensitive traffic utilizing priority queue and fast failover as most video traffic are loss tolerant. Moreover, TED can recompute the paths, weights and allocated bandwidth quickly to further recover the whole network maximally.
The detailed explanation of each phase are as follows.
Compute A Group of Maximum Number of s-t Edge-Disjoint Paths
------------------------------------------------------------
Select paths for demands: $(s_i,t_i,d_i), 1 \leq i \leq n*(n-1)$. Finding all groups of $s-t$ edge-disjoint paths is too complicated and requires exponential time. Therefore, we look for a group of edge-disjoint paths with maximum total number.
A lot of fast algorithms have been developed to solve the maximum flow problem for unit capacity, undirected networks [@ref:dinicmit]. Papers by Karzanov [@ref:karza] and Even and Tarjan [@ref:flowbook] showed that, for unit capacity networks (directed or undirected), a method called blocking flows invented by Dinic [@ref:dinic], solves the maximum flow problem in $O(m$ min $({n^{2/3}, m^{1/2}, v}))$ time (v is the value of the maximum flow).
We extend Dinic’s algorithm [^1] [@ref:dinic] to compute a group of maximum number of edge disjoint paths for our undirected graph with unit capacity (the weight of each edge is 1).
### Problem description
Given an undirected graph and two nodes (i.e., the source $s$ and the destination $t$) in it, the problem is to find out the maximum number of edge-disjoint paths from the source to the destination. Two paths are said edge-disjoint if they don’t share any edge.
This problem can be solved by reducing it to maximum flow problem. Following are steps:
1. Transform the original undirected graph into a symmetric directed graph;
2. Assign unit capacity to each edge;
3. Consider the given source and destination as source and sink in a flow network;
4. Run classic maximum flow algorithms to find the maximum flow from source to sink;
5. The maximum flow is equal to the maximum number of edge-disjoint paths.
### Path construction
Now we get the final flow matrix $f^*$ from the maximum flow algorithm. Note that multiple optimal solutions may exist, but the maximum flow value is the same.
With $f^*$, we are next to construct edge-disjoint paths. Note that even for a single flow matrix $f^*$, different path sets may be constructed.
The process of the path construction is quite simple:
1. Starting from the source, for each node $u$, find the next *unvisited* edge $(u,v)$ such that $f(u,v)=1$, and then move forward to $v$. Once the sink is reached, one path is constructed.
2. Repeat the previous process until all edges are visited.
Note that we can not use the augmenting paths directly, because sometimes, two augmenting paths may pass through the positive and negative direction of the same undirected edge as shown in Fig. \[fig:alg\] (a). For the max flow network which has the skew symmetry property is shown in Fig. \[fig:alg\] (b).
![The augmenting paths used in Dinic’s algorithm to compute the max flow of $1 \rightarrow 6$ for original graph (a) and the max flow network of $1 \rightarrow 6$ (b) in which the edge $(4,5)$ (whose two directions are both used) is counteracted.[]{data-label="fig:alg"}](algexample.pdf){width="0.6\linewidth"}
When $disp_i=2$, we can use Suurballe algorithm [@ref:suurballe] to find the shortest pairs of edge disjoint paths which has minimum total length. Notice that our path computation method can not guarantee the minimum total length.
Select Paths
------------
As the limitation of switch memory (e.g., TCAM) and the fairness among each $s-t$ pair, we select paths based on path budget [@ref:smore]. First, we sort the paths based on their weights. For our maximum number of edge disjoint paths, we use the path length as the weight in order to reduce the latency. Second, different from previous work [@ref:smore], we get the maximum path budget $K$ under flow entries’ limitation with time complexity $O(n^2)$. Third, we select the first $min(K,X)$ paths for each $s-t$ pair ($X$ stands for the maximum number of edge disjoint $s-t$ paths).
For the second step, considering normally $K$ is small, we can check from $i=1$ path for each $s-t$ pair and record the used number of flow entries of each switch to a vector. Judge whether they are under the flow entries’ limitation. If no, stop and $K=i-1$, otherwise, $i++$, go to next cycle. To be faster, in each cycle, we can reuse the previously recorded vector and only need to add the used flow entries of $i^{th}$ path for each $s-t$ pair to that vector.
We also design a two-step path selection method (“program") to make full use of limited flow entries. Step 1. find the result $F$ of maximize minimum $nop_{s,t}$ (means number of paths between $s$ and $t$) shown in Opt. .
Variable Description
------------------------------------- -------------------------------------------------------------------------------
$\mathbb{G}(\mathbb{V},\mathbb{E})$ network with vertices $\mathbb{V}$ and directed edges $\mathbb{E}$
$c_{e_k}$ capacity of $k^{\text{th}}$ edge, $e_k \in \mathbb{E}$
$\mathbb{P}$ a set of selected edge disjoint paths
$\mathbb{TM}$ bandwidth demand matrix
$h_{v_o}$ limitation of number of flow entries in $o^{\text{th}}$ switch
$\mathbb{D}$ bandwidth demand set, $\forall (s,t) \in \mathbb{D}, \mathbb{TM}[s,t] \neq 0$
$(s_i,t_i)$ $i^{\text{th}}$ ingress-egress switch pair in $\mathbb{D}$
$\mathbb{P}(s_i,t_i)$ edge disjoint paths between $s_i$ and $t_i$
$L[p_j,e_k]$ if path $p_j$ passed through $e_k$, $L[p_j,e_k]=1$, else 0
$R[p_j,v_o]$ if path $p_j$ passed through $v_o$, $R[p_j,v_o]=1$, else 0
$d_{s_i,t_i}$ bandwidth demand of $(s_i,t_i)$, $d_{s_i,t_i}=\mathbb{D}(s_i,t_i)$
$l_{e_k}$ overall used bandwidth of link $e_k$
$q_{v_o}$ used flow entries of $o^{\text{th}}$ switch
$a_{s_i,t_i}^{p_j}$ 1 for selecting $j^{\text{th}}$ path of $(s_i,t_i)$, else 0
$b_{s_i,t_i}^{p_j}$ allocated bandwidth for path $p_j$ of $(s_i,t_i)$
$w_{s_i,t_i}^{p_j}$ weight for path $p_j$ of $(s_i,t_i)$
: Summary of notation[]{data-label="table:symbols"}
$$\tag{1}\label{opt:maxminnopst}
\begin{aligned}
&\text{max} & & min_{(s_i,t_i) \in \mathbb{D}}{(nop_{s_i,t_i})} \\
& \text{s.t.} & & a_{s_i,t_i}^{p_j} \in \{0,1\}\\
&&& q_{v_o}=\sum_{(s_i,t_i) \in \mathbb{D}}{\sum_{p_j \in \mathbb{P}(s_i,t_i)}{a_{s_i,t_i}^{p_j}*R[p_j,v_o]}} \leq h_{v_o}, \forall v_o \in \mathbb{V}\\
&&& nop_{s_i,t_i}=\sum_{p_j \in \mathbb{P}(s_i,t_i)}{a_{s_i,t_i}^{p_j}}, \forall (s_i,t_i) \in \mathbb{D}\\
\end{aligned}$$
$$ \tag{2}\label{opt:maxsumnopst}
\begin{aligned}
&\text{max} & & \sum_{(s_i,t_i) \in \mathbb{D}}{nop_{s_i,t_i}} \\
& \text{s.t.} & & a_{s_i,t_i}^{p_j} \in \{0,1\}\\
&&& q_{v_o}=\sum_{(s_i,t_i) \in \mathbb{D}}{\sum_{p_j \in \mathbb{P}(s_i,t_i)}{a_{s_i,t_i}^{p_j}*R[p_j,v_o]}} \leq h_{v_o}, \forall v_o \in \mathbb{V}\\
&&& F \leq nop_{s_i,t_i}=\sum_{p_j \in \mathbb{P}(s_i,t_i)}{a_{s_i,t_i}^{p_j}}, \forall (s_i,t_i) \in
\mathbb{D}
\end{aligned}$$
Step 2. input $F$ to find the path selection result of maximize sum of $nop_{s,t}$ with a new constrain $F \leq nop_{s,t}$ to guarantee each $s-t$ pair at least has $F$ paths (Opt. ). The symbols and their explanations are shown in Table \[table:symbols\].
TE Optimization Algorithm
-------------------------
We use linear programming to compute the bandwidth allocation result of TE. Its input includes the edge disjoint paths $\mathbb{P}$ computed by the above path selection subsystem, the bandwidth demand matrix $\mathbb{TM}$ and the network topology $\mathbb{G}(\mathbb{V},\mathbb{E})$. The output is allocated bandwidth $b_{s_i,t_i}^{p_j}$ for each path of each $s_i-t_i$ pair. We can obtain the weight $w_{s_i,t_i}^{p_j}$ after normalization ( $w_{s_i,t_i}^{p_j}=\frac{b_{s_i,t_i}^{p_j}}{b_{s_i,t_i}},\sum_{p_j \in \mathbb{P}(s_i,t_i)}{w_{s_i,t_i}^{p_j}}=1$).
We use minimize max($l_e/c_e$) (Opt. ) as the target of Phase III. If the bandwidth allocation result makes some $l_e>c_e$ (set $Z=max(l_e/c_e)$), Phase IV is triggered and we input $TM$ and $Z$ to the TE with target of maximize the whole throughput and the bandwidth constrain will be changed to $d_{s_i,t_i}/Z \leq b_{s_i,t_i} \leq d_{s_i,t_i}$ (Opt. ). Thus the bandwidth allocation result will also hold the property: max($l_e$)=1.
$$\tag{3}\label{opt:minmaxlu}
\begin{aligned}
&\text{min} & & max_{e_k \in \mathbb{E}}{(l_{e_k}/c_{e_k})} \\
& \text{s.t.} & & 0 \leq b_{s_i,t_i}^{p_j}, \; \forall (s_i,t_i) \in \mathbb{D}, p_j \in \mathbb{P}(s_i,t_i)\\
&&& b_{s_i,t_i}=\sum_{p_j \in \mathbb{P}(s_i,t_i)}{b_{s_i,t_i}^{p_j}}=d_{s_i,t_i}, \forall (s_i,t_i) \in \mathbb{D}\\
&&& l_{e_k}=\sum_{(s_i,t_i) \in \mathbb{D}}{\sum_{p_j \in \mathbb{P}(s_i,t_i)}{b_{s_i,t_i}^{p_j}*L[p_j,e_k]}}, \forall e_k \in \mathbb{E}\\
\end{aligned}$$
$$\tag{4}\label{opt:maxT}
\begin{aligned}
&\text{max} & & \sum_{(s_i,t_i) \in \mathbb{D}}{b_{s_i,t_i}} \\
& \text{s.t.} & & 0 \leq b_{s_i,t_i}^{p_j}, \; \forall (s_i,t_i) \in \mathbb{D}, p_j \in \mathbb{P}(s_i,t_i)\\
&&& b_{s_i,t_i}=\sum_{p_j \in \mathbb{P}(s_i,t_i)}{b_{s_i,t_i}^{p_j}} \leq d_{s_i,t_i}, \forall (s_i,t_i) \in \mathbb{D}\\
&&& d_{s_i,t_i}/Z \leq b_{s_i,t_i}, \forall (s_i,t_i) \in \mathbb{D}\\
&&& l_{e_k}=\sum_{(s_i,t_i) \in \mathbb{D}}{\sum_{p_j \in \mathbb{P}(s_i,t_i)}{b_{s_i,t_i}^{p_j}*L[p_j,e_k]}} \leq c_{e_k}, \forall e_k \in \mathbb{E}\\
\end{aligned}$$
Evaluation {#sec:evaluation}
==========
![Google G-Scale topology.[]{data-label="fig:gscale"}](GScalegraph.pdf){width="0.6\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
[0.325]{} {width="\linewidth"}
Setup
-----
*Methodology.* We vary demands, single link failures and topologies to compare path computation algorithms, path selection algorithms and TE optimizations.
*TM Generation.* We use gravity model [@ref:genTM] to generate 60 traffic matrices randomly.
*Topologies.* We select 20 topologies from topology zoo [@ref:topologyzoo], which are also used in Yates [@ref:yates].
*Path set computation algorithms.* Our algorithm based on Dinic to compute the maximum number of edge disjoint paths is called “Custom" in the figures (implemented in C++). We compare with Racke, Ksp, Vlb, Ecmp, Edksp which are implemented in OCaml by Yates[@ref:yates]. Racke stands for Racke’s oblivious routing algorithm used in Smore [@ref:smore]. Ksp stands for Yen’s algorithm to compute k-shortest paths which is commonly used in TE. Vlb [@ref:vlb] means Valiant Load Balancing which routes traffic via randomly selected intermediate nodes. Ecmp is widely used equal-cost multi-path routing. Edksp is short for edge-disjoint $k$-shortest paths.
*Path selection algorithms.* Our path selection algorithm is denoted as “hardnop" in the figures (select the maximum path budget that flow entries can hold). We compare it with another method designed by us, called “program". It is a two-step path selection method to find path selection result by making full use of all flow entries. We implement both of them in Julia, and apply the same flow entries’ limitation to both methods.
*TE implementation.* We compare TED with optimal MCF [@ref:optimalmcf], TE (TM) and TE (TM/Z, when $Z>1$). Their target is to minimize the maximum link utilization. Optimal MCF does not have path or flow entry limitation (does not need to input paths as it can use the whole network under the flow conservation constraint). TE (TM) is the unmodified TE with objective of minimizing the maximum link utilization used in Smore [@ref:smore]. TE (TM/Z, when $Z>1$) is the simple modified TE, when $Z>1$, to use $TM/Z$ as the bandwidth allocation result to guarantee no congestion. We implement them in Julia by calling Gurobi’s optimization solver.
We use the following metrics for performance evaluation. When $Zopt \leq 1$ ($Zopt$ means the maximum link utilization result of optimal MCF), we evaluate the performance ratio ($= Zalg/Zopt$) [@ref:perfratio]; and when $Zopt>1$, we use the throughput ratio ($= Talg/(sum(TM)/Zopt)$). $Zalg$ and $Talg$ is computed using TED’s architecture and using corresponding “alg" as the algorithm to compute the path set in Phase I. Performance ratio show that how far from the $Zalg$ to $Zopt$. Throughput ratio shows that how much improve of $Talg$ compared with $sum(TM)/Zopt$ (simple method to use $TM/Zopt$ guaranteeing no congestion when $Zopt>1$). The source code of our experiment is on .
Number of flow entries
----------------------
We use much less number of flow entries (Fig. \[fig:flow-gscale-1-hardnop-ccdf\]) than Vlb, Ksp and Racke, but more than Ecmp for Gscale. Later we will show our robustness and performance is competitive with using other algorithms and much better than Ecmp. This is important for real networks which may not have that much flow entries. Especially to notice that large networks to guarantee good all-to-all communication need more than twice flow entries for Ksp and Racke compared with our Custom (Fig. \[fig:flow-cernet-1-hardnop-ccdf\]).
Also, comparing with two-step path selection method, “program" (Fig. \[fig:flow-gscale-1-program-ccdf\]), Racke, Ksp and Vlb use more flow entries than our path selection method, “hardnop" (Fig. \[fig:flow-gscale-1-hardnop-ccdf\]). This is because “program" can make full use of the flow entries and for Racke, Ksp and Vlb, their path sets have more paths to select than Ecmp, Edksp and Custom. However, next we will show that “program" does not have better performance than “hardnop" for each path set computing algorithms at least for G-Scale-like topologies.
Efficiency
----------
Fig. \[fig:algtime\] shows the computation of various algorithms. We can see that our path computation algorithm Custom is faster than Racke and much faster than Edksp. It is especially important when failure happens, to compute new paths in order to guarantee packet loss only lasts for a short time.
![**Computation time vs. the scale of the topology**[]{data-label="fig:algtime"}](time.pdf){width="0.7\linewidth"}
Performance
-----------
Our TED’s overall throughput is same as Smore (Racke) when $Z \leq 1$. We also consider $Z>1$ in our Phase IV of Fig. \[fig:sysnew4\] which is not considered in Smore. Therefore, we set the bandwidth allocation to be $TM/Z$ when $Z>1$ for Smore in order to avoid congestion. As TED can make full use of the link capacity when $Z>1$ (Fig. \[fig:le-gscale-1-Custom-hardnop-cdfcomp1algTMZstep1-d48\]), it is better than Smore (TM/Z when $Z>1$). Also, Smore and TED using “program" as the path selection method do not show improvement in G-Scale compared with our TED’s path selection method, “hardnop". That is also the reason that we use “hardnop" instead of “program" for G-Scale like well connected topologies. The other reason is that “program" needs to solve two 0-1 integer linear programming (NP-complete problem).
No matter $Zopt \leq 1$ (Fig. \[fig:Z-gscale-1-hardnop-Zless1all\]) or $Zopt>1$ (Fig. \[fig:Z-gscale-1-hardnop-Zlarge1-TZall\]), performance of TED (Custom) is much better than Ecmp and competitive with other path set computing algorithms (all algorithms use TED’s architecture) while using less flow entries. The reason is that TED has higher path utilization shown in Fig. \[fig:Path-gscale-1-Zless1-hardnop-cdf\]. Note that if we remove the 40% almost unused paths for Racke and Ksp, although their used flow entries can be less, their performance after failures can be even worse. Also, Fig. \[fig:le-gscale-1-hardnop-cdfalld41\] shows that when $Zopt \approx 0.66$, the reason of Fig. \[fig:Z-gscale-1-hardnop-Zless1all\]’s peak ($Zecmp/Zopt$ is about 1.18) of Ecmp is $Zecmp \approx 0.78$ ($1.18=0.78/0.66$). Notice that about 20% Ecmp links have higher link utilization than $Zopt$ which is because Ecmp has less paths to balance the traffic.
Robustness
----------
To test robustness, we fail each link once. For each link failure, we re-run TED using the changed topology (includes all the Phases) and the path set is computed by each algorithm. Although we use less flow entries, our performance after failure is better than Racke, Ksp and much better than Ecmp (which is zero in Fig. \[fig:TradiosinglelfailZless1\], so we remove it) when $Zopt \leq 1$. It is because we use edge-disjoint paths so that any single link failure at most makes one path unavailable for the influenced $s-t$ pairs. We are competitive with Edksp which uses edge-disjoint $k$-shortest paths and Vlb which routes traffic via randomly selected intermediate nodes but remember that we are much faster than Edksp and use less flow entries than Vlb.
Path length
-----------
As shown in Fig. \[fig:Pathlength-gscale-1-Zless1-hardnop1-cdf\], the average length of used paths between each $s-t$ pair of TED (Custom) is only a slightly longer than Ecmp, Ksp and Racke. Custom, Edksp and Vlb only make less than 10% $s-t$ pairs has longer (less than 0.5 hop) average used path length than the longest path of Racke, Ecmp and Ksp for G-Scale.
Limitations and future works
----------------------------
One shortcoming of our edge disjoint paths is that Custom may not perform very well when some $s-t$ pairs in a topology have only one edge disjoint path. We believe that this situation is not common, because we found that the 40 topologies in topology zoo we evaluated have rich connections.
The other limitation is that for network with diverse link capacity, using only edge disjoint paths may not full utilize the capacity. In the future, we will explore how to select robust paths under such condition.
We would also point out some interesting results which need further investigation. For example, Fig. \[fig:Pathnum-gscale-1-Zless1new-hardnop1-cdf\] shows that for all the path computation methods, at least more than 30% $s-t$ pairs have only one path with allocated bandwidth, although on average each $s-t$ pair has three paths inputted to the TE optimization in G-Scale. We will attempt to analyze the reason theoretically. Fig. \[fig:Path-Cernet-1-Zless1-hardnop-cdf\] shows that about 30% paths are used for all $TM$s for Custom and Edksp in Cernet but our performance ratio is still close to Racke using hardnop (Fig. \[fig:Z-Cernet-1-hardnop-Zless1all\]). One possible reason is that edge-disjoint paths can use less paths to cover more important links than Racke. In this case, we need to find and protect such important links specially. The other interesting finding is that combined with our program, Vlb performs near optimal for Cernet (Fig. \[fig:Z-Cernet-1-program-Zless1all\]) except the paths can be longer.
Related Work {#sec:related}
============
**Path selection for multi-commodity flow:** Multi-commodity flow problem [@ref:networkflow] has been studied for many years. Recently, Merlin [@ref:merlin] and SNAP [@ref:snap] use a mixed-integer linear program to select one path for each ingress-egress switch pair which is not enough for WANs. Danna et al. [@ref:fairness] uses “water filling” related algorithm to allocate bandwidth for multi-commodity flow. It is very efficient and stable to the variance of demand. However, as it focuses on fairness, in their setting, each commodity has multiple possible paths to route its demand, how to select these paths is not explained. Caesar et al. [@ref:dyn] argues that decoupling failure recovery from path computation leads to networks that are inherently more efficient, more scalable and easier to manage. We were inspired by their work although they only propose a multi-path scheme that endpoint utilizes a fixed set of $k$ disjoint-as-possible available paths without mentioning how to select these paths, either.
**Improve reliability for data center WAN:** The most widely used approach to deal with network failures, including link or switch failures, is to re-compute a new TE solution based on the changed topology and re-program the switches [@ref:b4; @ref:swan]. However, re-computing a new TE plan and updating the forwarding rules across the entire network take at least minutes and are error-prone.
Several proactive approaches have been proposed to solve this important problem. Suchara et al. [@ref:joint] modifies the rescaling behavior of ingress switch by pre-computing and configuring forwarding rules based on the likelihood of different failure cases to prevent rescaling-induced congestion after a data plane fault. Although it achieves near-optimal load balancing, this approach can handle only a limited number of potential failure cases as there are exponential many of them to consider. SWAN [@ref:swan] develops a new technique that leverages a small amount of scratch capacity on links to apply updates in a provably congestion-free manner. FFC [@ref:ffc] is proposed to proactively protect a network from congestion and packet loss due to data and control plane faults. Although FFC spreads network traffic such that congestion-free property is guaranteed under arbitrary combinations of up to $k$ failures, the price is very high. About 5%-10% of the network capacity depending on $k$ has to be always left vacant to handle traffic from rescaling.
All of them either uses the k-shortest paths for each ingress-egress switch pair, or selects paths considering the failure probability. The SWAN’s [@ref:swan] allocation function allocates rate by invoking TE separately for classes in priority order. After a class is allocated, its allocation is removed from remaining link capacity. Doing like this, SWAN also ensures that higher priority traffic is likelier to use shorter paths. However, it still has not mentioned how to select better paths as the input of TE. Then Smore [@ref:smore] is proposed to use semi-oblivious traffic engineering. It works well for non-peak hours when flow tables are enough but works worse than TED when flow table limit is the same as TED. Smore does not consider how to set the suitable path budget and how to spread traffic when $TM$ is large. As we have shown in evaluation, Smore’s performance is worse than TED for heavily-loaded networks before and after failures and it takes longer time to compute the paths.
Conclusion {#sec:conclusion}
==========
This paper presents the motivation, design, and evaluation of TED, a scalable TE system for SDN. We present our four-phase approach to guarantee network performance and robustness, no matter how large the traffic matrix is, under the limitation of flow entries. TED uses the maximum number of edge disjoint paths between each $s-t$ pair as the path set. TED then select paths from the path set by computing the maximum path budget to guarantee paths are diverse enough to handle various traffic matrices. Next, we use minimizing maximum link utilization optimization to compute $Z$. If $Z \leq 1$, set the weights of paths. Otherwise, set the weights and allocated bandwidth re-computed by the maximizing whole throughput optimization under the $TM/Z \leq TM' \leq TM$ constraint (to guarantee each $s-t$ pair can at least get $d_{s,t}/Z$ bandwidth). Such process guarantees not to sacrifice performance or QoE to provide robustness. TED utilizes priority queue to guarantee the transmission of loss and delay sensitive traffic before and after rescaling [@ref:kuijia18]. TED is also fast to guarantee that once failure happens, after rescaling, it can re-compute the weights and bandwidth allocation immediately using the changed topology. Also, operators can change the flow entries’ limitation or capacity limitation to some percent to trade-off between performance and robustness.
\[ConcPage\]
[^1]: Dinic’s algorithm is a strongly polynomial algorithm for computing the maximum flow in a flow network. The algorithm runs in $O(n^2m)$ time and each augmenting path used in the algorithm is the shortest one.
|
---
abstract: 'Optical coherence tomography (OCT) is a noninvasive imaging modality which can be used to obtain depth images of the retina. The changing layer thicknesses can thus be quantified by analyzing these OCT images, moreover these changes have been shown to correlate with disease progression in multiple sclerosis. Recent automated retinal layer segmentation tools use machine learning methods to perform pixel-wise labeling and graph methods to guarantee the layer hierarchy or topology. However, graph parameters like distance and smoothness constraints must be experimentally assigned by retinal region and pathology, thus degrading the flexibility and time efficiency of the whole framework. In this paper, we develop cascaded deep networks to provide a topologically correct segmentation of the retinal layers in a single feed forward propagation. The first network (S-Net) performs pixel-wise labeling and the second regression network (R-Net) takes the topologically unconstrained S-Net results and outputs layer thicknesses for each layer and each position. Relu activation is used as the final operation of the R-Net which guarantees non-negativity of the output layer thickness. Since the segmentation boundary position is acquired by summing up the corresponding non-negative layer thicknesses, the layer ordering (i.e., topology) of the reconstructed boundaries is guaranteed even at the fovea where the distances between boundaries can be zero. The R-Net is trained using simulated masks and thus can be generalized to provide topology guaranteed segmentation for other layered structures. This deep network has achieved comparable mean absolute boundary error (2.82 $\mu$m) to state-of-the-art graph methods (2.83 $\mu$m).'
author:
- Yufan He
- Aaron Carass
- 'Bruno M. Jedynak'
- 'Sharon D. Solomon'
- Shiv Saidha
- 'Peter A. Calabresi'
- 'Jerry L. Prince'
bibliography:
- 'ml.bib'
- 'oct.bib'
title: Topology guaranteed segmentation of the human retina from OCT using convolutional neural networks
---
Introduction {#s:intro}
============
Optical coherence tomography (OCT) is a widely used non-invasive and non-ionizing modality for retina imaging which can obtain 3D retina images rapidly [@hee1995archo]. The depth information of the retina from OCT enables measurements of layer thicknesses, which are known to change with certain diseases [@medeiros2009iovs]. Fast automated retinal layer segmentation tools are crucial for large cohort studies of these diseases.
Automated methods for retinal layer segmentation have been well explored ([@rathke2014mia; @antony2014miccai]). State-of-the-art methods use machine learning (e.g, random forest (RF) [@lang2013boe]) for coarse pixel-wise labeling and then level set [@carass2014boe] or graph methods [@garvin2009tmi; @lang2013boe] to guarantee the segmentation topology (i.e., the anatomically correct retinal layer ordering) and obtain the final boundary surfaces. They are limited by the manually selected features for the pixel-wise labeling task and the manually tuned parameters of the graph. To build the graph, boundary distances and smoothness constraints which are spatially varying need to be experimentally assigned. The manually selected features and fine tuned graph parameters limit the application across cohorts.
Deep learning automatically extracts relevant image features from the training data and performs the segmentation in a feed forward fashion. The fully convolutional network (FCN) proposed by Long et al. [@long2015cvpr] is a successful deep learning segmentation method and the U-Net variant [@ronneberger2015miccai] is widely used for medical image segmentation. Both Roy et al. [@roy2017arxiv] and He et al. [@he2017towards] proposed FCNs for retinal layer segmentation (the former also included fluid segmentation). However, these FCN methods provide pixel-wise labeling without explicitly utilizing high level priors like shape, and neither guarantee the correct topology. Examples of FCNs giving anatomical infeasible results are shown in Fig. \[UN\_DNS\].
In order to obtain structured output directly from deep networks, Zheng et al. [@Zheng_2015_ICCV] implemented conditional random field as a recurrent neural network. This method can provide better label consistency but cannot guarantee global topology. BenTaieb et al. [@bentaieb2016miccai] proposed to explicitly integrate the topology priors into the loss function during training and Romero et al. [@romero2017image] used a second auto-encoder network to learn the output shape prior. Although those methods can improve the segmentation results by utilizing shape and topology priors, they still cannot guarantee the correct topology.
To obtain a topologically correct segmentation of the retinal layers from a deep network in a single feed forward propagation, we propose a cascaded FCN framework that transforms the layer segmentation problem from pixel labeling into a boundary position regression problem. Instead of outputting the boundary position directly, we use the network to output the distance between two boundaries, i.e, the layer thickness. The first network (S-Net) performs pixel labeling and the second regression network (R-Net) takes the topologically unconstrained S-Net results and outputs layer thicknesses for each layer and each position. Relu [@dahl2013improving] activation is used as the final operation of R-Net, which guarantees the non-negativity of the output layer thicknesses. Since the boundary position is acquired by summing up the corresponding non-negative layer thicknesses, the ordering of the reconstructed boundaries is guaranteed even at the fovea where the distances between boundaries can be zero.
Method
======
Fig. \[framework\] shows a schematic of our framework. We describe each step in our processing below.
**Preprocessing** A typical retinal B-scan is $496\times 1024$ which can require large amounts of memory if directly processed by a deep network. To address this, we approximately identify the Bruch’s membrane, flatten the retina and crop the image to remove the black background. Overlapped patches of size $128\times128$ are extracted and segmented by the deep network.
**Segmentation Network (S-Net) Overview** Our segmentation FCN (S-Net) is based on the U-Net [@ronneberger2015miccai]. It takes a $128\times 128$ image as input and the output is a $10 \times 128\times 128$ segmentation probability map which includes probability maps for the eight retinal layers and the background above and below the retina (vitreous and choroid, respectively). Fig. \[unet\] shows the details of S-Net; specifically, four $2\times 2$ max pooling and 19 convolution layers are used.
**Regression Net (R-Net) Overview** The R-Net consists of two parts: a U-Net identical to our S-Net (except for the input channels) and a dense layer. The input to the R-Net are the topologically unconstrained results from the S-Net. R-Net is applied to learn the shape and topology priors of the layer structures, while being resistant to the segmentation defects. (see Fig. \[UN\_DNS\] for examples). The dense layer of the R-Net uses Relu activation and thus guarantees a non-negative output vector. The size of this output is $128 \times 9$ = 1152, corresponding to the thicknesses of the 9 layers over the 128 A-scans being segmented.
Training
--------
We train our framework in two steps: S-Net is trained with a common pixel-wise labeling scheme because every pixel in the training data can be treated as an independent training sample and the total training data size is enlarged [@dou20173d]. R-Net is trained with augmented ground truth masks to learn the shape and topology prior. An alternative way to train the R-Net is to take the S-Net output as input and output the ground truth layer thickness. However, training in this manner would be sub-optimal as the S-Net output is not the ground truth mask. Thus, the training pairs of the S-Net output and the ground truth thicknesses are biased, which would bias the resultant R-Net. Therefore, we train both networks independently. We note that training the R-Net separately with simulated training masks allows this network to be generalized for use with other layered structures.
**S-Net training** \[s-train\] The S-Net is trained with a common pixel-wise labeling scheme, namely the cross-entropy loss function: $$\mathcal{L} = -\sum\limits_{x\in \Omega}
g_l(x)\textrm{log}(p_l(x;\theta)).
$$ Here, $g_l(x)$ is an indicator function on the ground truth label of pixel $x$ and $p_l(x;\theta)$ is the prediction probability from the deep network that the pixel $x$ belongs to layer $l$. Standard back-propagation is used to minimize the loss and update the network parameter $\theta$.
**R-Net training** \[r-train\] The purpose of the regression net is to find a mapping from the pixel-wise segmentation probability maps into layer thicknesses. We simulate topology defects with the ground truth mask and use R-Net to recover the correct layer thicknesses. The training of R-Net is based on minimizing the mean squared loss function below with standard back-propagation, $$\mathcal{L} = ||T(g(x)) - \mathcal{R}(g(x)+s(x);\theta)||_2^2.
$$ Here, $g(x)$ is the ground truth mask, $T(g(x))$ is the corresponding ground truth layer thickness, $\mathcal{R}$ is the prediction from the regression net, $s(x)$ is the simulated defects and Gaussian noise [@romero2017image] added to the ground truth mask. The simulated defects are random ellipses with magnitude ranging from $-1$ to $1$. Examples of the simulated input masks to R-net are shown in Fig. \[simulate\]. To prevent R-net from over-fitting the thickness values, we randomly move the position of the masks vertically and dilate or shrink the masks.
Experiments
===========
Ten fully delineated Spectralis Spectral Domain OCT (SD-OCT) scans (of size $496\times 1024\times 49$) were used for training. 20 overlapped patches were extracted within each B-Scan for training both networks, which yielded $9600$ samples for training. 20 SD-OCT macular OCT scans (of size $496\times 1024\times 49$) were acquired for validation. Ten data sets in our validation cohort were diagnosed with multiple sclerosis (MS) and the remaining ten were healthy controls.
Results
-------
Boundary segmentation accuracy was evaluated by comparing the automatic segmentation results with manual delineation along every A-scan. The mean absolute distance (MAD), root mean square error (RMSE), and mean signed difference (MSD) were calculated for the state-of-the-art RF + Graph method (RF+G) [@lang2013boe][^1] and our proposed deep networks (S-Net + R-Net). The Wilcoxon signed test was used to compare these two methods and the 95$\%$ quantile of the MSD is also reported. These results are shown in Table. \[b\_acc\]. The depth resolution is 3.9 $\mu$m. From the table, both methods have MAE and RMSE less than 1 pixel and our proposed method achieves similar or slightly better results than the state-of-the-art graph methods. The MSD and 95$\%$ quantile show that compared to our proposed method, the graph method is more biased. Figs. \[UN\_DNS\] and \[UN\_DNS\_DARK\] show some examples that when the image is of poor quality or the boundaries in the image are not clear, the S-Net results can be wrong whereas R-Net guarantees the correct topology while maintaining state-of-the-art accuracy.
The total segmentation time of our proposed deep network for one $496\times 1024\times 49$ scan is 10 s (preprocessing and reconstruction included), of which the deep network inference takes 5.85 s. The segmentation is performed with Python 3.6 and the preprocessing is performed in Matlab R2016b called directly from the Python environment. The RF+G method, had a total segmentation time of 100s in Matlab R2016b, of which RF classification was 62 s and the graph method took 20 s.
---------------- ---------- -- ------------- -- ------ -- ---------- -- ------------- -- ------
**Boundary** **RF+G** **S+R-Net** p **RF+G** **S+R-Net** p
**Vitre-RNFL** 2.24 2.22 0.92 2.88 2.81 0.78
**RNFL-GCL** 2.90 2.95 0.76 4.28 4.45 0.51
**IPL-INL** 3.10 2.99 0.82 4.42 3.97 0.16
**INL-OPL** 3.09 3.22 0.20 4.31 4.18 0.90
**OPL-ONL** 2.74 2.78 0.76 3.92 3.82 0.95
**ELM** 2.32 2.63 0.07 2.94 3.29 0.06
**IS-OS** 2.38 2.12 0.76 2.91 2.62 0.97
**OS-RPE** 3.34 3.44 0.74 4.43 4.39 0.92
**RPE** 3.33 3.02 0.84 3.94 3.72 0.90
**Overall** 2.83 2.82 0.42 3.78 3.69 0.56
---------------- ---------- -- ------------- -- ------ -- ---------- -- ------------- -- ------
: Boundary accuracy evaluated on 20 fully delineated scans for RF+G [@lang2013boe] and our proposed method, S-Net followed by R-Net (S+R-Net).(MAD – mean absolute distance; RMSE – root mean square error; MSD – mean signed difference; p – $p$-value.)[]{data-label="b_acc"}
---------------- ---------------------- ------ -----------------------
\[-0.7em\]
**Boundary** **RF+G** **S+R-Net**
**Vitre-RNFL** 1.02 (-3.92, 6.77) 0.32 (-5.06, 6.12)
**RNFL-GCL** -0.37 (-9.14, 7.26) -0.43 (-9.13, 7.21)
**IPL-INL** 0.68 (-8.26, 8.70) 0.67 (-7.03, 8.75)
**INL-OPL** -0.04 (-8.70, 8.00) -1.08 (-9.44, 6.89)
**OPL-ONL** 0.53 (-7.75, 7.98) 0.80 (-6.78, 8.42)
**ELM** 0.13 (-5.69, 6.34) -0.94 (-7.51, 5.66)
**IS-OS** 1.53 (-3.09, 7.31) -0.07 (-4.70, 5.83)
**OS-RPE** 1.15 (-6.85, 12.20) -0.06 (-9.26, 10.39)
**RPE** 2.53 (-3.27, 12.27) 1.10 (-5.67, 10.72)
**Overall** 0.80 (-6.62, 9.08) 0.03 (-7.54, 8.17)
---------------- ---------------------- ------ -----------------------
: Boundary accuracy evaluated on 20 fully delineated scans for RF+G [@lang2013boe] and our proposed method, S-Net followed by R-Net (S+R-Net).(MAD – mean absolute distance; RMSE – root mean square error; MSD – mean signed difference; p – $p$-value.)[]{data-label="b_acc"}
-- --
-- --
--------- ---------
**(a)** **(b)**
**(c)** **(d)**
--------- ---------
Discussion and conclusion
=========================
In this paper, we presented a fast topology guaranteed deep learning method for retinal OCT segmentation. Our method adds a thickness regression network after a conventional pixel-wise labeling network and utilizes the Relu activation to guarantee the non-negativity of the output and thus guarantee the topology. Since the R-Net is trained on masks that can be easily generated, our proposed framework can provide a topology guaranteed segmentation solution for other layered structures.
Acknowledgments
===============
This work was supported by the NIH/NEI under grant R01-EY024655.
[^1]:
|
---
abstract: 'In this paper we study the nearest neighbor Ising model with ferromagnetic interactions in the presence of a space dependent magnetic field which vanishes as $|x|^{-\alpha}$, $\alpha >0$, as $|x|\to \infty$. We prove that in dimensions $d\geq 2$ for all $\beta$ large enough if $\alpha>1$ there is a phase transition while if $\alpha<1$ there is a unique DLR state.'
author:
- |
Rodrigo Bissacot\
\
\
\
Marzio Cassandro\
\
\
Leandro Cioletti\
\
\
\
\
Errico Presutti\
\
\
title: Phase Transitions in Ferromagnetic Ising Models with spatially dependent magnetic fields
---
Introduction {#intro}
============
The Ising Model is one of the most studied subjects in Statistical Physics and will complete a century in a few years[^1]. The literature about ferromagnetic Ising models on $\mathbb{Z}^d$, $d\ge 2$, is mainly focused on cases where the external magnetic field is constant. We will study ferromagnetic nearest neighbor hamiltonians of the form $$\begin{aligned}
\label{hamilton}
H^{w}_{\L}(\s)= - J\sum_{|x-y|=1, x,y \in \L}\s(x)\s(y) - \sum_{x\in \L} h(x)\s(x)
- J \sum_{|x-y|=1, x \in \L, y \notin \L} \s(x)w(y)\end{aligned}$$ where $\Lambda$ is any finite subset of $\mathbb{Z}^d$, $\s \in \{-1,1\}^{\Lambda}$ is a spin configuration in $\Lambda$, $w\in \{-1,1\}^{\Lambda^c}$ a boundary condition and $J >0$ the interaction strength.
When the magnetic field $h(\cdot)$ is constant, that is $h(x)=h$ for all $x \in \mathbb{Z}^d$ and $h=0$, then the classical Peierls’ argument guarantees the existence of a phase transition. If instead $h \neq 0$ at all temperatures there is a unique DLR measure, as it follows from the Lee-Yang Theory and GHS inequalities. The absence of phase transitions comes from the differentiability of the free energy with respect to the parameter $h$.
Alternating signs fields on the lattice $\mathbb{Z}^2$ are considered in [@NOZ], constant fields on semi-infinite lattices are studied in [@Ba; @FP]. The magnetic field in all these models has some spatial symmetry. The challenging case of i.i.d. random magnetic fields on $\mathbb{Z}^d$ with zero mean has been studied in [@AW; @Bo; @BK; @COP1; @COP2] and the case with positive mean in [@NF]. Some deterministic and not spatially symmetric fields have been considered in [@BC].
In this paper we study the hamiltonian in $\mathbb{Z}^d$, $d\ge 2$, with a non negative, space dependent magnetic field $h(\cdot)$ of the form $$\label{0.1}
h(x) = \begin{cases}\frac{h^* }{|x|^{\alpha}}& x\ne 0\\
h^* & x=0\end{cases},\quad \alpha >0, h^*>0$$ where if $x=(x_1,\ldots,x_d)$ then $|x|= \sum_{i=1}^{d}|x_i|$. Calling $ Z^w_{\beta, h(\cdot),\L}$ the corresponding partition function one can easily check that (along van Hove sequences) $$\lim_{{\Lambda}\to \mathbb Z^d} \frac{\log Z^w_{\beta, h(\cdot),\L}}{\beta |{\Lambda}|}
= p_{\beta}$$ independently of the boundary conditions $w$. The limit $p_{\beta}$ is equal to the thermodynamic pressure without magnetic fields (i.e. $h^*=0$). This indicates that the presence of $h(\cdot)$ does not change the thermodynamics thus suggesting that a phase transition may occur for $\beta$ large, just as when the magnetic field is absent. However surface effects are relevant in the analysis of phase transitions and indeed we shall prove in Theorem \[thm3.1\] that when $\alpha<1$ there is a unique DLR measure, while when $\alpha >1$ there is a phase transition for $\beta$ large enough, see Theorem \[thm2.1\].
The existence of phase transitions at $\alpha>1$ is based on the validity of the Peierls bounds for contours. The proof of uniqueness when $\alpha<1$ at low temperatures is more involved and it is based on an iterative scheme introduced in [@BMPZ]. For $\alpha = 1$ we have partial results but not a complete characterization.
Existence of phase transitions {#sec:2}
==============================
In this section we shall prove:
\[thm2.1\] Let $h(\cdot)$ be as in with $\alpha>1$. Then for $\beta$ large enough there is a phase transition, namely the plus and minus Gibbs measures $\mu_{\beta, h(\cdot),{\Lambda}}^{\pm}$ converge weakly as ${\Lambda}\to \mathbb Z ^d$ to mutually distinct DLR measures.
As we shall see the result extends to $\alpha=1$ under the additional assumption that $h^*$ is small enough and to non negative magnetic fields which are “local perturbations” of (by this we mean that the $L^1$ norm of the difference is finite). We shall first prove the theorem under a stronger assumption on the magnetic field, see below, which allows to reproduce the Peierls’ argument. We need some geometric notation that will be used extensively throughout the paper.
\[defin2.1\] Two sites $x$ and $y$ in $\mathbb Z^d$ are connected iff they are nearest neighbors. Given a finite set $K$ in $\mathbb Z^d$ we call $\bar K$ its complement, $\delta_{\rm out}(K)$ the sites $y\in \bar K$ which are connected to sites $x\in K$ and $\delta_{\rm in}(K)$ those in $K$ connected to sites in $\bar K$. $|\partial K|$ denotes the number of connected pairs $x,y$ with $x\in \delta_{\rm in}(K)$ and $y\in \delta_{\rm out}(K)$.
\[lemma2.3ME\]
Let $h(\cdot)$ be any non negative magnetic field such that $$\label{2.5}
J |\partial \Delta| > 2 \sum_{x\in \Delta} h(x)$$ for all finite regions $\Delta \subset \mathbb{Z}^d$. Then for all $\beta$ large enough there is a phase transition.
[**Proof.**]{} We shall use to prove the validity of the Peierls bounds, see below. Then for all $\beta$ large enough the weak limits of the Gibbs measures with plus and minus boundary conditions are distinct DLR measures $\mu_{\beta, h(\cdot)}^{\pm}$. We thus have a phase transition hence the lemma. We shall use later that $\mu_{\beta, h(\cdot)}^{\pm}$ have trivial $\s$-algebra at infinity so that they have disjoint support, see for instance the Georgii book, [@G].
Proof of the Peierls bounds. Contours are geometric objects in the dual lattice $\mathbb Z^d_*$, namely call $C_x$, $x\in \mathbb Z^d$, the closed unit cube in $\mathbb R^d$ with center $x$, then $\mathbb Z^d_*$ is the union over all n.n. pairs $x,y$ of the faces $C_x\cap C_y$. Given a spin configuration $\s$ its contours $\g$ are the maximal connected (in the sense of non void intersection) components of the union of all faces $C_x\cap C_y$ with $\s(x)\ne\s(y)$.
Let $\g$ be a contour and $I(\g)$ the interior of $\g$, i.e. the points which are connected to $\infty$ only via paths which cross $\g$. Suppose $\g$ is a minus contour i.e. $\s(y)=-1$ on $\delta_{\rm out}(I(\g)$). Denote by $Z_{I(\g);h(\cdot)}^-
(\s_{I(\g)}(x)=1, x\in \delta_{\rm in}(I(\g)) $ the partition function in $I(\g)$ with magnetic field $h(\cdot)$, minus boundary conditions and with the constraint that $\s_{I(\g)}(x)=1$ for all $ x\in \delta_{\rm in}(I(\g)$. Then $$\begin{aligned}
&& Z_{I(\g);h(\cdot)}^-
(\s_{I(\g)}(x)=1, x\in \delta_{\rm in}(I(\g)))
\\&& \hskip2cm \le
e^{\beta \sum_{x\in I(\g)} h_x}\;
Z_{I(\g); h\equiv 0}^-(\s_{I(\g)}(x)=1, x\in \delta_{\rm in}(I(\g)))
\\&& \hskip2cm \le
e^{-2\beta J |\partial I(\g)|} e^{\beta \sum_{x\in I(\g)} h_x} Z_{I(\g); h\equiv 0}^-(\s_{I(\g)}(x)=-1, x\in \delta_{\rm in}(I(\g)))
\\&& \hskip2cm \le
e^{-2\beta J |\partial I(\g)|} e^{2\beta \sum_{x\in I(\g)} h_x}
Z_{I(\g);h(\cdot)}^-(\s_{I(\g)}(x)=-1, x\in \delta_{\rm in}(I(\g))).
\end{aligned}$$ Thus by the weight of the contour $\g$ is bounded by $$\label{2.7}
\frac{Z_{I(\g);h(\cdot)}^-
(\s_{I(\g)}(x)=1, x\in \delta_{\rm in}(I(\g)))}
{Z_{I(\g);h(\cdot)}^-(\s_{I(\g)}(x)=-1, x\in \delta_{\rm in}(I(\g)))} \le
e^{-\beta J |\partial I(\g)|}.$$ Same bound holds for the plus contours.
The proof of Theorem \[thm2.1\] will be obtained by reducing to magnetic fields for which is satisfied, a task that will be achieved via a few lemmas where we shall extensively use the Isoperimetric Inequality (see [@LP] for a proof): for any finite $\Delta\subset \mathbb{Z}^d$ $(d\geq 2)$ $$|\Delta|^{\frac{d-1}{d}}\leq \frac{|\partial \Delta|}{2d}.$$
\[lemma2.1ME\] Let $h(\cdot)$ be as in with $\alpha>1$. Then there is $C\equiv C(h^*,\alpha,d,J)>0$ so that holds for all finite regions $\Delta$ such that $|\Delta|>C$.
[**Proof.**]{} Since $h(x)$ is a non increasing function of $|x|$, calling ${B}(0,R):= \{ x: |x| \le R\}$ we have $$\sum_{x\in \Delta} h(x) \le \sum_{x\in {B}(0,R)}h(x),\quad \text{ for $R$ such that }\; |{B}(0,R)| \ge|\Delta|$$
We claim that the condition $ |{B}(0,R)| \ge|\Delta|$ is satisfied if $$\label{raio}
R= \text{ smallest integer } \ge c |\partial \Delta|^{\frac{1}{d-1}}$$ with $c$ large enough. In fact, recalling that $|\partial {B}(0,n)| = 2d\cdot n^{d-1}$, we have $|{B}(0,R)| \ge a R^d$, $a>0$ small enough, hence using the isoperimetric inequality $$|{B}(0,R)| \ge a R^d \ge ac^d |\partial \Delta|^{\frac{d}{d-1}}
\ge ac^d (2d)^{\frac{d}{d-1}}|\Delta| \ge|\Delta|$$ for $c$ large enough.
Thus the lemma will be proved once we show that $$\lim_{R\to \infty} \frac 1{R^{d-1}} \sum_{|x|\le R} h(x)=0.$$ Recalling that $|\partial {B}(0,n)| = 2d\cdot n^{d-1}$ this is implied by $$\lim_{R\to \infty}\sum_{n=1}^{R} \frac {n^{d-1}}{R^{d-1}}\frac 1{n^{\alpha}} =0$$ whose validity follows from the Lebesgue dominated convergence theorem. The lemma is thus proved.
.5cm
Observe that when $\alpha =1$ and $h^*$ is small enough then holds again for all finite regions $\Delta$ large enough. The proof is analogous except at the end as we only have $$\limsup_{R\to \infty} \frac 1{R^{d-1}} \sum_{|x|\le R} \frac 1{|x|^{\alpha}}\le c$$
.5cm
\[lemma2.2ME\]
Let $h(\cdot)$ be as in with $\alpha>1$, then there is $R$ so that holds for all finite $\Delta$ when the magnetic field is $\hat h$: $$\hat h (x)= \begin{cases} 0 & \text{if $|x|\le R$}\\
h(x)&\text{if $|x|> R$}\end{cases}$$
[**Proof.**]{} Suppose $|\Delta| > C$, $C$ the constant in Lemma \[lemma2.1ME\], then $$2\sum_{x\in \Delta} \hat h(x) \le
2\sum_{x\in \Delta} h(x) \le J |\partial \Delta|$$ Suppose next $|\Delta| \leq C$, then by the Isoperimetric Inequality, $$\begin{aligned}
\sum_{x\in \Delta} \hat h(x) &=& \sum_{x\in \Delta; |x| > R} \hat h(x)
\\ &\leq & \frac{h^*|\Delta| }{R^{\alpha}} \leq \frac{h^*|\partial \Delta|^{\frac{d}{d-1}} }{R^{\alpha}(2d)^{\frac{d}{d-1}}}
\leq \frac{h^*C^{\frac{1}{d-1}}|\partial \Delta| }{R^{\alpha}(2d)^{\frac{d}{d-1}}}\end{aligned}$$ which is $\le J|\partial \Delta|$ for $R$ sufficiently large.
.5cm
[**Proof of Theorem \[thm2.1\].**]{} Let $h(\cdot)$ be as in with $\alpha>1$. By Lemma \[lemma2.3ME\] and \[lemma2.2ME\] for $\beta$ large enough there is a phase transition for the system with magnetic field $\hat h(\cdot)$, let $\mu_{\beta, \hat h(\cdot)}^{\pm}$ the corresponding DLR measures obtained as limit of the Gibbs measures with plus respectively minus boundary conditions. Call $ \phi(x):= h(x) -\hat h(x)= \mathbf 1_{|x|<R} h(x)$ and define the probability measures $$\label{2.6}
d\nu_{\beta, h(\cdot)}^{\pm}(\s) := C_{\pm}e^{\beta \sum \phi(x) \s(x)}
d \mu_{\beta, \hat h(\cdot)}^{\pm}(\s)$$ ($C_{\pm}$ the normalization constants). We shall first check that they are DLR measures with magnetic field $h(\cdot)$. To have lighter notation we drop super and subscripts writing just $\nu$, $\mu$ and $C$. We need to show that for any finite cube ${\Lambda}$ large enough (we need below that ${\Lambda}\supset B(0,R)$) the $\nu$ conditional probability given $\s_{\bar {\Lambda}}$ is the Gibbs measure with magnetic field $h(\cdot)$. By the DLR property for $\mu$ we have $$d\nu(\s)= Ce^{\beta \sum \phi(x) \s(x)}
\frac{ e^{-\beta \hat H(\s_{\Lambda}|\s_{\bar {\Lambda}})}}{\hat Z_{\Lambda}(\s_{\bar {\Lambda}})}
d\mu_{\bar{\Lambda}}(\s_{\bar {\Lambda}})$$ where $d\mu_{\bar{\Lambda}}(\s_{\bar {\Lambda}})$ is the marginal of $\mu$ on the spin configurations in $\bar {\Lambda}$. We then have $$d\nu(\s)= C
\frac{ e^{-\beta H(\s_{\Lambda}|\s_{\bar {\Lambda}})}}{ Z_{\Lambda}(\s_{\bar {\Lambda}})}
\frac{ Z_{\Lambda}(\s_{\bar {\Lambda}})}{\hat Z_{\Lambda}(\s_{\bar {\Lambda}})}
d\mu_{\bar{\Lambda}}(\s_{\bar {\Lambda}})$$ By integrating over $\s_{\Lambda}$ we get $$d\nu_{\bar{\Lambda}}(\s_{\bar{\Lambda}})= C
\frac{ Z_{\Lambda}(\s_{\bar {\Lambda}})}{\hat Z_{\Lambda}(\s_{\bar {\Lambda}})}
d\mu_{\bar{\Lambda}}(\s_{\bar {\Lambda}})$$ hence $$d\nu(\s)=
\frac{ e^{-\beta H(\s_{\Lambda}|\s_{\bar {\Lambda}})}}{ Z_{\Lambda}(\s_{\bar {\Lambda}})}
d\nu_{\bar{\Lambda}}(\s_{\bar{\Lambda}})$$ which proves the DLR property. Thus $ d\nu_{\beta, h(\cdot)}^{\pm}(\s)$ are DLR measures with magnetic field $h(\cdot)$ and are absolutely continuous w.r.t. $\mu_{\beta, \hat h(\cdot)}^{\pm}$. Hence they also have disjoint supports and are therefore distinct. Theorem \[thm2.1\] is proved.
Restricted ensembles and contour partition functions {#sec:3}
====================================================
We fix hereafter $h(x)$ as in and we shall prove that
\[thm3.1\]
Let $h(\cdot)$ as in , then for any $\beta$ large enough there is a unique DLR measure.
.5cm
In this section we shall prove some crucial estimates which will be used in the next section to prove Theorem \[thm3.1\] but which have an interest in their own right. Observe that when $h(\cdot)$ is given by the condition may fail for some $\Delta$ for instance a large ball centered at the origin.
With this in mind we classify the contours $\g$ by saying that $\g$ is “slim” if $$\label{b3.2}
J |\partial I(\g)| > 2 \sum_{x\in I(\g)} h(x)$$ see the proof of Lemma \[lemma2.3ME\] for notation. We call “fat” the contours which do not satisfy . Following Pirogov-Sinai we then introduce plus-minus restricted ensembles where spin configurations are restricted in such a way that there are only slim contours. We thus define for any bounded region ${\Lambda}$ the plus-minus restricted partition functions $$\label{b3.3}
Z^{\pm,{\rm slim}}_{\Lambda}: = \sum_{\s_{\Lambda}:\text{all contours are slim}} e^{-\beta H(\s_{\Lambda}| \pm \mathbf 1_{{\Lambda}^c})}.$$ Obviously the pressures in the plus and minus ensembles are equal but the Pirogov-Sinai theory requires for the existence of a phase transition finer conditions on the finite volume corrections to the pressure namely that the latter differs from the limit pressure by a surface term. In our case the correction is larger than a surface term because $\alpha<1$ as shown by the following:
\[thmb3.2\]
For any $\beta$ large enough there are positive constants $c_1$ and $c_2$ so that $$\label{b3.4}
Z^{-,{\rm slim}}_{\Lambda}\le c_1 e^{-\beta c_2 \sum_{x\in {\Lambda}} h(x)} Z^{+,{\rm slim}}_{\Lambda}.$$
[**Proof.**]{} By repeating the proof of Theorem \[thm2.1\] and denoting by $E^{-,{\rm slim}}_{\Lambda}$ the expectation w.r.t. the Gibbs measure in the minus restricted ensemble, we have for any $x\in {\Lambda}$: $$\label{b3.5}
E^{-,{\rm slim}}_{\Lambda}(\s(x))\le -1 + 2 \sum_{\g: I(\g)\ni 0} e^{-\beta J|\partial I(\gamma)|} = -m^*,\quad m^*>0$$ for $\beta$ large enough. Then $$\label{b3.6}
\mu_{\beta, h(\cdot),{\Lambda}}^{-,{\rm slim}} \Big[ \frac{\sum_{x\in {\Lambda}} h(x)\s_{\Lambda}(x)}{\sum_{x\in {\Lambda}} h(x)} \le -\frac{m^*}2\Big]
\ge \frac{m^*}{2-m^*} $$ To prove let $X$ be a random variable with values in $[-1,1]$ and $P$ its law. Suppose that $E(X) \le - m^*$ and call $p:= P[X\ge - m^*/2]$, then $$-m^* \ge -1(1-p) -\frac{m^*}2 p,\quad (1-\frac{m^*}2)p \le (1-m^*),\quad (1-p) \ge
\frac{m^*}{2-m^*}$$ hence .
Calling $Z^{-,{\rm slim}}_{\Lambda}(A)$ the partition function with the constraint $A$, we can rewrite as: $$\begin{aligned}
Z^{-,{\rm slim}}_{\Lambda}&\le& \frac{2-m^*}{m^*} Z^{-,{\rm slim}}_{\Lambda}\Big( \frac{\sum_{x\in {\Lambda}} h(x)\s_{\Lambda}(x)}{\sum_{x\in {\Lambda}} h(x)} \le -\frac{m^*}2\Big)\\
&\le& \frac{2-m^*}{m^*} e^{-\beta \frac{m^*}2\sum_{x\in {\Lambda}} h(x)}
Z^{-,{\rm slim}}_{{\Lambda}, h\equiv 0}
\\
&=& \frac{2-m^*}{m^*} e^{-\beta \frac{m^*}2\sum_{x\in {\Lambda}} h(x)}
Z^{+,{\rm slim}}_{{\Lambda}, h\equiv 0}.
\end{aligned}$$ By repeating the previous argument we get $$Z^{+,{\rm slim}}_{{\Lambda}, h\equiv 0} \le \frac{2-m^*}{m^*} e^{-\beta \frac{m^*}2\sum_{x\in {\Lambda}} h(x)}
Z^{+,{\rm slim}}_{{\Lambda}}$$ where $ Z^{+,{\rm slim}}_{{\Lambda}}$ is the partition function with the contribution of the magnetic field $h(\cdot)$. This concludes the proof of the theorem.
.5cm In the next section we shall use a corollary of Theorem \[thmb3.2\] that we state after introducing some notation. The geometry is as follows:
${\Lambda}$ is a cube with center the origin, $\Delta$ a subset of ${\Lambda}$ and $K$ a subset of $\Delta$ which is union of disjoint connected set $K_i$ where for each $i$ the complement $\bar K_i$ of $K_i$ has a unique maximally connected component (i.e. there are no “holes” in $K_i$). We also suppose that each $K_i$ is fat, i.e. $$J |\partial K_i| \le 2 \sum_{x\in K_i} h(x)$$ and that $\delta_{\rm out} K \subset \Delta$, see Definition \[defin2.1\]. With ${\Lambda}$, $\Delta$ and $K$ as above we denote by $\mathcal X_{{\Lambda},\Delta,K,M}$, $M \subset \delta_{\rm out} \Delta$, the set of all configuration $\s_{\Lambda}$ which have the following properties.
- $\s_{\Lambda}=-1$ on $\delta_{\rm in} \Delta$, $\s_{\Lambda}=-1$ on $M \subset \delta_{\rm out} \Delta$ and $\s_{\Lambda}=+1$ on $\delta_{\rm out} \Delta\setminus M$.
- $\s_{\Lambda}=-1$ on $\delta_{\rm } K$ and $\s_{\Lambda}=+1$ on $\delta_{\rm in} K$.
- $\s_{\Lambda}$ has only slim contours in $ \Delta\setminus K$
We denote by $Z_{\Lambda}^\omega(\mathcal X_{{\Lambda},\Delta,K,M})$ the partition function in ${\Lambda}$ with constraint $\mathcal X_{{\Lambda},\Delta,K,M}$ and boundary conditions $\omega$. Then:
\[corob3.1\]
Under the same assumptions of Theorem \[thmb3.2\] $$\label{b3.6.1}
Z_{\Lambda}^\omega(\mathcal X_{{\Lambda},\Delta,K,M}) \le c_1 e^{-\beta c_2 \sum_{x\in \Delta\setminus K} h(x)}
e^{-2\beta J |\partial K |}e^{ -2\beta J |\partial\Delta| + 4\beta J |M|} Z_{\Lambda}^\omega$$
In the applications of the next section the connected components of $\Delta$ should intersect some given set and this will enable to control the sum over $\Delta$ via the bound $e^{ -2\beta J |\partial \Delta|}$.
The sum over $K$ is instead controlled as follows. We introduce the fat-contours partition function on the whole $\mathbb Z^d$ as $$\label{b3.7}
Z^{{\rm fat}} : = \sum_{n=0}^\infty \sum^*_{\g_1,..,\g_n} e^{-\beta J \sum |\partial I(\g_i)|}$$ where the sum $*$ refers to a sum over only fat contours such that $I(\g_i)\cap
I(\g_j)=\emptyset$ for all $i\ne j$.
\[thmb3.3\]
For any $\beta$ large enough there is a positive constant $c_3$ so that $$\label{b3.8}
Z^{{\rm fat}} \le c_3$$
[**Proof.**]{} We order the points of $\mathbb Z^d$ in a way which respects the distance from the origin and given a contour $\g$ we denote by $X(\g)$ the minimal point in $\g$ with the given order. By the definition of fat contours and supposing $X(\g)\ne 0$, $$J|\partial I(\g)| \le 2 \sum_{x\in I(\g)} h(x) \le \frac{2h^*}{|X(\g)|^\alpha} |I(\g)|
\le \frac{2h^*C_p}{|X(\g)|^\alpha}|\partial I(\g)|^{\frac d{d-1}}$$ where $C_p$ is the isoperimetric constant. Hence $$\label{b3.9}
|\partial I(\g)| \ge (\frac{J}{2C_p h^*})^{d-1} |X(\g)|^{\alpha(d-1)},\quad X(\g)\ne 0$$ We write $$\begin{aligned}
Z^{{\rm fat}} &=& \sum_n\sum_{x_1,..,x_n} \sum^*_{\g_1,..,\g_n} \prod_{i=1}^n
\mathbf 1_{X(\g_i)=x_i}e^{-\beta J |\partial I(\g_i)|}\\&\le &
\prod_{x\in \mathbb Z^d}\Big (1+ \sum_{\g \; {\rm fat}: X(\g)=x} e^{-\beta J|\partial I(\g)|}\Big )
\\&= & (1+ \sum_{\g \; {\rm fat}: X(\g)=0} e^{-\beta J|\partial I(\g)|}\Big )
\prod_{x \ne 0}\Big (1+ \sum_{\g \; {\rm fat}: X(\g)=x} e^{-\beta J|\partial I(\g)|}\Big )
\end{aligned}$$ which using proves .
Before moving to the next section with the proof of Theorem \[thm3.1\] we point out that by the Dobrushin’s Uniqueness Theorem there is a unique DLR state also at high temperatures and since the system is ferromagnetic, uniqueness may be expected to hold at all temperatures. However the proof of such a statement when the external field is zero does not seem to extend easily to our case, see [@CMR] and [@Hag].
Uniqueness at low temperatures {#sec:4}
==============================
In this section we prove Theorem \[thm3.1\]. For any positive integer $n$ we denote by ${\Lambda}_n$ the cube with center the origin and side $2n+1$. We fix a positive integer $L$, eventually $L\to \infty$, and arbitrarily the spins outside ${\Lambda}_{L}$, denoting by $\mu_L$ the Gibbs measure on $\{-1,1\}^{{\Lambda}_{L}}$ with the given boundary conditions and external magnetic field as in .
[**Definitions.**]{}
- Given $\s_{{\Lambda}_L}$, $\Delta \subset {\Lambda}_L$, $B: B\cap \Delta=\emptyset$ we say that $x\in \Delta$ is $-$ connected in $\Delta$ to $B$ if there is $X\subset \Delta$ such that: $x\in X$, $X$ is connected to $B$ and $\s_{\Lambda}\equiv -1$ on $X$.
- Let $\mathfrak{C}_L$ be the random set of sites $x\in {\Lambda}_{L}$ which are $-$ connected in ${\Lambda}_{L}$ to ${\Lambda}_{L+1}\setminus {\Lambda}_L$ and let $\mathfrak{M}_k =
\mathfrak{C}_L \cap {\Lambda}_{k+1}\setminus {\Lambda}_k$, $k<L$; $\mathfrak{M}_L= {\Lambda}_{L+1}\setminus {\Lambda}_L$.
- Given $k\le L$ and $M\subset {\Lambda}_{k+1}\setminus {\Lambda}_k$ we define $\mathfrak{C}_{k,M}(\s_{{\Lambda}_L})$ as the set of all $x\in {\Lambda}_k$ which are $-$ connected in ${\Lambda}_k$ to $M$. In particular $\mathfrak{C}_{L,M}=\mathfrak{C}_{L}$ if $M={\Lambda}_{L+1}\setminus {\Lambda}_L$.
Suppose $\mathfrak{C}_L=C$ then the spins in
$\delta_{\rm out}(C\cup \bar{\Lambda}_L)$ are all equal to $+1$. Moreover if we change the configuration $\s_{\Lambda}$ leaving unchanged the spins in $C':=(C\cup \bar{\Lambda}_L) \cup \delta_{\rm out}(C\cup \bar{\Lambda}_L)$ we still have $\mathfrak{C}_L=C$.
Thus the spins in ${\Lambda}_L\setminus C'$ are distributed with Gibbs measure with plus boundary conditions. We shall prove that there exists $b^*<1$ so that $$\label{3.2}
\lim_{L\to \infty} \mu_{L}\Big[ \mathfrak{C}_L \cap {\Lambda}_{L(1-b^*)} =\emptyset\Big]=1$$ which then proves that $\mu_{L}$ converges weakly to the plus DLR measure, which is the weak limit of Gibbs measures with plus boundary conditions. Thus any DLR measure is equal to the plus DLR measure and Theorem \[thm3.1\] is proved. We are therefore reduced to the proof of which uses an iterative argument introduced in [@BMPZ].
It readily follows from the definitions that for $k<L$: $$\label{b4.3}
\mathfrak{C}_L \cap {\Lambda}_{k} = \mathfrak{C}_{k,\mathfrak{M}_k}, \;\;\;\; \mathfrak{M}_k=
\mathfrak{C}_L \cap ( {\Lambda}_{k+1}\setminus {\Lambda}_k).$$ The next property will be used to establish a connection with Corollary \[corob3.1\], it is therefore crucial in the proof of Theorem \[thm3.1\]. We claim that: $$\label{b4.3nn}
\s_{{\Lambda}_L}(x) = 1\, \text{ for all $x$ in $\delta_{\rm out}(\mathfrak{C}_{k,\mathfrak{M}_k})\setminus \mathfrak{M}_k$}$$ Proof: By definition $\s_{{\Lambda}_L}(x) = 1$ for all $x$ as in which are in ${\Lambda}_k$. It remains to consider all $x$ as in which are in ${\Lambda}_{k+1}\setminus {\Lambda}_k$. We argue by contradiction supposing $\s_{{\Lambda}_L}(x)=-1$. In such a case there is a path with all minuses which starts at $x$ and ends in $\mathfrak{M}_k$. Since $\mathfrak{M}_k \subset \mathfrak{C}_L$ and $\mathfrak{C}_L$ is connected, then $x\in \mathfrak{C}_L$ which implies (since $x\in{\Lambda}_{k+1}\setminus {\Lambda}_k$) that $x\in \mathfrak{M}_k$, hence the contradiction. is proved.
Before proceeding we need some extra notation:
[**Notation.**]{} We decompose $\mathfrak{C}_{k,M}$ into maximally connected components, each one of them is a connected set whose complement has an unbounded maximally connected component and maybe several maximally connected finite components. The latter are distinguished into fat and slim and we call $\bar {\mathfrak{C}}_{k,M}^{\rm fat}$ and $\bar {\mathfrak{C}}_{k,M}^{\rm slim}$ the union of all the fat, respectively slim ones.
It then follows directly from that $$\label{b4.3z}
\{ \mathfrak{M}_k = M \} \cap \{ \bar {\mathfrak{C}}_{k,M}^{\rm fat} = K \} \cap
\{ \mathfrak{C}_{k,M} \cup \bar {\mathfrak{C}}_{k,M}^{\rm slim} = \Delta \}
\subset \mathcal X_{{\Lambda},\Delta,K,M}$$ $\mathcal X_{{\Lambda},\Delta,K,M}$ the set considered in Corollary \[corob3.1\].
We are now ready for the proof of Theorem \[thm3.1\]. The basic point is that if $|\mathfrak{M}_{k_0}|$ is small for some $k_0$ then (with large probability) there is $k>k_0$ with $|\mathfrak{M}_{k}|$ even smaller. Iterating the argument we will then find a $k$ where $|\mathfrak{M}_{k}|=0$. The heuristic idea behind the proof of such properties is the following.
Suppose that $|\mathfrak{M}_{k_0}|= L^a$, $a>0$, $k_0$ a fraction of $L$. Let $0<a'<a$, fix a constant $b<1$ suitably small and distinguish two cases: $$|\mathfrak{M}_k| \le L^{a'}\quad \text{for some $k\in [ k_0-bL,k_0)$}$$ and the complement where $$\label{b4.3nnn}
|\mathfrak{M}_k| > L^{a'}\quad \text{for all $k\in [k_0-bL,k_0)$}$$ We argue that the event has vanishing probability as $L\to \infty$. To this end we use (with $k=k_0$) and Corollary \[corob3.1\] observing that (with the above notation) $|\Delta| \ge |\mathfrak{C}_{k_0,M}| \ge bL L^{a'}$, by . In we then have a dangerous term $e^{4\beta J L^a}$ (which comes from $M =\mathfrak{M}_{k_0}$, $|M|\le L^a$), while the contribution of the magnetic field is bounded by $e^{ -\beta c_2 (h^*L^{-\alpha})bL L^{a'}}$. If $$L^{1+a' -\alpha} > L^{a}$$ the magnetic field wins against the dangerous term. We need a lengthy counting argument to sum over all possible values of $\Delta$, $K$ and $M$ which will be given in the end of the section and which will prove that with probability going to 1 as $L\to \infty$ we can reduce to the case $|\mathfrak{M}_k| \le L^{a'}$ for some $k\in [ k_0-bL,k_0)$.
We can satisfy the previous inequality with $a' = a - \frac{1-\alpha}2$ and then iterate the argument to prove that after finitely many steps we get $\mathfrak{M}_{k}=\emptyset$ and thus conclude the proof.
With this in mind we introduce the sequence $a_n$, $n\ge 0$, by setting $$\label{3.3}
a_0=d-1,\;\; a_{n+1}=a_n - \frac{1-\alpha}2$$ and call $n^*$ the largest integer such that $a_{n^*}\ge 0$. Let $s_0=L$ and define recursively $s_n$ for $1\le n\le n^*$ by setting $$\label{3.4}
\text{$s_n$ the largest $k$ not larger than $s_{n-1}$ such that $|\mathfrak{M}_k| \le L^{a_n}$}$$ and, if there is no $k$ as in , we then set $s_n=0$ and stop the sequence. Observe that if $|\mathfrak{M}_{s_{n-1}}|=0$ then $s_n=s_{n-1}$. If not stopped earlier we define $s_{n^*+1}$ as $$\label{3.5}
\text{$s_{n^*+1}$ is the largest $k$ not larger than $s_{n^*}$ such that $|\mathfrak{M}_k| =0$}$$ setting $s_{n^*+1}=0$ if $k$ does not exist.
Let $b>0$ be such that $$\label{3.6}
b n^* < \frac 1{100}$$ Then $\mathfrak{C}_L \cap {\Lambda}_{L(1-b^*)} =\emptyset$ in the set $$\label{3.7}
\mathcal G := \bigcap_{1\le n\le n^*+1} \{ s_{n-1}-s_n \le bL\}$$ provided $b^*>1/2$ so that will follow once we prove that $$\label{b4.8}
\lim_{L\to \infty} \mu_{L}\Big[ \mathcal G\Big]=1.$$
We shall prove that for any $1\le p\le n^*+1$ $$\label{4.8}
\lim_{L\to \infty} \mu_L\Big[ s_{p+1} < s_{p}-bL \; ;\; s_{p} \ge L-pbL\Big]=0$$ which yields .
We write $\mu_L\big[ s_{p+1} < s_{p}-bL \; ;\; s_{p} \ge L-pbL\big]$ as the ratio of two partition functions, the one in the denominator is the full partition function $Z_{{\Lambda}_L}^\omega$, $\omega$ the boundary conditions outside ${\Lambda}_L$, while the one in the numerator will be simply called $Z$ and it will be the object of our analysis. We decompose the configurations according to the value $k$ of $s_p$ and $M$ of $\mathfrak{M}_k$. If $|M|=0$ we do not have to prove anything so that in the sequel we tacitly suppose $|M|>0$. We have $$\begin{aligned}
\label{4.9}
Z &\le & \sum_{L\ge k \ge L-pbL} \;\;\;\sum_{M\subset {\Lambda}_{k+1}\setminus {\Lambda}_k: |M|\le L^{a_p}} \sum_{C_{k,M}\subset {\Lambda}_k: |C_{k,M}|\ge bL^{1+a_{p+1}}} \nonumber\\ && \hskip3cm
Z_{{\Lambda}_L}\Big(\mathfrak{M}_k = M;\mathfrak{C}_{k,M}=C_{k,M}\Big)
\end{aligned}$$ The sets $K =\bar {\mathfrak{C}}_{k,M}^{\rm fat}$ and $\bar {C}_{k,M}^{\rm slim} = \bar {\mathfrak{C}}_{k,M}^{\rm slim}$ are uniquely determined by $C_{k,M}$ and we can rewrite as $$\begin{aligned}
\label{4.9bis}
Z &\le & \sum_{L\ge k \ge L-pbL} \;\;\;\sum_{M\subset {\Lambda}_{k+1}\setminus {\Lambda}_k: |M|\le L^{a_p}} \sum_{K,\Delta : |\Delta|\ge bL^{1+a_{p+1}}} \nonumber\\ && \hskip1cm
Z_{{\Lambda}_L}\Big(\mathfrak{M}_k = M; \bar{\mathfrak{C}}_{k,M}^{\rm fat} = K;
\mathfrak{C}_{k,M}\cup\bar {C}_{k,M}^{\rm slim}=\Delta\Big)
\end{aligned}$$ observing that $\Delta\subset {\Lambda}_k$ is the union of a finite number of disjoint connected sets (without “holes”, see Section \[sec:3\]), say $\Delta_1$,..,$\Delta_n$, each one connected to $M$. $K$ is the union of fat connected sets without holes each one contained in $\Delta$. When we add a $*$ to the sum over $K$ and $\Delta$ we mean that the sum is over sets with such a restriction. We then get from after using and $$\begin{aligned}
\label{4.9t}
Z &\le & \sum_{L\ge k \ge L-pbL} \;\;\;\sum_{M\subset {\Lambda}_{k+1}\setminus {\Lambda}_k: |M|\le L^{a_p}} \sum^*_{K,\Delta : |\Delta\setminus K|\ge bL^{1+a_{p+1}}} \nonumber\\ && \hskip1cm
c_1 e^{-\beta c_2 h^* L^{-\alpha} bL^{1+a_{p+1}}}
e^{-2\beta J |\partial K |}e^{ -2\beta J |\partial \Delta| + 4\beta J |M|} Z_{{\Lambda}_L}^\omega
\end{aligned}$$ where $Z_{{\Lambda}_L}^\omega$ is the full partition function. We next specify the maximal connected components of $\Delta$, called $\Delta_1,..,\Delta_n$, and use Theorem \[thmb3.3\] and to perform the sum over $K$ then getting $$\begin{aligned}
\label{4.9tt}
\frac{Z}{Z_{{\Lambda}_L}^\omega} &\le & \sum_{L\ge k \ge L-pbL} \;\;\;\sum_{M\subset {\Lambda}_{k+1}\setminus {\Lambda}_k: |M|\le L^{a_p}} \sum_{n\ge 1}\sum^*_{\Delta_1,..,\Delta_n } \nonumber\\ && \hskip1cm
c_1 e^{-\beta c_2 h^* L^{-\alpha} bL^{1+a_{p+1}}}
c_3^ne^{ -2\beta J |\partial \Delta| + 4\beta J L^{a_p}}
\end{aligned}$$ where the $*$ recalls that $\Delta_1,..,\Delta_n$ are mutually disjoint connected sets without holes each one connected to $M$, this implies that the sum is over $n \le |M|\le L^{a_p}$. Each $\Delta_i$ is then in one to one correspondence with $\delta_{\rm out}(\Delta_i)$, which is a $*$connected set which intersects $M$.
Thus we can bound the $*$ sum by summing over $n\le |M|$ disjoint $*$connected sets which intersect $M$. Hence $$\label{4.13}
\sum_{n\ge 1}\sum^*_{\Delta_1,..,\Delta_n }
e^{ -2\beta J |\partial \Delta|} \le \sum_{n=1}^{|M|} \frac{M!}{n!(M-n)!} e^{-\beta c_4 n}
\le \Big(1+e^{-\beta c_4}\Big)^{|M|}$$ where $c_4$ is such that $$\label{4.14}
e^{-\beta c_4 } \ge\;\;\; \sum_{D \ni 0, D * {\rm connected}}\;\;\; e^{ -2\beta J |D|}$$ holds for $\beta$ large enough, see for instance Lemma 3.1.2.4 in [@presutti].
Then recalling $$\begin{aligned}
\label{4.14t}
\frac{Z}{Z_{{\Lambda}_L}^\omega} &\le & \sum_{L\ge k \ge L-pbL} \;\;\;\sum_{M\subset {\Lambda}_{k+1}\setminus {\Lambda}_k: |M|\le L^{a_p}} \nonumber\\ && \hskip1cm
c_1 e^{-\beta c_2 h^* L^{-\alpha} bL^{1+a_{p+1}}}
c_3^{ L^{a_p}}e^{ 4\beta J L^{a_p}} \Big(1+e^{-\beta c_4}\Big)^{L^{a_p}}
\end{aligned}$$ We can now perform the sum over $M$ which using the Stirling formula is bounded by $ e^{ c_5 L^{a_p} \log L}$, $c_5$ a suitable constant and thus get $$\begin{aligned}
\label{4.14tt}
\frac{Z}{Z_{{\Lambda}_L}^\omega} &\le & L e^{ c_5 L^{a_p} \log L}
c_1 e^{-\beta c_2 h^* L^{-\alpha} bL^{1+a_{p+1}}}
c_3^{ L^{a_p}}e^{ 4\beta J L^{a_p}} \Big(1+e^{-\beta c_4}\Big)^{L^{a_p}}
\end{aligned}$$ which recalling the definition of $a_n$ proves that $$\label{4.15}
\mu_L\Big[ s_{p+1} < s_{p}-bL \; ;\; s_{p} \ge L-pbL\Big] \le c_6 e^{-\beta \frac{c_2}2 bL^{1+a_{p+1}-\alpha}}$$ thus proving and hence .
Concluding remarks {#sec:5}
==================
We have proved that when the magnetic field is given by for all $\beta$ large enough there is a phase transition when $\alpha >1$ while, if $\alpha<1$, there is a unique DLR state. It seems plausible that uniqueness extends to all $\beta$ but we do not have a proof. Using the random cluster representation uniqueness is related to the absence of percolation (see [@CMR]), perhaps this can be useful to deal with this question. When $\alpha =1 $ and $h^*$ small enough the proof of Section \[sec:2\] applies and we thus have a phase transition. However, our proof of uniqueness does not extend to the case $\alpha=1$ no matter how large is $h^*$ and a different approach should be used maybe related to an extension of Minlos-Sinai or the Wulff shape problem.\
[**Acknowledgments**]{}\
The authors thank Aernout van Enter for fruitful discussions. Rodrigo Bissacot is supported by the Grant 2011/22423-5 from FAPESP and Grant 308583/2012-4 from CNPq. Leandro Cioletti is supported by FEMAT. The authors thank Maria Eulália Vares and the organizers of the XVII Brazilian School of Probability during which the $\alpha >1$ case was proved. Rodrigo Bissacot acknowledges very kind hospitality and support from GSSI in L’Aquila, and from the Mathematics Department of the University of Brasília.
M. Aizenman and J. Wehr. [*Rounding effects of quenched randomness on first-order phase transitions*]{}. Communications in Mathematical Physics. Vol. 130, n. 3, 441-631, (1990).
A.G. Basuev. [*Ising Model in Half-Space: A Series of Phase Transitions in Low Magnetic Fields*]{}. Theoretical and Mathematical Physics Vol. 153, 1539-1574, (2007).
R. Bissacot and L. M. Cioletti. [*Phase Transition in Ferromagnetic Ising Models with Non-uniform External Magnetic Fields*]{}. Journal of Statistical Physics, Vol. 139, n. 5, pp 769-778, (2010).
A. Bovier: Statistical Mechanics of Disordered Systems A Mathematical Perspective. Cambridge University Press (2012).
A. Bovier, I. Merola, E. Presutti, M. Zahradnik. [*On the Gibbs phase rule in the Pirogov-Sinai regime*]{}. Journal of Statistical Physics, Vol. 114, pp 1235-1267, (2004).
J. Bricmont and A. Kupiainen. [*Phase transition in the 3d random field Ising model*]{} Communications in Mathematical Physics. Vol. 116, n. 4, 529-700, (1988).
M.Cassandro, E.Orlandi, P.Picco. [*Phase transitions in the 1D random field Ising model with Long range interactions*]{}. Communications in Mathematical Physics. Vol. 288, 731 (2009).
M.Cassandro, E.Orlandi, P.Picco. [*Typical Gibbs configurations for the 1d random field Ising model with long range interactions*]{}. Communications in Mathematical Physics. Vol. 309 , 229 (2012).
L. Chayes, J. Machta, and O. Redner. [*Graphical Representations For Ising Systems In External Fields*]{}. Journal of Statistical Physics, Vol. 93, pp 17-32, (1998).
L. R. G. Fontes and E. J. Neves . [*Phase uniqueness and correlation length in field diluted Ising models*]{}. Journal of Statistical Physics, v. 80, p. 1327-1339, (1995).
J. Fröhlich and C.E. Pfister. [*Semi-Infinite Ising Model II. The Wetting and Layering Transitions*]{}. Communications in Mathematical Physics. Vol 112, 51-74, (1987).
H.-O. Georgii: Gibbs Measures and Phase Transitions. de Gruyter, Berlin, (1988).
O. Häggström. [*A Note on (Non-)Monotonicity in Temperature for the Ising Model*]{}. Markov Processes and Related Fields 2, 529-537, (1996).
R. Lyons and Y. Peres: [*Probability on Trees and Networks*]{}. Cambridge University Press. In preparation. Current version available at [http://mypage.iu.edu/ rdlyons/]{}, (2014).
F.R. Nardi, E. Olivieri, and M. Zahradnik [*On the Ising Model with Strongly Anisotropic External Field*]{}. Journal of Statistical Physics, Vol. 97, pp. 87-144, (1999).
E. Presutti. [*Scaling limits in statistical mechanics and microstructures in continuum mechanics*]{}. Theoretical and mathematical physics. Springer (2009).
[^1]: Wilhelm Lenz introduced the model in 1920.
|
---
abstract: 'If a small compact object orbits a black hole, it is known that it can excite the black hole’s quasinormal modes (QNMs), leading to high-frequency oscillations (“wiggles”) in the radiated field at ${\mathcal{J}}^+$, and in the radiation-reaction self-force acting on the object after its orbit passes through periapsis. Here we survey the phenomenology of these wiggles across a range of black hole spins and equatorial orbits. In both the scalar-field and gravitational cases we find that wiggles are a generic feature across a wide range of parameter space, and that they are observable in field perturbations at fixed spatial positions, in the self-force, and in radiated fields at ${\mathcal{J}}^+$. For a given charge or mass of the small body, the QNM excitations have the highest amplitudes for systems with a highly spinning central black hole, a prograde orbit with high eccentricity, and an orbital periapsis close to the light ring. The QNM amplitudes depend smoothly on the orbital parameters, with only very small amplitude changes when the orbit’s (discrete) frequency spectrum is tuned to match QNM frequencies. The association of wiggles with QNM excitations suggest that they represent a situation where the *nonlocal* nature of the self-force is particularly apparent, with the wiggles arising as a result of QNM excitation by the compact object near periapsis, and then encountered later in the orbit. Astrophysically, the effects of wiggles at ${\mathcal{J}}^+$ might allow direct observation of Kerr QNMs in extreme-mass-ratio inspiral (EMRI) binary black hole systems, potentially enabling new tests of general relativity.'
author:
- Jonathan Thornburg
- Barry Wardell
- Maarten van de Meent
bibliography:
- 'journalshortnames.bib'
- 'commongsf.bib'
- 'jt-new.bib'
- 'aei-references.bib'
- 'meent.bib'
- 'wardell.bib'
title: 'Excitation of [K]{}err quasinormal modes in extreme–mass-ratio inspirals'
---
Introduction {#sect:introduction}
============
Consider a small (compact) body of mass $\mu M$ (with $0 < \mu \ll 1$) moving freely near a Schwarzschild or Kerr black hole of mass $M$. This system emits gravitational radiation, and there is a corresponding radiation-reaction influence on the small body’s motion. Calculating the resulting perturbed spacetime (including the small body’s motion and the emitted gravitational radiation) is a long-standing research question in general relativity.
There is also an astrophysical motivation for this calculation: If a neutron star or stellar-mass black hole of mass ${\sim}\, 1$–$100 M_{\odot}$ orbits a massive black hole of mass ${\sim}\, 10^5$–$10^7 M_{\odot}$,[^1] the resulting “extreme–mass-ratio inspiral” (EMRI) system is expected to be a strong astrophysical gravitational-wave (GW) source detectable by the planned Laser Interferometer Space Antenna (LISA) space-based gravitational-wave detector. LISA is expected to observe many such systems, some of them at quite high signal-to-noise ratios ([@Gair-etal-2004:LISA-EMRI-event-rates; @Barack-Cutler-2004; @Amaro-Seoane-etal-2007:LISA-IMRI-and-EMRI-review; @Gair-2009:LISA-EMRI-event-rates]). The data analysis for, and indeed the detection of, such systems will generally require matched-filtering the detector data stream against appropriate precomputed GW templates. The problem of computing such templates provides an astrophysical motivation for EMRI modelling.
In the test-particle limit it has long been known that an unbound (scattering) flyby can excite quasinormal modes (QNMs) of the background black hole. Kojima and Nakamura [@Kojima-Nakamura-1984] studied this process, finding that “A scattered particle excites the quasi-normal mode under the condition that twice the angular velocity at the periapsis is greater than the real part of the frequency of the quasi-normal mode”. Their figure 3(b) shows an example of the QNM oscillations in the radiated gravitational waves at ${\mathcal{J}}^+$.
Burko and Khanna [@Burko-Khanna-2007] found small oscillations in the total radiated energy flux from a test particle making a parabolic (unbound) flyby of a Kerr black hole. They attributed these oscillations to the particle encountering scattered gravitational waves emitted during the particle’s inbound motion.
O’Sullivan and Hughes [@OSullivan:2015lni] observed “small-amplitude high-frequency oscillations” in their calculations of the horizon shear of a Kerr black hole orbited by a test particle. Because they did not find corresponding oscillations in the horizon’s tidal distortion field, and their measured oscillation frequencies did not match known Kerr QNM frequencies, they concluded that the horizon-shear oscillations they observed “cannot be related to the \[Kerr black\] hole’s quasi-normal modes”.
Thornburg and Wardell [@Thornburg-Wardell-2017:Kerr-scalar-self-force] (hereinafter TW) calculated the scalar-field self-force for eccentric equatorial particle orbits in Kerr spacetime. For some systems where the Kerr black hole was highly spinning and the particle orbit was prograde and highly eccentric, TW found that the self-force exhibits large oscillations (“wiggles”) on the outgoing leg of the orbit shortly after periapsis passage. TW suggested that wiggles “are in some way *caused* by the particle’s close passage by the large black hole”. Thornburg [@Thornburg-2016-Capra-Meudon-talk; @Thornburg-2017-Capra-Chapel-Hill-talk] presented fits of damped-sinusoid models to these wiggles for a range of Kerr spins and particle orbits, found close agreement of the model complex-frequencies with those of known Kerr QNMs, and argued that this agreement shows that wiggles are, in fact, caused by Kerr QNMs excited by the particle’s close periapsis passage.
Nasipak, Osburn, and Evans [@Nasipak-Osburn-Evans-2019:Kerr-scalar-self-force-and-wiggles] calculated the scalar-field self-force and the radiated field at ${\mathcal{J}}^+$ for eccentric inclined particle orbits in Kerr spacetime. For one particular (equatorial) orbit configuration they confirmed TW’s finding of wiggles in the self-force and also found wiggles in the radiated scalar field at ${\mathcal{J}}^+$, fitting these to a superposition of $\ell\,{=}\,m\,{=}\,1$, $\ell\,{=}\,m\,{=}\,2$, $\ell\,{=}\,m\,{=}\,3$, and $\ell\,{=}\,m\,{=}\,4$ Kerr quasinormal modes (QNMs). They concluded that wiggles are caused by Kerr QNMs excited by the particle’s close periapsis passage.
Rifat, Khanna, and Burko [@Rifat-Khanna-Burko-2019:wiggles-in-near-extremal-Kerr] recently studied wiggles for EMRIs where the central Kerr BH is nearly extremal (dimensionless spin up to $0.999\,99$), finding that in such systems many Kerr QNMs can be simultaneously excited.
Here we extend Refs. [@Thornburg-2016-Capra-Meudon-talk; @Thornburg-2017-Capra-Chapel-Hill-talk] and survey wiggles’ phenomenology over a wide range in parameter space for eccentric equatorial orbits in Kerr spacetime, for both the scalar-field model and the full gravitational field. We focus on leading-order radiation and radiation-reaction effects computed via 1st-order perturbations of Kerr spacetime, i.e., (for the gravitational case) ${\mathcal{O}}(\mu)$ field perturbations near the Kerr black hole, ${\mathcal{O}}(\mu)$ radiation at ${\mathcal{J}}^+$, and ${\mathcal{O}}(\mu^2)$ radiation-reaction “self-forces” acting on the small body. We fit models of Kerr QNMs to all these diagnostics.
Our focus is the case where the perturbing body’s orbit is highly relativistic, so post-Newtonian methods (see, for example, [@Damour-in-Hawking-Israel-1987 Section 6.10]; [@Poisson-Will-book-2014; @Will-TEGP-2nd-Ed; @Blanchet-2014-living-review; @Futamase-Itoh-2007:PN-review; @Blanchet-2009:PN-review; @Schaefer-2009:PN-review] and references therein) are not reliably accurate. Since the timescale for radiation reaction to shrink the orbit is very long ($\sim \mu^{-1} M$) while the required resolution near the small body is very high ($\sim \mu M$), a direct “numerical relativity” integration of the Einstein equations (see, for example, [@Pretorius-2007:2BH-review; @Hannam-etal-2009:Samurai-project; @Hannam-2009:2BH-review; @Hannam-Hawke-2010:2BH-in-era-of-Einstein-telescope-review; @Campanelli-etal-2010:2BH-numrel-review] and references therein) would be prohibitively expensive (and probably insufficiently accurate) for this problem.[^2] Instead, we use black hole perturbation theory, treating the small body as an ${\mathcal{O}}(\mu)$ perturbation on the background spacetime.
We present results obtained from two separate numerical codes which were programmed independently, using completely different theoretical formalisms and numerical methods:
- Our scalar-field results were obtained using TW’s code [@Wardell-etal-2012; @Thornburg-Wardell-2017:Kerr-scalar-self-force] extended to compute additional field diagnostics. This code uses an effective-source regularization followed by an $e^{im\phi}$ Fourier-mode decomposition and a separate $2{+}1$-dimensional time-domain numerical evolution of each Fourier mode. The main outputs of this code are the regularized scalar field at selected (fixed) spatial positions, the regularized scalar field and self-force at the particle, and the physical radiated scalar field at ${\mathcal{J}}^+$.
- Our gravitational-field results were obtained using van de Meent’s code [@vandeMeent:2015lxa; @vandeMeent:2016pee; @vandeMeent:2016hel; @vandeMeent:2017bcc]. This code obtains the local metric perturbation in the frequency domain by reconstructing the metric perturbation from the Weyl scalar $\Psi_4$, followed by $\ell$-mode regularization to obtain the regular part. The main outputs of this code are the regularized outgoing–radiation-gauge metric perturbation and self-force at the particle, and the physical radiated $\Psi_4$ at ${\mathcal{J}}^+$ and selected fixed positions in the spacetime.
For both codes we take the particle orbit to be a bound equatorial geodesic; we do not consider changes in the orbit induced by the self-force.
To briefly summarize our main results, we observe wiggles across a wide range of Kerr spins and particle orbits. Wiggles are present in all of our field diagnostics in the strong-field region and at ${\mathcal{J}}^+$. Except for a few anomalous results for near-extremal Kerr spacetimes (dimensionless Kerr spins ${\gtrsim}0.9999$), our results are all consistent with the interpretation of wiggles as Kerr QNMs. Wiggles are stronger and more readily observable for high Kerr spins, highly eccentric prograde particle orbits, and particle periapsis radii close to the light ring.
The remainder of this paper is organized as follows:
Section \[sect:introduction/notation-etal\] summarizes our notation and our parameterization of bound geodesic orbits in Kerr spacetime. Section \[sect:calculations-of-Kerr-perturbations\] briefly summarizes our calculations of scalar-field (section \[sect:calculations-of-Kerr-perturbations/scalar-field\]) and gravitational (section \[sect:calculations-of-Kerr-perturbations/gravitational\]) perturbations of Kerr spacetime. Section \[sect:field-diagnostics\] describes our local- and radiated-field diagnostics. Section \[sect:QNM-models-and-fits\] describes our QNM models and how we fit these to time series of our field diagnostics. Section \[sect:data-and-QNM-fits\] presents our data for scalar-field and gravitational perturbations of Kerr spacetime, and QNM-model fits to this data. Finally, section \[sect:discussion-and-conclusions\] discusses our results and presents our conclusions.
Notation and parameterization of [K]{}err geodesics {#sect:introduction/notation-etal}
---------------------------------------------------
We generally follow the sign and notation conventions of Wald [@Wald-1984], with $G = c = 1$ units and a $(-,+,+,+)$ metric signature. We use the Penrose abstract-index notation, with indices $abcd$ running over spacetime coordinates, and $ijk$ running over the spatial coordinates. ${\nabla}_a$ is the (spacetime) covariant derivative operator and $g$ is the determinant of the spacetime metric. $X := Y$ means that $X$ is defined to be $Y$. ${\Box}:= {\nabla}_a {\nabla}^a$ is the 4-dimensional (scalar) wave operator [@Brill-etal-1972; @Teukolsky73].
We use Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ on Kerr spacetime, defined by the line element $$\begin{aligned}
ds^2 = {} &
- \left( 1 - \frac{2Mr}{\Sigma} \right) \, dt^2
- 4M^2 \tilde{a} \frac{r \sin^2\theta}{\Sigma} \, dt \, d\phi
\nonumber\\*
&
+ \frac{\Sigma}{\Delta} \, dr^2
+ \Sigma \, d\theta^2
\nonumber\\*
&
+ \left(
r^2 + M^2 \tilde{a}^2 + 2M^3 \tilde{a}^2 \frac{r \sin^2\theta}{\Sigma}
\right) \sin^2\theta \, d\phi^2
\text{~,}
\label{eqn:Kerr-Boyer-Lindquist-coords}\end{aligned}$$ where $M$ is the black hole’s mass, $\tilde{a} = J/M^2$ is the black hole’s dimensionless spin (limited to $|\tilde{a}| < 1$), $\Sigma := r^2 + M^2 \tilde{a}^2 \cos^2\theta$, and $\Delta := r^2 - 2Mr + M^2 \tilde{a}^2$. We also define $a := M\tilde{a}$ (this is unrelated to the use of $a$ as an abstract tensor index). In these coordinates the event horizon is the coordinate sphere $r = r_+ = M \left(1 + \sqrt{1 - \tilde{a}^2}\right)$ and the inner horizon is the coordinate sphere $r = r_- = M \left(1 - \sqrt{1 - \tilde{a}^2}\right)$. (See footnote \[footnote:sf-compactification\] for a discussion of TW’s coordinate compactification near the event horizon and ${\mathcal{J}}^+$.)
Following Ref. [@Sundararajan-Khanna-Hughes-2007], we define the tortoise coordinate $r_*$ by $$\frac{dr_*}{dr} = \frac{r^2+M^2 \tilde{a}^2}{\Delta}.
\label{eqn:rstar-defn}$$ and fix the additive constant by choosing $$\begin{aligned}
r_* = {}& r
+ 2M\frac{r_+}{r_+ - r_-} \ln\left( \frac{r - r_+}{2M} \right)
\nonumber\\
& \phantom{r}
- 2M\frac{r_-}{r_+ - r_-} \ln\left( \frac{r - r_-}{2M} \right)
\text{~.}
\label{eqn:rstar(r)}\end{aligned}$$ $u := t - r_*$ is thus an outgoing null coordinate.
The particle’s worldline is $x^i = x^i_{\text{particle}}(t)$; we consider this to be known in advance, i.e., we do *not* consider changes to the particle’s worldline induced by the self-force. For present purposes we consider only particle orbits in the Kerr spacetime’s equatorial plane; this restriction is for computational convenience and is not fundamental. We take the particle to orbit in the $d\phi/dt > 0$ direction, with $\tilde{a} > 0$ for prograde orbits and $\tilde{a} < 0$ for retrograde orbits.
We define $r_{\min}$ and $r_{\max}$ to be the particle’s periapsis and apoapsis $r$ coordinates, respectively. We parameterize bound geodesic equatorial particle orbits by the usual (dimensionless) semi-latus rectum $p$ and eccentricity $e$ (defined by $r_{\min} = pM\big/(1+e)$ and $r_{\max} = pM\big/(1-e)$), so that the particle orbit is given by $$r_{\text{particle}}(t) = \frac{pM}{1 + e \cos \chi_r(t)}
\text{~,}$$ for a suitable orbital-phase function $\chi_r$. We refer to the combination of a spacetime and a (bound geodesic) particle orbit as a “configuration”, and parameterize it with the triplet $(\tilde{a},p,e)$. We define $T_r$ to be the coordinate-time period of the particle’s radial motion; we usually refer to $T_r$ as the particle’s “orbital period”.
Calculations of scalar-field and gravitational perturbations of [K]{}err spacetime {#sect:calculations-of-Kerr-perturbations}
==================================================================================
Scalar-field perturbations of [K]{}err spacetime {#sect:calculations-of-Kerr-perturbations/scalar-field}
------------------------------------------------
In this section we summarize TW’s scalar-field calculations [@Wardell-etal-2012; @Thornburg-Wardell-2017:Kerr-scalar-self-force]. These authors consider a real scalar field $\Phi_{\text{physical}}$ satisfying the wave equation in Kerr spacetime, $${\Box}\Phi_{\text{physical}}= -4 \pi q
\int \frac{\delta^4 \bigl( x^a - x^a_{\text{particle}}(t) \bigr)}{\sqrt{-g}}
\, d\tau
\label{eqn:scalar-field-wave-eqn}
\text{~,}$$ sourced by a point “particle” of scalar charge $q$ which is taken to move on a (pre-specified) equatorial geodesic orbit. $\Phi_{\text{physical}}$ satisfies outgoing-radiation (retarded) boundary conditions at ${\mathcal{J}}^+$. This system provides a toy model of the full gravitational perturbation problem without the complexity of gauge choice.
Because $\Phi_{\text{physical}}$ diverges on the particle worldline, must be regularized. TW use an effective-source regularization of the type introduced by Barack and Golbourn [@Barack-Golbourn-2007] and Vega and Detweiler [@Vega-Detweiler-2008:self-force-regularization] (see [@Vega-Wardell-Diener-2011:effective-source-for-self-force] for a review), using the puncture function and effective source described by Wardell [[*et al.*]{}]{} [@Wardell-etal-2012]. In a neighborhood of the particle worldline, TW define a (real) regularized scalar field $\Phi_{\text{regularized}}= \Phi_{\text{physical}}- \Phi_{\text{puncture}}$, where $\Phi_{\text{puncture}}$ is a suitably-chosen approximation to the Detweiler-Whiting singular field [@Detweiler-Whiting-2003]. The (4-vector) self-force acting on the particle is then given by $$F_a = \bigl. ({\nabla}_a \Phi_{\text{regularized}}) \bigr| _{x^i = x^i_{\text{particle}}(t)}
\label{eqn:scalar-field-self-force}
\text{~.}$$
TW make an azimuthal Fourier decompositions of all the spacetime scalar fields into complex $e^{im\tilde{\phi}}$ modes, $$\Phi(t,r,\theta,\phi)
= \sum_{m=-\infty}^\infty
\dfrac{1}{r} \varphi_m(t,r,\theta) e^{im\tilde{\phi}}
\label{eqn:scalar-field-mode-sum}
\text{~,}$$ where the extra factor of $1/r$ is introduced for computational convenience and where $\tilde{\phi} := \phi + f(r)$ is an “untwisted” azimuthal coordinate, with $$f(r) = \frac{\tilde{a}}{2\sqrt{1-\tilde{a}^2}}
\ln \left| \frac{r-r_+}{r-r_-} \right|
\text{~.}$$
For each $m$-mode, TW introduce a finite worldtube surrounding the particle worldline in $(t,r,\theta)$ space. For particle orbits with significant eccentricity ($e {\gtrsim}0.2$) the worldtube (now viewed as a region in $(r,\theta)$ in each $t={\text{constant}}$ slice) moves inward and outward in $r$ during each orbit so as to always contain the particle. All the results reported here were obtained using a worldtube which is rectangular in $(r,\theta)$, with size $10M$ in $r_*$ by approximately $0.25$ radians in $\theta$, symmetric about the equatorial plane, and kept centered on the particle to within $0.25M$ in $r_*$ as the particle moves.
TW numerically solve for the piecewise function $$\bigl. (\varphi_m) \bigr. _{\text{numerical}}= \begin{cases}
\bigl. (\varphi_m) \bigr. _{\text{regularized}}& \text{inside the worldtube} \\
\bigl. (\varphi_m) \bigr. _{\text{physical}}& \text{outside the worldtube} \end{cases}
\text{~.}$$ using a time-domain $2{+}1$-dimensional finite-difference numerical evolution with mesh refinement. Because the (Kerr) background spacetime is axisymmetric, the Fourier modes $\varphi_m$ evolve independently – there is no mixing of the modes. Because the physical scalar field $\Phi$ is real, only the $m \ge 0$ modes need to be explicitly computed; the $m < 0$ modes may be obtained by symmetry.
Corresponding to the Fourier decomposition , the self-force can be written as a similar sum of $e^{im\tilde{\phi}}$ modes, $$F_a = q \sum_{m=0}^\infty F_a^{(m)}
\label{eqn:scalar-self-force-mode-sum}
\text{~,}$$ where each $F_a^{(m)}$ may be computed from the corresponding $\bigl. (\varphi_m) \bigr. _{\text{regularized}}$ field near the particle. TW compute a finite set of modes (typically $0 \le m \le 20$) and estimate the $m > 20$ contributions to the sum via a large-$m$ asymptotic series.
TW use a Zenginoğlu-type hyperboloidal compactification [@Zenginoglu-2008:hyperboloidal-foliations-and-scri-fixing; @Zenginoglu-2008:hyperboloidal-evolution-with-Einstein-eqns; @Zenginoglu-2011:hyperboloidal-layers-j-comp-phys; @Zenginoglu-Khanna-2011:Kerr-EMRI-waveforms-via-Teukolsky-evolution; @Zenginoglu-Kidder-2010:hyperboloidal-evolution-of-scalar-field-on-Schw; @Zenginoglu-Tiglio-2009:spacelike-matching-to-null-infinity; @Bernuzzi-Nagar-Zenginoglu-2011:Schw-EMRI-waveforms-via-EOB-evolution; @Bernuzzi-Nagar-Zenginoglu-2012:Schw-EMRI-horizon-absorption-effects] so they also have direct access to far-field quantities at ${\mathcal{J}}^+$ (where the coordinate $t$ becomes a Bondi-type retarded time coordinate).[^3]
Gravitational perturbations of [K]{}err spacetime {#sect:calculations-of-Kerr-perturbations/gravitational}
-------------------------------------------------
In this section we summarize the metric reconstruction approach used by van de Meent [@vandeMeent:2015lxa; @vandeMeent:2016pee; @vandeMeent:2016hel; @vandeMeent:2017bcc] to calculate gravitational perturbation of Kerr spacetime generated by particles on bound geodesic orbits. This approach starts from the the spin-(-2) Teukolsky variable, $$\Phi_{-2} := (r-i a\cos\theta)^4\Psi_4,$$ where $\Psi_4$ is one of the Weyl scalars. As shown by Teukolsky [@Teukolsky73], linear perturbations to this variable satisfy an equation of motion that decouples from all other degrees of freedom. Moreover, unlike the linearized Einstein equation, solutions to the Teukolsky equation can be decomposed into Fourier-harmonic modes, $$\Phi_{-2} = \sum_{{\mathfrak{l}}m \omega}
R_{{\mathfrak{l}}m \omega}(r) S_{{\mathfrak{l}}m\omega}(\theta)
e^{i m \phi- i \omega t}
\text{~,}$$ where the $R_{{\mathfrak{l}}m \omega}$ are solutions of the radial Teukolsky equation, the $S_{{\mathfrak{l}}m\omega}$ are spin-weight spheroidal harmonics, and ${\mathfrak{l}}$ is the spheroidal mode number. In van de Meent’s code the radial Teukolsky equation can be solved to arbitrarily high precision using a numerical implementation of the semi-analytical methods of Mano, Suzuki, and Takasugi [@Mano:1996gn; @Mano:1996vt].
Remarkably, $\Phi_{-2}$ contains almost all information about the corresponding perturbation of the metric [@Wald:1973], and in vacuum it is possible to reconstruct the metric perturbation in a radiation gauge [@Cohen:1974cm; @Kegeles:1979an; @Chrzanowski:1975wv; @Wald:1978vm]. As detailed in Refs. [@vandeMeent:2015lxa; @vandeMeent:2016pee; @vandeMeent:2016hel; @vandeMeent:2017bcc], this procedure can be used to calculate the backreaction of the metric perturbation on the particle, the gravitational self-force, which then takes the form $$F_a
= \sum_{\substack{{\mathfrak{l}}m \omega\\ nk\pm}}
\mathcal{C}^{\pm}_{a m \omega nk}
R_{{\mathfrak{l}}m \omega}^{(n)\pm}(r_0)
S_{{\mathfrak{l}}m\omega}^{(k)}(\theta_0)
e^{i m \phi_0 - i \omega t_0}
\text{~,}$$ where the $ R_{{\mathfrak{l}}m \omega}^{\pm}(r_0)$ are vacuum solutions of the radial Teukolsky equation analytically extended to the particle position $r_0$ from either infinity ($+$) or the black hole horizon ($-$) (method of extended homogeneous solutions [@Barack:2008ms]), and the $(n)$ and $(k)$ superscripts on a function denote derivatives with respect to the argument. The indices $n$ and $k$ run from 0 to 3.
Although each individual term in the sum above is finite, the sum as a whole does not converge. This is a simple consequence of the fact that it was built from the retarded field perturbation rather than the Detweiler-Whiting regular field. To obtain the regular field we still need to subtract the Detweiler-Whiting singular field. In principle this can be done mode-by-mode. To match previous analytical calculations of the large $\ell$-behavior of the singular field [@Barack:2002mh; @Barack:2009ux], we need to re-expand the spheroidal ${\mathfrak{l}}$-modes to spherical $\ell$-modes, $$F_a^{(\ell)} =
\sum_{\substack{{\mathfrak{l}}m \omega\\ n\pm}}
\tilde{\mathcal{C}}^{\ell \pm}_{a{\mathfrak{l}}m \omega n}
R_{{\mathfrak{l}}m \omega}^{(n)\pm}(r_0)
Y_{\ell m}(\theta_0)
e^{i m \phi_0 - i \omega t_0}
\text{~.}$$ With this re-expansion, the local gravitational self force is given by $$F_a = \sum_\ell F_a^{(\ell)} - B_a
\text{~,}$$ where, as shown in [@Pound:2013faa], we can use the Lorenz-gauge $B_a$ parameter given in [@Barack:2002mh; @Barack:2009ux].
The metric reconstruction formalism can only recover parts of the metric that contribute to $\Psi_4$. This means that the reconstructed metric carries an ambiguity, which can be shown [@Wald:1973] to consist of perturbations of the background within the class of Kerr metrics and pure gauge contributions. These ambiguities can be uniquely fixed based on physical considerations [@Merlin:2016boc; @vandeMeent:2017fqk]. The corrections needed to fix these ambiguities are known and straightforward to add. They contribute only to the low frequency envelope of the self-force. Hence, to facilitate identification and extraction of the wiggles in the gravitational self-force, we omit them in this work.
Frequency domain calculations of the type used here are ideally suited for calculating metric perturbations with a sparse discrete frequency spectrum, such as those sourced by a particle on a low eccentricity geodesic. That spectrum becomes denser at higher eccentricities, necessitating the calculation of more frequency modes and making the calculation more time-consuming. Moreover, as discussed in detail in Ref. [@vandeMeent:2016pee], the method of extended homogeneous solutions leads to large cancellations between different (low) frequency modes for high-$\ell$ modes, causing a large loss of precision. In this work we tackle this problem by harnessing the full power of the arbitrary precision implementation of our code and simply throw more precision at the computation than we lose in the cancellation.
For this work we calculated the full gravitational self-force for orbits with eccentricities up to $e = 0.8$, which involves dealing with cancellations of around 30 orders of magnitude. These calculations are fairly resource intensive, taking ${\mathcal{O}}(10^4)$ CPU hours (or a few days on 400 CPUs) to compute.
However, for many aspects of our investigation here, we do not need the full local regular metric perturbation. If we want to look for the dominant low-${\mathfrak{l}}$ QNMs, then these will contribute (mostly) to the low-${\mathfrak{l}}$ modes of the gravitational metric perturbation. For this purpose, we define the individual ${\mathfrak{l}}m$ modes of the Teukolsky variable $$\Phi_{-2}^{({\mathfrak{l}}m)}
= \sum_{\omega}
R_{{\mathfrak{l}}m \omega}(r) S_{{\mathfrak{l}}m\omega}(\theta)
e^{i m \phi - i \omega t},$$ and the gravitational self-force $$F_a^{({\mathfrak{l}}m)} :=
\sum_{\substack{\omega\\ nk\pm}}
\mathcal{C}^{\pm}_{a m \omega n k} R_{{\mathfrak{l}}m \omega}^{(n)\pm}(r_0)
S_{{\mathfrak{l}}m\omega}^{(k)}(\theta_0) e^{i m \phi_0 - i \omega t_0}.$$ These are much easier to compute, and for low ${\mathfrak{l}}$ do not suffer from the large loss of precision due to the method of extended homogeneous solutions, thus allowing for very high accuracy calculations without excessive computational cost.
Field diagnostics {#sect:field-diagnostics}
=================
We consider several different types of local- and radiated-field diagnostics, and attempt to fit the wiggles in these diagnostics to QNM models. Clearly the presence of wiggles in the physical scalar field or metric perturbation implies the presence of wiggles in some or all of the $e^{im\tilde{\phi}}$ scalar-field modes or $({\mathfrak{l}}m)$ metric-perturbation modes (respectively), and vice versa.[^4] Because many fewer QNMs are present at significant amplitude (usually only one, or in a few cases two), it is much simpler to analyze wiggles in the individual modes.
Table \[tab:field-diagnostics\] summarizes our local- and radiated-field diagnostics for studying wiggles. We consider (time series of) diagnostics at three locations in spacetime:
- *Diagnostics of the local field at selected fixed spatial “watchpoint” coordinate positions $(r,\theta,\phi) = {\text{constant}}$.* These diagnostics directly sample any QNMs that may be present, but the diagnostics may be contaminated by the direct field when the particle is close to the watchpoint position.
For the scalar-field case, we avoid any such possible contamination by considering the regularized field mode $\bigl. (\varphi_m) \bigr. _{\text{regularized}}$. However, this is only defined within the worldtube, so for orbits with significant eccentricity (where the worldtube moves in $(r,\theta)$ during the particle orbit) any given watchpoint may lie outside the worldtube (and thus leave $\bigl. (\varphi_m) \bigr. _{\text{regularized}}$ undefined) during some parts of the orbit. To minimize this effect, for many of the analyses reported here we use watchpoint positions which are near the orbit’s apoapsis, where the particle (and hence the worldtube) motion is relatively slow and hence a suitable watchpoint can remain within the worldtube for a relatively long time in each orbit. All our scalar-field watchpoints are in the equatorial plane.
For the gravitational case, the regularized field is not readily available, so instead we have the code output the retarded $\Phi_{-2}^{({\mathfrak{l}}m)}$ on the symmetry axis of the background Kerr spacetime ($\theta=0$) and the equatorial plane $(\theta=\pi/2)$ at coordinate radii corresponding to the particle’s periapsis and apoapsis distances.
- *Diagnostics of the local field at the particle.* Here we consider the $e^{im\tilde{\phi}}$ (scalar-field) and $({\mathfrak{l}}m)$ (gravitational) modes of the self-force itself. The main complication here is that these diagnostics sample the field perturbation at a *time-dependent* position (the particle position), so our fitting model for the QNM effects must include corrections for the spatial variation of the QNM eigenfunctions as the particle (sampling point) moves. For the azimuthal ($\phi$) particle motion this is straightforward (described in section \[sect:QNM-models-and-fits\]) but for the radial ($r$) motion we include this correction only approximately.
- *Diagnostics of the radiated field at ${\mathcal{J}}^+$.* These have the advantage of being physically observable and of allowing the $e^{im\tilde{\phi}}$ (scalar-field) and $({\mathfrak{l}}m)$ (gravitational) mode decompositions to be defined in a weak-field region (for the gravitational case, this also avoids any gauge dependence).
At ${\mathcal{J}}^+$ we only have the physical (retarded) fields available, so it is more difficult to separate wiggles from the overall radiation pattern. To help in making this separation, we observe that wiggles are of relatively high (temporal) frequency relative to other major features in the radiated fields, so that taking time derivatives of the radiated fields increases the wiggles’ amplitude relative to that of the other features. We have found that good results are obtained by using as diagnostics the second time derivatives $\partial_{tt} \left( \bigl. (\varphi_m) \bigr. _{\text{physical}}\right)$ evaluated in the equatorial plane (scalar field)[^5] and $\Psi_4$ evaluated either on the z axis or in the equatorial plane (gravitational).
----------------------------------------------------------------------------------------------------------------------------------------
Field perturbation at … Scalar field Gravitational
------------------------- --------------------------------------------------------------------------- ----------------------------------
fixed spatial position $\bigl. (\varphi_m) \bigr. _{\text{regularized}}$ $\Phi_{-2}^{({\mathfrak{l}}m)}$
(strong-field)
particle position $F_a^{(m)}$ ${}_{-2}F_a^{({\mathfrak{l}}m)}$
${\mathcal{J}}^+$ $\partial_{tt} $\Psi_4$
\left( \bigl. (\varphi_m) \bigr. _{\text{physical}}\right)$
----------------------------------------------------------------------------------------------------------------------------------------
: \[tab:field-diagnostics\] This table shows the local- and radiated-field diagnostics in which we study wiggles.
Quasinormal-mode models and fits {#sect:QNM-models-and-fits}
================================
Scalar-field perturbations {#sect:QNM-models-and-fits/scalar-field}
--------------------------
### Perturbations at a fixed spatial position {#sect:QNM-models-and-fits/scalar-field/watchpoint}
Consider first the case of wiggles in an individual $e^{im\phi}$ Fourier mode of the regularized scalar field, observed at a fixed “watchpoint” spatial position in the strong-field region. We consider the model
$$\bigl. (\varphi_m) \bigr. _{\text{regularized}}= B(t)
+ \sum_k A^{(k)}
\exp \left( - \lambda^{(k)} (t - t_{\text{ref}}) \right)
\,
\sin \left(
2\pi \frac{t - t_{\text{ref}}}{P^{(k)}}
+ \eta^{(k)}
\right)
\,\, \text{,}
\label{eqn:scalar-field=bg+sum-of-damped-sinusoids}$$
where $B$ is a spline function representing the slowly-varying “background” variation of the scalar field, $k$ indexes the damped-sinusoids included in the model, $A^{(k)}$, $\lambda^{(k)}$, $P^{(k)}$, and $\eta^{(k)}$ are respectively the amplitude, damping rate, period, and phase offset of each damped-sinusoid, and the subscript ${}_{\text{ref}}$ denotes a “reference” time chosen for convenience. To avoid degeneracy between the spline and the damped-sinusoid we require that the spacing in $t$ between adjacent spline control points always be at least $1.5 P^{(\max)}$, where $P^{(\max)} := \displaystyle\max_k P^{(k)}$ is the period of the longest-period damped-sinusoid in the model.
### Perturbations at the particle position {#sect:QNM-models-and-fits/scalar-field/self-force}
Consider next the case of wiggles in the radiation-reaction self-force (which is implicitly defined at the particle position). This introduces two complications: the self-force is a 4-vector (with nontrivial $t$, $r$, and $\phi$ components for our equatorial orbits), and the field perturbation is being sampled at a time-varying position. Analogously to , we thus consider the model
$$F_a(u) = \frac{B_a(u)}{r^3_{\text{particle}}(u)}
+ \sum_k \frac{A^{(k)}_a}{r_{\text{particle}}(u)}
\exp \left( - \lambda^{(k)} (u - u_{\text{ref}}) \right)
\sin \left(
2\pi \frac{u - u_{\text{ref}}}{P^{(k)}}
- m \bigl( \phi_{\text{particle}}(u) - \phi_{\text{ref}}\bigr)
+ \eta^{(k)}_a
\right)
\,\, \text{,}
\label{eqn:scalar-F-a=bg+sum-of-damped-sinusoids}$$
where we now parameterize the particle’s motion using the retarded time coordinate $u$,[^6] $B_a$ is now a 4-vector spline function representing the background variation of the self-force along the particle worldline, $k$ again indexes the damped-sinusoids included in the model, $A_a^{(k)}$, $\lambda^{(k)}$, $P^{(k)}$, and $\eta_a^{(k)}$ are now respectively the 4-vector amplitude, damping rate, period, and 4-vector phase offset of each damped-sinusoid, and the subscript ${}_{\text{ref}}$ again denotes a “reference” time chosen for convenience. The non-degeneracy condition on the background spline now applies to the spacing in $u$ between adjacent spline control points.
Notice that the damping rate and oscillation period of each damped-sinusoid are common across all tensor components of the self-force. The $- m \phi_{\text{particle}}(u)$ term models the variation in oscillation phase due to particle’s (i.e., the sampling point’s) motion in $\phi$. The $1/r^3_{\text{particle}}(u)$ and $1/r_{\text{particle}}(u)$ factors model the expected far-field variation in self-force and in the oscillation eigenfunction amplitude with position. (Actual QNM eigenfunctions have a much more complicated variation of amplitude with spatial position, but for simplicity we omit this from our model.)
### Perturbations at ${\mathcal{J}}^+$ {#sect:QNM-models-and-fits/scalar-field/Scri}
Finally, consider the case of wiggles in the radiated (physical) field at ${\mathcal{J}}^+$. Because of the hyperboloidal time slices, we observe these at finite coordinate time $t$ (the ${\mathcal{J}}^+$ time has an arbitrary offset relative to the strong-field coordinate time). As noted in section \[sect:field-diagnostics\], here it is somewhat difficult to separate wiggles from the overall radiation pattern, so we consider second time derivatives of the physical scalar-field modes. Analogously to and , we thus consider the model
$$\biggl.
\partial_{tt} \left( \bigl. (\varphi_m) \bigr. _{\text{regularized}}\right)
\biggr|_{{\mathcal{J}}^+}
= B(t)
+ \sum_k A^{(k)}
\exp \left( - \lambda^{(k)} (t - t_{\text{ref}}) \right)
\,
\sin \left(
2\pi \frac{t - t_{\text{ref}}}{P^{(k)}}
+ \eta^{(k)}
\right)
\label{eqn:scalar-field@Scri=bg+sum-of-damped-sinusoids}
\,\, \text{,}$$
where the meanings of all terms (and the non-degeneracy condition on the background spline) are the same as in .
### Fitting the models {#sect:QNM-models-and-fits/scalar-field/fits}
For each of the models , , and , we visually inspect plots of our time-series data to identify a suitable range of the independent variable for fitting and to choose initial guesses for the background and wiggle parameters, then use the nonlinear least-squares fitting subroutine from the library [@MINPACK] to fit the model to the data. To make the model closer to linear (which improves the convergence of the nonlinear fitting), we fit the wiggle amplitudes and phases as cosine- and sine-component amplitudes (i.e., $A \sin(X + \eta)$ is actually fitted as $A^{(\cos)} \cos(X) + A^{(\sin)} \sin(X)$). In most cases we used uniform weighting for the fits, but in a few cases we used weights proportional to $r^3$ so as to improve the fit at late times (close to apoapsis).
### Uncertainties in the fitted wiggle parameters {#sect:QNM-models-and-fits/scalar-field/Monte-Carlo}
The residuals from our wiggle fits are *not* random, but rather are dominated by low-amplitude oscillations of similar frequency to the wiggles themselves (this can be seen in figures \[fig:sf-a99p3e8-m1-fits-and-residuals\] and \[fig:sf-a99p3e8-m4-fits-and-residuals\]). This means that formal uncertainty estimates for the fitted parameters $\bigl( P^{(k)}, \tau^{(k)} \bigr)$ (derived assuming uncorrelated Gaussian residuals) are not realistic. Because of the oscillatory nature of the residuals, the fitted parameters are slightly dependent on the precise choice of fitting interval; this is, in fact, usually the dominant uncertainty in the fitted parameters.
We use a Monte-Carlo procedure to estimate realistic uncertainties in the fitted parameters: Given a fit of one of the above wiggle models to our data in some interval $I$ (in either $t$ or $u$) of length $L_{\text{fit}}\ge 4 P^{(\max)}$, we randomly choose $N_{\text{trial}} = 300$ subintervals of $I$ (randomly sampling each lower and upper interval endpoint from a uniform distribution) subject to the constraint that each subinterval must have a minimum length of $L_{\min} \,{=}\, 3 P^{(\max)}$.[^7] Then we repeat the wiggle-model fit for each subinterval.[^8] The ensemble of the $N_{\text{trial}}$ sets of Monte-Carlo–trial fitted parameters then provides an estimate of the uncertainty in the fitted parameters from the full-interval fit.
After allowing initial transients to decay, our numerical calculations extend over a number of particle orbits. Because the particle orbit precesses strongly, each orbit places the particle in a different position with respect to any fixed watchpoint or ${\mathcal{J}}^+$ observer. For each orbit we repeat the entire fitting procedure (including the full set of Monte-Carlo sub-interval trials). Our final estimate for the uncertainty in the fitted parameters is obtained from the union of all the Monte-Carlo trials over several (typically 3) distinct orbits.
This procedure has two main limitations:
- The procedure is not applicable to cases where the overall fitting interval is too short (length $L_{\text{fit}}< 4P^{(\max)}$). (Footnote \[footnote:sf-fits-minimum-length-requirements\] outlines the reasons for this.)
- If a wiggle is rapidly damped, then the wiggle amplitude becomes very small at the right (large $t$ or $u$) end of a long fitting interval, so a subinterval of near-minimum length ($3 P^{(\max)}$) which is close to the right end of the overall fitting interval will have a poorly-constrained fit, yielding a large scatter in the fitted parameters.
These limitations are most severe when the wiggles have low amplitude and are rapidly damped, as is the case for low Kerr spins.
### Other error sources {#sect:QNM-models-and-fits/scalar-field/other-error-sources}
There are a number of other error sources not taken into account in our Monte-Carlo error estimates:
- Our (TW’s) numerical code only computes the diagnostics to finite accuracy. Comparing diagnostics between calculations done with different numerical resolutions, we have generally excluded any data where the diagnostic computed at our highest resolution (that shown in table \[tab:sf-configurations\]) differs from that computed at the next-lower resolution listed in table \[tab:sf-grids\] by more than a few percent.
- Wiggles are not perfectly separable from the background variation of the diagnostics. That is, the actual frequency spectra of the diagnostics are almost certainly continuous, and cannot be unambiguously separated into low-frequency (background) and high-frequency (wiggle) components.
- Our models for the background variation are imperfect. Our constraint that background spline control points must be spaced at least $1.5$ wiggle periods apart keeps the background and wiggles from being degenerate, but at the cost of leaving the background model unable to accurately fit some non-wiggle variations, particularly for small-$m$ (longer-period) wiggles where the spline control points are forced to be quite far apart.
- For wiggles in $F_a$, our wiggle model doesn’t accurately include the actual spatial variation of the wiggle (QNM) eigenfunctions.
- There may be multiple wiggle modes present simultaneously in the diagnostics for a single $m$. Although our wiggle models and fitting software support simultaneously fitting an arbitrary number of wiggles, we have generally not done this, i.e., we have generally only attempted to fit a single-wiggle model for each diagnostic time series.[^9]
We believe that all these other error sources are small, but it is difficult to quantify them.
For each wiggle fit we visually assess the fit residuals to look for obvious systematics. For all results reported here the fit residuals are at least a factor of $10$ smaller than the wiggle amplitude; in most cases they are a factor of $30$ to $100$ smaller. This suggests that our fits are indeed accurately modelling at least the dominant wiggle features of the diagnostics.
Gravitational perturbations {#sect:QNM-models-and-fits/gravitational}
---------------------------
In the gravitational case we search for the QNM frequencies in the individual (spheroidal) $({\mathfrak{l}}m)$-modes. By looking at individual $({\mathfrak{l}}m)$-modes we minimize the number of QNMs that need to be modelled (fitted) simultaneously. Since QNMs appear naturally as spheroidal modes, using spheroidal modes minimizes the amount of “crosstalk” mixed in from neighboring modes. Note that since the spheroidicity of the spheroidal harmonics in the Teukolsky equation depends on the frequency of the modes, the QNMs have complex spheroidicity and will not project perfectly on the corresponding (real) spheroidal modes that appear in the field solutions. Consequently, “crosstalk” between the modes cannot be fully eliminated. Nonetheless, the crosstalk in the spheroidal modes should be significantly smaller than if one were to use the spherical $\ell$-modes.
### Fit models
The fit models used in the gravitational case are very similar to the ones used in the scalar case. For the “watchpoint” diagnostics we use
$$\Phi_{-2}^{({\mathfrak{l}}m)} =\sum_n B^{(n)} t^n + \sum_k\left\{
A_s^{(k)} \sin {\left[ \omega_k (t-t_{\text{ref}})\right]}
+
A_c^{(k)} \cos {\left[ \omega_k (t-t_{\text{ref}})\right]}
\right\}
e^{-\alpha_k (t-t_{\text{ref}})}.$$
In this case the smooth background of the signal is modelled by a simple polynomial in $t$. We maximize the number of linear fit parameters by writing the model as a sum of sines and cosines.
Similarly, the $({\mathfrak{l}}m)$-modes contributing to the local gravitational self-force are modelled by $$F_a^{({\mathfrak{l}}m)} =\sum_n B^{(n)}_a u^n + \sum_k\left\{
A_{s,a}^{(k)} \sin {\left[ \omega_k (u-u_{\text{ref}})-m\phi_{\text{particle}}\right]}
+
A_{c,a}^{(k)} \cos {\left[ \omega_k (u-u_{\text{ref}})-m\phi_{\text{particle}}\right]}
\right\}\frac{e^{-\alpha_k (u-u_{\text{ref}})}}{r_{\text{particle}}}.$$ As in the scalar case, the main shortcoming in this model is the inaccurate modelling of the QNMs’ radial profiles.
Finally, the model for the gravitational waveform at ${\mathcal{J}}^+$ is very similar to the model for the watchpoints, $$\label{eq:gravscriplusmodel}
\lim_{r\to\infty} r\Psi_{4}^{({\mathfrak{l}}m)} =\sum_n B^{(n)} u^n + \sum_k\left\{
A_s^{(k)} \sin {\left[ \omega_k (u-u_{\text{ref}})\right]}
+
A_c^{(k)} \cos {\left[ \omega_k (u-u_{\text{ref}})\right]}
\right\}e^{-\alpha_k (u-u_{\text{ref}})}
\,\, \text{.}$$
### Fitting procedure
The only non-linear parameters in the above models are the QNM frequencies $\omega_k$ and decay constants $\alpha_k$. Consequently, for fixed $\omega_k$ and $\alpha_k$ the remaining parameters can be determined efficiently through a linear least squares procedure. This is implemented by using [[<span style="font-variant:small-caps;">Mathematica</span>]{}]{}’s routine for each diagnostic on a suitable time window of data. To reduce the impact of un-modelled higher order QNMs, these fits are weighted by $\exp(2\alpha_1 t)$. Typically, the fits include around 20 terms in the background model and up to 8 QNMs.
The $\omega_k$ and $\alpha_k$ are then determined by maximizing the sum of the adjusted $R^2$ values of all the component fits. This is implemented using [[<span style="font-variant:small-caps;">Mathematica</span>]{}]{}’s with the method. The initial values for $\omega_k$ and $\alpha_k$ are set by the numerically known corresponding QNMs offset by a random ${\mathcal{O}}(1\%)$ amount. An indication of the modelling error is obtained by varying the fit window and number of background terms, and determining the spread of the best fits.
Data and quasinormal-mode fits {#sect:data-and-QNM-fits}
==============================
Scalar field {#sect:data-and-QNM-fits/scalar-field}
------------
We have surveyed a large number of configurations for Kerr spin $\tilde{a} = 0.99$, together with a smaller number of configurations for other Kerr spins; for selected configurations we have fitted (or attempted to fit) wiggle models as described in section \[sect:QNM-models-and-fits/scalar-field\]. Tables \[tab:sf-configurations\] and \[tab:sf-wiggle-symbols-key\] describe all the configurations surveyed here, and figure \[fig:sf-a99-phase-space-rmin-e\] shows the $(\text{periapsis radius}, \text{orbital eccentricity})$ phase space of the $\tilde{a} = 0.99$ configurations.
[l@d@d@d d@dcdr c@c@cc@c@cc@c@cc@c@cc@c@cc@c@c]{} Name & & & & & & $\dot{\phi}_{{\text{periapsis}}}$ & & & & & & & &\
& & & & & & ($M^{-1}$) & & & w & $F$ & ${\mathcal{J}}$ & w & $F$ & ${\mathcal{J}}$ & w & $F$ & ${\mathcal{J}}$ & w & $F$ & ${\mathcal{J}}$ & w & $F$ & ${\mathcal{J}}$ & w & $F$ & ${\mathcal{J}}$\
w9x5-161 & 0.99999 & 2.918315 & 0.807519 & 230.442 & 1.615 & 0.344414 & 11.810 & dro12-96 & & & & & & & & & & & & & & & & & &\
w9x4-368 & 0.9999 & 7.0 & 0.9 & 1513.112 & 3.684 & 0.145444 & 46.304 & dro8-64 & & & & & & & & & & & & & & & & & &\
w999-278 & 0.999 & 5.0 & 0.8 & 400.508 & 2.778 & 0.199135 & 21.448 & dro10-80 & & & & & & & & & & & & & & & & & &\
w999-368 & 0.999 & 7.0 & 0.9 & 1513.179 & 3.684 & 0.145443 & & dro8-64 & & & & & & & & & & & & & & & & & &\
ze98a & 0.99 & 2.3981 & 0.98 & 3414.259 & 1.211 & 0.430498 & 61.223 & dro6-48 & & & & & & & & & & & & & & & & & &\
ze98 & 0.99 & 2.4 & 0.98 & 3304.620 & 1.212 & 0.430300 & & dro10-80 & & & & & & & & & & & & & & & & & &\
w99-125a & 0.99 & 2.4375 & 0.95 & 957.757 & 1.25 & 0.421924 & 36.314 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-125b & 0.99 & 2.375 & 0.9 & 432.084 & 1.25 & 0.421555 & 20.543 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-125c & 0.99 & 2.25 & 0.8 & 247.845 & 1.25 & 0.420811 & 8.595 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-125d & 0.99 & 2.0 & 0.6 & 228.354 & 1.25 & 0.419300 & 3.003 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-139 & 0.99 & 2.5 & 0.8 & 211.605 & 1.389 & 0.389231 & & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-139b & 0.99 & 2.222222 & 0.6 & 133.939 & 1.389 & 0.386760 & 3.926 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-139d & 0.99 & 1.944444 & 0.4 & 126.071 & 1.389 & 0.384220 & 1.984 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-139c & 0.99 & 1.805556 & 0.3 & 135.092 & 1.389 & 0.382960 & 1.743 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-167m & 0.99 & 3.25 & 0.95 & 1329.680 & 1.667 & 0.336304 & 47.740 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-167 & 0.99 & 3.0 & 0.8 & 230.442 & 1.667 & 0.333682 & 11.810 & dro10-80 & & & & & & & & & & & & & & & & & &\
w99-167k & 0.99 & 2.333333 & 0.4 & 100.014 & 1.667 & 0.326143 & 2.332 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-167d & 0.99 & 2.166667 & 0.3 & 97.092 & 1.667 & 0.324141 & 2.332 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-167j & 0.99 & 2 & 0.2 & 98.818 & 1.667 & 0.322115 & 1.984 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-200d & 0.99 & 3.6 & 0.8 & 273.551 & 2.0 & 0.281145 & 14.784 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-200 & 0.99 & 3.0 & 0.5 & 111.575 & 2.000 & 0.274441 & & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-200b & 0.99 & 2.8 & 0.4 & 99.000 & 2.000 & 0.272062 & 3.003 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-200c & 0.99 & 2.6 & 0.3 & 91.817 & 2.000 & 0.269607 & 2.636 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-222b & 0.99 & 3.333333 & 0.5 & 117.908 & 2.222 & 0.245832 & 4.193 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-222 & 0.99 & 3.111111 & 0.4 & 102.677 & 2.222 & 0.243319 & 3.926 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-222c & 0.99 & 2.888889 & 0.3 & 93.395 & 2.222 & 0.240719 & 3.003 & dro6-48 & & & & & & & & & & & & & & & & & &\
e95 & 0.99 & 5.0 & 0.95 & 2436.050 & 2.564 & 0.220775 & & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-278 & 0.99 & 5.0 & 0.8 & 401.302 & 2.778 & 0.199076 & 20.543 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-278b & 0.99 & 4.166667 & 0.5 & 139.595 & 2.778 & 0.191854 & 5.712 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-278c & 0.99 & 3.888889 & 0.4 & 117.842 & 2.778 & 0.189278 & 4.193 & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-278d & 0.99 & 3.611111 & 0.3 & 103.835 & 2.778 & 0.186607 & 3.672 & dro6-48 & & & & & & & & & & & & & & & & & &\
s99 & 0.99 & 5.850762 & 0.861866 & 771.968 & 3.142 & 0.174409 & & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-357 & 0.99 & 5.0 & 0.4 & 146.751 & 3.571 & 0.139748 & & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-360c & 0.99 & 6.48 & 0.8 & 560.918 & 3.6 & 0.147411 & & dro6-48 & & & & & & & & & & & & & & & &\
w99-360b & 0.99 & 6.12 & 0.7 & 330.647 & 3.6 & 0.145260 & & dro6-48 & & & & & & & & & & & & & & & & & &\
w99-360a & 0.99 & 5.76 & 0.6 & 232.413 & 3.6 & 0.143040 & 11.810 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-360j & 0.99 & 5.4 & 0.5 & 179.842 & 3.6 & 0.140745 & 7.835 & dro6-48 & & & & & & & & & & & & &\
e9 & 0.99 & 7.0 & 0.9 & 1513.855 & 3.684 & 0.145429 & 25.105 & dro8-64 & & & & & & & & & & & & & & & & & &\
w99-444 & 0.99 & 8.0 & 0.8 & 745.170 & 4.444 & 0.113763 & & dro6-48 & & & & & & & & & & & & & & & & & &\
w95-368 & 0.95 & 7.0 & 0.9 & 1516.962 & 3.684 & 0.145349 & & dro6-48 & & & & & & & & & & & & & & & & & &\
n96 & 0.9 & 5.5 & 0.6 & 227.038 & 3.4375 & 0.151199 & & dro8-64 & & & & & & & & & & & & & & & & & &\
w9-368 & 0.9 & 7.0 & 0.9 & 1521.097 & 3.684 & 0.145202 & & dro6-48 & & & & & & & & & & & & & & & & & &\
n95 & 0.9 & 10.0 & 0.5 & 378.408 & 6.667 & 0.062994 & & dro8-64 & & & & & & & & & & & & & & & & & &\
w8-368 & 0.8 & 7.0 & 0.9 & 1530.314 & 3.684 & 0.144751 & 18.300 & dro6-48 & & & & & & & & & & & & & & & & & &\
e8b & 0.8 & 8.0 & 0.8 & 756.641 & 4.444 & 0.113578 & & dro6-48 & & & & & & & & & & & & & & & & & &\
e8 & 0.6 & 8.0 & 0.8 & 771.968 & 4.444 & 0.113000 & & dro8-64 & & & & & & & & & & & & & & & & & &\
w4-368 & 0.4 & 7.0 & 0.9 & 1588.133 & 3.684 & 0.140816 & & dro6-48 & & & & & & & & & & & & & & & & & &\
w2-368 & 0.2 & 7.0 & 0.9 & 1712.163 & 3.684 & 0.137622 & & dro6-48 & & & & & & & & & & & & & & & & & &\
ze4 & 0.2 & 6.15 & 0.4 & 354.628 & 4.393 & 0.106691 & & dro6-48 & & & & & & & & & & & & & & & & & &\
zze9 & 0.0 & 7.800001 & 0.9 & 2224.815 & 4.105 & 0.120223 & & dro8-64 & & & & & & & & & & & & & & & & & &\
ze9 & 0.0 & 7.8001 & 0.9 & 2112.079 & 4.105 & 0.120222 & & dro6-48 & & & & & & & & & & & & & & & & & &\
ns5 & 0.0 & 7.2 & 0.5 & 405.662 & 4.8 & 0.095855 & & dro8-64 & & & & & & & & & & & & & & & & & &\
s0 & 0.0 & 10.695207 & 0.708941 & 771.968 & 6.258 & 0.070830 & & dro4-32 & & & & & & & & & & & & & & & & & &\
n-55 & -0.5 & 10.0 & 0.5 & 505.428 & 6.667 & 0.062012 & & dro6-48 & & & & & & & & & & & & & & & & & &\
s-6 & -0.6 & 13.083066 & 0.609412 & 771.968 & 8.129 & 0.048498 & & dro4-32 & & & & & & & & & & & & & & & & & &\
wm8-631 & -0.8 & 10.1 & 0.6 & 747.545 & 6.313 & 0.066625 & & dro4-32 & & & & & & & & & & & & & & & & & &\
wm99-605 & -0.99 & 11.5 & 0.9 & 3401.251 & 6.053 & 0.072312 & & dro4-32 & & & & & & & & & & & & & & & & & &\
s-99 & -0.99 & 14.542929 & 0.534714 & 771.968 & 9.476 & 0.038531 & & dro4-32 & & & & & & & & & & & & & & & & & &
Symbol Meaning
----------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
we observed oscillations in the diagnostics, successfully fit the appropriate wiggle model described in section \[sect:QNM-models-and-fits/scalar-field\] over a $t$ or $u$ range of $\ge 4$ wiggle periods, and performed the Monte-Carlo error analysis described in section \[sect:QNM-models-and-fits/scalar-field/Monte-Carlo\].
\[1ex\] we observed oscillations in the diagnostics and successfully fit the appropriate wiggle model described in section \[sect:QNM-models-and-fits/scalar-field\], but the model was fitted over too short a $t$ or $u$ range ($< 4$ wiggle periods) for the Monte-Carlo error analysis described in section \[sect:QNM-models-and-fits/scalar-field/Monte-Carlo\] to be used
\[1ex\] we observed oscillations in the diagnostics which visually appeared to be wiggles, with physically reasonable period and decay rate, but we did not attempt to quantitatively fit a wiggle model
\[1ex\] we observed oscillations in the diagnostics which might have been wiggles, but these oscillations were not clearly separated from the background variation in the diagnostics
\[1ex\] we did not observe wiggle-like oscillations in the diagnostics; this could mean either that no wiggles are present, or alternatively that wiggles were present but they were at too low an amplitude and/or insufficiently separated from the background variation to be observed
\[1ex\] (blank) diagnostics were not computed, not computed sufficiently accurately for studying wiggles, or were computed but not assessed
: \[tab:sf-wiggle-symbols-key\] This table explains the meanings of the “wiggle symbols” used in table \[tab:sf-configurations\] and figure \[fig:sf-a99-phase-space-rmin-e\].
[lD[/]{}[/]{}[-1]{}D[/]{}[/]{}[-1]{}D[/]{}[/]{}[-1]{}D[/]{}[/]{}[-1]{}]{} & &\
& & & &\
& & & &\
dro12-96& 1/12 & /216 & 1/96 & /1728\
dro10-80& 1/10 & /180 & 1/80 & /1440\
dro8-64 & 1/8 & /144 & 1/64 & /1152\
dro6-48 & 1/6 & /108 & 1/48 & /864\
dro4-32 & 1/4 & /72 & 1/32 & /576
![\[fig:sf-a99-phase-space-rmin-e\] This figure shows the phase space of the Kerr spin $\tilde{a} = 0.99$ scalar-field configurations presented here, plotted in terms of the periapsis radius $r_{\min}$ and the eccentricity $e$. The shaded region at the left shows orbits with periapsis inside the horizon. The light ring, innermost bound circular orbit (IBCO), innermost stable circular orbit (ISCO), and the locus of marginally stable orbits are also shown. The meanings of the plot symbols are described in detail in Table \[tab:sf-wiggle-symbols-key\]. ](sf-a99-phase-space-rmin-e.pdf){width="\columnwidth"}
Figures \[fig:sf-a99p3e8-m1-wiggles-overview\] and \[fig:sf-a99p3e8-m4-wiggles-overview\] show the wiggles in the scalar-field diagnostics for the $(\tilde{a},p,e) = (0.99,3,0.8)$ configuration, for $m=1$ and $m=4$ respectively. Notice that the wiggles are visible in *all* the field diagnostics. Notice also the much higher frequency and smaller amplitude of the $m=4$ wiggles.
{width="\textwidth"}
{width="\textwidth"}
Figures \[fig:sf-a99p3e8-m1-fits-and-residuals\] and \[fig:sf-a99p3e8-m4-fits-and-residuals\] show our model fits to these wiggles for $m=1$ and $m=4$ respectively. Notice that in each case the spline control points span a wider range of $t$ or $u$ than the range over which the model is fitted. The $y$ coordinates at the spline control points outside the model-fitting range are still adjusted by the least-squares fitting algorithm, but have only small influences on the model within the fitting range.
{width="\textwidth"}
{width="\textwidth"}
Figures \[fig:sf-a99p3e8-all-QNMs-and-m1-6-MC-frequencies\]–\[fig:sf-other-spins-all-QNMs-and-m1-4-MC-frequencies\] show the fitted complex frequencies and their Monte-Carlo error estimates, compared to Kerr QNM frequencies calculated by Berti, Cardoso, and Starinets [@Berti-Cardoso-Starinets-2009:BH-etal-QNM-review; @Berti06b].[^10] In each case the fitted frequencies agree with the calculated QNM frequencies, lending further support to the identification of wiggles with QNMs (more precisely, QNMs sampled at the observation points).
![\[fig:sf-a99p3e8-all-QNMs-and-m1-6-MC-frequencies\] This figure shows all the Kerr QNM frequencies in the region $\bigl({\mathop{\text{Re}}\left(\omega\right)}, {\mathop{\text{Im}}\left(\omega\right)} \bigr)
\in \bigl( [0,3] \times [-0.085,0] \bigr) M^{-1}$ for Kerr spin $\tilde{a} \,{=}\, 0.99$, together with our fitted complex frequencies’ Monte-Carlo error estimates for the $m \,{=}\, 1$ through $m \,{=}\, 6$ modes of the $(\tilde{a},p,e) = (0.99,3,0.8)$ configuration. (Some of these QNM frequencies and all of the Monte-Carlo error estimates are also plotted at different scales in figure \[fig:sf-a99p3e8-least-damped-QNMs-and-m1-6-MC-frequencies-zoomed\].) The apparent anisotropy of the Monte-Carlo “point clouds” in this plot is a visual illusion due to the anisotropic plot scale; the point clouds are actually approximately isotropic in $\bigl( {\mathop{\text{Re}}\left(\omega\right)}, {\mathop{\text{Im}}\left(\omega\right)} \bigr)$. ](sf-a99p3e8-all-QNMs-and-m1-6-MC-frequencies.pdf){width="\columnwidth"}
{width="\textwidth"}
{width="\textwidth"}
{width="\textwidth"} {width="\textwidth"} {width="\textwidth"}
{width="\textwidth"} {width="\textwidth"} {width="\textwidth"}
Gravitational field {#sect:data-and-QNM-fits/gravitational-field}
-------------------
We now turn our attention to gravitational perturbations. We first consider the same $(\tilde{a},p,e) = (0.99, 3, 0.8)$ configuration studied in figures \[fig:sf-a99p3e8-m1-wiggles-overview\]–\[fig:sf-a99p3e8-least-damped-QNMs-and-m1-6-MC-frequencies-zoomed\] in the scalar field case. Figure \[fig:grava99p3e8-full\] displays both the gravitational self-force at the particle location and the waveform observed at ${\mathcal{J}}^+$. When looking at the local self-force the wiggles are most pronounced in the $F_\phi$ component. However, faint traces of wiggles can be found by zooming in on the $F_t$ and $F_r$ components. We note that the relative amplitudes of the wiggles in the gravitational self-force are much smaller than those in the scalar case for the same orbit (shown in figures \[fig:sf-a99p3e8-m1-wiggles-overview\] and \[fig:sf-a99p3e8-m4-wiggles-overview\]).
The waveform observed at ${\mathcal{J}}^+$ depends on the viewing angle. When the system is viewed “face on” (middle panel of figure \[fig:grava99p3e8-full\]) the waveform is determined by the $m=2$ modes with the $m={\mathfrak{l}}=2$ dominating. In this case the wiggles appear as a clear exponentially damped sinusoid. When the system is viewed “edge on” (bottom panel of figure \[fig:grava99p3e8-full\]), the wiggles have a much more irregular shape, consistent with a much larger collection of $m$’s and ${\mathfrak{l}}$’s contributing to the wiggles. Also note that while the overall waveform has a much larger amplitude when viewed edge on (due to contributions from higher modes), the observed wiggles are actually stronger when the system is viewed “face on”. This is consistent with the wiggles being dominated by the ${\mathfrak{l}}=m=2$ mode.
One of the advantages of using a frequency domain approach is that we can easily isolate individual ${\mathfrak{l}}m$-modes (as defined in section \[sect:calculations-of-Kerr-perturbations/gravitational\]). Figure \[fig:grava99p3e8-l2m2\] shows different aspects of the ${\mathfrak{l}}=m=2$ mode of the gravitational perturbation generated by a particle on our standard $(\tilde{a},p,e) = (0.99, 3, 0.8)$ configuration. The ${\mathfrak{l}}=m=2$ mode of the gravitational self-force (top panel) shows the same qualitative features as the full GSF; the $F_\phi$ components show the most obvious wiggles with weak wiggles visible in the other components. The ${\mathfrak{l}}=m=2$ mode of the field observed at ${\mathcal{J}}^+$ shows a clean exponentially decaying sinusoid wiggle just as the full field. In addition, the bottom panel of figure \[fig:grava99p3e8-l2m2\] shows the local Teukolsky variable $\Phi^{(22)}_{-2}$ at two watch points located on the background Kerr spacetime’s symmetry axis at radii corresponding to the periapsis and apoapsis of the particle orbit. These show the cleanest wiggles of any of our diagnostics.
To test our hypothesis that the observed wiggles are, in fact, QNM excitations, we perform a global fit of our three field diagnostics (local gravitational self-force, field at ${\mathcal{J}}^+$, and field at watchpoints) following the methodology set out in section \[sect:QNM-models-and-fits/gravitational\]. Table \[tab:gravqnmfits\] summarizes the results for some low order ${\mathfrak{l}}m$-modes. In each, case we recover the principal QNM frequency and damping time of the gravitational field within the estimated numerical precision of the fits. This provides yet more evidence for our hypothesis that the observed wiggles are QNM excitations. Note that while our fits include multiple QNMs, we do not conclusively recover any of the modes beyond the principal mode. (More precisely, we find that the estimated numerical errors of the recovered complex frequencies are comparable to the variation of the initial seed for the optimization.) Not including the higher modes, however, led to observable bias in the recovery of the principal QNMs.
[ccll]{} & & &\
Dependence on orbital parameters
--------------------------------
![\[fig:spinseries\] Dependence of the the fitted QNM amplitude $|A_1|$ of the lowest damped ${\mathfrak{l}}=m=2$ QNM as a function of the primary spin $a$ for a sequence of orbits with fixed $e=0.8$ and ratio $\Omega_\phi/\Omega_r = 2.684\,694\,379\dots$ (the frequency ratio for the $(\tilde{a},p,e) = (0.99,3,0.8)$ configuration). The parameter $u_{\text{ref}}$ is fixed to coincide with the particle passing through apoapsis. The data points are shaded according to the degree of alignment $\delta$ of the particle spectrum with the QNM frequency $\omega^{(1)}$.](spinseriesplotv2.pdf){width="\columnwidth"}
![\[fig:eccseries\] Dependence of the the fitted QNM amplitude $|A_1|$ of the lowest damped ${\mathfrak{l}}=m=2$ QNM as a function of the eccentricity $e$ for a sequence of orbits with fixed primary spin $a=0.95M$ and periapsis distance $r_{\min} = 1.85M$. The parameter $u_{\text{ref}}$ is fixed to coincide with the particle passing through periapsis. The data points are shaded according to the degree of alignment $\delta$ of the particle spectrum with the QNM frequency $\omega^{(1)}$. The lower panel shows the difference between the amplitude and a fitted quintic polynomial in $e$.](eccseriesplot.pdf){width="\columnwidth"}
![\[fig:paseries\] Dependence of the the fitted QNM amplitude $|A_1|$ of the lowest damped ${\mathfrak{l}}=m=2$ QNM as a function of the inverse periapsis distance $M/r_{\min}$ for a sequence of orbits with fixed primary spin $a=0.95M$ and eccentricity $e = 0.8$. The parameter $u_{\text{ref}}$ is fixed to coincide with the particle passing through periapsis. The data points are shaded according to the degree of alignment $\delta$ of the particle spectrum with the QNM frequency $\omega^{(1)}$.](paseriesplot.pdf){width="\columnwidth"}
In this section we study how the strength of the QNM excitations in the gravitational field depends on the parameters of the orbit. For this investigation we leverage the ease with which the $({\mathfrak{l}}m)$ modes of the gravitational field can be computed using our frequency domain code. As a measure of the strength of the excitations we take the amplitudes $A^{(k)}_c$ and $A^{(k)}_s$ in , which we combine to define $$\begin{aligned}
{\lvertA^{(k)}\rvert} :=\sqrt{{\left(A^{(k)}_c\right) }^{2}+{\left(A^{(k)}_s\right) }^2}.\end{aligned}$$ For the purpose of this investigation we assume that the observed wiggles are indeed QNM excitations, we therefore determine the ${\lvertA^{(k)}\rvert}$ by a linear fit, keeping the frequencies and decay rates fixed at the exact QNM values. The value of ${\lvertA^{(k)}\rvert}$ depends on the choice of $u_{\text{ref}}$ in . In this section we choose $u_{\text{ref}}$ to coincide with the particle passing through either the periapsis or apoapsis of the orbit.
We are particularly interested in whether excitation of the QNMs exhibits a strong dependence on the alignment of the discrete orbital frequency spectrum of the orbit with the QNM mode. If strong localized “resonances” between the orbit frequency spectrum and QNM excitations were to exist, these could have significant impact on waveform modelling strategies, as they would hamper an attempt to apply reduced order modelling to build efficient waveforms. On the other hand, such a phenomenon might lead to interesting and rich dynamics. To quantify the alignment between a QNM and the particle orbit we define $$\delta^{(k)} :=
\min_{n\in{\mathbbm{Z}}}
\frac{{\lvert\omega^{(k)} - m \Omega_\phi - n\Omega_r\rvert}}{\Omega_r},$$ i.e., $\delta^{(k)}$ is the distance between the QNM frequency $\omega^{(k)}$ and the nearest line in the orbit’s frequency spectrum, normalized such that $\delta^{(k)} \,{=}\, 0$ corresponds to maximal alignment and $\delta^{(k)} \,{=}\, 1/2$ to maximal misalignment.
In figure \[fig:spinseries\] we examine the dependence of the QNM amplitude (for the least-damped ($k=1$) ${\mathfrak{l}}=m=2$ QNM) on the spin $a$ of the background Kerr spacetime while keeping eccentricity and ratio of the orbital frequencies (matching the eccentricity and frequency ratio for the $(\tilde{a},p,e) = (0.99,3,0.8)$ configuration). We see that the dependence of the QNM excitation amplitude on the Kerr spin $a$ is very smooth, with no noticeable dependence on the spectrum misalignment parameter $\delta$. Notice that the QNM excitations persist for negative spins (i.e., retrograde orbits), although they become exceedingly weak.
Figure \[fig:eccseries\] explores the dependence of the amplitude ${\lvertA^{(1)}\rvert}$ of the least damped ($k=1$) ${\mathfrak{l}}=m=2$ QNM on the particle’s orbital eccentricity. For this exploration we keep the spin of the background Kerr spacetime fixed at $a=0.95M$, and we fix the periapsis distance at $r_{\min} = 1.85M$. The relationship between ${\lvertA^{(1)}\rvert}$ and $e$ appears almost linear by eye, with slight deviations both at high and low eccentricity. If we subtract off the dominant trend in the form of a quintic fit in $e$, we see what appears to be a systematic trend where orbits with $\delta^{(1)}=0$ have a slightly larger amplitude ${\lvertA^{(1)}\rvert}$ than orbits with $\delta^{(1)}=1/2$. This difference becomes stronger for low eccentricity orbits. This latter effect is consistent with the frequency spectrum of the orbit becoming sparser at lower eccentricities. However, we stress that this effect is very small, with the variation of the QNM amplitude ${\lvertA^{(1)}\rvert}$ due to changing $\delta^{(1)}$ being only about 1 part in $10^3$.
Finally, figure \[fig:paseries\] explores the relation between the amplitude ${\lvertA^{(1)}\rvert}$ of the least damped ($k=1$) ${\mathfrak{l}}=m=2$ QNM and the particle’s inverse periapsis distance $M/ r_{\min}$, keeping the Kerr spin ($a = 0.95 M$) and particle eccentricity ($e = 0.8$) fixed. As is to be expected, the amplitude ${\lvertA^{(1)}\rvert}$ drops off sharply as we increase the particle periapsis radius.
We emphasize that the overall shape of the plots in figures \[fig:spinseries\]–\[fig:paseries\] depends sensitively on the choice of $u_{\text{ref}}$, hence one should not read too much into the shapes themselves. However, there are three main lessons that we learn from this investigation that do not depend on the choice of $u_{\text{ref}}$:
- The amplitudes of the wiggles depend smoothly on the Kerr spin (figure \[fig:spinseries\]) and orbital parameters (figures \[fig:eccseries\] and \[fig:paseries\]). In particular, no fine-tuning is needed for wiggles to appear.
- The wiggles are strongest for high spin and prograde particle orbits with high eccentricity and low periapsis distance. However, there is no indication that they will completely disappear in any region of the parameter space (although they may become very difficult to separate from the rest of the field due to low amplitudes, high damping rates, and/or longer periods).
- The effect of aligning the orbital frequencies with the QNM frequencies is very small, and decreases still further when the orbital spectrum becomes denser for more eccentric orbits.
Discussion and Conclusions {#sect:discussion-and-conclusions}
==========================
![\[fig:WTF\] The ${\mathfrak{l}}=m=2$ mode of the gravitational waveform at ${\mathcal{J}}^{+}$ for the configuration $(\tilde{a},p,e) = (0.999\,99,2.918,0.807)$. The observed wiggles are surprising because the real part of the frequency (in the highlighted area) lies between $0.93 M^{-1}$ and $0.98 M^{-1}$, whereas the QNM frequencies are bunched up near $0.995 M^{-1}$.](WFa99999.pdf){width="\columnwidth"}
In this paper we study an interesting class of features first observed in the scalar self-force for point particles in orbit in Kerr spacetime [@Thornburg-Wardell-2017:Kerr-scalar-self-force]. That study identified the feature, introduced the term “wiggles”, and argued that it was in some (unspecified) manner “*caused* by the particle’s close passage by the large black hole”, but did not attempt to attribute it to any particular physical origin. More recently Refs. [@Thornburg-2016-Capra-Meudon-talk; @Thornburg-2017-Capra-Chapel-Hill-talk; @Nasipak-Osburn-Evans-2019:Kerr-scalar-self-force-and-wiggles] have shown further examples of wiggles, demonstrated that wiggles’ complex frequencies agree with known Kerr quasinormal-mode (QNM) frequencies, and concluded that wiggles are in fact “just” a sampling at the measurement point(s) of Kerr QNMs excited by the particle.
Here we survey the phenomenology of wiggles for both the scalar-field and gravitational cases, across a range of Kerr spins and particle orbits. In both the scalar-field and gravitational cases we find that wiggles are essentially a generic phenomenon, i.e., they occur over a wide range of configuration space without any “fine-tuning” of parameters. Wiggles are observable in field perturbations at fixed spatial positions, in the radiation-reaction “self-force”, and in the radiated fields at ${\mathcal{J}}^+$.
In both the scalar-field and gravitational cases we find that at all observed locations in spacetime, wiggles can be quantitatively fit by models of QNMs sampled at the observation points. In particular, in both the scalar-field and gravitational cases our fitted wiggle frequencies agree well \[in both real (oscillatory) and imaginary (damping) parts\] with Kerr QNM frequencies calculated by Berti, Cardoso, and Starinets [@Berti-Cardoso-Starinets-2009:BH-etal-QNM-review; @Berti06b] (figures \[fig:sf-a99p3e8-all-QNMs-and-m1-6-MC-frequencies\]–\[fig:sf-other-spins-all-QNMs-and-m1-4-MC-frequencies\] and table \[tab:gravqnmfits\]).
The appearance of pronounced wiggles appears to rely on three key aspects of the configuration of the system:
- A highly spinning central (Kerr) black hole (the closer to $\tilde{a}=1$, the more pronounced the effect).
- A highly eccentric prograde orbit for the particle (the closer to $e=1$, the more pronounced the effect).
- A close periapsis passage by the particle (the closer to the light ring, the more pronounced the effect).[^11]
This is not surprising: highly spinning black holes have much longer-lived QNMs than those with low spin. Increasing eccentricity of the particle orbit does three things: it increases the strength of the perturbation at the periapsis, it widens the frequency spectrum of the perturbation (increasing the overlap with the QNMs), and it provides a natural “quiet” period when the particle approaches its apoapsis, during which the QNMs can more easily be observed. Finally, bringing the particle periapsis closer to the light ring allows the perturbation to deposit more energy in the QNMs. (QNMs in Kerr spacetime are readily excited by orbits near the light ring [@Goebel-1972:BH-QNM-as-GWs-at-light-ring; @Berti-Cardoso-Starinets-2009:BH-etal-QNM-review; @Khanna-Price-2017].)
Interestingly, we find that the amplitude with which wiggles are excited does *not* depend sensitively on the particle’s precise orbital motion near periapsis. Notably, we find that the wiggle amplitude varies smoothly and monotonically with the particle periapsis radius and orbital eccentricity (and with the Kerr spin).
We have not attempted to carefully delineate the exact boundaries of the region in configuration space where wiggles occur (even assuming that there are, in fact, configurations with *no* QNM excitation, which is not obvious). It is likely that different modelling/fitting schemes could observe and fit low-amplitude and/or rapidly-damped wiggles even in some cases where we fail to observe them (e.g., the cases in table \[tab:sf-configurations\]). For example, figure \[fig:spinseries\] strongly suggests that although the wiggle amplitude is very small in some cases, wiggles are present for *all* Kerr spins along this sequence of orbits, including retrograde as well as prograde orbits.
Our scalar-field wiggle modelling/fitting scheme is (deliberately) quite conservative in requiring visual observation of a wiggle in a time-series plot of the original diagnostic. This requirement reduces the risk of false positives (where we would misidentify a fitting or background-spline artifact as a wiggle), at the cost of reducing our sensitivity to low-amplitude and/or rapidly-damped wiggles.
An interesting example of these factors at play is the $(\tilde{a},p,e) = (0.99,8,0.8)$ scalar-field configuration, for which Nasipak, Osburn, and Evans [@Nasipak-Osburn-Evans-2019:Kerr-scalar-self-force-and-wiggles] observed and fitted $\ell\,{=}\,m\,{=}\,1$, $\ell\,{=}\,m\,{=}\,2$, $\ell\,{=}\,m\,{=}\,3$, and $\ell\,{=}\,m\,{=}\,4$ wiggle (QNM) modes. Their figures 8 and 9 show the $\ell\,{=}\,m\,{=}\,4$ wiggle as having an amplitude approximately $10^9$ times smaller than the $\ell\,{=}\,m\,{=}\,1$ wiggle; this is only detectable by virtue of the high accuracy and low numerical noise level of their frequency-domain code. In contrast, for this configuration we observed wiggles for $m \,{=}\,1$ but not for $m \,{\ge}\, 2$; this is likely because even the $m \,{=}\,2$ wiggles are already too low in amplitude to be visually observable in the original time series.
Existing astrophysical models of extreme mass ratio binaries [@Miller-etal:probing-stellar-dynamics-in-galactic-nuclei; @Hopman-Alexander-2005] and observations of highly spinning black holes [@Brenneman:2013oba] suggest that it is quite reasonable to expect some fraction of EMRIs to fall within the region of parameter space where wiggles are excited with significant amplitude. (Both the magnitude of this fraction and the absolute numbers of such systems are still very uncertain.)
Given that the QNM excitations appear not just in the local self-force, but also in the gravitational waveform, a natural question is whether they could be experimentally observed by LISA or other detectors. While this is certainly possible in principle, there are two considerations which make it less likely in practice. Most importantly, the effect is quite weak in all but the most extreme cases. In most of the gravitational-field cases investigated here, it was necessary to zoom in on plots in order to see the wiggles visually, reflecting the fact that their magnitude represents at most a few percent of the total signal. A second consideration in terms of detectability is that the dynamical evolution of EMRIs may tend to avoid the wiggles region of parameter space (e.g., if most EMRIs evolve to low orbital eccentricities while still at relatively large periastron radii). This would imply that the event rate for *detectable* EMRI wiggles would be quite low.
Despite these concerns, it would be worthwhile to conduct a more thorough study to quantitatively address the question of detectability of QNM wiggles by LISA or future next-generation gravitational-wave detectors. It may even be the case that advanced data analysis techniques could be used to boost the detectability. For example, although an individual wiggle is weak, it will repeat for each orbit throughout the entire inspiral. As noted by Ref. [@Nasipak-Osburn-Evans-2019:Kerr-scalar-self-force-and-wiggles], wiggles will appear with almost the *same frequency* throughout the inspiral (the QNM frequency only depends on the mass and spin parameters of the larger black hole, and these change very little during the inspiral). Moreover, this frequency is much higher than than the main orbital frequency, potentially making it easier to separate these signal components in data analysis.
The analysis done here has been somewhat post-hoc, in that we first identified a feature in the signal and then fit this feature to a damped sinusoid representing a QNM ringdown. Our intuitive interpretation of this QNM ringdown is that it is a result of strong QNM excitation near periapsis, which is then encountered over an extended period later in the orbit. The self-force in curved spacetimes arises from *nonlocal* interactions of the object with its self-field, which was generated in the object’s past and scattered off the spacetime curvature. The association of wiggles with QNM excitations suggest that they represent a situation where this nonlocal nature of the self-force is particularly apparent. To more explicitly develop this interpretation, it may be informative to attempt the analysis in the other direction, first by starting with a model for a QNM excitation from a burst of radiation generated near a periapsis passage, and then comparing such a model to the observed signal. This approach would allow one to pinpoint where in the orbit the QNM excitation occurs, would give a deeper understanding of the effect, and may even provide a link to geometric features such as caustics and the propagation of waves on black hole spacetimes. A Green function approach [@Casals:2013mpa; @Wardell-etal-2014:self-force-via-Green-fn] would be a natural choice for such a study, but is quite distinct from the methodology used in this paper so we leave it for future work.
In this work we have focused mostly on systems with somewhat realistic Kerr spins $J/M^2 \lesssim 0.999$. Initial investigations of the near-extremal regime suggest a rich phenomenology, involving many different QNMs at fixed ${\mathfrak{l}}$ and $m$. One puzzling result is the ${\mathfrak{l}}=m=2$ mode generated at ${\mathcal{J}}^{+}$ by a particle orbiting a black hole with spin $J/M^2=0.999\,99$ and the same orbital frequencies as the $(\tilde{a},p,e) = (0.99,3,0.8)$ orbit, shown in figure \[fig:WTF\]. One of the puzzling aspects of this waveform is that the decaying wiggles in the highlighted area have a frequency between $0.93 M^{-1}$ and $0.98 M^{-1}$ (depending on where it is measured), while the nearest QNMs all have frequency close to $0.995 M^{-1}$. Whether this is the result of some complicated collective behaviour of the QNMs or some new physical effect is currently unclear, and should be investigated in future works.[^12]
Acknowledgments {#sect:ack}
===============
We thank Leor Barack for invaluable discussions throughout the course of this research. We thank Richard Brito for useful discussions about QNMs. We also thank Dan Kennefick, Scott Hughes, Marc Casals, Peter Zimmerman, Conor O’Toole and Adrian Ottewill for useful discussions on the results of this paper.
JT thanks the Alexander von Humboldt Foundation for fellowship funding for my stay at the Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institute), and the AEI (Division of Astrophysical and Cosmological Relativity) and Indiana University (Office of the Vice Provost for Research, Center for Spacetime Symmetries, and Department of Astronomy) for additional funding.
MvdM was supported by European Union’s Horizon 2020 research and innovation programme under grant agreement No. 705229. Some of the numerical results in this paper were obtained using the IRIDIS High Performance Computing Facility at the University of Southampton and the Karst and Data Capacitor facilities at Indiana University (supported by the U.S. National Science Foundation under Grant No. CNS-0521433, by Lilly Endowment, Inc. through its support for the IU Pervasive Technology Institute, and by the Indiana Metabolomics and Cytomics (METACyt) Initiative).
[^1]: $M_{\odot}$ denotes the solar mass.
[^2]: A number of researchers have attempted direct numerical-relativity binary black hole simulations for systems with “intermediate” mass ratios up to $100\,{:}\,1$ ($\mu = 0.01$), (see, for example, [@Bishop-etal-2003; @Bishop-etal-2005; @Sopuerta-etal-2006; @Sopuerta-Laguna-2006; @Lousto-etal-2010:intermediate-mass-2BH-numrel-Lazarus; @Lousto-Zlochower-2011:100-to-1-mass-ratio-2BH; @Husa-etal-2015:frequency-domain-GW-model-q-up-to-18]). However, it has not (yet) been possible to extend these results to the extreme-mass-ratio case nor to accurately evolve any systems with mass ratios more extreme than $18\,{:}\,1$ for a radiation-reaction time scale [@Husa-etal-2015:frequency-domain-GW-model-q-up-to-18] (Hinder [@Hinder-pers-comm-2019:32:1-2BH-in-progress] reports ongoing efforts to extend this to $32\,{:}\,1$).
[^3]: \[footnote:sf-compactification\] More precisely, TW define compactified coordinates $(T,R_*)$ which are identical to (respectively) the Boyer-Lindquist $t$ and the tortoise coordinate $r_*$ throughout a neighborhood of the region $r_{\min} \le r \le r_{\max}$ containing the particle orbit, but which are compactified near the event horizon and ${\mathcal{J}}^+$. $T$ is a Bondi-type retarded time coordinate at ${\mathcal{J}}^+$. For present purposes the details of the compactification are not important, so for convenience of exposition we refer to $T$ as $t$ hereinafter when describing our diagnostics at ${\mathcal{J}}^+$.
[^4]: While it would be theoretically possible for multiple modes to have wiggles of the same frequency whose amplitudes sum to approximately zero (leading to an absence of wiggles in the physical fields), in practice we have never observed this.
[^5]: Recall (cf. footnote \[footnote:sf-compactification\]) that in our scalar-field computations, $t$ is a Bondi-type retarded time coordinate at ${\mathcal{J}}^+$.
[^6]: Heuristically, the choice of $u$ rather than $t$ as an independent variable in the model is suggested by the QNM signals propagating outward along light cones after being excited near periapsis. However, it is not clear that this is a good approximation to actual QNM dynamics, so we experimented with models using both $t$ and $u$ as independent variables. We found the latter to give better fits to our numerical data.
[^7]: \[footnote:sf-fits-minimum-length-requirements\] The minimum-length requirement for the subintervals ensures that each subinterval is long enough to allow a reasonable estimate of the wiggle decay rate and period (the fitting errors should scale roughly inversely with $L_{\min}$). The minimum-length requirement for the full fitting interval ensures that different subintervals can sample significantly different regions of the data.
[^8]: Each subinterval fit uses a subset of the original fit’s background spline control points which just span that subinterval, plus one point outside the subinterval on each of the interval’s left and right endpoints.
[^9]: In a few cases where our best-fitting single-wiggle model’s residuals showed strong systematics, we then proceeded to fit 2-wiggle models. These improved the residuals by at least an order of magnitude. However, all the results presented in section \[sect:data-and-QNM-fits/scalar-field\] are based on single-wiggle fits.
[^10]: Data tables downloaded from on 19 April 2019.
[^11]: We refer here to the *prograde* light ring; we observe only very small QNM excitation when the particle periapsis is close to the retrograde light ring.
[^12]: While we were making final revisions to this manuscript Rifat, Khanna, and Burko [@Rifat-Khanna-Burko-2019:wiggles-in-near-extremal-Kerr] reported a detailed study of wiggles in near-extremal Kerr spacetimes, particularly the $(\tilde{a},p,e) = (0.999\,99,2.918,0.807)$ system. Their results are consistent with ours; they find that the anomolous wiggle frequencies are an intermediate-time effect caused by the superposition of many simultaneously-excited Kerr QNMs.
|
---
abstract: 'The task of maximizing coverage using multiple robots has several applications such as surveillance, exploration, and environmental monitoring. A major challenge of deploying such multi-robot systems in a practical scenario is to ensure resilience against robot failures. A recent work [@lz] introduced the Resilient Coverage Maximization (*RCM*) problem where the goal is to maximize a submodular coverage utility when the robots are subject to adversarial attacks and/or failures. The *RCM* problem is known to be NP-hard. The state-of-the-art solution of the *RCM* problem [@lz] employs a greedy approximation strategy with theoretical performance guarantees. In this paper, we propose two approximation algorithms for the *RCM* problem, both of which empirically outperform the existing solution in terms of accuracy and/or execution time. To demonstrate the effectiveness of our proposed solution, we empirically compare our proposed algorithms with the existing solution and a brute force optimum algorithm.'
author:
-
-
title: |
Improved Resilient Coverage Maximization\
with Multiple Robots\
---
Coverage Maximization, Multi-Robot Systems, Resilience
[00]{} L. Zhou, V. Tzoumas, G. J. Pappas, and P. Tokekar, “Resilient active target tracking with multiple robots,” IEEE Robotics and Automation Letters, vol. 4, pp. 129–136, 2018.
P. Tokekar, E. Branson, J. Vander Hook, and V. Isler, “Tracking aquatic invaders: Autonomous robots for monitoring invasive fish,” IEEE Robotics & Automation Magazine, vol. 20, no. 3, pp. 33–41, 2013.
B. Grocholsky, J. Keller, V. Kumar, and G. Pappas, “Cooperative air and ground surveillance,” IEEE Robotics & Automation Magazine, vol. 13, no. 3, pp. 16–25, 2006.
V. Kumar and N. Michael, “Opportunities and challenges with autonomous micro aerial vehicles,” The International Journal of Robotics Research, vol. 31, no. 11, pp. 1279–1291, 2012.
N. Atanasov, J. Le Ny, K. Daniilidis, and G. J. Pappas, “Information acquisition with sensing robots,” in IEEE International Conference on Robotics and Automation, 2014, pp. 6447–6454.
C. Robin and S. Lacroix, “Multi-robot target detection and tracking: taxonomy and survey,” Autonomous Robots, vol. 40, pp. 729–760, 2016.
J. R. Spletzer and C. J. Taylor, “Dynamic sensor planning and control for optimally tracking targets,” The International Journal of Robotics Research, vol. 22, no. 1, pp. 7–20, 2003.
A. Pierson, Z. Wang, and M. Schwager, “Intercepting rogue robots: An algorithm for capturing multiple evaders with multiple pursuers,” IEEE Robotics and Automation Letters, vol. 2, no. 2, pp. 530–537, 2017.
P. Tokekar, V. Isler, and A. Franchi, “Multi-target visual tracking with aerial robots,” in IEEE/RSJ International Conference on Intelligent Robots and Systems, 2014, pp. 3067–3072.
E. Sless, N. Agmon, and S. Kraus, “Multi-robot adversarial patrolling: Facing coordinated attacks,” in International Conference on Autonomous Agents and Multi-agent Systems, 2014, pp. 1093–1100.
H. H. González-Banos, C.-Y. Lee, and J.-C. Latombe, “Real-time combinatorial tracking of a target moving unpredictably among obstacles,” in Robotics and Automation, 2002. Proceedings. ICRA’02. IEEE International Conference on, vol. 2. IEEE, 2002, pp. 1683–1690.
S. I. Roumeliotis, G. S. Sukhatme, and G. A. Bekey, “Sensor fault detection and identification in a mobile robot,” in Intelligent Robots and Systems, 1998. Proceedings., 1998 IEEE/RSJ International Conference on, vol. 3. IEEE, 1998, pp. 1383–1388.
V. Tzoumas, A. Jadbabaie, and G. J. Pappas, “Resilient Non-Submodular Maximization over Matroid Constraints,” arXiv: 1804.01013, 2018.
B. Schlotfeldt, V. Tzoumas, D. Thakur, and G. J. Pappas, “Resilient active information gathering with mobile robots,” IEEE/RSJ International Conference on Intelligent Robots and Systems, 2018.
D. S. Hochbaum and A. Pathria, “Analysis of the greedy approach in problems of maximum k-coverage,” Naval Research Logistics, vol. 45, no. 6, pp. 615–627, 1998.
M. Conforti and G. Cornuéjols, “Submodular set functions, matroids and the greedy algorithm,” Discrete Applied Mathematics, vol. 7, no. 3, pp. 251–274, 1984.
R. Iyer, S. Jegelka, and J. Bilmes, “Fast semidifferential-based submodular function optimization,” in International Conference on Machine Learning, 2013, pp. 855–863.
A. Krause, and D. Golovin, “Submodular function maximization,” 2014.
|
---
abstract: 'We present a new approach to the modelling of stress propagation in static granular media, focussing on the conical sandpile constructed from a point source. We view the medium as consisting of cohesionless hard particles held up by static frictional forces; these are subject to microscopic indeterminacy which corresponds macroscopically to the fact that the equations of stress continuity are incomplete – no strain variable can be defined. We propose that in general the continuity equations should be closed by means of a constitutive relation (or relations) between different components of the (mesoscopically averaged) stress tensor. The primary constitutive relation relates radial and vertical shear and normal stresses (in two dimensions, this is all one needs). We argue that the constitutive relation(s) should be local, and should [*encode the construction history of the pile*]{}: this history determines the organization of the grains at a mesoscopic scale, and thereby the local relationship between stresses. To the accuracy of published experiments, the pattern of stresses beneath a pile shows a scaling between piles of different heights (RSF scaling) which severely limits the form the constitutive relation can take; various asymptotic features of the stress patterns can be predicted on the basis of this scaling alone. To proceed further, one requires an explicit choice of constitutive relation; we review some from the literature and present two new proposals. The first, the FPA (fixed principal axes) model, assumes that the eigendirections (but not the eigenvalues) of the stress tensor are determined forever when a material element is first buried. (This assumes, among other things, that subsequent loadings are not so large as to produce slip deep inside the pile.) A macroscopic consequence of this mesoscopic assumption is that the principal axes have fixed orientation [*in space*]{}: the major axis everywhere bisects the vertical and the free surface. As a result of this, stresses propagate along a nested set of archlike structures within the pile, resulting in a [*minimum*]{} of the vertical normal stress between the apex of the pile, as seen experimentally (“the dip"). This experiment has not been explained within previous continuum approaches; the appearance of arches within our model corroborates earlier physical arguments (of S. F. Edwards and others) as to the origin of the dip, and places them on a more secure mathematical footing. The second model is that of “oriented stress linearity" (OSL) which contains an adjustable parameter (one value of which corresponds to FPA). For the general OSL case, the simple interpretation in terms of nested arches does not apply, though a dip is again found over a finite parameter range. In three dimensions, the choice for the primary constitutive relation must be supplemented by a secondary one; we have tried several, and find that the results for the stresses in a three dimensional (conical) pile do not depend much on which secondary closure is chosen. Three dimensional results for the FPA model are in good semiquantitative agreement with published experimental data on conical piles (including the dip); the data does not exclude, but nor does it support, OSL parameters somewhat different from FPA. The modelling strategy we adopt, based on local, history-dependent constitutive relations among stresses, leads to nontrivial predictions for piles which are prepared with a different construction history from the normal one. We consider several such histories in which a pile is prepared and parts of it then removed and/or tilted. Experiments along these lines could provide a searching test of the theory.'
author:
- |
J. P. Wittmer and M. E. Cates\
Department of Physics and Astronomy\
University of Edinburgh, JCMB King’s Buildings\
Mayfield Road, Edinburgh EH9 3JZ, UK\
\
P. Claudin$^*$\
Cavendish Laboratory, Madingley Road\
Cambridge CB3 OHE, UK\
$^*$ Present Address:\
Service de Physique de l’Etat Condensé, CEA\
Ormes des Merisiers, 91191 Gif-sur-Yvette, Cedex France
date:
- 1 May 1996
-
title: Stress Propagation and Arching in Static Sandpiles
---
=15.5cm =-2.0cm =5.5cm =.5cm =-.5cm
PACS numbers 46.10.+z, 46.30.-i, 81.35.+k, 83.70 Fn
Introduction {#Introduction}
============
A sandpile is normally constructed by pouring sand from a stationary point source, as shown in Fig.1(a). Each element of sand arrives at the apex of the pile, rolls down the slopes, comes to rest, and is finally buried. The final (static) sandpile then consists of a symmetrical cone whose surface is at the angle of repose of the material. Some of the simplest questions one can ask about this system concern the distribution of stresses in the pile. Specifically, it is possible experimentally to measure the downward force on the supporting surface at different positions under the pile [@jokati; @smid]. (Throughout the paper we assume this to be a high friction surface so that slip does not occur at it.) Intuitively one can guess that the maximum force would be recorded directly beneath the apex of the pile; but in fact, the experiments show a pronounced [*dip*]{} in the force beneath the apex. This counterintuitive result has stimulated various theoretical [@Hong; @Huntley; @EO; @EM; @BCC] and computational [@bagster; @liffmann] studies; but so far, there has been no clear consensus on the origin of the dip.
In the present work we pursue a continuum mechanics approach based on the equations of stress continuity in a cohesionless granular medium [@nedderman; @BCC; @sokolovskii; @drescher]. This approach immediately encounters the problem of [*indeterminacy*]{}: even in the simplest case of a two-dimensional pile (which we consider in detail), the continuity of stress does not lead to a closed set of equations. In elastic materials, this deficiency would be rectified by invoking the usual [*constitutive relation*]{} between stress and strain (Hooke’s law). For granular media, however, there is no clear definition of “strain". Rather, it is widely assumed that the physics of granular media can be understood purely in terms of rigid particles packed together in frictional contact (so that no strain variables can be defined). The indeterminacy of the stress equations then has a clear origin: for two rigid particles in frictional contact with a specified normal force, the coefficient of static friction defines only the [*maximum*]{} shear force that may be present. Our continuum mechanics approach, like some previous ones [@BCC] assumes that, despite this local indeterminacy, there emerges on length scales much larger than the grain size some definite relation between the [*average*]{} frictional and normal forces. Thus we assume the existence of one or more constitutive relations, not between stress and strain, but among the various components of the stress tensor itself. In two dimensions, one such relation is enough to close the equations; in three dimensions (subject to certain symmetry assumptions) two are needed.
A basic tenet of our approach is that the constitutive relations are [*local*]{}: we assume that that these relations between stresses do not depend on distant perturbations, although the stresses themselves certainly do. Clearly, the constitutive relation (or relations – we suppress the plural in what follows), between stresses in some material element, must reflect the packing arrangement of grains in that element. This raises the possibility that the constitutive behaviour could vary from place to place in the pile. More generally, we believe that in principle the constitutive relation of a given material element should encode its entire [*construction history*]{}. We shall assume, however, that the important part of this history comes to an end at the moment where the element is buried: at later times, although the stresses passing through the element may vary, the constitutive relation between them cannot. We call this the assumption of [*perfect memory*]{}. We show below (Sec. 2.3) that the perfect memory and locality assumptions, when combined with a simple and experimentally motivated scaling assumption (called RSF scaling), drastically limit the form the constitutive equation can take.
A consequence of perfect memory is that the stresses in an element, once buried, respond reversibly (though not necessarily linearly) to any subsequent additional loading. Such loadings can be brought about either by adding more material to the pile, or by putting a small weight on its surface, for example. Obviously our perfect memory assumption, and indeed that of locality, may fail if the load added is so large as to lead to rearrangement of grains within the element (that is, slip). >From our viewpoint, however, if slip does occur, this represents a change in construction history which must explicitly be taken into account. It turns out that for most of the models and geometries considered in this paper, perturbative loadings of the pile do not, in fact, cause slip except at the surface of the pile.
Our assumption of a local, history-dependent constitutive relation among stresses is not widely accepted as a modelling strategy for sandpiles. (Indeed, we have not seen any really clear exposition of this strategy in the previous literature.) Many would argue the necessity of explicitly invoking the deformability of particles (allowing a strain variable to enter); others would argue that infinitesimal distant loads should cause rearrangement of a network of contacts among hard particles (leading to intrinsically nonlocal stress propagation). This paper aims to explore in detail the kinds of prediction that can be made within our overall modelling strategy, and to introduce some physically plausible candidates for constitutive relations.
Previous continuum approaches
-----------------------------
Much existing work on the static continuum mechanics of cohensionless granular materials has invoked, as an implicit constitutive relation, the assumption of [*Incipient Failure Everywhere*]{} (IFE). That this is indeed an [assumption]{}, is not always made clear in the engineering literature [@nedderman; @sokolovskii; @drescher; @Wood]. The IFE model supposes that the material is everywhere just on the point of slip failure: all frictional forces are “fully mobilized”. Thus an appropriately chosen (local) [*yield criterion*]{} [@nedderman; @drescher] of the material provides the missing constitutive equation.
The physics of this assumption is dubious. When a pile is made from a point source (Fig.1(a)) there is a continuous series of landslides at the surface; we therefore accept that an incipient failure condition is maintained [*at the surface*]{}. Even in saying this, we ignore the distinction between the angle of repose (that of the free surface just after a landslide) and the maximum angle of stability (that just before). These differ by the Bagnold hysteresis angle, which is small compared to the repose angle itself [@Bagnold; @BCPE]; we neglect this hysteresis effect from now on. However, the validity of the incipient failure condition at the surface, which contains material elements just at the point of burial, does not mean that the same condition still holds for an element long afterwards. Such an element lies deep beneath the surface, and has since burial been loaded by adding more material to the pile above. In fact, the IFE assumption can clearly be ruled out on experimental grounds: as we show below, it fails to account for the dip.
The IFE closure implies that two of the [*principal stresses*]{} are proportional [@nedderman]. In contrast, proportionality of the [*horizontal and vertical normal stresses*]{} was proposed as a constitutive relation recently by Bouchaud, Cates and Claudin [@BCC] (we refer to this closure as the BCC model). In two dimensions it was found that the stress continuity equation then has a convenient analytic property: it becomes a wave equation. However [@BCC], the BCC model predicts a stress plateau, rather than a dip, at the centre of a sandpile. Attempts to explain the dip by introducing various nonlinear terms in the constitutive relation proved unconvincing, at least for small nonlinearities, whose perturbative inclusion showed a [*hump*]{} instead of a dip. The BCC paper (Sec.3.2 of Ref. [@BCC]) in fact included a brief discussion of certain strongly nonlinear models [@language], which were also argued to give a hump. This conclusion turns out to be incorrect, for subtle reasons that we discuss below (Sec. 2.9). In one sense, the “oriented stress linearity" (OSL) model, which we study in detail in this paper, can be viewed as an extreme limit of this type of model. The OSL model, like BCC, has a [*linear relation between normal stresses*]{} but now in a coordinate system that is tilted (through a constant angle) with respect to the vertical.
An apparently completely different approach to describing the stress distribution in sandpiles was proposed by Edwards and Oakeshott [@EO]. These authors considered a pile consisting of a stack of nested [*arches*]{}, Fig.1(b). (A recent modification considers a model of platelike granules and allows curvature of the arches [@EM]). Each arch supports only its own weight, and consequently the vertical stress decreases for the smaller arches near the centre of the pile. This approach provides a very appealing physical picture of why there is a dip. In the arching mechanism, the load in an element is transmitted unevenly to those below. The central part of the pile is thereby “screened" from additional loadings which are supported instead by the outer regions.
However, there are some obvious drawbacks with this approach. Firstly, the dip is greatly [*overpredicted*]{}: by construction there is no downward force whatever at the centre of the pile, while experiments show a finite value. Secondly, unless the arches are parallel to the free surface, the outermost “arches" are incomplete. It is mechanically impossible for one of these to transmit its weight purely along its own length: there is an unbalanced couple about the base of such an arch which would cause it to fall over. If, instead, the arches are parallel to the free surface then the model is stable, but it predicts an abrupt discontinuity in the downward force at the edge of the pile, which is not observed.
These difficulties arise, at least in part, from an inconsistent attempt to treat the vertical normal stress (“weight") independently of the other stress components. This is rectified in the OSL models introduced below. Among these is a special case, the “fixed principal axes" (FPA) model, which is very close in its physical content to the picture of nested arches originally suggested by Edwards and Oakeshott. As its name suggests, in the FPA model, the principal axes of the stress tensor have a constant angle of inclination to the vertical. These axes turn out to coincide with the stress-propagation characteristics, which resemble a set of Edwards arches (Sec. 2.7). In fact, we believe that our FPA model gives, for the first time, a fully consistent continuum mechanics implementation of Edwards’ arching picture. As shown in Sec. 3, this model gives good agreement with experimental data in three-dimensional sandpiles.
The discussion above (like others in the recent physics literature on sandpiles) attributes the idea of arching to Edwards and Oakeshott [@EO]. However, the same basic picture has a longer pedigree in the rock mechanics literature, and can be traced at least as far back as the pioneering work of Trollope in the 1950’s [@Trollope50s]. (The latter is very clearly reviewed in [@Trollope].) Something very like the Edwards-Oakeshott model is called by Trollope the “full arching limit" and something very like the BCC model turns out to be the “no arching limit" of Trollope’s theory (although his predictions based on the latter do not take proper account of the Coulomb yield criterion, as was done by BCC). Trollope also developed a “systematic arching theory" to provide an interpolation between these two limits. This model, though it does not provide a systematic theory of arching, is quite interesting, and we discuss it in more detail in Section 2.10. It is based on quite different physical principles from our own work, partly because Trollope attributed the arching phenomenon to [*small displacements of the supporting surface*]{} under a wedge. This idea, based on Trollope’s own experiments (on wedges whose construction history we have been unable to find out) is clearly at odds with our own explanation, and appears to be contradicted by the more recent experimental data on sandpiles constructed from a point source [@jokati; @smid; @deflection].
Related modelling work
----------------------
Other approaches to the problem of stress propagation in static sandpiles include particle packing models [@Hong; @Huntley], where one considers a regular packing of (usually spherical) grains, with simple transmission laws for the downward force between one layer of particles and those below. These models show a flat stress plateau in two dimensions, a feature shared with BCC [@BCC] which can be viewed as a (slightly generalized) continuum limit of such models. An important and related class of discrete models address the propagation of [*noise effects*]{} in sandpiles; in these the transmission of forces between particles is stochastic [@Liu; @degennes]. The relation between these models and our own (noise-free) continuum approach will be explored in detail elsewhere [@ournoise].
More elaborate discrete models are increasingly being studied. That examined numerically by Bagster and Kirk [@bagster] invokes nontrivial force propagation rules locally, and for some parameter ranges shows a dip in the stress. However, it is not clear whether the physics included in this model is that of real sandpiles; and so far, the relation to any continuum description is uncertain. A widespread numerical approach is that of discrete element modelling [@discrete]; however, as discussed by Buchholtz and Poeschel [@volkard1], many such methods cannot so far even reproduce the fact that the repose angle of a pile is independent of its size. Various improved algorithms have been suggested [@goddard]; we do not know whether the same is true for these. In a future paper [@volkard2] we will present results from a simulation approach involving nominally frictionless, but nonspherical, slightly deformable particles, following Ref.[@volkard1]. These offer some promise of confirming, or at least testing, the ideas put forward in the present work.
The model of Liffman [*et al.*]{} [@liffmann] invokes a more specific mechanism, based on size-segregation, to explain the dip in the stress. When confronted with the experimental data [@jokati; @smid] one sees a serious drawback of this explanation: the data show a scaling behaviour which the model cannot support. The observed scaling (called RSF scaling, see Sec. 2.3 below) indicates that there is [*no characteristic length-scale*]{} intrinsic to the granular medium of which a pile is made. In general, segregation effects introduce such a scale by setting up gradients in the material properties of the pile [@footdistrib], and hence violate the observed scaling. It is notable that a finite deformability of the particles would also introduce a characteristic length, and is therefore also ruled out by RSF scaling (we discuss this further in Sec. 2.3).
The IFE model, defined above, represents one limiting case of a more general group of elasto-plastic continuum theories; some of these are highly developed and widely used within a finite element framework (though usually in the context of hoppers rather than sandpiles) [@karlsruhe]. The physical basis of these models for dry cohesionless granular media is not always clear (many are based on models developed earlier for wet soil [@Wood]). In any case, to whatever extent elasticity is invoked, such models are again in violation of RSF scaling.
The idea that the properties of a granular medium depend on its construction history is central to our work. This concept is not new, and plays a strong role in the recent experimental literature on granular media in hoppers (for example the exit flow from a hopper depends on how it was filled) [@rotter1]. Indeed, this is part of the reason why standardized shear and triaxial tests are used to measure the internal friction coefficient of a granular medium; the repose angle, which in the simplest theories is completely equivalent [@nedderman], in reality depends appreciably on construction history, as do other mechanical properties [@rotter1] (this is discussed further in Section 4). Moreover, it has long been argued that the manner in which the construction history enters is via the local packing geometry of the grains. This forms part of the idea of “granular fabric", in which one constructs a local tensor that parameterizes the distribution of particle-particle contacts [@fabric]. The concept of the fabric tensor has usually been developed in an elasticity context, rather than one in which constitutive relations directly among stresses are assumed. Nonetheless, the orientational memory effects embodied in the new models described below can certainly be viewed in terms of a local tensorial property of the medium (see Sec. 2.7). However, no specific interpretation of this quantity (in terms of the contact distribution) appears to be required.
The present work
----------------
In Section \[sec2D\] we give a coarse-grained continuum description of the two-dimensional symmetrical sandpile. Instead of assuming in advance a particular constitutive relation, we first approach the problem systematically by exploiting the implications of symmetry, and of the boundary condition of incipient failure at the free surface (IFS). We discuss, with reference to the construction history of the pile, the scaling ansatz of a [*radial stress field*]{} (RSF). This ansatz transforms the partial differential equations into ordinary differential equations, which can be solved easily for all the closures considered later. Using this, and the idea of [*perfect memory*]{} mentioned above, the range of possible constitutive relations is greatly reduced.
After this, we will solve our equations for four specific closures in two dimensions; these are incipient failure everywhere (IFE), Bouchaud-Cates-Claudin (BCC), fixed principal axes (FPA), and finally the family of oriented stress-linearity (OSL) closures, which includes BCC and FPA as special cases. All of these comply with our modelling strategy of seeking local constitutive relations among stress components. For the OSL model, stresses propagate along straight characteristics which can be interpreted by analogy with wave propagation along “light-rays". We thereby arrive at a very simple geometrical picture of stress propagation, from which the forms of the stress profiles can be swiftly deduced. For the OSL model, the effect of a small perturbation (adding a little extra weight somewhere) is studied, and an appropriate Green function described. This helps sharpen the idea of stress being carried along arches.
In Section \[sec3D\] we extend our calculations to the three dimensional conical sandpile. Because of the larger number of stress components (some of which can be eliminated by symmetry) a second constitutive equation is now required. We study several possible choices for this secondary closure, and find that all of these lead to qualitatively similar stress profiles. A comparison with the experimental results of Smid and Novosad [@smid] is then made. This shows, firstly, that the RSF scaling assumption is well-verified, and, secondly, that the data is fit rather well by the FPA model, without adjustable parameters. Viewed alternatively as a comparison with the OSL model, which does have an adjustable parameter (the tilt angle), the evidence suggests that parameter values close to the FPA case must be chosen. The data thereby presents strong evidence for the arching picture as an explanation of the dip.
Until the end of Section 3, we will have considered only the case where the free surface of the pile is at the angle of repose. However, it is possible experimentally to achieve piles which are flatter than this. We discuss this and a number of related problems in Section 4, where, for simplicity, we restrict attention to the FPA model in two dimensions. We show that it matters how a sandpile is made: for example, if a flattened pile is created by slicing wedges off the top of a steeper one, the stresses should differ from those found by choosing a material with a lower repose angle to begin with. It is in geometries such as this, that the dependence of the constitutive equation on the construction history of the pile can be probed.
Section 5 contains a brief summary of our approach and a concluding discussion. Our calculations for the stress propagation in two-dimensional sandpiles, using the OSL and FPA models, are new; as are our three dimensional results for these models, and for BCC, although some of the FPA results were outlined elsewhere [@nature]. For the IFE model, which is more classical, the corresponding results may exist in the literature (though we have not found them); in any case we include them for comparison.
The two dimensional symmetrical sandpile {#sec2D}
========================================
Indeterminacy of stress continuity equation {#secMCE}
-------------------------------------------
As mentioned previously, the continuum approach to calculating the stress distribution in a static sandpile immediately encounters an indeterminacy. Indeed, the stress continuity equation in two dimensions reads (componentwise) $$\begin{aligned}
\partial_r {\mbox{$\sigma_{rr}$}}+ \partial_z {\mbox{$\sigma_{rz}$}}& = & 0 \label{eqpde}\\\nonumber
\partial_r {\mbox{$\sigma_{rz}$}}+ \partial_z {\mbox{$\sigma_{zz}$}}& = & g\end{aligned}$$ which, clearly, provides only two relations between the three independent elements of the stress tensor , and ${\mbox{$\sigma_{rz}$}}={\mbox{$\sigma_{zr}$}}$.
To ease the later generalization to three dimensions (Section 3) we here use cylindrical polar coordinates, with $z$ measured [*downward*]{} from the apex of the pile and $r$ a radial coordinate from the symmetry axis – see Fig.1(c). (In two dimensions, $r= |x|$, with $x$ a cartesian coordinate.) We have assumed that the granular medium has [*constant density*]{}, $\rho$, thereby excluding segregation effects (see Section 1.2 above), and have chosen units where $\rho=1$. The acceleration due to gravity is denoted $g$; because it enters linearly, this could also be set to unity, but we retain it for clarity. The stress tensor $\sigma_{ij}$ is defined to be symmetric in $i,j$, as usual in the physics literature.
In our cylindrical polar coordinate system the stress tensor is a function of $z$ and $r$, where $r \ge 0$ by definition. However, in terms of cartesians $(z,x)$, one would have both positive and negative $x$; in this case, the normal stresses would be even functions of $x$ and the shear stress an odd function [@BCC]. (The latter holds because the unit vector along $r$ reverses sign at the symmetry axis.) Confusion can be reduced by restricting attention to the left half of the pile (positive $x$) for which the two coordinate systems coincide. In any case, on the symmetry axis itself ($r=0$), the shear stress must vanish by symmetry, and the $z$ and $r$ directions are both principal axes there.
As well as the three stress components , and , it will be useful to consider the following three quantities: the average stress $P = ({\mbox{$\sigma_{rr}$}}+{\mbox{$\sigma_{zz}$}})/2$, the “radius of Mohr’s circle” $R$ defined via $R^2 = ({\mbox{$\sigma_{zz}$}}-{\mbox{$\sigma_{rr}$}})^{2}/4 + {\mbox{$\sigma_{zr}$}}^2 $ [@Mohr], and the (positive) angle of inclination $\Psi$ between the $z$-axis to the major principal axis of the stress tensor (see Fig.1(c)). In terms of these, $$\begin{aligned}
{\mbox{$\sigma_{rr}$}}& = & P-R \cos(2 \Psi) \label{eqPRPsitosigma}\\ \nonumber
{\mbox{$\sigma_{zz}$}}& = & P+R \cos(2 \Psi) \\ \nonumber
{\mbox{$\sigma_{zr}$}}& = & R \sin(2 \Psi)\end{aligned}$$ Following the usage of the engineering literature [@nedderman] we define a material point to be in an [*active*]{} state if the normal stress in the direction of the external compressive force (here gravity) is larger that the stress perpendicular to it. On the symmetry axis ($r=0$), the $z$-axis is the major principle axis and $\Psi=0$ in the case of an active state, whereas for a passive state the major axis is $r$, and $\Psi=\pm\pi/2$.
The simplest model of a granular medium is known as the [*ideal cohesionless Coulomb material*]{}. The Coulomb model plays the same role in the study of granular materials as the Newtonian fluid does in viscous flow, and we will use it here. Plastic failure occurs in a given material element if there exist a plane defined by a unit normal $\bf n$ (or angle of inclination $\tau =
\sin^{-1}({\bf n.\hat z}$)) through this element, on which the shear forces exceeds a given fraction of the normal force across the plane [@coulomb]. Conversely the element is stable if, for all such planes, $$|{\mbox{$\sigma_{nm}$}}| \leq \tan(\phi) {\mbox{$\sigma_{nn}$}}\label{eqCoulomb1}$$ For a material with [*cohesion*]{}, a constant $c$ is added to the right hand side; we treat only the cohesionless case. The Coulomb yield criterion can then alternatively be expressed as $${\mbox{$\Upsilon$}}\equiv {R\over P\sin(\phi)} \leq 1
\label{eqCoulomb2}$$ (Put differently, “the yield locus must not cut Mohr’s circle" [@Mohr].) The coefficient $\tan(\phi)$ in eqn.(\[eqCoulomb1\]) is the coefficient of static friction of the material; elementary arguments show that $\phi$ is then the angle of repose [@nedderman] (defined, as usual, as the inclination of the free surface to the horizontal; see Fig.1(c)).
IFS boundary conditions
-----------------------
We now use the yield criterion to specify the stresses on the surface of the pile. In doing this, we neglect the small Bagnold hysteresis angle (as mentioned in Sec. 1.1) and demand that the [surface]{} of a pile, constructed from a point source and at its angle of repose, is in a state of incipient slip. (Sandpiles constructed differently, for which this is not the case, are considered in Section 4).
First we note that (in two dimensions) all stress components have to vanish on the surface: $${\mbox{$\sigma_{rr}$}}(S=1)={\mbox{$\sigma_{zz}$}}(S=1)={\mbox{$\sigma_{zr}$}}(S= 1)=0
\label{eqby1}$$ Here we have introduced, for reasons that will be clarified later, a scaling variable $S=r/(cz)$ with $c = \cot(\phi)$. (Hence the equation of the free surface is $r=cz$, or $S=1$.) The vanishing of the stresses is a direct consequence of the yield criterion, as we now show by considering the stress components in a rotated coordinate system $(n,m)$ (see Fig.1(c)). For a system inclined at angle $\tau$ to the vertical, one has $$\begin{aligned}
{\mbox{$\sigma_{nn}$}}& = & \cos^2(\tau) {\mbox{$\sigma_{rr}$}}+ \sin^2(\tau) {\mbox{$\sigma_{zz}$}}-
2 \sin(\tau) \cos(\tau) {\mbox{$\sigma_{zr}$}}\label{eqrot}\\ \nonumber
{\mbox{$\sigma_{mm}$}}& = & \sin^2(\tau) {\mbox{$\sigma_{rr}$}}+ \cos^2(\tau) {\mbox{$\sigma_{zz}$}}+
2 \sin(\tau) \cos(\tau) {\mbox{$\sigma_{zr}$}}\\ \nonumber
{\mbox{$\sigma_{nm}$}}& = & - \sin(\tau) \cos(\tau) ({\mbox{$\sigma_{zz}$}}-{\mbox{$\sigma_{rr}$}}) +
(\cos^2(\tau) - \sin^2(\tau)) {\mbox{$\sigma_{zr}$}}\end{aligned}$$ Now choosing $\tau = \pi/2-\phi$, so that $n$ is normal to the surface, we require that the normal stress at the free surface has to vanish (this assumes that no external forces act there). The yield criterion eqn. (\[eqCoulomb2\]) then requires $${\mbox{$\sigma_{mm}$}}^2 \cos^2(\phi) + 4 {\mbox{$\sigma_{nm}$}}^2 \leq 0 \label{another}$$ Accordingly, the remaining two stress components and must also vanish, and the stress tensor is zero in the $(n,m)$, and hence in the $(r,z)$, coordinate system.
The criterion that the surface of the pile is a slip plane, not only implies that the stresses on the surface vanish, but also fixes their ratios in its immediate neighbourhood. Demanding equality in eqn. (\[eqCoulomb1\]) as the surface is approached, we obtain the condition $$\hbox{\rm lim}_{S\to 1}{{\mbox{$\sigma_{nm}$}}(S)\over {\mbox{$\sigma_{nn}$}}(S)} =-\tan(\phi)
\label{eqby2a}$$ (the sign can be confirmed from eqn. (\[eqrot\])). Applying also eqn.(\[another\]) (with equality) in this limit gives a second condition: $$\hbox{\rm lim}_{S\to 1} {{\mbox{$\sigma_{mm}$}}(S)\over{\mbox{$\sigma_{nn}$}}(S)}=1 +
2\tan^2(\phi)\equiv 1/\eta_0
\label{eqby2b}$$ where the final notation will prove convenient later. By rotating using eqn. (\[eqrot\]) these can be written in the $(r,z)$-system as $$\begin{aligned}
\hbox{\rm lim}_{S\to 1}{{\mbox{$\sigma_{zr}$}}(S)\over {\mbox{$\sigma_{rr}$}}(S)} & = & \tan(\phi)
\label{barmy}\\ \nonumber
\hbox{\rm lim}_{S\to 1}{{\mbox{$\sigma_{rr}$}}(S)\over {\mbox{$\sigma_{zz}$}}(S)} & = & \eta_0\end{aligned}$$ (The results in $(m,n)$ and in $(r,z)$ coordinates look rather similar because, as it turns out, the two frames are related by a reflection through the major principal axis [@bisect].)
The requirements expressed by equations (\[eqby1\], \[barmy\]), represent a set of “boundary conditions" which we denote IFS (incipient failure at the surface). Along with the stress continuity equation, these form the boundary value problem for determining the stress profile of a sandpile. At first sight there may appear to be more boundary conditions than required; however (as emphasised before) to close the problem in two dimensions, we will need a constitutive relation between stress components, which is yet to be chosen. One can therefore view any extra “boundary conditions" as constraints limiting this choice. We next develop some general scaling arguments which, combined with some other physically motivated simplifications, further restrict the choice of constitutive equation.
Scaling analysis {#secScaling}
----------------
The basis of our scaling approach is to assume that the macroscopic material properties of a granular medium (under gravity) are independent of length scale. Obviously, any such medium has a characteristic length associated with the grain size, but in a continuum description this should not be important. The scaling hypothesis supposes that not only this length, but also any other characteristic length-scale that the medium possesses, is either extremely small, or else extremely large, compared to the scales probed in a macroscopic sandpile experiment. As mentioned previously, and shown below, our scaling assumption is well-verified in the experiments of Ref.[@smid]. However, those experiments are on piles of a limited size range (20 to 60 cm high). It is possible that for smaller or larger piles our scaling assumptions would break down, due to a characteristic length arising from size segregation [@liffmann], for example. A corollary of our scaling assumption is that no relevant intrinsic scale exists for stresses: otherwise, this scale could be compared with the gravitational stress to give a length. Thus we exclude, for example, deformable particles whose elastic modulus sets a stress scale, and thereby a “deformation length" (which for rigid particles is infinite). These simpifications, though guided by experiment, are not physically obvious [*a priori*]{}; the problem deserves more careful experimental study to determine the limits to the scaling regime for real granular materials.
Assuming that the medium indeed has no intrinsic characteristic length, the stress distributions in all piles formed the same way (of the same material) should be similar. Hence we search for a scaling solution of eqn. (\[eqpde\]) of the form: $$\sigma_{ij} = g z s_{ij}(S)
\label{eqonlyH}$$ with the scaling variable $S=r/cz$ was introduced previously. In anticipation of this ansatz, our earlier discussion of the boundary conditions was couched in terms of $S$. These boundary conditions impose $s_{ij}(1)=0$, and also fix ratios of derivatives of $s_{ij}$ (see eqns.(\[eqby1\],\[barmy\]) above). The functions $s_{ij}$ have to be continuous everywhere; however, we are dealing with hyperbolic equations, and the stresses need not be differentiable [@discfoot].
The scaling ansatz, eqn. (\[eqonlyH\]), reduces the partial differential equations for stress continuity, eqns. (\[eqpde\]), to the following ordinary differential equations: $$\begin{aligned}
{\mbox{$s_{rr}$}}'/c + {\mbox{$s_{zr}$}}- S {\mbox{$s_{zr}$}}' & = & 0 \label{eqDE}\\ \nonumber
{\mbox{$s_{zr}$}}'/c + {\mbox{$s_{zz}$}}- S {\mbox{$s_{zz}$}}' & = & 1\end{aligned}$$ The primes denote derivatives with respect to the scaling variable $S$; recall that $c=\cot(\phi)$. Solutions of eqn. (\[eqDE\]) are usually called “[*radial stress fields*]{}” [@radialstressfield] and we refer to (\[eqonlyH\]) as RSF scaling. >From the scaling behaviour of $s_{ij}$ follows a corresponding scaling of the mean stress, $P(r,z) = g z
\tilde{P}(S)$, the radius of Mohr’s circle, $R(r,z) = g z \tilde{R}(S)$, and the angle of the major principle axis $\Psi(r,z)={\Psi}(S)$.
We now ask, what are reasonable closure relations consistent with the radial stress field form of eqn. (\[eqonlyH\])? On physical grounds we first impose the requirement of [*locality*]{}: the unknown stress component in a material element depends on the known ones in that element, and not elsewhere. (As mentioned previously, this would fail if distant loads were able to rearrange the grains themselves in a given neighbourhood.) The most general form consistent with our scaling ansatz is then $${\mbox{$\sigma_{rr}$}}/{\mbox{$\sigma_{zz}$}}= {\mbox{${\large C}$}}({\mbox{$\sigma_{zr}$}}/{\mbox{$\sigma_{zz}$}},S)
\label{eqnointrinsic}$$ where the absence of an intrinsic stress scale, noted above, means that only [dimensionless ratios]{} of stresses can enter.
To restrict further, we now invoke the assumption of [*perfect memory*]{} mentioned in the introduction: that the constitutive relation in a material element is determined at the time of its burial and is not subsequently altered. Now, it is clear that each material element is buried while just at the surface of the pile ($S=1$), after which it experiences continually decreasing values of $S$ as the pile gets larger. Hence any explicit dependence of on $S$ would violate the perfect memory assumption. Accordingly, we must have $${\mbox{$\sigma_{rr}$}}/{\mbox{$\sigma_{zz}$}}= {\mbox{${\large C}$}}(U)
\label{eqmemory}$$ where we have defined the reduced shear stress $$U(S)={\mbox{$\sigma_{zr}$}}/{\mbox{$\sigma_{zz}$}}={\mbox{$s_{zr}$}}/{\mbox{$s_{zz}$}}\label{udef}$$
An exception to the above argument should be made for material elements on the central axis, which are buried, and remain forever, with a value $S=0$. In two dimensions the centreline divides grains which have rolled to the left from those which have rolled to the right, and which therefore have had qualitatively different histories [@leftright]. Accordingly, there is no requirement that behave smoothly at the origin; and although the constitutive models studied below all appear analytic when expressed in polar ($z,r$) coordinates, for some of them does become singular on the symmetry axis, when a cartesian ($z,x$) coordinate system is employed. (The models with this property are the OSL models, including FPA, but with the exception of BCC.)
We can now reformulate the radial stress field eqns. (\[eqDE\]) as $$\begin{aligned}
\frac{d{\mbox{$s_{zz}$}}}{dS} & = & \frac{{\mbox{$\hat{c}$}}{\mbox{$\hat{e}$}}-{\mbox{$\hat{b}$}}{\mbox{$\hat{f}$}}}{{\mbox{$\hat{b}$}}{\mbox{$\hat{d}$}}- {\mbox{$\hat{a}$}}{\mbox{$\hat{e}$}}}
\label{eqRK}\\ \nonumber
\frac{d{\mbox{$s_{zr}$}}}{dS} & = & \frac{{\mbox{$\hat{a}$}}{\mbox{$\hat{f}$}}-{\mbox{$\hat{c}$}}{\mbox{$\hat{d}$}}}{{\mbox{$\hat{b}$}}{\mbox{$\hat{d}$}}- {\mbox{$\hat{a}$}}{\mbox{$\hat{e}$}}}\end{aligned}$$ Here we have introduced the notations $$\begin{aligned}
{\mbox{$\hat{a}$}}& = & -\frac{1}{c} \frac{U^2 d({\mbox{${\large C}$}}(U)/U)}{dU} \label{eqabcdef}\\
\nonumber
{\mbox{$\hat{b}$}}& = & -S + \frac{1}{c} \frac{d{\mbox{${\large C}$}}(U)}{dU} \\ \nonumber
{\mbox{$\hat{c}$}}& = & {\mbox{$s_{zr}$}}(S) \\ \nonumber
{\mbox{$\hat{d}$}}& = & -S \\ \nonumber
{\mbox{$\hat{e}$}}& = & 1/c \\ \nonumber
{\mbox{$\hat{f}$}}& = & {\mbox{$s_{zz}$}}(S) -1\end{aligned}$$ where only the first two functions explicitly involve the closure relation. Similarly we can also reformulate the Coulomb yield criterion, eqn. (\[eqCoulomb2\]) as: $$\begin{aligned}
1 \geq {\mbox{$\Upsilon$}}(U)^2 =
\frac{(1-{\mbox{${\large C}$}}(U))^{2}+4 U^{2}}{(1+{\mbox{${\large C}$}}(U))^{2} \sin^2(\phi)}
\label{eqCoulomb3}\end{aligned}$$ eqns. (\[eqRK\]) give a systematic procedure for solving (at least numerically) the boundary problem for any specified constitutive relation ; eqn. (\[eqCoulomb3\]) then allows one to check its stability. The latter step is necessary to ensure that the yield criterion (marginally satisfied at the surface) is not violated deep inside the pile; any closure for which such violations arise must be rejected.
Asymptotic behaviours {#secSM}
---------------------
Without specifying further, we can now use the scaled continuity equations, eqn. (\[eqDE\]), to examine the possible asymptotic behaviours close to the free surface, and close to the symmetry axis of the pile.
As mentioned previously, all the stresses vanish at the free surface ($S \to
1$). Asymptotically we expect them to vanish as power laws ${\mbox{$s_{rr}$}}= a_1 (1-S)^\alpha$, ${\mbox{$s_{zz}$}}= b_1 (1-S)^\beta$, ${\mbox{$s_{zr}$}}= d_1 (1-S)^\delta$ ($a_1,b_1,d_1,\alpha,\beta,\delta > 0$). The IFS condition, eqn. (\[barmy\]), requires that and have to vanish with the same power, $\alpha=\beta$, since their ratio approaches a constant. More generally one finds by substituting the power law forms into eqn. (\[eqDE\]) that the stresses on the surface have to vanish [*linearly*]{} : $\alpha=\beta=\delta=1$. This applies for any choice of the closure . (For the models solved below, this linear behaviour is visible in Fig.2(a).) Only one of the coefficients $a_1,b_1,d_1$ then remains free; using (\[eqDE\]) we find $d_1 = a_1/c$ and $-d_1/c +b_1 = 1$. Using again the IFS boundary condition, eqn. (\[barmy\]), we obtain finally [@overdetermined] $$\begin{aligned}
a_1 & = & 1/(1 + \tan^2(\phi)) \label{eqabdonslope} \\ \nonumber
b_1 & = & (1+ 2\tan^2(\phi))/(1 + \tan^2(\phi)) \\ \nonumber
d_1 & = & \tan(\phi)/(1+\tan^2(\phi))\end{aligned}$$ This completes the specification of the asymptotic behaviour near the free surface. The average reduced stress on the surface is $\tilde P=1-S$ and the reduced Mohr’s radius $\tilde R=\sin(\phi) (1-S)$. Since, from eqn. (\[eqPRPsitosigma\]), $\cot(2\Psi)=\left| (b_1-a_1)/2d_1 \right|$ we can solve for the direction of the major principal axis at the free surface $$\Psi(1) = \psi \equiv (\pi-2\phi)/4
\label{eqPsionslope}$$ This asymptote for $\Psi(S\to1)$ bisects the angle between the vertical and the free surface itself.
A similar analysis can be made of the stresses close to the symmetry axis of the pile ($S \to 0$). Again without knowing details of , we can establish for $S\to 0$ a solution involving a linearly vanishing shear stress ${\mbox{$s_{zr}$}}=d_0 S$, a vertical normal stress of the form ${\mbox{$s_{zz}$}}= b_0 + b_{cusp} S$, and a flat horizontal normal stress ${\mbox{$s_{rr}$}}= \eta b_0$. Here $\eta \equiv {\mbox{${\large C}$}}(0)$ is the ratio of horizontal and vertical stresses in the middle of the pile. One can also show that $b_0=1-d_0/c$, with $b_0, d_0$ positive. The existence of a cusp, $b_{cusp}
\neq 0$, is associated with a breakdown in the smoothness of the closure relation on the symmetry axis; as discussed above, this is physically permissible, and is a distinguishing feature of the new models introduced in this paper.
The results of this and the preceding section were obtained by combining RSF scaling with the perfect memory assumption, without further restriction on . In the next few sections we finally consider various model constitutive relations with which to close the equations and thereby calculate explicit stress profiles for the sandpile.
The IFE model {#secIFE}
-------------
The traditional [@drescher; @nedderman; @sokolovskii] assumption of Incipient Failure Everywhere (IFE) means that the granular material is everywhere marginally unstable; the frictional forces are fully mobilized and ${\mbox{$\Upsilon$}}= 1$. (Accordingly the two principal stresses are everywhere in fixed ratio.) Indeed, we can solve eqn. (\[eqCoulomb3\]) with equality to find: $${\mbox{${\large C}$}}(U) = \frac{1}{\cos^{2}(\phi)}
\left( (\sin^{2}(\phi)+1) \pm 2 \sin(\phi) \sqrt{1- (\cot(\phi) U)^2} \right)
\label{eqgrrIFE}$$ where $U(S)={\mbox{$s_{zr}$}}/{\mbox{$s_{zz}$}}$ is the reduced shear stress introduced previously. Here the negative sign must be chosen: this corresponds to requiring downward (rather than upward!) incipient slip of the grains at the free surface. The resulting IFE constitutive equation then fixes the two functions ${\mbox{$\hat{a}$}}(S),{\mbox{$\hat{b}$}}(S)$ defined earlier, as follows: $$\begin{aligned}
{\mbox{$\hat{a}$}}& = &
\frac{1}{\cos^2(\phi)} \left(1+\sin^2(\phi)-2 \sin(\phi)
\sqrt{1-(\cot(\phi)U)^2} \right) \label{eqabIFE}\\ \nonumber
{\mbox{$\hat{b}$}}& = & -s + \frac{2 U}{c \sin(\phi) \sqrt{1-(\cot(\phi)U)^2}}\end{aligned}$$ thereby closing the the radial stress field equations (\[eqRK\]). We have not obtained an analytical solution for this model, but a numerical solution was readily found by a standard Runge-Kutta procedure (by shooting from the middle of the pile to the boundary conditions on the surface). The reduced stresses , and are shown for the case $\phi=30^o$ in Fig.2(a). (Also shown are results from the other models discussed below.) The IFE model give a smooth “hump" in $s_{zz}$.
The BCC model {#secBCC}
-------------
In place of the IFE assumption that the [*principal*]{} stresses are proportional, the BCC model [@BCC] assumes instead the proportionality of vertical and horizontal normal stress components: $${\sigma_{rr}\over \sigma_{zz}}\equiv {\mbox{${\large C}$}}(U) = \eta
\label{bccclos}$$ This assumption, which is perhaps the simplest possible, is related (but not identical) to one made in the classical work of Janssen [@janssen; @nedderman; @drescher]. Invoking also the IFS boundary conditions, complete results were obtained analytically for the two dimensional sandpile [@BCC]. These results for the stresses, the yield function ${\mbox{$\Upsilon$}}$ and the orientation angle $\Psi$ of the major principal axis are shown (alongside those of IFE and FPA) in Figs.2(a-c).
The IFS condition (eqs.\[eqby1\],\[eqby2a\],\[eqby2b\]) in fact requires $\eta=\eta_0$ (where $\eta_0$ is defined in eqn.\[eqby2b\]). The stresses are piecewise linear with a cusp at $S = S_0={\mbox{$c_{0}$}}/c$, where ${\mbox{$c_{0}$}}= \sqrt\eta_0$ and, as shown by BCC, ${\mbox{$c_{0}$}}/c$ is strictly less than unity. The inequalities ${\mbox{$c_{0}$}}/c \le S \le
1$ define an “outer region" of the pile in which the stresses obey $s_{zz} =
(1-S)/(1 - ({\mbox{$c_{0}$}}/c)^2)$ and $s_{zr} = (1-S){\mbox{$c_{0}$}}^2 c/(c^2 -{\mbox{$c_{0}$}}^2)$ . These match at $S=S_0$ onto an “inner region" ($0
\le S \le {\mbox{$c_{0}$}}/c$), in which there is a plateau for the vertical normal stress, $s_{zz} = 1/(1 + {\mbox{$c_{0}$}}/c)$, while the shear stress vanishes linearly on the central axis, $s_{zr} =
S {\mbox{$c_{0}$}}/(1+{\mbox{$c_{0}$}}/c)$. As shown in eqn.(\[eqPsionslope\]) above, the inclination angle of the major axis obeys at the surface $\Psi(1) = \psi$. For the BCC closure in two dimensions, this value is maintained throughout the outer region, while in the inner region $\Psi(S)$ obeys $\tan(2\Psi) = 2 {\mbox{$c_{0}$}}S/({\mbox{$c_{0}$}}^2 -1)$. Hence $\Psi(S)$ vanishes smoothly at $S=0$.
The Fixed Principal Axis (FPA) model {#secFPA}
------------------------------------
Neither the IFE nor the BCC closure gives a“dip" in two dimensions (nor in three as shown below). We therefore propose a new hypothesis [@nature] which appears to capture, within a fully consistent continuum theory, the physics of arching (as expounded by Edwards and coworkers [@EO; @EM]). Specifically, we postulate that the major principal axis of the stress tensor has a [*fixed angle of inclination*]{} to the downward vertical: $\Psi(S)$ is constant. We first describe the results and afterwards discuss in more detail the physical content of this model.
The FPA hypothesis provides a local constitutive equation by assuming that the principal stress axes in a material element have constant orientation fixed at the time of its burial. However, with the exception of those lying right on the symmetry axis, all such elements were first buried at the surface of the pile ($S=1$). Since the IFS boundary condition already fixes $\Psi(1) = \psi \equiv (\pi-2\phi)/4$, our FPA model requires $$\Psi = \psi$$ everywhere. Using the results of Sec. 2.1, one finds immediately that this is equivalent to the following constitutive relation: $${{\mbox{$\sigma_{rr}$}}\over {\mbox{$\sigma_{zz}$}}} \equiv
{\mbox{${\large C}$}}(U) = 1 -2\tan(\phi)\, U \label{fixedpa}$$ Note that if $r$ is replaced by $x$ (cartesian coordinates) this becomes $${s_{xx}\over {\mbox{$\sigma_{zz}$}}} \equiv
{\mbox{${\large C}$}}(U) = 1 -2\,{\hbox{\rm sign}}(x)\tan(\phi)\, U \label{fixedpaxz}$$ The ${\hbox{\rm sign}}(x)$ factor is a reminder that, in the FPA model (unlike BCC) our ${\mbox{${\large C}$}}(U)$ is nonanalytic at the symmetry axis of the pile. A compact way to write eqn. (\[fixedpaxz\]) is [@nature] ${\mbox{${\large C}$}}(U) = 1 -2\tan(\phi)\, |U|$, though (see Section 4 below), this version is not equivalent for all construction histories.
Since the repose angle $\phi$ (and thereby $\psi$) is a material parameter fixed by experiment, the FPA model gives a complete closure of the $d=2$ sandpile problem. The resulting equations are linear. Their structure is clearest when written in terms of the [unscaled]{} stress components $\sigma_{ij}$; substituting eqn.(\[fixedpa\]) and eqn.(\[udef\]) into eqn.(\[eqpde\]) (taking $\sigma_{zz}$ and $\sigma_{rz}$ as the independent variables) gives: $$\begin{aligned}
\partial_r {\mbox{$\sigma_{rz}$}}+ \partial_z {\mbox{$\sigma_{zz}$}}& = & g \\
\partial_r ({\mbox{$\sigma_{zz}$}}-2\tan\phi \,{\mbox{$\sigma_{rz}$}}) + \partial_z {\mbox{$\sigma_{rz}$}}& = & 0\end{aligned}$$ which can be rewritten $$(\partial_z-c_1\partial_r) (\partial_z-c_2\partial_r)\sigma_{ij} = 0
\label{FPAwave}$$ with $$\begin{aligned}
c_1+c_2 & = & -2\tan(\phi) \\
c_1c_2 & = & -1\end{aligned}$$ where we $c_1$ ($c_2$) to denote the positive (negative) roots. A little manipulation then shows that $c_1 = \tan(\psi)$ and $c_2 = -\cot(\psi)$. A similar equation for the stresses, but with $c_1=-c_2 = \sqrt{\eta_0}$, was obtained for the BCC model [@BCC], in which case eqn.(\[FPAwave\]) becomes the wave equation in two dimensions. We shall call (\[FPAwave\]) a “wave equation" even when $c_1+c_2\neq 0$; under these conditions, it becomes an ordinary wave equation (with equal velocities) if tilted coordinate axes are chosen [@tilt].
A complete solution is readily found and is given explicitly (in the context of the more general OSL model) in the next Section. As with BCC, one finds for $s_{zz}(S)$ a piecewise linear function, with inner and outer regions. The material in the outer region again saturates the yield criterion (\[eqCoulomb3\]) whereas the inner part does not; these regions are separated by a cusp at $S=S_0 = c_1/c$. For the FPA model, there is always a dip in $s_{zz}$ at the centre of the pile. The dip takes the form of a cusp at $S=0$ and is connected with the nonanalyticity of $C(U)$, which reflects the sudden change in the direction of the major principal axis on passing through the central axis of the pile. The maximum vertical stress (at $S=S_0$) is a factor $(1+2\tan^2(\phi))$ times larger than the value at $S=0$ (the latter is always finite). These results are compared with the BCC and IFE models in Fig.2(a-c).
At first sight the requirement of fixed orientation of the stress tensor ($\Psi=\psi$) is at odds with the fact that, on the centre line of the pile, there are no shear stresses and so the horizontal and vertical directions must be principal axes ($\Psi=0$). This paradox can be resolved by noting that on the centre line the stress in the FPA model is actually isotropic, thus satisfying both criteria at once.
The correspondence between the FPA model and the Edwards arching picture becomes clearer on considering the characteristics of the wave equation, (\[FPAwave\]), which are straight lines of slope $c_1$ and $c_2$ (as discussed further in Section 2.9 below), Fig.3. Since for the FPA model $c_1 =
\tan(\psi)$ and $c_2 = -\cot(\psi)$, the characteristics are at rightangles to one another; moreover, [*they coincide with the principal axes of the stress tensor*]{}. It is this special property of the FPA model that we believe embodies Edwards’ physical picture of arches [@EO; @EM]. The stress arising from the weight of an element of sand propagates along two straight characteristics, one at $\psi$ to the vertical (which we identify as the “arch direction", coincident with the major principal axis) and the other at rightangles. As shown in Section 2.9 below, the [*majority*]{} of the stress is carried by the outward characteristic (slope $c_1$). The material can therefore be viewed as a set of nested arches (Fig.1(b)) down which most of the stress propagates. (This ties in with Edwards’ idea of “lines of force" [@EM].) However, [*a minority*]{} of the stress is transmitted instead from one arch to its inner neighbour; this transfer imparts mechanical stability to the outer, incomplete, arches. Since the principal axes and the characteristics coincide, there are no shear forces acting at the interface between successive arches, which are therefore effectively in [*frictionless*]{} contact with one another. This seems to be as close as one can get, within a consistent continuum theory, to the intuitive picture of arches as [independent]{} load-bearing structures.
We emphasize that for a sandpile constructed from a point source, the FPA model can be viewed in two ways; either as a [*macroscopic*]{} hypothesis concerning the transmission of stresses at the scale of a pile (principal axes fixed in space), or as a [*microscopic*]{} hypothesis concerning the way the growth history of the pile is locally encoded. Our own modelling approach, based on local constitutive relations among stresses, corresponds to the latter view. Indeed, after making the assumption of perfect memory, we have to choose one scale-free property (the constitutive equation) to be “remembered" by any element from the moment of its burial: and the choice made by FPA corresponds to remembering the orientation of the stress tensor (principal axes fixed in time). Note that, once the basic FPA assumption is made, there is no free parameter left in the theory (at least, not in two dimensions), since $\phi$ is fixed by experiment.
It seem plausible that the construction of the pile (by a series of avalanches at the surface) imparts to the local packing of grains a permanent sense of direction. If so, the FPA constitutive relation is perhaps the simplest model for how this “orientiational memory" within an element could determine the constitutive relation among stresses arising there subsequently. A possible (though not a necessary) interpretation of this directional memory is in terms of a fabric tensor $\lambda_{ij}$ [@fabric]. For example, one could postulate that $\lambda_{ij}$ was constant throughout the medium (when expressed in cylindrical polar coordinates) and moreover had a principal axis $\Psi$ bisecting the free surface and the vertical. If this were true, the FPA constitutive relation would reduce to the commutation requirement $\sigma_{ij}\lambda_{jk} =
\lambda_{ij}\sigma_{jk}$.
Apart from its appealing simplicity, however, we have no detailed mechanistic justification for the FPA model in terms of the fabric tensor or any similar quantity: why should the orientation of the principal axes be remembered, rather than something else? (For example, in the IFE model each element “remembers" instead that it was at the critical threshold for slip when buried, and remains so forever after.) We therefore propose the FPA model as a phenomenological hypothesis to be tested against experiment. It is interesting, in that context, to consider alternative closures; we do this next, by embedding the FPA model within a broader scheme.
The Oriented Stress Linearity (OSL) model {#secOSL}
-----------------------------------------
A sandpile is formed by layerwise deposition of particles that have rolled down its free surface. Thus the grains of sand may end up arranged in a packing that locally distinguishes the directions toward and away from the central axis. An arching effect can arise if this anisotropy tends to direct stresses outward from the centre, thus “screening" the central part of the pile from the added load of new layers. This offers a possible way to explain the dip; and indeed the FPA model can be viewed in exactly these terms. However, it is not unique in this respect.
A more general approach can be generated by an adaptation of the BCC model, in which it was assumed that $\sigma_{rr} = \eta\sigma_{zz}$. BCC thus singles out for special treatment the vertical and horizontal normal stresses. We now define the oriented stress linearity (OSL) model by assuming a similar linear relationship between normal stresses, not in a ($z,r$) coordinate system, but in a tilted one ($n,m$). The latter system is now characterized by an [*arbitrary*]{} (but constant) tilt angle $\tau$ to the vertical, and is related to ($z,r$) via the transformation equation (\[eqrot\]). (In general this $n,m$ system does [*not*]{} coincide with the one used earlier to discuss the IFS boundary condition.) In the tilted coordinates, we now require (following BCC) that the two normal stresses, and are proportional: $${\mbox{$\sigma_{nn}$}}= K {\mbox{$\sigma_{mm}$}}$$ (see Fig.1(c)). Despite its formal similarity, the OSL closure differs critically from BCC in that it violates the assumption, tacitly made by BCC, that the properties of the medium vary smoothly as one passes through the centre line of the pile; it thereby allows a cusp (dip or hump) to arise in $s_{zz}$.
Leaving the angle $\tau$ and the constant $K$ free for the moment, we use the rotation eqn. (\[eqrot\]) to obtain, in ($z,r$) coordinates, the OSL constitutive relation $${\sigma_{rr}\over\sigma_{zz}} ={\mbox{${\large C}$}}(U) = \eta + \mu \, U
\label{eqlinclosure}$$ (where, as always, $U = \sigma_{rz}/\sigma_{zz}$). As with FPA, in cartesians ($z,x$) this becomes $${\sigma_{xx}\over\sigma_{zz}} ={\mbox{${\large C}$}}(U) = \eta + \mu\, {\hbox{\rm sign}}(x)\, U
\label{eqlinclosurexz}$$ whereby the singularity on the centreline becomes apparent.
The constants $\eta$ and $\mu$ obey: $$\begin{aligned}
\eta & = & \frac{K-\tan^2(\tau)}{1-K \tan^2(\tau)} \label{eqetamuKtau}\\
\nonumber
\mu & = & \frac{2 (K+1) \tan(\tau)}{1-K \tan^2(\tau)}.\end{aligned}$$ Clearly the BCC model corresponds to $\eta = \eta_0(\phi)$ (defined in eqn.\[eqby2b\]) and $\mu=0$, whereas the FPA model, eqn. (\[fixedpa\]), is obtained by setting $\eta = 1$, $\mu = -2\tan(\phi)$. Both are thereby special cases of OSL [@ifebcclink]. Note that pairs of OSL coordinate systems inclined through angles $\tau$ and $\tau + \pi/2$ (or $\tau-\pi/2$) are identical, subject to interchanging the $m$ and $n$ axes; they give the same values of $\eta$ and $\mu$ in (\[eqlinclosure\]) and hence the same stress profiles in the pile.
The coefficients $\eta$ and $\mu$ (or equivalently $K$ and $\tau$) in the OSL model are not independent: an equation between them can be found from the IFS boundary condition as formulated in eqns. (\[barmy\]). For a given repose angle $\phi$, this condition restricts the OSL parameters to the “IFS line" : $$\eta= \eta_0 (1 - \mu \tan(\phi))
\label{eqstabline}$$ Hence the OSL model has one remaining free parameter (unlike BCC or FPA, which have none). The IFS lines in the $(\mu,\eta)$-plane are shown for two values of the friction angle $\phi$ in Fig.4.
The OSL constitutive relation eqn.(\[eqlinclosure\]) can be substituted into the stress continuity equations (\[eqpde\]) to give $$\begin{aligned}
\partial_r {\mbox{$\sigma_{rz}$}}+ \partial_z {\mbox{$\sigma_{zz}$}}& = & g \label{oslform}\\ \nonumber
\partial_r (\eta{\mbox{$\sigma_{zz}$}}+\mu {\mbox{$\sigma_{rz}$}}) + \partial_z {\mbox{$\sigma_{rz}$}}& = & 0\end{aligned}$$ from which we can obtain a wave equation of the form (\[FPAwave\]), as discussed already in the context of the FPA model. In this more general case, however, $c_1$ and $c_2$ are the positive and negative roots respectively of $$c_{1,2} = \frac{1}{2}(\mu \pm \sqrt{\mu^2 + 4 \eta})
\label{eqvelocities}$$ The propagation velocities become equal in magnitude if coordinate axes are rotated by the tilt angle $\tau$.
The resulting stress propagation equations can be solved without difficulty. As with the FPA model, there are inner and outer regions which meet at $S
=S_0=c_1/c$; in the outer region we obtain $$\begin{aligned}
{\mbox{$s_{zz}$}}& = & s_{*} (c-\mu) (1-S) \label{eqoutside}\\ \nonumber
{\mbox{$s_{rr}$}}& = & s_{*} \eta c (1-S)\\ \nonumber
{\mbox{$s_{zr}$}}& = & s_{*} \eta (1-S)\end{aligned}$$ where we have introduced the constant $$s_{*} = \frac{c}{c^2-\mu c -\eta} = \frac{c c_1}{(c c_1 + \eta)(c -c_1)}$$ In the inner region ($0 \le S \leq c_1/c$) we find $$\begin{aligned}
{\mbox{$s_{zz}$}}& = & s_{*} (c-c_1)/c \ (c_1 - \mu S) \label{eqinside}\\ \nonumber
{\mbox{$s_{rr}$}}& = & s_{*} \eta c_1 (c-c_1)/c \\ \nonumber
{\mbox{$s_{zr}$}}& = & s_{*} \eta (c-c_1)/c \ S\end{aligned}$$ As stated already for FPA and BCC, we thus obtain stress profiles that are piecewise linear functions of $S$. We see from (\[eqinside\]) that a dip in $s_{zz}$ is present so long as $\mu < 0$. This applies for OSL models on the part of the IFS line which lies in the left hand half plane, Fig.4; such models are separated by the BCC model ($\mu = 0$) from OSL models with a hump ($\mu > 0$).
As described earlier in connection with eqn. (\[eqCoulomb3\]), a further check on the consistency of the model should now in general be made: we require that the yield threshold is not exceeded within the pile. The above equations show that the threshold is exactly saturated, not only at the surface (IFS) but throughout the outer region. However, there is also the possibility of yield in the inner region; when this happens, it first occurs at the very centre of the pile [@midslip]. In this neighbourhood, where shear stresses are negligible, the Coulomb criterion eqn. (\[eqCoulomb3\]) simplifies to $$\begin{aligned}
\eta_{min} \leq \eta \leq \eta_{max} \label{limits}\\
\nonumber
\eta_{min} = {1-\sin\phi\over1+\sin\phi} = \eta_{max}^{-1}\end{aligned}$$ Where $\eta_{min}$ is known as“Rankine’s coefficient of active earth pressure" [@nedderman]. Thus, for a given repose angle $\phi$, acceptable OSL parameters lie on the the segment of the IFS line bounded by eqn.(\[limits\]) (dash-dotted lines in Fig.4). Outside this range, there is either too deep a dip or too high a hump, leading respectively to passive or active failure of the material at the centre of the pile.
The FPA model lies in the ($\eta,\mu$) plane at the point where the IFS line crosses $\eta = 1$. It divides those OSL models which, on the central axis of the pile, have active behaviour ($\Psi(0)=0$), from those which are passive there ($\Psi(0)=\pm\pi/2$). For the OSL models generally, the orientation of the principal axes varies smoothly as one passes from left to right through the centre of the pile (though the constitutive equation is nonanalytic there). The sole exception to this is FPA, which has instead a discontinuity in $\Psi$ at $S=0$, for the reasons discussed in the previous section. This highlights the fact that FPA is the only model in the OSL family for which the geometrical picture of “nested arches" (Fig.1(b) and Sec. 2.6 above) can definitely be said to apply.
Linear models and “light-rays" {#secLinear}
------------------------------
We have seen that in the OSL model, the stresses propagate with a wave equation in which $c_1$ and $c_2$ are the positive and negative roots of eqn.(\[eqvelocities\]). The characteristics of this hyperbolic equation are thus straight lines of slopes $c_1$ and $c_2$. This means that, if a perturbation is made at some point in the pile (for example, increasing the weight of a certain element of sand), the resulting information travels along two “light rays" (together called a “light cone" in Ref.[@BCC]). Since the stress propagation equations are linear, the entire stress distribution can be constructed by summing the contributions from all elements of sand propatated along suitable rays; this offers an instructive geometric insight into the problem.
First we consider the Green function which describes the stress perturbation arising from a point source of weight. Such a source term violates the left-right (“cylindrical") symmetry of our two-dimensional system; to deal with it we must introduce cartesians ($z,x$) as opposed to the polar coordinates ($z,r$) used so far. In such coordinates, eqn. (\[FPAwave\]) is virtually unchanged: $$(\partial_z\pm c_1\partial_x) (\partial_z\pm c_2\partial_x)\sigma_{ij} = 0
\label{greenwave}$$ where the $+$ signs apply for $x>0$ and $-$ for $x<0$. Our source term then consists of adding $\Delta g(z,x) =\delta(x-x_0)\delta(z-z_0)$ to the right hand side of the second member of the stress continuity equation (\[eqpde\]), in which $x$ now replaces $r$. This yields an inhomogeneous form of (\[greenwave\]) with derivatives of the delta-function on the right hand side. The algebraic form of the Green function is complicated (we do not write it out explicitly here) but its geometric interpretation is relatively simple, as shown in Fig.5(a).
Of the stress $\sigma_{zz}$ contributed by a small element of sand, a fraction $A_1$ propagates along the outward light ray and $A_2$ along the inner ray. (Note that $A_1+A_2 = 1$: the vertical normal stress is a conserved quantity in $z$.) A ray of amplitude $A_i$, with velocity $c_i$, also carries shear and horizontal normal stresses ${\mbox{$\sigma_{xz}$}}= c_i{\mbox{$\sigma_{zz}$}}$ and ${\mbox{$\sigma_{xx}$}}= c_i^2{\mbox{$\sigma_{zz}$}}$. (Here $i=1,2$; these relations may be confirmed by direct application of the wave equations). Because shear stress is also conserved in $z$ (for a point source of weight, the $x$-integral of ${\mbox{$\sigma_{xz}$}}$ is zero), one has $A_1/A_2 =
|c_2/c_1|$.
Since the wave velocities $c_1$ and $c_2$ become reversed as one crosses the centreline, in all cases (except for the BCC model where $c_1=-c_2$), this line forms a boundary between two different wave media and any ray impinging on it undergoes both reflection and refraction. For simplicity we now move the origin of our $z,x$ coordinates to the point where the ray meets the centreline, in which case an incident ray emanating from our point source corresponds to a disturbance $\sigma_{zz} = A_2
\delta(x-c_2z)$, whereas the reflected ray obeys $\sigma_{zz} = RA_2 \delta(x-c_1z)$, and the transmitted ray $\sigma_{zz} = TA_2 \delta(x+c_1z)$ (this incorporates the sign change of $c_1$ on crossing the centreline). The factors $R$ and $T$ can be deduced as follows. First, one imposes the conservation law for ${\mbox{$\sigma_{zz}$}}$ defined above; the total weight supported by the reflected and transmitted rays is the same as that in the incident ray. This yields immediately $R+T=1$. Secondly, by considering the force on a small element, one finds that not only the shear stress but also the horizontal normal stress ${\mbox{$\sigma_{xx}$}}$ must be continuous across the centreline. (Note that the same does [*not*]{} apply, in general, to the vertical normal stress.) Imposing this for the normal stresses, we equate ${\mbox{$\sigma_{xx}$}}(x=0^+) = A_2 c_2^2\delta(x-c_2z) + A_2 R c_1^2\delta(x-c_1z)$ with ${\mbox{$\sigma_{xx}$}}(x=0^-) = A_2 T c_1^2\delta(x+c_1z)$. Using also the fact that $\delta(x-cz) = |c|^{-1}\delta(x/c - z)$, we find a second relation, $|c_1/c_2| + R = T$. Thus we obtain the results $$\begin{aligned}
T &=& \left(|c_2/c_1|+1\right)/2 \label{reftrans}\\ \nonumber
R &=&
\left(1-|c_2/c_1|\right)/2\end{aligned}$$ which completes our analysis of the reflection/refraction processes. Note that for OSL models with dip ($\mu < 0$) the reflected ray factor $R$ has to be negative.
The above argument shows that the stress response at height $z$ to a point source above this level in the pile consists of either two delta functions (amplitudes $A_1$ and $A_2$) or three delta-functions (amplitudes $A_1, A_2R$ and $A_2T$) according to whether or not the inward-going ray from the source has met the centre-line. Using this information we can construct a geometrical solution of the wave equations for each stress component; for the vertical normal stress $\sigma_{zz}(x)$ at a point $x$ on the base (say), this is done as follows (Fig.5(b)). From each of two points separated by a small distance $\Delta x$ centred on $x$, construct the backward light rays (allowing for any reflection at $x=0$). This defines two strips of material, one of length $L_1$ and the other of length $L_2$, with a third and fourth each of length $L_3$ if there is a reflected/refracted ray. The corresponding widths $w_1,w_2$ (and $w_3=w_4$) are as shown in the figure. The total vertical normal force between our two points now obeys $$\sigma_{zz}(x)\Delta x = g\left(A_1 (w_1 L_1 + R w_3L_3) +
A_2(w_2L_2+T w_3L_3)\right)
\label{rayweight}$$ In other words, one adds the stress contributions of all the material elements which have a light ray ending in the given interval $\Delta x$ on the base. The above construction provides a formula for $\sigma_{zz}$ which is, of course, identical to eqns.(\[eqoutside\],\[eqinside\]) derived earlier. Since the other stress components also obey a wave equation, each of these can be constructed similarly, as a weighted sum of the three $L$’s. The fact that each stress component is a piecewise linear function of $x$ then follows from the elementary geometry of triangles.
The Green function construction shown in Fig.5 gives some direct insight into the role of the arching concept in describing stress propagation in granular media. The relation between the characteristic slopes $c_{1,2}$ and the arching effect is far from intuitive, however. Specifically we can ask how, starting from the BCC model ($c_1+c_2=0$) one should adjust the tilt parameter ($\tau$) of the OSL model so as to obtain an arching effect, and thereby a dip in the stress. In a slightly different language[@language], this was addressed briefly in Ref.[@BCC], where it was suggested that to get an arching behaviour the light rays emanating from an element would have to be tilted [*outwards*]{} ($\tau > 0$, or $c_1+c_2>0$), thus transporting the load away from the centre. This suggestion, though at first sight reasonable enough, is actually wrong. In the FPA model, and all other OSL models giving a dip, the light rays are actually tilted [*inward*]{} relative to the BCC model: their average slope, $(c_1+c_2)/2$ is negative ($\tau < 0$). This would, at first sight, appear to carry the weight of the grains [*toward the centre of the pile*]{}. The paradox is resolved by realizing that, on tilting the rays inwards, the amplitudes $A_1,A_2$ defined above, adjust so that $A_1$ becomes larger than $A_2$. This means that a higher fraction of the weight of a grain is carried along the outward ray, and away from the centre of the pile; this redistribution is more than enough to compensate for the average inward tilt of the two rays.
Trollope’s Model
----------------
As mentioned in the Introduction, Trollope [@Trollope50s; @Trollope] proposed a model which yields, in effect, Edwards’ arches and BCC as its two limiting cases. The relation between this model and our own work is most clearly seen in terms of the above analysis using rays; we therefore discuss it now.
In his model, Trollope, without invoking any differential formulation of the problem, directly assumed that the stress could be constructed in a manner similar to that above, but using [*three*]{} rays; two with equal and opposite velocities $c_1=-c_2$ (just as in the BCC model), and a third, horizontal ray ($c_3=-\infty$). The parameter $c_1$ was taken as fixed globally by the type of packing; however, a second parameter $k$ was introduced. This $k$ represents an imposed amplitude ratio $A_2/A_1=k$ for the outward and inward components of the BCC-like propagation; as $k$ is varied, the amplitude $A_3$ of the third ray also changes (in a manner that can be deduced from stress continuity). It is interesting that Trollope already realized the importance of singular behaviour on the centreline; for $k\neq 1$, his model has this property.
In the limit $k=1$ (no arching, symmetric propagation), $A_3$ vanishes and one recovers a BCC-like picture (though unlike BCC, Trollope did not connect $c_1$ to the repose angle). In the limit $k=0$ one again has two rays, one of which is now horizontal. This limit does not correspond to any OSL model however: within the OSL model an infinite $c_2$ (representing the horizontal ray) automatically has zero amplitude: $A_2=0$. Trollope’s horizontal ray enables stress continuity to be satisfied while giving a maximal dip (zero normal stress ${\mbox{$\sigma_{zz}$}}$ at $x=0$), reminiscent of the Edwards approach. (The Coulomb yield criterion is violated, however.)
By use of the third ray, Trollope managed to interpolate these limits in what he called the “systematic arching theory". However, the introduction of this extra ray seems extremely ad-hoc, which is perhaps why the model is not more widely used today. Mathematically its presence means that rays can no longer be identified as characteristics of a partial differential equation in two dimensions; therefore Trollope’s construction cannot correspond to any local constitutive equation among stresses. (All such closures must lead to hyperbolic equations for which a formal solution using characteristics is available, even if the characteristics are curved – as happens, for example, in the IFE model [@nedderman].) We conclude that Trollope’s systematic arching theory must be rejected as unphysical – a view tacitly shared by most of the recent sandpile literature. However, many of the physical ideas behind the model, including the emphasis on discontinuities in propagation across the centreline, remain highly pertinent to the present work.
The conical sandpile {#sec3D}
====================
We now extend our continuum modelling approach to the three dimensional conical sandpile.
One additional missing constitutive equation {#secDmissing}
--------------------------------------------
The conical pile is as shown in Fig.1(d); in addition to the cylindrical coordinates $(z,r)$ introduced before, an azimuthal coordinate $\chi$ is required. Since we have axial symmetry around the $z$-axis, the principal axes of the stress tensor must include the azimuthal ($\chi\chi$) direction. (The orientation of this tensor can thus be fully specified, as before, by the inclination angle $\Psi$ to the vertical of the major principal axis in the $r,z$ plane.) Recalling that the stress tensor is symmetric, we therefore have ${\mbox{$\sigma_{r \chi}$}}={\mbox{$\sigma_{\chi r}$}}={\mbox{$\sigma_{z \chi}$}}={\mbox{$\sigma_{\chi z}$}}=0$. Hence the three dimensional conical pile has only [*one*]{} additional independent stress component compared to the two dimensional case [@BCC]. The stress continuity equation for a conical sandpile is $$\begin{aligned}
\partial_r {\mbox{$\sigma_{rr}$}}+ \partial_z {\mbox{$\sigma_{rz}$}}& = & \frac{{\mbox{$\sigma_{\chi \chi}$}}-{\mbox{$\sigma_{rr}$}}}{r} \label{eqpdeD3}\\
\nonumber
\partial_r {\mbox{$\sigma_{rz}$}}+ \partial_z {\mbox{$\sigma_{zz}$}}& = & g - \frac{{\mbox{$\sigma_{zr}$}}}{r} \\ \nonumber
\partial_{\chi} \sigma_{ij} & = & 0\end{aligned}$$ The first two equations differ from those found earlier in two dimensions by additional “source terms", $({{\mbox{$\sigma_{\chi \chi}$}}-{\mbox{$\sigma_{rr}$}}}/{r})$ and $- {{\mbox{$\sigma_{zr}$}}}/{r}$ respectively, on the right hand side.
Because of the high symmetry of the conical pile, closure of these equations requires only that we find [*two*]{} constitutive equations which together should determine any two of the independent stress components in terms of the remaining two. Choosing the latter as before (${\mbox{$\sigma_{rz}$}}$ and ${\mbox{$\sigma_{zz}$}}$) we refer to the resulting equation for ${\mbox{$\sigma_{rr}$}}$ as the primary, and that for ${\mbox{$\sigma_{\chi \chi}$}}$ as the secondary constitutive equation. Note that symmetry requires also $$\begin{aligned}
{\mbox{$\sigma_{zr}$}}(r=0) & = & 0\label{eqcylindrical}\\ \nonumber
{\mbox{$\sigma_{rr}$}}(r=0) &= &{\mbox{$\sigma_{\chi \chi}$}}(r=0)\end{aligned}$$
As our yield criterion for plastic failure of the granular material we retain the Coulomb criterion [@Conical] for cohesionless granular materials (which becomes a relation between principal stresses in the $r,z$ plane). However, the Coulomb yield criterion is essentially two-dimensional in character and gives no explicit information on the circumferential stress ${\mbox{$\sigma_{\chi \chi}$}}$. In common with previous authors [@footstress] we argue nonetheless that this should vanish on the free surface, as all the other stress components do: $${\mbox{$s_{\chi \chi}$}}(S=1)=0.
\label{eqby3}$$ Note that if we were to use instead the Conical Yield criterion [@nedderman; @Wood; @Conical] or a similar (fully three dimensional) condition at the surface, eqn. (\[eqby3\]) would not be an extra assumption.
The form of the new source terms in eqn. (\[eqpdeD3\]) is of interest. If these remain relatively small everywhere, one can expect to find (independent of the form of chosen for the secondary closure relation) qualitatively similar results to those obtained earlier in the two dimensional case. This scenario is indeed fulfilled for the various different secondary closures tried below. In any case, given that all stresses vanish at the surface (as just described), these source terms become strictly negligible near the free surface of the pile, which may therefore be viewed locally as having a planar two-dimensional geometry. Accordingly, the IFS boundary condition is completely unaffected; the stresses on the surface of a pile obeying IFS are still given by eqns.(\[eqby2a\],\[eqby2b\],\[barmy\]). It follows that (subject to the usual scaling assumptions, see below) eqns. (\[eqabdonslope\]) and (\[eqPsionslope\]) still govern the asymptotic behaviour near the free surface. Thus the relation between the repose angle $\phi$ and parameters in the primary constitutive equation (such as the tilt angle $\Psi$ in the FPA model, or the $\eta$ and $\mu$ parameters in OSL) remain as they were in two dimensions.
Scaling analysis {#scalingassumpt}
----------------
As for the two dimensional case, by invoking the absence of an intrinsic length scale we may demand that solutions of of the stress continuity equation take the RSF scaling form, eqn. (\[eqonlyH\]). Substituting this into eqn. (\[eqpdeD3\]) gives a set of ordinary differential equations: $$\begin{aligned}
{\mbox{$s_{rr}$}}'/c + {\mbox{$s_{rr}$}}+ {\mbox{$s_{zr}$}}- {\mbox{$s_{\chi \chi}$}}- s {\mbox{$s_{zr}$}}' & = & 0 \label{eqDED3}\\ \nonumber
{\mbox{$s_{zr}$}}'/c + {\mbox{$s_{zr}$}}+ {\mbox{$s_{zz}$}}- s {\mbox{$s_{zz}$}}' & = & 1\end{aligned}$$ The asymptotic analysis in Section 2.4 for the stresses near the surface carries over to the case of three dimensions, as mentioned already above. The source terms in eqn. (\[eqpdeD3\]) can in principle affect the the asymptotic behaviour given in Section 2.4 near the centre of the pile (i.e., small $S$) but [*qualitative*]{} changes arise only if either ${\mbox{$\sigma_{zr}$}}/r$ or $({\mbox{$\sigma_{rr}$}}-{\mbox{$\sigma_{\chi \chi}$}})/r$ become large in this limit. This does not occur for any of the models studied below.
Following the arguments made earlier in two dimensions, based on our assumptions of RSF scaling and “perfect memory”, we now propose local forms for both the primary and secondary constitutive relations, which must be as follows: $$\begin{aligned}
{\mbox{$s_{rr}$}}/{\mbox{$s_{zz}$}}& = & {\mbox{${\large C}$}}(U) \label{eqClosureD3}\\ \nonumber
{\mbox{$s_{\chi \chi}$}}/{\mbox{$s_{zz}$}}& = & {\mbox{${\large D}$}}(U)\end{aligned}$$ where we have set $U={\mbox{$s_{zr}$}}/{\mbox{$s_{zz}$}}$ as usual.
The form of eqn. (\[eqRK\]) for the RSF scaling solution remains basically unchanged, except that in eqn. (\[eqabcdef\]) the terms ${\mbox{$\hat{c}$}}={\mbox{$s_{zr}$}}+{\mbox{$s_{rr}$}}-{\mbox{$s_{\chi \chi}$}}$ and ${\mbox{$\hat{f}$}}=
{\mbox{$s_{zz}$}}+{\mbox{$s_{rz}$}}-1$ are somewhat modified. The resulting stress profiles can readily be calculated numerically from eqns. (\[eqRK\],\[eqabcdef\]) for any choice of the closure relations. However, no analytic solution appears to be obtainable even for those models which, in two dimensions, reduce to wavelike propagation. The problem can, of course, still be viewed as quasi two-dimesional (one spacelike, one timelike variable), but if so the extra “source terms" make the solution complicated. Alternatively these models can be formulated in terms of wave propagation in two spacelike dimensions (the $r,\chi$ plane) with $z$ as a timelike variable. However, the Green function for such waves is itself surprisingly complicated (there is no sharp “light-cone" [@BCC]) and not directly amenable to the simple geometrical interpretations offered earlier.
Choice of constitutive equations
--------------------------------
For the primary constitutive equation, we can choose among those discussed earlier, namely IFE and OSL, with the latter including both BCC and FPA as special cases. We continue to require that the IFS boundary condition is obeyed at the surface (which again fixes the OSL parameters $\mu$ and $\eta$ to lie on the IFS line) and that the Coulomb yield criterion is not violated in the interior of the pile.
For the secondary constitutive equation, we have investigated three ways of selecting the function $D(U)$. The first is to insist that $D(U)$ coincides with $C(U)$ so that ${\mbox{$\sigma_{\chi \chi}$}}= {\mbox{$\sigma_{rr}$}}$ everywhere in the pile: $${\mbox{${\large D}_1$}}(U) \equiv {\mbox{${\large C}$}}(U)
\label{Closure1}$$ This has the merit of simplicity. (Note that by symmetry this relation must hold anyway at the central axis of the pile, but not necessarily elsewhere.) A second choice is suggested by the observation that the $\chi\chi$ direction is a minor principle axis for a conical sandpile with axial symmetry. Generalizing slightly an assumption often made the context of conical hoppers [@Wood; @nedderman], one could then choose as the secondary closure ${\mbox{$\sigma_{\chi \chi}$}}=P-R$ (the Haar - von Karman hypothesis [@Wood]). This implies $${\mbox{${\large D}_2$}}(U)=(1+{\mbox{${\large C}$}}(U))/2-\sqrt{(1-{\mbox{${\large C}$}}(U))^{2}/4+U^2}
\label{Closure2}$$ Although the motivation for this choice in the sandpile context is not very clear, we have tried it out for comparison. Our third choice of secondary closure, unlike the first two, does not explicitly depend involve the primary closure $C(U)$; it is the linear relation $${\mbox{${\large D}_3$}}(U)=\eta + \tilde{\mu} U
\label{Closure3}$$ which should be compared with the OSL primary closure, eqn.(\[eqlinclosure\]). In fact, for the OSL model the constant term $\eta$ has to be identical to that chosen in the primary closure, to meet the second requirement of eqn. (\[eqcylindrical\]). The coefficient $\tilde{\mu}$ is in principle free. In practice, however, we have found that the requirement that the Coulomb criterion ${\mbox{$\Upsilon$}}\leq 1$ holds in the interior of the pile means that values of $\tilde{\mu}$ close to $\mu$ are required; hence for OSL models the closure $D_3(U)$ is never very different from $D_1(U)$.
We have investigated these three closure relations $D_1,D_2,D_3$ for all the different primary closures already discussed in Section \[sec2D\], for various values of the repose angle $\phi$ (mainly in a range around $\phi =
30^o$). For all the parameters we tried, the extra “source terms" led mainly to smoothing of the two dimensional curves without qualitatively altering the presence or absence of the dip. Since these source terms do not have a dramatic effect, it follows that the choice made for $D$, at least among those investigated here, itself does not qualitatively change the stress profiles.
Results
-------
Rather than provide a catalogue of curves for various combinations of primary and secondary closure, we will focus attention on the FPA model. In Fig.6 we compare the stress curves for the three-dimensional FPA with closures $D_1$ and $D_2$. As mentioned previously, the choice of secondary closure proves quantitatively but not qualitatively important. Also shown are the experimental results of Ref. [@smid] for piles of height 20-60 cm. The stresses are normalized by the total weight of the pile; notice the good scaling collapse of curves from piles of varying heights. This confirms that the RSF scaling hypothesis made in this paper is obeyed to experimental accuracy, at least for the materials and pile sizes studied in Ref. [@smid]. The agreement between experiment and FPA theory is generally satisfactory, although there is a significant error near the maximum of the vertical normal stress. Obviously it would be helpful to have more data for small values of $S$, but there are sufficient data points at the origin to clearly establish the presence and magnitude of the dip. The experimental data shown are for two different media both with repose angles close to $\phi =
33^o$. The resulting curves differ by an amount similar to the difference between the two choices of secondary closure, with $D_1$ giving slightly better results for “quartz sand" and $D_2$ for “NPK-1 fertilizer". (We do not attach any significance to this.) As mentioned previously, the FPA model has no adjustable parameters once $D$ is chosen and $\phi$ is fixed by experiment.
In Fig.7 we show the same predictions for the FPA model with closure $D_1$ alongside those for several other models with the same secondary closure. These models are BCC and IFE (neither showing a dip); and two parameter choices for the OSL model ($\eta = 0.8$ and $\eta = 1.2$) which bracket the FPA case ($\eta =1$). This comparison shows a clear preference of FPA over those other models that have no adjustable parameter. It is conceivable that the data could be fit better by choosing an OSL model with $\eta$ slightly different from unity. However, we do not believe the improvement is enough to justify the adoption of an extra fitting parameter, although further careful experiments might reveal this to be necessary.
Finally in Fig.8 we plot the yield parameter $\Upsilon(S)$ for the BCC, FPA and IFE models. By definition, $\Upsilon(S) = 1$ everywhere in the IFE model; it also obeys $\Upsilon(1)
=1$ in all models obeying the IFS boundary condition. In two dimensions it is also unity throughout the outer regime of the pile for all OSL models. In three dimensions this is not the case, and in fact for OSL models the material is clearly below the yield criterion throughout the bulk of the pile. This underlies the important distinction between the classical IFE assumption (fully mobilized friction, $\Upsilon = 1$) and the new models adopted in this paper.
Role of construction history {#secalpha}
============================
So far we have only considered the stress profiles of idealized sandpiles constructed by pouring sand from a stationary pipe, for which the boundary condition of incipient failure at the surface (IFS) was assumed. The slope $\alpha$ to the horizontal of the free surface is by definition given by the repose angle: $\alpha=\phi$. According to our approach, however, the constitutive equation encodes the construction history, and piles built differently can behave differently. In discussing this issue, we restrict attention to the FPA constitutive model in two dimensions.
We consider first the following hypothetical experiment: a material with $\phi = \phi_0$ is formed into a pile by the usual method. The pile is then reduced to a flatter (symmetrical) one of angle $\alpha$ by simply taking away the upper section, grain by grain, without disturbing any material below (Fig.9(a)). According to our model, the constitutive equation remains that of a pile with the larger repose angle, though the stresses are of course altered. The resulting stress pattern is shown in Fig.9(b) in comparison to that of a pile of repose angle $\phi_1=\alpha$ which has the same final geometry. The first of the two piles has the larger dip. We can now ask the following: if a pile of $\alpha < \phi$ is tilted from the base through an angle $t$ (Fig.9(c)) how large may $t$ become before an avalanche occurs? A classical answer, based on the view that the repose angle $\phi$ is a material property [*independent of construction history*]{}, is that one would be able to tilt until $t_{max}+\alpha$ is again equal to $\phi$. (This ignores, as we have done throughout this paper, the small hysteresis effects associated with the Bagnold angle [@Bagnold]). However, in our approach this should not quite be true, since the inclination angle of the principal axes in a pile at this condition is different (by an angle $t$) from that of a pile created by the normal method at its repose angle. It turns out, however (Fig.9(d)) that unless the pile is substantially flattened ($\phi-\alpha \simeq 10^o$ or more), the difference between $t_{max}+\alpha$ and $\phi$ is very slight, at least for repose angles in the usual range ($\phi < 45^o$).
This calculation can, with caution, be proposed as a model for what happens when a sandpile, built normally, is suddenly tilted through a finite angle $t$. Of course, in this case an avalanche does occur: however, if this happens by removal of a wedge [*without significant reorganization of the remaining part of the pile*]{}, leaving the new surface in a state of incipient failure, the above calculation can be applied (except that, for simplicity, we have contrived a version in which the pile remains symmetrical). In principal, there should then be a change in the resulting repose angle if $t$ is large enough. However, the assumption that an avalanche occurs with no rearrangement of the remaining grains, is, for large $t$, highly dubious.
Another critical test of our ideas is the following: a triangular pile is constructed as usual and then a large part of it removed (grain by grain) leaving a pile whose left hand slope is at the angle of repose $\phi$, and whose right hand slope is at angle $\beta$ (say) to the horizontal. The geometry is chosen so that all of the material in the new pile was [*originally in the left half*]{} of the parent pile (Fig.10(a)). To describe this situation, we have to use the FPA constitutive equation in the form (\[fixedpaxz\]), in a coordinate system where $x=0$ denotes the centreline of the original pile. With the modified construction history just described, the singularity on the line $x=0$ lies outside the newly created pile, and the characteristics of stress propagation should be identical on both sides of its apex. Throughout the pile, a majority of the stress is carried down the [*leftmost*]{} (rather than [*outermost*]{}) characteristic. An interesting question now is, what is the maximum angle $\beta$ that we can choose for the right hand slope? This can be found from the usual stability criterion ${\mbox{$\Upsilon$}}\le 1$; the marginal case has equality at the free surface on the right and, for the FPA model, this gives (after some algebra) the condition $$\tan(\psi - \beta) \ge \tan^3(\psi)
\label{tangent}$$ The maximum $\beta$ for which the new pile is stable is shown as a function of $\phi$ in Fig.10(b); for $\phi = 30^o$, one has $\beta_{max}
= 19.1^o$. This is a very interesting result, since it predicts that the repose angle of the right hand part of a pile built this way is quite different from the usual value, which prevails on the left.
This prediction must, of course, be interpreted with caution since its extension to a fully three dimensional geometry is not obvious. Perhaps the simplest three dimensional analogue is to build a pile and then open a hole directly below the vertex, allowing sand to flow out leaving a “volcano crater" [@entov]; according to this prediction, the angle of repose on the inner side of the crater may differ substantially from that on the outer slope. This possibility deserves careful experimental study; a significant difference is not ruled out [@Rotter]. The situation is again complicated by the fact that the experiment will set up a flow which may rearrange the grains that remain in the pile. Indeed, the removal by avalanche of the right hand part of the pile may set up a large region in which the grains have slipped down to the right, for which the constitutive equation may revert to that of the right hand part of a normal pile. (This could be true even if the actual particle displacements are extremely small.) If so, the measured repose angle could again approach $\phi$, rather than $\beta_{max}$ which applies only when the removal of sand does not perturb the remainder. We show in Fig.10(c) the stress distribution in a pile made in this careful fashion (with $\phi = 30^o$ and $\beta = 15^o$). As one might expect, there is now no dip but a (lopsided) maximum in the vertical normal stress. The maximum lies to the left of the new apex (at the point where an outgoing ray from this apex strikes the base).
Note that quite different predictions for this geometry could have been obtained by writing the FPA closure in a somewhat different form, which is, [*for a symmetrical pile only*]{} equivalent to eqn. (\[fixedpaxz\]) [@nature], as discussed in Section 2.7: $$s_{xx}/s_{zz}
= 1 - 2\tan\phi\, |U|$$ Here the explicit dependence on construction history via the ${\hbox{\rm sign}}(x)$ factor has been replaced by an $x$-independent but highly nonlinear constitutive relation among the stresses (in the spirit of some of the models discussed in Ref.[@BCC]). Using this form, one could find a solution, with $\beta = \phi$, for our asymmetrically constructed pile that would precisely coincide with the usual symmetrical case. (This possibility arises because the sign of $U$, though not of $x$, can change on the centreline of the new pile.) Since we only have data for symmetric piles, we cannot on the existing facts rule out this rather different version of the FPA model, although it does not correspond to our assumption that the principal axes of a material element are fixed at the time of burial. Accordingly experiments on asymmetric piles would be a strong test of the theory.
A somewhat different experiment would be to start with a symmetrical pile and then remove parts of it (grain by grain) so that the remainder forms an asymmetrical pile whose apex has not moved from the line $x=0$. Shown in Fig.11 is the stress distribution in such a pile with left and right slopes $\alpha_1 = \phi = 30^o$ and $\alpha_2 = 12^o$. An interesting feature is visible under the apex, where the vertical normal stress $\sigma_{zz}$ (and therefore also the yield function ${\mbox{$\Upsilon$}}$) is discontinuous. This behaviour is in fact a generic feature of sandpiles that include the line $x=0$ but are asymmetric about it. It stems from the fact that this normal stress is not continuous in the geometry of incident, reflected and transmitted rays considered earlier (Section 2.9). An exception to this rule is if an asymmetrical pile is made by pouring sand onto a sloping base plate. In this case, we expect the apex of the pile to move slightly relative to the source so that more material rolls down the “long" side of the pile and the repose angle in the two directions remain equal to $\phi$. The constitutive equation must then (given RSF scaling) be the same as for a pile formed normally, and the stress exerted on the supporting plate is the same as that on an inclined plane inscribed through a normal pile. (In two dimensions, this can be found easily from our earlier results.) Though asymmetric, this stress distribution will not show any discontinuity beneath the apex.
As a last example of a sandpile constructed normally and then manipulated, in Fig.12 we show the stress distribution in a pile whose top section has simply been cut off. Though grain-by-grain removal of sections of a pile may impose experimental difficulties, this is perhaps the simplest geometry for which it could be achieved. As shown, the dip is gradually diminished and replaced by a plateau as larger and larger upper sections are taken away.
Finally, we note that a sandpile could be made by first distributing sand uniformly (not from a point source) in a retaining bin, from which the side walls are then removed. As with some of the examples studied above, the predictions depend crucially on whether significant slip occurs within the part of the pile that finally remains. The initial loading of the bin is likely to produce principal axes with vertical and horizontal orientations ($\Psi = 0$), so that if no slip occurs, we would expect the BCC model to apply (no dip). However, if slip does occur so the remaining pile has been sheared downwards, the FPA picture should be more appropriate.
The various types of experiment discussed above, in which the construction history of the pile is deliberately manipulated, provide a strong test of our basic modelling hypothesis that the constitutive equation encodes the construction history. For some of these geometries, the theoretical predictions challenge the “classical" assumption, maintained in the recent physics literature on sandpiles, that for cohesionless granular media (of a single grain size [@nedderman]) the repose angle is the same for all types of pile of a given material. (As mentioned in Sec. 1.2, this assumption has long been avoided in the engineering literature on hoppers [@Rotter; @rotter1].) Of course, the repose angle remains a genuine material parameter in that the angle of a “normal" pile (built from a point source) will differ for different cohesionless materials; and for our purposes this can be taken as the unique definition of $\phi$. It does not necessarily follow, however, that the repose angle taken up by a pile of the same material with a different construction history, will always be exactly the same. In any case, our modelling approach leads to a clear expectation that the [*stresses*]{} in such piles can be different (even when the repose angles are not significantly different). This is a readily testable prediction which we believe deserves urgent experimental attention.
Conclusion
==========
This paper is a long one. It therefore seems useful to provide a brief summary of our modelling strategy, in the form of a list of contentions for which more detailed arguments can be found in the text above. We stress, however, that several points on the list have no first-principles justification: they are hypotheses whose value can at present only be judged by comparison with experiment. We also stress that several of these ideas have a long history (which is not the same as saying that they are widely agreed upon).
A manifesto for sandpile modelling
----------------------------------
Our modelling strategy is based on the following claims:
\(1) There is a construction history, $\cal H$. This determines the arrangement of grains. We define the “normal" history to be the construction from a point source of a pile at its repose angle.
\(2) There is a stress tensor $\sigma_{ij}$ which is well-defined as a local (mesoscopic) average over many grains.
\(3) For hard particles (of infinite elastic modulus), no strain variables exist; static frictional forces are indeterminate. Stress continuity requires one supplementary equation for closure in two dimensions, and two for a conical pile in three dimensions.
\(4) Scaling behaviour (RSF scaling) is observed, to experimental accuracy. Hence there is no characteristic length scale in a sandpile under gravity. Particle deformability would provide such a length; so would size segregation.
\(5) The limit of uniform nondeformable, cohesionless, particles presumably therefore exists, and should describe those experiments for which RSF scaling is observed.
\(6) We should therefore seek as closure a scale-free, local constitutive relation among stresses. Formally: there is a function $\cal C$ such that $${\cal C}(\sigma_{ij}(r,z), {\cal H}) = 0$$ The constitutive relation depends on the local packing and therefore on the construction history: $\cal C$ encodes $\cal H$.
\(7) $\cal C$ for a material element is “frozen in" at the time of burial (perfect memory assumption). Combined with RSF scaling, this means that for a sandpile constructed from a point source, $\cal C$ is independent of position when expressed in cylindrical polar coordinates, though it may be singular on the central axis.
\(8) The boundary conditions for a pile constructed normally are IFS: incipient failure at the free surface. This means that [*at the surface*]{}, the major principal stress axis bisects the free surface and the downward vertical. (Here and elsewhere, hysteresis effects associated with the Bagnold angle are ignored.)
\(9) The search for a constitutive relation ${\cal C}(\sigma_{ij}, {\cal H})$ may legitimately entail (a) making simplified hypotheses to compare with experiment; (b) microphysical modelling from first principles. We pursue the former in this paper, the latter elsewhere [@volkard2].
\(10) A classical choice of $\cal C$ is incipient failure everywhere (IFE); this is hard to defend physically. It does not predict a dip in the stress beneath the apex of a pile.
\(11) A physically more plausible (but by no means unique) choice for $\cal C$ is provided by the FPA hypothesis. According to this, each element of material is impressed at burial with a sense of direction, which fixes forever the orientation of the stress tensor ellipsoid that the element can support. The model predicts a dip in two dimensions.
\(12) In three dimensions, a secondary closure relation is needed. Among the more obvious choices, it makes relatively little difference which is chosen. Even the simplest choice (${\mbox{$\sigma_{\chi \chi}$}}= {\mbox{$\sigma_{rr}$}}$), combined with FPA, gives a reasonably good fit to the data of Ref.[@smid], without adjustable parameters.
\(13) A generalized model (OSL), of which FPA is a special case, can be introduced. This has an adjustable parameter, the introduction of which is not demanded by the present data.
\(14) The above modelling approach, though initially set up for static sandpiles constructed from a point source, can also be used for more complex construction histories (at least in some cases). For piles constructed normally and then modified by careful removal of grains, this approach predicts a nontrivial dependence of the repose angle $\phi$, and of the stress distribution, on the way a pile is made.
Discussion
----------
Of the models considered in this paper, it is clear that the FPA model has some especially attractive features. This model leads directly to an arch-like stress-propagation, with the major part of any load being carried down the arch direction. The latter coincides with the major principal axis of the stress tensor; this [*everywhere*]{} bisects the free surface and the downward vertical. The predictions of the FPA constitutive relation thereby describe similar physics to the arching model of Edwards [@EO] (and indeed the earlier “full arching theory" of Trollope [@Trollope]). Like such models, the FPA hypothesis can be viewed as a direct macroscopic ansatz of how stresses propagate: one assumes that the principal axes are fixed in space. Viewed this way, we believe that the FPA model provides the first description of the arching picture within a fully consistent continuum mechanics framework. Its experimental success strongly suggests that the presence of a macroscopic arching structure in sandpiles is the correct explanation for the observed minimum in the vertical normal stress below the apex of the pile.
However, unlike previous arching models, the FPA hypothesis can also be interpreted as providing a local, history-dependent constitutive relation among stresses. In this context, it is among the simplest such equations that can plausibly be devised: we assume that the principal axes of a material element are fixed at the time of its burial. Viewed as such, the FPA hypothesis contains no assumption of any macroscopic arching structure; rather, it provides a plausible microscopic explanation for how such structures arise. Its experimental success offers strong support for a modelling strategy cast in terms of such constitutive relations. For parameter values other than FPA, which is a special case, the more general OSL model predicts, within the same modelling framework, a more complex pattern of stress propagation. (The principal axes and the propagation characteristics no longer coincide.) The extra fitting parameter provided by OSL is probably not justified by the existing data. One feature of OSL models which stands out strongly (at least in two dimensions) is the presence of reflection and refraction of stress-paths at the central axis of the pile (see Sec. 2.9). Careful experiments on the effect of small perturbing loads could reveal whether or not this really occurs, providing a strong test of this class of model.
In view of its attractive physical features, and of its experimental success, we currently favour the FPA hypothesis as the simplest starting point for more refined theories of sandpiles. It also forms a promising basis for future study of stress propagation in static granular media of geometries quite different from the normal conical pile. The richness of this area is amply illustrated by the handful of examples studied above in Section 4. Consideration of these and other geometries could allow stringent experimental tests of both the FPA model, and the overall modelling strategy we have proposed. Within this framework, there is, no doubt, scope for much more sophisticated models of how the construction history of a pile determines the local constitutive behaviour, but further efforts in this direction may require much more experimental input. The validity of the framework itself deserves close experimental scrutiny, particularly concerning the degree to which RSF scaling is obeyed. Our assumption of scale-free (RSF) behaviour offers an immense simplification, but closer experimental investigation may reveal that this is not quantitative except under some limiting conditions. Despite these uncertainties, we feel that the modelling framework presented above has significant potential to provide improved physical theories of stress propagation granular media.
In future work [@ournoise] we will explore the close connection between our OSL model and a recent discrete stochastic models for stress propagation in sandpiles [@degennes] (see also [@Liu]), of which OSL can be viewed as the (mean-field) continuum limit. An important concept arising from the stochastic models and from experiment [@Liu] (see also [@nagel]) is that of [*stress paths*]{}; these are pathways through the medium along which most of the load is locally transmitted. The noise-free models considered in this paper can be viewed as making hypothetical statements about the [*average*]{} orientation and load-bearing properties of these paths (see the discussion of characteristics in Sec. 2.9). Such statements are testable, if not directly, then at least in simulation studies. Ongoing work [@volkard2] suggests a promising correspondence between the average alignment of these paths and the orientation angle $\Psi$ arising in the OSL model.
$\,$
[**Acknowledgement:**]{} The authors are indebted to J.-P. Bouchaud for a series of essential discussions which laid the foundations for this work. We also thank S. F. Edwards, V. Entov, M. Evans, J. Goddard, T.C.B. McLeish, R. M. Nedderman and J. M. Rotter for illuminating discussions. JPW thanks C. E. Lester, C. S. and M. J. Cowperthwaite for supporting enthusiastically this research. MEC acknowledges the hospitality of the Isaac Newton Institute for Mathematical Sciences (Cambridge) where part of this work was done. This work was funded in part by EPSRC under Grant GR/K56223 and in part by the Colloid Technology programme.
[99]{}
T. Jotaki and R. Moriyama, [*J. Soc. Powder Technol. Jpn.*]{}, [**60**]{}, 184 (1979).
J. Smid and J. Novosad, in [*Proc. of 1981 Powtech Conference, Ind. Chem. Eng. Symp.*]{} [**63**]{}, D3/V/1-D3/V/12 (1981).
D. C. Hong, [*Phys. Rev. E.*]{}, [**47**]{}, 760 (1993).
J. Huntley, [*Phys. Rev. E.*]{}, [**48**]{}, 4099 (1993).
S. F. Edwards and R. B. Oakeshott, [*Physica D*]{}, [**38**]{}, 88 (1989).
S. F. Edwards, C. C. Mounfield, [*Physica A*]{}, [**226**]{}, 1,12,25 (1996).
J.-P. Bouchaud, M. E. Cates, P. Claudin, [*J. Physique II*]{}, [**5**]{}, 639 (1995).
D. F. Bagster and R. Kirk, [*J. Powder Bulk Solids Tech.*]{}, [**1**]{}, 19, (1985).
K. Liffman, D. Y. Chan and B. D. Hughes, [*Powder Technology*]{} [**72**]{}, 225 (1992); [*Powder Technology*]{} [**78**]{}, 263 (1994).
R. .M. Nedderman, [*Statics and Kinematics of Granular Materials*]{}, Cambridge University Press (1992).
V. V. Sokolovskii, [*Statics of Granular Materials*]{}, Pergamon, Oxford (1965).
A. Drescher, [*Analytical Methods in Bin-Load Analysis*]{}, Elsevier, Amsterdam (1991).
D. M. Wood, [*Soil Behaviour and Critical State Soil Mechanics*]{}, Cambridge University Press (1990).
R. A. Bagnold, [*The Physics of Blown Sand and Desert Dunes*]{}, Methuen, London (1941).
J.-P. Bouchaud, M. E. Cates, J. R. Prakash and S. F. Edwards, [*J. Physique I*]{}, [**4**]{}, 1383 (1994); [*Phys. Rev. Lett.*]{} [**74**]{}, 1982 (1995).
In Ref.[@BCC], a family of nonlinear models are produced in which a velocity-like parameter $V$ is considered to be a function of the shear stress. The OSL model can be viewed as the limit in which $V(\sigma_{zx})$ approaches a step function (giving a positive or negative constant value according to the sign of $x$). A finite slope of this function at the origin ($V'(0)\neq 0$) would lead to rounding of the cusp at the centre of the dip on some characteristic length scale (in violation of RSF scaling).
D. H. Trollope, [*The Stability of Wedges of Granular Materials*]{}, Ph.D. Thesis, University of Melbourne (1956); [*Proc. 4th Int. Conf. S. M. and F. E.*]{}, London [**2**]{} 383 (1957). See also, C. F. Jenkin, [*Proc. Roy. Soc. A*]{} [**131**]{}, 53 (1931).
D. H. Trollope, in [*Rock Mechanics in Engineering Practice*]{}, (Eds. K. G. Stagg and O. C. Zienkiewicz), Ch.9. John Wiley, NY (1968).
If support deflection were important, one would not expect RSF scaling as defined below. (The elasticity of the support introduces a length scale just as that of particles would do.)
C. H. Liu, S. R. Nagel, D. A. Schecter, S. N. Coppersmith, S. Majumdar, O. Narayan, T. A. Witten, [*Science*]{}, [**269**]{}, 513 (1995).
P. G. de Gennes, private communication.
P. Claudin, J. P. Wittmer, J.-P. Bouchaud and M. E. Cates, to be published.
P. A. Cundall and O. D. Strack, [*Geotechnique*]{} [**29**]{}, 47 (1975); P. A. Cundall, in [*Micromechanics of Granular Materials*]{}, M. Satake and J. T. Jenkins, Eds., Elsevier (Amsterdam) 1987; C. Thornton and D. J. Barnes, [*Acta Mech.*]{}, [**64**]{}, 46 (1986).
V. Buchholtz and T. Poeschel, [*Physica A*]{}, [**202**]{}, 390 (1994); T. Poeschel and V. Buchholtz, [*Phys. Rev. Lett.*]{}, [**71**]{}, 3963 (1993).
X. Zhuang, A. D. Didwania and J. D. Goddard, [*J. Com. Phys.*]{}, [**121**]{}, 331 (1995).
V. Buchholtz, J.P. Wittmer and M. E. Cates, in preparation.
In principle one could construct piles in which the scaling was nonetheless obeyed, by demanding that the [*relative*]{} variation of the material properties between (say) an element at the apex, and an element directly beneath the apex at the base, must be the same for all piles. However, this would require the spatial gradient of the size distribution, for the element at the base, to vary inversely with the height of the pile. Clearly, the size distribution (and its gradient) in this element is fixed forever long before the eventual height of the pile is decided. We conclude that, although size segregation may arise, models invoking this to explain the dip are inconsistent with the experiments of Ref. [@smid].
See, e.g., J. Eibl, H. Landahl, U. Haussler and W. Gladen, [*Beton-und Stahlbetonbau*]{} [**77**]{}, 104 (1982); J. Eibl and F. Dahlhaus, Proc. 3rd Euro. Symp.: Storage and Flow of Particulate Solids, PARTEC Nurnberg, 219 (1995); J. Y. Ooi and J. M. Rotter, [*Computers and Structures*]{}, [**37**]{}, 361 (1990). J. Y. Ooi, J. F. Chen, R. A Lohnes and J. M. Rotter, [*Construction and Building Materials*]{}, [**10**]{}, 109 (1996); J. M. Aribert and E. Ragneau; Second European Symposium on Stress and Strain in Particulate Solids, CHISA 90, Paper No 1669, Prague (1990).
See, e.g., J. M. Rotter, J. Y. Ooi, J. F. Chen, P. J. Tiley, I. Mackintosh and F. R. Bennet, [*Flow Pattern Measurement in Full Scale Silos*]{}, British Materials Handling Board Publication, 230 pp (1995) (ISBN 094 6637 091); J. Y. Ooi and J. M. Rotter, Proc. Int. Conf. Bulk Materials – Towards the Year 2000, p. 181 (Inst. Mech. Eng., London, 1991); J. Y. Ooi, W. C. Soh, Z. Zhong and J. M. Rotter, Proc. Int. Symp.: Reliable Flow of Particulate Solids II, EFchE Pubs. Ser. No. 96, p.75 (1993).
See e.g.: Y. C. Chen, I. Ishibashi and J. T. Jenkins, [*Geotechnique*]{}, [**38**]{}, 25, 33 (1988) and references therein; [*IUTAM Symp. on Deformation and Failure of Granular Materials*]{}, P. A. Vermeer and H. J. Luger (Eds.), Balkema, Rotterdam (1982); L. Rothenberg, R. J. Bathurst and M. B. Duesseault, in [*Powders and Grains*]{}, Biarez and Gourves, (Eds.), Balkema, Rotterdam (1989).
J. P. Wittmer, P. Claudin, M. E. Cates and J.-P. Bouchaud, [*An explanation of the central stress minimum in sandpiles*]{}, Nature, submitted.
A simple method for determining the stress components in two dimensions on a particular plane through a point and comparing them with the yield criterion is known as [*Mohr’s circle*]{} [@nedderman]. Unfortunately, this method forces an unusual sign convention we do not wish to introduce here.
C. A. Coulomb, [*Mem. de Math. de l’Acad. Royale des Sciences*]{} [**7**]{}, 343 (1776).
The tilt axis $\Psi$ exactly bisects the angle between the free surface and the vertical (as proved in Sec. 2.2).
It turns out that discontinuous derivatives can arise for three values of $S$: at the outer slope ($S= 1$); on the central axis of the pile ($S=0$); and the point $S=S_0$ where the stress information from the apex of the pile reaches the bottom; examples can be seen in Fig.2(a).
Obviously one can equally write the scaling ansatz eqn. (\[eqonlyH\]) as $\sigma = g R Q(\theta)$ where $R$ is the radius measured from the top of the pile and $\theta$ is the angle measured clockwise from the (downwards pointing) $z$-axis (see Fig.1(c)). This yields the trivial identification $Q(\theta) = g(S)/\sqrt{1+(cS)^2}$.
Note that the RSF scaling, while allowing (in two dimensions) a distinction between the construction histories of elements in the right and left halves of the pile, does not permit any distinction between the elements [*within*]{} one half.
It might seem that the problem is overdetermined, but two equations (the first of the eqns. (\[eqabdonslope\]) and eqn. (\[eqby2a\])) are automatically identical in a radial stress field.
H. A. Janssen, [*Z. Vert. Dt. Ing.*]{} [**39**]{}, 1045 (1895) .
In the FPA model the tilted axes must be chosen so as to coincide with the principal axes themselves.
The IFE model is [*not*]{} a special case of OSL; however it turns out that the term $\cot(\phi)U$ in eqn. (\[eqgrrIFE\]) can be checked [*a posteriori*]{} to be rather small for all $S$ and $\phi$. Accordingly, the IFE is well approximated by ${\mbox{${\large C}$}}(U) = \eta + \lambda U^2
$ with no term linear in $U$, and $\lambda >0$ (leading to a quadratic hump, as discussed by BCC). This means that IFE is in some sense “close" to the BCC model on the $\eta,\mu$ plane (hence its smooth behaviour at the central axis) even though it lies outside the OSL parameterization.
Though an analytic proof should be possible, we have so far only checked this numerically.
The “Coulomb yield criterion” is the frictional analogue of Tresca’s criterion [@Wood]. The frictional analogue of von Mises’ criterion is the “Conical yield criterion”. For a two dimensional problem both criteria are identical. In three dimensions, the failure conditions predicted by these two criteria only differ slightly and the stresses (calculated for instance with an IFE assumption) are predicted to be similar [@nedderman].
One argument for this is that grains at the surface cannot be pressed together more in $\chi$ direction than in $r$-direction: ${\mbox{$s_{rr}$}}(1) \geq
{\mbox{$s_{\chi \chi}$}}(1)$.
We are grateful to Prof. V. Entov for suggesting this thought-experiment.
J. M. Rotter, private communication.
S. R. Nagel, [*Rev. Mod. Phys*]{},. [**64**]{}, 321 (1992).
|
---
abstract: |
We construct a certain $\F_2$-valued analogue of the mixed volume of lattice polytopes. This 2-mixed volume cannot be defined as a polarization of any kind of an additive measure, or characterized by any kind of its monotonicity properties, because neither of the two makes sense over $\F_2$. In this sense, the convex-geometric nature of the 2-mixed volume remains unclear.
On the other hand, the 2-mixed volume seems to be no less natural and useful than the classical mixed volume – in particular, it also plays an important role in algebraic geometry. As an illustration of this role, we obtain a closed-form expression in terms of the 2-mixed volume to compute the signs of the leading coefficients of the resultant, which were by now explicitly computed only for some special cases.
author:
- 'Arina Arkhipova[^1], Alexander Esterov'
title: Signs of the Leading Coefficients of the Resultant
---
[**Key words:** Convex geometry, algebraic geometry, tropical geometry, mixed volumes, Newton polyhedra, resultants]{}\
[**Introduction**]{}\
The classical resultant was initially studied by Sylvester (1853), and later extended to the case of a system of $n$ homogeneous polynomials in $n$ variables by Cayley (1948) and Macaulay (1902). In the 1990s, the advances in several fields, such as symbolic algebra and multivariate hypergeometric functions, revived the interest in resultants. Sparse resultants were introduced and studied by Agrachev, Gelfand, Kapranov, Zelevinsky, and Sturmfels (see e.g. [@gkz]). In particular, in [@sturmfels], Sturmfels gives an explicit combinatorial construction of the Newton polytope of the sparse resultant, and proves that the leading coefficient of the resultant with respect to an arbitrary monomial order is equal to $\pm1$. However, the signs of such coefficients have been computed explicitly only for some special cases so far, although the general answer might be useful for the purposes of real algebraic geometry.
In our work, we construct the [*2-mixed volume*]{} (Definition \[defmv2\]), which is an analogue of the classical mixed volume of convex lattice polytopes taking values in $\F_2$. Besides that, we express the signs of the leading coefficients of the sparse resultant in terms of the 2-mixed volume of certain tuples of polytopes (Theorem \[restheo\]).
The 2-mixed volume is a symmetric and multilinear function of lattice polytopes (Proposition \[propmv2\]). However, its convex-geometric nature remains unclear, because we cannot define it as a polarization of any kind of an additive measure, or characterize it by any kind of its monotonicity properties.
Our explicit formula for the 2-mixed volume employs the so-called 2-determinant, that is, the unique nonzero multilinear function of $n+1$ vectors in the $n$-dimensional vector space over the field $\F_2$ which ranges in $\F_2$, remains invariant under all linear transformations, and equals zero whenever the rank of the $n+1$ vectors is less than $n$. This function implicitly appeared in the context of the class field theory for multidimensional local fields by Parshin and Kato (see e.g. Remark 1 in Section 3.1 of [@parshin], which is probably the first occurence of the 2-determinant in the literature). Later this notion was explicitly introduced in full generality by A.Khovanskii in [@det2] for the purpose of his multidimensional version of the Vieta formula (i.e. the computation of the product in the group $\CC^n$ of all the roots for a system of $n$ polynomial equations with sufficiently generic Newton polytopes).
The algebro-geometric part of our work is an extension of related results by A. Khovanskii. In particular, our notion of the 2-mixed volume is the result of our effort to provide an invariant interpretation of the sign in Khovanskii’s multivariate version of the Vieta formula, and to relax the genericity assumptions on the Newton polytopes in this formula.
The convex-geometric part of our work employs the techniques of tropical geometry to prove the existence of the 2-mixed volume (Theorem \[main\]).\
[**Structure of the paper**]{}\
The paper is organized as follows. In Section 1, we recall some necessary facts and notation concerning convex and tropical geometry.
Section 2 is devoted to the notion of the [*2-mixed volume*]{}. First, we recall the definition and the basic properties of the [*2-determinant*]{}, and use it to define the so-called [*2-intersection number*]{} of tropical hypersurfaces. Then we show that the 2-intersection number depends only on the Newton polytopes of the hypersurfaces, which yields a well-defined function of lattice polytopes — the so-called [*2-mixed volume*]{}.
Section 3 concerns the multivariate Vieta’s formula which expresses the product of roots for a polynomial system of equations in terms of the 2-mixed volume of its Newton polytopes.
In Section 4, we first recall the definition of the sparse mixed resultant, then we compute the signs of the leading coefficients of the resultant reducing this problem to finding the product of roots for a certain system of equations (see Subsection \[SignsRes\], Theorem \[restheo\]).\
Preliminaries
=============
Some Definitions, Notation and Basic Facts
------------------------------------------
Here we introduce some basic notation, definitions and facts that will be used throughout this paper. For more details, we refer the reader to the works [@bernstein], [@EsKhov] and [@tropsturmfels].
### Laurent Polynomials and Newton Polytopes
A [*Laurent polynomial*]{} $f(x_1,\ldots, x_n)$ in the variables $x_1,\ldots, x_n$ over a field $\F$ is a formal expression $$f(x_1,\ldots, x_n)=\sum_{(a_1,\ldots, a_n)\in\Z^n} c_{a_1,\ldots, a_n} x_1^{a_1}\ldots x_n^{a_n},$$ where the coefficients $c_{a_1,\ldots, a_n}$ belong to $\F$ and only finitely many of them are non-zero. We denote the ring of Laurent polynomials in $n$ variables with coefficients in $\F$ by $\F[x_1^{\pm 1},\ldots, x_n^{\pm 1}]$.
Throughout this paper, we use the multi-index notation, i.e., for $a=(a_1,\ldots, a_n)\in \Z^n$, instead of $c_{a_1,\ldots, a_n} x_1^{a_1}\ldots x_n^{a_n}$, we will use the expression $c_a x^a$.
Let $f(x)=\sum_{a\in \Z^n} c_a x^a$ be a Laurent polynomial. The [*support of $f$*]{} is the set ${\mathop{\rm supp}\nolimits}(f)\subset \Z^n$ which consists of all points $a\in \Z^n$ such that the corresponding coefficient $c_a$ of the polynomial $f$ is non-zero.
The [*Newton polytope of $f(x)$*]{} is the convex hull of ${\mathop{\rm supp}\nolimits}(f)$ in $\R^n$, i.e., the minimal convex lattice polytope in $\R^n$ containing the set ${\mathop{\rm supp}\nolimits}(f)$. We denote the Newton polytope of a polynomial $f(x)$ by $\newton(f)$.
\[minkowski\] For a pair of subsets $A, B\subset\R^n$, their [*Minkowski sum*]{} is defined to be the set $A+B=\{a+b\mid a\in A, b\in B\}$.
The following important fact provides a connection between the operations of the Minkowski addition and multiplication in the ring of Laurent polynomials.
For a pair of Laurent polynomials $f,g$, we have the following equality: $$\newton(fg)=\newton(f)+\newton(g).$$
\[suppface\] Let $A\subset\R^m$ be a convex lattice polytope and $\ell\in(\R^*)^m$ be a covector. Consider $\ell$ as a linear function, and denote by $\ell\mid_A$ its restriction to the polytope $A$. The function $\ell\mid_A$ attains its maximum at some face $\Gamma\subset A$. This face is called [*the support face*]{} of the covector $\ell$ and is denoted by $A^{\ell}$.
### Cones and Fans
A [*polyhedral cone*]{} $C$ in $\R^n$ is the positive hull of a finite subset $S=(s_1,\ldots, s_m)\subset\R^n$, i.e., $C=\{\sum_{i=1}^m \lambda_i s_i\mid \lambda_i\geqslant 0\}$.
Let $C\subset\R^n$ be a polyhedral cone. A face of $C$ is the intersection $\{l=0\}\cap C$ for a linear function $l:\R^n\to\R$ such that $C\subset \{l\geqslant0\}$.
A [*polyhedral fan*]{} $\Sigma$ in $\R^n$ is a collection of polyhedral cones which satisfies the following properties: every face of every cone from $\Sigma$ is an element of $\Sigma$, and for any pair of cones $S_1, S_2\in\Sigma$, $S_1\cap S_2$ is a face for both $S_1$ and $S_2$.
Throughout this paper, will mostly deal with the following special case of a polyhedral fan.
For a convex polytope $A\subset R^m$ and a face $\Gamma\subset A$, we define the [*normal cone*]{} $N(\Gamma)$ to be the union of all the covectors $\ell\in (\R^*)^m$ such that the support face $A^{\ell}\subset A$ (see Definition \[suppface\]) contains $\Gamma$.
Then, the [*normal fan*]{} of the polytope $A$ is the collection $N(A)=\{N(\Gamma)\,|\,\Gamma\subset A\}$ over all the faces $\Gamma\subset A$.
Let $\Sigma$ be a polyhedral fan in $\R^n$. The [*support*]{} ${\mathop{\rm supp}\nolimits}(\Sigma)$ of $\Sigma$ is the union of all of its cones.
Let $\Sigma_1, \Sigma_2$ be polyhedral fans. The [*common refinement*]{} $\Sigma_1\wedge \Sigma_2$ is defined to be the fan consisting of all the intersections $C_1\cap C_2$, where $C_i\in \Sigma_i$.
Let $P=(P_1,\ldots, P_m)$ be a tuple of polytopes in $\R^n$. A fan $\Sigma$ is said to be [*compatible*]{} with the tuple $P$, if each of its cones is contained in a cone of the common refinement ${N(P_1)\wedge N(P_2)\wedge\ldots\wedge N(P_m)}$ of the normal fans of the polytopes in $P$.
The following statement relates the Minkowski sums (see Definition \[minkowski\]) to normal fans.
Let $P, Q\subset\R^n$ be polytopes. Then the following equality holds: $$N(P)\wedge N(Q)=N(P+Q).$$
### The Mixed Volume and the Bernstein–Kushnirenko Formula
Let $\gamma\neq 0$ in $(\R^*)^n$ be a covector and $f(x)$ be a Laurent polynomial with the Newton polytope $\newton(f)$. The [*truncation*]{} of $f(x)$ with respect to $\gamma$ is the polynomial $f^{\gamma}(x)$ that can be obtained from $f(x)$ by omitting the sum of monomials which are not contained in the support face ${\newton(f)^{\gamma}}$.
It is easy to show that for a system of equations $\{f_1(x)=\ldots=f_n(x)=0\}$ and an arbitrary covector $\gamma\neq 0$, the system $\{f_1^{\gamma}(x)=\ldots=f_n^{\gamma}(x)=0\}$ by a monomial change of variables can be reduced to a system in $n-1$ variables at most. Therefore, for the systems with coefficients in general position, the “truncated" systems are inconsistent in $\CC^n$.
Let $(f_1,\ldots, f_n)$ be a tuple of Laurent polynomials. In the same notation as above, the system ${f_1(x)=\ldots=f_n(x)=0}$ is called [*degenerate at infinity*]{}, if there exists a covector $\gamma\neq 0$ such that the system $\{f_1^{\gamma}(x)=\ldots=f_n^{\gamma}(x)=0\}$ is consistent in $\CC^n$.
\[mv\] Let $\mathscr P$ be the semigroup of all convex polytopes in $\R^n$ with respect to the Minkowski addition (see Definition \[minkowski\]). The [*mixed volume*]{} is a unique function $${\mathop{\rm MV}\nolimits}\colon\underbrace{\mathscr{P}\times\ldots\times\mathscr{P}}_{\mbox{$n$ times}}\to\R$$ which symmetric, multilinear (with respect to the Minkowski addition) and which satisfies the following property: the equality $MV(P,\ldots, P)={\mathop{\rm Vol}\nolimits}(P)$ holds for every polytope ${P\in \mathscr P}$.
The following theorem allows to compute the number of roots for a non-degenerate polynomial system of equations in terms of the mixed volume of its polynomials.
The number of roots for a polynomial system of equations $\{f_1(x)=\ldots=f_n(x)=0\}$ in $\CC^n$ that is non-degenerate at infinity counted with multiplicities is equal to $n!{\mathop{\rm MV}\nolimits}(\newton(f_1),\ldots, \newton(f_n))$.
A Very Little Bit of Tropical Geometry
--------------------------------------
We define the [*tropical semifield*]{} $\T=\R \cup \{-\infty\}$ to be the set of real numbers with $-\infty$ equipped with the following arithmetic operations: $$\alpha\oplus \beta=\begin{cases}
\max(\alpha,\beta),&\text{if $\alpha\neq \beta$;}\\
[-\infty,\alpha],&\text{if $\alpha=\beta$;}
\end{cases}$$ $$\alpha\odot \beta = \alpha+\beta.$$
Tropical addition and multiplication have the identity elements: ${{\mbox{\bfseries \large 0}}}=-\infty$ and ${{\mbox{\bfseries\large 1}}}=0$.
It is easy to check the following facts:
- $\T$ is a commutative multivalued semigroup (a hypersemigroup) with respect to addition;
- $\T\setminus\{-\infty\}$ is a commutative group with respect to multiplication;
- in $\T$ we have the distribution law: $\forall \alpha,\beta,\gamma \in \T~ \alpha\odot (\beta\oplus \gamma) = \alpha\odot\beta\oplus\alpha\odot\gamma$.
Having defined tropical arithmetic operations, we can consider tropical polynomials.
\[troppol\] Let $A \subset \Z^n$ be finite and ${\forall a \in A} ~c_a \in \T$. Then a [*tropical polynomial*]{} is given by $$f(x) =\bigoplus_{a\in A} c_a\odot x^{\odot a},$$ where $x\in \T^n$.
In the notation of \[troppol\], the [*support*]{} of the polynomial $f(x)$ is the set ${\mathop{\rm supp}\nolimits}(f)=\{a\in A\mid c_a\neq {{\mbox{\bfseries \large 0}}}\}$ (see \[troppol\]). The [*Newton polytope*]{} $\newton(f)$ of $f(x)$ is defined as the convex hull of ${\mathop{\rm supp}\nolimits}(f)$ in $\R^n$. We denote by $\vert supp(f)\vert$ and $|\newton(f)|$ the cardinality of the set ${\mathop{\rm supp}\nolimits}(f)$ and $\newton(f)\cap\Z^n$, respectively.
We denote by $=_{\mathbb T}$ [*the multivalued equality sign*]{}: $f(x_0)=_{\mathbb T}{{\mbox{\bfseries \large 0}}}$ for some $x_0\in \T^n$ means that ${{{\mbox{\bfseries \large 0}}}=-\infty}$ belongs to the image $f(x_0)$.
\[weight1\] Let $f(x)=\bigoplus_{a\in A} c_a\odot x^{\odot a}$ be a tropical polynomial. Consider the set ${H=\{x_0\in \T^n\mid f(x_0) =_{\T}{{\mbox{\bfseries \large 0}}}\}}$ (or, equivalently, the set of points $x_0\in \T^n$ such that there exist $a_1\neq a_2 \in A$ satisfying $c_{a_1}+x_0\cdot a_1=c_{a_2}+x_0\cdot a_2=\max f(x_0)$).
A point ${x_0\in~H}$ is said to be [*smooth*]{} if the points $a_1$ and $a_2$ above are uniquely defined (up to the transposition). In this case, the [*weight*]{} of $H$ at $x_0$ is defined as the integer length of the vector $a_1-a_2$ (i.e. the g.c.d. of its coordinates).
\[trophyp1\] The [*tropical hypersurface*]{} defined by $f$ is the set $H$ whose facets (i.e., the connected components of the smooth part of $H$) are equipped with weights (see \[weight1\]).
As a set, the tropical hypersurface $f =_{\T}{{\mbox{\bfseries \large 0}}}$ is just the set of points where $f$ is not smooth, i. e., the corner locus of this convex piecewise linear function.
See, for example, [@trop1] and [@tropsturmfels] for a more detailed introduction into tropical geometry.
The local support set of a polynomial $f(x)=\bigoplus_{a\in A} c_a\odot x^{\odot a}$ at a point $x_0$ is the set ${\mathop{\rm supp}\nolimits}_{x_0}(f)$ of all $a$ such that $c_{a}+x_0\cdot a=\max f(x_0)$. The local Newton polytope $\newton_{x_0}(f)$ is the convex hull of the local support set ${\mathop{\rm supp}\nolimits}_{x_0}(f)$.
Note that $x_0$ belongs to the hypersurface $H=\{f=_{\T}{{\mbox{\bfseries \large 0}}}\}$ if and only if ${\mathop{\rm supp}\nolimits}_{x_0}(f)$ consists of more than one point, and is smooth if ${\mathop{\rm supp}\nolimits}_{x_0}(f)$ consists of two points. Also note that the local Newton polytope (in contrast to the local support set) depends only on the hypersurface $H$, and not on its defining equations $f$, so we shall also denote it by $\newton_{x_0}(H)$.
2-mixed volume {#2-volume}
==============
This Section is devoted to the notion of the [*2-mixed volume*]{} of lattice polytopes. In Subsection \[defdet2\], the definition and some basic properties of the 2-determinant are provided. Subsection \[2ind\] concerns the definition of the [*2-intersection number*]{} for a tuple of tropical hypersurfaces. In Subsections \[ideamain\], \[buildwall\] and \[passwall\] we prove Theorem \[main\] stating that under some assumptions on the Newton polytopes of the tropical hypersurfaces, the 2-intersection number depends not on the hypersurfaces, but on their Newton polytopes, which yields a well-defined function of lattice polytopes that takes values in $\mathbb F_2$ – the [*2-mixed volume*]{}.
Analog of the determinant for $n+1$ vectors in an $n$-dimensional space over the field $\mathbb F_2$ {#defdet2}
----------------------------------------------------------------------------------------------------
We define $\det_2$ to be the function of $n+1$ vectors in an $n$-dimensional linear space over $\mathbb F_2$, that takes values in $\mathbb F_2$ and satisfies the following properties:
- $\det_2(k_1,\ldots,k_{n+1})$ is equal to zero, if the rank of the collection of vectors $k_1,\ldots,k_{n+1}$ is smaller than $n$;
- $\det_2(k_1,\ldots,k_{n+1})$ is equal to $\lambda^1+\ldots+\lambda^{n+1}+1$, if the vectors $k_1,\ldots,k_{n+1}$ are related by the unique relation $\lambda^1k_1+\ldots+\lambda^{n+1}k_{n+1}=0$.
\[det2\] The function $\det_2$
1. is $\mathrm GL_n(\mathbb F_2)$-invariant, i.e. for any linear transformation $A\in\mathrm GL_n(\mathbb F_2)$ the equality $\det_2(k_1,\ldots,k_{n+1})=\det_2(Ak_1,\ldots,Ak_{n+1})$ holds;
2. is multilinear.
[@det2] There exists a unique nonzero function $\det_2$ which satisfies the properties of g \[det2\].
[@det2]\[formuladet2\] In coordinates the function $\det_2$ can be expressed by the formula $${{\mathop{\rm {det}}\nolimits}_2}(k_1,\ldots,k_{n+1})=\sum_{j>i}\Delta_{ij},$$ where $\Delta_{ij}$ is the determinant of the $n \times n$ matrix whose first $n-1$ columns represent the sequence of vectors $k_1,\ldots k_{n+1}$ from which the vectors with the indices $i$ and $j$ are deleted, and the last column is the coordinate-wise product of the vectors $k_i$ and $k_j$.
Let $k_1,\ldots, k_{n+1}$ be a tuple of vectors such that ${\mathop{\rm rk}\nolimits}(k_1,\ldots, k_{m+1})=m$ for some $m<n$. Then, there exists a natural projection $\pi\colon\Z^n\to\Z^n/\langle k_1,\ldots, k_{m+1}\rangle$. The statement below easily follows from Theorem \[formuladet2\] and the well-known formula for computing the upper triangular block matrix determinant.
\[blockdet2\] In the same notation as above, the following equality holds: $${\mathop{\rm {det}}\nolimits}_2(k_1,\ldots, k_{n+1})={\mathop{\rm {det}}\nolimits}_2(k_1,\ldots,k_{m+1})\cdot\det(\pi(k_{m+2}),\ldots,\pi(k_{n+1})).$$
2-intersection number {#2ind}
---------------------
Let $H_1, \ldots, H_n$ be tropical hypersurfaces. We say that $H_1, \ldots, H_n$ intersect [*transversely*]{} (denote by $H_1\pitchfork \ldots\pitchfork H_n$), if $|H_1\cap H_2\cap\ldots \cap H_n|<\infty$ and all the points $x\in H_1\cap H_2\cap \ldots \cap H_n$ are smooth for every $H_i$ (see Definition \[weight1\]).
\[1-index\] Let $H_1\pitchfork \ldots\pitchfork H_n$ be a transverse tuple of tropical hypersurfaces. The [*intersection number*]{} $\iota(H_1,\ldots, H_n)\in \Z$ is the sum $$\label{eq2ind}
\iota(H_1,\ldots, H_n)\stackrel{\mathrm{def}}{=}\sum\limits_{x\in H_1\cap H_2\cap \ldots \cap H_n} {\mathrm{det}}(\newton_x(H_1),\ldots \newton_x(H_n)).$$
It is well known that the intersection number of tropical hypersurfaces depends only on their Newton polytopes (and coincides with the mixed volume of the Newton polytopes). This fact is often referred to as the tropical Bernstein–Kushnirenko formula. We shall need the following $\F_2$-verison of the intersection number.
\[2-index\] Consider an arbirtary point $\zeta\in~\mathbb Z^n$. Let $H_1\pitchfork \ldots\pitchfork H_n$ be a transverse tuple of tropical hypersurfaces. We define the [*2-intersection number*]{} $\iota_2(H_1,\ldots, H_n;\zeta)\in \F_2$ as follows: $$\label{eq2ind}
\iota_2(H_1,\ldots, H_n;\zeta)\stackrel{\mathrm{def}}{=}\sum\limits_{x\in H_1\cap H_2\cap \ldots \cap H_n} {\mathrm{det}_2}(\newton_x(H_1),\ldots \newton_x(H_n),\zeta).$$
Unfortunately, in general, the 2-intersection number does depend on the tropical hypersurfaces, and not only on their Newton polytopes. However, this dependence disappears if the Newton polytopes themselves are in general position in a sense that we describe below.
Let $P\subset \mathbb R^n$ be a polytope or a finite set. We define the [*support face*]{} of a covector $v\in (\R^n)^*$ to be the maximal subset of $P$ on which $v\mid_P$ attains its maximum. We shall denote this face by $P^v$.
A finite set $P\subset \Z^n$ is called a [*$2$-vertex*]{}, if for any pair of points $p_1, p_2\in P,~p_1\equiv p_2 {\ (\text{mod}\ 2)}$ (i.e., the corresponding coordinates of the points $p_1, p_2$ are of the same parity). A lattice polytope is called a [*$2$-vertex*]{}, if the set of its vertices is a [*$2$-vertex*]{}.
\[devwrt\] Let $P_1, \ldots, P_n$ be convex lattice polytopes in $\R^n$ or finite sets in $\Z^n$, and $\zeta$ be a point in $\Z^n$. The tuple $P_1,\ldots, P_n$ is said to be [*2-developed with respect to*]{} $\zeta$ if, for any covector $v\in(\Z^n)^*$ such that $v(\zeta)\not\equiv 0 \mod 2$, there exists $i\in \{1,\ldots, n\}$ such that the support face $P_i^v$ is a $2$-vertex.
\[prickly\] A tuple $P=(P_1,\ldots, P_n)$ of convex lattice polytopes is said to be [*$\zeta$–prickly*]{}, if for any covector $v\in (\R^*)^n$ such that $v(\zeta)\neq 0$, there exists $i\in \{1,\ldots, n\}$ such that the support face $P_i^{v}$ is a vertex.
Obviously, if a tuple $P$ is $\zeta$–prickly, then it is $2$–developed with respect to $\zeta$.
\[main\] Consider a point $\zeta\in\Z^n$ and finite lattice sets $P_1, \ldots, P_n$. Suppose that $P_1, \ldots, P_n$ are 2-developed with respect to $\zeta$. Then for any two tuples $(H_1,\ldots, H_n)$ and $(H'_1,\ldots, H'_n)$ of tropical hypersurfaces, whose equations are supported at $P_1, \ldots, P_n$, the 2-intersection numbers $\iota_2(H_1,\ldots, H_n;\zeta)$ and $\iota_2(H'_1,\ldots, H'_n;\zeta)$ coincide.
\[defmv2\] For a tuple of polytopes $P_1,\ldots,P_n$, 2-developed with respect to $\zeta\in\Z^n$, consider generic tropical hypersurfaces $H_1,\ldots,H_n$, such that the equation of $H_i$ is supported at the set of vertices of $P_i$. Then the function ${{\mathop{\rm MV}\nolimits}_2\colon(P_1,\ldots,P_n;\zeta)\mapsto\iota_2(H_1,\ldots, H_n;\zeta)}$ is well-defined. We call it the [*2-mixed volume*]{}.
\[propmv2\] The function ${\mathop{\rm MV}\nolimits}_2$ is symmetric and multiplinear with respect to the Minkowski summation of the arguments.
$\vartriangleleft$ The symmetry is obvious. In order to prove the additivity ${\mathop{\rm MV}\nolimits}_2(P,P_2,\ldots,P_n;\zeta)+{\mathop{\rm MV}\nolimits}_2(Q,P_2,\ldots,P_n;\zeta)={\mathop{\rm MV}\nolimits}_2(P+Q,P_2,\ldots,P_n;\zeta)$ whenever the summands make sense, chose generic tropical polynomials $p,q,p_2,\ldots,p_n$ with the Newton polytopes $P,Q,P_2,\ldots,P_n$ respectively. Then the 2-intersection numbers in the tautological equality $\iota_2(p=0,p_2=0,\ldots,p_n=0;\zeta)+\iota_2(q=0,p_2=0,\ldots,p_n=0;\zeta)=\iota_2(p\cdot q=0,p_2=0,\ldots,p_n=0;\zeta)$ make sense and equal the corresponding 2-mixed volumes. $\vartriangleright$
The Idea of the Proof of Theorem \[main\] {#ideamain}
-----------------------------------------
A tropical polynomial $\varphi$ corresponds to a point in $\T^{\vert{\mathop{\rm supp}\nolimits}(\varphi)\vert}$. Namely, to every tropical polynomial we associate the collection of its coefficients. Therefore, tropical hypersurfaces defined by tropical polynomials with some fixed support $A\subset\Z^n$ can be considered as points in $\T^{\vert A\vert}$.
Thus, given a tuple $(A_1,\ldots, A_n)$ of finite sets in $\Z^n$, we can consider tuples $(H_1,\ldots,H_n)$ of tropical hypersurfaces defined by tuples of tropical polynomials $(\varphi_1, \ldots, \varphi_n)$ such that ${\mathop{\rm supp}\nolimits}(\varphi_i)=A_i, 1\leqslant i\leqslant n,$ as points in the space $$\EuScript{M}=\prod_1^n \T^{\vert A_i\vert}.$$
By $\EuScript{S}_0\subset\EuScript{M}$ denote the set of all transverse tuples of hypersurfaces. Obviously, the set $\EuScript{S}_0$ is open and everywhere dense.
\[constS0\] The 2-intersection number $\iota_2(H_1,\ldots, H_n;\zeta)$ defined in $\ref{2-index}$ is constant on the connected components of $\EuScript{S}_0$.
$\vartriangleleft$ Suppose that the points $(H_1,\ldots, H_n)$ and $(H'_1,\ldots, H'_n)$ belong to the same connected component of $\EuScript{S}_0$. Then, there exists a one-to-one correspondence between the sets ${H_1\cap H_2\cap \ldots \cap H_n}$ and $H'_1\cap H'_2\cap \ldots \cap H'_n$, which maps every $x\in H_1\cap H_2\cap \ldots \cap H_n$ to the point $x'\in H'_1\cap H'_2\cap \ldots \cap H'_n$ such that $\newton_x(H_i)=\newton_{x'}(H'_i)$ for every $i\in\{1,\ldots, n\}$. Therefore, the corresponding $2$-determinants in the right-hand side of (\[eq2ind\]) coincide, which implies the sought equality $\iota_2(H_1,\ldots, H_n;\zeta)=\iota_2(H'_1,\ldots, H'_n;\zeta)$. $\vartriangleright$
The next step is to construct a codimension 1 set $\EuScript{S}_1\subset\EuScript{M}\setminus\EuScript{S}_0$ of “almost transverse” tuples of tropical hypersurfaces such that passing from one connected component of $\EuScript{S}_0$ to another through the points of $\EuScript{S}_1$ does not change the $2$-intersection number, and ${\mathop{\rm codim}\nolimits}(\EuScript{M}\setminus(\EuScript{S}_0\cup\EuScript{S}_1))\geqslant 2$. Theorem \[main\] will then follow, since the complement to a codimension 2 subset is connected.
We shall prove below that the pieces of the sought set $\EuScript{S}_1$ are in one–to– one correspondence with certain combinatorial structures of the form $(I_i,\, i\in I)$, $I\subset\{1,\ldots,n\}, I_i\subset A_i$, that we call elementary obstacles.
A collection of numbers $I\subset\{1,\ldots,n\}$ and pairs of points $I_i\subset A_i,\, i\in I$, is called an [*elementary obstacle of type 1*]{}, if the convex hull of the Minkowski sum $\sum_{i\in I} I_i$ has dimension $|I|-1$, and no proper subcollection $I'\subset I,\, I_i,\, i\in I'$, is an elementary obstacle of type 1.
A collection of numbers $I\subset\{1,\ldots,n\}$, a triple of points $I_j\in A_j,\, j\in I$, and pairs of points $I_i\subset A_i,\, i\in I\setminus\{j\}$, is called an [*elementary obstacle of type 2*]{}, if the convex hull of $I_j$ is a triangle, the convex hull of the Minkowski sum $\sum_{i\in I} I_i$ has dimension $|I|$, and no proper subcollection $I'\subset I,\, I_i,\, i\in I'$, is an elementary obstacle of type 1 or 2.
A one element collection $I=\{i\}$ and a triple of points $I_i\subset A_i$ is called an [*elementary obstacle of type 3*]{}, if the convex hull of $I_i$ is a segment.
Below are shown all possible elementary obstacles in dimension 2.
(-2,-1)–(-2,1); (-0.5,-1)–(-0.5,1); (-2,1) circle\[radius=0.15\]; (-2,-1) circle\[radius=0.15\]; (-0.5,1) circle\[radius=0.15\]; (-0.5,-1) circle\[radius=0.15\]; (3,-1)–(3,1)–(5,-1)–(3,-1); (3,1) circle\[radius=0.15\]; (3,-1) circle\[radius=0.15\]; (5,-1) circle\[radius=0.15\]; (6.5,-1)–(8.5,1); (6.5,-1) circle\[radius=0.15\]; (8.5,1) circle\[radius=0.15\]; (12,0)–(16,0); (12,0) circle\[radius=0.15\]; (14,0) circle\[radius=0.15\]; (16,0) circle\[radius=0.15\];
Figure 1. Elementary obstacles of types 1, 2 and 3 in dimension 2.
We say that tropical hypersurfaces $H_1,\ldots,H_n$ defined by tropical polynomials $\varphi_1,\ldots,\varphi_n$ have a non-transversality of type $k$, if there exists an elementary obstacle $I_i\subset A_i,\, i\in I$, of type $k$, such that for some point $x\in H_1\cap\ldots\cap H_n$ we have ${\mathop{\rm supp}\nolimits}_x(\varphi_i)=I_i,\, i\in I$.
The rest of Section 2 will be spent to observe that generic tuples of hypersurfaces with a non-transversality of one of the three types form the sought set $\EuScript{S}_1$. In other words, if we travel between two tuples of transversal hypersurfaces along a generic path in $\EuScript{M}$, then we shall encounter finitely many generic tuples with a non-transversality of type $k$, and passing through them will not change the 2-intersection number. Thus, passing through generic tuples with a non-transversality of type $k$ plays the role of Reidemeister moves in knot theory. The figure below shows all such “Reidemeister moves” in dimension two.
(-2,0)–(0,0)–(0,-2); (0,0)–(2,2);at (1.5,1.5) [ $C_1$]{}; (-1,2)–(0,1)–(2,-1);at (-1.5,1.2) [ $C_2$]{}; (0.5,0.5) circle\[radius=0.1\]; (-3,2)–(-4,0)–(-3,-2); (-4,0)–(-8.5,0); (-4.5,2.5)–(-5.5,1)–(-4.8,-0.8); (-5.5,1)–(-7,1); (-8,2.5)–(-7,1)–(-8,-1); (-9.5,1.5)–(-8.5,0)–(-9.5,-2); (-5.13,0) circle\[radius=0.1\]; (-7.5,0) circle\[radius=0.1\]; at (-4,1.7) [$C_2$]{}; at (-7.7, 2.2) [$C_1$]{}; (4,0)–(8,0); (6,-2)–(6,2); (6,0) circle\[radius=0.1\]; at (6,1.5) [$C_2$]{}; at (4.5,0) [$C_1$]{};
(-2,0)–(0,0)–(0,-2); (0,0)–(2,2);at (1.5,1.5) [ $C_1$]{}; (-2,2)–(0,0)–(2,-2);at (-1.5,1.5) [ $C_2$]{}; (0,0) circle\[radius=0.1\]; (-3,2)–(-4,0)–(-3,-2); (-4,0)–(-5.5,0); (-4.5,1.5)–(-5.5,0)–(-5,-1.7); (-5.5,0)–(-7,0); (-8,1.5)–(-7,0)–(-7.7,-1.7); (-7,0)–(-8.5,0); (-9.5,1.5)–(-8.5,0)–(-9.5,-2); (-5.5,0) circle\[radius=0.1\]; (-7,0) circle\[radius=0.1\]; at (-4,1.7) [$C_2$]{}; at (-8, 1.5) [$C_1$]{}; (4,0)–(8,0); (6,-2)–(6,2); (6,0) circle\[radius=0.1\]; at (6,1.5) [$C_2$]{}; at (4.5,0) [$C_1$]{};
(-2,0)–(0,0)–(0,-2); (0,0)–(2,2);at (1.5,1.5) [ $C_1$]{}; (-2,1)–(0,-1)–(1,-2);at (-2,1) [ $C_2$]{}; (0,-1) circle\[radius=0.1\]; (-1,0) circle\[radius=0.1\]; (-3,2)–(-4,0)–(-3,-2); (-4,0)–(-8.5,0); (-4.5,0.8)–(-5.5,-0.7)–(-5,-2.4); (-5.5,-0.7)–(-7,-0.7); (-8,0.8)–(-7,-0.7)–(-7.7,-2.4); (-9.5,1.5)–(-8.5,0)–(-9.5,-2); (-5.075,0) circle\[radius=0.1\]; (-7.43,0) circle\[radius=0.1\]; at (-4,1.7) [$C_2$]{}; at (-7.9, 0.5) [$C_1$]{}; (4,0)–(8,0); (5,-2)–(5,2); (7,-2)–(7,2); (5,0) circle\[radius=0.1\]; (7,0) circle\[radius=0.1\]; at (7,1.5) [$C_2$]{}; at (4.5,0) [$C_1$]{};
Figure 2. “Reidemeister moves” of types 1, 2 and 3 in dimension 2.
Building the Walls {#buildwall}
------------------
Here we consider tuples of finite sets in $\Z^n$. Thus the words “simplex” and “interval” mean a set of all vertices of a simplex or an interval, respectively.
In the notation of Theorem \[main\], consider a $n$-tuple $A=(A_1,\ldots, A_n)$ of finite sets in $\Z^n$ such that ${\mathop{\rm conv}\nolimits}(A_i)=P_i, 1\leqslant i\leqslant n$ and the polytopes $P_1,\ldots, P_n$ are 2-developed with respect to a point $\zeta\in\Z^n$. Let $B=(B_1,\ldots, B_n)$ be an arbitrary subtuple $B=(B_1,\ldots, B_n), B_i\subset A_i$ of simplices.
[@sturmfels]\[codim\] For an arbitrary subset $I\subset\{0,\ldots,n\}$, we define its [*codimension*]{}, which we denote by ${\mathop{\rm codim}\nolimits}(I)$, as follows: $${\mathop{\rm codim}\nolimits}(I)=\dim({\mathop{\rm conv}\nolimits}(\sum_{i\in I} B_i))-|I|.$$ The [*codimension of the tuple*]{} $B$ is defined by the following equality: $${\mathop{\rm codim}\nolimits}(B)=\min_{I\subset\{0,\ldots,n\}}{\mathop{\rm codim}\nolimits}(I).$$
Let $\varphi=(\varphi_1,\ldots, \varphi_n)$ be a $n$-tuple of tropical polynomials, where ${\varphi_i=\bigoplus_{a\in A_i} c_{a,i}\odot x^{\odot a}}$. To every point $b\in B_i$ one can associate the point $b^{\varphi}=(b,c_{b,i})\in\Z^n \times\T$. We denote by $B^{\varphi}$ the tuple $(B_1^{\varphi_1}, \ldots, B_n^{\varphi_n})$, where $B_i^{\varphi_i}=\{b^{\varphi_i}\mid b\in B_i\}$.
In the previous notation, given a tuple $B$, by ${L_B}$ we denote the affine subspace consisting of all the points $\varphi\in\EuScript{M}$ such that the following equality holds: $\dim({\mathop{\rm conv}\nolimits}(\sum_1^n B_i^{\varphi_i}))=\dim({\mathop{\rm conv}\nolimits}(\sum_1^n B_i))$.
A subtuple $B\subset A$ is called a [*trouble*]{}, if $L_B\neq \EuScript{M}$ and $|B_i|>1$ for all $i$. We say that a trouble $B$ is an [*obstacle*]{}, if ${\mathop{\rm codim}\nolimits}(L_B)=1$. In this case, we call the hyperplane $L_B$ a [*wall*]{}.
The sets $L_B$ corresponding to each of the troubles $B\subset A$ cover the set $\EuScript{M}\setminus\EuScript{S}_0$. In these terms, the sought codimension 1 set $\EuScript{S}_1\subset\EuScript{M}$ is a set, which contains all the points $\varphi=(\varphi_1,\ldots, \varphi_n)$ such that all the walls containing $\varphi$ coincide and for every trouble $B\subset A$, ${\mathop{\rm codim}\nolimits}(L_B)>1$ implies $\varphi\notin L_B$. In order to obtain the explicit description of the set $\EuScript{S}_1$, we need first to describe and classify the obstacles $B\subset A$.
Let $B\subset A$ be a trouble. It is obvious, that ${\mathop{\rm codim}\nolimits}(B)\leqslant 0$. Consider the following cases:
1. ${\mathop{\rm codim}\nolimits}(B)\leqslant -1$;
2. ${\mathop{\rm codim}\nolimits}(B)=0$.
[**Case 1.**]{} Consider a subtuple $B'\subset B$ of intervals $B'_i\subset B_i$. For every set ${I\subset\{0,\ldots,n\}}$, the inequality $\dim({\mathop{\rm conv}\nolimits}(\sum_{i\in I} B'_i))\leqslant \dim({\mathop{\rm conv}\nolimits}(\sum_{i\in I} B_i))$ holds, therefore ${\mathop{\rm codim}\nolimits}(B')\leqslant{\mathop{\rm codim}\nolimits}(B)$. Let $I_{\min}$ be the minimal subset $I\subset \{1,\ldots, n\}$ such that ${\mathop{\rm codim}\nolimits}(B')={\mathop{\rm codim}\nolimits}(I)$.
The set $I_{\min}$ is non-empty, because ${\mathop{\rm codim}\nolimits}(\varnothing)=0$,\
while ${\mathop{\rm codim}\nolimits}(I_{\min})\leqslant -1$.
By $B''$ we denote the subtuple $(B'_i, i\in I_{\min})$.
It is easy to show that ${\mathop{\rm codim}\nolimits}(B')={\mathop{\rm codim}\nolimits}(B'')$. We obviously have ${\mathop{\rm codim}\nolimits}(B'')\leqslant{\mathop{\rm codim}\nolimits}(B')$. Assume that ${\mathop{\rm codim}\nolimits}(B'')<{\mathop{\rm codim}\nolimits}(B')$. Then, there exists a subset $J\subsetneq I_{\min}$ of such that ${\mathop{\rm codim}\nolimits}(J)<{\mathop{\rm codim}\nolimits}(I_{\min})$, which contradicts with the choice of the subset $I_{\min}$.
\[codim1\] The following equality holds: ${\mathop{\rm codim}\nolimits}(L_{B''})=-{\mathop{\rm codim}\nolimits}(B'')$.
$\vartriangleleft$ Without loss of generality, suppose that $I_{min}={1,\ldots,m}$ for some $m\leqslant n$. Then the tuple $B''$ is a $m$-tuple $B'_1,\ldots, B'_m$ of intervals in $\Z^n$. For every $1\leqslant i\leqslant m$, by $v_i$ we denote the vector $\overrightarrow{B'_i}\in\Z^n$. By $v_{ij}$ we denote the $j$-th component of the coordinate vector of $v_i$ in the standard basis. Consider an arbitrary point $\varphi=(\varphi_1,\ldots, \varphi_n)\in\EuScript{M}$. Let $\alpha_i$ and $\beta_i$ be the coefficients of $\varphi_i$ corresponding to each of the endpoints of the interval $B'_i$. In these terms, the set $L_{B''}$ consists of the points $\varphi\in\EuScript{M}$ such that the following equality holds: $$\label{rankcodim1}
m+{\mathop{\rm codim}\nolimits}(B'')={\mathop{\rm rk}\nolimits}\begin{pmatrix}
v_{11} & \dots & v_{1n} \\
\vdots & \ddots & \vdots \\
v_{m1} & \dots & v_{mn}
\end{pmatrix}={\mathop{\rm rk}\nolimits}\begin{pmatrix}
v_{11} & \dots & v_{1n} & (\beta_1-\alpha_1) \\
\vdots & \ddots & \vdots \\
v_{m1} & \dots & v_{mn} & (\beta_m-\alpha_m)
\end{pmatrix}$$ We can suppose without loss of generality that the first $m+{\mathop{\rm codim}\nolimits}(B'')$ columns span the column space of the first matrix. Then the equality (\[rankcodim1\]) means that the last column of the second matrix can be expressed as their linear combination. Thus the sought codimension of the plane $L_{B''}$ equals ${m-m-{\mathop{\rm codim}\nolimits}(B'')=-{\mathop{\rm codim}\nolimits}(B'')}$. $\vartriangleright$
If ${\mathop{\rm codim}\nolimits}(B)\leqslant -2$, then ${\mathop{\rm codim}\nolimits}(L_B)\geqslant 2$.
Obviously, $L_B\subset L_{B''}$. Therefore, ${\mathop{\rm codim}\nolimits}(L_B)\geqslant {\mathop{\rm codim}\nolimits}(L_{B''})$. Applying \[codim1\], we obtain ${{\mathop{\rm codim}\nolimits}(L_B)\geqslant {\mathop{\rm codim}\nolimits}(L_{B''})=-{\mathop{\rm codim}\nolimits}(B'')\geqslant -{\mathop{\rm codim}\nolimits}(B)\geqslant 2}$.
In the previous notation, if $B$ is an obstacle, and ${\mathop{\rm codim}\nolimits}(B)=-1$, then $L_B=L_{B''}$. In this case, we call $B''$ an [*elementary obstacle*]{} corresponding to the obstacle $B$.
The equalities ${\mathop{\rm codim}\nolimits}(B)=-1$ and ${\mathop{\rm codim}\nolimits}(L_B)=1$ imply\
that $1\leqslant -{\mathop{\rm codim}\nolimits}(B'')={\mathop{\rm codim}\nolimits}(L_{B''})\leqslant 1$. So, ${\mathop{\rm codim}\nolimits}(L_{B''})=-{\mathop{\rm codim}\nolimits}(B'')=1$. Therefore, $1={\mathop{\rm codim}\nolimits}(L_B)\geqslant{\mathop{\rm codim}\nolimits}(L_{B''})=1$, which finishes the proof.
[**Case 2a.**]{} Suppose ${\mathop{\rm codim}\nolimits}(B)=0$ and the convex hull of at least one of $B_i$’s is not a segment. Then we can choose $B'\subset B$ to be a subtuple consisting of a triangle and $(n-1)$ intervals $B'_i\subset B_i$. Let $M_{B'}=\min{\mathop{\rm codim}\nolimits}(I)$, where the minimum is taken over all the subsets $I\subset\{1,\ldots, n\}$ such that the tuple $(B'_i, i\in I)$ contains the triangle. By $I_{\min}$ denote the minimal set such that ${\mathop{\rm codim}\nolimits}(I_{\min})=M_{B'}$ and the tuple $B''=(B'_i, i\in I_{\min})$ contains the triangle.
\[codim2\] In the previous notation, the following equality holds:\
${\mathop{\rm codim}\nolimits}(L_{B''})=-M_{B'}+1.$
$\vartriangleleft$ The proof is almost the same as the one of Proposition \[codim1\]. The only difference is that the triangle gives rise to two vectors instead of one, thus, we will deal with $(m+1)\times n$ and $(m+1)\times (n+1)$-matrices of rank $m+M_{B'}$. Therefore, in this case, the sought codimension equals $m+1-m-M_{B'}=-M_{B'}+1$. $\vartriangleright$
If ${\mathop{\rm codim}\nolimits}(B'')\leqslant -1$, then ${\mathop{\rm codim}\nolimits}(L_B)\geqslant 2$.
$\vartriangleleft$ Using \[codim2\], we have\
${{\mathop{\rm codim}\nolimits}(L_B)\geqslant{\mathop{\rm codim}\nolimits}(L_{B''})
=-{\mathop{\rm codim}\nolimits}(B'')+1\geqslant 2}$, which finishes the proof. $\vartriangleright$
If $B$ is an obstacle, and ${\mathop{\rm codim}\nolimits}(B)=0$, $L_B=L_{B''}$. In this case, $B''$ is called an [*elementary obstacle*]{} corresponding to the obstacle $B$.
$\vartriangleleft$ The equalities ${\mathop{\rm codim}\nolimits}(B)=0$ and ${\mathop{\rm codim}\nolimits}(L_B)=1$ imply that\
$1=-{\mathop{\rm codim}\nolimits}(B)+1\leqslant-M_{B'}+1={\mathop{\rm codim}\nolimits}(L_{B''})\leqslant
{\mathop{\rm codim}\nolimits}(L_B)=1$, which finishes the proof.
Moreover, from the proof, it follows that $M_{B'}=0$. $\vartriangleright$
[**Case 2b.**]{} Suppose ${\mathop{\rm codim}\nolimits}(B)=0$ and the convex hull of each of $B_i$’s is not a segment. If $|B_i|=2$ for all $i$, then ${\mathop{\rm codim}\nolimits}L_B=0$, and if $|B_i|>3$ for some $i$, then ${\mathop{\rm codim}\nolimits}L_B>1$. So, if $B$ is an obstacle, then $|B_i|=3$ for some $i$. In this case we denote the one element subtuple $(B_i)$ by $B''$ and observe that $L_B=L_{B''}$ provided that ${\mathop{\rm codim}\nolimits}L_B=1$.
\[elemobst\] An [*elementary obstacle*]{} is a subtuple $\mathsf K=(K_i\mid i\in I)$ of subsets $K_i\subset A_i$, where $I\subset\{1,\ldots,n\}$, belonging to one of the following types:
[**Type 1:**]{} A codimension $-1$ tuple of intervals such that $I_{\min}=I$;
[**Type 2:**]{} A codimension $0$ tuple consisting of one triangle and $|I|-1$ intervals which satisfies the following properties:
- $I_{\min}=I$;
- there exists no elementary obstacle $\mathsf M=(M_j\mid j\in J\subset I)$ of Type 1 such that $ M_j\subset K_j$ for every $j\in J$.
[**Type 3:**]{} $I=\{i\}$, and $K_i$ consists of three points on a line.
Let $\mathsf{K}$ be an elementary obstacle. Without loss of generality, suppose that $I=\{1,\ldots, m\}$ for some $m\leqslant n$. If $\mathsf{K}$ is of the first type, then for every $1\leqslant i\leqslant m$, by $v_i$ we denote the vector $\overrightarrow{K_i}\in\Z^n$. Otherwise, suppose that $K_1$ is a triangle. Then we set $v_0=\overrightarrow{K'_0}$, and $v_1=\overrightarrow{K'_1}$ for any two edges $K'_0$ and $K'_1$ of $K_1$ and $v_i=\overrightarrow{K_i}$ for every $2\leqslant i\leqslant m$.
By $v_{ij}$ we denote the $j$-th component of the coordinate vector of $v_i$ in the standard basis. Consider an arbitrary point $\varphi=(\varphi_1,\ldots, \varphi_n)\in\EuScript{M}$. Let $\alpha_i$ and $\beta_i$ be the coefficients of $\varphi_i$ corresponding to each of the endpoints of the intervals $K_i$ and $K'_i$.
In these terms, if the obstacle $\mathsf{K}$ is of the first type, then the wall $L_{\mathsf{K}}$ consists of the points $\varphi\in\EuScript{M}$ such that the following equality holds: $$m-1={\mathop{\rm rk}\nolimits}\begin{pmatrix}
v_{11} & \dots & v_{1n} \\
\vdots & \ddots & \vdots \\
v_{m1} & \dots & v_{mn}
\end{pmatrix}={\mathop{\rm rk}\nolimits}\begin{pmatrix}
v_{11} & \dots & v_{1n} & (\beta_1-\alpha_1) \\
\vdots & \ddots & \vdots \\
v_{m1} & \dots & v_{mn} & (\beta_m-\alpha_m)
\end{pmatrix}$$
Therefore, $L_{\mathsf{K}}$ is defined by the equations of the following form: $$\label{defeq1}
\det \begin{pmatrix}
v_{1{j_1}} & \dots & v_{1{j_{m-1}}} & (\beta_1-\alpha_1) \\
\vdots & \ddots & \vdots \\
v_{m{j_1}} & \dots & v_{m{j_{m-1}}} & (\beta_m-\alpha_m)
\end{pmatrix}=0$$
Thus the defining equations of $L_{\mathsf{K}}$ are linear and employ only the coefficients $\alpha_i$ and $\beta_i$ as variables. Moreover, for every $1\leqslant i\leqslant m$, $\alpha_i$ and $\beta_i$ occur at least in one of the defining equations (\[defeq1\]) with nonzero coefficients, since otherwise, we would find a $(m-1)$-tuple of linearly dependent vectors $(v_1,\ldots,\hat{v_i}, v_m)$, which would mean that $\mathsf{K}$ is not an elementary obstacle.
The same arguments work for the case of $\mathsf{K}$ being an elementary obstacle of type 2. In this case, the wall $L_{\mathsf{K}}$ consists of the points $\varphi\in\EuScript{M}$ such that the following equality holds:
$$m={\mathop{\rm rk}\nolimits}\begin{pmatrix}
v_{01} & \dots & v_{0n} \\
\vdots & \ddots & \vdots \\
v_{m1} & \dots & v_{mn}
\end{pmatrix}={\mathop{\rm rk}\nolimits}\begin{pmatrix}
v_{01} & \dots & v_{0n} & (\beta_0-\alpha_0) \\
\vdots & \ddots & \vdots \\
v_{m1} & \dots & v_{mn} & (\beta_m-\alpha_m)
\end{pmatrix}$$
Therefore, the wall $L_{\mathsf{K}}$ is defined by the equations of the following form: $$\label{defeq2}
\det \begin{pmatrix}
v_{0{j_1}} & \dots & v_{0{j_{m}}} & (\beta_0-\alpha_0) \\
\vdots & \ddots & \vdots \\
v_{0{j_1}} & \dots & v_{m{j_{m}}} & (\beta_m-\alpha_m)
\end{pmatrix}=0$$ The same arguments as above imply that the equations \[defeq2\] are linear and employ only the coefficients $\alpha_i$ and $\beta_i$ as variables, moreover, each of the coefficients $\alpha_i$ and $\beta_i$ occurs at least in one of the equations with a nonzero coefficient.
The same is obviously valid for elementary obstacles of type 3, so, we have proved the following
\[elementobst\] In the previous notation, let $\mathsf K_1$ and $\mathsf K_2$ be elementary obstacles. Then $L_{\mathsf K_1}=L_{\mathsf K_2}$ if and only if $\mathsf K_1=\mathsf K_2$.
The following statements easily follow from Lemma \[elementobst\].
Each obstacle $B\subset A$ has a unique elementary obstacle ${K\subset B}$.
Let $B_1,B_2\subset A$ be obstacles with the elementary obstacles $K_1$ and $K_2$ respectively. Then, the walls $L_{B_1}$ and $L_{B_2}$ coincide if and only if $K_1=K_2$.
The set $\EuScript{S}_1$ of [*almost transverse*]{} tropical hyperplanes is defined to be the set of all $\varphi=(\varphi_1,\ldots,\varphi_n)\in \EuScript{M}\setminus\EuScript{S}_0$ such that the following conditions are satisfied:
- there exists a wall containing $\varphi$;
- all the walls containing $\varphi$ coincide;
- for every trouble $B$, ${\mathop{\rm codim}\nolimits}(L_B)\geqslant 2$ implies that $\varphi\not\in L_B$.
For every point $\varphi\in\EuScript{S}_1$ there exists a unique elementary obstacle $\mathsf{K}$ such that $\varphi\in L_{\mathsf{K}}$. Moreover, in a small neighborhood of the point $\varphi$, we have $\EuScript{S}_1=L_{\mathsf K}$.
The following inequality holds: ${\mathop{\rm codim}\nolimits}(\EuScript{M}\setminus(\EuScript{S}_0\cup\EuScript{S}_1))\geqslant 2$.
The set $\EuScript{S}_0\cup\EuScript{S}_1$ is connected.
We now explicitly describe every tuple $\varphi=(\varphi_1,\ldots,\varphi_n)\in\EuScript{S}_1$.
A [*generic extension*]{} of an elementary obstacle ${\mathsf K}=(K_i,\, i\in I)$ is a collection ${\mathsf K}'=(K'_1,\ldots,K'_n),\, K'_i\subset A_i$, satisfying the following properties:
- Assume that ${\mathsf K}$ is of type 1. Then there are three possibilities for $K'$ (1) $K'_i=K_i$ for $i\in I$, and otherwise $K'_i$ is a pair of points such that the convex hull of the Minkowski sum $\sum_i K'_i$ has codimension 1. (2) The same as (1), but $K'_i\supset K_i$ is a triangle for one $i\in I$, and the Minkowski sum $\sum_i K'_i$ is not contained in a hyperplane. (3) The same as (1), but $K'_i$ is a triangle for one $i\notin I$, and the Minkowski sum $\sum_i K'_i$ is not contained in a hyperplane.
- Assume that ${\mathsf K}$ is of type 2. Then $K'_i=K_i$ for $i\in I$, and otherwise $K'_i$ is a pair of points such that the Minkowski sum $\sum_i K'_i$ is not contained in an affine hyperplane.
- Assume that ${\mathsf K}$ is of type 3. Then $K'_i=K_i$ for $i\in I$, and otherwise $K'_i$ is a pair of points such that the Minkowski sum $\sum_i K'_i$ is not contained in an affine hyperplane.
\[badinters\] According to the classification of the elementary obstacles $\mathsf{K}\subset A$ given in \[elemobst\], each point $\varphi=(\varphi_1,\ldots,\varphi_n)\in\EuScript{S}_1\cap L_{\mathsf K}$ defines a tuple $H=(H_1,\ldots, H_n)$ of tropical hypersurfaces that intersects as follows: at every non-transversal point $x$ of the intersection $H_1\cap\ldots\cap H_n$, the local support sets ${\mathop{\rm supp}\nolimits}_x(\varphi_i)$ form a generic extension of ${\mathsf K}$.
In particular, if the elementary obstacle ${\mathsf K}=(K_i,\, i\in I)$ is of
[**type 1:**]{} the intersection $C$ of the $|I|$ facets $C_i$ of the tropical hypersurfaces $(H_i, i\in I)$ that are dual to the intervals $(K_i, i\in I)$ is ${(n-|I|+1)}$-dimensional. The intersection of $C$ with the rest of $H_i,~i\notin I$, is a graph with vertices of degree 3 and 1. All the other intersections of the hypersurfaces $(H_1,\ldots,H_n)$ are transverse;
[**type 2:**]{} [**exactly**]{} one of the intersection points $x\in H_1\cap\ldots\cap H_n$ is not smooth, since $\newton_x(H_i)$ is a triangle for some $i\in I$. All the other intersections of the hypersurfaces $(H_1,\ldots,H_n)$ are transverse;
[**type 3:**]{} The intersection $H_1\cap\ldots\cap H_n$ is transversal in the sense that at each of its finitely many points $x$ the local Newton polytopes $\newton_x(H_i)$ are transversal segments, but at some of these points the support set ${\mathop{\rm supp}\nolimits}_x(\varphi_i),\, i\in I,$ consists of three points instead of two.
Our next step is to show that the 2-intersection number $\iota_2$ is constant on $\EuScript{S}_0\cup\EuScript{S}_1$.
Passing through the walls {#passwall}
-------------------------
We say that a point $x\in\T^n\times\{(H_1,\ldots,H_n)\}\subset\T^n\times\EuScript{M}$ belongs to the [*stable intersection*]{} of tropical hypersurfaces $H_1,\ldots,H_n$ (denote $H_1\cap_{st}\ldots\cap_{st}H_n$), if for every $\varepsilon>0$ there exists $\delta>0$ such that for any tuple of translations $(T_{\overrightarrow{v_1}},\ldots, T_{\overrightarrow{v_n}})$, where $|{\overrightarrow{v_i}}|<\delta$, there exists a point $x'\in T_{\overrightarrow{v_1}}(H_1)\cap\ldots\cap T_{\overrightarrow{v_n}}(H_n)$ such that $|x-x'|<\varepsilon$.
If $H_1\pitchfork\ldots\pitchfork H_n$, then $H_1\cap_{st}\ldots\cap_{st}H_n =H_1\cap\ldots\cap H_n$.
\[epsilon\] For every open $U\subset\T^n$ such that ${U\times\{(H_1,\ldots,H_n)\}\subset\T^n\times\EuScript{M}}$ contains the stable intersection $H_1\cap_{st}\ldots\cap_{st}H_n$, there exists an open ${V\subset\EuScript{M}}$ satisfying the following properties: $(H_1,\ldots,H_n)\in V$ and for every ${(H'_1,\ldots,H'_n)\in V}$ the stable intersection ${H'_1\cap_{st}\ldots\cap_{st}H'_n}$ is contained in the set ${U\times\{(H'_1,\ldots,H'_n)\}}$.
$\vartriangleleft$ Consider the projection $\pi\colon\T^n\times\EuScript{M}\to\EuScript{M}$. It suffices to show that for any element $\varphi\in\EuScript{M}$ and for any open $\pi^{-1}(\varphi)\subset U\subset\T^n\times\{\varphi\}$ there exists an open $V\subset\EuScript{M}$ such that $\varphi\in V$ and for every $\varphi'\in V$ the preimage $\pi^{-1}(\varphi')$ is contained in $U\times V$. Choose an arbitrary neighbourhood $I\ni\varphi$ and consider the set $K= \pi(\overline{U\times I}\setminus\pi^{-1}(I))$. This set is compact and does not contain the point $\varphi$, so there exists a neighbourhood $V\ni \varphi$ such that $K\cap V=\varnothing$, which finishes the proof of the lemma. $\vartriangleright$
Consider a point $(H_1,\ldots,H_n)\in \EuScript{S}_1$. Besides the non-transverse intersection points mentioned in \[badinters\], the hypersurfaces $H_1,\ldots,H_n$ have finitely many transverse intersection points $Q_1, \ldots, Q_l$. The following statement is obvious.
\[goodpoints\] For every transverse intersection $Q_j\in H_1\cap\ldots \cap H_n$, there exist an open ${U_j\ni Q_j}$ and an open ${V_j\ni (H_1,\ldots,H_n)}$ such that for every ${(H'_1,\ldots, H'_n)\in V_j}$ all the intersection points of the hypersurfaces $H'_1,\ldots,H'_n$ are transverse in $U_j$.
\[calculS1\] For every $(H_1,\ldots,H_n)\in\EuScript{S}_1$, $\iota_2$ is constant on ${V_0=\bigcap_{j=1}^m (V_j\cap V)\subset\EuScript{M}}$.
In Subsection \[buildwall\], we obtained the following classification of the walls in $\EuScript M$. Every wall is the hyperplane $L_{\mathsf K}$ corresponding to an elementary obstacle $\mathsf K\subset A$, i.e., a subtuple $\mathsf K=(K_i\mid i\in I)$, where $K_i\subset A_i$ and $I\subset\{1,\ldots,n\}$, belonging to one of the following types:
[**Type 1:**]{} A codimension $-1$ tuple of intervals such that $I_{\min}=I$;
[**Type 2:**]{} A codimension $0$ tuple consisting of one triangle and $|I|-1$ intervals such that $I_{\min}=I$.
[**Type 3:**]{} One triple of points on a line.
For each of the types of walls $L_{\mathsf K}$, we will show that passing through points $\varphi\in L_{\mathsf K}$ does not change the 2-intersection number. Namely, if connected components $S$ and $S'$ of the set $\EuScript{S}_0$ are separated by a wall $L_{\mathsf K}$, then for any points $(H_1,\ldots, H_n)\in S$ and $(H'_1,\ldots, H'_n)\in S'$, there exists a path $\gamma\colon [0,1]\to \EuScript{S}_0\cup\EuScript{S}_1$ with the endpoints $(H_1,\ldots, H_n)$ and $(H'_1,\ldots, H'_n)$ such that $\iota_2(\gamma(0);\zeta)=\iota_2(\gamma(1);\zeta)$.
[**Case 1.**]{} Consider a point $(H_1,\ldots, H_n)\in\EuScript{S}_1$ that belongs to a wall $L_{\mathsf K}$ corresponding to the elementary obstacle $\mathsf K=(K_i\mid i\in I)$ of type 1. Moreover, we may assume that the point $(H_1,\ldots, H_n)$ is generic in $L_{\mathsf K}$.
Without loss of generality, assume that $I=\{1,\ldots,m\}$ for some $m\leqslant n$. By $C_i$ we denote the facet of $H_i$ which is dual to the interval $K_i$, and by $C$ we denote the intersection $C_1\cap\ldots\cap C_m$.
In these terms, for every point $x\in C\cap H_{m+1}\cap\ldots\cap H_n$, the tuple $\mathsf K$ is the elementary obstacle corresponding to the obstacle $B_x=(\newton_x(H_1),\ldots, \newton_x(H_n))$. It immediately follows from Corollary \[badinters\] that for every $i\in I$, we have\
${H_1\pitchfork\ldots\pitchfork H_{i-1}\pitchfork\hat{H_i}\pitchfork H_{i+1}\pitchfork\ldots\pitchfork H_n}$ (where $\hat{H_i}$ means that the hypersurface $H_i$ is omitted), i.e., these hypersurfaces intersect in a tropical curve. Without loss of generality, we take $i=1$, and by $\Sigma$ we denote the tropical curve $\Sigma=H_2\cap\ldots\cap H_n$.
In the previous notation, we have $\Sigma\cap H_1=(\Sigma\cap C)\cup(Q_1\cup\ldots\cup Q_l)$, where $Q_j$ are the transverse intersection points of the hypersurfaces $H_1,\ldots, H_n$. Pick a normal $\overrightarrow{w}$ to the hyperplane containing the facet $C_1$ of the tropical hypersurface $H_1$. Thus we obtain a family of hypersurfaces $H_1(\varepsilon)=T_{\varepsilon\overrightarrow{w}}(H_1)$ (where $T_{\overrightarrow{v}}$ stands for the shift by a vector $\overrightarrow{v}$) parametrized by $\varepsilon\in\R$. By $C_1(\varepsilon)$ we denote the image of the facet $C_1$ of the hypersurface $H_1$.
The neighbourhood $V_0\ni (H_1,\ldots, H_n)$ constructed above contains an open ball $\EuScript B\ni (H_1,\ldots, H_n)$, therefore, there exists $\varepsilon_0>0$ such that for every $\varepsilon \in (-\varepsilon_0, \varepsilon_0)$ the point $(H_1(\varepsilon), H_2,\ldots, H_n)$ belongs to $\EuScript B$. Fix an arbitrary ${0<\varepsilon_1<\varepsilon_0}$. For every $\varepsilon\neq 0$ in $[-\varepsilon_1, \varepsilon_1]$, we have $H_1(\varepsilon)\pitchfork H_2\pitchfork\ldots\pitchfork H_n$. So, our next step is to compare the 2-intersection numbers $\iota_2(H_1(-\varepsilon_1), H_2, \ldots, H_n;\zeta)$ and $\iota_2(H_1(\varepsilon_1), H_2, \ldots, H_n;\zeta)$ for the tuples $H_1(\pm\varepsilon_1), H_2, \ldots, H_n$ which are contained in different connected components of the set $\mathsf S_0$ separated by the wall $L_{\mathsf K}$.
From Proposition \[goodpoints\], it follows that for every transverse intersection point\
${Q_j\in H_1\cap H_2\cap\ldots\cap H_n}$, there exists a neighbourhood ${U_j\ni Q_j}$, such that for every ${0\neq\varepsilon\in [-\varepsilon_1, \varepsilon_1]}$, the neighbourhood $U_j$ contains a unique transverse intersection point ${Q_j(\varepsilon)\in H_1(\varepsilon)\cap H_2\cap\ldots\cap H_n}$. Therefore, for every ${0\neq\varepsilon\in [-\varepsilon_1, \varepsilon_1]}$, we have: $$\begin{gathered}
\label{iotasum}
\iota_2(H_1(\varepsilon), \newton_{x}(H_2),\ldots, H_n;\zeta)= \sum\limits_{x\in C_1(\varepsilon)\cap\Sigma}{\mathrm{det}_2}(\newton_x(H_1(\varepsilon)), H_2,\ldots \newton_x(H_n),\zeta) \\+\sum_{j=1}^{j=l}{\mathrm{det}_2}(\newton_{Q_j(\varepsilon)}(H_1(\varepsilon)), \newton_{Q_j(\varepsilon)}(H_2),\ldots, \newton_{Q_j(\varepsilon)}(H_n),\zeta). \end{gathered}$$
Moreover, from the construction of the interval $[-\varepsilon_1, \varepsilon_1]$, it follows that in the summands ${\mathrm{det}_2}(\newton_{Q_j(-\varepsilon_1)}(H_1(-\varepsilon_1)),\ldots \newton_{Q_j(-\varepsilon_1)}(H_n),\zeta)$\
and ${\mathrm{det}_2}(\newton_{Q_j(\varepsilon_1)}(H_1(\varepsilon_1)),\ldots \newton_{Q_j(\varepsilon_1)}(H_n),\zeta)$ from the right-hand side of (\[iotasum\]), the same Newton intervals occur, thus the two sums over the corresponding transverse intersection points coincide. Therefore, it suffices to deduce the following equality: $$\begin{gathered}
\sum\limits_{x\in C_1(-\varepsilon_1)\cap\Sigma}{\mathrm{det}_2}(\newton_x(H_1(-\varepsilon_1)), \newton_{x}(H_2),\ldots \newton_x(H_n),\zeta)=\\ \sum\limits_{x\in C_1(\varepsilon_1)\cap\Sigma}{\mathrm{det}_2}(\newton_x(H_1(\varepsilon_1)), \newton_{x}(H_2),\ldots \newton_x(H_n),\zeta)\end{gathered}$$
\[genericity\] Recall that the point $\varphi=(H_1,\ldots, H_n)\in L_{\mathsf K}$ is chosen to be generic (i.e., the corresponding dual subdivisions of the Newton polytopes $\newton(H_i)$ are simplicial for all $i\in\{1,\ldots,n\}$), thus, we can also assume that it satisfies the following conditions:
1. the curve $\Sigma$ is non-singular, i.e., all of its vertices are of valence 3;
2. for every $x\in H_1\cap\ldots\cap H_n$ which belongs to $\partial(C)\cap\Sigma$, the tuple $B_x=(\newton_x(H_1),\ldots, \newton_x(H_n))$ consists of a triangle and $n-1$ intervals.
\[interseps\] For an arbitrary $0\neq\varepsilon\in[-\varepsilon_1, \epsilon_1]$, by $\mathscr W(\varepsilon)$ we denote the finite set ${(C_1(\varepsilon)\cap H_2\cap\ldots\cap H_n)\setminus\{Q_1(\varepsilon),\ldots, Q_l(\varepsilon)\}}$.
Consider the sets $\mathscr W(-\varepsilon_1)=\{x_1(-\varepsilon_1),\ldots, x_p(-\varepsilon_1)\}$ and $\mathscr W(\varepsilon_1)=\{x_1(\varepsilon_1),\ldots, x_q(\varepsilon_1)\}$. We define $$\mathscr W(0)=\{\lim_{\varepsilon\to -0}(x_1(\varepsilon)), \ldots, \lim_{\varepsilon\to -0}(x_p(\varepsilon))\}\cup\{\lim_{\varepsilon\to +0}(x_1(\varepsilon)), \ldots, \lim_{\varepsilon\to +0}(x_q(\varepsilon))\}\subset (C\cap \Sigma).$$
It is straightforward to show that the sets $\mathscr W(0)$ and $\partial(C)\cap\Sigma$ coincide.
Condition 2 from Remark \[genericity\] implies that for any point $x\in \mathscr W(0)=\partial(C)\cap\Sigma,$ there exist exactly two points $y_1=x_i(\pm\varepsilon_1), y_2=x_j(\pm\varepsilon_1)\in\mathscr W(-\varepsilon_1)\cup \mathscr W(\varepsilon_1)$ such that\
$x=\lim_{\varepsilon\to\pm 0}(x_i(\varepsilon))=\lim_{\varepsilon\to\pm0}(x_j(\varepsilon))$.
We have the following equality: $$\begin{gathered}
{\mathrm{det}_2}(\newton_{y_1}(H_1(\pm\varepsilon_1)), \newton_{y_1}(H_2),\ldots\newton_{y_1}(H_n),\zeta)+\\{\mathrm{det}_2}(\newton_{y_2}(H_1(\pm\varepsilon_1)), \newton_{y_2}(H_2),\ldots\newton_{y_2}(H_n),\zeta)+\\{\mathrm{det}_2}(\newton_x(H_1),\ldots \newton_x(H_n),\zeta)=0.\end{gathered}$$
$\vartriangleleft$ Consider the tuple $B_x=(\newton_x(H_1),\ldots, \newton_x(H_n))$. By the genericity assumption, it consists of a triangle and $n-1$ intervals. Moreover, $\mathsf K$ is the elementary obstacle corresponding to $B_x$. Since $x$ belongs to $\partial(C)\cap\Sigma$, by definition of the boundary of $C$, it follows that $\newton_x(H_i)$ is a triangle containing $K_i$ as an edge for some $1\leqslant i\leqslant m$, while for every $j\neq i$ in $\{1,\ldots, m\}$ we have $\newton_x(H_j)=K_j$. Moreover, if $i=1$, then the other edges of the triangle are $\newton_{y_1}(H_1(\pm\varepsilon_1))$ and $\newton_{y_2}(H_1(\pm\varepsilon_1))$. If $i\neq 1$, then these edges are $\newton_{y_1}(H_i)$ and $\newton_{y_2}(H_i)$. Therefore, the sought equality follows from the linearity property of the 2-determinant. Indeed, by the genericity assumption, for $j\neq i$ the intervals $K_j, \newton_{y_1}(H_j), \newton_{y_2}(H_j)$ (or $K_1, \newton_{y_1}(H_1(\pm\varepsilon_1)), \newton_{y_1}(H_1(\pm\varepsilon_1))$, if $1=j\neq i$) coincide. Moreover, it is obvious that for $m+1\leqslant j\leqslant n$ the intervals $\newton_x(H_j), \newton_{y_1}(H_j)$ and $\newton_{y_2}(H_j)$ coincide, therefore, we have $$\begin{gathered}
{\mathrm{det}_2}(\newton_{y_1}(H_1(\pm\varepsilon_1)), \newton_{y_1}(H_2),\ldots\newton_{y_1}(H_n),\zeta)+\\{\mathrm{det}_2}(\newton_{y_2}(H_1(\pm\varepsilon_1)), \newton_{y_2}(H_2),\ldots\newton_{y_2}(H_n),\zeta)+{\mathrm{det}_2}(\newton_x(H_1),\ldots \newton_x(H_n),\zeta)=\\ {\mathrm{det}_2}(K_1,\ldots, K_{i-1}, \newton_{y_1}(H_i), K_{i+1}, \ldots, K_m, \newton_{y_1}(H_{m+1}), \ldots, \newton_{y_1}(H_n), \zeta)+\\{\mathrm{det}_2}(K_1,\ldots, K_{i-1}, \newton_{y_2}(H_i), K_{i+1}, \ldots, K_m, \newton_{y_2}(H_{m+1}), \ldots, \newton_{y_2}(H_n), \zeta)+\\{\mathrm{det}_2}(K_1,\ldots, K_{i-1}, K_i, K_{i+1}, \ldots, K_m, \newton_{x}(H_{m+1}), \ldots, \newton_{x}(H_n), \zeta)=0.\end{gathered}$$ Exactly the same argument works in case $i=1$. $\vartriangleright$
So, in order to prove Case 1 of Theorem \[calculS1\], it suffices to prove
The following equality holds: $$\sum\limits_{x\in \mathscr W(0)}{\mathrm{det}_2}(K_1,\ldots, K_{i-1}, K_i, K_{i+1}, \ldots, K_m, \newton_{x}(H_{m+1}), \ldots, \newton_{x}(H_n), \zeta)=0.$$
$\vartriangleleft$ Each of the tuples $(K_1,\ldots, K_{i-1}, K_i, K_{i+1}, \ldots, K_m, \newton_{x}(H_{m+1}), \ldots, \newton_{x}(H_n), \zeta)$ has a subtuple $(K_1,\ldots, K_m)$ of rank $m-1$. There exists a natural projection $\pi\colon\Z^n\to\Z^n/\langle K_1,\ldots, K_{m}\rangle$. From Corollary \[blockdet2\], it follows that $$\begin{gathered}
\sum\limits_{x\in \mathscr W(0)}{\mathrm{det}_2}(K_1,\ldots, K_{i-1}, K_i, K_{i+1}, \ldots, K_m, \newton_{x}(H_{m+1}), \ldots, \newton_{x}(H_n), \zeta)=\\ {\mathrm{det}_2}(K_1,\ldots, K_m)(\sum\limits_{x\in \mathscr W(0)}\det(\pi(\newton_{x}(H_{m+1})), \ldots, \pi(\newton_{x}(H_n)), \pi(\zeta))). \end{gathered}$$ In order to show that the sum $\sum\limits_{x\in \mathscr W(0)}\det(\pi(\newton_{x}(H_{m+1})), \ldots, \pi(\newton_{x}(H_n)), \pi(\zeta))$ is equal to zero, we use Condition 1 from Remark \[genericity\] and the assumption on the Newton Polytopes $\newton(H_1),\ldots, \newton(H_n)$ being 2-developed with respect to the point $\zeta$. The genericity assumption allows us to use the balancing condition at each of the vertices of the curve $\Sigma\cap C$. Fix an arbitrary smooth point $\mu$ on every edge of the curve $\Sigma\cap C$. For each of the vertices $a$ of this curve, consider the sum $M_a$ of the summands $\det(\pi(\newton_{\mu}(H_{m+1})), \ldots, \pi(\newton_{\mu}(H_n)), \pi(\zeta))$ over the edges adjacent to $a$. On the one hand, it immediately follows from the balancing condition that all such sums are equal to $0$. On the other hand, if we take the sum of $M_a$ over all the vertices of the curve $\Sigma\cap C$, we will obtain the sought sum. Indeed, since the summands over the edges with two endpoints in this sum are taken twice, they are cancelled. Thus, we obtain that the sought sum over the points $x\in\mathscr W(0)$ equals the sum over all rays (the edges with a single endpoint) of the classical intersection numbers of $C\cap\Sigma$ with the tropical hypersurface $S$ dual to the interval $E=\{0, \pi(\zeta)\}\subset\R^{n-m+1}$. This sum is always zero over $\mathbb F_2$.
Indeed, pick any smooth point $x$ on a ray of the curve $\Sigma$, and denote the generating vector of this ray by $v$. By definition of a tuple of polytopes 2-developed with respect to a point (see Definition \[devwrt\]), it follows that either $v(\zeta)=0$ over $\mathbb F_2$, or, for some $1\leqslant i\leqslant n$, the local Newton interval $\newton_x(\Sigma)$ has the endpoints of the same parity. In the first case, all the arguments of the sought determinant ${\mathrm{det}_2}(K_1,\ldots, K_{i-1}, K_i, K_{i+1}, \ldots, K_m, \newton_{x}(H_{m+1}), \ldots, \newton_{x}(H_n), \zeta)$ are contained in the same hyperplane $v^\perp$ in ${\mathbb F_2}^n$. In the second case, one of the arguments of the sougnh determinant is even, which finishes the proof of the lemma. $\vartriangleright$
Case 1 of Theorem \[calculS1\] is proved.
[**Case 2.**]{} Consider a generic point $(H_1,\ldots, H_n)\in\EuScript{S}_1$ that belongs to a wall $L_{\mathsf K}$ corresponding to the elementary obstacle $\mathsf K=(K_i\mid i\in I)$ of Type 2. Without loss of generality suppose that $I=\{1,\ldots, m\}$ for some $m\leqslant n$. The elementary obstacle $K$ consists of a triangle and $m-1$ intervals. Moreover, we can assume that $K_1$ is the triangle. By $\Sigma$ denote the tropical curve obtained as the intersection $H_2\cap\ldots\cap H_n$.
The set $H_1\cap\ldots\cap H_n$ consists of a finite set $\{Q_1,\ldots, Q_l\}$ of the transverse intersection points and the non-smooth point $x\in H_1$. Note that $x$ is a smooth point of the curve $\Sigma$. By $\overrightarrow{u}$ denote the primitive vector of the edge of $\Sigma$ which contains $x$. Pick any edge $K'_1$ of the triangle $K_1$ and consider the face $C_1$ of the hypersurface $H_1$ which is dual to $K'_1$. In analogy with the proof of Case 1, take any normal vector $\overrightarrow{w}$ to the hyperplane containing the face $C_1$. Moreover, we can assume that the vectors $\overrightarrow{u}$ and $\overrightarrow{w}$ are linearly independent, since we can always choose an edge $K'_1$ such that this condition is satisfied.
Thus, we obtain the family of hypersurfaces $H_1(\varepsilon)=T_{\varepsilon\overrightarrow{w}}(H_1)$ parametrized by $\varepsilon\in\R$. By $C_1(\varepsilon)$ we denote the image of the face $C_1$ of the hypersurface $H_1$.
The neighbourhood $V_0\ni (H_1,\ldots, H_n)$ constructed above contains an open ball $\EuScript B\ni (H_1,\ldots, H_n)$, therefore, there exists $\varepsilon_0>0$ such that for every $\varepsilon \in (-\varepsilon_0, \varepsilon_0)$ the point $(H_1(\varepsilon), H_2,\ldots, H_n)$ belongs to $\EuScript B$. Fix an arbitrary ${0<\varepsilon_1<\varepsilon_0}$. For every $\varepsilon\neq 0$ in $[-\varepsilon_1, \varepsilon_1]$, we have $H_1(\varepsilon)\pitchfork H_2\pitchfork\ldots\pitchfork H_n$. So, our next step is to compare the 2-intersection numbers $\iota_2(H_1(-\varepsilon_1), H_2, \ldots, H_n;\zeta)$ and $\iota_2(H_1(\varepsilon_1), H_2, \ldots, H_n;\zeta)$.
From Proposition \[goodpoints\], it follows that for every transverse intersection point\
${Q_j\in H_1\cap H_2\cap\ldots\cap H_n}$, there exists a neighbourhood ${U_j\ni Q_j}$, such that for every ${0\neq\varepsilon\in [-\varepsilon_1, \varepsilon_1]}$, the neighbourhood $U_j$ contains a unique transverse intersection point ${Q_j(\varepsilon)\in H_1(\varepsilon)\cap H_2\cap\ldots\cap H_n}$.
Besides the transverse intersection points $Q_j(\pm\varepsilon_1)$, the union of the sets $\mathscr W(-\varepsilon_1)$ and $\mathscr W(\varepsilon_1)$ (see Definition \[interseps\]) contains the smooth intersection points $y_1, y_2, y_3$ which appear as the result of the $\pm\varepsilon_1$-deformation of the non-smooth intersection\
${x\in H_1\cap\ldots\cap H_n}$.
Therefore, for $\varepsilon=\pm\varepsilon_1$ we have the following equality: $$\begin{gathered}
\label{iotasum2}
\iota_2(H_1(\varepsilon), H_2,\ldots, H_n;\zeta)= \\\sum_{j=1}^{j=l}{\mathrm{det}_2}(\newton_{Q_j(\varepsilon)}(H_1(\varepsilon)), \newton_{Q_j(\varepsilon)}(H_2),\ldots, \newton_{Q_j(\varepsilon)}(H_n),\zeta)+\\\sum\limits_{y\in \mathscr W(\varepsilon)}{\mathrm{det}_2}(\newton_y(H_1(\varepsilon)), H_2,\ldots \newton_y(H_n),\zeta). \end{gathered}$$
By the construction of the interval $[-\varepsilon_1,\varepsilon_1]$, the tuples of local Newton intervals $B_{Q_j(-\varepsilon_1)}$ and $B_{Q_j(\varepsilon_1)}$ coincide for every $1\leqslant j\leqslant n$. Thus, to prove Case 2 of Theorem \[calculS1\], it suffices to prove the following
In the previous notation, the following equality holds: $$\begin{gathered}
\sum\limits_{y\in \mathscr W(-\varepsilon_1)}{\mathrm{det}_2}(\newton_y(H_1(-\varepsilon_1)), \newton_{y}(H_2),\ldots \newton_y(H_n),\zeta)+ \\\sum\limits_{y\in \mathscr W(\varepsilon_1)}{\mathrm{det}_2}(\newton_y(H_1(\varepsilon_1)), \newton_{y}(H_2),\ldots \newton_y(H_n),\zeta)=0.\end{gathered}$$
$\vartriangleleft$ Rewrite the sought equality in terms of the intersection points $y_1,y_2, y_3$ defined above: $$\begin{gathered}
{\mathrm{det}_2}(\newton_{y_1}H_1(\pm\varepsilon_1)), \newton_{y_1}(H_2),\ldots \newton_{y_1}(H_n),\zeta)+\\{\mathrm{det}_2}(\newton_{y_2}H_1(\pm\varepsilon_1)), \newton_{y_2}(H_2),\ldots \newton_{y_2}(H_n),\zeta)+\\{\mathrm{det}_2}(\newton_{y_3}H_1(\pm\varepsilon_1)),\newton_{y_3}(H_2),\ldots \newton_{y_3}(H_n),\zeta)=0\end{gathered}$$ Note that the intervals $\newton_{y_1}H_1(\pm\varepsilon_1))$, $\newton_{y_2}H_1(\pm\varepsilon_1))$, $\newton_{y_3}H_1(\pm\varepsilon_1))$ are exactly the edges of the triangle $K_1$. Moreover, obviously, the other Newton intervals $\newton_{y_i}(H_j)$ coincide with $\newton_{x}(H_j)$ for all $i\in\{1,2,3\}$ and $j\in\{2,\ldots,n\}$. Therefore, the sought equality follows from the linearity property of the 2-determinant, which finishes the proof of the lemma. $\vartriangleright$
So, Cases 1 and 2 of Theorem \[calculS1\] are proved. The case of a tuple $H=(H_1,\ldots,H_n)$, corresponding to an elementary obstacle $K_1=\{a,b,c\}$ of type 3, is obvious: every intersection point of multiplicity ${\mathrm{det}_2}(a-c, \newton_{y_1}(H_2),\ldots \newton_{y_1}(H_n),\zeta)$ splits into two intersection points of multiplicites ${\mathrm{det}_2}(a-b, \newton_{y_1}(H_2),\ldots \newton_{y_1}(H_n),\zeta)$ and ${\mathrm{det}_2}(b-c, \newton_{y_1}(H_2),\ldots \newton_{y_1}(H_n),\zeta)$ as the tuple $H$ perturbs.
Theorem \[calculS1\] together with Proposition \[constS0\] imply Theorem \[main\], therefore, we obtained a well-defined function ${{\mathop{\rm MV}\nolimits}_2\colon(P_1,\ldots,P_n;\zeta)\mapsto\iota_2(H_1,\ldots, H_n;\zeta)}$ — the so-called [*2-mixed volume*]{}.
Multivariate Vieta’s Formula {#multVieta}
============================
In this Section, we show that for a certain class of multivariate polynomial systems equations, the product of their roots can be expressed in terms of the 2-volume of their Newton polytopes. The Section is organised as follows. First we obtain such a formula for binomial systems (see Subsection \[BinomVieta\]). In Subsection \[DefVieta\], we provide all the necessary definitions and notation crucial to formulating Theorem \[VietaMain\] and conducting its proof. Subsection \[multvietaproof\] is devoted to the statement and proof of the multivariate Vieta’s formula (see Theorem \[VietaMain\]).
Multivariate Vieta’s Formula for Binomial Systems {#BinomVieta}
-------------------------------------------------
\[VietaBin\] Let $f_1,\ldots, f_n$ be binomials such that all the coefficients of $f_i, 1\leqslant i\leqslant n,$ are equal to 1 and the Newton intervals $\newton(f_1),\ldots,\newton(f_n)\subset\Z^n$ are affinely independent.Fix an arbitrary point $a\in \mathbb Z^n$. Then the following equality holds: $$\prod_{f_1(x)=\ldots=f_n(x)=0}x^a=(-1)^{{\mathop{\rm MV}\nolimits}_2(\newton(f_1),\ldots, \newton(f_n); a)}$$
$\vartriangleleft$ We will prove this lemma by induction on $n$. The base $n=1$ follows from the classical Vieta’s formula for the product of roots for a polynomial in one variable. Now suppose that the statement is true for $k=n-1$. We will now deduce it in the case $k=n$.
Since the product of roots for a system of equations is invariant under invertible monomial changes of variables and the Newton intervals of the polynomials of the system are affinely independent, we can assume without loss of generality, that the system $\{f_1=\ldots=f_n=0\}$ is of the following form:
$$\label{binsys}
\begin{cases}
{x_1}^{v_{1,1}}{x_2}^{v_{2,1}}\ldots {x_{n-1}}^{v_{{n-1},1}}+1=0\\
{x_1}^{v_{1,2}}{x_2}^{v_{2,2}}\ldots {x_{n-1}}^{v_{{n-1},2}}+1=0\\
\ldots \\
{x_1}^{v_{1,{n-1}}}{x_2}^{v_{2,{n-1}}}\ldots {x_{n-1}}^{v_{{n-1},{n-1}}}+1=0\\
{x_1}^{w_1} {x_2}^{w_2}\ldots{x_n}^{w_n}+1=0
\end{cases}$$
Thus, the statement of Lemma \[VietaBin\] for the system (\[binsys\]) can be reformulated as follows: $$\prod_{f_1(x)=\ldots=f_n(x)=0}x^a=(-1)^{\det_2\left(\begin{smallmatrix}
v_{1,1} & \cdots & {v_{1,{n-1}}} & w_1 & a_1\\
v_{2,1} & \cdots & {v_{2,{n-1}}} & w_2 & a_2 \\
\vdots & \ddots & \vdots & \vdots & \vdots \\
v_{{n-1},1} & \cdots & {v_{{n-1},{n-1}}} & w_{n-1} & a_{n-1} \\
0 & \cdots & 0 & w_n & a_{n}
\end{smallmatrix}\right)}\label{binVieta}$$
By the induction hypothesis, we have the following equalities for $1\leqslant i\leqslant n-1$:
$$\label{prodsmalli}
\prod_{f_1(x)=\ldots=f_{n-1}(x)=0} x_i=(-1)^{\det_2\left(\begin{smallmatrix}
v_{1,1} & \cdots & {v_{1,{n-1}}} & 0\\
\vdots & \ddots & \vdots & \vdots \\
v_{{i-1},1} & \cdots & {v_{{i-1},{n-1}}} & 0 \\
v_{i,1} & \cdots & {v_{i,{n-1}}} & 1 \\
v_{{i+1},1} & \cdots & {v_{{i+1},{n-1}}} & 0 \\
\vdots & \ddots & \vdots & \vdots \\
v_{{n-1},1} & \cdots & {v_{{n-1},{n-1}}} & 0
\end{smallmatrix}\right)}$$
Using these equalities, it is easy to compute the product $\prod_{f_1(x)=\ldots=f_n(x)=0} x_i$. Indeed, each of the roots for the system $\{f_1(x)=\ldots=f_n(x)=0\}$ is obtained from substituting the roots for the smaller system $\{f_1(x)=\ldots=f_{n-1}(x)=0\}$ into the last equation $f_n(x)=0$ and solving this equation in the variable $x_n$. Thus, we have $w_n$ solutions for the system $\{f_1(x)=\ldots=f_n(x)=0\}$ corresponding to each of the roots $(\alpha_1, \ldots, \alpha_{n-1})$ for the system $\{f_1(x)=\ldots=f_{n-1}(x)=0\}$. Therefore, the sought product equals exactly the $w_n$-th power of the product (\[prodsmalli\]). A straightforward computation consisting in applying the explicit formula for the 2-determinant (see Theorem \[formuladet2\]) shows that the following equality holds for every $i\in\{1,\ldots, n-1\}$: $$w_n {\mathrm{\det}_2\left(\begin{smallmatrix}
v_{1,1} & \cdots & {v_{1,{n-1}}} & 0\\
\vdots & \ddots & \vdots & \vdots \\
v_{{i-1},1} & \cdots & {v_{{i-1},{n-1}}} & 0 \\
v_{i,1} & \cdots & {v_{i,{n-1}}} & 1 \\
v_{{i+1},1} & \cdots & {v_{{i+1},{n-1}}} & 0 \\
\vdots & \ddots & \vdots & \vdots \\
v_{{n-1},1} & \cdots & {v_{{n-1},{n-1}}} & 0
\end{smallmatrix}\right)}=\mathrm{\det}_2 \left(\begin{smallmatrix}
v_{1,1} & \cdots & {v_{1,{n-1}}} & w_1 & 0\\
\vdots & \ddots & \vdots & \vdots & \vdots \\
v_{{i-1},1} & \cdots & {v_{{i-1},{n-1}}} & w_{i-1} & 0 \\
v_{i,1} & \cdots & {v_{i,{n-1}}} & w_i & 1 \\
v_{{i+1},1} & \cdots & {v_{{i+1},{n-1}}} & w_{i+1} & 0 \\
\vdots & \ddots & \vdots & \vdots \vdots \\
v_{{n-1},1} & \cdots & {v_{{n-1},{n-1}}} & w_{n-1} & 0 \\
0 & \cdots & 0 & w_n & 0
\end{smallmatrix}\right)$$
Now let us compute the product $\prod_{f_1(x)=\ldots =f_n(x)=0} x_n$. This can be easily done using the classical Vieta’s formula. Consider a root $(\alpha_1,\ldots, \alpha_{n-1})$ for the system $\{f_1(x)=\ldots=f_{n-1}(x)=0\}$. Substituting it into the last equation $f_n(x)=0$ we obtain $w_n$ roots $(\alpha_1, \ldots, \alpha_{n-1}, \beta_j), 1\leqslant j\leqslant w_n,$ for the system $\{f_1(x)=\ldots =f_n(x)=0\}$. The product $\prod_{j=1}^{w_n} \beta_j$ equals $(-1)^{w_n}\alpha_1^{-w_1}\ldots\alpha_{n-1}^{-w_{n-1}}$. Therefore, we obtain that the sought product equals $(-1)^M$, where $$M=w_n \vert\{x\mid f_1(x)=\ldots =f_{n-1}(x)=0\}\vert+{\prod_{f_1(x)=\ldots=f_{n-1}(x)=0} x_1^{-w_1}\ldots x_{n-1}^{-w_{n-1}}}.$$ From the Bernstein-Kushnirenko theorem (see [@bernstein]) and the induction hypothesis, we have $$\label{binvietform}
M=w_n\det\left(\begin{smallmatrix}
v_{1,1} & \cdots & {v_{1,{n-1}}}\\
\vdots & \ddots & \vdots \\
v_{{n-1},1} & \cdots & {v_{{n-1},{n-1}}}
\end{smallmatrix}\right)+\mathrm{\det}_2 \left(\begin{smallmatrix}
v_{1,1} & \cdots & {v_{1,{n-1}}} & w_1\\
\vdots & \ddots & \vdots & \vdots \\
v_{{n-1},1} & \cdots & {v_{{n-1},{n-1}}} & w_{n-1}\\
\end{smallmatrix}\right);$$ one can easily check, that the expression in the right-hand side of the equality (\[binvietform\]) equals exactly the following 2-determinant: $$M=\mathrm{\det}_2 \left(\begin{smallmatrix}
v_{1,1} & \cdots & {v_{1,{n-1}}} & w_1 & 0\\
\vdots & \ddots & \vdots & \vdots & \vdots \\
v_{{n-1},1} & \cdots & {v_{{n-1},{n-1}}} & w_{n-1} & 0 \\
0 & \cdots & 0 & w_n & 1
\end{smallmatrix}\right).$$ Lemma \[VietaBin\] now follows, since the 2-determinant is multilinear and $\mathrm GL_n(\mathbb F_2)$-invariant. $\vartriangleright$
The rest of the Section is devoted to a generalization of this result to a richer class of multivariate polynomial systems of equations.
Some Necessary Notation and Definitions {#DefVieta}
---------------------------------------
Take an arbirtary point $0\neq a\in Z^n$ and consider an $a$–prickly tuple $P=(P_1,\ldots, P_n)$ of convex lattice polytopes in $\R^n$ (see Definition \[prickly\]). By $\C_1^{P_i}$ we denote the set of all polynomials $f=\sum_{p\in P_i} c_p x^p$ such that $\newton(f)=P_i$ and if $p\in P_i$ is a vertex, then $c_p\ne 0$. Consider the set $\C_1^{P}=\C_1^{P_1}\times\ldots\times\C_1^{P_n}$.
Let $\gamma\neq 0$ in $(\R^*)^n$ be a covector. Let $f(x)$ be a Laurent polynomial with the Newton polytope $\newton(f)$. The [*truncation*]{} of $f(x)$ with respect to $\gamma$ is the polynomial $f^{\gamma}(x)$ that can be obtained from $f(x)$ by omitting the sum of monomials which are not contained in the support face ${\newton(f)^{\gamma}}$.
It is easy to show that for a system of polynomial equations $\{f_1(x)=\ldots=f_n(x)=0\}$ and an arbitrary covector $\gamma\neq 0$, the “truncated" system $\{f_1^{\gamma}(x)=\ldots=f_n^{\gamma}(x)=0\}$ by a monomial change of variables can be reduced to a system in $n-1$ variables at most. Therefore, for the systems with coefficients in general position, the “truncated" systems are inconsistent in $\CC^n$.
Let $(f_1,\ldots, f_n)$ be a tuple of polynomials in $\C_1^P$. In the same notation as above, the system ${f_1(x)=\ldots=f_n(x)=0}$ is called [*degenerate at infinity*]{}, if there exists a covector $\gamma\neq 0$ such that the system $\{f_1^{\gamma}(x)=\ldots=f_n^{\gamma}(x)=0\}$ is consistent in $\CC^n$.
Let $\mathscr H\subset\C_1^P$ be the set of all systems that are degenerate at infinity. Consider $\mathscr H'\subset \mathscr H$ — the set of the systems degenerate at infinity, which satisfy the following property: there exists a covector $\gamma\in(\R^*)^n$ such that the system $\{f_1^{\gamma}(x)=\ldots=f_n^{\gamma}(x)=0\}$ is consistent in $(\C^*)^n$ and the Minkowski sum of the support faces ${\newton(f_i)^{\gamma}}, 1\leqslant i\leqslant n$ is of codimension greater than $1$. One can easily see that ${\mathop{\rm codim}\nolimits}(\mathscr H')>1$.
We denote by $\mathscr D$ the discriminant hypersurface in $\C_1^P$ (i.e. the closure of the set of all systems that have a multiple root in $\CC^n$), and $\mathscr D'\subset\mathscr D$ stands for the set of all $(f_1,\ldots f_n)\in\C_1^P$ such that the system $\{f_1(x)=\ldots=f_n(x)=0\}$ has non-isolated roots. It is easy to show that ${\mathop{\rm codim}\nolimits}(\mathscr D')>1$.
The system $\{f_1(x)=\ldots=f_n(x)=0\}$ with $F=(f_1,\ldots, f_n)\in\C_1^P$ is called [*non-degenerate*]{}, if $F\in\C_1^P\setminus(\mathscr H\cup\mathscr D)$.
The Multivariate Vieta’s Formula: Statement and Proof {#multvietaproof}
-----------------------------------------------------
Fix an arbitrary point $0\neq a\in\Z^n$ and an $a$-prickly tuple $P=(P_1,\ldots, P_n)$ of polytopes. In the notation of Subsection \[DefVieta\], the multivariate Vieta’s formula expresses the product of the monomials $x^a$ over all the roots $x$ for a system of polynomial equations $f_1(x)=\ldots=f_n(x)$, where $F=(f_1,\ldots,f_n)\in\C_1^P$ and the coefficients of $f_i$ at the vertices of its Newton polytope are equal to 1, in terms of the 2-mixed volume (see Section \[2-volume\]) of the polytopes $P_1,\ldots, P_n$ and the point $a$. This Subsection consists of two parts. First, we obtain a holomorphic everywhere defined function $\Phi\colon\C_1^P\to\C$ with no zeroes and poles, which maps a point $(f_1,\ldots,f_n)\in\C_1^P$ to the abovementioned product of monomials. It immediately follows that the function $\Phi$ is constant on tuples $(f_1,\ldots,f_n)$ such that the coefficients of $f_i$ at the vertices of its Newton polytope are equal to 1. Secondly, we compute this constant, reducing this problem to the case of binomial systems of equations, which was studied in Subsection \[BinomVieta\].
In the notation of Subsection \[DefVieta\], consider the function $\Phi_0\colon\C_1^P\setminus(\mathscr H\cup\mathscr D)\to\C$, defined as follows: $$\Phi_0\colon (f_1,\ldots, f_n)\mapsto\prod_{f_1(x)=\ldots=f_n(x), x\in\CC^n} x^a.$$
\[const\] The function $\Phi_0$ is a monomial in the coefficients of $f_1,\ldots,f_n$ at the vertices of their Newton polytopes.
Obviously, $\Phi_0$ is a well-defined holomorphic function with no zeroes and poles in the open dense subset $\C_1^P\setminus(\mathscr H\cup\mathscr D)$. Our goal is to show that this function can be extended to a holomorphic function $\Phi\colon\C_1^P\to\C$ with no zeroes and poles and, therefore, is a monomial in the coefficients of $f_1,\ldots,f_n$ at the vertices of their Newton polytopes. We will construct this extension in several steps.
There exists a holomorphic extension of the function $\Phi_0$ on the set ${\C_1^P\setminus(\mathscr H'\cup\mathscr D)}$.
$\vartriangleleft$ Since the product of roots is invariant under monomial changes of variables, we can assume without loss of generality that $a=(0,\ldots,0,\alpha)\in\Z^n$. Consider the projection $\pi\colon\Z^n\to\Z^{n-1}$ along the radius vector of the point $a$ and the tuple $Q=(Q_1,\ldots,Q_n)$ of convex lattice polytopes, where $Q_i={\mathop{\rm conv}\nolimits}(\pi(P_i))\subset\R^{n-1}$. Take a simple fan $\Omega$ compatible with the tuple $Q$ of polytopes and consider the smooth projective toric variety $X_{\Omega}$. We obtain an inclusion $\CC^{n-1}\times\CC\hookrightarrow X_{\Omega}\times\CC$.
Let $\Phi_1\colon\C_1^p\setminus(\mathscr H'\cup\mathscr D)\to\C$ be the function defined as follows: $$\label{rootprod}
\Phi_1\colon (f_1,\ldots, f_n)\mapsto \prod_{f_1(x)=\ldots=f_n(x), x\in X_{\Omega}\times\CC} x^a.$$ This function is well-defined, since at a point $x=(y,t)\in X_{\Omega}\times\CC$, the monomial $x^a$ equals $t^{\alpha}\in\CC$, therefore, the function $\Phi_1$ is the sought extension of $\Phi_0$. $\vartriangleright$
The function $\Phi_1$ can be regularly extended to a function on the set ${\C_1^p}$.
$\vartriangleleft$
\[killsing\] Let $\Psi\colon\C^N\setminus\Sigma\to\mathbb C$, where ${\mathop{\rm codim}\nolimits}(\Sigma)=1$, be a holomorphic function. If $\Sigma'\subset \Sigma$ is such that ${\mathop{\rm codim}\nolimits}(\Sigma')>1$ and $\Psi$ can be continuously extended to $\tilde{\Psi}\colon\C^N\setminus\Sigma'\to\C$, then there exists a holomorphic extension $\bar{\Psi}\colon\mathbb{C}^N\to\mathbb C$ of $\Psi$.
Take an isolated multiple root $q$ for the system $\{f_1=\ldots=f_n=0\}$, where $(f_1,\ldots, f_n)\in\C_1^P\setminus(\mathscr H'\cup \mathscr D')$. Its multiplicity equals the degree of the map $F\colon\mathscr U\to\C^n$, where $\mathscr U\subset \C^n$ is a small neighborhood of the point $q$, $F\colon x\mapsto (f_1(x),\ldots, f_n(x))$. For almost all points $\varepsilon=(\varepsilon_1,\ldots, \varepsilon_n)$ in a sufficiently small neighborhood $0\in\mathscr V\subset\C^n$, the number of preimages $F^{-1}(\varepsilon)$ equals the same number $k$ of multiplicity $1$ roots $q_1(\varepsilon),\ldots, q_k(\varepsilon)$ of the system $\{f_1(x)=\varepsilon_1,\ldots f_n(x)=\varepsilon_n\}$, which are contained in $\mathscr U$. Moreover, as $\varepsilon_i\rightarrow 0, 1\leqslant i\leqslant n$, we have $q_j(\varepsilon)\rightarrow q, 1\leqslant j\leqslant k$. Therefore, for the product of monomials, we have $\prod_1^k (q_k(\varepsilon))^a\rightarrow (q^k)^a$. Thus, letting the monomials $x^a$ enter the product the number of times equal to the multiplicity of the corresponding root $x$, we obtain a continuous extension $\tilde{\Phi}_1\colon\C_1^P\setminus (\mathscr H'\cup\mathscr D')$.
Using Proposition \[killsing\] for $\Psi=\Phi_1$, $\Sigma=\mathscr H\cup\mathscr D$ and $\Sigma'=\mathscr H'\cup\mathscr D'$, we obtain the desired holomorphic extension $\Phi\colon\C_1^P\to\C$. $\vartriangleright$
Note that the function $\Phi$ has no zeroes and poles in $\C_1^P$, therefore, it is a monomial, which finishes the proof of the theorem.
It follows from Theorem \[const\] that there exists a well-defined function $\Phi(P_1,\ldots, P_n; a)$, which maps $a\in\Z^n$ is a point and $P=(P_1,\ldots, P_n)$ is an $a$-prickly tuple of polytopes in $\R^n$ to the product (\[rootprod\]) of monomials over the roots for a polynomial system of equations ${\{f_1(x)=\ldots=f_n(x)=0\},}$ where ${(f_1,\ldots, f_n)\in\C_1^P}$ is an arbitrary point such that all coefficients of $f_i$ at the vertices of its Newton polytope are equal to 1.
Now we are ready to state the multivariate Vieta’s formula.
\[VietaMain\] Under the same assumptions as above, we have $$\Phi(P_1,\ldots, P_n; a)=(-1)^{{\mathop{\rm MV}\nolimits}_2(P_1,\ldots, P_n; a)}. \label{Vieta}$$
The main idea of the proof is to introduce a new variable, a parameter $t$, in such a way that as $t\rightarrow\infty$, the system of equations asymptotically breaks down into a union of binomial systems, which were considered in Subsection \[BinomVieta\].
Consider an arbitrary point $(f_1,\ldots, f_n)\in\C_1^P$, where $f_i(x)=\sum_{k\in P_i\cap\Z^n} c_{i,k}x^k$. With the system $\{f_1(x)=\ldots=f_n(x)=0\}$ we associate a perturbed system $\{\tilde{f}_1(x,t)=\ldots=\tilde{f}_n(x,t)=0\}$ of equations of the following form: $$\tilde{f}_i(x,t)=\sum_{k\in P_i\cap\Z^n} c_{i,k}t^{n_{i,k}}x^k,$$ where $n_{i,k}$ are non-negative integers. The Newton polytopes of $\tilde{f}_i, 1\leqslant i\leqslant n,$ are denoted by $\tilde{P}_i$. Note that $\tilde{P}_i\subset\R^n\times \R$ are lattice polytopes lying over the polytopes $P_i$. By $\rho\colon\R^{n}\times\R\to\R^n$ we denote the projection forgetting the last coordinate.
\[uppface\] A face $\tilde{\Gamma}\subset\tilde{P}_i$ is said to be an [*upper face*]{}, if there exists a covector $v\in(\R^*)^n+1$ with the strictly positive last coordinate such that $\tilde{\Gamma}=\tilde{P}_i^v$.
Each of the polytopes $\tilde{P}_i$ defines a convex subdivision $\Delta_i$ of the polytope $P_i$. Namely, each cell $\Gamma\in\Delta_i$ is equal to $\rho(\tilde{\Gamma})$ for some upper face $\tilde{\Gamma}\subset\tilde{P}_i$. In the same way, the Minkowski sum $\EuScript{\tilde{P}}=\sum \tilde{P}_i$ defines a convex subdivision $\Delta$ of the polytope $\EuScript{P}=\sum P_i$. Each cell $\Gamma\in\Delta$ can be uniquely represented as a sum $\Gamma=\Gamma_1+\ldots+\Gamma_n$, where $\Gamma_i\in\Delta_i$.
To the tuple of polynomials $\tilde{f}_1,\ldots, \tilde{f}_n, \tilde{f}_i=\sum_{k\in P_i\cap\Z^n} c_k x^k t^{n_{i,k}}$, we associate tuples of polynomials $g_1,\ldots, g_n$ and $\tilde{g}_1,\ldots, \tilde{g}_n$ defined as follows: $$g_i=\sum_{k\in P_i\cap\Z^n} a_k x^k,$$ where $a_k$ equals $1$, if $k$ is a vertex of a cell of $\Delta_i$, and $0$ otherwise, and $\tilde{g}_i=\sum a_k x^k t^n_k, 1\leqslant i\leqslant n$. From Theorem \[const\], it follows that the sought product of monomials does not depend on the choice of the system of equations, so, it suffices to compute it for the system $\{g_1(x)=\ldots=g_n(x)=0\}$.
\[consistface\] A tuple of faces $\Gamma_1,\ldots,\Gamma_n, \Gamma_i\subset\tilde{P}_i,$ is said to be [*consistent*]{}, if there exists a covector $v\in(\R^*)^n$ such that $\Gamma_i=\tilde{P}_i^{v}$.
[@det2]\[defaffin\] A tuple of consistent faces $\Gamma_1,\ldots,\Gamma_n, \Gamma_i\subset\tilde{P}_i,$ is said to be [*affinely independent*]{}, if for the Minkowski sum $\Gamma=\Gamma_1+\ldots+\Gamma_n$, the equality $\dim(\Gamma)=\dim(\Gamma_1)+\ldots+\dim(\Gamma_n)$ holds.
Clearly, if the faces $\Gamma_1,\ldots, \Gamma_n$ are affinely independent, then either one of these faces is a point, or these faces are all segments.
[@det2]\[affind\] In the previous notation, we can choose the polytopes ${\tilde{P_i},1\leqslant i\leqslant n,}$ in such a way that any consistent tuple $\Gamma_1,\ldots, \Gamma_n$ of faces is affinely independent.
It follows from Proposition \[affind\] that without loss of generality we can assume that all the consistent tuples $\Gamma_1,\ldots, \Gamma_n$ are affinely independent.
By $\K$ we denote the field $\C\{\{t\}\}$ of Puiseux series with the standard valuation function ${\mathop{\rm val}\nolimits}\colon(\C\{\{t\}\}\setminus 0)\to\R$. If $0\neq b\in \K$ and ${\mathop{\rm val}\nolimits}(b)=q$, then we write $b=\beta t^q+\ldots$ to distinguish the leading term. Note also that we identify the space $\R^n$ with its dual by means of the standard inner product.
For almost all the values $\tau$ of the parameter $t$, the systems ${\{\tilde{g}_1(x,\tau)=\ldots=\tilde{g}_n(x,\tau)=0\}}$ have the same finite number of roots $z_1(\tau),\ldots, z_d(\tau)$. It follows from the Bernstein–Kushnirenko formula that $d={\mathop{\rm MV}\nolimits}(P_1,\ldots, P_n)$, see [@bernstein] for the details. Each of the roots $z_j(t)$ can be considered as an algebraic function $z_j\colon\CC\to\CC^n$ in the variable $t$. The Puiseux series of the curve $z_j$ at infinity is of the following form: $$\label{roott}
z_j(t)=(\beta_1t^{v_1},\ldots,\beta_n t^{v_n})+\mathrm{componentwise~lower~ terms}.$$ For simplicity of notation, we shall write the expression (\[roott\]) in the following form: $z_j(t)=(B_j t^{V_j})+\ldots$, where $B_j=(\beta_1,\ldots, \beta_n)$ and $V_j$ is the valuation vector $(v_1,\ldots, v_n)$.
In the previous notation, consider the valuation vectors $V_j=(v_1\ldots, v_n)$ and $W_j=(v_1, \ldots, v_n, 1)$ of the roots $z_j(t)$ and $Z_j=(z_j(t), t)$ respectively. The function $\langle W_j, \cdot \rangle$ attains its maximum at some faces $\Gamma_1^{W_j}, \ldots,\Gamma_n^{W_j}, \Gamma_i^{W_j}\subset \tilde{P}_i$, which are, according to our assumption, affinely independent, and, therefore, if none of them is a vertex, then they are all segments. By $G_i^{W_j}$ we denote the set $\rho(\Gamma_i^{W_j})\cap \Z^n$. Note that if $\Gamma_i^{W_j}$ is a segment, then $\vert G_i^{W_j} \vert=2$ (those 2 elements are exactly the endpoints of the segment $\rho(\Gamma_i^{W_j})$).
Now, let us substitute the root $Z_j=(z_j(t),t)$ into the equations $\tilde{g}_1(x,t), \ldots, \tilde{g}_1(x,t)$. On one hand, what we obtain is nothing but $0$. On the other hand, the result is a Puiseux series in the variable $t$: $$\label{rootpuiseux}
\tilde{g}_i(z_j(t),t)=\sum_{k\in P_i\cap\Z^n} a_k (\prod_{m=1}^n \beta_m^{v_m}) t^{\langle V_j, k \rangle + n_{i,k}}+\ldots=\sum_{k\in P_i\cap\Z^n} a_k B_j^{V_j} t^{\langle V_j, k \rangle + n_{i,k}}+\ldots.$$
\[binreduce\] It is obvious that the leading coefficient of the series (\[rootpuiseux\]) equals the sum $$h_i(B_j)=\sum_{k\in G_i^w} a_k B_j^k,$$ while $\tilde{g}_i(z_j(t),t)=0$. Therefore, $B_j=(\beta_1, \ldots, \beta_n)$ is a root of the system $\{h_1(x)=\ldots=h_n(x)=0\}$, where each of the polynomials $h_i$ is obtained by omitting the terms that are not contained in $G_i^{W_j}$.
If the tuple $G_i^{W_j}, 1\leqslant j\leqslant n,$ contains a singleton $G_m^{W_j}$ for some $m$, then, the corresponding system $\{h_1(x)=\ldots=h_n(x)=0\}$ is inconsisent in $\CC^n$, since the polynomial $h_m$ is a monomial. Therefore, all the roots $B_j$ mentioned in Remark \[binreduce\] are the roots for the corresponding binomial system $\{h_1(x)=\ldots=h_n(x)=0\}$. The number of the roots for such a system equals ${\mathop{\rm MV}\nolimits}(G_1^{W_j}, \ldots, G_n^{W_j})$, by the Bernstein–Kushnirenko theorem, see [@bernstein].
It follows from Remark \[binreduce\] and Lemma \[VietaBin\] that the product of monomials from Theorem \[VietaMain\] equals the sign of the limit $$\lim_{t\rightarrow+\infty}(\prod_{j=1}^d(Z_j(t))^a)=\lim_{t\rightarrow+\infty}(\prod_{j=1}^d (B_j t^{V_j})^a+\ldots)=\lim_{t\rightarrow+\infty}(\prod_{j=1}^d B_j^a t^{\langle V_j,a\rangle})+\ldots).$$
Note that by Lemma \[VietaBin\] and the definition of the 2-mixed volume (see Definition \[2-volume\]), the product $\prod_{j=1}^d(B_j)^a$ equals exactly the 2-mixed volume $(-1)^{{\mathop{\rm MV}\nolimits}_2(P_1,\ldots, P_n; a)}$, so, the sought product can be expressed as the limit $$\label{rootmove}
\lim_{t\rightarrow+\infty}(\prod_{j=1}^d(Z_j(t))^a)=(-1)^{{\mathop{\rm MV}\nolimits}_2(P_1,\ldots, P_n; a)}\lim_{t\rightarrow+\infty}(t^{\sum^{d}_{j=1} \langle V_j,a\rangle}+\ldots).$$
From Theorem \[const\], it follows that the sought product of monomials is a monomial. Therefore, it suffices to compute the sign of the limit (\[rootmove\]) as $t\rightarrow\infty, t\in \R_{>0}$, which obviously equals $(-1)^{{\mathop{\rm MV}\nolimits}_2(P_1,\ldots, P_n; a)}$.
Signs of the Leading Coefficients of the Resultant {#resultant}
==================================================
In this Section, we show how to reduce the computation of the leading coefficients of the resultant to finding the product of roots for a certain system of equations. In Subsection \[NewtPolRes\], we recall the definition of the [*sparse mixed resultant*]{}. Then, using the multivariate Vieta’s formula (see Theorem \[VietaMain\]), we obtain a closed-form expression in terms of the 2-mixed volume to compute the signs of the leading coefficients of the sparse mixed resultant.
The Sparse Mixed Resultant and Its Newton Polytope {#NewtPolRes}
--------------------------------------------------
Recall the definition of the codimension of a tuple of finite sets in $\Z^n$ given in Subsection \[buildwall\].
Let $A=(A_0,\ldots, A_n)$ be a $n$-tuple of finite sets in $\Z^n$. For an arbitrary subset $I\subset\{0,\ldots,n\}$, we define its [*codimension*]{}, which we denote by ${\mathop{\rm codim}\nolimits}(I)$, as follows: $${\mathop{\rm codim}\nolimits}(I)=\dim(\sum_{i\in I} A_i)-|I|.$$ The [*codimension of the tuple*]{} $A$ is defined by the following equality: $${\mathop{\rm codim}\nolimits}(A)=\min_{I\subset\{0,\ldots,n\}}{\mathop{\rm codim}\nolimits}(I).$$
[@sturmfels]\[resuldef\] Consider a tuple $A=(A_0,\ldots, A_n)$ of finite sets in $\Z^n$ such that ${\mathop{\rm codim}\nolimits}(A)=-1$ and the sets $A_i$ jointly generate the affine lattice $\Z^n$. Then the [*sparse mixed resultant*]{} $\resul(A)$ is a unique (up to scaling) irreducible polynomial in $\sum_0^n|A_i|$ variables $c_{i,a}$ which vanishes whenever the Laurent polynomials $f_i(x)=\sum_{a\in A_i} c_{i,a}x^a$ have a common zero in $\CC^n$.
Here we provide an explicit description of the vertices of the Newton polytope of the sparse mixed resultant $\resul(A)$. For more details and proofs we refer the reader to the paper [@sturmfels].
Let $Q=(Q_0,\ldots, Q_n)$ be the tuple of convex hulls of the sets $A_i$. Let $\omega\colon\cup_{i=0}^n Q_i\to \R_{\geqslant 0}$ be an arbitrary function. By $\tilde{Q}=(\tilde{Q}_0,\ldots, \tilde{Q}_n)$ denote the tuple of convex hulls of the sets $Q_i(\omega)=\{(a, \omega(a))\mid a\in Q_i\}$ and by $\rho$ the standard projection $\R^{n+1}\to\R^n$ forgetting the last coordinate. The polytopes $\tilde{Q}_i, 0\leqslant i\leqslant n,$ define convex subdivisions $\Delta_i$ and $\Delta$ of polytopes $Q_i, 0\leqslant i\leqslant n,$ and $\EuScript{Q}=Q_0+Q_1+\ldots+Q_n$, respectively. Namely, each cell $\Gamma\in\Delta_i$ is equal to $\rho(\tilde{\Gamma})$ for some upper face $\tilde{\Gamma}\subset\tilde{Q}_i$. In the same way, the Minkowski sum $\EuScript{\tilde{Q}}=\sum \tilde{Q}_i$ defines a convex subdivision $\Delta$ of the polytope $\EuScript{Q}=\sum Q_i$.
In the previous notation, a tuple of subdivisions $\Delta_i$ of the polytopes $Q_i$ is called a [*mixed subdivision*]{} of the polytopes $Q_0,\ldots, Q_n$.
Note that each upper face $\tilde{\Gamma}\subset\EuScript{\tilde{Q}}$ (see Definition \[uppface\]) can be uniquely represented as a sum ${\tilde{\Gamma}=\tilde{\Gamma}_0+\ldots+\tilde{
\Gamma}_n}$, where $\tilde{ \Gamma}_i\subset\tilde{Q}_i$. Proposition \[affind\] implies that without loss of genericity, we can assume that the all the tuples of consistent faces $(\tilde{\Gamma}_0,\ldots, \tilde{\Gamma}_n)$ are affinely independent (see Definitions \[consistface\] and \[defaffin\]). By the Dirichlet principle, we have that for every upper face ${\tilde{\Gamma}}\subset \EuScript{\tilde{Q}}$, one of the faces, say, $\tilde{ \Gamma}_j$, employed in the decomposition ${\tilde{\Gamma}=\tilde{\Gamma}_0+\ldots+\tilde{
\Gamma}_n}$ is a vertex. By $c_{\Gamma}$ we denote the corresponding coefficient $c_{j, \rho{\tilde{\Gamma}}}$ of the polynomial $f_j$ of the system $\{f_0(x)=\ldots=f_n(x)=0\}$.
To each of the mixed subdivisions we can associate a vertex of the Newton polytope $\newton{\resul(A)}$ as follows. The leading term of $\resul(A)$ which corresponds to a given mixed subdivision $\Delta_0,\ldots,\Delta_n$ is the product over all the facets $\Gamma\in \Delta$ (i.e., faces of codimension 1) in the subdivision $\Delta$ of the Minkowski sum $\EuScript{Q}$ of the multiples equal to $c_{\Gamma}^{{\mathop{\rm Vol}\nolimits}(\Gamma)}$.
The construction given above provides a bijection between the set of all the mixed subdivisions of the tuple $Q=(Q_0,\ldots, Q_n)$ and the vertices of the Newton polytope $\newton(\resul(A))$ of the resultant.
Take $A=(A_0, A_1)$, where $A_0=\{0,1\}, A_1=\{0,1,2\}\subset\Z$. Our aim is construct the Newton polytope of the polynomial $\resul(A)\in\C[a_0, a_1, b_0, b_1, b_2]$ which vanishes whenever the system $\{a_0+a_1x=b_0+b_1x+b_2x^2=0\}$ is consistent. In the same notation as above, we have $Q_0=[0,1], Q_1=[0,2]$. The following figure describes one of the three possible mixed subdivisions of the intervals $Q_0$ and $Q_1$ which yields the vertex $(1,1,0,1,0)\in\newton(\resul(A))\subset\R^5$.
(0,0) grid (2,2); (3,0) grid (5,2); (7,0) grid (11,4); (7,0)–(8,0); (8,0)–(9,0); (9,0)–(10,0); (3,0) circle\[radius=0.07\]; (5,0) circle\[radius=0.07\]; (7,0) circle\[radius=0.07\]; (8,0) circle\[radius=0.07\]; (9,0) circle\[radius=0.07\]; (10,0) circle\[radius=0.07\]; (0,0)–(0,1); (1,0)–(1,1); (3,0)–(3,1); (4,0)–(4,2); (5,0)–(5,1); (7,0)–(7,2); (8,0)–(8,3); (9,0)–(9,3); (10,0)–(10,2); (7,2)–(8,3); (8,3)–(9,3); (9,3)–(10,2); (7,2) circle\[radius=0.07\]; (8,3) circle\[radius=0.07\]; (9,3) circle\[radius=0.07\]; (10,2) circle\[radius=0.07\]; (3,1)–(4,2); (0,1)–(1,1); (4,2)–(5,1); (3,0)–(4,0); (0,0)–(1,0); (4,0)–(5,0); at (0,-0.1) [$a_0$]{}; at (1,-0.1) [$a_1$]{}; at (3,-0.1) [$b_0$]{}; at (4,-0.1) [$b_1$]{}; at (5,-0.1) [$b_2$]{}; (0,1) circle\[radius=0.1\]; (4,2) circle\[radius=0.1\]; (1,1) circle\[radius=0.1\]; (3,1) circle\[radius=0.07\]; (5,1) circle\[radius=0.07\]; (0,0) circle\[radius=0.14\]; (1,0) circle\[radius=0.14\]; (4,0) circle\[radius=0.14\]; (12,1.5) – (13.5,1.5); at (14,1.5) [$a_0^1$]{}; at (14.4,1.5) [$a_1^1$]{}; at (14.8,1.5) [$b_1^1$]{}; at (7, -1.2) [Figure 3. The bijection between the mixed subdivisions of $A$ and the vertices of $\newton(\resul(A))$.]{};
Computing the Signs of the Leading Coefficients of the Resultant {#SignsRes}
----------------------------------------------------------------
In the previous notation, consider a tuple $A=(A_0,\ldots, A_n)$ of finite sets in $\Z^n$ satisfying the properties given in Definition \[resuldef\]. Recall that by $|A_i|$ we denote the cardinality of the set $A_i\subset \Z^n$, and $|A|$ stands for the sum $\sum_0^n|A_i|$. For simplicity of notation, by $\resul$ we denote the sparse mixed resultant $\resul(A)$.
Consider the Newton polytope $\newton(\resul)$ of the resultant $\resul(A)$ (see [@sturmfels] for its explicit description). Suppose that we are given a pair of gradings ${\gamma=(\alpha_{i,a}\mid i\in\{0,\ldots,n\}, a\in A_i)}$ and ${\sigma=(\beta_{j,b}\mid j\in\{0,\ldots,n\}, b\in A_i)\in (\Z^*)^{|A|}}$ with strictly positive coordinates such that the support faces $\newton(\resul)^{\gamma}$ and $\newton(\resul)^{\sigma}$ are $0$-dimensional. We will now compute the quotient of the coefficients ${r_{\gamma}}$ and ${r_{\sigma}}$ of $\resul$ which are leading with respect to the gradings $\gamma$ and $\sigma$ respectively, by reducing this problem to the multivariate Vieta’s formula (see Theorem \[VietaMain\]).
To the covectors $\gamma, \sigma$ one can associate the tuple $P^{\gamma,\sigma}=(P_0^{\gamma,\sigma},\ldots, P_n^{\gamma,\sigma})$ of polytopes in $\R^{n+1}$ such that\
$${P_i^{\gamma,\sigma}={\mathop{\rm conv}\nolimits}(\{(a, \alpha_{i,a})\mid a\in A_i\}\cup\{(a, -\beta_{i,a})\mid a\in A_i\})}.$$
\[exa1\] Let $A=(A_0, A_1)$, where $A_0=\{0,1\}, A_1=\{0,1,2\}\subset\Z$. The Newton polytope $\newton(\resul(A))$ is a triangle with vertices $\bar{\gamma}=(2,0,0,0,1)$, $\bar{\sigma}=(0,2,1,0,0)$ and $\bar{\delta}=(1,1,0,1,0)$. Consider the covectors $\gamma=(2,1,1,1,2)$, $\sigma=(1,2,2,1,1)$, and $\delta=(2,2,1,2,1)$, whose support faces are the vertices $\bar{\gamma}, \bar{\sigma}, \bar{\delta}$. Thus, we obtain the the polygons $P_0^{\gamma,\sigma}$ and $P_1^{\gamma,\sigma}$ (see fig. 4a) and the polygons $P_0^{\gamma,\delta}$ and $P_1^{\gamma,\delta}$ (see fig. 4b).
(-5,0)–(-4,0); (-5,0) circle\[radius=0.06\]; (-4,0) circle\[radius=0.06\]; (-5,-1)–(-5,2); (-5,-1) circle\[radius=0.1\]; (-5,2) circle\[radius=0.1\]; (-4,-2)–(-4,1); (-4,-2) circle\[radius=0.1\]; (-4,1) circle\[radius=0.1\]; (-5,2)–(-4,1); (-5,-1)–(-4,-2); (-1,0)–(1,0); (-1,0) circle\[radius=0.06\]; (0,0) circle\[radius=0.06\]; (1,0) circle\[radius=0.06\]; (-1,-2)–(-1,1); (-0,-1)–(-0,1); (1,-1)–(1,2); (-1,1)–(1,2); (-1,-2)–(1,-1); (1,-1) circle\[radius=0.1\]; (1,2) circle\[radius=0.1\]; (-1,-2) circle\[radius=0.1\]; (-1,1) circle\[radius=0.1\]; (-5,1) circle\[radius=0.06\]; (-4,-1) circle\[radius=0.06\]; (-1,-1) circle\[radius=0.06\]; (-1,0) circle\[radius=0.06\]; (1,1) circle\[radius=0.06\]; (0,-1) circle\[radius=0.06\]; (0,1) circle\[radius=0.06\]; at (-2.5,-2.5) [Figure 4a.]{}; at (-2,-3) [The polygons $P_0^{\gamma,\sigma}$ and $P_1^{\gamma,\sigma}$]{}; (3,0)–(4,0); (3,0) circle\[radius=0.06\]; (4,0) circle\[radius=0.06\]; (3,-2)–(3,2); (3,-2) circle\[radius=0.1\]; (3,2) circle\[radius=0.1\]; (4,-2)–(4,1); (4,-2) circle\[radius=0.1\]; (4,1) circle\[radius=0.1\]; (3,2)–(4,1); (3,-2)–(4,-2); (7,0)–(9,0); (7,0) circle\[radius=0.06\]; (8,0) circle\[radius=0.06\]; (9,0) circle\[radius=0.06\]; (7,-1)–(7,1); (8,-2)–(8,1); (9,-1)–(9,2); (7,1)–(9,2); (7,-1)–(8,-2)–(9,-1); (9,-1) circle\[radius=0.1\]; (9,2) circle\[radius=0.1\]; (8,-2) circle\[radius=0.1\]; (7,-1) circle\[radius=0.1\]; (7,1) circle\[radius=0.1\]; (3,1) circle\[radius=0.06\]; (3,-1) circle\[radius=0.06\]; (4,-1) circle\[radius=0.06\]; (8,-1) circle\[radius=0.06\]; (8,1) circle\[radius=0.06\]; (9,1) circle\[radius=0.06\]; at (6,-2.5) [Figure 4b.]{}; at (6 ,-3) [The polygons $P_0^{\gamma,\delta}$ and $P_1^{\gamma,\delta}$]{};
\[restheo\] Let $A=(A_0,\ldots, A_n)$ be a tuple of finite sets in $\Z^n$ satisfying the properties given in Definition \[resuldef\] and $\gamma,\sigma\in(\Z^*)^{|A|}$ be a pair of gradings with strictly positive coordinates and $0$-dimensional support faces $\newton(\resul)_{\gamma}$ and $\newton(\resul)_{\sigma}$. Then the quotient of the coefficients ${r_{\gamma}}$ and ${r_{\sigma}}$ of $\resul(A)$ that are leading with respect to the gradings $\gamma$ and $\sigma$ respectively can be computed as follows: $$\label{formula_res}
\frac{r_{\gamma}}{r_{\sigma}}=(-1)^{{\mathop{\rm MV}\nolimits}(P_0^{\gamma,\sigma},\ldots, P_n^{\gamma,\sigma})}(-1)^{{\mathop{\rm MV}\nolimits}_2(P_0^{\gamma,\sigma},\ldots, P_n^{\gamma,\sigma},\bigl(\begin{smallmatrix}{0}\\{1}\end{smallmatrix}\bigr))}.$$
Using Theorem \[restheo\], let us compute the quotient of the coefficients ${r_{\gamma}}$ and ${r_{\sigma}}$ corresponding to the vertices $\bar\gamma$ and $\bar\sigma$ which were considered in \[exa1\]:
$$\frac{r_{\gamma}}{r_{\sigma}}=1\cdot (-1)^{\det_2\bigl(\begin{smallmatrix} {1} & {0} & {0}\\{1} & {1} & {1} \end{smallmatrix}\bigr)+\det_2\bigl(\begin{smallmatrix} {0} & {0} & {0}\\{1} & {1} & {1} \end{smallmatrix}\bigr)+\det_2\bigl(\begin{smallmatrix} {1} & {0} & {0}\\{1} & {0} & {1} \end{smallmatrix}\bigr)}=(-1)^0=1.$$
For the coefficients ${r_{\gamma}}$ and ${r_{\delta}}$ corresponding to the vertices $\bar\gamma$ and $\bar\delta$, we obtain
$$\frac{r_{\gamma}}{r_{\delta}}=1\cdot (-1)^{\det_2\bigl(\begin{smallmatrix} {0} & {1} & {0}\\{1} & {1} & {1} \end{smallmatrix}\bigr)+\det_2\bigl(\begin{smallmatrix} {0} & {0} & {0}\\{1} & {1} & {1} \end{smallmatrix}\bigr)+\det_2\bigl(\begin{smallmatrix} {1} & {0} & {0}\\{0} & {0} & {1} \end{smallmatrix}\bigr)+\det_2\bigl(\begin{smallmatrix} {1} & {1} & {0}\\{0} & {1} & {1} \end{smallmatrix}\bigr)}=(-1)^1=-1.$$
Thus, we obtain the well-known formula for the resultant $\resul=\resul(f,g)$ of the polynomials $f=a_0+a_1x, g= b_0+b_1x+b_2x^2$: we have $\resul=\pm(a_0^2b_2+a_1^2b_0-a_0a_1b_1)$, just as expected.
The rest of this Subsection is devoted to the proof of Theorem \[restheo\].
To the gradings $\gamma, \sigma$, we can associate the [*Khovanskii curve*]{} $\mathscr{C}^{\gamma,\sigma}\subset\C^{|A|}$ parametrized by the complex parameter $t\neq 0$ and defined by the following equations: $z_{i,a}=t^{\alpha_{i,a}}+t^{-\beta_{i,a}}$, where $i\in\{0,\ldots,n\}$ and $a\in A_i$.
Restricting the resultant $\resul$ to the Khovanskii curve $\mathscr{C}^{\gamma,\sigma}$, we obtain a Laurent polynomial in the variable $t$, which we denote by $\phi(t)$. The following statements are obvious.
The coefficient of the leading (lowest) term of $\phi(t)$ equals $r_{\gamma}$ ($r_{\sigma}$, respectively).
\[zerophi\] The equality $\phi(t_0)=0$ holds if and only if the point with coordinates $(t_0^{\alpha_{i,a}}+t_0^{-\beta_{i,a}}\mid i\in\{0,\ldots,n\}, a\in A_i)$ belongs to the set $\{\resul=0\}\cap\mathscr C^{\gamma,\sigma}$.
\[res\_t\] Note that the polytopes $P_0^{\gamma,\sigma},\ldots, P_n^{\gamma,\sigma}$ are exactly the Newton polytopes of the Laurent polynomials $g_0(x,t),\ldots, g_n(x,t)$, where $$g_i(x,t)=\sum_{a\in A_i}(t^{\alpha_{i,a}}+t^{-\beta_{i,a}})x^a.$$
Remark \[res\_t\] implies that the equality (\[formula\_res\]) can be rewritten as follows: $$\frac{r_{\gamma}}{r_{\sigma}}=(-1)^{{\mathop{\rm MV}\nolimits}(\newton(g_0),\ldots,\newton(g_n))}(-1)^{{\mathop{\rm MV}\nolimits}_2(\newton(g_0),\ldots,\newton(g_n),\bigl(\begin{smallmatrix}{0}\\{1}\end{smallmatrix}\bigr))}.$$
At the same time, using the classical Vieta’s formula, we obtain $$\frac{r_{\sigma}}{r_{\gamma}}=\prod_{\phi(t)=0}t.$$
It follows from Proposition \[zerophi\] and the Bernstein theorem (see [@bernstein] for the details) that $$\prod_{\phi(t)=0}t=(-1)^{|\{\resul=0\}\cap\mathscr C^{\gamma,\sigma}|}\prod_{g_0(x,t)=\ldots=g_n(x,t)=0}t=(-1)^{{\mathop{\rm MV}\nolimits}(\newton(g_0),\ldots,\newton(g_n))}\prod_{g_0(x,t)=\ldots=g_n(x,t)=0}t.$$ Then, applying the multivariate Vieta’s formula (see Theorem \[VietaMain\]), we have $$(-1)^{{\mathop{\rm MV}\nolimits}(\newton(g_0),\ldots,\newton(g_n))}\prod_{g_0(x,t)=\ldots=g_n(x,t)=0}t=(-1)^{{\mathop{\rm MV}\nolimits}(\newton(g_0),\ldots,\newton(g_n))}(-1)^{{\mathop{\rm MV}\nolimits}_2(\newton(g_0),\ldots,\newton(g_n),\bigl(\begin{smallmatrix}{0}\\{1}\end{smallmatrix}\bigr))},$$ which finishes the proof of the theorem.
[99]{} Bernstein D.N., 1975: [*The number of roots of a system of equations*]{}, Funct Anal Appl 9:183–185. Brugallé E., Shaw K., 2013: [*A bit of tropical geometry*]{}, arXiv:1311.2360v3 \[math.AG\]. Gelfand I.M., Kapranov M.M., Zelevinsky A.V., 1994: [*Discriminants, resultants, and multidimensional determinants*]{}, Math. Theory Appl., Birkhauser Boston, Boston, MA. Khovanskii A.G., Esterov A.I., 2008: [*Elimination theory and Newton polytopes*]{}, Funct. Anal. Other Math., no.1, 45–71. Khovanskii A.G., 1999: [*Newton polyhedra, a new formula for mixed volume, product of roots of a system of equations*]{}, The Arnoldfest, Toronto, 1997, Fields inst commun, vol. 24. Am. Math. Soc., Providence, pp 325–364. Parshin A.N., 1985: [*Local class field theory*]{}, Proceedings of the Steklov Institute of Mathematics, 165, 157–-185. Maclagan D., Sturmfels B., 2015: [*Introduction to Tropical Geometry*]{}, Graduate Studies in Mathematics, vol. 161. Am. Math. Soc., Providence, RI. Sturmfels B., 1994: [*On the Newton polytope of the resultant*]{}, J. Algebraic Combin. 3 (1994), no. 2, 207–236.
[^1]: National Research University Higher School of Economics The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2016-2017(grant N 16-01-0069) and supported within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.
|
---
abstract: 'We prove that any $k$-uniform hypergraph on $n$ vertices with minimum degree at least $\frac{n}{2(k-1)}+o(n)$ contains a loose Hamilton cycle. The proof strategy is similar to that used by Kühn and Osthus for the 3-uniform case. Though some additional difficulties arise in the $k$-uniform case, our argument here is considerably simplified by applying the recent hypergraph blow-up lemma of Keevash.'
author:
- Peter Keevash
- Daniela Kühn
- Richard Mycroft
- Deryk Osthus
title: Loose Hamilton Cycles in Hypergraphs
---
Introduction {#intro}
============
A fundamental theorem of Dirac [@DIRAC] states that any graph on $n$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle. A natural question is whether this theorem can be extended to hypergraphs.
For this, we first need to extend the notions of minimum degree and of Hamilton cycles to hypergraphs. A *$k$-uniform hypergraph* or *$k$-graph* $H$ consists of a vertex set $V$ and a set of edges each consisting of $k$ vertices. We will often identify $H$ with its edge set and write $e \in H$ if $e$ is an edge of $H$. Given a $k$-graph $H$, we say that a set of $k-1$ vertices $T \in \binom{V}{k-1}$ has *neighbourhood* $N_H(T) = \{x \in V: \{x\} \cup T \in H \}$. The *degree of $T$* is $d_{k-1}(T) = |N_H(T)|$. The *minimum degree* of $H$ is the minimum size of such a neighbourhood, that is, $\delta_{k-1}(H) = \min \{d_{k-1}(T) : T \in \binom{V}{k-1}\}$.
We say that a $k$-graph $C$ is a *cycle of order $n$* if its vertices can be given a cyclic ordering $v_1,\dots, v_n$ so that every consecutive pair $v_i, v_{i+1}$ lies in an edge of $C$ and every edge of $C$ consists of $k$ consecutive vertices. A cycle of order $n$ is *tight* if every set of $k$ consecutive vertices forms an edge; it is *loose* if every pair of adjacent edges intersects in a single vertex, with the possible exception of one pair of edges, which may intersect in more than one vertex. This final condition allows us to consider loose cycles whose order is not a multiple of $k-1$. Figure \[fig:cycles\] shows the structure of each of these cycle types. A *Hamilton cycle* in a $k$-graph $H$ is a sub-$k$-graph of $H$ which is a cycle containing every vertex of $H$.
Rödl, Ruciński and Szemerédi [@RRS; @RRS2] showed that for any $\eta>0$ there is an $n_0$ so that if $n>n_0$ then any $k$-graph $H$ on $n$ vertices with minimum degree $\delta_{k-1}(H)\ge n/2+ \eta n$ contains a tight Hamilton cycle (this improved an earlier bound by Katona and Kierstead [@KK99]). They gave a construction which shows that this result is best possible up to the error term $\eta n$. In this paper, we prove the analogous result for loose Hamilton cycles.
\[main\] For all $k\geq 3$ and any $\eta>0$ there exists $n_0$ so that if $n>n_0$ then any $k$-graph $H$ on $n$ vertices with $\delta_{k-1}(H) > ( \frac{1}{2(k-1)} + \eta)n$ contains a loose Hamilton cycle.
The case when $k=3$ was proved by Kühn and Osthus [@KO]. We will use a similar method of proof for general $k$-graphs, but this will be greatly simplified by the use of the recent blow-up lemma of Keevash [@K2].
Proposition \[bestpos\] shows that Theorem \[main\] is best possible up to the error term $\eta n$. In fact, Proposition \[bestpos\] actually tells us more than this, namely that up to the error term, this minimum degree condition is best possible to ensure the existence of any (not necessarily loose) Hamilton cycle in $H$. This means that the minimum degree needed to find a Hamilton cycle in a $k$-graph of order $n$ is $\frac{n}{2(k-1)} + o(n)$.
Whilst finalizing this paper we learnt that Hàn and Schacht [@HS] independently and simultaneously proved Theorem \[main\], using a different approach. The result in [@HS] also covers the notion of a $k$-uniform $\ell$-cycle for $\ell<k/2$ (here one requires consecutive edges to intersect in precisely $\ell$ vertices). More recently Kühn, Mycroft and Osthus [@KMO] further developed the method of Hàn and Schacht to include all $\ell$ such that $k-\ell \nmid k$ (the remaining values of $\ell$ are covered by the results of Rödl, Ruciński and Szemerédi [@RRS; @RRS2]).
There is also the notion of a *Berge-cycle*, which consists of a sequence of vertices where each pair of consecutive vertices is contained in a common edge. This is less restrictive than the cycles considered in this paper. Hamiltonian Berge-cycles were studied in [@bermond].
\[fig:cycles\] {width="0.4\columnwidth"}
Extremal example and outline of the proof {#sec:extremal}
=========================================
The next proposition shows that Theorem \[main\] is best possible, up to the error term $\eta n$.
\[bestpos\] For all integers $k \geq 3$ and $n\ge 2k-1$, there exists a $k$-graph $H$ on $n$ vertices such that $\delta_{k-1}(H) \geq \lceil\frac{n}{2k-2}\rceil-1$ but $H$ does not contain a Hamilton cycle.
[[**Proof.**]{}]{} Let $V_1$ and $V_2$ be disjoint sets of size $\lceil\frac{n}{2k-2}\rceil-1$ and $n - \lceil\frac{n}{2k-2}\rceil+1$ respectively. Let $H$ be the $k$-graph on the vertex set $V = V_1 \cup V_2$, with $e \in \binom{V}{k}$ an edge if and only if $e \cap V_1 \neq \emptyset$, that is, if $e$ contains at least one vertex from $V_1$. Then $H$ has minimum degree $\delta_{k-1}(H) = \lceil\frac{n}{2k-2}\rceil-1$. However, any cyclic ordering of the vertices of $H$ must contain $2k-2$ consecutive vertices $v_1,\dots,v_{2k-2}$ from $V_2$, but then $v_{k-1}$ and $v_k$ cannot be contained in a common edge consisting of $k$ consecutive vertices, and so $H$ cannot contain a Hamilton cycle.
In our proof of Theorem \[main\] we construct the loose Hamilton cycle by finding several paths and joining them into a spanning cycle. Here a $k$-graph $P$ is a *path* if its vertices can be given a linear ordering such that every edge of $P$ consists of $k$ consecutive vertices, and so that every pair of consecutive vertices of $P$ lie in an edge of $P$. Similarly as for cycles, we say that a path $P$ is *loose* if edges of $P$ intersect in at most one vertex. The ordering of the vertices of $P$ naturally gives an ordering of the edges of $P$. We say that any vertex of $P$ which lies in the initial edge of $P$, but not the second edge of $P$, is an *initial vertex*. Similarly, any vertex of $P$ which lies in the final edge of $P$ but not the penultimate edge is a *final vertex*. Also, we refer to vertices of $P$ which lie in more than one edge of $P$ as *link vertices*. Thus, for example, a loose path $P$ has $k-1$ initial vertices, $k-1$ final vertices, and one link vertex in each pair of consecutive edges.
In Section \[sec:regularity\], we shall introduce various ideas we will need in the proof of Theorem \[main\]. In particular, we will state a version of the hypergraph regularity lemma due to Rödl and Schacht [@RS] and Theorem \[robust-universal\] due to Keevash [@K2]. The latter provides a useful way of applying the hypergraph blow-up lemma. In Section \[sec:prelims\], we shall prove various auxiliary results, including a result on finding loose paths in complete $k$-partite $k$-graphs, and an approximate minimum degree condition to guarantee a near-perfect packing of $H$ with a particular $k$-graph $\mathcal{A}_k$. Finally, in Section \[proof\] we shall prove Theorem \[main\] as follows.
Imposing structure on $H$.
--------------------------
In Section \[structure\] we use the hypergraph regularity lemma to split $H$ into $k$-partite $k$-graphs $H^i$ on disjoint vertex sets $X^i$. These $k$-graphs $H^i$ will be suitable for embedding almost spanning loose paths, and all the vertices of $H$ not contained in any of the $X^i$ will be included in an ‘exceptional’ loose path $L_e$ (actually, if $|V(H)|$ is not divisible by $k-1$, then $L_e$ will contain two consecutive edges which intersect in more than one vertex). The requirement that $H^i$ contains an almost spanning loose path means that the vertex classes of the $H^i$ must have suitable size. We achieve this by first defining a suitable ‘reduced $k$-graph’ $R$ of $H$. Then we cover almost all vertices of $R$ by copies of a suitable auxiliary $k$-graph ${\mathcal A}_k$. For each copy of ${\mathcal A}_k$, the corresponding sub-$k$-graph of $H$ is then split into the same number of disjoint $H^i$.
The linking strategy.
---------------------
In Section \[linking\] we shall use the structure imposed on $H$ to find a Hamilton cycle in $H$ by the following process.
- The $k$-graphs $H^i$ are connected by means of a walk $W = e_1, \dots, e_\ell$ in the ‘supplementary graph’. This graph (which we will define in Section \[supp\]) has vertices $1,\dots,t'$ corresponding to the $k$-graphs $H^i$.
- Using Lemma \[interpath\], each edge $e_j$ of $W$ is used to create a short ‘connecting’ loose path $L_j$ in $H$ joining two different $H^i$s.
- $L_e$ and the paths $L_j$ are extended to ‘prepaths’ (these can be thought of as a path minus an initial vertex and a final vertex) $L_e^* = I_0L_eF_0$ and $L_j^* = I_jL_jF_j$, where $I_0, F_0$ and all $I_j, F_j$ are sets of size $k-2$. These prepaths have the property that there are large sets $I'_j$ and $F'_j$ such that $L_j^*$ can be extended to a loose path by adding any vertex of $I'_j$ as an initial vertex and any vertex of $F'_j$ as a final vertex. Similarly there are large sets $I'_{\ell+1}$ and $F'_0$ so that $L_e^*$ can be extended to a path by adding any vertex of $I'_{\ell+1}$ as an initial vertex and any vertex of $F'_0$ as a final vertex. $I'_{j+1}$ and $F'_j$ both lie in the same $H^i$ (for all $j=0,\dots,\ell$).
- For each $H^i$ and for all those pairs $I'_{j+1},F'_j$ which lie in $H^i$, we choose a loose path $L'_{j+1}$ inside $H^i$ from $F'_j$ to $I'_{j+1}$. For each $i$, we will use the hypergraph blow-up lemma (in the form of Theorem \[robust-universal\]) to ensure that together all those $L'_j$ which lie in $H^i$ use all the remaining vertices of $H^i$.
- The loose Hamilton cycle is then the concatenation $L_e^*L_1'L_1^*\dots L'_\ell L_\ell^* L'_{\ell+1}$.
Controlling divisibility.
-------------------------
Note that the number of vertices of a loose path is $1$ modulo $k-1$. So in order to apply Theorem \[robust-universal\] to obtain spanning loose paths in a subgraph of $H^i$, we need this subgraph to satisfy this condition. So we choose our paths sequentially to satisfy the following congruences modulo $k-1$.
- $L_e$ is chosen with $|V(H){\setminus}V(L_e)| \equiv -1$.
- Let $X^i(j-1)$ be the subset of $X^i$ obtained by removing $V(L_1),\dots,V(L_{j-1})$. (All the $X^i$ will be disjoint from $V(L_e)$.) Let $d_i$ be the number of times that $W$ visits $H^i$. When choosing $L_j$, for every $X^i$ it traverses (except the final one) we arrange to intersect $X^i(j-1)$ in a set of size $\equiv t_i(j) \equiv |X^i(j-1)|+d_i$ (the size modulo $k-1$ of the intersection of $L_j$ with the final $X^i$ it traverses is then determined by the sizes of the other intersections). The choice of $L_e$ in (a) ensures that after all $L_j$ have been picked, the remaining part $X^i(\ell)$ of $X^i$ has size $\equiv -d_i$.
- Each $L_j$ is extended to a prepath $L_j^*$ by adding $I_j$ and $F_j$. Similarly, $L_e$ is extended into a prepath $L_e^*$ by adding $I_0$ and $F_0$. Now the remaining part of $X^i$ has size $\equiv d_i$.
- It remains to select $d_i$ paths $L'_j$ within each $X^i$: each uses $\equiv 1$ vertices, so the divisibility conditions are satisfied.
Regularity and the Blow-up Lemma {#sec:regularity}
================================
Graphs and complexes
--------------------
We begin with some notation. By $[r]$ we denote the set of integers from 1 to $r$. For a set $A$, we use $\binom{A}{k}$ to denote the collection of subsets of $A$ of size $k$, and similarly $\binom{A}{\leq k}$ to denote the collection of non-empty subsets of $A$ of size at most $k$. We write $x = y \pm z$ to mean that $y-z \leq x \leq y+z$. We shall omit floors and ceilings throughout this paper whenever they do not affect the argument.
A *hypergraph* $H$ consists of a vertex set $V(H)$ and an edge set, such that each edge $e$ of the hypergraph satisfies $e \subseteq V(H)$. So a $k$-graph as defined in Section \[intro\] is a hypergraph in which all the edges are of size $k$. We say that a hypergraph $H$ is a *$k$-complex* if every edge has size at most $k$ and $H$ forms a simplicial complex, that is, if $e_1 \in H$ and $e_2 \subseteq e_1$ then $e_2 \in H$. As for $k$-graphs we identify a hypergraph $H$ with the set of its edges. So $|H|$ is the number of edges in $H$, and if $G$ and $H$ are hypergraphs then $G {\setminus}H$ is formed by removing from $G$ any edge which also lies in $H$. If $H$ is a hypergraph with vertex set $V$ then for any $V' \subseteq V$ the *restriction $H[V']$ of $H$ to $V'$* is defined to have vertex set $V'$ and all edges of $H$ which are contained in $V'$ as edges. Also, for any hypergraphs $G$ and $H$ we define $G - H$ to be the hypergraph $G[V(G) {\setminus}V(H)]$.
We say that a hypergraph $H$ is *$r$-partite* if its vertex set $X$ is divided into $r$ pairwise-disjoint parts $X_1, \dots, X_r$, in such a way that for any edge $e \in H$, $|e \cap X_i| \leq 1$ for each $i$. We call the $X_i$ the *vertex classes* of $H$ and say that the partition $X_1, \dots, X_r$ of $X$ is *equitable* if all the $X_i$ have the same size. We say that a set $A \subseteq X$ is *$r$-partite* if $|A \cap X_i| \leq 1$ for each $i$. So every edge of an $r$-partite hypergraph is $r$-partite. In the same way we may also speak of $r$-partite $k$-graphs and $r$-partite $k$-complexes. Given a $k$-graph $H$, we define a $k$-complex $H^\leq = \{e_1 \colon e_1 \subseteq e_2$ and $e_2 \in H\}$ and a $(k-1)$-complex $H^< = \{e_1 \colon e_1 \subset e_2$ and $e_2 \in H\}$. Conversely, for a $k$-complex $H$ we define the $k$-graph $H_=$ to be the ‘top level’ of $H$, i.e. $H_= = \{e \in H \colon |e| = k\}$. (Here $V(H)=V(H^\le)=V(H^<)=V(H_=)$.)
Given a $k$-graph $G$ and a set $W$ of vertices of $G$, we denote by $G[W]$ the sub-$k$-graph of $G$ obtained by removing all vertices and edges not contained in $W$ (in this case, we say $G$ is *restricted to $W$*). For a $k$-graph $G$ and a sub-$k$-graph $H \subseteq G$ write $G - H$ for $G[V(G){\setminus}V(H)]$.
Let $X_1, \dots, X_r$ be pairwise-disjoint sets of vertices, and let $X = X_1 \cup \dots \cup X_r$. Given $A \in \binom{[r]}{\leq k}$, we write $K_A(X)$ for the complete $|A |$-partite $|A|$-graph whose vertex classes are all the $X_i$ with $i\in A$. The *index* of an $r$-partite subset $S$ of $X$ is $i(S) = \{i \in [r]: S \cap X_i \neq \emptyset\}$. Furthermore, given any set $B\subseteq i(S)$, we write $S_B = S \cap \bigcup_{i\in B} X_i $. Similarly, given $A \in \binom{[r]}{\leq k}$ and an $r$-partite $k$-graph or $k$-complex $H$ on the vertex set $X$ we write $H_A$ for the collection of edges in $H$ of index $A$ and let $H_\emptyset=\{\emptyset\}$. In particular, if $H$ is a $k$-complex then $H_{\{i\}}$ is the set of all those vertices in $X_i$ which lie in an edge of $H$ (and thus form a (singleton) edge of $H$). In general, we will often view $H_A$ as an $r$-partite $|A|$-graph with vertex set $X$. Also, given a $k$-complex $H$ we similarly write $H_{A^\leq} = \bigcup_{B \subseteq A} H_B$ and $H_{A^<} = \bigcup_{B \subset A} H_B$. We write $H^*_A$ for the $|A|$-graph whose edges are those $r$-partite sets $S\subseteq X$ of index $A$ for which all proper subsets of $S$ belong to $H$. (In other words, a set $S$ with index $A$ satisfies $S \in H^*_A$ if and only if for all $j < |A|$ the edges of $H$ which have size $j$ and are subsets of $S$ form a complete $j$-graph on $|S|$ vertices.) Then the *relative density of $H$ at index $A$* is $d_A(H) = |H_A|/|H_A^*|$. The *absolute density of $H_A$* is $d(H_A) = |H_A|/|K_A(X)|$. (Note that $|K_A(X)|=\prod_{i\in A} |X_i|$.) If $H$ is a $k$-partite $k$-complex we may simply write $d(H)$ for $d(H_{[k]})$. Similarly, the *density* of a $k$-partite $k$-graph $H$ on $X=X_1 \cup \dots \cup X_k$ is $d(H)=|H|/|K_{[k]}(X)|$.
Finally, for any vertex $v$ of a hypergraph $H$, we define the *vertex degree* $d(v)$ of $v$ to be the number of edges of $H$ which contain $v$. Note that this is not the same as the degree defined earlier, which was for sets of $k-1$ vertices. The *maximum vertex degree* of $H$ is then the maximum of $d(v)$ taken over all vertices $v \in V(H)$. The *vertex neighbourhood* $VN(v)$ of $v$ is the set of all vertices $u \in V(H)$ for which there is an edge of $H$ containing both $u$ and $v$. For a $k$-partite $k$-complex $H$ on the vertex set $X_1 \cup \dots \cup X_k$ we also define the *neighbourhood complex* $H(v)$ of a vertex $v \in X_i$ for some $i$ to be the $(k-1)$-partite $(k-1)$-complex with vertex set $\bigcup_{j\neq i} X_j$ and edge set $\{e \in H: e \cup \{x\} \in H\}$.
Regular complexes {#regcomplexes}
-----------------
In this subsection we shall define the concept of regular complexes (which was first introduced in the $k$-uniform case by Rödl and Skokan [@RSk]) in the form used by Rödl and Schacht [@RS; @RS2]. This is a generalization of the standard concept of regularity in graphs, where we say that a bipartite graph $B$ on vertex classes $U$ and $V$ forms an ${\epsilon}$-regular pair if for any $U' \subseteq U$ and $V' \subseteq V$ with $|U'| > {\epsilon}|U|$ and $|V'| > {\epsilon}|V|$ we have $d(B[U' \cup V']) = d(B)\pm {\epsilon}$.
In the same way, we say that a $k$-complex $G$ is regular if the restriction of $G$ to any large subcomplex of lower rank has similar densities to $G$. More precisely, let $G$ be an $r$-partite $k$-complex on the vertex set $X = X_1 \cup \dots \cup X_r$. For any $A \in \binom{[r]}{\leq k}$, we say that $G_A$ is *${\epsilon}$-regular* if for any $H\subseteq G_{A^<}$ with $|H^*_A| \geq {\epsilon}|G^*_A|$ we have $$\frac{|G_A \cap H_A^*|}{|H_A^*|} = d_A(G) \pm {\epsilon}.$$ We say $G$ is *${\epsilon}$-regular* if $G_A$ is ${\epsilon}$-regular for every $A \in \binom{[r]}{\leq k}$. Note that if $G$ is a graph without isolated vertices, then the definition in the previous paragraph is equivalent to the 2-complex $G^\le$ being ${\epsilon}$-regular. To illustrate the definition for $k=3$, suppose that $A=[3]$. Then for instance the top level of $G_{[2]}$ is the bipartite subgraph of $G$ induced by $X_1$ and $X_2$ and $G_A^*$ is the set of (graph) triangles in $G$. So roughly speaking, the regularity condition states that if we consider a subgraph of $G_{[2]} \cup G_{\{1,3\}} \cup G_{\{2,3\}}$ which spans a large number of triangles, then the proportion of these which also form an edge of $G_A$ is close to $d_A(G)$, i.e. close to the proportion of (graph) triangles in $G$ between $X_1$, $X_2$ and $X_3$ which form an edge of $G$.
Roughly speaking, the hypergraph regularity lemma states that an arbitrary $k$-graph can be split into pieces, each of which forms a regular $k$-complex. The version of the regularity lemma we shall use also involves the notion of a ‘partition complex’, which is a certain partition of the edges of a complete $k$-complex. As before, let $X = X_1 \cup \dots \cup X_r$ be an $r$-partite vertex set. A *partition $k$-system $P$ on $X$* consists of a partition $P_A$ of the edges of $K_A(X)$ for each $A \in \binom{[r]}{\leq k}$. We refer to the partition classes of $P_A$ as *cells*. So every edge of $K_A(X)$ is contained in precisely one cell of $P_A$. $P$ is a *partition $k$-complex on $X$* if it also has the property that whenever $S, S' \in K_A(X)$ lie in the same cell of $P_A$, we have that $S_B$ and $S_B'$ lie in the same cell of $P_B$ for any $B \subseteq A$. This property of $S, S'$ forms an equivalence relation on the edges of $K_A(X)$, which we refer to as *strong equivalence*. To illustrate this, again suppose that $k=3$ and $A=[3]$. Then if $P$ is a partition $k$-complex, $P_{\{1\}}$, $P_{\{2\}}$ and $P_{\{3\}}$ together yield a vertex partition $Q_1$ refining $X_1,X_2,X_3$. $Q_1$ naturally induces a partition $Q_2$ of the $3$ complete bipartite graphs induced by the pairs $X_i,X_j$. $P_{\{1,2\}}$, $P_{\{2,3\}}$ and $P_{\{1,3\}}$ also yield a partition $Q'_2$ of these complete bipartite graphs. The requirement of strong equivalence now implies that $Q'_2$ is a refinement of $Q_2$. At the next level, $Q'_2$ naturally induces a partition $Q_3$ of the set of triples induced by $X_1,X_2$ and $X_3$. As before, strong equivalence implies that the partition $P_{\{1,2,3\}}$ of these triples is a refinement of $Q_3$.
Let $P$ be a partition $k$-complex on $X=X_1 \cup \dots \cup X_r$. For $i\in [k]$, the cells of $P_{ \{i \} }$ are called *clusters* (so each cluster is a subset of some $X_i$). We say that $P$ is *vertex-equitable* if all clusters have the same size. $P$ is *$a$-bounded* if $|P_A| \leq a$ for every $A$ (i.e. if $K_A(X)$ is divided into at most $a$ cells by the partition $P_A$). Also, for any $r$-partite set $Q\in \binom{X}{\le k}$, we write $C_Q$ for the set of all edges lying in the same cell of $P$ as $Q$, and write $C_{Q^\leq}$ for the $r$-partite $k$-complex whose vertex set is $X$ and whose edge set is $\bigcup_{Q' \subseteq Q} C_{Q'}$. (Since $P$ is a partition $k$-complex, $C_{Q^\le }$ is indeed a complex.) The partition $k$-complex $P$ is *${\epsilon}$-regular* if $C_{Q^\leq}$ is ${\epsilon}$-regular for every $r$-partite $Q \in \binom{X}{\leq k}$.
Given a partition $(k-1)$-complex $P$ on $X$ and $A \in \binom{[r]}{k}$, we can define an equivalence relation on the edges of $K_A(X)$, namely that $S, S' \in K_A(X)$ are equivalent if and only if $S_B$ and $S_B'$ lie in the same cell of $P$ for any strict subset $B \subset A$. We refer to this as *weak equivalence*. Note that if the partition complex $P$ is $a$-bounded, then $K_A(X)$ is divided into at most $a^k$ classes by weak equivalence . If we let $G$ be an $r$-partite $k$-graph on $X$, then we can use weak equivalence to refine the partition $\{G_A, K_A(X) {\setminus}G_A\}$ of $K_A(X)$ (i.e. two edges of $G_A$ are in the same cell if they are weakly equivalent and similarly for the edges not in $G_A$). Together with $P$, this yields a partition $k$-complex which we denote by $G[P]$. If $G[P]$ is ${\epsilon}$-regular then we say that $G$ is *perfectly ${\epsilon}$-regular with respect to $P$*. Note that if $G[P]$ is ${\epsilon}$-regular then $P$ must be ${\epsilon}$-regular too.
Finally, we say that $r$-partite $k$-graphs $G$ and $H$ on $X$ are *$\nu$-close* if $|G_A \triangle H_A| < \nu |K_A(X)|$ for every $A \in \binom{[r]}{k}$, that is, if there are few edges contained in $G$ but not in $H$ and vice versa.
We can now present the version of the regularity lemma we shall use to split our $k$-graph $H$ into regular $k$-complexes. It actually states that there is some $k$-graph $G$ which is close to $H$ and which is regular with respect to some partition complex. This will be sufficient for our purposes, as we shall avoid the use of any edges in $G {\setminus}H$, so every edge used will lie in both $G$ and $H$. There are various other forms of the regularity lemma for $k$-graphs which give information on $H$ itself (the first of these were proved in [@RSk; @G1]) but these do not have the hierarchy of densities necessary for the application of the blow-up lemma (see [@K2] for a fuller discussion of this point). The version below is due to Rödl and Schacht [@RS] (actually it is a very slight restatement of their result).
\[eq-partition\] Suppose integers $n,a,r,k$ and reals ${\epsilon}, \nu$ satisfy $1/n \ll{\epsilon}\ll 1/a \ll \nu, 1/r, 1/k$ and where $a!r$ divides $n$. Suppose also that $H$ is an $r$-partite $k$-graph whose vertex classes $X_1, \dots, X_r$ form an equitable partition of its vertex set $X$, where $|X|=n$. Then there is an $a$-bounded ${\epsilon}$-regular vertex-equitable partition $(k-1)$-complex $P$ on $X$ and an $r$-partite $k$-graph $G$ on $X$ that is $\nu$-close to $H$ and perfectly ${\epsilon}$-regular with respect to $P$.
Here (and later on) we write $0<a_1 \ll a_2 \ll a_3 \ll a_4\leq 1$ to mean that we can choose the constants $a_1,\dots,a_4$ from right to left. More precisely, there are increasing functions $f_1,f_2,f_3$ such that, given $a_4$, whenever we choose some $a_3 \leq f_3(a_4)$, $a_2\le f_2(a_3)$ and $a_1 \leq f_1(a_2)$, all calculations needed in the proof of the subsequent statement are valid. Hierarchies with more constants are defined similarly.
One important property of regular complexes is that they remain regular when restricted to a large subset of their vertex set. For regular $k$-partite $k$-complexes this property is formalised by the following lemma, a special case of Lemma 6.18 in [@K2].
\[restrict\] Suppose ${\epsilon}\ll {\epsilon}' \ll d \ll c \ll 1/k$, and that $G$ is an ${\epsilon}$-regular $k$-partite $k$-complex on the vertex set $X = X_1 \cup \cdots \cup X_k$ such that $G_{\{i\}} = X_i$ for each $i$ and $d(G) > d$. Let $W$ be a subset of $X$ such that $|W\cap X_i| \geq c|X_i|$ for each $i$. Then the restriction $G[W]$ of $G$ to $W$ is ${\epsilon}'$-regular, with $d(G[W]) > d(G)/2$ and $d_{[k]}(G[W]) > d_{[k]}(G)/2$.
Robustly universal complexes.
-----------------------------
Apart from Theorem \[eq-partition\], the other main tool we shall use in the proof of Theorem \[main\] is the recent hypergraph blow-up lemma of Keevash. This result involves not only a $k$-complex $G$, but also a $k$-graph $M$ of ‘marked’ edges on the same vertex set. If the pair $(G,M)$ is ‘super-regular’, then this blow-up lemma can be applied to embed any spanning bounded-degree $k$-complex in $G {\setminus}M$, that is, within $G$ but avoiding any marked edges. We will apply this with $M=G \setminus H$ where $G$ is the $k$-graph given by Theorem \[eq-partition\]. Super-regularity is a stronger notion than regularity. A result in [@K2] states that every ${\epsilon}$-regular $k$-complex can be made super-regular by deleting a few of its vertices. Unfortunately, the notion of hypergraph super-regularity is very technical, but the following definition from [@K2] avoids many of these technicalities. Let $J'$ be a $k$-partite $k$-complex. Roughly speaking, we say that $J'$ is robustly $D$-universal if the following holds: even after the deletion of many vertices of $J'$, the resulting complex $J$ has the property that one can find in $J$ a copy of any $k$-partite $k$-complex $L$ which has vertex degree at most $D$ and whose vertex classes are the same as those of $J$. Condition (i) puts a natural restriction on the number of vertices we are allowed to delete from the neighbourhood complex of a vertex of $J$ and condition (iii) states that for a few vertices $u$ of $L$ we can even prescribe a ‘target set’ in $V(J)$ into which $u$ will be embedded.
[[**Definition. (Robustly universal complexes)**]{}]{} Suppose that $J'$ is a $k$-partite $k$-complex on $V' = V'_1 \cup\dots \cup V'_k$ with $J'_{\{i\}} = V'_i$ for each $i \in [k]$. We say that $J'$ is *$(c,c_0)$-robustly $D$-universal* if whenever
- $V_j {\subseteq}V'_j$ are sets with $|V_j| \ge c|V'_j|$ for all $j \in [k]$, such that writing $V = \bigcup_{j \in [k]} V_j$ and $J=J'[V]$ we have $|J(v)_=| \ge c|J'(v)_=|$ for any $j \in [k]$ and $v \in V_j$,
- $L$ is a $k$-partite $k$-complex of maximum vertex degree at most $D$ on some vertex set $U = U_1 \cup \dots \cup U_k$ with $|U_j|=|V_j|$ for all $j \in [k]$,
- $U_* \subseteq U$ satisfies $|U_* \cap U_j| \le c_0|U_j|$ for every $j \in [k]$, and sets $Z_u \subseteq V_{i(u)}$ satisfy $|Z_u| \ge c|V_{i(u)}|$ for each $u \in U_*$, where for each $u$ we let $i(u)$ be such that $u \in U_{i(u)}$,
then $J$ contains a copy of $L$, in which for each $j \in [k]$ the vertices of $U_j$ correspond to the vertices of $V_j$, and $u$ corresponds to a vertex of $Z_u$ for every $u \in U_*$.
So our use of the blow-up lemma will be hidden through this definition. Of course, we shall also need to obtain robustly universal complexes. This is the purpose of the next theorem, which states that given a regular $k$-partite $k$-complex $G$ with sufficient density, and a $k$-partite $k$-graph $M$ on the same vertex set which is small relative to $G$, we can delete a small number of vertices from their common vertex set so that $G {\setminus}M$ is robustly universal. It is a special case of Theorem 6.32 in [@K2].
\[robust-universal\] Suppose that $1/n \ll {\epsilon}\ll c_0 \ll d^* \ll d_a \ll \theta \ll d, c, 1/k, 1/D, 1/C$, $G$ is a $k$-partite $k$-complex on $V = V_1 \cup \dots \cup V_k$ with $n \leq |G_{\{j\}}| = |V_j| \leq Cn$ for every $j \in [k]$, $G$ is ${\epsilon}$-regular with $d_{[k]}(G) \ge d$ and $d(G_{[k]}) \ge d_a$, and $M {\subseteq}G_=$ with $|M| \le \theta |G_=|$. Then we can delete at most $2\theta^{1/3} |V_j|$ vertices from each $V_j$ to obtain $V' = V_1' \cup \dots \cup V_k'$, $G' = G[V']$ and $M' = M[V']$ such that
- $d(G') > d^*$ and $|G'(v)_=| > d^*|G'_=|/|V'_i|$ for every $v \in V'_i$, and
- $G' {\setminus}M'$ is $(c,c_0)$-robustly $D$-universal.
Preliminary results {#sec:prelims}
===================
In this section we will collect the preliminary results we need to prove Theorem \[main\]. In order to apply Theorem \[robust-universal\], we need to know under what conditions we can find particular loose paths in complete $k$-partite $k$-graphs, which is the topic of the next subsection.
Loose paths in complete graphs
------------------------------
The problem of when we can find particular loose paths in a complete $k$-partite $k$-graph can be reformulated in terms of the question of which strings satisfying certain adjacency conditions can be produced from a fixed character set; the following lemma is the result we will need.
\[strings\] Let $\ell$ and $a_1,\dots,a_k$ be integers such that $0 \leq a_i < \ell/2$ for all $i$, and $\ell = \sum_{i=1}^k a_i$. Then for any $s,t \in [k]$ there exists a string of length $\ell$ on alphabet $x_1,\dots,x_k$ such that the following properties hold:
1. no two consecutive characters are equal,
2. the first character is not $x_s$ and the final character is not $x_t$,
3. the number of occurrences of character $x_i$ is $a_i$.
[[**Proof.**]{}]{} Note that the conditions on $\ell$ and the $a_i$ imply that $\ell\ge 3$. We will construct the required string by starting with an ‘empty string’ of $\ell$ blank positions, and for each $i$ inserting precisely $a_i$ copies of character $x_i$. This ensures that condition (3) will be satisfied. We shall fill the empty positions in the following order: first the first position, then the third, and so on through the odd-numbered positions, until we reach either position $\ell$ or position $\ell-1$ (dependent on whether $\ell$ is odd or even). We then fill the second position, then the fourth, and so on until all positions are filled. Note that if we proceed by inserting all copies of one character, then all the copies of another character, and so forth, then condition (1) must be satisfied. This is because to get two consecutive copies of $x_i$, we must have inserted a copy of $x_i$ at some odd position $p$, then $p+2$, $p+4$, and so on until reaching $\ell$ or $\ell-1$, and then filled even positions $2,4,6,\dots,p-1$. However, this would imply that we had inserted at least $\ell/2$ copies of character $x_i$, contradicting the fact that $a_i < \ell/2$.
We therefore only need to determine an order to insert the different characters so as to satisfy (2). We first consider the case $s \neq t$, say $s=1$ and $t=2$. In this case we insert $x_2$ first, $x_1$ last, and the remaining character blocks in any order in between. Clearly this prevents the first character from being $x_1$ and the last from being $x_2$, and so (2) is satisfied. Now we may assume $s=t$, say $s=t=1$. Then if $\ell$ is odd, we insert the characters in the following order: $x_2, x_3, \dots, x_k, x_1$. Then all the copies of $x_1$ must be in even positions (since $a_1 < \ell/2$), and so (2) is satisfied. Alternatively, if $\ell$ is even, we insert first $x_i$ for some $i \neq 1$ with $a_i > 0$, then $x_1$, and then the remaining blocks of characters in any order. (Note that these include at least one character other than $x_1$ and $x_i$ since $\ell \ge 3$ and $a_j < \ell/2$ imply that at least three $j$ have $a_j \geq 1$.) So neither the first nor last character can be $x_1$, and so (2) is again satisfied.
The next lemma is the result we were aiming for in this section, giving information about which loose paths can be found in complete $k$-partite $k$-graphs. Note that the maximum vertex degree of a loose path is two, and so this lemma will tell us when we can find a loose path in a robustly universal $k$-complex.
\[loosepath\] Let $G$ be a complete $k$-partite $k$-graph on the vertex set $V_1\cup \dots\cup V_k$. Let $b_1,\dots,b_k$ be integers with $0\le b_i \leq |V_i|$ for each $i$. Suppose that
- $n := \frac{1}{k-1}((\sum_{i=1}^{k} b_i) -1)$ is an integer, and
- $\frac{n}{2}+1 \leq b_i \leq n$ for all $i$.
Then for any $s,t \in [k]$, there exists a loose path in $G$ with an initial vertex in $V_s$, a final vertex in $V_t$, and containing $b_i$ vertices from $V_i$ for each $i\in [k]$.
[[**Proof.**]{}]{} Note first that $n$ is the number of edges such a path must contain. Let $a_i = n - b_i$ for each $i$, so that $0 \leq a_i < (n-1)/2$. By Lemma \[strings\] we can find a string $S$ of length $n-1$ on the alphabet $V_1, V_2,\dots, V_k$ such that $V_i$ appears $a_i$ times, no two consecutive characters are identical, the first character is not $V_s$ and the final character is not $V_t$. Let $S_i$ be the $i$th character of $S$. To construct a loose path $P$ in $G$, first choose any vertex from $V_s$ to be the initial vertex of $P$, and any vertex from $V_t$ to be the final vertex of $P$. We also use $S$ to choose the link vertices of $P$: choose the $i$th link vertex (i.e. the vertex lying in the intersection of the $i$th and $(i+1)$th edges of $P$) to be any member of $S_i$ not yet chosen. We have now assigned two vertices to each edge of $P$. Finally, we complete $P$ by assigning to each edge one as yet unchosen vertex from each of the $k-2$ classes not yet represented in that edge. This is possible since precisely $a_i$ link vertices are from the class $V_i$ and so the total number of vertices used from $V_i$ is $n-a_i = b_i$. Since $G$ is complete we know that each edge of $P$ is an edge of $G$, and so $P$ is a loose path satisfying all the conditions of the lemma.
Walks and connectedness in $k$-graphs {#walks}
-------------------------------------
A *walk $W$* in a hypergraph $H$ consists of a sequence of edges $e_1, \dots, e_\ell$ of $H$ and a sequence $x_0, \dots, x_{\ell}$ of (not necessarily distinct) vertices of $H$, satisfying $x_{i-1} \neq x_i$ for all $i \in [\ell]$, and also $x_0 \in e_1$, $x_\ell \in e_\ell$ and $x_i \in e_i \cap e_{i+1}$ for all $i \in [\ell-1]$. The *length of $W$* is the number of its edges. We say that $x_0$ is the *initial vertex of $W$*, $x_\ell$ is the *final vertex of $W$*, and that $x_1,\dots, x_{\ell-1}$ are the *link vertices of $W$*. By a *walk from $x$ to $y$* we mean a walk with initial vertex $x$ and final vertex $y$.
Note that the vertices of a hypergraph $H$ can be partitioned using the equivalence relation $\sim$, where $x \sim y$ if and only if either $x=y$ or there exists a walk from $x$ to $y$. We call the equivalence classes of this relation *components* of $H$. We say that $H$ is *connected* if it has precisely one component. Observe that all vertices of an edge of $H$ must lie in the same component. Finally, note that if $H$ is a connected hypergraph of order $n$, then for any two vertices $x,y$ of $H$ we can find in a walk from $x$ to $y$ of length at most $n$ in $H$.
Random splitting
----------------
In this section we shall obtain, with high probability, a lower bound on the density of a subgraph of a $k$-partite $k$-graph chosen uniformly at random. We will use Azuma’s inequality on the deviation of a martingale from its mean.
\[azuma\] Suppose $Z_0,\dots, Z_m$ is a martingale, i.e. a sequence of random variables satisfying $\mathbb{E}(Z_{i+1} \mid Z_0, \dots, Z_i) = Z_i$, and that $|Z_i-Z_{i-1}| \leq c_i$ for some constants $c_i$ and all $i\in [m]$. Then for any $t \geq 0$, $$\mathbb{P}(|Z_m - Z_0| \geq t) \leq 2 \exp \left( -\frac{t^2}{2 \sum_{i=1}^{m} c_i^2}\right).$$
\[randomsplit\] Suppose $1/n \ll c,\beta, 1/k, 1/b < 1$, and that $H$ is a $k$-partite $k$-graph on the vertex set $X = X_1 \cup \dots \cup X_k$, where $n \leq |X_i| \leq bn$ for each $i\in [k]$. Suppose also that $H$ has density $d(H) \ge c$ and that for each $i$ we have $\beta |X_i| \leq t_i \leq |X_i|$. If we choose a subset $W_i \subseteq X_i$ with $|W_i| = t_i$ uniformly at random and independently for each $i$, and let $W = W_1 \cup \dots \cup W_k$, then the probability that $H[W]$ has density $d(H[W]) > c/2$ is at least $1-1/n^2$. Moreover, the same holds if we choose $W_i$ by including each vertex of $X_i$ independently with probability $t_i/|X_i|$.
[[**Proof.**]{}]{} Let $m = |X|$. To prove the first assertion, we obtain our subsets $W_i \subseteq X_i$ through the following two-stage random process, independently for each $i$. First we assign the vertices of each $X_i$ into sets $X_i^1$ and $X_i^2$ independently at random, with each vertex being assigned to $X_i^1$ with probability $t_i/|X_i|$, and assigned to $X_i^2$ otherwise. Then, in the (highly probable) event that we have $|X_i^1| \neq t_i$ we shall select uniformly at random a set of vertices to transfer between $X_i^1$ and $X_i^2$ to obtain from $X_i^1$ the set $W_i$ with $|W_i| = t_i$. For each $i$, no subset $W_i \subseteq X_i$ of size $t_i$ is more likely to result from this process than any other, so we have chosen each $W_i$ uniformly at random. It remains to show that $H[W]$ is likely to have high density. We do this by noting that $H[X^1]$ is likely to have high density (where $X^1 = X^1_1 \cup \dots \cup X^1_k$) and that with high probability we will only need to transfer a small number of vertices to form $W=W_1\cup\dots\cup W_k$, which can have only a limited effect on the density.
More precisely, let $x_1, \dots, x_m$ be an ordering of the vertices of $X$, and for each $i\in [m]$ let the random variable $Y_i$ take the value 1 if $x_i \in X^1$, and 0 otherwise. Recall that we write $|H|$ to denote the number of edges of a $k$-graph $H$. For all $i=0,\dots,m$ we now define random variables $Z_i$ by $Z_i = \mathbb{E}(|H[X^1]| \mid Y_1, \dots, Y_i)$. Then the sequence $Z_0, \dots, Z_m$ is a martingale, $Z_m = |H[X^1]|$, and as we formed each $X_i^1$ by assigning vertices of $X_i$ independently at random into $X_i^1$ and $X_i^2$, we have $Z_0 = \mathbb{E}(|H[X^1]|) \ge c \prod_{i=1}^k t_i$. Also, for any vertex $x_i$, let $f(i)$ be such that $x_i \in X_{f(i)}$ (i.e. $f(i)$ is the index of $x_i$). Then $|Z_i - Z_{i-1}| \leq \prod_{j \neq f(i)} |X_{j}| \leq (bn)^{k-1}$ for all $i\in [m]$. Thus we can apply Lemma \[azuma\] to obtain $$\mathbb{P}\left( |Z_m - Z_0| \geq \frac{c\prod_{i=1}^k t_i}{4} \right) \leq
2 \exp \left( -\frac{c^2 \prod_{i=1}^k t_i^2}{32m b^{2k-2} n^{2k-2}}\right) \le \frac{1}{n^3}.$$ Therefore the event that $d(H[X^1]) > 3c/4$ has probability at least $1-1/n^3$. Also, by a standard Chernoff bound, for each $i\in [k]$ the event that $|X_i^1| = t_i \pm |X_i|^{2/3}$ has probability at least $1-1/n^3$. Thus with probability at least $1-1/n^2$ all of these events will happen. Now, if $|X_i^1| > t_i$, we choose a set of $|X_i^1| - t_i$ vertices of $X_i^1$ uniformly at random and move these vertices from $X_i^1$ to $X_i^2$. Similarly, if $|X_i^1| < t_i$, then we choose a set of $t_i - |X_i^1|$ vertices of $X_i^2$ uniformly at random and move these vertices to $X_i^1$. In either case, for any $i$ this action can decrease $d(H[X^1])$ by at most $||X_i^1|-t_i|/|X_i^1| \ll c $. Thus if we let $W$ be the set obtained from $X^1$ in this way, we have $d(H[W]) > c/2$, proving the first part of the lemma.
The proof of the ‘moreover part’ is the same except that we can omit the ‘transfer’ step at the end of the proof.
Decomposition of $G$ into copies of ${\mathcal{A}}_k$
-----------------------------------------------------
Let ${\mathcal{A}}_k$ denote the $k$-graph whose vertex set $V({\mathcal{A}}_k)$ is the union of $2k-2$ disjoint sets $U_0, U_1, U_2, \dots, U_{2k-3}$ of size $k-1$ and whose edges consist of all $k$-tuples of the form $U_i \cup \{x\}$, with $i>0$ and $x \in U_0$ (see Figure 2).
\[fig:a\_kdiag\] {width="0.3\columnwidth"}
So $|V({\mathcal{A}}_k)|=2(k-1)^2$. An *${\mathcal{A}}_k$-packing* in a $k$-graph $G$ is a collection of pairwise vertex-disjoint copies of ${\mathcal{A}}_k$ in $G$.
\[ak-pack-lem\] Suppose $1/m \ll \theta \ll \psi\ll 1/k$, and that $G$ is a $k$-graph on $[m]$ such that $|N_G(S)| > (\frac{1}{2(k-1)} + \theta)m$ for all but at most $\theta m^{k-1}$ sets $S \in \binom{[m]}{k-1}$. Then $G$ has an ${\mathcal{A}}_k$-packing which covers more than $(1-\psi)m$ vertices of $G$.
[[**Proof.**]{}]{} Let $A_1, \dots, A_t$ be an ${\mathcal{A}}_k$-packing of $G$ of maximum size, so $t \le m/(2(k-1)^2)$. Let $X$ be the set of uncovered vertices, and suppose that $|X|>\psi m$. Let $b = \theta |X|$. Our first aim is to choose disjoint sets $S_1,\dots,S_b$ in $\binom{X}{k-1}$ so that $|N_G(S_i)| > (1/(2(k-1)) + \theta)m$ and $|N_G(S_i) \cap X| < \theta m/2$ for all $i \in [b]$. Note that $\theta \ll \psi$ implies that $\binom{|X|-2b(k-1)}{k-1} \gg\theta m^{k-1}$. So we can greedily choose disjoint $S_1, \dots, S_{2b}\in \binom{X}{k-1}$ such that $|N_G(S_i)| > (1/(2(k-1)) + \theta )m$ for all $i\in [2b]$. Let $T = \{i \in [2b]: |N_G(S_i) \cap X| \geq \theta m/2\}$. We claim that $|T| \le b$. Otherwise, consider the bipartite graph $B$ with vertex classes $T$ and $X$, where we join $i \in T$ to $x \in X$ if $S_i \cup \{x\}$ is an edge of $G$. Note that $B$ cannot contain a complete bipartite graph with $2k-3$ vertices in $T$ and $k-1$ vertices in $X$, as this would correspond to a copy of ${\mathcal{A}}_k$ contained in $X$, which is impossible as $A_1, \dots, A_t$ is a maximum size ${\mathcal{A}}_k$-packing. However, by definition of $T$ we have $d_B(i) \geq \theta m/2$ for every $i \in T$, and double-counting pairs $(i,P)$ with $i \in T$ and $P \in \binom{N_B(i)}{k-1}$ gives $$|T| \binom{\theta m/2}{k-1} \leq \#\{(i,P)\} < (2k-3)\binom{|X|}{k-1},$$ a contradiction. This proves the claim, and by relabelling the $S_i$ we can assume that $|N_G(S_i)| > (1/(2(k-1)) + \theta )m$ and $|N_G(S_i) \cap X| < \theta m/2$ for all $i \in [b]$.
Now we show how to enlarge the ${\mathcal{A}}_k$-packing $A_1,\dots,A_t$. For $i \in [b]$ let $$F_i = \{j\in [t]: |N_G(S_i) \cap V(A_j)| \ge k \}.$$ Since $|V(A_i)|= 2(k-1)^2$ for each $i\in [b]$ we have $$\begin{aligned}
\left(\frac{1}{2(k-1)} + \frac{\theta}{2} \right)m
& < |N_G(S_i) {\setminus}X| = \sum_{j=1}^t |N_G(S_i) \cap V(A_j)|\\ & \le |F_i| \cdot 2(k-1)^2 + (t-|F_i|) \cdot (k-1)
< 2(k-1)^2|F_i| + \frac{(k-1)m}{2(k-1)^2},\end{aligned}$$ and so $|F_i| > \theta m/(4(k-1)^2)$. We now double-count pairs $(i,Q)$ with $i \in [b]$ and $Q \in \binom{F_i}{k-1}$. The number of such pairs is $$\sum_{i=1}^b \binom{|F_i|}{k-1} > \theta \psi m \binom{\frac{\theta m}{4(k-1)^2}}{k-1} >
\sqrt{m} \binom{t}{k-1}.$$ So we can find some $Q \in \binom{[t]}{k-1}$ and $R \subseteq [b]$ with $|R| > \sqrt{m}$ such that $Q \in \binom{F_r}{k-1}$ for every $r \in R$. For each $r \in R$ and each $q \in Q$ fix some $k$-set $K^{r,q} \subseteq N_G(S_r) \cap V(A_q)$ (which is possible by definition of $F_r$). Then we can choose $R' \subseteq R$ with $|R'|=k(2k-3)$ so that $K^{r,q} = K^{r',q}$ for all $r, r' \in R'$ and every $q \in Q$. For each $q \in Q$ we write $K^q$ for $K^{r,q}$ with $r \in R'$.
We will now use the $K^q$ to find $k$ new copies of ${\mathcal{A}}_k$ that only intersect $k-1$ of the copies in our packing. We arbitrarily divide $R'$ into $k$ sets $R'_1, \dots, R'_k$ of size $2k-3$ and label $V(K^q) = \{v_{q,1}, \dots, v_{q,k}\}$ for all $q \in Q$. The new copies $A'_1,\dots,A'_k$ of ${\mathcal{A}}_k$ are obtained for each $i \in [k]$ by identifying $U_1,\dots,U_{2k-3}$ with $\{S_r: r \in R'_i\}$ and $U_0$ with $\{v_{q,i}\}_{q \in Q}$. Replacing the copies $\{A_q: q \in Q\}$ by $A'_1,\dots,A'_k$ we obtain a larger ${\mathcal{A}}_k$-packing. This contradiction completes the proof.
\[ak-pack\] Lemma \[ak-pack-lem\] still holds if we insist that the sub-$k$-graph of $G$ induced by the vertices covered by the ${\mathcal{A}}_k$-packing must be connected.
[[**Proof.**]{}]{} Apply Lemma \[ak-pack-lem\] to obtain an ${\mathcal{A}}_k$-packing $A_1,\dots,A_\ell$ in $G$ with $m_0:= |\bigcup_{i=1}^{\ell} V(A_i)| > (1- \psi/2)m$, and let $A$ be the sub-$k$-graph of $G$ induced by $\bigcup_{i=1}^{\ell} V(A_i)$. By hypothesis at most $\theta m^{k-1}$ sets $S \in \binom{[m]}{k-1}$ have fewer than $m/(2(k-1))$ neighbours in $G$ and so at most $\theta m^{k-1}$ sets $T \in \binom{V(A)}{k-1}$ have no neighbours in $V(A)$. By the definition of a component, no edges of $A$ contain vertices from different components of $A$. Therefore the largest component $C$ of $A$ must contain at least $(1-\psi)m$ vertices. Indeed, if not then there are at least $\binom{m_0}{k-2}(\psi m/2)/(k-1)\gg \theta m^{k-1}$ sets $T \in \binom{V(A)}{k-1}$ which meet at least two components of $A$ and thus have no neighbours in $A$, a contradiction (we can obtain such a set $T$ by choosing $k-2$ vertices arbitrarily in $V(A)$ and then choosing the final vertex in a different component of $A$ than the first vertex). Thus we may take the ${\mathcal{A}}_k$-packing consisting of all those copies $A_i$ of ${\mathcal{A}}_k$ with $V(A_i)\subseteq V(C)$.
Proof of Theorem \[main\] {#proof}
=========================
In our proof we will use constants that satisfy the hierarchy $$\frac{1}{n} \ll {\epsilon}\ll d^*\ll d_a \ll \frac{1}{a} \ll \nu,\frac{1}{r} \ll \theta \ll d \ll c \ll \phi \ll \delta \ll \eta \ll \frac{1}{k}.$$ Furthermore, for any of these constants $\alpha$, we use $\alpha \ll \alpha' \ll \alpha '' \ll \dots$ and assume that the above hierarchy also extends to the additional constants, e.g. $d''\ll c\ll c''\ll \phi$.
Imposing structure on $H$ {#structure}
-------------------------
### Step 1. Applying the regularity lemma {#regsection}
Let $H_1$ be the sub-$k$-graph obtained from $H$ by removing up to $a!r$ vertices so that $|V(H_1)|$ is divisible by $a!r$. Let $T = T_1 \cup \dots \cup T_r$ be an equitable $r$-partition of the vertices of $H_1$, and let $H_2$ consist of all those edges of $H_1$ that are $r$-partite sets in $T$. Then $H_2$ is an $r$-partite $k$-graph with order divisible by $a!r$, and so we may apply the regularity lemma (Theorem \[eq-partition\]), which yields an $a$-bounded ${\epsilon}$-regular vertex-equitable partition $(k-1)$-complex $P$ on $T$ and an $r$-partite $k$-graph $G$ on $T$ that is $\nu$-close to $H_2$ and perfectly ${\epsilon}$-regular with respect to $P$.
Let $M = G {\setminus}H_2$. So any edge of $G{\setminus}M$ is also an edge of $H$. Let $V_1,\dots, V_m$ be the clusters of $P$. So $T=V_1\cup \dots\cup V_m$ and $G$ is $m$-partite with vertex classes $V_1\cup \dots\cup V_m$. Note that $m \leq ar$ since $P$ is $a$-bounded. Moreover, since $P$ is vertex-equitable, each $V_i$ has the same size. So let $n_1 = |V_i| =|T|/m$.
As is usual in regularity arguments, we shall consider a reduced $k$-graph, whose vertices correspond to the clusters $V_i$, and whose edges indicate that within the cells of $P$ corresponding to the edge we can find a subcomplex to which we can apply Theorem \[robust-universal\]. For this we would like $G$ to have high density in these cells, and $M$ to have low density. Thus we define the *reduced $k$-graph $R$* on $[m]$ as follows: a $k$-tuple $S$ of vertices of $R$ corresponds to the $k$-partite union $S' = \bigcup_{i \in S} V_i$ of clusters. The edges of $R$ are precisely those $S\in \binom{[m]}{k}$ for which $G[S']$ has density at least $c''$ (i.e. $|G[S']| > c'' |K_S(S')|$) and for which $M[S']$ has density at most $\nu^{1/2}$ (i.e. $|M[S']| < \nu^{1/2}|K_S(S')|$).
Now, the edges in the reduced graph are useful in the following way. Given an edge $S\in R$, let $S' = \bigcup_{i \in S} V_i$ again. Using weak equivalence (defined in Section \[regcomplexes\]), the cells of $P$ induce a partition $C^{S,1},\dots,C^{S,m_S}$ of the edges of $K_S(S')$. Recall that $m_S\le a^k$. Therefore at most $c''|K_S(S')|/3$ edges of $K_S(S')$ can lie in sets $C^{S,i}$ with $|C^{S,i}| \leq c''|K_S(S')|/(3a^k)$. Furthermore, $|M[S']| < \nu^{1/2} |K_S(S')|$ (as $S\in R$) and so at most $\nu^{1/4}|K_S(S')|$ edges of $K_S(S')$ can lie in sets $C^{S,i}$ with $|M \cap C^{S,i}| \geq \nu^{1/4}|C^{S,i}|$. Together with the fact that $|G[S']| > c'' |K_S(S')|$ this now implies that more than $c''|K_S(S')|/2$ edges of $G[S']$ lie in sets $C^{S,i}$ with $|C^{S,i}| > c''|K_S(S')|/(3a^k)$ and $|M \cap C^{S,i}| < \nu^{1/4}|C^{S,i}|$. Thus there must exist such a set $C^{S,i}$ that also satisfies $|G \cap C^{S,i}| >c''|C^{S,i}|/2$. Fix such a choice of $C^{S,i}$ and denote it by $C^S$. Let $G^S$ be the $k$-partite $k$-complex on the vertex set $S'$ consisting of $G \cap C^S$ and the cells of $P$ that ‘underlie’ $C^S$, i.e. for any edge $Q\in G\cap C^S$ we have $$\label{eqGS}
G^S=(G\cap C^S)\cup \bigcup_{Q'\subset Q} C_{Q'}.$$ (Recall that $C_{Q'}$ was defined in Section \[regcomplexes\].) We also define the $k$-partite $k$-graph $M^S = G^S \cap M$ on the vertex set $S'$. Then the following properties hold:
- $G^S$ is ${\epsilon}$-regular.
- $G^S$ has $k$-th level relative density $d_{[k]}(G^S) \geq d'$.
- $G^S$ has absolute density $d(G^S) \geq d_a'$.
- $M^S$ satisfies $|M^S| < 2\nu^{1/4}|(G^S)_=|/c''$.
- $(G^S)_{\{i\}} = V_i$ for any $i \in S$.
Indeed, (A1) follows from (\[eqGS\]) since $G$ is perfectly ${\epsilon}$-regular with respect to $P$. To see (A2), note that $(G^S_{[k]})^*=C^S$ and so $d_{[k]}(G^S) = |G^S_{[k]}|/|(G^S_{[k]})^*|=|G^S \cap C^S|/|C^S| > c''/2$ by our choice of $C^S$. Similarly, (A3) follows from our choice of $C^S$ since $$d(G^S) = \frac{|G^S_{[k]}|}{|K_S(S')|} = \frac{|G^S \cap C^S|}{|C^S|} \cdot \frac{|C^S|}{|K_S(S')|} > \frac{(c'')^2}{6a^k}>d'_a.$$ (A4) holds since $|(G^S)_=| = |G \cap C^S| >c''|C^S|/2$ and $|M^S| \leq |M \cap C^S| < \nu^{1/4}|C^S|$. Finally, (A5) follows from (\[eqGS\]) and the fact that $C_{\{v\}}=V_i$ for all $v\in V_i$.
### Step 2. Choosing an ${\mathcal{A}}_k$-packing of $R$
The next step in our proof is to use Corollary \[ak-pack\] to find an ${\mathcal{A}}_k$-packing in the reduced $k$-graph $R$. For this we shall need an approximate minimum degree condition for $R$. Let $$J = \left\{I \in \binom{[m]}{k-1}: |N_R(I)| \le
\left(\frac{1}{2(k-1)} + \phi \right)m \right\}.$$ We shall show that $J$ is small, that is, that almost all $(k-1)$-tuples of vertices of $R$ have degree at least $(1/(2(k-1))+\phi)m$ in $R$. Consider how many edges of $H$ do not belong to $G[S']$ for some edge $S \in R$. (Recall that $S'=\bigcup_{i\in S} V_i$.) There are three possible reasons why an edge $e\in H$ does not belong to such a restriction:
- $e$ is not an edge of $G$. This could be because $e$ lies in $H$ but not $H_1$, in $H_1$ but not $H_2$, or in $H_2$ but not $G$. There are at most $a!rn^{k-1}$ edges of the first type, at most $n^k/r$ of the second type, and at most $\nu n^k$ of the third type.
- $e \in G$ contains vertices from $V_{i_1}, \dots, V_{i_k}$ such that the restriction of $M$ to $S'=\bigcup_{i\in S} V_i$ satisfies $|M[S']| \geq \nu^{1/2}|K_S[S']|$, where $S=\{i_1,\dots,i_k\}$. (Note that since $G$ and thus $M$ is $m$-partite, $i_1, \dots, i_k$ are all distinct.) Since $G$ and $H_2$ are $\nu$-close and thus $|M|\le \nu n^k$ there are at most $\nu^{1/2} n^k$ edges of this type.
- $e \in G$ contains vertices from $V_{i_1}, \dots, V_{i_k}$ such that the restriction of $G$ to $\bigcup_{i\in S} V_i$ has density less than $c''$. There are at most $c''n^k$ edges of this type.
Therefore there are fewer than $2c''n^k$ edges of $H$ that do not belong to the restriction of $G$ to $S'$ for some $S \in R$, and so we have $$\begin{aligned}
|J| n_1^{k-1} \cdot \left( \frac{1}{2(k-1)} + \eta \right)n
& < \sum_{I \in J} \sum_{x_i \in V_i, i \in I} |N_H(\{x_i : i \in I\})| \\
< 2c''k n^k + \sum_{I \in J} |N_R(I)|n_1^k
&\leq 2c'' kn^k + |J|\left(\frac{1}{2(k-1)} + \phi \right)m n_1^k.\end{aligned}$$ Since $n-a!r\le mn_1\le n$ we deduce that $|J|n_1^{k-1}(\eta-\phi)n < 2c''kn^k<3c''k(mn_1)^{k-1}n$, and so $|J| < \phi m^{k-1}$ (since $c'' \ll \phi \ll \eta$). This allows us to apply Corollary \[ak-pack\] (with $G=R$) to obtain an ${\mathcal{A}}_k$-packing $A_1, \dots, A_t$ in $R$ with $|\bigcup_{i=1}^t A_i| > (1-\delta)m$, such that the sub-$k$-graph of $R$ induced by $\bigcup_{i=1}^t V(A_i)$ is connected. For each $i\in [t]$, let the vertex set of $A_i$ be $U^i_0 \cup U^i_1 \cup \dots \cup U^i_{2k-3}$, with each $U^i_j$ of size $k-1$, so that the edge set is $\{ U^i_j \cup \{x\}: j\in [2k-3], x \in U^i_0\}$.
### Step 3. Forming the exceptional path.
Given a sub-$k$-graph $R'$ of $R$ and a cluster $V_i$, we say that *$V_i$ belongs to $R'$* if $i\in V(R')$. Let $V'_0$ contain the at most $a!r$ vertices of $H$ we removed at the start of the proof, and also the vertices in all those clusters not belonging to some copy of ${\mathcal{A}}_k$ in our packing (there are at most $\delta n$ of the latter). We will incorporate these vertices into a path $L_e$ which will later form part of our loose Hamilton cycle. We also include in $V'_0$ an arbitrary choice of $\delta n_1$ vertices from each $V_y$ for which $y \in U_j^i$ for some $j\in [2k-3]$ and some $i\in [t]$ (we do not modify any of the $V_y$ for which $y \in U_0^i$). We add up to $k-3$ more vertices from $U^1_1$ (say) to $V'_0$ so that $|V'_0| \equiv 0 \mod k-2$. We delete all these vertices from the clusters they belonged to and still write $V_y$ for the subcluster of a cluster $V_y$ obtained in this way. This gives $|V'_0| \leq 5\delta n/2$.
Now, we shall construct a path $L_e$ in $H$, which will contain all the vertices in $V_0'$ and avoid all the clusters $V_y$ with $y \in U_0^i$. Let $V_{>0} = \bigcup\{V_y: y \in U_j^i, j\in [2k-3], i\in [t]\}$. So we shall use only vertices from $V'_0$ and $V_{>0}$ in forming $L_e$. Recall that if $|V(H)|$ is not a multiple of $k-1$, then a loose Hamilton cycle contains a single pair of edges which intersect in more than one vertex: we shall make allowance for this here. Choose $A,B \subseteq V_{>0}$ satisfying $|A|=|B| = k-1$, $|A \cap B| \equiv 1-|V(H)|\mod k-1$ and $1 \leq |A \cap B| \leq k-1$. Now choose distinct $x_0, x_1 \in V_{>0}\setminus (A\cup B)$ such that $\{x_0\} \cup A \in H$ and $\{x_1\} \cup B \in H$ (we shall see in a moment that such $x_0,x_1$ exist). These edges will be the first 2 edges of $L_e$. To complete $L_e$, let $Z_1,\dots,Z_s$ be any partition of the vertices of $V'_0$ into sets of size $k-2$. We proceed greedily in forming $L_e$: for each $i=1,\dots,s$ choose any $x_{i+1}\in V_{>0}\setminus (A\cup B)$ such that $Z_i \cup \{x_i,x_{i+1}\} \in H$ (where the $x_i$ are all chosen to be distinct).
Let us now check that there will always be such a vertex available. Indeed, every set in $\binom{V(H)}{k-1}$ has at least $(1/(2(k-1))+\eta)n$ neighbours and we can choose any such neighbour which lies in $V_{>0}$ and has not already been used. But $|V(H) {\setminus}V_{>0}| \leq n/(2(k-1)) +|V_0'|$ and at most $|V_0'|+2k\le 3\delta n$ vertices have been used before. Thus (since $\delta \ll \eta$) for each choice of an $x_i$ we have at least $\eta n/2$ vertices of $V_{>0}$ to choose from. Moreover, these vertices must be contained in at least $\eta n/(2n_1)$ different $V_y$ such that $y \in U_j^{i'}$ ($j>0$). Thus we can avoid choosing a vertex from any single $V_y$ more than $6 \delta n_1/\eta \le \delta' n_1/2$ times. The path $L_e$ thus formed has edges $\{x_0\} \cup A$, $B \cup \{x_1\}$ and $\{x_i, x_{i+1}\} \cup Z_i$ for all $i \in [s]$. So all the vertices of $V'_0$ are included in $L_e$. For each cluster $V_y$, we still denote the subset of $V_y$ lying in $V(H-L_e)$ by $V_y$. Then each $V_y$ with $y \in U_0^i$ for some $i$ still satisfies $|V_y| = n_1$, and each $V_y$ with $y \in U_j^i$ for some $j>0$ satisfies $$\label{Vy}
(1-\delta')n_1 \leq \left(1-\delta-\frac{\delta'}{2}\right)n_1-(k-3)\le |V_y|\leq (1-\delta)n_1.$$ In addition $$\label{exceppathsize}
|V(H) {\setminus}V(L_e)| \equiv |V(H)|-|A \cup B \cup \{x_0,x_1 \}| \equiv -1 \mod k-1.$$ Note that $L_e$ need not be a loose path, but that even if it is not it may still form part of a loose Hamilton cycle. Also observe that $|V(L_e)| \leq 6 \delta n$.
### Step 4. Splitting our copies of ${\mathcal{A}}_k$.
The next step of the proof will be to split the copies $A_1,\dots,A_t$ of ${\mathcal{A}}_k$ (more precisely the clusters belonging to the $A_i$) into sub-$k$-complexes of $G$ that we shall later use to embed spanning loose paths. Consider any $A_i$. For convenient notation we identify each $U^i_j$ in $A_i$ with $[k-1]$ (but recall that they are disjoint sets). For each $y \in U^i_0 = [k-1]$ we have $|V_y| = n_1$, and so we can partition $V_y$ uniformly at random into $2k-3$ pairwise disjoint subsets $S^i_{y,1}, \dots, S^i_{y,2k-3}$, each of size $\frac{n_1}{2k-3}$. Similarly, given $z\in U^i_j=[k-1]$ with $j\in [2k-3]$, (\[Vy\]) and the fact that $\delta' \ll \eta$ imply that we can partition $V_z$ uniformly at random into $k-1$ pairwise disjoint subsets $T^i_{j,z}$ and $\{U^i_{j,z,w}\}_{w \in [k-1] {\setminus}\{z\}}$ so that $\frac{n_1}{2k-3} \leq |T^i_{j,z}| \leq \frac{(1-\eta)2n_1}{2k-3}$ and $|U^i_{j,z,w}| = \frac{(1-\eta)2n_1}{2k-3}$ for all $w \in [k-1] {\setminus}\{z\}$. Figure 3 shows how we do this in the case $k=3$.
\[fig:split\] {width="0.4\columnwidth"}
We arrange these pieces into $(k-1)(2k-3)$ collections of $k$ sets as follows: for each $y \in U^i_0$ and each $j \in [2k-3]$ we have a collection consisting of $S^i_{y,j}$, $T^i_{j,y}$ and $\{U^i_{j,z,y}\}_{z \ne y}$. ($3$ of these collections are illustrated in Figure 3.) For convenient notation we relabel these collections as $\{ X_{i,1}, \dots, X_{i,k} \}$ with $1 \le i \le t' = (k-1)(2k-3)t$, where for all $i \in [t']$ we have $$\label{splitsizes1}
|X_{i,1}| = \frac{n_1}{2k-3},\ \frac{n_1}{2k-3} \leq |X_{i,2}| \leq
\frac{(1-\eta)2n_1}{2k-3} \textrm{ and } |X_{i,j}| = \frac{(1-\eta)2n_1}{2k-3}
\ \mbox{ for } 3 \le j \le k,$$ and $$\label{splitsizes2}
(1-\delta')n_1 \le \sum_{j=2}^k |X_{i,j}| \le (1-\delta)n_1$$ ((\[splitsizes2\]) follows from (\[Vy\]) using the fact that all the $U^{i'}_{j',z,w}$ have equal size.) Let $X_i = \bigcup_{j \in [k]} X_{i,j}$, so each $X_i$ is a $k$-partite set, on which we shall now find a sub-$k$-complex $G_i$ of $G$ that is suitable for applying Theorem \[robust-universal\].
Consider any copy $A_{i'}$ in our ${\mathcal{A}}_k$-packing. Note that for each of the $(k-1)(2k-3)$ collections $\{X_{i,1}, \dots, X_{i,k} \}$ obtained by splitting up the clusters belonging to $A_{i'}$ there is an edge $S(i)\in A_{i'}$ such that each $X_{i,j}$ lies in a cluster belonging to $S(i)$ (and these clusters are distinct for each of $X_{i,1},\dots,X_{i,k}$). Recall that $S'(i)$ denotes the union $\bigcup_{\ell\in S(i)} V_\ell$ of all the clusters belonging to $S(i)$. Let $G_i$ denote the restriction of the $k$-partite $k$-complex $G^{S(i)}$ (which was defined in Section \[regsection\]) to $X_i$, i.e. $G_i=G^{S(i)}[X_i]$. Let $M_i = M \cap G_i=M^{S(i)}[X_i]$. We claim that we may choose the above collections $\{ X_{i,1}, \dots, X_{i,k} \}$ such that $$\label{desityHG}
d(H[X_i])\ge \frac{c''}{4} \ \ \text{for all } i\in [t'].$$ Indeed, since $S(i)\in R$, $G[S'(i)]$ has absolute density at least $c''$ and $M[S'(i)]$ has density at most $\nu^{1/2}$. Since $G{\setminus}M\subseteq H$ and $\nu \ll c''$ this shows that $H[S'(i)]$ has density at least $c''/2$. Lemma \[randomsplit\] now implies that each $H[X_i]$ has density at least $c''/4$ with probability $1-1/n_1^2$, and so with non-zero probability this is true for all $i\in [t']$.
Lemma \[restrict\] and properties (A1)–(A3) and (A5) imply that $G_i$ is an ${\epsilon}'$-regular $k$-partite $k$-complex on the vertex set $X_i$, with absolute density $d(G_i) \geq d(G^{S(i)})/2\ge d_a$, relative density $d_{[k]}(G_i) \geq d$, and $(G_i)_{\{j\}} = X_{i,j}$ for each $j$. Moreover, using $\nu \ll \theta \ll c$, property (A4) and the fact that $d(G_i) \geq d(G^{S(i)})/2$ we see that $$|M_i| \le |M^{S(i)}|<\frac{2\nu^{1/4}|(G^{S(i)})_=|}{c''}\le \theta |(G_i)_=|.$$ So by Theorem \[robust-universal\] we can delete at most $\theta' |X_{i,j}|$ vertices from each $X_{i,j}$ so that if we let $X'_{i,j} \subseteq X_{i,j}$ consist of the undeleted vertices, and let $X'_i := \bigcup_{j=1}^k X'_{i,j}$, $G_i' := G_i[X'_i]$ and $M_i' := M_i[X'_i]$, then $G_i' {\setminus}M_i'$ is $(c,{\epsilon}'')$-robustly $2^k$-universal, $d(G_i') > d^*$ and $|G_i'(v)_=| > d^*|(G_i')_=|/|X'_{i,j}|$ for every $v \in X'_{i,j}$. In particular, the latter two conditions together imply that $d(G_i'(v)_=) > (d^*)^2$ for every $v \in X_i'$. Let $X''$ denote the set of vertices deleted from any $X_{i,j}$, so $|X''|\le \theta' n$. By deleting up to $k-3$ more vertices if necessary, we may assume that $|X''|$ is divisible by $k-2$. The latter will help us to extend $L_e$ into a path which contains all the vertices in $X''$.
### Step 5. Extending the exceptional path $L_e$.
When extending $L_e$ in order to incorporate $X''$, we shall have to remove some more vertices from some of the $X'_{i,j}$, and we wish to do this so that the remainder satisfies (i) in the definition of robust universality. For this reason, we partition each $X'_{i,j}$ into two parts $AX'_{i,j}$ and $BX'_{i,j}$ as follows (where we write $BX'_i$ for $\bigcup_{j\in [k]} BX'_{i,j}$):
- For all $i,j$ and every $v \in X'_{i,j}$ we have $|(G_i'(v)[ BX'_i])_= | \ge 2c|G_i'(v)_=|$.
- Every set of $k-1$ vertices of $H$ has at least $n/(4k)$ neighbours in $\bigcup_{i,j} AX'_{i,j}$.
(Recall that for a $(k-1)$-complex $F$, $F_=$ denotes the ‘$(k-1)$th level’ of $F$.) To see that such a partition exists, consider a partition obtained by assigning each vertex to a part with probability $1/2$ independently of all other vertices. (B2) is then satisfied with high probability by a standard Chernoff bound. Now consider (B1). The ‘moreover’ part of Lemma \[randomsplit\] implies that with high probability we have for all $i, j$ and for all $v \in X'_{i,j}$ that $d((G_i'(v)[ BX'_i])_=) \ge d(G_i'(v)_=)/2$. Also, a standard Chernoff bound implies that with high probability $|BX'_{i,j'}|\ge |X'_{i,j'}|/3$ for all $j'\in [k]$. Thus $$|(G_i'(v)[ BX'_i])_= |=d((G_i'(v)[ BX'_i])_=)\prod_{j'\neq j}|BX'_{i,j'}|\ge
\frac{d(G_i'(v)_=)}{2}\prod_{j'\neq j}\frac{|X'_{i,j'}|}{3}\ge 2c|G_i'(v)_=|.$$ Now, we shall extend our path $L_e$ to include the vertices in $X''$, using only vertices from $\bigcup_{i,j} AX'_{i,j}$. We proceed similarly to when constructing $L_e$. So we split $X''$ into sets $Z_1,...,Z_{s'}$ of size $k-2$ (so $s' \leq \theta' n$). Letting $x_0$ be a final vertex of $L_e$, for $i\in [s']$, we successively choose $x_i$ to be a neighbour of the $(k-1)$-tuple $Z_i \cup \{x_{i-1}\}$ contained in some $AX'_{i',j'}$ and not already included in $L_e$, and extend $L_e$ by the edge $Z_i \cup \{x_{i-1}, x_i\}$, continuing to denote the extended path by $L_e$. Recall that $L_e$ originally contained at most $6 \delta n$ vertices. Since $|X''|\le \theta' n$, after each extension of $L_e$ we shall have $|V(L_e)| < \eta n$. So (B2) implies that for each choice of $x_i$ we have at least $n/(5k)$ suitable vertices and hence at least $t'/(5k)$ of the sets $AX'_{i'}$ contain such a suitable vertex. This shows that we can choose the $x_i$ in such a way that at most $\theta'' n_1$ vertices are chosen from any single $AX'_{i'}$.
For each $i\in [t']$ let $X^i = X^i_1 \cup \dots \cup X^i_k$ be the vertices remaining after the removal from $X_i'$ of the at most $\theta'' n_1$ vertices used in extending $L_e$, let $G^i = G'_i[X^i]$, and let $M^i = M'_i[X^i]$. By (\[desityHG\]) there are at least $cn$ vertices $v \in V(H)$ such that $v$ lies in some $X^i$ for which at least $|H[X^i]|/(2|X^i|)$ edges of $H[X^i]$ contain $v$. So we may add two further edges of $H$ to $L_e$ (one at each end) so that the new path $L_e$ has an initial vertex $x_e$ and a final vertex $y_e$ which each lie in at least $|H[X^i]|/(2|X^i|)$ edges of their respective $H[X^i]$. (We also delete the vertices of these additional two edges from their $X^i$, $G^i$ and $M^i$). Note that $x_e$ may be contained in some $BX'_{i,j}$ (and the same is true of $y_e$), but by (B2) we may choose these two additional edges so that all other vertices used lie in some $AX_{i,j}'$.
We claim that the above steps give us the following useful structure: a path $L_e$ which is ready to form part of a loose Hamilton cycle, and disjoint $k$-partite vertex sets $X^i = X^i_1 \cup \dots \cup X^i_k$ supporting $k$-complexes $G^i$ and $k$-graphs $M^i$ for each $i \in [t']$ which satisfy the following properties.
- Every vertex of $H$ lies in either the path $L_e$ or precisely one of the $k$-partite sets $X^i$.
- For each $i$, $G^i$ is a $k$-partite sub-$k$-complex of $G$ on the vertex set $X^i$. $M^i$ is the $
k$-partite $k$-graph $M \cap G^i$, and $G^i {\setminus}M^i \subseteq H$. Clearly these statements remain true after the deletion of up to ${\epsilon}n_1$ vertices of $X^i$.
- Even after the deletion of up to ${\epsilon}n_1$ vertices of $X^i$, the following statement holds. Let $L$ be a $k$-partite $k$-complex on the vertex set $U = U_1 \cup \dots \cup U_k$, where $|U_j| = |X^i_j|$ for each $j$, and let $L$ have maximum vertex degree at most $2^k$. Let $\ell \le 2(t')^2$ and suppose we have $u_1, \dots, u_\ell \in U$ and sets $Z_s \subseteq X^i_{j(u_s)}$ with $|Z_s| \geq c|X^i_{j(u_s)}|$ for each $s \in [\ell]$ (where $j(u_s)$ is such that $u_s \in U_{j(u_s)}$). Then $G^i {\setminus}M^i$ contains a copy of $L$, in which for each $j$ the vertices of $U_j$ correspond to the vertices of $X^i_j$, and each $u_s$ corresponds to a vertex in $Z_s$.
- For each $i$, $H^i = H[X^i]$ has density at least $c'$, even after the deletion of up to ${\epsilon}n_1$ vertices of $X^i$.
- If we delete up to ${\epsilon}n_1$ vertices from any $X^i$, and let $t_j = |X^i_j|$ for each $j\in [k]$ after these deletions, and let $n'_i = \frac{(\sum t_j)-1}{k-1}$, then $n'_i/2+1 \leq t_j \leq n'_i$ for all $j$.
- The initial vertex $x_e$ of $L_e$ lies in at least $|H[X^i]|/(2|X^i|)$ edges of $H[X^i]$, where $i$ is such that $x_e\in X^i$. The analogue holds for the final vertex $y_e$ of $L_e$.
(When we talk of removing a vertex of $X^i$ we implicitly mean that $G^i$, $M^i$ and $H^i$ are all restricted to the remaining vertices of $X^i$.) These properties hold for the following reasons. (C1) holds as every vertex deleted from an $X_i$ has been added to $L_e$, whilst (C2) is clear as whenever we deleted vertices we simply restricted $G$ and $M$ to the remaining vertices. For (C3), recall that $G_i' {\setminus}M_i'$ was $(c,{\epsilon}'')$-robustly $2^k$-universal. Moreover, for all $i\in [t']$ and all $j \in [k]$ we have $|X^i_j| \geq |X_{i,j}'|/2 \ge c|X_{i,j}'|$, since we ensured that we only deleted $\theta ''n_1$ vertices from any single $AX'_{i}$ (and at most two from $BX'_i$). Furthermore by (B1) we know that $|G^i(v)_=| \ge |(G_i'(v)[ BX'_i])_= | \ge c|G'_i(v)_=|$ for any $v \in X^i$. (Also, even if we had arbitrarily deleted a further ${\epsilon}n_1$ vertices from $X'_i$ when obtaining $X^i$, $G^i$ and $M^i$, these bounds would still hold.) So $G^i {\setminus}M^i$ satisfies (i) in the definition of a robustly universal complex (where $X_j^i$ plays the role of $V_j$). The sets $Z_s$ satisfy (iii) in the definition and so we can find the required copy of $L$ (even after the deletion of up to ${\epsilon}n_1$ more vertices of $X^i$). (C4) follows from (\[desityHG\]) and the fact that $X^i$ was formed by deleting at most $(\theta'+\theta'') n_1 \ll c' |X_i|$ vertices from $X_i$. Similarly, for (C5) note that (even after up to ${\epsilon}n_1$ more deletions) we have deleted at most $2\theta'' n_1$ vertices from each $X_i$ since we split the clusters to form the $X_i$. So by (\[splitsizes1\]), after these deletions we must have
- $\frac{n_1}{2k-3} - 2\theta'' n_1 \leq |X^i_1| \leq \frac{n_1}{2k-3}$,
- $\frac{n_1}{2k-3} - 2\theta'' n_1 \leq |X^i_2| \leq \frac{(1-\eta)2n_1}{2k-3}$, and
- $ \frac{(1-\eta)2n_1}{2k-3} - 2\theta'' n_1 \leq |X^i_j| \leq \frac{(1-\eta)2n_1}{2k-3}$ for $3 \le j \le k$,
and by (\[splitsizes2\]) we must have
- $
(1-\delta')n_1 - 2(k-1)\theta'' n_1 \le \sum_{j=2}^k |X_{i,j}| \le (1-\delta)n_1.$
Since $\theta'' \ll \delta \ll \delta' \ll \delta'' \ll \eta$, we deduce that
- $n_i' \ge \frac{1}{k-1} \left( n_1 \left( 1-\delta' + \frac{1}{2k-3}-2k\theta'' \right)-1 \right)
\ge \frac{(1-\eta)2n_1}{2k-3}$, and
- $n'_i \leq \frac{n_1}{k-1} \left( 1-\delta + \frac{1}{2k-3} \right) \le \frac{(2-\delta)n_1}{2k-3}.$
So property (C5) follows. Finally, (C6) follows from the final step in the construction of $L_e$, in which we added an extra edge to each end of $L_e$ so that (C6) would be satisfied.
The supplementary graph {#supp}
-----------------------
Roughly speaking, our aim is to find a spanning loose path in each $G^i\setminus M^i$ (and thus in $H^i$) such that all these paths together with $L_e$ form a loose Hamilton cycle in $H$. So we have to ensure that the complete $k$-partite $k$-graph on $X^i$ contains a spanning loose path (for this, we will need $|X^i| \equiv 1\mod k-1$) and we need to join up all the loose paths we find in the $H^i$. The purpose of this section is to find the ‘connecting loose paths’ which join up the $X^i$ in such a way that the divisibility problems are dealt with as well. To do this, we first define a supplementary hypergraph $R^*$ whose vertices correspond to the $X^i$. We will show that $R^*$ is connected and that ‘along’ edges of $R^*$ we can find our loose paths in $H$ which join up all the $X^i$.
The vertex set of the *supplementary hypergraph $R^*$* is $[t']$. A subset $e\subseteq [t']$ of size at least 2 is an edge of $R^*$ if there exists an edge $S_e\in R$ such that for all $j\in S_e$ there are $i_j\in e$ and $\ell_j\in [k]$ with $X^{i_j}_{\ell_j}\subseteq V_j$ and $e=\{i_j: j\in S_e\}$. (We fix one such edge $S_e$ for every $e\in R^*$.) Then every edge of $R^*$ has size at most $k$. We say that $X^i$ *belongs* to an edge $e\in R^*$ if $i\in e$. Similarly, $X^i$ *belongs to* some subhypergraph $R'\subseteq R^*$ if $i\in V(R')$.
\[suppgraphconnected\] The supplementary graph $R^*$ is connected.
[[**Proof.**]{}]{} Recall that we chose the copies $A_\ell$ of ${\mathcal{A}}_k$ in such a way that the sub-$k$-graph $A$ of $R$ induced by $\bigcup_{\ell=1}^t A_\ell$ is connected. Suppose that $R^*$ is not connected. Let $R^*_1$ be a component of $R^*$ and let $R^*_2 = R^* -R^*_1$. Let $R_1 = \{j\in[m]: X^i_s \subseteq V_j$ for some $i \in V(R^*_1), s \in [k]\}$. So $R_1$ corresponds to the set of all those clusters which meet some $X^i$ belonging to $R^*_1$. Define $R_2$ similarly. Then $R_1 \cup R_2 = V(A)$ and thus $A$ contains some edge $S$ intersecting both $R_1$ and $R_2$. But then $S$ corresponds to an edge of $R^*$ intersecting both $V(R^*_1)$ and $V(R^*_2)$, a contradiction.
The next lemma shows that within the $X^i$ belonging to an edge of $R^*$, we can find a reasonably short loose path in $H$ and we may choose (modulo $k-1$) how many vertices this path uses from each $X^i$. Using the connectedness of $R^*$, this will allow us to find the connecting loose paths which join up the $X^i$ whilst having control over the divisibility properties. We shall also insist that the path in Lemma \[interpath\] avoids a number of ‘forbidden vertices’, to enable us to ensure that our connecting loose paths are disjoint, and that the endvertices of these paths lie in many edges of the relevant $H^i$.
\[interpath\] Suppose that $e\in R^*$ and that for every $i\in e$ there is an integer $t_i$ such that $0 \leq t_i \leq k-1$ and $\sum_{i\in e} t_i \equiv 1 \mod k-1$. Let $i',i''\in e$ be distinct. Moreover, suppose that $Z$ is a set of at most $100(t')^2k^3$ ‘forbidden’ vertices of $H$. Then in the sub-$k$-graph of $H$ induced by $\bigcup_{i\in e} X^i$ we can find a loose path $L$ with the following properties.
- $L$ contains at most $4k^3$ vertices.
- $L$ has an initial vertex $u$ in $X^{i'}$ and a final vertex $v$ in $X^{i''}$.
- $|V(L) \cap X^{i}| \equiv t_i \mod k-1$ for each $i\in e$.
- $L$ contains no forbidden vertices, i.e. $V(L) \cap Z = \emptyset$.
- $u$ lies in at least $|H^{i'}|/(2|X^{i'}|)$ edges of $H^{i'}$, and $v$ lies in at least $|H^{i''}|/(2|X^{i''}|)$ edges of $H^{i''}$.
[[**Proof.**]{}]{} Recall that in Section \[regsection\] we assigned a $k$-partite $k$-complex $G^S$ to every edge $S\in R$ such that (A1)–(A5) are satisfied. To simplify notation, we write $S$ for the edge $S_e\in R$ corresponding to $e$ and suppose that $S=[k]$. For each $j\in S=[k]$ choose $i_j\in e$ and $\ell_j\in [k]$ such that $X^{i_j}_{\ell_j}\subseteq V_j$ and such that $e=\{i_j: j\in S=[k]\}$. To simplify notation we write $Y_j$ for $X^{i_j}_{\ell_j}{\setminus}Z$, $Y=\bigcup_{j\in [k]}Y_j$ and assume that $i'=i_1$ and $i''=i_k$. For each $i\in e$ let $J_i$ be the set of all $j\in S=[k]$ with $i_j=i$. So the sets $J_i$ are disjoint and their union is $[k]$. Pick some $j\in J_i$ and let $t'_j=t_i$ and $t'_s=0$ for all $s\in J_i{\setminus}\{j\}$. Our path $L$ will consist of $t'_j$ vertices from each $Y_j$ (modulo $k-1$) and thus of $t_i$ vertices from each $X^i$ (modulo $k-1$).
Since $G^S$ satisfies (A1)–(A3) and (A5), Lemma \[restrict\] implies that the restriction $G^S[Y]$ is ${\epsilon}'$-regular, with absolute density at least $d(G^S)/2\ge d_a$, relative density at index $[k]$ at least $d$ and $(G^S)_{\{j\}}[Y]=Y_j$. Furthermore, (A4) together with the fact that $d(G^S[Y])\ge d(G^S)/2$ imply that $$|M^S[Y]| < |M^S|< \frac{2\nu^{1/4}|G^S|}{c''}\le \theta |G^S[Y]|.$$ Thus Theorem \[robust-universal\] implies that we can delete $\theta'|Y_j|$ vertices from each $Y_j$ to obtain subsets $Y'_j$ such that $G^S[Y'] {\setminus}M^S[Y']$ is $(c,{\epsilon}'')$-robustly $2^k$-universal, where $Y'=\bigcup_{j\in [k]} Y'_j$.
Now, let $v_j = (k+2)(k-1)+t'_j$. Then $\sum v_j \equiv 1 \mod k-1$ and so $n' = ((\sum v_i) - 1)/(k-1)$ is an integer. Furthermore, $k(k+2) \leq n' \leq k(k+3)$, and so $n'/2+1 \leq v_j \leq n'$ for each $j$. Thus by Lemma \[loosepath\] we can find a loose path in the complete $k$-partite $k$-graph on the vertex set $Y'$, beginning in $Y_1'$, finishing in $Y_k'$ and using $v_j$ vertices from each $Y_j'$. Since $G^S[Y'] {\setminus}M^S[Y']$ is $(c, {\epsilon}'')$-robustly $2^k$-universal, we can find such a loose path $L$ in $G^S[Y']{\setminus}M$ and hence in $H-Z$. (Indeed, we can do this by finding the complex $L^\leq$, which has maximum vertex degree at most $2^k$. Note that we use the definition with $J=G'$ in (i)). Note that $L$ contains at most $k(k-1)(k+3) \leq 4k^3$ vertices.
To see that we can insist on the final condition of the lemma, recall that $d(H^i)\ge c'$ by (C4). Thus for all $j\in [k]$ at least $c'|X^i_j|/2$ vertices of $X^i_j$ lie in at least $|H^i|/(2|X^i_j|)$ edges of $H^i$, and so we may restrict the initial and final vertices of $L$ to these sets of vertices (minus the vertices of $Z$) by (iii) in the definition of robust universality.
Constructing the loose Hamilton cycle {#linking}
-------------------------------------
As discussed before, our Hamilton cycle in $H$ will consist of $L_e$ and paths in each $H^i$ as well as paths connecting the $X^i$. However, we need to make sure that all these paths join up nicely, motivating the following definition. Suppose $L$ is a path in some $k$-graph $K$ with initial vertex $x'$ and final vertex $y'$. Also, let $I, F\subseteq V(K){\setminus}V(L)$ be disjoint sets of size $k-2$. Then $L^*=I \cup F \cup V(L)$ is a *prepath*. Note that $L^*$ is not (the vertex set of) a $k$-graph, but that if we can find vertices $x, y \in V(K){\setminus}L^*$ such that $\{x,x'\} \cup I, \{y, y'\} \cup F \in K$, then adding $x$ and $y$ to $L^*$ gives another path. We refer to all such vertices $x \in V(K)$ as *possible initial vertices* of $L^*$ and to all such vertices $y \in V(K)$ as *possible final vertices*. If $L$, $L'$ and $L''$ are disjoint loose paths, $I,F,x,y$ are as before, $x$ is also the final vertex of $L'$ and $y$ is also the initial vertex of $L''$ then $I$ and $F$ together with $L',L,L''$ form a single loose path, illustrating how we shall join paths together.
We start by converting our exceptional path $L_e$ into a prepath. Recall that $|V(L_e)|<\eta n$ and that the initial vertex $x_e$ of $L_e$ and its final vertex $y_e$ satisfy (C6). Let $a\in [t']$ and $u_a\in [k]$ be such that $x_e\in X^{a}_{u_a}$. Pick any $u'_a\in [k]$ with $u_a\neq u'_a$. (C4) and (C6) together imply that there is a set $I_0\subseteq X^{a}\setminus (X^{a}_{u_a}\cup X^{a}_{u'_a})$ for which $X^{a}_{u'_a}$ contains at least $c|X^{a}|$ vertices $v$ which form an edge of $H^a$ together with $I_0\cup \{x_e\}$. Let $I'_0\subseteq X^{a}_{u'_a}$ be such a set of vertices. Similarly, letting $b\in [t']$, $u_b\neq u'_b\in [k]$ be such that $y_e\in X^b_{u_b}$, there is a set $F_0\subseteq X^{b}\setminus (X^{b}_{u_b}\cup X^{b}_{u'_b}\cup I_0)$ for which $X^{b}_{u'_b}$ contains at least $c|X^{b}|$ vertices $v$ which form an edge of $H^b$ together with $F_0\cup \{y_e\}$. Let $F'_0\subseteq X^{b}_{u'_b}$ be such a set of vertices. Let $L^*_e$ be the prepath $I_0\cup F_0\cup V(L_e)$. Then $I'_0$ is a set of possible initial vertices of $L_e^*$ and $F'_0$ is a set of possible final vertices. (We do not remove $I_0$ from $X^a$ and $F_0$ from $X^b$ at this stage.)
Since by Lemma \[suppgraphconnected\] the supplementary graph $R^*$ is connected, we can find a walk $W$ from $b$ to $a$ in $R^*$ such that every $i\in [t']=V(R^*)$ appears as an initial, link or final vertex in $W$ (these vertices were defined in Section \[walks\]) and such that $W$ has length $\ell \leq (t')^2$. Let $e_1,\dots,e_\ell$ be the edges of this walk, let $r_1 = b, r_{\ell+1} = a$, and let $r_2, \dots, r_{\ell}$ be the link vertices of the walk. For each $i\in [t']$, let $d_i = |\{j\in [\ell+1]: r_j = i\}|$, that is, the number of times $i$ appears as an initial, link or final vertex in $W$. So $d_i > 0$ for every $i$ and $\sum d_i = \ell+1 $.
Our next aim is to apply Lemma \[interpath\] to each edge $e_j$ in order to find a loose path $L_j$ in $H$, which we will extend to a prepath $L_j^*$ with many possible initial vertices in $X^{r_j}$ and many possible final vertices in $X^{r_{j+1}}$. We shall do this for each $e_1,\dots,e_\ell$ in turn. So suppose that $s\in [\ell]$ and that for all $j=1,\dots,s-1$ we have defined loose paths $L_j$ in $H$ as well as sets $I_j,F_j$ extending $L_j$ to a prepath $L^*_j$ which satisfy the following properties:
- $L_j$ lies in the sub-$k$-graph of $H$ induced by $\bigcup_{i\in e_j} X^i$ and contains at most $4k^3$ vertices.
- The initial vertex $x_j$ of $L_j$ lies in $X^{r_j}$ and its final vertex $y_j$ lies in $X^{r_{j+1}}$.
- $I_j\subseteq X^{r_j}$ and $F_j\subseteq X^{r_{j+1}}$.
- There is a set $I'_j\subseteq X^{r_j}$ of at least $c|X^{r_j}|$ possible initial vertices for $L^*_j$. Similarly, there is a set $F'_j\subseteq X^{r_{j+1}}$ of at least $c|X^{r_{j+1}}|$ possible final vertices for $L^*_j$.
- All the prepaths $L^*_e,L^*_1,\dots,L^*_{s-1}$ are disjoint.
- For each $i\in [t']$ and all $j=0,\dots,s-1$ let $X^i(j) = X^i{\setminus}(V(L_1)\cup\dots\cup V(L_j))$, where $X^i(0)=X^i$. For each $j\in [s-1]$ set $t_i(j) = |X^i(j-1)|+d_i$. Then for every $i\in e_j$ with $i\neq r_{j+1}$ we have $|V(L_j)\cap X^i|\equiv t_i(j) \mod k-1$. Moreover $|V(L_j)\cap X^{r_{j+1}}|\equiv 1-\sum_{i\in e_j,\ i\neq r_{j+1}} t_i(j) \mod k-1$.
Let us now show how to find $L_s$, $I_s$ and $F_s$. Apply Lemma \[interpath\] with $e=e_s$, $i'=r_s$, $i''=r_{s+1}$ and with $Z=L^*_1\cup\dots L^*_{s-1}\cup I_0\cup F_0$ to find a loose path $L_s$ which satisfies (D1), (D2), (D6) and is disjoint from $L^*_e,L^*_1,\dots,L^*_{s-1}$. Moreover, the initial vertex $x_s$ of $L_s$ lies in at least $|H^{r_s}|/(2|X^{r_s}|)$ edges of $H^{r_s}$, and the final vertex $y_s$ of $L_s$ lies in at least $|H^{r_{s+1}}|/(2|X^{r_{s+1}}|)$ edges of $H^{r_{s+1}}$. We can now use the latter property to choose sets $I_s$ and $F_s$ which extend $L_s$ to a prepath $L^*_s$ satisfying (D3)–(D5). The argument for this is similar to that for the extension of $L_e$ to $L^*_e$. Altogether this shows that we can find prepaths $L^*_1,\dots,L^*_\ell$ satisfying (D1)–(D6).
For each $i \in [t']$ we let $j_i$ be the maximal integer such that $i \in e_{j_i}$. Thus $X^i(\ell) = X^i(j_i) = X^i(j_i-1) {\setminus}V(L_{j_i})$ by (D1). But if $i \neq r_{\ell+1}$ then (D5) and (D6) together imply that $$|V(L_{j_i}) \cap X^i(j_i-1)|=|V(L_{j_i}) \cap X^i| \equiv t_i(j_i) \equiv |X^i(j_i-1)|+d_i \mod k-1$$ and so $|X^i(\ell)| \equiv -d_i \mod k-1$. We claim that this also holds if $i = r_{\ell+1}$. To see this, recall that since $L_j$ is loose, we have $|V(L_j)|\equiv 1\mod k-1$ for each $j\in [\ell]$. Hence $$\begin{aligned}
|X^{r_{\ell+1}}(\ell)| & = & |V(H) {\setminus}V(L_e)|-\sum_{j\in [\ell]} |V(L_j)|-
\sum_{i\in [t'],\ i\neq r_{\ell+1}}|X^i(\ell)|\\
& \stackrel{(\ref{exceppathsize})}{\equiv} & -1-\ell+\sum_{i\in [t'] \ i\neq r_{\ell+1}} d_i
\equiv -d_{r_{\ell+1}} \mod k-1\end{aligned}$$ as $\ell+1=\sum_{i\in [t']} d_i$. Let $Y^i=X^i{\setminus}(L^*_e\cup L^*_1\cup\dots\cup L^*_\ell)$. Since by (D3) for each $i\in [t']$ there are exactly $2(k-2)d_i$ vertices of $X^i$ which lie in $L^*_e,L^*_1,\dots,L^*_\ell$ but not in $L_e,L_1,\dots,L_\ell$, this in turn implies that $$\label{Yi}
|Y^i|\equiv -d_i-2(k-2)d_i \equiv d_i \mod k-1.$$ Let $x_{\ell+1}=x_e$, $y_0=y_e$, $L^*_0=L^*_e$, $I_{\ell+1}=I_0$ and $I'_{\ell+1}=I'_0$. In order to complete our prepaths $L^*_0,\dots,L^*_\ell$ to a Hamilton cycle we wish to choose $d_i$ disjoint loose paths $L^i_1,\dots,L^i_{d_i}$ within each $H[Y^i]$ which together contain all the vertices in $Y^i$ and which ‘connect’ successive prepaths $L_j^*$. We achieve this as follows. Let $J_i$ be the set of all $j\in [\ell+1]$ with $r_j=i$. So $J_i$ is the set of positions at which $i$ occurs as an initial, final or link vertex in our walk $W$ and $|J_i|=d_i$. Let $j_1\le \dots\le j_{d_i}$ be the elements of $J_i$. Then we choose the $L^i_s$ ($s\in [d_i]$) in such a way that the initial vertex of $L^i_s$ lies in $F'_{j_s-1}$ and its final vertex lies in $I'_{j_s}$, all the $L^i_s$ are disjoint and together they cover all the vertices in $Y^i$. To see that this can be done, first note that $|X^i{\setminus}Y^i|\le \ell (4k^3+2(k-2))+2(k-2) \ll {\epsilon}n_1$. So using Lemma \[loosepath\] together with (C5) and (\[Yi\]) it is easy to check that the complete $k$-partite $k$-graph on $Y^i$ contains such paths (e.g. first choose $L^i_1,\dots,L^i_{d_i-1}$, each consisting of precisely 2 edges, and then apply (C5) and Lemma \[loosepath\] to find a loose path $L^i_s$ containing all the remaining vertices of $Y^i$). Now (C3) and (D4) together imply that $G^i[Y^i]{\setminus}M^i[Y^i]$ contains the $k$-complexes induced by these paths (i.e. it contains $(L^i_1)^\le,\dots,(L^i_{d_i})^\le$). But this means that we can find the required paths $L^i_1,\dots,L^i_{d_i}$ in each $H[Y^i]$.
Finally, for each $s\in [d_i]$ write $L'_{j_s}$ for $L^i_s$ and $x'_{j_s}$ for its initial and $y'_{j_s}$ for its final vertex (where $j_s$ is as defined in the previous paragraph). To obtain our Hamilton cycle in $H$ we first traverse $L_0=L_e$, then we use the edge $F_0\cup \{y_0,x'_1\}$ in order to move to the initial vertex $x'_1$ of $L'_1$. (This is possible since $x'_1\in F'_0$.) Now we traverse $L'_1$ and use the edge $I_1\cup \{y'_1,x_1\}$ to get to $x_1$. (Again, this is possible since $y'_1\in I'_1$.) Next we traverse $L_1$ and use the edge $F_1\cup \{y_1,x'_2\}$ to move to $x'_2$. We continue in this way until we have reached the initial vertex $x_{\ell+1}=x_e$ of $L_0=L_e$ again. (So in the last step we traversed $L'_{\ell+1}$ and used the edge $I_{\ell+1}\cup \{y'_{\ell+1},x_{\ell+1}\}$.) This completes the proof of Theorem \[main\].
[99]{}
K. Azuma, Weighted sums of certain dependant random variables, [*Tôhoku Math. J.*]{} [**19**]{} (1967), 357–367.
J.C. Bermond, A. Germa, M.C. Heydemann and D. Sotteau, Hypergraphes hamiltoniens, *Prob. Comb. Théorie graph Orsay* [**260**]{} (1976), 39–43.
G.A. Dirac, Some theorems on abstract graphs, [*Proc. London. Math. Soc.*]{} [**2**]{} (1952), 69–81.
W.T. Gowers, Hypergraph Regularity and the multidimensional Szemerédi Theorem, [*Annals of Math.*]{} [**166**]{} (2007), 897–946.
H. Hàn and M. Schacht, Dirac-type results for loose Hamilton cycles in uniform hypergraphs, [*J. Combinatorial Theory B*]{} [**100**]{} (2010), 332–346.
G.Y. Katona and H.A. Kierstead, Hamiltonian chains in hypergraphs, [*J. Graph Theory*]{} [**30**]{} (1999), 205–212.
P. Keevash, A hypergraph blow-up lemma, [*Random Structures and Algorithms*]{}, to appear.
Y. Kohayakawa, V. Rödl and J. Skokan, Hypergraphs, quasi-randomness, and conditions for regularity, *J. Combinatorial Theory A* **97** (2002), 307–352.
D. Kühn and D. Osthus, Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree, [*J. Combinatorial Theory B*]{} [**96**]{} (2006), 767–821.
D. Kühn, R. Mycroft and D. Osthus, Hamilton $\ell$-cycles in uniform hypergraphs, [*J. Combinatorial Theory A*]{} [**117**]{} (2010), 910–927.
V. Rödl, A. Ruciński and E. Szemerédi, A Dirac-type theorem for 3-uniform hypergraphs, [*Combin. Probab. Comput.*]{} [**15**]{} (2006), 229–251.
V. Rödl, A. Ruciński and E. Szemerédi, An approximate Dirac-type theorem for $k$-uniform hypergraphs, [*Combinatorica*]{} [**28**]{} (2008), 229–260.
V. Rödl and M. Schacht, Regular partitions of hypergraphs: regularity lemmas, [*Combin. Probab. Comput.*]{} [**16**]{} (2007), 833–885.
V. Rödl and M. Schacht, Regular partitions of hypergraphs: counting lemmas, [*Combin. Probab. Comput.*]{} [**16**]{} (2007), 887–901.
V. Rödl and J. Skokan, Regularity lemma for uniform hypergraphs, [*Random Structures & Algorithms*]{} [**25**]{} (2004), 1–42.
Peter Keevash, Richard Mycroft, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, United Kingdom, {[p.keevash,r.mycroft]{}}[@qmul.ac.uk ]{}
Daniela Kühn, Deryk Osthus, School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom, {[kuehn,osthus]{}}[@maths.bham.ac.uk ]{}
|
---
abstract: 'This paper discusses several methods for describing the dynamics of open quantum systems, where the environment of the open system is infinite-dimensional. These are purifications, phase space forms, master equation and liouville equation forms. The main contribution is in using Feynman-Kac formalisms to describe the infinite-demsional components.'
author:
- 'J. Gough[^1], T.S. Ratiu[^2], O.G. Smolyanov[^3]'
title: 'Feynman, Wigner, and Hamiltonian Structures Describing the Dynamics of Open Quantum Systems'
---
This paper discusses several approaches for describing the dynamics of open quantum systems. Open quantum systems play an important role in modelling physical systems coupled to their environment and, in particular, for the emerging field of quantum feedback control theory (see \[12\]). Thus, in studying coherent quantum feedback, models consisting of a quantum system related to quantum control system, so that each of these systems turns out to be open, are considered.
Generally the master equation is presented in the theoretical physics literature as the central description of an open quantum system. In practice, however, it is these solutions that are important for potential applications, rather than the master equations themselves. Our goal is to solve the exact master equations, which describe the reduced dynamics of subsystems of certain large systems generated by the dynamics of these large systems: these master equation arise from a number of different approaches which we will consider.
In fact, we examine four approaches for describing subsystems dynamics, and in each case we exploit Feynman type formulas (see \[1, 2\]). Moreover, we assume that the quantum systems under consideration are obtained by the Schrödinger quantization \[3\] of classical Hamiltonian systems.
1. Our first approach is based on a representation of mixed states as random pure ones, where the dynamics of a subsystem of the isolated quantum system is described by a random process taking values in the Hilbert space of the subsystem. The random process is then defined by using the Feynman formula for the solution of the Schrödinger equation for the united system.
2. The second approach uses the Wigner function \[4\] and also its infinite dimensional analogue, the Wigner measure, which was introduced in \[5\]. If the phase space of the classical Hamiltonian system generating the quantum system under consideration is finite dimensional, then the density of the Wigner measure with respect to the standard Lebesgue measure coincides with the Wigner function. As shown in \[5\], the evolution of the Wigner measure of a closed quantum system is described by a Liouville-Moyal type equation; in order to obtain a solution of the master equation for the dynamics of the Wigner measure (or function) of a subsystem of the initial system from a solution of this equation represented by using a Feynman type formula, it suffices to integrate this representation over the coordinates of the phase space of the corresponding classical subsystem. Another approach for describing the evolution of the Wigner function of a subsystem is discussed in \[6\].
3. The third approach again uses the Feynman formulas for the Schrödinger equation for a quantum system and its environment, but this time these formulas are used to describe the evolution of the density operator of each part; it is given by the corresponding partial trace of the evolving density operator of the united system.
4. The final approach considered here is based on the representation of any state of the quantum system by a probability measure on its Hilbert space. In the case of a closed system, the evolution of this measure is described by the Liouville equation generated by the Hamiltonian structure, and here the Hamiltonian equation coincides with the Schrödinger equation (see \[7-9\]). At the same time, the correlation operator of this measure coincides with the density operator \[10\], so knowing the evolution of the density operator of the subsystem allows us to obtain the evolution of the probability measure on its Hilbert space and hence solve the master equation generated by the associated Liouville equation. It should perhaps be emphasized that the technique for treating Wigner measures by employing a suitable projection of the (pseudo)measure defined on the space of the combined classical system, is not applicable in this situation because the Hilbert space of the united system is the tensor product, rather than the Cartesian product.
This paper focuses on the algebraic structures related to the problems under consideration, and we do not explicitly state the analytical details.
STATES OF OPEN QUANTUM SYSTEMS
==============================
Let $\mathfrak{H}_{1}$ and $\mathfrak{H}_{2}$ be the Hilbert spaces of states of two quantum systems. In what follows, we refer to the first system (as well as to its classical counterpart) as the *open system* and to the second as *the environment*, respectively. These two systems form a composite system, whose Hilbert space is the Hilbert tensor product \[3\] $\mathfrak{H}=\mathfrak{H}_{1}\otimes \mathfrak{H}_{2}$.
Let $\mathscr{Q}_{1}$ and $\mathscr{Q}_{2}$ be the configuration spaces of the corresponding classical open system and the environment, respectively. We assume that $\mathscr{Q}_{1}$ and $\mathscr{Q}_{2}$ are real separable Hilbert spaces. In both cases we shall assume that we have measures $\nu_j$ ($j=1,2$) defined on the $\sigma$-algebra of Borel subsets of the corresponding spaces. We then set $$\mathfrak{H}_{1}=L^{2}(\mathscr{Q}_{1},\nu_1) , \quad \mathfrak{H}_{2}=L^{2}(\mathscr{Q}_{2},\nu_2)$$ and in particular the composite Hilbert space is then $$\mathfrak{H}=\mathfrak{H}_{1}\otimes \mathfrak{H}_{2} \cong L^{2}(\mathscr{Q}_{1}\times \mathscr{Q}_{2},\nu
_{L}\otimes \nu_2).$$
The open system we wish to describe will be quantum mechanical, so we have dim$\, \mathscr{Q}
_{1}<\infty $, and fix $\nu_1$ to be standard Lebesgue measure on $\mathscr{Q}
_{1}$ for definiteness.
The dimension of $\mathscr{Q}_{2}$ will typically be infinite, in this case, according to the well-known result of Weil, there does not exist a non-zero $\sigma $-finite countably additive locally finite Borel measure on $\mathscr{Q}_{2}$. Instead, we fix a Gaussian measure $\nu_2$ on $\mathscr{Q}_{2}$ (this is a matter of convenience, however, and non-Gaussian measures may be used as well).
If $\varphi \in L^{2}(\mathscr{Q}_{1}\times \mathscr{Q}_{2},\nu_1\otimes
\nu_2)$ is normalized, that is, $$\int_{\mathscr{Q}_{1}\times \mathscr{Q}_{2}}\left| \varphi \left( q_{1},q_{2}\right) \right| ^{2} \nu_1(dq_{1})\nu_2(dq_{2})=1,$$ then the *marginal distributions* $\rho _{k}$ are defined by $$\begin{aligned}
\rho _{1}(q_{1})\triangleq \int_{\mathscr{Q}_{2}}\left| \varphi \left( q_{1},q_{2}\right) \right| ^{2}\nu_2(dq_{2}) \\
\rho _{2}(q_{2})\triangleq \int_{\mathscr{Q}_{1}}\left| \varphi \left( q_{1},q_{2}\right) \right| ^{2}\nu_1(dq_{1})\end{aligned}$$ A probability measure $\mathbb{P}_{2}$ on $\mathscr{Q}_{2}$ is then defined by $$\begin{aligned}
\mathbb{P}_{2 } (dq_2)= \rho_2 (q_2 ) \, \nu_2 (dq_2 ),\end{aligned}$$ so $\mathbb{P}_2$ is absolutely continuous with respect $\nu_2$ with Radon-Nikodym density $\rho_2$, and describes the results of measurements of the environment coordinates. The pair $(\mathscr{Q}_{2},\mathbb{P}_{2})$ is a Kolmogorov probability space.
Taking the fixed pure state $\varphi$, we may define a $\mathfrak{H}_1 (=L^{2}(\mathscr{Q}_{1},\nu_1))$-valued random variable on $(\mathscr{Q}_{2},\mathbb{P}_{2})$ by $$\begin{aligned}
\Psi_1: q_{2}\mapsto \varphi
\left( \cdot ,q_{2}\right) .
\label{eq:F}\end{aligned}$$
If we fix an operator $\hat{A}_1$ on $\mathfrak{H}_1$, then $$\begin{aligned}
\mathbb{E}_2 \left[ \frac{ \langle \Psi_1
\vert \hat{A}_1 \vert \Psi_1 \rangle_{\mathfrak{H}_1} }
{\langle \Psi_1
\vert \Psi_1 \rangle_{\mathfrak{H}_1}} \right]
&=& \int_{\mathscr{Q}_2} \frac{ \langle \Psi_1
\vert \hat{A}_1 \vert \Psi_1 \rangle_{\mathfrak{H}_1} (q_2) }
{\langle \Psi_1
\vert \Psi_1 \rangle_{\mathfrak{H}_1} (q_2) }
\, \mathbb{P}_2 (dq_2)
\\
&=& \int_{\mathscr{Q}_2} \frac{ \int_{\mathscr{Q}_1} \varphi (q_1 ,q_2 )^\ast (\hat{A}_1 \otimes I_2 \, \varphi ) (q_1,q_2) \nu_1 (dq_1)}
{\int_{\mathscr{Q}_1} \vert \varphi (q^\prime_1 ,q_2 ) \vert^ 2 \nu_1 (dq^\prime_1) } \mathbb{P}_2 (dq_2) \\
&=& \int_{\mathscr{Q}_1} \int_{\mathscr{Q}_2} \varphi (q_1 ,q_2 )^\ast (\hat{A}_1 \otimes I_2 \, \varphi ) (q_1,q_2) \nu_1 (dq_1)
\nu_2 (dq_2) \\
&\equiv& \langle \varphi \vert \hat{A}_1 \otimes I_2 \vert \varphi \rangle_{\mathfrak{H}_1 \otimes \mathfrak{H}_2} \\
& \equiv & \mathrm{tr}_{\mathfrak{H}_1} [ \hat{\varrho}_1 \hat{A}_1 ].\end{aligned}$$ where $\hat{\varrho}_1$ is the von Neumann density operator corresponding the the marginal state of the open system.
**Proposition 1**. *The correlation operator of the probability measure on* $L^{2}(\mathscr{Q}_{1},\nu_1)$*, which is the distribution of results of measurements of the random pure state* $\Psi_1$ *given in (\[eq:F\]), coincides with the von Neumann density operator*.
In Dirac notation, we may write $\ langle q_1 \vert \Psi \rangle $ for the complex-valued random variable $\langle q_1 \vert \Psi : q_2 \mapsto \varphi (q_1 , q_2 )$, then $$\begin{aligned}
\mathbb{E}_2 \left[ \frac{ \langle q_1 \vert \Psi_1 \rangle
\langle \Psi_1 \vert q^\prime_1 \rangle }
{\langle \Psi_1
\vert \Psi_1 \rangle_{\mathfrak{H}_1}} \right]
= \varrho( q_1 , q^\prime_1 )
,\end{aligned}$$ where $\varrho( q_1 , q^\prime_1 )$ is the kernel operator of $\hat{\varrho}_1$.
**Remark 1**. It is useful to compare the following two approaches for calculating the probability distribution of the results of measurements of the coordinate $q_{1}$ of the first system, one of which directly uses the function $\varphi \in L^{2}(\mathscr{Q}_{1}\times \mathscr{Q}_{2},\nu_1
\otimes \nu_2)$, which represents the pure state of the composite system, and the other one uses the $L^{2}(\mathscr{Q}_{1},\nu_1)$-valued random variable $\Psi_1 $. In the former case, the marginal probability density $\rho _{1}$ giving the results of measurements of the coordinate $q_{1}$. In the second approach, the density $\rho _{1}$ can be obtained by using the Chapman-Kolmogorov formula, the random variable $F$, and the probability $\mathbb{P}_{2}$ as $$\begin{aligned}
\rho _{1}(q_{1}) &= &\int_{\mathscr{Q}_{2}}\rho _{1}(q_{1}|q_{2})\mathbb{P}_{2}(dq_{2}) \\
&=&\int_{\mathscr{Q}_{2}}\frac{\left| \varphi \left( q_{1},q_{2}\right)
\right| ^{2}}{\int_{\mathscr{Q}_{1}}\left| \varphi \left( q_{1}^{\prime
},q_{2}\right) \right| ^{2}\nu_1(dq_{1}^{\prime })}\int_{\mathscr{Q}_{1}}\left| \varphi \left( q_{1}^{\prime \prime },q_{2}\right) \right|
^{2}\nu_1(dq_{1}^{\prime \prime })\nu_2(dq_{2}) \\
&\equiv &\int_{\mathscr{Q}_{2}}\left| \varphi \left( q_{1},q_{2}\right)
\right| ^{2}\nu_2(dq_{2}),\end{aligned}$$ where the conditional probability density $\rho _{1}(q_{1}|q_{2})$ is defined by $\rho _{1}(q_{1}|q_{2})=\left| \varphi \left( q_{1},q_{2}\right)
\right| ^{2}/\int_{\mathscr{Q}_{1}}\left| \varphi \left( q_{1}^{\prime
},q_{2}\right) \right| ^{2}\nu_1(dq_{1}^{\prime })$.
**Remark 2**. Of course, the Hilbert-valued random variable representing a mixed state of the open system is not uniquely determined; e.g., instead of the *coordinate representation* $L^{2}(\mathscr{Q}_{1},\nu_1)$ of the Hilbert space $\mathfrak{H}_{1}$, we can use a *momentum representation* of this Hilbert space.
**Remark 3**. It also follows from the above considerations that the evolution of the open system can be described by a random process taking value in the same Hilbert space. However, this process is not uniquely determined either; thus, the corresponding master equation (which is an equation with a time dependent random coefficient) is not uniquely determined.
RANDOM PROCESSES DESCRIBING\
THE EVOLUTION OF OPEN SYSTEMS
=============================
We recall that the Feynman formulas are representations of Schrödinger groups or semigroups as limits of integrals over finite Cartesian products of some space $\mathscr{X}$, (see, e.g., \[1, 2\]). If $\mathscr{X}$ coincides with the domain $\Omega $ of functions from the space on which these groups or semigroups act and $\Omega \subset \mathscr{Q}=\mathscr{Q}_{1}\times \mathscr{Q}_{2}$ then the corresponding Feynman formula is said to be *Lagrangian*; if $\mathscr{X}=\Omega \times \mathscr{P}$, where $\mathscr{P}=\mathscr{P}_{1}\times \mathscr{P}_{2}$ is the momentum space of the classical version of the quantum system under consideration, then the Feynman formula is said to be Hamiltonian (not all Feynman formulas belong to one of these two classes)[^4].
We identify (see \[4, 10\]) $\mathscr{P}_{j}$ with $\mathscr{Q}_{j}^{\ast }$and $\mathscr{Q}_{j}$ with $\mathscr{P}_{j}^{\ast }$ $(j=1,2)$. These identifications generate isomorphisms (cf. \[4\]) $$J:\mathscr{Q}_{j}\times \mathscr{P}_{j}\ni \left( q,p\right) \mapsto \left(
p,q\right) \in \left( \mathscr{Q}_{j}\times \mathscr{P}_{j}\right) ^{\ast }$$ and a similar isomorphism between the spaces $$\mathscr{Q}\times \mathscr{P}\triangleq \left( \mathscr{Q}_{1}\times \mathscr{Q}_{2}\right) \times \left( \mathscr{P}_{1}\times \mathscr{P}_{2}\right)$$ and $\left( \mathscr{Q}\times \mathscr{P}\right) ^{\ast }$.
Let $\psi _{1}\in \mathfrak{H}_{1}\left( =L^{2}(\mathscr{Q}_{1},\nu_1\right) $ be the initial state of the open system, and let $\psi _{2}$ be the initial state of the environment, which is called the *reference state*. We have $\left( \psi _{1}\otimes \psi _{2}\right) \left( q_{1},q_{2}\right)
=\psi _{1}(q_{1})\psi _{2}(q_{2})$. Suppose that a classical Hamiltonian function $H:\mathscr{Q}\times \mathscr{P}\mapsto \mathbb{R}$ is defined by $$H\left( q_{1},p_{1},q_{2},p_{2}\right) \triangleq H_{1}\left(
q_{1},p_{1}\right) +H_{2}\left( q_{2},p_{2}\right) +H_{12}\left(
q_{1},p_{1},q_{2},p_{2}\right)$$
The Hamiltonian observable $\hat{H}$ governing the evolution of the composite system may be written as $$\hat{H}=\hat{H}_{1}\otimes I_{2}+I_{1}\otimes \hat{H}_{2}+\hat{H}_{12},$$ where $\hat{H}_{j}$ is the pseudo-differential operator on $\mathfrak{H}_{j}$ with Weyl symbol $H_{j}$ for $j=1,2$ and $\hat{H}_{12}$ is the pseudo-differential operator on with Weyl symbol $H_{12}$ (for the definition of pseudo-differential operators on spaces of functions square integrable with respect to a measure different from the Lebesgue measure, see \[2, 10\]). It is useful to assume that $\hat{H}_{1}$ governs the internal dynamics of the open system, $\hat{H}_{2}$ governs the internal dynamics of the environment, and $\hat{H}_{12}$ describes the interaction.
**Theorem 1**. *Suppose that, for each* $t=0$*,* $\varphi
(t)\in H_{1}\otimes H_{2}$* denotes the state of the composite system at the moment* $t$*. Then, for all* $(q_{1},q_{2})\in \mathscr{Q}_{1}\times \mathscr{Q}_{2}$*,* $$\begin{aligned}
\varphi (t)\left( q_{1},q_{2}\right) &=&\left( e^{it\hat{H}}\,\psi
_{1}\otimes \psi _{2}\right) \left( q_{1},q_{2}\right) =\lim_{n\rightarrow
\infty }\left( \widehat{e^{i\frac{t}{n}H}}\right) ^{n}\,\psi _{1}\otimes
\psi _{2}\left( q_{1},q_{2}\right) \\
&=&\lim_{n\rightarrow \infty }\left( \widehat{e^{i\frac{t}{n}H_{1}\otimes
I_{2}}}\circ \widehat{e^{i\frac{t}{n}I_{1}\otimes H}}\circ \widehat{e^{i\frac{t}{n}H_{12}}}\right) ^{n}\,\psi _{1}\otimes \psi _{2}\left(
q_{1},q_{2}\right) .\end{aligned}$$
The proof is based on Chernoff’s theorem \[11\].
**Remark 4**. The substitution of the explicit expressions for the pseudo-differential operators on the right-hand side of the last relation turns this relation into a Feynman type formula.
We now define two random processes describing the dynamics of the open quantum system. Suppose that, for each $t\geq 0$, $\mathbb{P}_{t}$ is a probability measure on a copy $\mathscr{Q}_{2}^{t}$ of the space $\mathscr{Q}_{2}$ whose density $\rho _{t}\left( \cdot \right) $ with respect to $\nu_2$ is defined as $$\rho _{t}(q_{2})\triangleq \int_{\mathscr{Q}_{1}}\left| \lim_{n\rightarrow
\infty }\left( \widehat{e^{i\frac{t}{n}H}}\right) ^{n}\,\psi _{1}\otimes
\psi _{2}\left( q_{1},q_{2}\right) \right|^2 \nu_1(dq_{1}),$$ $\mathbb{P}$ is the probability measure on the product space $\mathscr{X}$ of the family of spaces $\left\{ \mathscr{Q}_{2}^{t}:t\geq 0\right\} $ defined as the product of the measures $\mathbb{P}_{t}$, and $\psi ^{\mathbb{P}}:[0,\infty )\times \left( \mathscr{X},\mathbb{P}\right) \mapsto L^{2}(\mathscr{Q}_{1})$ is the $L^{2}(\mathscr{Q}_{1})$-valued random process defined by $$\psi ^{\mathbb{P}}\left( t,q\right) =\psi _{t}^{\mathbb{P}}(q)\triangleq
\varphi \left( t\right) \left( \cdot ,q(t)\right)$$ where $q(=q(\cdot ))\in \mathscr{X}$ and $\varphi $ is the pure state function appearing in Theorem 1. Suppose also that, for the same $t$, $\gamma (t)$ is a bijection between $\mathscr{Q}_{2}$ and $\mathscr{Q}_{2}^{t}$, which determines an isomorphism between the measure space $(\mathscr{Q}_{2},\nu_2)$ and the measure space $(\mathscr{Q}_{2}^{t},\mathbb{P}_{t})$, and $\psi
_{v}:[0,\infty )\times \left( \mathscr{Q}_{2},\nu_2\right) \mapsto L^{2}(\mathscr{Q}_{1}, \nu_1 )$ is the random process defined by $$\begin{aligned}
\psi _{\nu
}(t,q)\triangleq \varphi \left( t\right) \left( \cdot ,\gamma (t)\left(
q\right) \right) .\end{aligned}$$
**Theorem 2**. *Under the above assumptions, the state of the open quantum system at a moment of time* $t$* is described by the* $L_{2}(\mathscr{Q}_{1}, \nu_1)$*-valued random variables* $\psi^{\mathbb{P}} (t, \cdot )$ *(on* $(\mathscr{X},\mathbb{P})$ *) and* $\psi_{\nu_2}(t,q)$ *(on (*$\mathscr{Q}_{2},\nu_2)$ *).*
THE WIGNER EVOLUTION FUNCTIONS OF THE OPEN QUANTUM SYSTEM
=========================================================
Given a density operator $T$ on $\mathfrak{H}$, the Weyl function generated by $T $ is the function $W_{T}:\mathscr{Q}\times \mathscr{P}\mapsto \mathbb{R}$ defined by $$W_{T}(H)\triangleq \mathrm{tr}\left\{ Te^{-i\hat{H}}\right\} ,$$ where $\hat{H}$ is the pseudo-differential operator on $\mathfrak{H}=L^{2}(\mathscr{Q},\nu_1\otimes \nu_2)$ with symbol $JH\in \mathscr{Q}^{\ast
}\times \mathscr{P}^{\ast }$ \[5\]. The *Wigner measure* on $\mathscr{Q}\times \mathscr{P}$ generated by the density operator $T$ is defined by $$\int_{\mathscr{Q}\times \mathscr{P}}e^{i\left( \mathfrak{p}_{1}\mathfrak{q}_{2}-\mathfrak{q}_{1}\mathfrak{p}_{2}\right) }W_{T}^{M}\left( d\mathfrak{q}_{1},d\mathfrak{p}_{1}\right) =W_{T}\left( \mathfrak{q}_{1},\mathfrak{p}_{2}\right) ,$$ with $\left( \mathfrak{q},\mathfrak{p}\right) \in \mathscr{Q}\times \mathscr{P}$, cf. \[5\].
The Wigner measure $W_{T_{1}}^{M}$ on $\mathscr{Q}_{1}\times \mathscr{P}_{1}$ generated by a density operator $T_{1}$ on $\mathfrak{H}_{1}$ is defined in a similar way. The density of the measure $W_{T_{1}}^{M}$ with respect to $\nu_1$ coincides with the classical Wigner function (see \[5\]).
**Theorem 3**. *If* $T$* is a density operator on* $H$* and* $T_{1}$* is the corresponding reduced density operator on* $H_{1}$*, then* $$W_{T_{1}}^{M}\left( \cdot \right) =\int_{\mathscr{Q}_{2}\times \mathscr{P}_{2}}W_{T}^{M}\left( \cdot ,dq_{2},dp_{2}\right) .$$
Using this theorem and the Feynman formula for the solution of the Moyal type equation which describes the evolution of the Wigner measure on $\mathscr{Q}\times \mathscr{P}$, we can obtain a formula describing the evolution of the Wigner measure (and, thereby, the Wigner function) on $\mathscr{Q}_{1}\times \ \mathscr{P}_{1}$.
HAMILTONIAN STRUCTURES
======================
This section considers the third and the fourth approach for describing the dynamics of open quantum systems, which are closely related to each other. We assume that $\psi _{1}$, $\psi _{2}$, and $\mathfrak{H}$ are the same as above and $T(\cdot )$ is a function describing the dynamics of the open system, whose values are density operators on $\mathfrak{H}_{1}$.
**Theorem 4**. *If, for each* $t>0$*,* $k_{T}(t)$* is the integral kernel of a trace-class operator* $T(t)$* on* $H_{1} $*, then* $$\begin{aligned}
k_{T}\left( t,q_{1},q_{2}\right) &=&\int_{\mathscr{Q}_{2}}\left[
\lim_{n\rightarrow \infty }\left( \widehat{e^{i\frac{t}{n}H}}\right)
^{n}\,\psi _{1}\otimes \psi _{2}\left( q_{1},q\right) \right] \\
&& \times \left[
\lim_{n\rightarrow \infty }\left( \widehat{e^{i\frac{t}{n}H}}\right)
^{n}\,\psi _{1}\otimes \psi _{2}\left( q_{2},q\right) \right] \nu_2(dq).\end{aligned}$$
**Theorem 5**. *Let* $\nu_2(\cdot )$* be a function of a real variable such that, for each* $t$*,* $\nu_{2,t}$* is a Gaussian measure on* $\mathfrak{H}_{1}$* with correlation operator* $T(t)$*. Then the function* $\nu_{2,t}$* satisfies the master (Liouville) equation.*
*ACKNOWLEDGMENTS*
This work was supported by the Government of Russian Federation, state contract no. 11.G34.31.0054. T.S. Ratiu acknowledges the support of the Switzerland National Scientific Foundation, Swiss NSF grant no. 200021140238. O.G. Smolyanov acknowledges the support of his visit to Aberystwyth by London Mathematical Society.
[99]{} O. G. Smolyanov, in Quantum Probability and White Noise Analysis (World Sci., Singapore, 2013), Vol. 30, pp. 301-314.
T. S. Ratiu and O. G. Smolyanov, Dokl. Math. 87, 289-292 (2013).
V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis, 2nd ed. (Izhevsk, 2011) \[in Russian\].
V. V. Kozlov and O. G. Smolyanov, Teor. Veroyatn. Ee Primen. 51 (1), 114 (2006).
V. V. Kozlov and O. G. Smolyanov, Dokl. Math. 84, 571-575 (2011).
J. Kupsch and O. G. Smolyanov, Russ. J. Math. Phys. 12 (6), 205-214 (2005).
P. R. Chernoff and J. E. Marsden, Properties of Infinite Dimensional Hamiltonian Systems (SpringerVerlag, Berlin, 1974).
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. (Bejamin, Reading, Mass., 1978).
J. E. Marsden and T. S. Ratiu, Intoruction to Mechanics and Symmetry, 2nd ed. (Springer, New York, 1994).
V. V. Kozlov and O. G. Smolyanov, Dokl. Math. 85, 416-420 (2012).
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations (Springer, New York, 2000).
J. Gough, V.P. Belavkin, O.G. Smolyanov, J. Opt. B: Quantum Semiclass. Opt. 7, S237-S244 (2005).
[^1]: Institute of Mathematics, Physics, and Computer Sciences, Aberystwyth University, Aberystwyth, United Kingdom, email: `jug@aber.ac.uk`
[^2]: Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, Lausanne, CH 1015 Switzerland, email: `tudor.ratiu@epfl.ch`
[^3]: Mechanics and Mathematics Faculty, Moscow State University, Moscow, 119991 Russia, email: `smolyanov@yandex.ru`
[^4]: The Feynman-Kac formulas are representations of the same groups and semigroups as integrals over the space consisting of functions of a real variable taking values in the same space $\mathscr{X}$. The multiple integrals in the Feynman formulas approximate the (infinite dimensional) integrals in the Feynman-Kac formulas. In the case of the Schrödinger semigroups generated by Hamiltonians quadratic in momenta, the infinite dimensional integrals in the Feynman-Kac formulas turn out to be integrals with respect to probability measures; however, in the case of Schrödinger groups, there appear integrals with respect to the so-called Feynman pseudo-measures or their analogues in the Feynman-Kac formulas (in many realistic situations, integrals with respect to pseudo-measures are defined as the limits of appropriate sequences of finitely many integrals).
|
---
abstract: 'The fragment production in multifragmentation of finite nuclei is affected by the critical temperature of nuclear matter. We show that this temperature can be determined on the basis of the statistical multifragmentation model (SMM) by analyzing the evolution of fragment distributions with the excitation energy. This method can reveal a decrease of the critical temperature that, e.g., is expected for neutron-rich matter. The influence of isospin on fragment distributions is also discussed.'
---
-1cm 16.0cm
[**The critical temperature of nuclear matter and fragment distributions in multifragmentation of finite nuclei.**]{}
[R. Ogul$^{1,2}$ and A.S. Botvina$^{1,3}$]{}
*$^{1}$Gesellschaft fur Schwerionenforschung, D–64291 Darmstadt, Germany.*
$^{2}$Department of Physics, University of Selcuk, 42079 Kampus, Konya, Turkey.
$^{3}$Institute for Nuclear Research, Russian Academy of Science, 117312 Moscow, Russia.
PACS numbers: 25.70.Pq, 21.65.+f
Properties of nuclear matter have been under investigation for several decades (see e.g. [@baym]). Besides their general interest for nuclear physics, these studies are very important for our understanding astrophysical objects, such as neutron stars. The information about nuclear matter in its ground state and at low temperatures is usually obtained as a theoretical extrapolation, based on nuclear models designed to describe the structure of real nuclei.
It is instructive to investigate the thermodynamical properties of neutron–rich matter under extreme conditions of low densities and high temperatures. This situation is expected, for example, at supernova II explosions and during the formation of neutron stars. We believe that the liquid-gas type phase transition is manifested, in this case, in the forms of instabilities leading to fragment production. The models used for extracting nuclear matter properties in the phase transition region should be capable of describing the disintegration of homogeneous matter into fragments. It is also important that they should allow to be tested in nuclear reactions leading to the total disintegration of real nuclei at high excitation energy. The multifragmentation reactions, which started to be investigated experimentally nearly 20 years ago (see reviews [@hufner85; @bondorf95]), suit perfectly for this purpose.
Naturally, a thermodynamical model involved in this kind of analysis has to include the ingredients necessary for the description of nuclear matter and to provide a good reproduction of experimental data. So far, the statistical multifragmentation model (SMM) [@bondorf95] satisfies these requirements. The SMM has been designed to describe fragmentation and multifragmentation of excited finite nuclei [@bondorf85; @botvina85]. It includes a liquid-drop approximation for individual fragments which corresponds to the liquid-drop description of nuclear matter [@baym; @ravenhall83]. This is an essential difference of the SMM from other multifragmentation models, e.g. [@gross; @randrup], which do not take the nuclear matter properties explicitly into account. Examples of very successful applications of the SMM for the description of different experimental data can be found in \[3,9–17\]. Furthermore, the descriptions of data with statistical models confirm that multifragmentation of nuclei, despite of being a very fast process, proceeds under a high degree of thermalization.
Details of the SMM can be found in [@bondorf95], here we concentrate on parts of the model which are important for the following discussion. The model describes the fragment formation at a low-density freeze-out ($\rho {\,\raisebox{-.5ex}{$\stackrel{<}{\scriptstyle\sim}$}\,}1/3 \rho_0$, $\rho_0 \approx 0.15$fm$^{-3}$ is the normal nuclear density), where the nuclear liquid and gas phases coexist. The SMM phase diagram has already been under intensive investigations (see e.g. Ref. [@bugaev]). The liquid-drop approximation suggests that the fragmentation process is accompanied by an increase of the surface of nuclear drops. The surface entropy contributes essentially to the statistical partition sum. We should point out that the surface free energy depends on the ratio of the temperature $T$ to the critical temperature of nuclear matter $T_c$. In the SMM the surface tension $\sigma (T)$ is given by $$\sigma (T) = \sigma (0) \Biggl(\frac{T_c^2-T^2}{T_c^2+T^2}\Biggr)^{5/4}.$$ This formula is obtained as a parameterization of the calculations of thermodynamical properties of the interface between two phases (liquid and gas) of nearly symmetric nuclear matter, which were performed with the Thomas–Fermi and Hartree–Fock methods by using the Skyrme forces [@ravenhall83]. In addition the scaling properties in the vicinity of the critical point (see [@landau]) were taken into account. At the critical temperature $T_c$ for the liquid-gas phase transition, the isotherm in the phase diagram has an inflection point. The surface tension vanishes at $T_c$ and only the gas phase is possible above this temperature. We emphasize that our analysis is based on this general effect and that our conclusions will remain qualitatively true in the case of other parametrizations satisfying this condition. This surface effect provides an effective way to study the influence of $T_c$ on fragment production in the multi-fragment decay of hot finite nuclei.
As was established by numerous studies (see e.g. \[3–5,9,15,16\]) the mass (charge) distribution of fragments produced in the disintegration of nuclei evolves with the excitation energy. At low temperatures ($T{\,\raisebox{-.5ex}{$\stackrel{<}{\scriptstyle\sim}$}\,}5$ MeV), there is a so-called $U$-shape distribution corresponding to partitions with few small fragments and one big residual fragment. This distribution looks like a result of an evaporative emission. At high temperatures ($T{\,\raisebox{-.5ex}{$\stackrel{>}{\scriptstyle\sim}$}\,}6$ MeV) the big fragments disappear, and there is an exponential-like fall of the mass distribution with mass number $A$. In the transition region $T\approx 5-6$ MeV, however, there is a smooth transformation of the first distribution into the second one. The mass distribution of intermediate mass fragments (IMFs, fragments with $A=5-40$) can be approximated by a power law $A^{-\tau}$ [@hufner85; @bondorf95; @goodman]. The $\tau$ parameter decreases with the temperature, goes through the minimum at $T\approx 5-6$ MeV, and then increases again. The small values of $\tau$ indicate that the probability for survival of the biggest fragment decreases drastically with the temperature. This behavior may be associated with a phase transition in finite systems. It has been shown in many studies (see e.g. [@bondorf95; @dagostino99; @srivastava] and references therein), that there are numerous peculiarities in this region, such as a plateau-like behavior of the caloric curve, large fluctuations of the temperature and of the number of the produced fragments, scaling laws for fragment yields, and other phenomena expected for critical behavior. Therefore, the temperature characterizing these phenomena is sometimes called the critical temperature for finite systems, and $\tau$ is considered as one of the critical exponents. The SMM can describe the critical behavior observed in the experiments \[15–17\]. However, in the present work we use the $\tau$ parametrization only for the characterization of shapes of the fragment mass distributions. In order to avoid any confusion with the standard definition of the critical temperature for nuclear matter, we note, in the following, the temperature corresponding to the critical phenomena as a break-up temperature for the disintegration of finite nuclei [@botvina85].
The decrease of the surface energy with increasing $T/T_c$ (see formula (1)) influences the fragment production and, therefore, can be observed in the fragment distributions. In this paper we show that this effect can be used for the evaluation of $T_c$ by finding the minimum $\tau$ parameter $\tau_{min}$ and the corresponding temperature $T_{min}$. The physics behind the phenomenon is quite transparent: If the contribution of the surface energy is rapidly decreasing, a nucleus prefers to disintegrate into small fragments already at low temperatures. Simultaneously, fluctuations of size of the fragments increase considerably. As a result the mass distribution becomes flatter in the transition region, and this leads to a decrease of $\tau$.
The SMM calculations were carried out for the $Au$ nucleus ($A_0$=197, $Z_0$=79) at different excitation energies and at a freeze-out density of one-third of the normal nuclear density. This choice is justified by the previous descriptions of the experimental data obtained for peripheral collisions \[9,12–16\]. Below we present results as a function of both the temperature and the excitation energy, since they are related quantities [@bondorf95].
We have started by using the standard value of the critical temperature implemented in the SMM, $T_c$=18 MeV. This value is consistent with many theoretical studies [@ravenhall83; @ogul]. In Fig. 1 we show typical mass and IMF charge distributions, $\langle N_A \rangle \sim A^{-\tau}$ and $\langle N_Z \rangle \sim Z^{-\tau_z}$, at an excitation energy $E_{x}$=7 MeV/nucleon. One can see from this figure that the extracted $\tau$ and $\tau_z$ values are very close to each other, since the neutron-to-proton ratio of produced IMFs changes very little within their narrow charge range [@botvina95; @botvina01]. As seen from Fig. 2, the dependences of these parameters versus excitation energy are nearly the same. The parameters obtained for primary hot fragments (excited nuclear matter drops) and after their secondary de-excitation (measured cold fragments) are also shown in this figure. One can see that the difference between the two cases is smallest around the minimum $\tau$ parameter. Therefore, the $\tau_{min}$ point is weakly affected by secondary processes.
The critical temperature reflects the properties of nuclear matter, however, these properties depend on the composition of this matter. For example, $T_c$ tends to decrease for neutron rich matter [@muller]. As it was discussed by many authors, see e.g. [@bonche], the critical temperature can be traced back towards neutron rich matter by studies of the disassembly of nuclei far from stability. We consider $T_c$ as a free parameter in the SMM and analyze how $\tau$ and the break-up temperature can change. In Fig. 3 we show the results for $T_c=$10 and 30 MeV. In these cases the evolution of $\tau$ with the excitation energy is similar to the one shown in Fig. 2. However, the values of $T_{min}$ and $\tau_{min}$ are essentially different. This reflects a considerable change of masses for the dominating fragments. These values are plotted in Fig. 4 versus the critical temperature. It is seen that both parameters increase with $T_c$ and that they tend to saturate at $T_c \rightarrow \infty$ corresponding to the case of the temperature-independent surface. This behavior is expected, since in this case only the translational and bulk entropies of fragments, but not the surface entropy, influence the probability for the fragment formation.
In the case of neutron-rich matter, the contribution of the symmetry energy increases considerably. It is necessary to take into account the standard dependence of the $\tau$ parameter on the isospin of the source, while searching for $T_c$. We performed SMM calculations of multifragmentation of $^{124}Sn$ and $^{124}La$ nuclei, which can be used in experiments [@kezzar], and compare them with the results obtained for $Au$ nuclei. This $Sn$ nucleus is nearly as neutron-rich as the gold nucleus, while the $La$ nucleus is neutron-poor. We have used the same model parameters as for the $Au$ case, with the standard $T_c=$18 MeV. It is seen from Fig. 5 that the neutron-poor source results in slightly lower microcanonical temperatures in the transition region ($E_x \approx 3-5$ MeV/nucleon). In Fig. 5 we show also the evolution of the $\tau$ parameter with the excitation energy. The results for $Au$ and $Sn$ are very similar and different from those obtained for the $La$. One can conclude, that IMF distributions approximately scale with the size of the sources, and that they depend on the neutron-to-proton (N/Z) ratios of the sources. This is because the symmetry energy still dominates over the Coulomb interaction energy for these intermediate-size sources. One can see from Fig. 5 that the source with the lower N/Z ratio leads to smaller $\tau$ parameters, i.e. to the flatter fragment distribution. We can explain this as an effect of the isospin (i.e. the symmetry energy) on fragment formation: A high N/Z ratio of the source favors the production of big clusters, since they have a large isospin. Therefore, in the transition region, partitions consisting of small IMFs and a big cluster dominate. This leads to a very prominent U-shape distribution with large $\tau$. When the N/Z ratio is low, the probability for a big cluster to survive is small and the system can disintegrate into IMFs, which have a favorable isospin in this case. The dominant fragment partitions tend to include IMFs of different sizes, and the fragment distribution is characterized by small $\tau$. Since the difference in the $\tau$ parameters between the neutron-rich and neutron-poor sources is quite large, it can be easily identified. The evolutions of fragment mass distributions caused by decreasing $T_c$ and changing N/Z ratio can interfere and, therefore, the influence of the critical temperature can be separated only after the comparison of experimental data with calculations.
In view of these theoretical findings it is instructive to demonstrate the possibility of the application of such an approach for the analysis of experimental data. Presently, there are several experimental analyses aimed at the extraction of the critical exponent $\tau$ in reactions with $Au$ nuclei [@dagostino99; @srivastava; @elliott]. Those methods are not equivalent to the one suggested above, however, they are related to the fragment distributions, and the critical exponents can be used for an estimation of $\tau_{min}$ and $T_c$. The extracted break-up excitation energies $E_x$ vary within 3.8 to 4.5 MeV/nucleon, while $\tau$ is in the range from 2.12 to 2.18. As seen from Figs. 2 and 3, for $T_c \geq$ 18, at these excitation energies $\tau$ is larger than $\tau_{min}$. We have performed an interpolation of the SMM calculations for the Au sources and found that they fit the experimental values of $E_x$ and $\tau$ if $T_c$ is in the range between 18 and 22 MeV. We have also seen from our analysis that $\tau_{min}$ is lower than the extracted $\tau$ by around 15%. The $T_c$ estimated in this way is very close to the standard SMM parametrization. This conclusion is supported by the analyses of Refs. \[15–17\] showing that the standard SMM reproduces both the experimental critical exponents and other characteristics of produced fragments. It is interesting that in Ref. [@srivastava] a small critical exponent $\tau \approx$ 1.88 is reported for multifragmentation of $Kr$ nuclei, which have N/Z ratios lower than $Au$ nuclei. This is an indication of the importance of the isospin effects, as discussed above. It is worth noting that recently a very close critical temperature ($T_c \approx 16.6$ MeV) was extracted from analysis of the break-up (“limiting”) temperatures in Ref. [@natowitz]. It is also in agreement with the temperature $T_c \approx 20$ MeV obtained in the experiment of Ref. [@karnaukhov]. However, it would be important to identify regular changes of $T_c$, which needs involving new sources with different isospins.
In summary, we have pointed out that within the SMM the critical temperature of nuclear matter can influence the fragment production in multifragmentation of nuclei through the surface energy. We have suggested that this influence can be observed in the $A^{-\tau}$ parameterization of the fragment yields by finding the minimum $\tau$ parameter. In the experiments the measured values of the parameters are consistent with the standard SMM assumption and slightly higher values of the critical temperature, $T_c \approx 18-22$ MeV. The SMM predicts that variations of $\tau_{min}$ are especially large in the region of low $T_c$. Therefore, there is a possibility to investigate the decrease of $T_c$ for nuclear matter under extreme conditions, by studying the evolution of $\tau_{min}$ and $T_{min}$ in the multifragmentation of finite nuclei. This could be realized in the case of the neutron-rich nuclei delivered by current accelerators with radioactive heavy-ion beams [@kezzar]. The isospin of the source influences also the fragment production through the symmetry energy. We have demonstrated how the isospin affects the fragment distributions, that should be taken into account in these studies.
The authors thank GSI for hospitality and support. We appreciate stimulating discussions with V.A.Karnaukhov and I.N.Mishustin. We are very indebted to W. Trautmann and J. Lukasik for discussions and help in preparation of the manuscript. R. Ogul acknowledges financial support of TUBITAK-DFG cooperation.
G. Baym, H. Bethe and C.J. Pethick, Nucl. Phys. [**A175**]{}, 225 (1971). J. Hüfner, Phys. Rep. [**125**]{}, 129 (1985). J.P. Bondorf, A.S. Botvina, A.S. Iljinov, I.N. Mishustin, and K. Sneppen, Phys. Rep. [**257**]{}, 133 (1995). J.P. Bondorf [*et al.*]{}, Nucl. Phys. [**A443**]{}, 321 (1985). A.S. Botvina, A.S. Il’inov and I.N. Mishustin, Sov. J. Nucl. Phys. [**42**]{}, 712 (1985). D.G. Ravenhall, C.J. Pethick and J.M. Lattimer, Nucl. Phys. [**A407**]{}, 571 (1983). D.H.E. Gross, Rep. Progr. Phys. [**53**]{}, 605 (1990). S.E. Koonin and J. Randrup, Nucl. Phys. [**A474**]{}, 173 (1987). A.S. Botvina [*et al.*]{}, Nucl. Phys. [**A584**]{}, 737 (1995). M. D’Agostino [*et al.*]{}, Phys. Lett. [**B371**]{}, 175 (1996). C. Williams [*et al.*]{}, Phys. Rev. [**C55**]{}, R2132 (1997). S.P. Avdeyev, [*et al.*]{}, Eur. Phys. J. [**A3**]{}, 75 (1998). W. Trautmann, Adv. Nucl. Dynam. [**4**]{}, 349 (1998). G. Wang [*et al.*]{}, Phys. Rev. [**C60**]{}, 014603 (1999). M. D’Agostino, [*et al.*]{}, Nucl. Phys. [**A650**]{}, 329 (1999). R.P. Scharenberg [*et al.*]{}, Phys. Rev. [**C64**]{}, 054602 (2001). B.K. Srivastava [*et al.*]{}, Phys. Rev. [**C65**]{}, 054617 (2002) K.A. Bugaev [*et al.*]{}, Phys. Rev. [**C62**]{}, 044320 (2000). L.D. Landau and E.M.Lifshitz, Statistical physics, Pergamon, Oxford, 1980. A.L. Goodman, J.I. Kapusta and A.Z. Mekjian, Phys. Rev. [**C30**]{}, 851 (1984). R. Ogul, Int. J. Mod. Phys. [**E 7(3)**]{}, 419 (1998). A.S. Botvina and I.N. Mishustin, Phys. Rev. [**C63**]{}, 061601(R) (2001). H. Müller and B.D. Serot, Phys. Rev. [**C52**]{}, 2072 (1995). P. Bonche, S. Levit and D. Vautherin, Nucl. Phys. [**A436**]{}, 265 (1985). K. Kezzar [*et al.*]{}, Proposal of experiment S254 at SIS (GSI), http://www-kp3.gsi.de/www/kp3/proposals.html . J.B. Elliott [*et al.*]{}, Phys. Rev. Lett. [**88**]{}, 042701 (2002). J.B. Natowitz [*et al.*]{}, submitted to Phys. Rev. Lett., ArXiv: nucl-ex/0204015. V.A. Karnaukhov [*et al.*]{}, submitted to Phys. Rev. [**C**]{}.
[**[Figure captions]{}**]{}\
[**Fig.1:**]{} [Average fragment mass and charge yields $\langle N_A \rangle$ and $\langle N_Z \rangle$, after multifragmentation of $Au$ nuclei at an excitation energy of 7 MeV/nucleon. Solids lines are $\sim A^{-\tau}$ and $\sim Z^{-\tau_z}$ fits of the IMF yields. ]{}
[**Fig.2:**]{} [Evolution of $\tau_z$ (top panel) and $\tau$ (bottom panel) parameters with the excitation energy $E_x$ of $Au$ sources calculated with the standard SMM parameterization. The open circles are for hot primary fragments and the full squares are for observed cold fragments. ]{}
[**Fig.3:**]{} [Evolution of $\tau$ parameters in the SMM for cold fragments with the excitation energy $E_x$. The top panel is for the critical temperature $T_c$=10 MeV, the bottom panel for $T_c$=30 MeV. ]{}
[**Fig.4:**]{} [The minimum $\tau_{min}$ parameter for cold fragments (top panel) and the corresponding temperature $T_{min}$ (bottom panel) as function of the critical temperature $T_c$ of nuclear matter. ]{}
[**Fig.5:**]{} [ The temperature (top panel) and $\tau$ parameters for cold fragments (bottom panel) versus excitation energy in multifragmentation. The solid, dashed and dotted lines are SMM calculations performed for sources with different sizes or isospin (see the figure) with the standard $T_c$=18 MeV. ]{}
|
---
abstract: 'We develop by example a type of index theory for non-Fredholm operators. A general framework using cyclic homology for this notion of index was introduced in a separate article [@CaKa:TIH] where it may be seen to generalise earlier ideas of Carey-Pincus and Gesztesy-Simon on this problem. Motivated by an example in two dimensions in [@BGGSS:WKS] we introduce in this paper a class of examples of Dirac type operators on $\rr^{2n}$ that provide non-trivial examples of our homological approach. Our examples may be seen as extending old ideas about the notion of anomaly introduced by physicists to handle topological terms in quantum action principles with an important difference, namely we are dealing with purely geometric data that can be seen to arise from the continuous spectrum of our Dirac type operators.'
address:
- 'Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia'
- 'Department of Physics, University of Vienna, Boltzmanngasse, Vienna'
- 'International School of Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy'
author:
- Alan Carey
- Harald Grosse
- Jens Kaad
title: Anomalies of Dirac type operators on Euclidean space
---
[^1]
Introduction
============
Background
----------
In two interesting papers from the latter part of the previous century R. W. Carey and J. Pincus in [@CaPi:IOG], and F. Gesztesy and B. Simon in [@GeSi:TIW] made a start on an ‘index theory’ for non-Fredholm operators. These two papers study different aspects of the problem using related techniques. In both papers this ‘index’ is expressed in terms of the Krein spectral shift function from scattering theory.[^2] A comprehensive list of papers on the Witten index may be found in [@CGPST:WSS].
In a companion paper [@CaKa:TIH] we developed a formalism based on cyclic homology in which this previous work can be seen to fit as a special case. We termed the spectral invariant constructed in [@CaKa:TIH] the ‘homological index’. The Gesztesy-Simon index is related to a scaling limit of the homological index which is a functional defined on the homology of a certain bicomplex constructed using techniques from cyclic theory [@Lod:CH]. However it is not at all clear whether there are interesting examples of this formalism except in the very simplest case studied in [@BGGSS:WKS].
In noncommutative geometry, from the spectral point of view, we start with an unbounded selfadjoint operator $\C D$ densely defined on a Hilbert space $\C H$. Then we introduce a subalgebra of the algebra of bounded operators $\sL(\C H)$ on $\C H$ and probe the structure of this algebra using Kasparov theory (computing in particular the topological index as a pairing in $K$-theory, [@CoMo:LIF]).
The homological formalism developed in our companion paper suggests a variant on the conventional approach via spectral triples. This variant is also suggested by the concrete picture developed in the early papers of Bollé et al, [@BGGSS:WKS], and of Gesztesy and Simon, [@GeSi:TIW]. Loosely speaking, whereas the topological interpretation of the Fredholm index is given by K-theory we view cyclic homology as the appropriate tool to replace K-theory in the case of non-Fredholm operators. The main results presented here are about finding non-trivial examples of our homological index. These examples arise from the study of Dirac type operators on the manifold $\mathbb R^{2n}$.
Our motivation stems partly from the desire to understand higher dimensional examples of the operators considered in [@BGGSS:WKS], and partly from an ambition to probe the meaning of the homological index from the spectral point of view for Dirac type operators on general non-compact manifolds, There is also motivation from magnetic Hamiltonians in dimensions greater than two and we will take this up elsewhere.
To explain our results we need some notation which we now present.
Unbounded operators {#s:unb}
-------------------
In this paper we will work with the following framework. First we double our Hilbert space setting $\C H^{(2)} := \C H \op \C H$, introducing a grading operator $\ga = \ma{cc}{1 & 0 \\ 0 & -1}$. We let $\dir_+$ be a closed densely defined operator on $\C H$ and form the odd selfadjoint operator $\dir := \ma{cc}{0 & \dir_- \\ \dir_+ & 0}$, where $\dir_- = (\dir_+)^*$. We will study a class of perturbations of $\dir$ of the form $$\C D := \ma{cc}{
0 & \dir_- + A^* \\
\dir_+ + A & 0
},$$ where $A$ is a bounded operator on $\C H$. We want to study an invariant of $\C D$ by mapping to bounded operators using the Riesz map $$\C D \mapsto T = (\dir_+ + A)\big(1 + (\dir_-+A^*)(\dir_+ + A) \big)^{-1/2}.$$ These bounded operators generate an algebra to which our homological theory [@CaKa:TIH] applies. To see how natural constraints arise on $T$ and its adjoint $T^*$ note that we have the identities $$\C D^2 = \ma{cc}{
(\dir_- + A^*)(\dir_+ + A) & 0 \\
0 & (\dir_+ + A)(\dir_- + A^*)
}$$ and $$\label{eq:boures}
\begin{split}
1 - T T^* = ( 1 + \C D_+ \C D_- )^{-1} \q \T{and} \q
1 - T^* T = ( 1 + \C D_- \C D_+ )^{-1},
\end{split}$$ where $\C D_+ := \dir_+ + A$ and $\C D_- = (\C D_+)^* = \dir_- + A^*$.
We will be interested in the case where the following conditions hold:
\[a:unb\]
1. The unbounded operator $\dir_+$ is normal, thus $\dir_+ \dir_- = \dir_- \dir_+$.
2. The bounded operators $A$ and $A^*$ have the domain of $\dir_+$ as an invariant subspace.
3. The sum of commutators $[\dir_+,A^*] + [A,\dir_-] : \T{Dom}(\dir_-) \to \C H$ extends to a bounded operator on $\C H$.
Remark that the normality of $\dir_+$ entails that $\T{Dom}(\dir_+) = \T{Dom}(\dir_-)$ by [@Rud:FA Theorem 13.32]. Under the conditions in Assumption \[a:unb\] we also obtain $$\T{Dom}( \C D_+ \C D_- ) = \T{Dom}(\dir_+ \dir_-) = \T{Dom}(\dir_- \dir_+) = \T{Dom}( \C D_- \C D_+ )$$ and furthermore that the identity $$\big[ (\dir_+ + A),(\dir_- + A^*)\big](\xi) = [\dir_+,A^*](\xi) - [\dir_-,A](\xi) + [A,A^*](\xi)$$ holds for each vector $\xi \in \T{Dom}(\dir_+ \dir_-)$. This entails that $[\C D_+, \C D_-] : \T{Dom}(\dir_+\dir_-) \to \C H$ extends to a bounded operator on $\C H$.
Let $F \in \sL(\C H)$ denote the bounded extension of the commutator $[\C D_+, \C D_-]$. We may then impose compactness conditions on the difference $$(1 - T^* T) - (1 - T T^*) = ( 1 + \C D_+ \C D_- )^{-1} F ( 1 + \C D_- \C D_+ )^{-1}.$$ In fact we will show in Section \[s:anodir\] that there is a class of Dirac-type operators on $\rr^{2n}$, $n \in \nn$ which satisfy our hypotheses and lead to the condition $$\label{summ}
(1 - T^* T)^n - (1 - T T^*)^n \in \sL^1(\C H),$$ where $\sL^1(\C H) \su \sL(\C H)$ denotes the ideal of trace class operators. The connection between the dimension of the underlying space $\rr^{2n}$ and the condition (\[summ\]) is not evident in the earlier work [@Cal:AIO; @BGGSS:WKS] but is natural from the point of view of spectral geometry.
The next definition is fundamental for the present text:
Suppose that there exists an $n \in \nn$ such that $(1 - T^* T)^n - (1 - T T^*)^n$ is of trace class. By the *homological index* of $T$ in degree $n$ we will understand the trace $\T{Tr}\big( (1 - T^* T)^n - (1 - T T^*)^n \big) \in \rr$. The homological index in degree $n$ is denoted by $\T{H-Ind}_n(T)$. When $T=\C D_+(1+\C D_-\C D_+)^{-1/2}$ we say that the homological index associated to $\C D_+$ is $\T{H-Ind}_n(T)$.
Stability
---------
One of the central problems to study concerns the stability or invariance properties of the homological index. More precisely, given a bounded operator $B : \C H \to \C H$, when can we say that the homological index (in degree $n$) associated to $\dir_+ + A + B$ exists and agrees with the homological index (in degree $n$) of $\dir_+ + A$? Since we are dealing with a genuinely non-compact situation, it would be naive to expect that such an invariance result holds for any bounded operator $B$, and this is one of the main differences between the homological index and the Fredholm index. Indeed it is known from the examples in [@BGGSS:WKS], in connection with the Witten index, that in degree $n=1$ the homological index cannot be stable under general bounded perturbations $B$. In fact a stability result for the Witten index, that gives the flavour of the complexity of the issue, is proved in [@GeSi:TIW].
In the examples we are considering in this paper we must impose decay conditions at infinity on the bounded operator $B$, and this is naturally done using Schatten ideals. In our companion paper we gave a careful treatment of the invariance problem and, for the convenience of the reader, we state a simplified version of the main result here. We would like to emphasize though that the invariance result proved in the companion paper is more general (and for this reason we use distinct notation there) and allows us to deal with unbounded perturbations as well.
Suppose that $$(1 + \C D_+ \C D_-)^{-j} B (1 + \C D_- \C D_+)^{-k-1/2} \, \, , \, \, (1 + \C D_- \C D_+)^{-j} B^* (1 + \C D_+ \C D_-)^{-k-1/2} \in \sL^{n/(j + k)}(\C H)$$ for all $j,k \in \{0,\ldots,n\}$ with $1 \leq j + k \leq n$. Suppose furthermore that there exists an $\ep \in (0,1/2)$ such that $$B \cd (1 + \C D_- \C D_+)^{-n-1/2 + \ep} \, \, , \, \, B^* \cd (1 + \C D_+ \C D_-)^{-n-1/2 + \ep} \in \sL^1(\C H)$$
Then the homological index of $T_B := (\C D_+ + B)(1 + (\C D_- + B^*)(\C D_+ + B))^{-1/2}$ exists in degree $n \in \nn$ if and only if the homological index of $T := \C D_+(1 + \C D_- \C D_+)^{-1/2}$ exists in degree $n \in \nn$. And in this case we have that $$\T{H-Ind}_n(T_B) = \T{H-Ind}_n(T)$$
For a proof of this invariance result we refer to [@CaKa:TIH Theorem 8.1]. We remark also that there is, in [@CaKa:TIH Section 5], a discussion of the homotopy invariance of the homological index.
The reader may be puzzled by the complexity of the statement of this preceding theorem. This can be understood in part in terms of the topological meaning of the homological index. In the notation of the previous theorem we know from the examples in [@BGGSS:WKS] that the homological index cannot be invariant under compact perturbations of $T$. This is the first indication that, unlike the Fredholm index, the homological index is not directly associated to topological K-theory. In our companion paper we show that the homological index is defined as a functional on certain homology groups of the algebra generated by $T$ and $T^*$. It is thus stable under perturbations that do not change the homology class and the hypotheses of the previous theorem stem from this fact.
On general manifolds we expect to see that the homological index depends not only on the topology, but in fact is a spectral invariant that is much finer, depending also on the geometry of the underlying space. This geometric dependence may be seen from our main result of this paper (which we explain in the next subsection) although we remark that the precise nature of the information that can be obtained from the homological index remains to be determined.
In our companion paper, where the homological index was introduced, we did not address the question of non-triviality. The main results of this paper can be seen to be a proof that the homological index is not trivial. Most importantly, the homological viewpoint allows the systematic development of a higher dimensional theory into which the early work [@BGGSS:WKS; @CaPi:IOG; @GeSi:TIW] fits as the lowest degree case.
We now describe our main method.
Scaling limits of the homological index
---------------------------------------
Let $\dir_+ : \T{Dom}(\dir_+) \to \C H$ be a closed unbounded operator and let $A \in \sL(\C H)$ be a bounded operator. We will impose the conditions of Assumption \[a:unb\] throughout this section.
Let $\la \in (0,\infty)$ and consider the scaled operator $\la^{-1/2} \C D_+$. Let $$T_\la := \la^{-1/2}\C D_+(1 + \la^{-1} \C D_- \C D_+)^{-1/2}.$$ As in we obtain $$1 - T_\la T_\la^* = \la \cd (\la + \C D_+ \C D_-)^{-1} \q \T{and} \q
1 - T_\la^* T_\la = \la \cd (\la + \C D_- \C D_+)^{-1}.$$ We may then consider the difference of powers: $$(1- T_\la^* T_\la)^n - (1-T_\la T_\la^*)^n = \la^n \cd \big( (\la + \C D_- \C D_+)^{-n} - (\la + \C D_+ \C D_-)^{-n} \big)$$
*We are, in this paper, interested in the scaling limits of the homological index as the parameter $\la$ goes to either zero or infinity.*
The scaling limit at zero is called the ‘Witten index’ in the case $n=1$ in [@BGGSS:WKS] and [@GeSi:TIW]. What is known about this limit can be found in these papers, the Carey-Pincus article and the recent preprint [@CGPST:WSS]. In particular we remark that when the operator $\C D$ is Fredholm and the scaling limit at zero exists then these articles demonstrate that it coincides with the Fredholm index. The arguments in those papers can be adapted to establish an analogous result about the scaling limit at zero for $n>1$ in the Fredholm case. However in the non-Fredholm case little is known for $n>1$ and this is the subject of a future investigation. Here our main focus will lie in the scaling limit at infinity. We will use the following terminology motivated by the examples in [@BGGSS:WKS; @Cal:AIO].
Suppose that there exists an $n \in \nn$ such that $(1- T_\la^* T_\la)^n - (1-T_\la T_\la^*)^n \in \sL^1(\C H)$ for all $\la \in (0,\infty)$. The *anomaly* in degree $n$ of the perturbed operator $\C D$ is then defined as the scaling limit $$\T{Ano}_n(\C D) := \lim_{\la \to \infty} \T{H-Ind}_n(T_\la)$$ whenever this limit exists.
If we were in a situation where the McKean-Singer formula for the Fredholm index of $\C D_+$ was defined then the usual approach to obtaining a local formula for the index is to calculate the small time asymptotics of the trace of the heat kernel. This local formula is often referred to as the ‘anomaly’ in the physics literature. It corresponds in our situation to calculating the scaling limit at infinity.
The use of the terminology ‘anomaly’ can also be traced back to Callias [@Cal:AIO] where a resolvent type formula, (that partly motivated the approach of [@GeSi:TIW]) is introduced for the index of certain Fredholm operators (of Dirac type) on $\mathbb R^{2n-1}$. His index formula spurred many developments in the late 1970s and 80s. It differs from ours in two ways, first it applies to the odd dimensional case and second, the constraint (\[summ\]) is absent: his formula depends only on the difference of resolvents. On the other hand the resolvent expansion methods of Callias are analogous to our approach for calculating the anomaly.
We remark that the result of Seeley which Callias states as Theorem 1 in [@Cal:AIO] may be used to demonstrate that the examples of Dirac type operators introduced in this paper are not Fredholm for general choices of connection. Despite the fact that we are dealing with the non-Fredholm situation, we are able to use the scaling limit at infinity (that is, the anomaly) as a means of obtaining information on the non-triviality of the homological index.
*In the present paper we prove two main results. First, we find conditions which ensure the existence of the homological index for a fixed degree $n \in \nn$. This will be carried out in Section \[ss:resana\] and is stated as Theorem \[t:tradifpow\].*
Second, and most importantly, in Theorem \[t:locano\] we give a [*local formula*]{} for the anomaly in terms of an integral of a $2n$-form on ${\mathbb R}^{2n}$ that is constructed as a function of the curvature of the connection coming from our Dirac type operator. The expression for the anomaly may be seen to be non-zero by the calculation presented in Section \[nont\].
The attentive reader will have noticed that our definitions of the anomaly, the Witten index (the scaling limit of the homological index at zero), as well as the homological index all depend on the degree $n \in \nn$. The precise way in which these spectral invariants depend on the degree is yet to be determined. What can be inferred from the examples in this paper is that there exists a “critical value” of the degree such that the invariants are not well-defined in degrees strictly less than the critical value. For the Dirac-type operators, we consider in this paper, this critical value is exactly half the dimension of the underlying smooth manifold. Our computation of the anomaly for the examples in question shows moreover that the anomaly is independent of the degree above this critical value. We do not know whether this kind of behaviour holds in general for all of the spectral invariants introduced in this paper.
We remark that we impose very strong conditions on the connections we consider to reduce the number and complexity of the estimates needed to compute the anomaly. It is, of course, important to establish the weakest conditions under which the anomaly exists, however, we have to leave this to another place.
Motivation from physics examples
--------------------------------
We have indicated above the motivation that comes from the work of Bollé [*et al*]{}, Callias, Gesztesy-Simon and Witten. The index problem posed in [@Cal:AIO] deals with non-self-adjoint operators on odd dimensional spaces. It fits into the framework of our companion paper [@CaKa:TIH] but we do not discuss this kind of example here.
A second source of motivating examples is the study of Dirac operators coupled to connections in odd dimensions (in particular, the three dimensional case is of interest in condensed matter theory). As we now explain these can be incorporated into our framework here. In odd dimensions the fundamental invariant is spectral flow. It is often claimed for models in condensed matter theory that spectral flow is a physically relevant invariant but in many cases it is not obvious that one is dealing with Fredhom operators. On the other hand spectral flow is known to be related directly to Krein’s spectral shift function and both can be expressed in terms of an index for a Dirac-type operator in even dimensions as explained in [@GLMST]. We outline the argument.
Starting from a pair of self-adjoint operators, denoted $A_{\pm}$ (in general unbounded), on a Hilbert space $\mathcal H$, we introduce a ‘flow parameter’ $s\in \mathbb R$ and a path of self adjoint operators $A(s)$ with $\lim_{s\to\pm\infty} A(s) =A_\pm$ (where the limit is taken in an appropriate topology). We then introduce a new operator acting on the ‘big Hilbert space’ $L^2(\mathbb R, \mathcal H\oplus \mathcal H)$ of the form $$\C D := \ma{cc}{
0 & \partial_s + A(s) \\
-\partial_s +A(s) & 0
} = \ma{cc}{0 & \C D_- \\ \C D_+ & 0}$$ Under various assumptions one can relate the index of $\C D_+$ to the spectral flow for the path $A(s)$ when the latter path consists of Fredholm operators. When the path $A(s), s\in \mathbb R$ consists of Dirac type operators on $\mathbb R^{2n-1}$ then $\C D$ is of Dirac type on $\mathbb R^{2n}$. One may then ask (in the non-Fredholm case) the question of whether the homological index for $T=\C D_+(1+\C D_-\C D_+)^{-1/2}$ is related to some generalised spectral flow, or spectral shift function, for the pair $A_{\pm}$. This kind of problem arises when $A_\pm$ are magnetic Dirac type operators. So for example if $A_\pm$ are operators in three dimensions then the degree $n=2$ homological index comes into play for $\C D_+$. We are currently investigating this application and will report on the outcome elsewhere.
Preliminaries
=============
Notation
--------
Let $\dir_+ : \T{Dom}(\dir_+) \to \C H$ be a closed unbounded operator and let $A \in \sL(\C H)$ be a bounded operator. The conditions in Assumption \[a:unb\] will be in effect.
\[n:Iresana\] We will use the following notation for various unbounded operators related to $A$ and $\dir_+$: $$\C D_+ := \dir_+ + A : \T{Dom}(\dir_+) \to \C H, \quad\quad
\C D_- := (\C D_+)^* = \dir_- + A^* : \T{Dom}(\dir_-) \to \C H$$ $$\De := \dir_+ \dir_- = \dir_- \dir_+ : \T{Dom}(\De) \to \C H,$$ $$\De_1 := \C D_+ \C D_- : \T{Dom}(\De) \to \C H, \quad
\De_2 := \C D_- \C D_+ : \T{Dom}(\De) \to \C H
$$ The bounded extension of the commutator $[\C D_+,\C D_-] : \T{Dom}(\De) \to \C H$ is denoted by $F \in \sL(\C H)$.
Remark that we have the expressions $$\begin{split}
\De_1 - \De & = A \dir_- + \dir_+ A^* + A A^* : \T{Dom}(\De) \to \C H, \\
\De_2 - \De & = A^* \dir_+ + \dir_- A + A^* A : \T{Dom}(\De) \to \C H.
\end{split}$$
\[n:IIresana\] Introduce the following notation $$\begin{split}
V_1 := A \dir_- + \dir_+ A^* + A A^* : \T{Dom}(\dir_+) \to \C H, \\
V_2 := A^* \dir_+ + \dir_- A + A^* A : \T{Dom}(\dir_+) \to \C H
\end{split}$$ for the extensions to $\T{Dom}(\dir_+)$ of $\De_1 - \De$ and $\De_2 - \De$, respectively. Furthermore, we let $$Y_1(\la) := V_1 \cd (\la + \De)^{-1} : \C H \to \C H \q Y_2(\la) := V_2 \cd (\la+\De)^{-1} : \C H \to \C H$$ for all $\la \in (0,\infty)$.
It is important to notice that the linear operators $Y_1(\la) : \C H \to \C H$ and $Y_2(\la) : \C H \to \C H$ are unbounded operators in general. This is due to the fact that the commutator $[\dir_-,A]$ does *not* always extend to a bounded operator. This condition is not even satisfied for the perturbations of the Dirac operator on $\rr^{2n}$ which we consider in Section \[s:anodir\]. We will however impose differentiability conditions on $A$ later on which imply that $Y_1(\la)$ and $Y_2(\la)$ are bounded. These differentiability conditions will be the subject of the next subsection.
Quantum differentiability {#ss:quadif}
-------------------------
While this subsection concerns classical pseudo-differential operators we anticipate noncommutative applications and hence frame the definitions accordingly. In the first part of this subsection we will consider a selfadjoint positive unbounded operator $\De : \T{Dom}(\De) \to \C H$ on the Hilbert space $\C H$. The functional calculus for selfadjoint unbounded operators provides us with a scale of dense subspaces $\C H^s := \T{Dom}(\De^{s/2}) \su \C H$, $s \in [0,\infty)$. Each subspace $\C H^s$ becomes a Hilbert space in its own right when equipped with the inner product $$\inn{\cd,\cd}_s : (\xi,\eta) \mapsto \inn{\xi,\eta} + \inn{\De^{s/2}\xi,\De^{s/2}\eta} \q \xi,\eta \in \C H^s.$$ Notice also that $\C H^s \su \C H^r$ whenever $s \geq r$. This scale of Hilbert spaces plays the role of Sobolev spaces in noncommutative geometry.
Following Connes and Moscovici, [@CoMo:LIF Appendix B], we consider the $*$-subalgebra $\T{OP}^0 \su \sL(\C H)$ consisting of bounded operators $T$ for which
1. The domain of $\De^{n/2}$ is an invariant subspace for all $n \in \nn$.
2. The iterated commutator $\de^n(T) : \T{Dom}(\De^{n/2}) \to \C H$ extends to a bounded operator on $\C H$ for all $n \in \nn$.
Here $\de^n(T) : \T{Dom}(\De^{n/2}) \to \C H$ is defined recursively by $\de(T) = [\De^{1/2},T]$ and $\de^n(T) = [\De^{1/2},\de^{n-1}(T)]$.
For each $\la \in (0,\infty)$ and each $m \in \nn_0$ we define the algebra automorphism $$\si_\la^m : \T{OP}^0 \to \T{OP}^0 \q \si_\la^m(T) := (\la + \De)^m T (\la + \De)^{-m}.$$ Notice that the inverse $\si_\la^{-m} : \T{OP}^0 \to \T{OP}^0$ of $\si_\la^m$ can be defined by $\si_\la^{-m}(T) := \big( \si_\la^m(T^*) \big)^*$. The element $\si_\la^{-m}(T) \in \T{OP}^0$ then agrees with the bounded extension of $(\la + \De)^{-m} T (\la + \De)^m : \T{Dom}(\De^m) \to \C H$.
We record the following:
\[l:difact\] Let $m \in \zz$. For each $T \in \T{OP}^0$ we have the estimate $$\| \si^m_\la(T) - T\| = O(\la^{-1/2}) \q \T{as }\, \la \to \infty$$
It is enough to prove the assertion for $m \in \nn_0$. The easiest proof then runs by induction. The statement is trivial for $m= 0$. Thus suppose that it holds for some $m_0 \in \nn_0$.
Let $T \in \T{OP}^0$. Then $$\begin{split}
\si^{m_0 + 1}_\la(T) & = (\la + \De)^{m_0 + 1} T (\la + \De)^{-m_0 - 1} \\
& = \si^{m_0}_\la(T) + (\la + \De)^{m_0} [\De,T](\la + \De)^{-m_0 - 1}.
\end{split}$$ The fact that $T \in \T{OP}^0$ implies that $[\De,T](1+ \De)^{-1/2}$ extends to an element $R(T) \in \T{OP}^0$. We thus have that $$(\la + \De)^{m_0} [\De,T] (\la + \De)^{-m_0 - 1} = \si_\la^{m_0}\big( R(T) \big) \cd (1+\De)^{1/2} \cd (\la + \De)^{-1}.$$
Since $\|(1+\De)^{1/2} \cd (\la + \De)^{-1}\| = O(\la^{-1/2})$ we obtain the estimate $\|\si^{m_0 + 1}_\la(T) - T\| = O(\la^{-1/2})$ by applying the induction hypothesis to $T$ and $R(T)$.
We will now return to the setup introduced in the beginning of subsection \[s:unb\]. The conditions on the closed operator $\dir_+$ and the bounded operator $A$ which are stated in Assumption \[a:unb\] will in particular be in effect. We will then consider the unbounded operators $Y_1(\la) = V_1 (\la + \De)^{-1}$ and $Y_2(\la) = V_2(\la + \De)^{-1}$ introduced in Notation \[n:IIresana\].
\[l:disanaest\] Suppose that $A \in \T{OP}^0$. Then $Y_1(\la), Y_2(\la) \in \T{OP}^0$ for all $\la \in (0,\infty)$. Furthermore, we have the estimate $$\|Y_1(\la)\| = \|Y_2(\la)\| = O(\la^{-1/2}) \q \T{as }\, \la \to \infty$$
Let $\la \in (0,\infty)$. To ease the notation, let $F_+(\la) := \dir_+ (\la + \De)^{-1}$ and $F_-(\la) := \dir_-(\la+\De)^{-1}$. Compute as follows, $$\begin{split}
V_1 \cd (\la + \De)^{-1} & = A \dir_- \cd (\la+\De)^{-1} + \dir_+ A^* \cd (\la + \De)^{-1} + AA^* \cd (\la+\De)^{-1} \\
& = A \cd F_-(\la) + AA^* \cd (\la+\De)^{-1} + F_+(\la) \cd \si_\la(A^*).
\end{split}$$ Since each of the bounded operators $A \cd F_-(\la)$, $AA^* \cd (\la+\De)^{-1}$ and $F_+(\la) \cd \si_\la(A^*)$ lies in $\T{OP}^0$ this shows that $Y_1(\la) \in \T{OP}^0$ as well.
The desired estimate on $\|Y_1(\la)\|$ follows from the above identities and Lemma \[l:disanaest\] since $\|F_+(\la)\| = \|F_-(\la)\| = O(\la^{-1/2})$ as $\la \to \infty$.
A similar argument proves the claims on $Y_2(\la)$ as well.
Recall that a unital $*$-subalgebra $\sA$ of a unital $C^*$-algebra $A$ is said to be *closed under the holomorphic functional calculus* when for each $x \in \sA$ and each holomorphic function $f$ on the spectrum of $x$ in $A$ we have that $f(x) \in \sA$.
The following result is well known but we prove it here for lack of an adequate reference:
\[l:resqua\] The unital $*$-subalgebra $\T{OP}^0 \su \sL(\C H)$ is closed under the holomorphic functional calculus.
For each $N \in \nn$, let $\T{OP}^0_N$ denote the unital $*$-algebra consisting of the bounded operators $T$ such that
1. The domain of $\De^{n/2}$ is an invariant subspace for all $n \in \{1,\ldots,N\}$.
2. The iterated commutator $\de^n(T) : \T{Dom}(\De^{n/2}) \to \C H$ extends to a bounded operator on $\C H$ for all $n \in \{1,\ldots,N\}$.
The unital $*$-algebra $\T{OP}^0_N$ becomes a unital Banach $*$-algebra when equipped with the norm $$\|\cd\|_N : T \mapsto \sum_{n=0}^N \frac{1}{n!} \|\de^n(T)\|.$$ It then follows from [@BlCu:DNS Proposition 3.12] that $\T{OP}^0_N$ is closed under holomorphic functional calculus. But this proves the lemma since $\T{OP}^0 = \bigcap_{N=1}^\infty \T{OP}^0_N$.
The next statement is now a consequence of Lemma \[l:disanaest\] and Lemma \[l:resqua\].
\[p:anainv\] Suppose that $A \in \T{OP}^0$. Then there exists a constant $C > 0$ such that $\big(1 + Y_1(\la) \big)^{-1}$ and $\big( 1 + Y_2(\la)\big)^{-1}$ are well-defined elements in $\T{OP}^0$ for all $\la \geq C$.
Resolvent expansions {#ss:resana}
====================
The object of this section is to find verifiable criteria which imply that the homological index exists. We will thus be studying expressions of the form $$(z + \De_2)^{-n} - (z+ \De_1)^{-n}$$ for a natural number $n \in \nn$ and a complex number $z \in \cc\sem (-\infty,0]$. The notation used throughout will be the one introduced in Notation \[n:Iresana\] and Notation \[n:IIresana\]. Furthermore, the conditions in Assumption \[a:unb\] will be in effect.
The main idea is to approximate the operators $\De_1 = \C D_+ \C D_-$ and $\De_2 = \C D_- \C D_+$ by the Laplacian $\De = \dir_+ \dir_-$ using perturbation theoretic techniques.
We start by stating some preliminary lemmas. The proofs are well-known and will not be repeated here.
\[l:difpowexp\] Let $n \in \nn$ and let $z \in \cc\sem (-\infty,0]$. Then $$(z + \De_2)^{-n} - (z + \De_1)^{-n} = \sum_{i=0}^{n-1} (z + \De_1)^{-i-1} \cd F \cd (z + \De_2)^{-(n-i)}.$$
\[l:resexp\] Let $H_1,H_2$ be selfadjoint positive unbounded operators on $\C H$ with $\T{Dom}(H_1) = \T{Dom}(H_2)$. Let $z,z_0 \in \cc\sem (-\infty,0]$ and suppose that the linear operator $(H_1 - H_2 + z - z_0)(z_0 + H_2)^{-1} : \C H \to \C H$ is bounded with $\|(H_1 - H_2 + z- z_0)(z_0 + H_2)^{-1}\| < 1$. We then have the identity $$(z + H_1)^{-1} = (z_0 + H_2)^{-1} \cd \big( 1 + (H_1 - H_2 + z-z_0)(z_0 + H_2)^{-1} \big)^{-1}.$$
The next result implies that the trace of the quantity $$z^n \big( (z+ \C D_- \C D_+)^{-n} - (z + \C D_+ \C D_-)^{-n} \big),$$ which is computing the homological index, depends holomorphically on the complex parameter $z \in \cc\sem (-\infty,0]$.
\[p:holhomind\] Let $n \in \nn$ and let $i \in \{0,\ldots,n-1\}$. Suppose that there exists a $z_0 \in \cc\sem (-\infty,0]$ such that $$(z_0 + \De_1)^{-i-1} \cd F \cd (z_0 + \De_2)^{-(n-i)} \in \sL^1(\C H).$$ Then the operator $(z + \De_1)^{-i-1} \cd F \cd (z + \De_2)^{-(n-i)}$ is of trace class for all $z \in \cc\sem (-\infty,0]$. Furthermore, the map $$\cc\sem (-\infty,0] \to \sL^1(\C H) \q
z \mapsto (z + \De_1)^{-i-1} \cd F \cd (z + \De_2)^{-(n-i)}$$ is holomorphic with respect to the trace norm on $\sL^1(\C H)$.
Let $z \in \cc\sem (-\infty,0]$. Remark then that $$\label{eq:powexp}
(z + \De_1)^{-i-1} = (z_0 + \De_1)^{i+1} \cd (z + \De_1)^{-i-1} \cd (z_0 + \De_1)^{-i-1}.$$ This shows that the map $z \mapsto (z + \De_1)^{-i-1}$ factorizes as a product $z \mapsto H(z) \cd (z_0 + \De_1)^{-i-1}$, where $z \mapsto H(z)$, $\cc\sem (-\infty,0] \to \sL(\C H)$ is holomorphic.
A similar computation shows that the map $z \mapsto (z + \De_2)^{-(n-i)}$ factorizes as a product $z \mapsto (z_0 + \De_2)^{-(n-i)} \cd K(z)$, where $z \mapsto K(z)$, $\cc\sem (-\infty,0] \to \sL(\C H)$, is holomorphic.
As a consequence, we obtain $$(z + \De_1)^{-i-1} \cd F \cd (z + \De_2)^{-(n-i)}
= H(z) \cd (z_0 + \De_1)^{-i-1} \cd F \cd (z_0 + \De_2)^{-(n-i)} \cd K(z)$$ for all $z \in \cc\sem (-\infty,0]$. This proves the proposition.
In the following proposition, we introduce an expansion of the quantity $$(\la + \De_2)^{-n} - (\la + \De_1)^{-n}.$$ This expansion will serve us in two ways. First, it allows us to identify a good criterion for the existence of the homological index and second, it will help us to give a concrete computation of this quantity (or more precisely of its scaling limit at infinity).
To ease the notation, let $X_1(\la) := (1 + Y_1(\la))^{-1}$ and $X_2(\la) := (1 + Y_2(\la))^{-1}$ whenever these quantities make sense as bounded operators.
\[p:resexprea\] Suppose that $A \in \T{OP}^0$ and let $n \in \nn$. Then there exists a constant $C > 0$ such that $$\begin{split}
& (\la + \De_1)^{-i-1} \cd F \cd (\la + \De_2)^{-(n-i)} \\
& \q = \si^{-1}_\la\big(X_1(\la)\big) \clc \si^{-i-1}_\la \big( X_1(\la) \big) \cd (\la + \De)^{-i-1} \cd F \\
& \qqq \cd (\la + \De)^{-(n-i)} \cd \si^{n-i-1}_\la \big(X_2(\la) \big) \clc X_2(\la)
\end{split}$$ for all $\la \geq C$ and all $i \in \{0,\ldots,n-1\}$
Choose the constant $C > 0$ as in Proposition \[p:anainv\] and let $\la \geq C$. It follows that $X_1(\la) = \big(1+Y_1(\la) \big)^{-1}$ and $X_2(\la) = \big(1 + Y_2(\la)\big)^{-1}$ lie in $\T{OP}^0$.
Let $i \in \{0,\ldots,n-1\}$. The result of Lemma \[l:resexp\] yields that $$\begin{split}\label{eq:firres}
& (\la + \De_1)^{-i-1} \cd F \cd (\la + \De_2)^{-(n-i)} \\
& \q = \big( (\la + \De)^{-1} X_1(\la) \big)^{i+1} \cd F \cd \big(
(\la + \De)^{-1} \cd X_2(\la) \big)^{n-i}.
\end{split}$$
But this implies the statement of the proposition since $$\begin{split}
& \big( (\la + \De)^{-1} \cd X_1(\la) \big)^{i+1} = \si^{-1}_\la\big( X_1(\la) \big) \clc \si^{-i-1}_\la \big( X_1(\la) \big) \cd (\la + \De)^{-i-1}
\q \T{and} \\
& \big( (\la + \De)^{-1} \cd X_2(\la) \big)^{n-i} = (\la + \De)^{-n + i} \si^{n-i-1}_\la \big( X_2(\la) \big) \cd \ldots \cd X_2(\la),
\end{split}$$ by the definition of the automorphisms $\si^m_\la : \T{OP}^0 \to \T{OP}^0$, $m \in \zz$.
We are now ready to state and prove the announced criterion for the existence of the homological index.
\[p:tradifpow\] Suppose that $A \in \T{OP}^0$ and that there exists an $n_0 \in \nn$ such that $$(1+ \De)^{-i-1} \cd F \cd (1+\De)^{-n_0+i} \in \sL^1(\C H) \q \T{for all }\, i \in \{0,1,\ldots,n_0-1\}.$$ Let $n \geq n_0$ and let $i \in \{0,1,\ldots,n-1\}$. Then $$(z+\De_1)^{-i-1} F (z+\De_2)^{-n+i} \in \sL^1(\C H) \q
\T{for all }\, z \in \cc\sem (-\infty,0].$$ Furthermore, the map $$\cc\sem (-\infty,0] \to \sL^1(\C H) \q z \mapsto (z+\De_1)^{-i-1} F (z+\De_2)^{-n+i}$$ is holomorphic in trace norm.
Choose a constant $C > 0$ as in Proposition \[p:resexprea\] and let $\la \geq C$. We obtain $$\begin{split}
& (\la + \De_1)^{-i-1} \cd F \cd (\la + \De_2)^{-(n-i)} \\
& \q = \si^{-1}_\la\big(X_1(\la)\big) \clc \si^{-i-1}_\la \big( X_1(\la) \big) \cd (\la + \De)^{-i-1} \cd F \\
& \qqq \cd (\la + \De)^{-(n-i)} \cd \si^{n-i-1}_\la \big(X_2(\la) \big) \clc X_2(\la)
\end{split}$$ This implies that $(\la + \De_1)^{-i-1} \cd F \cd (\la + \De_2)^{-(n-i)} \in \sL^1(\C H)$. Indeed, when $i \in \{0,\ldots,n_0 - 1\}$ we have that $$(\la + \De)^{-i-1} \cd F \cd (\la + \De)^{-(n-i)} = (\la + \De)^{-i-1} \cd F \cd (\la + \De)^{-n_0 + i} \cd (\la +\De)^{n_0 - n} \in \sL^1(\C H)$$ whereas when $i \in \{n_0,\ldots,n-1\}$ we have that $$\begin{split}
& (\la + \De)^{-i-1} \cd F \cd (\la + \De)^{-(n-i)}\\
& \q = (\la + \De)^{n_0 - i - 1} \cd (\la + \De)^{-n_0} \cd F \cd (\la + \De)^{-1} \cd (\la + \De)^{-n+1 + i}
\in \sL^1(\C H).
\end{split}$$ The statement of the proposition is now a consequence of Proposition \[p:holhomind\].
Let us summarize what we have proved thus far:
\[t:tradifpow\] Let $\dir_+ : \T{Dom}(\dir_+) \to \C H$ be a closed operator and let $A$ be a bounded operator that together satisfy the conditions in Assumption \[a:unb\]. Suppose that $A \in \T{OP}^0$ (with respect to the Laplacian $\De$) and that there exists an $n_0 \in \nn$ such that $(1+\De)^{-i-1} F (1+ \De)^{-n_0+i}$ is of trace class for all $i \in \{0,\ldots,n_0-1\}$.
Then the homological index $\T{H-Ind}_n(T_\la)$ exists for all $\la \in (0,\infty)$ and all $n \geq n_0$. Furthermore, for each $n \geq n_0$, the map $z \mapsto z^n \cd \T{Tr}\big( (z + \De_2)^{-n} - (z+ \De_1)^{-n} \big)$ provides a holomorphic extension of $\la \mapsto \T{H-Ind}_n(T_\la)$ to the open set $\cc\sem(-\infty,0]$.
Large scale expansions of the homological index {#s:larscaexp}
===============================================
As previously $\C H$ is a Hilbert space. Define the projections $$P_+ := \ma{cc}{
1 & 0 \\
0 & 0
} \, \T{ and }\, P_- := \ma{cc}{
0 & 0 \\
0 & 1
} : \C H \op \C H \to \C H \op \C H.$$
Consider the vector subspace $\sL^1_s(\C H) \su \sL(\C H \op \C H)$ consisting of square matrices $T = \ma{cc}{T_{11} & T_{12} \\ T_{21} & T_{22}}$ of bounded operators with $T_{11} - T_{22} \in \sL^1(\C H)$. This vector space becomes a Banach space when equipped with the norm $$\|\cd\|_1^s : \sL^1_s(\C H) \to [0,\infty) \q \|\cd\|_1^s : T \mapsto \|T\| + \|P_+ T P_+ - P_- T P_-\|_1.$$ The super trace $$\T{Tr}_s : \sL^1_s(\C H) \to \cc \q \T{Tr}_s : T \mapsto \T{Tr}\big( P_+ T P_+ - P_- T P_-\big)$$ defines a bounded linear functional on this Banach space.
Let now $A : \C H \to \C H$ be a bounded operator and let $\dir_+ : \T{Dom}(\dir_+) \to \C H$ be a closed unbounded operator. For the rest of this section the conditions in Assumption \[a:unb\] will be in effect. Furthermore, we will impose the following:
\[a:tra\] Assume that $A \in \T{OP}^0$ and that there exists an $n_0 \in \nn$ such that $(1 + \De)^{-i-1} \cd F \cd (1+\De)^{-n_0 +i+\ep}$ is of trace class for all $i \in \{-1,0,1,\ldots,n_0-1\}$ and all $\ep \in (0,1)$.
Let us fix a natural number $n \in \{n_0,n_0 + 1,\ldots\}$.
As a consequence of Theorem \[t:tradifpow\] we have a well-defined homological index $$\la^n \T{Tr}\big( (\la + \De_2)^{-n} - (\la + \De_1)^{-n} \big) = \T{H-Ind}_n(T_\la)$$ for all $\la \in (0,\infty)$. This quantity can be reinterpreted as follows. Consider the $n^{\T{th}}$ power of the resolvent of the scaled operator $\la^{-1}\C D^2$, $$\label{eq:ressqu}
\begin{split}
(1 + \la^{-1}\C D^2)^{-n}
& = \ma{cc}{
(1 + \la^{-1}\C D_- \C D_+)^{-n} & 0 \\
0 & (1 + \la^{-1}\C D_+\C D_-)^{-n}
} \\
& = \la^n \ma{cc}{
(\la + \De_2)^{-n} & 0 \\
0 & (\la + \De_1)^{-n}
}.
\end{split}$$ The homological index in degree $n$ is then obtained as the super trace of the operator in : $$\begin{split}
\T{H-Ind}_n(T_\la) & = \T{Tr}\big( P_+(1 + \la^{-1} \C D^2)^{-n} P_+ - P_-(1 + \la^{-1} \C D^2)^{-n}P_- \big) \\
& := \T{Tr}_s \big( (1 + \la^{-1}\C D^2 )^{-n} \big)
\end{split}$$ Remark that this super trace makes sense even though we do not assume that each term on the diagonal is of trace class.
The first aim of this section is to obtain an expansion of the resolvent $(1 + \la^{-1} \C D^2)^{-n} \in \sL^1_s(\C H)$ when the scaling parameter $\la > 0$ is large. The standard terms in this expansion have the following form:
\[n:omedeg\] For each $J = (j_1,\ldots,j_n) \in \nn_0^n$ and each $\la > 0$, define the bounded operators $$\begin{split}
\om_1(J,\la) & := (-1)^{j_1 \plp j_n}(\la + \De)^{-1} Y_1(\la)^{j_1} \clc (\la + \De)^{-1} Y_1(\la)^{j_n} \q \T{and} \\
\om_2(J,\la) & := (-1)^{j_1 \plp j_n}(\la + \De)^{-1} Y_2(\la)^{j_1} \clc (\la + \De)^{-1} Y_2(\la)^{j_n}
\end{split}$$ on the Hilbert space $\C H$.
The main technical result for obtaining the desired expansion of the resolvent is a trace norm estimate on the difference $\om_2(J,\la) - \om_1(J,\la)$, for $J \in \nn_0^n$ and large scaling parameters $\la$. This is contained in the next lemma.
\[l:omeest\] The diagonal operator $$\om(J,\la) := \ma{cc}{
\om_2(J,\la) & 0 \\
0 & \om_1(J,\la)
}$$ lies in the Banach space $\sL^1_s(\C H)$ for all $J \in \nn_0^n$ and all $\la >0$. Furthermore, for each $\ep \in (0,1)$, there exist constants $C,K>0$ such that $$\|\om(J,\la)\|_1^s \leq (j_1 \plp j_n + 1) \cd K^{j_1 \plp j_n} \cd \la^{n_0 - n -(j_1 \plp j_n)/2 + 1/2 - \ep}$$ for all $J = (j_1,\ldots,j_n) \in \nn_0^n$ and all $\la \geq C$.
Let $J \in \nn_0^n$ and let $\la > 0$. Let $k \in \{1,\ldots,n\}$ and let $l \in \{1,\ldots,j_k\}$. To show that $\om_2(J,\la) - \om_1(J,\la) \in \sL^1(\C H)$ it is enough to verify that $$\begin{split}
& (\la + \De)^{-1} Y_2(\la)^{j_1} \clc (\la + \De)^{-1} Y_2(\la)^{j_{k - 1}} \\
& \q \cd (\la + \De)^{-1} Y_2(\la)^{l-1} \cd \big( Y_2(\la) - Y_1(\la) \big) Y_1(\la)^{j_k -l} \\
& \qq \cd (\la + \De)^{-1} Y_1(\la)^{j_{k + 1}} \clc (\la + \De)^{-1} Y_1(\la)^{j_n} \in \sL^1(\C H).
\end{split}$$ Using Lemma \[l:disanaest\] this bounded operator can be rewritten as $$\begin{split}
& -\si_\la^{-1}\big(Y_2(\la)^{j_1} \big) \clc \si_\la^{-k+1}\big( Y_2(\la)^{j_{k-1}} \big)
\cd \si_\la^{-k}\big( Y_2(\la)^{l-1} \big) \\
& \q \cd (\la + \De)^{-k} \cd F \cd (\la + \De)^{-n+k-1} \\
& \qq \cd \si_\la^{n-k}\big( Y_1(\la)^{j_k - l} \big) \cd \si_\la^{n-k-1}\big( Y_1(\la)^{j_{k+1}} \big) \clc Y_1(\la)^{j_n},
\end{split}$$ where we recall that $F \cd (\la + \De)^{-1} = Y_1(\la) - Y_2(\la)$. This proves that $\om(J,\la) \in \sL^1_s(\C H)$ since the conditions of Assumption \[a:tra\] are in effect. See also the proof of Proposition \[p:tradifpow\].
To prove the norm estimate, we first note that there exist constants $C_1,K_1 > 0$ such that $$\label{eq:openorest}
\|\om_1(J,\la) \| \, , \, \|\om_2(J,\la)\| \leq \la^{-n} K_1^{j_1 \plp j_n} \cd \la^{-(j_1\plp j_n)/2}$$ for all $\la \geq C_1$ and all $J = (j_1,\ldots,j_n) \in \nn_0^n$. See Lemma \[l:disanaest\].
Let now $\ep \in (0,1)$. By our standing Assumption \[a:tra\] we may choose a constant $K_3 > 0$ such that $$\label{eq:tranorest0}
\| (\la + \De)^{-k} \cd F \cd (\la + \De)^{-n+k-1} \|_1 \leq K_3 \cd \la^{n_0 - n - \ep}$$ for all $k \in \{1,\ldots,n\}$ and all $\la \geq 1$.
Furthermore, by an application of Lemma \[l:disanaest\] and Lemma \[l:difact\] we may choose constants $C_2,K_2 > 0$ such that $$\label{eq:tranorest}
\begin{split}
& \big\| \si_\la^{-1}\big(Y_2(\la)^{j_1} \big) \clc \si_\la^{-k+1}\big( Y_2(\la)^{j_{k-1}} \big)
\cd \si_\la^{-k}\big( Y_2(\la)^{l-1} \big) \\
& \qq \cd (\la + \De)^{-k} \cd F \cd (\la + \De)^{-n+k-1} \\
& \qqq \cd \si_\la^{n-k}\big( Y_1(\la)^{j_k - l} \big) \cd \si_\la^{n-k-1}\big( Y_1(\la)^{j_{k+1}} \big) \clc Y_1(\la)^{j_n} \big\|_1 \\
& \q \leq K_2^{j_1 \plp j_n -1} \cd \la^{-(j_1 \plp j_n - 1)/2} \cd \| (\la + \De)^{-k} \cd F \cd (\la + \De)^{-n+k-1} \|_1
\end{split}$$ for all $J \in \nn_0^n$, $k \in \{1,\ldots,n\}$, $l \in \{1,\ldots,j_k\}$ and all $\la \geq C_2$.
A combination of , and proves the norm estimate on $\om(J,\la)$.
Let $\C B : \nn_0^n \to \nn_0$ denote the addition, $\C B : (j_1,\ldots,j_n) \mapsto j_1 \plp j_n$. The level sets will be written as $\C B_k := \C B^{-1}(\{k\})$, for all $k \in \nn_0$.
We are now ready to rewrite the homological index using an infinite series of operators of the form $\om(J,\la)$, where $J \in \nn_0^n$ and the scaling parameter $\la$ is large. This expansion will be the main tool for computing the anomaly of Dirac type operators in Section \[s:locano\].
\[p:degexp\] Let $\dir_+ : \T{Dom}(\dir_+) \to \C H$ be a closed operator and let $A : \C H \to \C H$ be a bounded operator which together satisfy the conditions in Assumption \[a:unb\] and Assumption \[a:tra\] for some $n_0 \in \nn$. Let $n \in \{n_0,n_0 +1,\ldots\}$.
Then there exists a constant $C > 0$ such that $$(1 + \la^{-1}\C D^2)^{-n} = \la^n \cd \sum_{k = 0}^\infty \sum_{J \in \C B_k} \om(J,\la)$$ for all $\la \geq C$, where the sum converges absolutely in $\sL^1_s(\C H)$. In particular, we have that $$\T{H-Ind}_n(T_\la) = \la^n \cd \sum_{k = 0}^\infty \sum_{J \in \C B_k} \T{Tr}_s\big( \om(J,\la) \big)$$ for all $\la \geq C$, where the sum converges absolutely.
Choose constants $C_1,K_1 > 0$ as in Lemma \[l:omeest\] (with $\ep = 3/4$). We then have that $$\begin{split}
\la^n \cd \sum_{k=0}^\infty \sum_{J \in \C B_k} \|\om(J,\la)\|_1^s
& \leq \sum_{k=0}^\infty \sum_{J \in \C B_k} (k+1) \cd K_1^k \cd \la^{n_0 -k/2 - 1/4} \\
& = \la^{n_0 - 1/4} \sum_{k=0}^\infty (k+1) \cd (K_1/\sqrt{\la})^k \cd {n + k-1 \choose n-1},
\end{split}$$ for all $\la \geq C_1$. This shows that the sum $\la^n \cd \sum_{k = 0}^\infty \sum_{J \in \C B_k} \om(J,\la)$ converges absolutely in $\sL^1_s(\C H)$ whenever $\la > \T{max}\{C_1,K^2_1\}$.
Now, by Lemma \[l:disanaest\] we may choose a constant $C_2 > 0$ such that $\|Y_1(\la)\| \, , \, \|Y_2(\la)\| < 1$, for all $\la \geq C_2$. An application of Lemma \[l:resexp\] then implies that $$\begin{split}
(\la + \De_l)^{-n} & = \Big( (\la + \De)^{-1} \big( 1 + Y_l(\la) \big)^{-1} \Big)^n
= \Big( (\la + \De)^{-1} \cd \sum_{j=0}^\infty (-1)^j Y_l(\la) \Big)^n \\
& = \sum_{J \in \nn_0^n} \om_l(J,\la) = \sum_{k=0}^\infty \sum_{J \in \C B_k} \om_l(J,\la),
\end{split}$$ for all $\la \geq C_2$ and $l \in \{1,2\}$, where the sums converge absolutely in operator norm. A combination of these observations proves the first part of the proposition since $$(1 + \la^{-1} \C D^2)^{-n} = \la^n \cd \ma{cc}{ (\la + \De_2)^{-n} & 0 \\
0 & (\la + \De_1)^{-n}
},$$ for all $\la > 0$.
The second statement of the proposition is now a consequence of the boundedness of the super trace $\T{Tr}_s : \sL^1_s(\C H) \to \cc$ since $\T{H-Ind}_n(T_\la) := \T{Tr}_s\big( (1 + \la^{-1} \C D^2)^{-n} \big)$ for all $\la > 0$.
As a first application of the above expansion we show that the anomaly, when it exists, only depends on finitely many of the terms $\om(J,\la)$.
\[p:verhig\] Let $\dir_+ : \T{Dom}(\dir_+) \to \C H$ be a closed operator and let $A : \C H \to \C H$ be a bounded operator which together satisfy the conditions in Assumption \[a:unb\] and Assumption \[a:tra\] for some $n_0 \in \nn$. Let $n \in \{n_0,n_0 + 1,\ldots\}$.
The anomaly of $\C D_+ := \dir_+ + A$ in degree $n$ then exists if and only if the limit $$\lim_{\la \to \infty} \la^n \cd \sum_{k=0}^{2n_0-1} \sum_{J \in \C B_k} \T{Tr}_s\big( \om(J,\la) \big)$$ exists. In this case we have that $$\T{Ano}(\C D) = \lim_{\la \to \infty} \la^n \cd \sum_{k=0}^{2n_0-1} \sum_{J \in \C B_k} \T{Tr}_s\big( \om(J,\la) \big).$$
By Proposition \[p:degexp\] it is enough to show that $$\lim_{\la \to \infty} \la^n \cd \sum_{k=2n_0}^\infty \sum_{J \in \C B_k}\T{Tr}_s\big( \om(J,\la) \big) = 0.$$
By Lemma \[l:omeest\] we may choose constants $C,K > 0$ (with $\ep = 3/4$) such that $$\la^n \|\om(J,\la)\|_1^s \leq \la^{n_0 - 1/4} \cd (\C B(J)+1) \cd (K/\sqrt{\la})^{\C B(J)}$$ for all $\la \geq C$ and all $J \in \nn_0^n$.
Let now $\la \geq \T{max}\{2K^2,C\}$. It then follows from the above estimate that $$\begin{split}
\big\| \la^n \cd \sum_{k=2n_0}^\infty \sum_{J \in \C B_k} \om(J,\la) \big\|_1^s & \leq \sum_{k = 2n_0}^\infty \sum_{J \in \C B_k} (k+1) \cd
\big( K/\sqrt{\la} \big)^k \cd \la^{n_0 -1/4} \\
& \leq K^{2n_0} \cd \la^{-1/4}
\cd \sum_{k = 2n_0}^\infty \sum_{J \in \C B_k} (1/2)^{k-2n_0}.
\end{split}$$ But this proves the theorem. In fact we obtain $$\Big| \la^n \cd \sum_{k=2n_0}^\infty \sum_{J \in \C B_k}\T{Tr}_s\big( \om(J,\la) \big) \Big| = O(\la^{-1/4})$$ as $\la \to \infty$.
The anomaly for Dirac operators on $\rr^{2n}$ {#s:anodir}
=============================================
For this example we start with the flat Dirac operator on $\rr^{2n}$ defined as follows. Let $\cc_{2n-1}$ denote the Clifford algebra over $\rr^{2n-1}$. Recall that this is the unital $*$-algebra over $\cc$ with $(2n-1)$-generators $e_1,\ldots,e_{2n-1} \in \cc_{2n-1}$ satisfying the relations $$e_j = e_j^* \q e_i e_j + e_j e_i = 2 \de_{ij} \q \T{for all }\, i,j \in \{1,\ldots,2n-1\}.$$ The unit will be denoted by $1 = e_0 \in \cc_{2n-1}$. We fix an irreducible representation $\pi_{2n-1} : \cc_{2n-1} \to \sL(\cc^{2^{n-1}})$. The flat Dirac operator on $\rr^{2n}$ is then given by $$\label{eq:dir}
\dir := \ma{cc}{0 & \dir_- \\ \dir_+ & 0} := \ma{cc}{0 & -\frac{\pa}{\pa x_{2n}} + i \sum_{j=1}^{2n-1} c_j \frac{\pa}{\pa x_j}
\\ \frac{\pa}{\pa x_{2n}} + i \sum_{j=1}^{2n-1} c_j \frac{\pa}{\pa x_j} & 0},$$ where $c_j := \pi_{2n-1}(e_j)$, for all $j \in \{1,\ldots,2n-1\}$. The domain of $\dir$ is a direct sum of the Sobolev space $H^1(\rr^{2n}) \ot \cc^{2^{n-1}}$ with itself, thus $\T{Dom}(\dir_+) = \T{Dom}(\dir_-) = H^1(\rr^{2n}) \ot \cc^{2^{n-1}}$. The Clifford matrices act on the second component of the tensor product and the partial differentiation operators act on the first component. We recall that the Sobolev space $H^1(\rr^{2n})$ consists of the elements in $L^2(\rr^{2n})$ which have all weak derivatives of first order in $L^2(\rr^{2n})$.
We note that $\dir_+$ is normal and that the unbounded operator $$\dir_- \dir_+ = -\sum_{j=1}^{2n} \frac{\pa^2}{\pa x_j^2} = \dir_+ \dir_- : H^2(\rr^{2n}) \ot \cc^{2^{n-1}} \to L^2(\rr^{2n}) \ot \cc^{2^{n-1}}$$ is a diagonal operator with the Laplacian on $\rr^{2n}$ in every entry on the diagonal. The notation $H^2(\rr^{2n})$ refers to the second Sobolev space on $\rr^{2n}$.
We let $C^\infty_b(\rr^{2n})$ denote the unital $*$-algebra of smooth complex valued functions with all derivatives bounded.
Let $N \in \nn$. Our strategy is to perturb the flat Dirac operator $\dir$ by a bounded operator corresponding to a $U(N)$-connection. More precisely, we define $$\label{eq:con}
\begin{split}
A : L^2(\rr^{2n}) \ot \cc^N \ot \cc^{2^{n-1}} \to L^2(\rr^{2n}) \ot \cc^N \ot \cc^{2^{n-1}} \q A := i a_{2n} - \sum_{j=1}^{2n-1} c_j a_j,
\end{split}$$ where $a_1,\ldots,a_{2n} : \rr^{2n} \to M_N(\cc)$ are selfadjoint elements in $C^\infty_b\big(\rr^{2n}, M_N(\cc)\big)$ acting by multiplication on $L^2(\rr^{2n}) \ot \cc^N$. We are thus interested in the unbounded operator $$\label{eq:perdir}
\C D := \ma{cc}{
0 & \dir_- + A^* \\
\dir_+ + A & 0
} : H^1(\rr^{2n}) \ot \cc^{2^n} \ot \cc^N \to L^2(\rr^{2n}) \ot \cc^{2^n} \ot \cc^N,$$ where $$\begin{split}
\dir_+ + A & = \big( \frac{\pa}{\pa x_{2n}} + i a_{2n} \big) + i \sum_{j=1}^{2n-1} c_j \big( \frac{\pa}{\pa x_j} + ia_j\big) \q \T{and} \\
\dir_- + A^* & = - \big( \frac{\pa}{\pa x_{2n}} + i a_{2n} \big) + i \sum_{j=1}^{2n-1} c_j \big( \frac{\pa}{\pa x_j} + ia_j\big)
\end{split}$$
We remark that the bounded operators $A$ and $A^*$ preserve the domain of $\dir_+$ (the first Sobolev space) and that the sum of commutators is given by $$\begin{split}
[\dir_+,A^*] + [A,\dir_-]
& = 2 \big[\frac{\pa}{\pa x_{2n}},-\sum_{j=1}^{2n-1} c_j a_j\big] + 2 \big[ i a_{2n},i \sum_{j=1}^{2n-1} c_j \frac{\pa}{\pa x_j} \big] \\
& = 2 \sum_{j=1}^{2n-1} c_j \Big( \frac{\pa a_{2n}}{\pa x_j} - \frac{\pa a_j}{\pa x_{2n}} \Big)
\end{split}$$ This shows that the conditions in Assumption \[a:unb\] are satisfied.
We note furthermore that $$[A, A^*] = 2\big[i a_{2n}, -\sum_{j=1}^{2n-1} c_j a_j \big] = 2i \sum_{j=1}^{2n-1}c_j [a_j,a_{2n}].$$ The bounded extension of the commutator $[\C D_+ ,\C D_-]$ is therefore given by the expression $$F := 2 \cd \sum_{j=1}^{2n-1} c_j \Big( \frac{\pa a_{2n}}{\pa x_j} - \frac{\pa a_j}{\pa x_{2n}} + i [a_j,a_{2n}] \Big),$$
The smoothness and boundedness conditions on the maps $a_1,\ldots,a_{2n} : \rr^{2n} \to M_N(\cc)$ imply that $A$ lies in $\T{OP}^0$ with respect to the Laplacian $\De : H^2(\rr^{2n}) \ot \cc^{2^{n-1}} \ot \cc^N \to L^2(\rr^{2n}) \ot \cc^{2^{n-1}} \ot \cc^N$.
To deal with the trace class condition on $(1+\De)^{-i-1} F (1 + \De)^{-n +i}$ for $i \in \{-1,0,\ldots,n-1\}$ we make the following standing assumption:
\[a:fct\] Assume that the maps $a_1,\ldots,a_{2n-1} : \rr^{2n} \to M_N(\cc)$ are selfadjoint elements in $C_b^\infty\big(\rr^{2n},M_N(\cc)\big)$ such that the partial derivative $$\frac{\pa a_j}{\pa x_{2n}} : \rr^{2n} \to M_N(\cc)$$ has compact support for each $j \in \{1,\ldots,2n-1\}$. Assume furthermore that $a_{2n} = 0$.
The above assumption implies that $$F = - 2 \cd \sum_{j=1}^{2n-1} c_j \frac{\pa a_j}{\pa x_{2n}} : \rr^{2n} \to M_{2^{n-1}}(\cc) \ot M_N(\cc)$$ is smooth and compactly supported. It then follows from [@Sim:TIA Theorem 4.1] and [@Sim:TIA Theorem 4.5] that $(1 + \De)^{-i-1} F (1 + \De)^{-n+i + \ep}$ is of trace class for all $i \in \{-1,0,\ldots,n-1\}$ and all $\ep \in (0,1)$.
We recollect what we have proved thus far in the following:
\[t:exihom\] Suppose that the maps $a_1,\ldots,a_{2n-1},a_{2n} : \rr^{2n} \to M_N(\cc)$ satisfy the conditions in Assumption \[a:fct\]. Then the pair consisting of the closed unbounded operator $\dir_+$ and the bounded operator $A$ as defined in and satisfies the conditions in Assumption \[a:unb\] and Assumption \[a:tra\] for $n_0 = n$. In particular, there is a well-defined homological index in degree $m$, $\T{H-Ind}_m(T_\la)$ for each $\la \in (0,\infty)$ and each $m \geq n$.
The unbounded operators $V_1 \T{ and } V_2 : \T{Dom}(\dir_+) \to L^2(\rr^{2n}) \ot \cc^{2^{n-1}}$ turn out to be first order differential operators. Indeed, we have that $$\begin{split}
V_1 & = A \dir_- + \dir_+ A^* + A A^* \\
& = -i \sum_{j,k=1}^{2n-1} c_j c_k a_j \frac{\pa}{\pa x_k} - i \sum_{j,k=1}^{2n-1} c_k c_j \frac{\pa}{\pa x_k} a_j
- \big[ \frac{\pa}{\pa x_{2n}}, \sum_{j=1}^{2n-1} c_j a_j \big] + \sum_{j,k = 1}^{2n-1} c_j c_k a_j a_k \\
& = i \sum_{1 \leq j < k \leq 2n-1} c_j c_k \big( \frac{\pa a_j}{\pa x_k} - \frac{\pa a_k}{\pa x_j} -i [a_j,a_k]\big) -i \cd \sum_{j= 1}^{2n-1} \big( a_j \frac{\pa}{\pa x_j} + \frac{\pa}{\pa x_j} a_j + i a_j^2\big) + \frac{1}{2} F
\end{split}$$ And it then follows from the identity $F(\xi) = (V_1 - V_2)(\xi)$ for all $\xi \in \T{Dom}(\dir_-)$ that $$V_2 = -i \sum_{j= 1}^{2n-1} \big( a_j \frac{\pa}{\pa x_j} + \frac{\pa}{\pa x_j} a_j + i a_j^2\big)
- \frac{1}{2} F + i \sum_{1 \leq j < k \leq 2n-1} c_j c_k \big( \frac{\pa a_j}{\pa x_k} - \frac{\pa a_k}{\pa x_j} -i [a_j,a_k] \big).$$
\[n:difope\] Define the first order differential operator $$S := -i \sum_{j= 1}^{2n-1} \big( 2 a_j \frac{\pa}{\pa x_j} + \frac{\pa a_j}{\pa x_j} + i a_j^2\big) : H^1(\rr^{2n}) \ot \cc^{N \cd 2^{n-1}} \to L^2(\rr^{2n}) \ot \cc^{N \cd 2^{n-1}}.$$ Define the multiplication operators $$\begin{split}
F & := - 2 \cd \sum_{j=1}^{2n-1} c_j \frac{\pa a_j}{\pa x_{2n}} : L^2(\rr^{2n}) \ot \cc^{N \cd 2^{n-1}} \to L^2(\rr^{2n}) \ot \cc^{N \cd 2^{n-1}} \q \T{and} \\
G & := i \sum_{1 \leq j < k \leq 2n-1} c_j c_k \big( \frac{\pa a_j}{\pa x_k} - \frac{\pa a_k}{\pa x_j} -i [a_j,a_k] \big) \\
& \qqqq : L^2(\rr^{2n}) \ot \cc^{N \cd 2^{n-1}} \to L^2(\rr^{2n}) \ot \cc^{N \cd 2^{n-1}}.
\end{split}$$
Using the above notation, we have that $V_1 = S + F/2 + G$ and $V_2 = S - F/2 + G$.
It is important, before continuing, to relate the bounded operators $F$ and $G$ to the $U(N)$-connection used for constructing the perturbation of the flat Dirac operator $\dir$.
Since $a_{2n} = 0$, the connection $1$-form is given by $$\om := i \sum_{j=1}^{2n-1} a_j dx_j \in \Om^1\big(\rr^{2n},M_N(\cc)\big).$$ And it follows that the curvature operator is induced by the $2$-form $$\begin{split}
R := \om \we \om + d\om & = i \sum_{1 \leq j < k \leq 2n-1} \Big( \frac{\pa a_k}{\pa x_j} - \frac{\pa a_j}{\pa x_k} + i [a_j,a_k] \Big) dx_j \we dx_k \\
& \qqq -i \sum_{j = 1}^{2n-1} \frac{\pa a_j}{\pa x_{2n}} dx_j \we dx_{2n}.
\end{split}$$ Let $\pi_{2n} : \cc_{2n} \to \sL(\cc^{2^n})$ denote the representation of the Clifford algebra with $(2n)$-generators $e_1,\ldots,e_{2n}$ used for the definition of our Dirac-type operator $\C D$. The Clifford contraction of the curvature form can then be written as $$\label{eq:confg}
\begin{split}
\G K & := \sum_{1 \leq j < k \leq 2n} \pi_{2n}(e_j e_k) R\big(\frac{\pa}{\pa x_j},\frac{\pa}{\pa x_k}\big) \\
& = i \sum_{1 \leq j < k \leq 2n-1} \pi_{2n}(e_j e_k) \Big( \frac{\pa a_k}{\pa x_j} - \frac{\pa a_j}{\pa x_k} + i [a_j,a_k] \Big) \\
& \qqq -i \sum_{j = 1}^{2n-1} \pi_{2n}(e_j e_{2n}) \frac{\pa a_j}{\pa x_{2n}} \\
& = \ma{cc}{
F/2 - G & 0 \\
0 & -F/2 - G
}.
\end{split}$$ We may thus rewrite the differential operator $V := \ma{cc}{V_2 & 0 \\ 0 & V_1}$ as $V = S - \G K$.
[*The goal of the next Section is to provide a local formula for the anomaly (in degree $m \geq n$) of $\C D_+ = \dir_+ + A$ in terms of the above Clifford contraction $\G K$*]{}.
A local formula for the anomaly {#s:locano}
===============================
Throughout this section we will operate in the context of Section \[s:anodir\]. The conditions in Assumption \[a:fct\] on the $U(N)$-connection will in particular be in effect. We are then aiming to compute the anomaly of the perturbed Dirac operator $\C D_+ = \dir_+ + A$ as the integral of a $(2n)$-form associated to the Clifford contraction of the curvature form, that is, in terms of a polynomial in the operators $F$ and $G$.
Our first concern is to eliminate a substantial amount of terms in the general resolvent expansion of the homological index provided in Proposition \[p:degexp\]. To do this we start with a short preliminary on Clifford algebras.
Preliminaries on Clifford algebras {#ss:precli}
----------------------------------
We define irreducible representations of the Clifford algebra $\cc_k$ for all $k \in \nn$ recursively as follows. For $k = 1$, let $$\pi_1 : \cc_1 \to \sL(\cc) \q \pi : e_1 \mapsto 1.$$ For $k = 2m$, let $\pi_{2m} : \cc_{2m} \to \sL(\cc^{2^m})$ be defined by $$\pi_{2m} : e_i \mapsto \fork{ccc}{
\pi_{2m-1}(e_i) \ot \ma{cc}{0 & 1 \\ 1 & 0} & \T{for} & i \in \{1,\ldots,2m-1\} \\
1 \ot \ma{cc}{0 & i \\ -i & 0} & \T{for} & i = 2m.
}$$ For $k = 2m+1 > 1$, let $\pi_{2m+1} : \cc_{2m+1} \to \sL(\cc^{2^m})$ be defined by $$\pi_{2m+1} : e_i \mapsto \fork{ccc}{
\pi_{2m}(e_i) & \T{for} & i \in \{1,\ldots,2m\} \\
\ma{cc}{
1_{2^{m-1}} & 0 \\
0 & - 1_{2^{m-1}}
} & \T{for} & i = 2m+1.
}$$
Let $k \in \nn$. For each subset $I = \{i_1,\ldots,i_j\} \su \{1,\ldots,k\}$ with $i_1 < \ldots < i_j$, define $$e_I := e_{i_1} \clc e_{i_j} \in \cc_k \q e_{\emptyset} := 1 \in \cc_k.$$ The odd part of $\cc_k$ is the vector subspace $\cc_{\T{odd}} \su \cc_k$ defined by $$\cc_{\T{odd}} := \T{span}_{\cc}\big\{ e_{i_1} \clc e_{i_j} \, | \, 1 \leq i_1 < \ldots < i_j \leq k, \, j \, \T{ odd}\big\}.$$ The even part of $\cc_k$ is the vector subspace $\cc_{\T{ev}} \su \cc_k$ defined by $$\cc_{\T{ev}} := \T{span}_{\cc}\big\{ e_{i_1} \clc e_{i_j} \, | \, 1 \leq i_1 < \ldots < i_j \leq k, \, j \, \T{ even}\big\}.$$
\[l:tracli\] Let $m \in \nn$. Let $I \su \{1,\ldots,2m-1\}$, then $$\T{Tr}\big( \pi_{2m-1}(e_I) \big) = \fork{ccc}{
2^{m-1} & \T{for} & I = \emptyset \\
(-2i)^{m-1} & \T{for} & I = \{1,\ldots,2m-1\} \\
0 & \T{for} & I \neq \emptyset \, , \, \{1,\ldots,2m-1\}.
}$$
The fact that $\T{Tr}(\pi_{2m-1}(1)) = 2^{m-1}$ is obvious. We may thus suppose that $I \neq \emptyset$.
Suppose that the number of elements $j$ in $I = \{i_1,\ldots,i_j\}$ is even. Then $$\T{Tr}\big( \pi_{2m-1}(e_I) \big) = - \T{Tr}\big( \pi_{2m-1}(e_{i_j} \cd e_{i_1} \clc e_{i_{j-1}}) \big)
= - \T{Tr}\big( \pi_{2m-1}(e_I) \big).$$ This shows that $\T{Tr}\big( \pi_{2m-1}(e_I) \big) = 0$ in this case.
We may thus suppose that the number of elements $j$ in $I$ is odd.
Suppose that $j < 2m-1$. Choose an element $k \in \{1,\ldots,2m-1\}\sem J$. The endomorphism $\pi_{2m-1}(e_I) : \cc^{2^{m-1}} \to \cc^{2^{m-1}}$ then anticommutes with the selfadjoint unitary operator $\pi_{2m-1}(e_k)$. This implies that $\T{Tr}\big( \pi_{2m-1}(e_I) \big) = 0$.
We may thus suppose that $I = \{1,\ldots,2m-1\}$. The identity $\T{Tr}\big( \pi_{2m-1}(e_I)\big) = (-2i)^{m-1}$ then follows by induction. Indeed, for $m > 1$ we have that $$\begin{split}
\pi_{2m-1}(e_I) & = \pi_{2m-3}(e_1 \clc e_{2m-3}) \ot \ma{cc}{0 & 1 \\ 1 & 0} \cd \ma{cc}{0 & i \\ -i & 0} \cd \ma{cc}{1 & 0 \\ 0 & - 1} \\
& = \pi_{2m-3}(e_1 \clc e_{2m-3}) \ot \ma{cc}{-i & 0 \\ 0 & -i},
\end{split}$$ which implies that $\T{Tr}\big( \pi_{2m-1}(e_I) \big) = (-2i) \cd \pi_{2m-3}(e_1 \clc e_{2m-3})$.
Vanishing of lower Clifford terms
---------------------------------
Let $m \in \{n,n+1,\ldots\}$. Recall from Proposition \[p:degexp\] that the homological index of $T_\la := \la^{-1/2} \C D_+ (1+ \la^{-1} \C D_- \C D_+)$ in degree $m$ can be written as $$\T{H-Ind}_m(T_\la) = \la^m \cd \sum_{k=0}^\infty \sum_{J \in \C B_k} \T{Tr}_s\big( \om(J,\la) \big),$$ where the sum converges absolutely whenever $\la \geq C$ for an appropriate constant $C > 0$. The index sets are given by $\C B_k := \big\{ J = (j_1,\ldots,j_m) \in \nn_0^m \, | \, j_1 \plp j_m = k\big\}$ for each $k \in \nn_0$ whereas the bounded operator $\om(J,\la) = \ma{cc}{\om_2(J,\la) & 0 \\ 0 & \om_1(J,\la)}$ is defined in Notation \[n:omedeg\] for each $J \in \nn_0^m$ and each $\la > 0$.
Since the irreducible representation $\pi_{2n} : \cc_{2n} \to M_{2^n}(\cc)$ is an isomorphism we have a vector space decomposition $M_{2^n}(\cc) \cong \bop_{I \su \{1,\ldots,2n\}} \cc \pi_{2n}(e_I)$. In particular we have an idempotent $$\label{eq:cliide}
E_I : \sL(L^2(\rr^{2n})) \ot M_N(\cc) \ot M_{2^n}(\cc) \to \sL(L^2(\rr^{2n})) \ot M_N(\cc) \ot M_{2^n}(\cc)$$ with image, $\T{Im}(E_I) = \sL(L^2(\rr^{2n})) \ot M_N(\cc) \ot \cc \pi_{2n}(e_I)$, for each $I \su \{1,\ldots,2n\}$.
We shall see in this section that the “lower order terms” do not contribute to the homological index. More precisely, we will prove that the super trace of the terms $\om(J,\la)$ is trivial whenever $J \in \C B_k$ for some $k \in \{0,\ldots,n-1\}$ and $\la > 0$.
\[l:vollef\] Let $J \in \nn_0^m$ and let $\la > 0$. Then $$\T{Tr}_s\big(\om(J,\la)\big) = \T{Tr}_s\big( E_{\{1,\ldots,2n\}}(\om(J,\la)) \big).$$
Let us define the super trace $\T{Tr}_s : M_{2^n}(\cc) \to \cc$, $\T{Tr}_s : T \mapsto \T{Tr}\big( T \cd \pi_{2n+1}(e_{2n+1}) \big)$, where $\T{Tr} : M_{2^n}(\cc) \to \cc$ denotes the usual matrix trace.
We may then rewrite the super trace $\T{Tr}_s : \sL^1_s\big( L^2(\rr^{2n}) \ot \cc^N \ot \cc^{2^n} \big) \to \cc$ as the composition $\T{Tr} \ci (1 \ot 1 \ot \T{Tr}_s)$, where $\T{Tr} : \sL^1\big( L^2(\rr^{2n}) \ot \cc^N \big) \to \cc$ denotes the usual operator trace.
An application of Lemma \[l:tracli\] now proves the lemma.
Before we continue, we make a short digression on the form of the bounded operator $\om(J,\la)$, for $J \in \nn_0^m$ and $\la > 0$. Recall from the discussion after Notation \[n:difope\] that the first order differential operator $V := \ma{cc}{V_2 & 0 \\ 0 & V_1}$ can be written as $V = S - \G K$, where $\G K = \ma{cc}{F/2 - G & 0 \\ 0 & -F/2 - G}$ denotes the Clifford contraction of the curvature form associated with our fixed $U(N)$-connection.
Notice next that the Clifford algebra $\cc_{2n}$ is a filtered algebra $\{0\} \su C_0 \su \ldots \su C_{2n-1} \su C_{2n} = \cc_{2n}$. The subspace $C_i \su \cc_{2n}$ is defined by $$\label{eq:filcli}
C_i := \T{span}_{\cc}\big\{ e_I \, | \, I \su \{1,\ldots,2n\} \, , \, |I| \leq i\big\}$$ for each $i \in \{0,\ldots,2n\}$, where $|I|$ denotes the number of elements of a subset $I \su \{1,\ldots,2n\}$.
Let now $\la > 0$ and let us apply the notation $$V_\la := V \cd (\la + \De)^{-1} \q S_\la := S \cd (\la + \De)^{-1} \q \G K_\la := \G K \cd (\la + \De)^{-1}.$$ The bounded operator $S_\la$ then lies in the subspace $\sL\big( L^2(\rr^{2n}) \ot \cc^N\big) \ot \cc 1_{2^n}$ and the bounded operator $\G K_\la$ lies in the subspace $\sL\big( L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_2)$ of $\sL\big( L^2(\rr^{2n}) \ot \cc^{N\cd 2^n} \big)$.
Let $J = (j_1,\ldots,j_m) \in \nn_0^m$. The above discussion then shows that $$\om(J,\la) = (\la + \De)^{-1} V_\la^{j_1} \clc (\la + \De)^{-1} V_\la^{j_m} \in \sL\big( L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_{2\cd \C B(J)}),$$ where we put $C_i := \cc_{2n}$ whenever $i \geq 2n$. Recall here that $\C B(J) = j_1 \plp j_m$.
A combination of these observations and Lemma \[l:vollef\] implies the following result. It provides a significant reduction of terms in the resolvent expansion for the homological index.
\[p:low\] Suppose that the maps $a_1,\ldots,a_{2n} : \rr^{2n} \to M_N(\cc)$ satisfy the conditions in Assumption \[a:fct\]. Then $$\T{Tr}_s\big( \om(J,\la) \big) = 0,$$ for all $\la > 0$ and all $J = (j_1,\ldots,j_m) \in \nn_0^m$ with $j_1 \plp j_m \leq n-1$.
Vanishing of higher order terms
-------------------------------
In this subsection we will continue reducing the number of terms which contribute to the anomaly of the Dirac-type operator $\C D$. We are thus interested in comparing the super trace of the operators $\la^m \cd \om(J,\la)$ with the size of the scaling parameter $\la > 0$ whenever $J \in \C B_k := \C B^{-1}(\{k\})$ for some $k \in \{n + 1,\ldots,2n - 1\}$.
Let $\C H := L^2(\rr^{2n}) \ot \cc^N \ot \cc^{2^{n-1}}$ and recall that $\sL^1_s(\C H) \su \sL(\C H \op \C H)$ denotes the Banach space of bounded operators $$T = \ma{cc}{
T_{11} & T_{12} \\
T_{21} & T_{22}
}$$ with $T_{11} -T_{22} \in \sL^1(\C H)$. The norm is given by $\|T\|_1^s := \|T\| + \|T_{11} - T_{22}\|_1$.
We will apply the notation $P_+ : \C H \op \C H \to \C H$ and $P_- : \C H \op \C H \to \C H$ for the orthogonal projections onto the first and the second component in $\C H \op \C H$, respectively.
Notice that the terms of the form $\la^m \om(J,\la)$ converges to zero in the Banach space $\sL^1_s(\C H)$ as $\la \to \infty$ whenever $J \in \C B_k := \C B^{-1}(\{k\})$ for some $k \in \{2n,2n+1,\ldots\}$. This is a consequence of Lemma \[l:omeest\].
In order to obtain the desired estimates on the terms $\la^m \cd \om(J,\la)$ for $J \in \C B_k$ for $k \in \{n+1,\ldots,2n-1\}$ we need to consider a more detailed expansion of the bounded operators $\om(J,\la)$. To this end we introduce the following:
Let $j \in \nn_0$. For any sequence $L = (l_1,\ldots,l_j) \in \{0,1\}^j$, define the bounded operator $$\ga_\la(L) := \ga_\la(l_1) \clc \ga_\la(l_j) \q \ga_\la(0) = S \cd (\la + \De)^{-1} \, , \, \ga_\la(1) = - \G K \cd (\la + \De)^{-1}.$$ When $j = 0$, we identify the set $\{0,1\}^j$ with a basepoint $*$ and put $\ga_\la(*) := 1$.
For any sequence $L = (l_1,\ldots,l_j) \in \{0,1\}^j$, let $D(L) = l_1 \plp l_j$. When $j = 0$, we put $D(*) = 0$.
For any $J = (j_1,\ldots,j_m) \in \nn_0^m$ and any $\la > 0$ we then have that $$\label{eq:omeexp}
\begin{split}
\om(J,\la) & = (\la + \De)^{-1} \big(\ga_\la(0) + \ga_\la(1) \big)^{j_1} \clc (\la + \De)^{-1} \big(\ga_\la(0) + \ga_\la(1) \big)^{j_m} \\
& = (\la + \De)^{-1} \Big( \sum_{L_1 \in \{0,1\}^{j_1}} \ga_\la(L_1) \Big) \clc (\la + \De)^{-1} \Big( \sum_{L_m \in \{0,1\}^{j_m}} \ga_\la(L_m) \Big).
\end{split}$$
In the next lemmas we will therefore prove a few estimates on operators of the type $\ga_\la(L)$.
\[l:actfin\] Let $i \in \zz$, let $j \in \nn_0$ and let $L \in \{0,1\}^j$. Then $$\big\| \si_\la^i\big( \ga_\la(L) \big) \big\| = O(\la^{-(D(L) + j)/2})$$ as $\la \to \infty$.
Since $\|(\la + \De)^{-1/2}\| = O(\la^{-1/2})$ as $\la \to \infty$, it is enough to show that $$\big\| \si_\la^i\big(S \cd (\la + \De)^{-1/2}\big) \big\| = O(1) \q \T{and} \q \big\|\si_\la^i(\G K) \big\| = O(1)$$ as $\la \to \infty$. But this follows easily from Lemma \[l:disanaest\].
\[l:galest\] Let $j \in \nn_0$ and let $L = (l_1,\ldots,l_j) \in \{0,1\}^j$. Then $$(\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \in \sL^1_s(\C H )$$ for all $i \in \{1,\ldots,m\}$ and we have the norm estimate $$\big\|(\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i}\big\|_1^s = O(\la^{n-m+1/4 - (D(L) + j)/2})$$ as $\la \to \infty$.
Let $i \in \{1,\ldots,m\}$.
Notice first that $$\big\| (\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \big\| = O(\la^{-m - (D(L)+j)/2})$$ as $\la \to \infty$ by an application of Lemma \[l:actfin\].
Suppose that $D(L) = 0$. We then have that $$(\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} = (\la + \De)^{-i} \cd S_\la^j \cd (\la + \De)^{-m+i}$$ This shows that $(\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \in \sL^1_s(\C H)$ with $$\begin{split}
\big\| (\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \big\|_1^s
& = \big\| (\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \big\| \\
& = O(\la^{-m - (D(L)+j)/2})
\end{split}$$ as $\la \to \infty$.
Suppose that $D(L) \geq 1$. Choose an $r \in \{1,\ldots,j\}$ such that $L_i = (l_1,\ldots,l_{r-1},1,l_{r+1},\ldots,l_j)$. We then have that $$\begin{split}
& (\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \\
& \q = - \si^{-i}_\la\big( \ga_\la(l_1) \clc \ga_\la(l_{r-1}) \big) \cd (\la + \De)^{-i} \G K (\la + \De)^{-m-1 + i} \\
& \qqq \cd \si^{m-i}_\la \big( \ga_\la(l_{r+1}) \clc \ga_\la(l_j) \big).
\end{split}$$ This shows that $(\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \in \sL^1_s(\C H)$. To prove the desired norm estimate, we recall from (with $\ep = 3/4$) that $$\big\| (\la + \De)^{-i} \G K (\la + \De)^{-m-1 + i} \big\|_1^s = O(\la^{n-m-3/4})$$ as $\la \to \infty$. An application of Lemma \[l:actfin\] now yields that $$\big\| (\la + \De)^{-i} \ga_\la(L) (\la + \De)^{-m+i} \big\|_1^s = O(\la^{n-m+1/4 - (D(L)+j)/2}).$$
This proves the lemma.
\[l:galestfin\] Let $J = (j_1,\ldots,j_m) \in \nn_0^m$ and let $L_1 \in \{0,1\}^{j_1}, \ldots, L_m \in \{0,1\}^{j_m}$. Then $$(\la + \De)^{-1} \ga_\la(L_1) \clc (\la + \De)^{-1} \ga_\la(L_m) \in \sL^1_s(\C H)$$ and we have the norm estimate $$\big\| (\la + \De)^{-1} \ga_\la(L_1) \clc (\la + \De)^{-1} \ga_\la(L_m) \big\|_1^s = O\big(\la^{n-m+1/4 - (D(L) + \C B(J))/2}\big)$$ as $\la \to \infty$, where $D(L) := D(L_1) \plp D(L_m)$ and $\C B(J) := j_1 \plp j_m$.
The proof follows the same pattern as the proof of Lemma \[l:galest\]. It is therefore left to the reader.
Recall now that $\cc_{2n}$ is a filtered algebra with filtration $\{0\} \su C_0 \su \ldots \su C_{2n-1} \su C_{2n} = \cc_{2n}$ given by the subspaces defined in . Notice that the Clifford contraction $\G K$ lies in $\sL\big(L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_2)$ and that the bounded operator $S \cd (\la + \De)^{-1}$ lies in $\sL\big(L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_0)$.
Let $j \in \nn_0$ and let $L = (l_1,\ldots,l_j) \in \{0,1\}^j$. It follows from the above observation that $$\ga_\la(L) \in \sL\big(L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_{2 \cd D(L)}),$$ where $D(L) := l_1 \plp l_j$.
We are now ready to prove the main result of this section.
\[p:hig\] Suppose that the maps $a_1,\ldots,a_{2n} : \rr^{2n} \to M_N(\cc)$ satisfy the conditions in Assumption \[a:fct\]. Let $k \in \{n+1,\ldots,2n-1\}$ and let $J = (j_1,\ldots,j_m) \in \C B_k$. Then $$\lim_{\la \to \infty} \la^m \cd \T{Tr}_s\big( \om(J,\la) \big) = 0.$$
Let $L_1 \in \{0,1\}^{j_1}, \ldots, L_m \in \{0,1\}^{j_m}$.
By Lemma \[l:galestfin\] and the identity in it is enough to show that $$\lim_{\la \to \infty} \la^m \T{Tr}_s\big( (\la + \De)^{-1} \ga_\la(L_1) \clc (\la + \De)^{-1} \ga_\la(L_m) \big) = 0.$$
As in Lemma \[l:vollef\] we have that $$\begin{split}
& \T{Tr}_s\big( (\la + \De)^{-1} \ga_\la(L_1) \clc (\la + \De)^{-1} \ga_\la(L_m) \big) \\
& \q = \T{Tr}_s\Big( E_{\{1,\ldots,2n\}} \big( (\la + \De)^{-1} \ga_\la(L_1) \clc (\la + \De)^{-1} \ga_\la(L_m) \big)\Big),
\end{split}$$ where $E_{\{1,\ldots,2n\}} : \sL\big(L^2(\rr^{2n}) \ot \cc^{N \cd 2^n}\big) \to \sL\big(L^2(\rr^{2n}) \ot \cc^{N \cd 2^n}\big)$ is the idempotent associated with the vector space decomposition $\cc_{2n} \cong \cc e_{\{1,\ldots,2n\}} \op C_{2n-1}$ of the Clifford algebra.
Notice that the operator $(\la + \De)^{-1} \ga_\la(L_1) \clc (\la + \De)^{-1} \ga_\la(L_m)$ lies in $\sL\big(L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_{2 \cd D(L)})$. We may thus suppose that $D(L) = D(L_1) \plp D(L_m) \geq n$.
By Lemma \[l:galestfin\] we can choose constants $C,K > 0$ such that $$\begin{split}
\big| \la^m \T{Tr}_s\big( (\la + \De)^{-1} \ga_\la(L_1) \clc (\la + \De)^{-1} \ga_\la(L_m) \big) \big|
& \leq K \cd \la^{n + 1/4 - (D(L) + \C B(J))/2} \\
& \leq K \cd \la^{-1/4},
\end{split}$$ for all $\la \geq C$, where we have used that $D(L) \geq n$ and $\C B(J) \geq n+1$.
This proves the proposition.
A local formula for the anomaly {#a-local-formula-for-the-anomaly}
-------------------------------
Let $m \in \{n,n+1,\ldots\}$. Throughout this section the conditions in Assumption \[a:fct\] on the Dirac type operator $\C D$ introduced in Section \[s:anodir\] will be in effect. Let us start by recollecting what we have proved thus far. It follows from Proposition \[p:verhig\], Proposition \[p:hig\] and Proposition \[p:low\] that the anomaly in degree $m$ of the Dirac type operator $\C D$ exists if and only if the limit $$\lim_{\la \to \infty} \la^m \cd \sum_{J \in \C B_n} \T{Tr}_s\big( \om(J,\la) \big)$$ exists and in this case, we have that $$\T{Ano}_m(\C D) = \lim_{\la \to \infty} \la^m \cd \sum_{J \in \C B_n} \T{Tr}_s\big( \om(J,\la) \big).$$ Recall here that $$\om(J,\la) := \ma{cc}{\om_2(J,\la) & 0 \\ 0 & \om_1(J,\la)} \in \sL(\C H)$$ where the bounded operators $\om_2(J,\la)$ and $\om_1(J,\la)$ have been defined in Notation \[n:omedeg\] for all $J \in \nn_0^m$ and all $\la > 0$. The index set $\C B_n$ consists of all $J \in \nn_0^m$ with $j_1 \plp j_m = n$.
\[l:onldeg\] The anomaly of $\C D$ in degree $m$ exists if and only if the limit $$\lim_{\la \to \infty} \la^m \cd \T{Tr}_s\big( \G K^n \cd (\la + \De)^{-n-m} \big)$$ exists and in this case $$\T{Ano}_m(\C D) = {m + n-1 \choose m-1} \cd (-1)^n \cd \lim_{\la \to \infty} \la^m \cd \T{Tr}_s\big( \G K^n \cd (\la + \De)^{-n-m} \big).$$
Let $J = (j_1,\ldots,j_m) \in \C B_n$ and let $\la > 0$. Recall from Lemma \[l:vollef\] that $$\T{Tr}_s\big( \om(J,\la) \big) = \T{Tr}_s\big( E_{\{1,\ldots,2n\}}\big( \om(J,\la) \big) \big),$$ where the idempotent $E_{\{1,\ldots,2n\}} : \sL(\C H) \to \sL(\C H)$ was introduced in .
Notice also that $\G K_\la := \G K (\la + \De)^{-1} \in \sL\big(L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_2)$ and $S_\la := S (\la + \De)^{-1} \in \sL\big(L^2(\rr^{2n}) \ot \cc^N\big) \ot \pi_{2n}(C_0)$, where the subspaces $C_i \su \cc_{2n}$, $i \in \{0,\ldots,2n\}$ were introduced in .
We therefore have that $$\label{eq:idedegn}
\begin{split}
& E_{\{1,\ldots,2n\}}\big( \om(J,\la)\big)
= (-1)^n \cd E_{\{1,\ldots,2n\}}\big( (\la + \De)^{-1}\G K_\la^{j_1} \clc (\la + \De)^{-1} \G K_\la^{j_m} \big) \\
& \q = (-1)^n \cd E_{\{1,\ldots,2n\}}\Big( \si_\la^{-1}(\G K) \clc \si_\la^{-j_1}(\G K) \cd \si_\la^{-j_1 -2}(\G K) \clc \si_\la^{-j_1 - j_2 -1}(\G K) \\
& \qq \clc \si_\la^{-j_1-\ldots - j_{m-1} -m}(\G K) \clc \si_\la^{-j_1 -\ldots - j_m -m+1}(\G K)
\cd (\la + \De)^{-n-m} \Big).
\end{split}$$
Recall now from (with $\ep = 3/4$) that $\big\| (\la + \De)^{-n-m+1} \G K (\la + \De)^{-1}\big\|_1^s = O(\la^{1/4-m})$ as $\la \to \infty$. It therefore follows from and Lemma \[l:difact\] that $$\label{eq:remact}
\la^m \cd \big\| (-1)^n E_{\{1,\ldots,2n\}}\big( (\la + \De)^{-n-m+1} \G K^n (\la + \De)^{-1}\big) - E_{\{1,\ldots,2n\}}\big( \om(J,\la) \big) \big\|_1^s = O(\la^{-1/4})$$ as $\la \to \infty$. An application of the cyclic property of the operator trace then yields that $$\lim_{\la \to \infty} \la^m \Big( \T{Tr}_s\big( \om(J,\la)\big) - (-1)^n \T{Tr}_s\big( \G K^n (\la + \De)^{-n-m} \big)\Big) = 0.$$
The discussion in the beginning of this section now entails the result of this lemma since the number of elements in the index set $\C B_n \su \nn_0^m$ is ${m+n-1 \choose m-1}$.
We will consider the matrix valued $(2n)$-form $\sA \in \Om^{2n}(\rr^{2n},M_N(\cc))$ given by $$\label{eq:fordef}
\sA : x \mapsto \si_{2n}(x)(\G K^n(x)) \q \T{for all } \, x \in \rr^{2n}$$ where $\si_{2n}(x) : M_{2^n}(\cc) \ot M_N(\cc) \to \La^{2n}_x(\rr^{2n}) \ot M_N(\cc)$ is defined by $$\si_{2n}(x)(\pi_{2n}(e_I) \ot M) := \fork{ccc}{
0 & \T{for} & I \neq \{1,\ldots,2n\} \\
(dx_1 \wlw dx_{2n})(x) \ot M & \T{for} & I = \{1,\ldots,2n\}.
}$$ for all $x \in \rr^{2n}$ and all $M \in M_N(\cc)$. Recall here that $\G K : \rr^{2n} \to M_N(\cc) \ot M_{2^n}(\cc)$ is the *Clifford contraction* of the curvature form associated with the $U(N)$-connection which determines our Dirac type operator, see the discussion in the end of Section \[s:anodir\]. The identity in shows that this Clifford contraction can be expressed in a simple way in terms of the bounded operators $F$ and $G$ which we introduced in Notation \[n:difope\].
The following theorem is the main result of this paper. We refer to the beginning of Section \[s:anodir\] for a definition of the Dirac type operator $\C D : H^1(\rr^{2n}) \ot \cc^{2^n \cd N} \to L^2(\rr^{2n}) \ot \cc^{2^n \cd N}$.
\[t:locano\] Let $a_1,\ldots,a_{2n} : \rr^{2n} \to M_N(\cc)$ be maps which satisfy the conditions in Assumption \[a:fct\]. Let $m \in \{n,n+1,\ldots\}$. Then the anomaly in degree $m$ associated with the Dirac type operator $\C D := \dir + \ma{cc}{0 & A^* \\ A & 0}$ exists and is given by the integral $$\T{Ano}_m(\C D) = \frac{(-1)^n}{(2 \pi i)^n \cd n!} \cd \int \T{Tr}(\sA)$$ of the $(2n)$-form $\sA$ defined in .
By Lemma \[l:onldeg\] it is enough to show that $$\begin{split}
& (-1)^n \cd {m+n-1 \choose m-1} \cd \lim_{\la \to \infty} \big( \la^m \cd \T{Tr}_s\big( \G K^n \cd (\la + \De)^{-n-m} \big) \big) \\
& \q = \frac{(-1)^n}{ (2 \pi i )^n \cd n!} \cd \int \T{Tr}(\sA).
\end{split}$$
Let $g \in C^\infty_c(\rr^{2n},M_N(\cc))$ denote the unique smooth compactly supported function such that $\sA = g \cd dx_1 \wlw dx_{2n}$. Let $\la > 0$. We then have that $$\begin{split}
\T{Tr}_s\big( \G K^n \cd (\la + \De)^{-n-m} \big) & = \T{Tr}_s\big( E_{\{1,\ldots,2n\}}(\G K^n) \cd (\la + \De)^{-n-m} \big) \\
& = \T{Tr}(g \cd (\la + \De)^{-n-m}) \cd (-2i)^n,
\end{split}$$ since the super trace of the Clifford matrix $\pi_{2n}(e_{\{1,\ldots,2n\}}) \in M_{2^n}(\cc)$ is the number $(-2i)^n$, see Lemma \[l:tracli\].
Let $j \in \{1,\ldots,N\}$ and let $g_j : \rr^{2n} \to \rr$ denote the matrix entry in position $(j,j)$ of the self adjoint element $g \in M_N\big(C^\infty_c(\rr^{2n})\big)$. The bounded operator $$g_j \cd (\la + \De)^{-n-m} : L^2(\rr^{2n}) \to L^2(\rr^{2n})$$ is unitarily equivalent (by the Fourier transform) to the integral operator $T_{K_j} : L^2(\rr^{2n}) \to L^2(\rr^{2n})$ with kernel $K_j \in C(\rr^{2n} \ti \rr^{2n}) \cap L^2(\rr^{2n} \ti \rr^{2n})$ given by $$K_j(x,y) = (2 \pi)^{-n} \cd \C F(g_j)(x-y) \cd (\la + \sum_{k = 1}^{2n} y_k^2)^{-n-m}.$$ Here $\C F(g_j)$ denotes the Fourier transform of $g_j \in C_c^\infty(\rr^{2n})$.
Since $g_j \cd (\la + \De)^{-n-m}$ is of trace class it follows (see for example [@GoKr:LNO Corollary 10.2]) that $$\begin{split}
& \T{Tr}\big( g_j \cd (\la + \De)^{-n-m} \big) = \int_{\rr^{2n}} K_j(x,x) dx \\
& \q = (2 \pi)^{-2n} \cd \int_{\rr^{2n}} g_j dx \cd \int_{\rr^{2n}} (\la + \sum_{k=1}^{2n} x_k^2)^{-n-m} dx \\
& \q = (2 \pi)^{-2n} \cd \int_{\rr^{2n}} g_j dx \cd \T{Vol}(S^{2n-1}) \cd \int_0^\infty r^{2n-1} (\la + r^2)^{-n-m} dr \\
& \q = (2 \pi)^{-2n} \cd \int_{\rr^{2n}} g_j dx \cd \T{Vol}(S^{2n-1}) \cd \la^{-m} \cd \int_0^\infty r^{2n-1} (1 + r^2)^{-n-m} dr \\
& \q = (2 \pi)^{-2n} \cd \int_{\rr^{2n}} g_j dx \cd \T{Vol}(S^{2n-1}) \cd \la^{-m} \cd \frac{1}{2} \cd \frac{(n-1)! \cd (m-1)!}{(m+n-1)!},
\end{split}$$ where $\T{Vol}(S^{2n-1}) = 2 \cd \pi^n /(n-1)!$ is the volume of the unit sphere in $\rr^{2n}$.
It follows from the above computations that $$\begin{split}
& (-1)^n \cd {m+n-1 \choose m-1} \cd \lim_{\la \to \infty} \big( \la^m \cd \T{Tr}_s\big( \G K^n \cd (\la + \De)^{-n-m} \big) \big) \\
& \q = (-1)^n \cd {m+n-1 \choose m-1} \cd (-2i)^n \cd \frac{1}{(4\pi)^n} \cd \frac{(m-1)!}{(m+n-1)!} \cd \int \T{Tr}(\sA) \\
& \q = \frac{(-1)^n}{(2 \pi i)^n \cd n!} \cd \int \T{Tr}(\sA).
\end{split}$$ This proves the theorem.
It is worthwhile to notice that the anomaly of the Dirac type operator $\C D$ does *not* depend on the degree $m \in \{n,n+1,\ldots\}$. This is can be seen immediately from the above theorem.
Non-triviality of the anomaly \[nont\]
======================================
The aim of this section is to demonstrate that the anomaly of the Dirac type operator $\C D : H^1(\rr^{2n}) \ot \cc^{2^n} \to L^2(\rr^{2n}) \ot \cc^{2^n}$ associated with a $U(1)$-connection can be non-trivial. We shall thus assume that the functions $a_1,\ldots,a_{2n} : \rr^{2n} \to \rr$ satisfy the conditions in Assumption \[a:fct\].
Recall from Theorem \[t:locano\] that the anomaly can be computed as the integral $$\T{Ano}_m(\C D) = \frac{(-1)^n}{(2 \pi i)^n \cd n!} \cd \int \sA \q m \in \{n,n+1,\ldots\},$$ where the compactly supported $(2n)$-form $\sA \in \Om^{2n}_c(\rr^{2n})$ was defined in .
*Suppose first that $n = 1$*. The $2$-form $\sA \in \Om^2_c(\rr^2)$ is then simply the curvature form of the $U(1)$-connection used for constructing the Dirac type operator $\C D$. See the end of Section \[s:anodir\]. This curvature form is given by $$d\om = -i \frac{\pa a_1}{\pa x_2} dx_1 \we dx_2$$ and the local formula can then be rewritten as follows: $$\T{Ano}_m(\C D) = \frac{1}{2\pi} \int \frac{\pa a_1}{\pa x_2} dx_1 \we dx_2 \q m \in \nn$$
Let now $\al, \be \in \rr$ and choose a smooth function $h : \rr \to \rr$ such that the derivative $\frac{dh}{dt} : \rr \to \rr$ has compact support and such that $\lim_{t \to \infty} h(t) = \be$ and $\lim_{t \to -\infty}h(t) = \al$. Choose a smooth compactly supported function $\phi : \rr \to [0,\infty)$.
Suppose then that $a_1 : \rr^2 \to \rr$ is defined by $a_1 : (x_1,x_2) \mapsto h(x_2) \cd \phi(x_1)$ for all $(x_1,x_2) \in \rr^2$. We then have that $$\begin{split}
\T{Ano}_m(\C D) = \frac{1}{2\pi} \cd \int_{-\infty}^\infty \frac{dh}{dt} dt \cd \int_{-\infty}^\infty \phi(x) dx
= \frac{(\be - \al)}{2\pi} \cd \int_{-\infty}^\infty \phi(x) dx,
\end{split}$$ for all $m \in \nn$.
*Suppose now that $n = 2$*. In this case, the curvature form is given by $d\om = \sum_{1 \leq i < j \leq 4} g_{i,j} dx_i \we dx_j$, where the coefficient functions are defined by $$g_{i,j} := i \cd \Big( \frac{\pa a_j}{\pa x_i} - \frac{\pa a_i}{\pa x_j} \Big) \in C^\infty_b(\rr^{2n}) \q 1 \leq i < j \leq 4.$$ The compactly supported $4$-form $\sA = \si_4(\G K^2) \in \Om^4_c(\rr^4)$ can therefore be written as $$\sA = 2 \cd (g_{1,2} \cd g_{3,4} - g_{1,3} \cd g_{2,4} + g_{1,4} \cd g_{2,3}) dx_1 \wlw dx_4.$$ And the local formula for the anomaly yields the identity $$\T{Ano}_m(\C D) = -\frac{1}{4 \pi^2} \cd \int \big( g_{1,2} \cd g_{3,4} - g_{1,3} \cd g_{2,4} + g_{1,4} \cd g_{2,3} \big) dx_1 \wlw dx_4,$$ for all $m \in \{2,3,\ldots\}$.
Let $h : \rr \to \rr$ be as above and let $\phi_1,\phi_3 : \rr^3 \to \rr$ be smooth compactly supported functions.
Suppose that $a_4 = a_2 = 0$ and that $a_1, a_3 : \rr^4 \to \rr$ are defined by $$\begin{split}
& a_1 : (x_1,\ldots,x_4) \mapsto h(x_4) \cd \phi_1(x_1,x_2,x_3) \q \T{and} \\
& a_3 : (x_1,\ldots,x_4) \mapsto h(x_4) \cd \phi_3(x_1,x_2,x_3)
\end{split}$$ for all $(x_1,\ldots,x_4) \in \rr^4$. Let $m \in \{2,3,\ldots\}$. The anomaly is then given by $$\begin{split}
\T{Ano}_m(\C D)
& = -\frac{1}{4 \pi^2} \cd \int h \cd \frac{dh}{dt} \, dt
\cd \int \Big( \phi_1 \cd \frac{\pa \phi_3}{\pa x_2} - \frac{\pa \phi_1}{\pa x_2} \cd \phi_3 \Big) dx_1 dx_2 dx_3 \\
& = -\frac{(\be^2 - \al^2)}{8 \pi^2} \cd \int \Big( \frac{\pa (\phi_1 \cd \phi_3)}{\pa x_2} - 2 \cd \frac{\pa \phi_1}{\pa x_2} \cd \phi_3 \Big) dx_1 dx_2 dx_3 \\
& = \frac{(\be^2 - \al^2)}{4 \pi^2} \cd \int \frac{\pa \phi_1}{\pa x_2} \cd \phi_3 dx_1 dx_2 dx_3,
\end{split}$$ where the integral $\int \frac{\pa (\phi_1 \phi_3)}{\pa x_2} dx_1 dx_2 dx_3$ is trivial since $\phi_1 \cd \phi_3$ has compact support.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[<span style="font-variant:small-caps;">BGG[[$^{+}$]{}]{}87</span>]{}
<span style="font-variant:small-caps;">D. Boll[é]{}</span>, <span style="font-variant:small-caps;">F. Gesztesy</span>, <span style="font-variant:small-caps;">H. Grosse</span>, <span style="font-variant:small-caps;">W. Schweiger</span>, and <span style="font-variant:small-caps;">B. Simon</span>, *Witten index, axial anomaly, and [K]{}reĭn’s spectral shift function in supersymmetric quantum mechanics*, J. Math. Phys. **28** (1987), no. 7, 1512–1525.
<span style="font-variant:small-caps;">B. Blackadar</span> and <span style="font-variant:small-caps;">J. Cuntz</span>, *Differential [B]{}anach algebra norms and smooth subalgebras of [$C\sp *$]{}-algebras*, J. Operator Theory **26** (1991), no. 2, 255–282.
<span style="font-variant:small-caps;">A. Carey</span> and <span style="font-variant:small-caps;">J. Kaad</span>, *Topological invariance of the homological index*, `arXiv:1402.0475 [math.KT]`.
<span style="font-variant:small-caps;">C. Callias</span>, *Axial anomalies and index theorems on open spaces*, Comm. Math. Phys. **62** (1978), no. 3, 213–234.
<span style="font-variant:small-caps;">R. W. Carey</span> and <span style="font-variant:small-caps;">J. D. Pincus</span>, *Index theory for operator ranges and geometric measure theory*, Geometric measure theory and the calculus of variations ([A]{}rcata, [C]{}alif., 1984), Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 149–161.
<span style="font-variant:small-caps;">A. Carey</span>, <span style="font-variant:small-caps;">F. Gesztesy</span>, <span style="font-variant:small-caps;">D. Potapov</span>, <span style="font-variant:small-caps;">F. Sukochev</span>, and <span style="font-variant:small-caps;">Y. Tomilov</span>, *On the Witten index in terms of spectral shift functions*, `arXiv:1404.0740 [math.SP]`.
<span style="font-variant:small-caps;">A. Connes</span> and <span style="font-variant:small-caps;">H. Moscovici</span>, *The local index formula in noncommutative geometry*, Geom. Funct. Anal. **5** (1995), no. 2, 174–243.
F.Gesztesy, Yu. Latushkin, K. Makarov, F. Sukochev, Yu. Tomilov, Yuri [*The index formula and the spectral shift function for relatively trace class perturbations,*]{} Adv. Math. [**227**]{} (2011), no. 1, 319–420.
<span style="font-variant:small-caps;">F. Gesztesy</span> and <span style="font-variant:small-caps;">B. Simon</span>, *Topological invariance of the [W]{}itten index*, J. Funct. Anal. **79** (1988), no. 1, 91–102.
<span style="font-variant:small-caps;">I. C. Gohberg</span> and <span style="font-variant:small-caps;">M. G. Kre[ĭ]{}n</span>, *Introduction to the theory of linear nonselfadjoint operators*, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969.
<span style="font-variant:small-caps;">J.-L. Loday</span>, *Cyclic homology*, second ed., Grundlehren der Mathematischen Wissenschaften \[Fundamental Principles of Mathematical Sciences\], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by Mar[í]{}a O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili.
<span style="font-variant:small-caps;">W. Rudin</span>, *Functional analysis*, McGraw-Hill Book Co., New York, 1973, McGraw-Hill Series in Higher Mathematics.
<span style="font-variant:small-caps;">B. Simon</span>, *Trace ideals and their applications*, second ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005.
<span style="font-variant:small-caps;">E. Witten</span>, *Constraints on supersymmetry breaking*, Nuclear Phys. B **202** (1982), no. 2, 253–316.
[^1]: All authors thank the referees for comments that have improved the exposition. The first author thanks the Alexander von Humboldt Stiftung and colleagues at the University of Münster and acknowledges the support of the Australian Research Council. The third author is supported by the Fondation Sciences Mathématiques de Paris (FSMP) and by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098). All authors are very appreciative of the support offered by the Erwin Schrödinger Institute where much of this research was carried out. We are also grateful for the advice and wisdom of Fritz Gesztesy while this investigation was proceeding.
[^2]: It is termed the Witten index in [@GeSi:TIW] but in fact Witten considered a scaling limit of a generalised McKean-Singer formula, that is, an expression in terms of heat semigroups, [@Wi:CSB]. It was Gesztesy-Simon who discovered the connection between Witten’s idea and the spectral shift function and hence to Carey-Pincus.
|
---
abstract: 'We interpret symplectic geometry as certain sheaf theory by constructing a sheaf of curved $A_\infty$ algebras which in some sense plays the role of a “structure sheaf" for symplectic manifolds. An interesting feature of this “structure sheaf" is that the symplectic form itself is part of its curvature term. Using this interpretation homological mirror symmetry can be understood by well-known duality theories in mathematics: Koszul duality or Fourier-Mukai transform. In this paper we perform the above constructions over a small open subset inside the smooth locus of a Lagrangian torus fibration. In a subsequent work we shall use the language of derived geometry to obtain a global theory over the whole smooth locus. However we do not know how to extend this construction to the singular locus. As an application of the local theory we prove a version of homological mirror symmetry between a toric symplectic manifold and its Landau-Ginzburg mirror.'
author:
- 'Junwu Tu[^1]'
title: ' Homological mirror symmetry and Fourier-Mukai transform '
---
Introduction
============
#### Backgrounds and histories.
Homological mirror symmetry conjecture was proposed by M. Kontsevich [@Kont] in an address to the 1994 International Congress of Mathematicians, aiming to give a mathematical framework to understand the mirror phenomenon originated from physics. Roughly speaking this conjecture predicts a quasi-equivalence between two $A_\infty$ triangulated categories ${\mathsf{Fuk}}(M)$ and ${{\mathsf D}_{\mathsf{coh}}^b}(M^{{\scriptscriptstyle\vee}})$ naturally associated to a symplectic manifold $M$ and a complex manifold $M^{{\scriptscriptstyle\vee}}$. Note that despite of the notation, the mirror manifold $M^{{\scriptscriptstyle\vee}}$ is not uniquely determined by $M$. After nearly two decades since Kontsevich’s proposal, his conjecture has been generalized and has been proven in a lot of deep and inspiring situations [@PZ], [@KS], [@Seidel], [@Sh], [@Seidel2], [@AKO], [@FLTZ], and definitely many more [^2].
In spite of our increasing knowledge of this conjecture, less is understood about the mathematical reason behind it. The first attempt towards a general mathematical theory to understand the mirror phenomenon is given by A. Strominger, S-T. Yau and E. Zaslow. In [@SYZ] they proposed a geometric picture to produce mirror pairs: mirror duality should arise between (special) Lagrangian torus fibrations and the associated dual fibrations. Indeed the SYZ proposal was successfully realized in the semiflat cases, showing beautiful interactions between special Lagrangians and stable holomorphic vector bundles [@LYZ], [@AP], [@CL]. In the toric Calabi-Yau case, see [@CLL]. To get interesting symplectic manifolds and complex manifolds (other than torus) one has to allow these fibrations to have singular fibers for which we refer to the massive work of M. Gross and B. Siebert on toric degenerations [@GS], [@GS2].
While this gives a nice theory to produce mirror pairs, it gives less hint on why mirror pairs produced from SYZ proposal should interchange symplectic geometry with complex geometry [^3]. The main purpose of the current paper is to suggest an answer to this question. Namely we show the mirror duality between symplectic geometry of a Lagrangian torus fibration and complex geometry of its dual fibration is in fact a well-known duality in mathematics: Koszul duality or its global version Fourier-Mukai transform.
The current paper grew out of understanding an algebraic framework for K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono’s series of papers [@FOOOtoric],[@FOOOtoric2],[@FOOOtoric3]. There are also inspiring works of Fukaya [@Fukaya2] and Seidel [@Seidel3].
#### An example.
We begin with a simple example illustrating the main ideas. Let ${\mathbb{R}}$ be endowed with a linear coordinate $x$, and let ${\mathbb{R}}^{{\scriptscriptstyle\vee}}$ be endowed with the dual coordinate $y^{{\scriptscriptstyle\vee}}$ (this choice of notation will be clear later). The cotangent bundle $T^*({\mathbb{R}})={\mathbb{R}}\times {\mathbb{R}}^{{\scriptscriptstyle\vee}}$ has a canonical symplectic form $\omega:=dx\wedge dy^{{\scriptscriptstyle\vee}}$. To have a Lagrangian torus fibration we consider the quotient space $M:={\mathbb{R}}\times ({\mathbb{R}}^{{\scriptscriptstyle\vee}}/{\mathbb{Z}}^{{\scriptscriptstyle\vee}})$. Since $\omega$ is translation invariant, it descends to a symplectic form on $M$. The projection map $\pi: M{\rightarrow}{\mathbb{R}}$ onto the first component defines a Lagrangian torus fibration.
Consider a complex vector bundle over the base ${\mathbb{R}}$ whose fiber over a point $u\in {\mathbb{R}}$ is the cohomology group $H^*(\pi^{-1}(u),{\mathbb{C}})$. Denote by ${{\mathscr H}}$ the corresponding sheaf of $C^\infty$-sections. It is well-known that the sheaf ${{\mathscr H}}$ is in fact a D-module endowed with the Gauss-Manin connection $\nabla$. Moreover the cup product on cohomology defines an algebra structure on ${{\mathscr H}}$ which is $\nabla$-flat. Thus its de Rham complex $\Omega_{\mathbb{R}}^*({{\mathscr H}})$ is a differential graded algebra. We remark that the construction of this differential graded algebra does not involve the symplectic structure on $M$.
To encode the symplectic structure into the algebra $\Omega_{\mathbb{R}}^*({{\mathscr H}})$, we can add a curvature term to it which is just the symplectic form $\omega$ itself (up to sign)! More precisely if we denote by $e=dy^{{\scriptscriptstyle\vee}}$ the $\nabla$-flat integral generator for ${{\mathscr H}}^1$ (degree one part), then $-\omega$ can be viewed as an element of $\Omega_{\mathbb{R}}^*({{\mathscr H}})$ by writing it as $e\otimes dx$. Note that this element $-\omega$ is of even degree, and it is closed in $\Omega_{\mathbb{R}}^*({{\mathscr H}})$. Since the differential graded algebra $\Omega^*({{\mathscr H}})$ is supercommutative any such element can be viewed as a curvature term. Thus we have obtained a sheaf of curved differential graded algebras over the base manifold ${\mathbb{R}}$ which we denote by ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$[^4] in the following. We may think of it as a structure sheaf of symplectic geometry of $M$.
#### From symplectic functions to holomorphic functions: Koszul duality.
To understand the mirror phenomenon which interchanges symplectic geometry with complex geometry, let us compute the Koszul dual algebra of ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$. We shall work over the base ring $\Omega^*_{{\mathbb{R}}}$, the complex valued de Rham complex of the base manifold ${\mathbb{R}}$. If we ignore the curvature term $-\omega=e\otimes dx$, the underlying differential graded algebra structure of ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ is simply the exterior algebra generated by $e$ in degree one over $\Omega^*_{{\mathbb{R}}^{{\scriptscriptstyle\vee}}}$. Hence its Koszul dual algebra is the symmetric algebra $\operatorname{sym}_{\Omega^*_{{\mathbb{R}}}}(y)$ generated by a variable $y:=e^{{\scriptscriptstyle\vee}}[-1]$ which is of degree zero (since $e$ has degree one).
For the curvature term observe that $-\omega=e\otimes dx$ is a linear curvature since we are working over $\Omega^*_{{\mathbb{R}}}$. It is well-known in Koszul dual theory how to deal with linear curvatures: they introduce additional differential in Koszul dual algebras. The resulting Koszul dual algebra of ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ is thus the symmetric algebra $\operatorname{sym}_{\Omega^*_{{\mathbb{R}}}}(y)\cong \operatorname{sym}(y)\otimes \Omega^*_{{\mathbb{R}}}$ endowed with a differential acting on it by the operator $(\partial_{x}+\partial_{y}) dx$. If we perform a change of variable by $$x\mapsto x, \;\; y\mapsto -\sqrt{-1} y, \;\; dx \mapsto d\overline{z};$$ this Koszul dual algebra is identified with the Dolbeault algebra resolving the structure sheaf of the complex manifold $T{\mathbb{R}}={\mathbb{R}}\times {\mathbb{R}}$ with its canonical complex structure[^5]. From this example we see that the algebraic reason that Koszul duality interchanges a linear curvature term with an additional differential in its Koszul dual, when applied to this situation, gives a direct link between the symplectic structure and its mirror complex structure.
#### Approaching HMS via duality of algebras.
From the previous example we see that the sheaf of symplectic functions is “Koszul dual" to the sheaf of holomorphic functions. In general we propose to understand the homological mirror symmetry conjecture in the following steps:
- Associated to any Lagrangian torus fibration $\pi: M{\rightarrow}B$ construct a sheaf of $A_\infty$ algebras ${{\mathscr O}}^\omega$ which plays the role of a structure sheaf in symplectic geometry;
- Construct another sheaf ${{\mathscr O}}^{{\mathsf{hol}}}$ over $B$, which in some sense is “Koszul dual" to ${{\mathscr O}}^\omega$;
- Associated to any Lagrangians (with unitary local systems) in $M$ construct a module over ${{\mathscr O}}^\omega$, and show that the Fukaya category ${\mathsf{Fuk}}(M)$ fully faithfully embeds into the category of modules over ${{\mathscr O}}^\omega$;
- Understand the module correspondence between the “Koszul dual" algebras ${{\mathscr O}}^\omega$ and ${{\mathscr O}}^{{\mathsf{hol}}}$.
Here the notion of a sheaf of $A_\infty$ algebras used in ${\expandafter\@slowromancap\romannumeral 1@}$ needs more explanation: locally over small open subsets in $B^{{\mathsf{int}}}$ we can construct honest sheaves of $A_\infty$ algebras, i.e. transition functions for gluing the underlying sheaves are in fact linear $A_\infty$ homomorphisms. Globally we need to take care of possible wall-crossing discontinuity by gluing these local sheaves of $A_\infty$ algebras up to coherent homotopy on intersections, triple intersections, quadruple intersections, and so forth, producing a *homotopy sheaf* of $A_\infty$ algebras.
The current paper contains partial results in all four steps mentioned above. Here “partial" mainly means that we work out the constructions over a small local open subset inside the smooth locus $B_0$ of a Lagrangian torus fibration. The global constructions of ${{\mathscr O}}^\omega$ and its mirror complex manifold over the whole smooth locus $B_0$ will appear in forthcoming works [@Tu], [@Tu2]. We should also confess that it is not clear at present how the singular locus $B^{{\mathsf{sing}}}$ should enter into this study.
There are lots of advantages in this new approach to homological mirror symmetry conjecture. The first advantage is a natural construction of (local) mirror functors (see Sections \[sec:koszul\] and \[sec:fourier\]) using classical Koszul duality theory of modules. Note that functors constructed in this way are $A_\infty$ functors with explicit formulas. It is also conceptually clearer in this approach how lots of ad-hoc constructions in mirror symmetry should enter into the theory. For instance the inclusion of a B-field to complexify symplectic moduli, or the appearance of quantum corrections in deforming the semi-flat complex structure in mirror complex manifold constructions. Finally we believe this approach offers a way to prove homological mirror symmetry conjecture in an abstract form (i.e. without computing both sides explicitly).
In the remaining part of the introduction we give an overview of materials contained in each section.
#### Section \[sec:symp\].
In this section we deal with step ${\expandafter\@slowromancap\romannumeral 1@}$ over a small open subset $U$ inside the smooth locus $B_0$ of a Lagrangian torus fibration $\pi: M{\rightarrow}B$. Denote again by $\pi: M(U){\rightarrow}U$ the projection map. Consider the sheaf ${{\mathscr H}}$ of $C^\infty$ sections of $R\pi_*\Lambda^\pi\otimes C^\infty_U$ where $\Lambda^\pi$ is certain relative Novikov ring (see Section \[sec:symp\] for its precise definition). The Bott-Morse Lagrangian Floer theory developed in [@FOOO] and [@Fukaya] endows for each Lagrangian fiber $L_u$ an $A_\infty$ algebra structure on $H^*(L_u,\Lambda^\pi)$. We shall call this $A_\infty$ algebra Fukaya algebra of $L_u$. For our purpose we need to consider this $A_\infty$ structure as a family over $U$. This construction has been taken care of by K. Fukaya in [@Fukaya] which we follow in this paper.
Just as in the one dimensional example, the sheaf ${{\mathscr H}}$ is a D-module with the Gauss-Manin connection. But this time the structure maps $m_k$ are not $\nabla$-flat in general. Our main observation is the following compatibility between the D-module structure on ${{\mathscr H}}$ and its $A_\infty$ structure. $$[\nabla, m_k]=\sum_{i=1}^{k+1} m_{k+1}(\cdots,\omega,\cdots)$$ This equation is a simple consequence of cyclic symmetry proved in [@Fukaya]. An immediate corollary of these compatibility equations is the following theorem.
There is a curved $A_\infty$ algebra structure on the de Rham complex $\Omega^*_U({{\mathscr H}})$ whose curvature is given by $m_0-\omega$.
We denote this sheaf of curved $A_\infty$ algebras over $U$ by ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}_{M(U)}$ (or simply ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$). The notation is due to the usage of the $A_\infty$ algebra $H^*(L_u,\Lambda^\pi)$ on each fiber of ${{\mathscr H}}$ which was called canonical models in [@FOOO2]. If we replace the canonical model $H^*(L_u,\Lambda^\pi)$ by the full de Rham complex $\Omega^*(L_u,\Lambda^\pi)$ on each fiber $L_u$, then same construction yields an $A_\infty$ structure on the relative de Rham complex of $\pi$. We shall denote this sheaf of $A_\infty$ algebras by ${{\mathscr O}}^\omega_{M(U)}$ (or simply ${{\mathscr O}}^\omega$). There is a subtle difference between the two sheaves of $A_\infty$ algebras ${{\mathscr O}}^\omega$ and ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ in view of mirror symmetry which is explained in Sections \[sec:koszul\] and \[sec:fourier\]. Roughly speaking the “mirror" of the ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ is the tangent bundle $TU$ while that of ${{\mathscr O}}^{\omega}$ is the dual torus bundle over $U$.
#### Section \[sec:lag\].
We show how to obtain modules over the sheaf of symplectic functions (${{\mathscr O}}^\omega$ or ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$) from Lagrangian branes. We need to assume the following assumption.
**Weak unobstructedness assumption:** *For any $u\in U$, the $m_0$ term of the curved $A_\infty$ algebra on $H^*(L_u,\Lambda^\pi)$ is a scalar multiple of ${{\mathbf 1}}$, the strict unit of $H^*(L_u,\Lambda^\pi)$.*
This assumption implies that pairs of the form $(L_u,\alpha)$, where $L_u$ is a Lagrangian torus fiber over a point $u\in U$ and $\alpha$ is a purely imaginary one form in $H^1(L_u,{\mathbb{C}})$, are Lagrangian branes in ${\mathsf{Fuk}}(M)$. Denote by ${\mathsf{Fuk}}^\pi(M)$ the full subcategory of ${\mathsf{Fuk}}(M)$ consisting of such Lagrangian branes.
\[intro:lag\] Assume the weak unobstructedness assumption. Then there exists a linear $A_\infty$ functor $P:{\mathsf{Fuk}}^\pi(M){\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^\omega)$ which is a quasi-equivalence onto its image. The same statement also holds for ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$.
#### Section \[sec:koszul\].
In this section we study mirror symmetry by Koszul duality theory. We continue to work with the above assumption. First we describe the Koszul dual algebra of the sheaf ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$. For this observe that the weak unobstructedness assumption implies that for any $b\in H^1(L_u,\Lambda^\pi)$ we have $$m_0+m_1(b)+m_2(b,b)+m_3(b,b,b)+\cdots+m_k(b,\cdots,b)+\cdots \equiv 0 \pmod 1$$ where $1$ is the strict unit for the $A_\infty$ algebra $H^*(L_u,\Lambda^\pi)$. Then as in [@FOOOtoric] we can define a potential function $W$ on the tangent bundle $TU$ by
$$W(u, b):= \mbox{ the coefficient of $1$ of the sum\;} \sum_{k=0}^\infty m_k(b^{\otimes k}).$$ We then show that that $W$ is in fact a “holomorphic" function on $TU$ with its natural complex structure. Define ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$ to be the “Dolbeault complex" of the structure sheaf of $TU$ endowed with a curvature term given by $W$ [^6]. Then ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$ is Koszul dual to ${{\mathscr O}}_{M(U)}^{\omega,{{\mathsf{can}}}}$ in the following sense.
\[intro:koszul\] There is a universal Maurer-Cartan element $\tau$ in the tensor product curved $A_\infty$ algebra $ {{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes_{\Omega^*_U}{{\mathscr O}}_{M(U)}^{\omega,{{\mathsf{can}}}}$.
This universal Maurer-Cartan element can be used to construct an $A_\infty$ functor $$\Phi^\tau:{{\mathsf{tw}}}({{\mathscr O}}^{\omega,{{\mathsf{can}}}}) {\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{TU})$$ which is also a quasi-equivalence onto its image. Pre-composing with the functor $P$ in Theorem \[intro:lag\] yields the following.
\[intro:hms\] The composition ${\mathsf{Fuk}}^\pi(M)\stackrel{P}{{\rightarrow}} {{\mathsf{tw}}}({{\mathscr O}}^{\omega,{{\mathsf{can}}}}) \stackrel{\Phi^\tau}{{\rightarrow}} {{\mathsf{tw}}}({{\mathscr O}}^{{{\mathsf{hol}}}})$ is a quasi-equivalence onto its image.
The image of this composition functor can also be described under certain assumptions.
\[thm:torus\] Assume that strictly negative Maslov index does not contribute to structure maps $m_k$, and assume further that the potential function $W\equiv 0$. Then the object $\Phi^\tau({{\mathcal L}}_u(\alpha))$ is quasi-isomorphic to a skyscraper sheaf $\Lambda^\pi(u,\alpha)$ supported at $u\in U$.
#### Section \[sec:fourier\].
This section is parallel to the previous section, but replacing ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ by ${{\mathscr O}}^{\omega}$. We also need to replace Koszul duality by its global version: Fourier-Mukai transform. Similar results as Theorem \[intro:hms\] and Theorem \[thm:torus\] are obtained. The universal Maurer-Cartan element $\tau$ in Theorem \[intro:koszul\] is replaced by a quantum version of the Poincar' e bundle. Again this construction yields (local) $A_\infty$ mirror functors. We refer to Section \[sec:fourier\] for more details.
#### Section \[sec:functor\].
As an application of the general theory, we prove a version of homological mirror symmetry between a compact toric symplectic manifold and its Landau-Ginzburg mirror. We summarize the main results in the following theorem.
\[intro:toric\] Let $M$ be a compact smooth toric symplectic manifold, and denote by $\pi:M(\Delta^{{\mathsf{int}}}){\rightarrow}\Delta^{{\mathsf{int}}}$ the Lagrangian torus fibration over the interior of the polytope of $M$. Then there exists an $A_\infty$ functor $\Psi: {\mathsf{Fuk}}^\pi(M) {\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{T(\Delta^{{\mathsf{int}}})})$ which is a quasi-equivalence onto its image. If furthermore $M$ is Fano of complex dimension less or equal to two, then image of $\Psi$ split generates ${{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{T(\Delta^{{\mathsf{int}}})})$ after reduction from $\Lambda^\pi$ to ${\mathbb{C}}$.
[**Remark:** ]{}The Fano assumption is to ensure convergence over ${\mathbb{C}}$ while the dimension assumption is more of technical nature. We expect the functor $\Psi$ to be always essentially surjective over Novikov ring as long as $W$ has isolated singularities.
#### Appendices \[app:modules\] and \[app:fm\].
We include materials on homological algebras of $A_\infty$ modules over curved $A_\infty$ algebras which might have “internal curvatures". We also interpret Koszul duality functors as an affine version of Fourier-Mukai transform. Materials in these appendices are well-known to experts, but are not easy to find in literature. We include them here for completeness. A detailed explanation of the sign conventions used in this paper is also included in the appendix \[app:fm\].
#### Acknowlegment.
The author is grateful to G. Alston, L. Amorim, K. Fukaya, and Y.-G. Oh for useful discussions on symplectic geometry; to A. Polishchuk for his help on essentially everything that involves a lattice. Thanks to D. Auroux and M. Abouzaid for sharing their comments on an earlier version of this paper. The author is also indebted to his advisor A. Căldăraru for teaching him classical Fourier-Mukai transform and providing generous support during graduate school. This work is done during the author’s transition from University of Wisconsin to University of Oregon, thanks to both universities for providing excellent research condition.
#### Notations and Conventions.
- $\Lambda^\pi$: relative Novikov ring, see Definition \[def:ring\];
- ${\mathsf{Fuk}}^\pi(M)$: the full $A_\infty$ subcategory of ${\mathsf{Fuk}}(M)$ consisting of Lagrangian torus fibers endowed with purely imaginary invariant one form;
- $x_1,\cdots,x_n$: local action coordinates on the base $U$ of a Lagrangian torus fibration;
- $y_1^{{\scriptscriptstyle\vee}},\cdots, y_n^{{\scriptscriptstyle\vee}}$: angle coordinates on Lagrangian torus fibers;
- $y_1,\cdots,y_n$: dual angle coordinates on dual torus fibers.
- $e_1:=dy_1^{{\scriptscriptstyle\vee}},\cdots, e_n:=dy_n^{{\scriptscriptstyle\vee}}$: the corresponding integral basis for the lattice $R^1\pi_*{\mathbb{Z}}$ over $U$;
- We use the following two signs frequently in this paper.
- $\epsilon_k:= \sum_{i=1}^{k-1} (k-i)|a_i|$ associated to homogeneous tensor products $a_1\otimes\cdots\otimes a_k$;
- $\eta_k=\sum_{i=1}^{k-1} |a_i|(|b_{i+1}|+\cdots+|b_k|)$ resulting from the permutation $$(a_1\otimes b_1)\otimes\cdots\otimes (a_k\otimes b_k) \mapsto (a_1\otimes\cdots\otimes a_k)\otimes (b_1\otimes\cdots\otimes b_k).$$
We will use two different sign conventions: one used by the authors of [@FOOO] in Lagrangian Floer theory, and the other one as in [@Keller]. Structure maps of the former will be denoted by $m_k$, and the latter by $m_k^\epsilon$. They are related by $$m_k(a_1\otimes\cdots\otimes a_k)=(-1)^{\epsilon_k} m_k^{\epsilon}(a_1\otimes\cdots\otimes a_k).$$
Symplectic functions {#sec:symp}
====================
Let $\pi:M(U){\rightarrow}U$ be a small (to be made precise below) and smooth local piece of a Lagrangian torus fibration $M{\rightarrow}B$. In this section we present a construction of a sheaf of curved $A_\infty$ algebras over $U$ encoding symplectic geometry of $M$. An interesting feature of our construction is that the symplectic form $\omega$ itself enters as part of the curvature term.
#### $A_\infty$ algebras associated to Lagrangian submanifolds.
Let $L$ be a relatively spin compact Lagrangian submanifold in a symplectic manifold $(M,\omega)$. In [@FOOO] and [@Fukaya] a curved $A_\infty$ algebra structure was constructed on $\Omega^*(L,\Lambda_0)$, the de Rham complex of $L$ with coefficients in certain Novikov ring $\Lambda_0$ over ${\mathbb{C}}$ [^7]. Here coefficient ring $\Lambda_0$ is defined by $$\Lambda_0:=\left\{\sum_{i=1}^\infty a_i T^{\lambda_i} \mid a_i\in {\mathbb{C}}, \lambda_i\in {\mathbb{R}}^{\geq 0}, \lim_{i{\rightarrow}\infty} \lambda_i= \infty\right\}$$ where $T$ is a formal parameter of degree zero. Note that the map ${{\mathsf{val}}}:\Lambda_0{\rightarrow}{\mathbb{R}}$ defined by ${{\mathsf{val}}}(\sum_{i=1}^\infty a_i T^{\lambda_i}):= \inf_{a_i\neq 0} \lambda_i$ endows $\Lambda_0$ with a valuation ring structure. Denote by $\Lambda_0^+$ the subset of $\Lambda_0$ consisting of elements with strictly positive valuation. This is the unique maximal ideal of $\Lambda_0$. In [@FOOO] there was an additional parameter $e$ to encode Maslov index to have a ${\mathbb{Z}}$-graded $A_\infty$ structure. If we do not use this parameter, we need to work with ${\mathbb{Z}}/2{\mathbb{Z}}$-graded $A_\infty$ algebras.
We briefly recall the construction of this $A_\infty$ structure on $\Omega^*(L,\Lambda_0)$. Let $\beta\in\pi_2(M,L)$ be a class in the relative homotopy group, and choose an almost complex structure $J$ compatible with (or simply tamed) $\omega$. Form ${{\mathscr M}}_{k+1,\beta}(M,L;J)$, the moduli space of stable $(k+1)$-marked $J$-holomorphic disks in $M$ with boundary lying in $L_0$ of homotopy class $\beta$ with suitable regularity condition in interior and on the boundary. The moduli space ${{\mathscr M}}_{k+1,\beta}(M,L;J)$ is of virtual dimension $d+k+\mu(\beta)-2$ (here $\mu(\beta)$ is the Maslov index of $\beta$).
There are $(k+1)$ evaluation maps $ev_i:{{\mathscr M}}_{k+1,\beta}(M,L;J){\rightarrow}L$ for $i=0,\cdots,k$ which can be used to define a map $m_{k,\beta}:(\Omega^*(L,{\mathbb{C}})^{\otimes k}) {\rightarrow}\Omega^*(L,{\mathbb{C}})$ of form degree $2-\mu(\beta)-k$ by formula $$m_{k,\beta} (\alpha_1,\cdots,\alpha_k):=(\operatorname{ev}_0)_!(\operatorname{ev}_1^*\alpha_1\wedge\cdots\wedge \operatorname{ev}_k^*\alpha_k).$$ To get an $A_\infty$ algebra structure we need to combine $m_{k,\beta}$ for different $\beta$’s. For this purpose we define a submonoid $G(L)$ of ${\mathbb{R}}^{\geq 0}\times 2{\mathbb{Z}}$ as the minimal one generated by the set $$\left\{(\int_\beta \omega,\mu(\beta))\in {\mathbb{R}}^{\geq 0}\times 2{\mathbb{Z}}\mid \beta\in \pi_2(M,L), {{\mathscr M}}_{0,\beta}(M,L;J)\neq \emptyset\right\}.$$ Then we can define the structure maps $m_k:(\Omega(L,\Lambda_0))^{\otimes k} {\rightarrow}\Omega(L,\Lambda_0)$ by $$m_k(\alpha_1,\cdots,\alpha_k):=\sum_{\beta\in G(L)} m_{k,\beta}(\alpha_1,\cdots,\alpha_k) T^{\int_\beta \omega}.$$ Note that we need to use the Novikov coefficients here since the above sum might not converge for a fixed value of $T$. The boundary stratas of ${{\mathscr M}}_{k+1,\beta}(M,L;J)$ are certain fiber products of the diagram $$\begin{CD}
{{\mathscr M}}_{i+1,\beta_1}(M,L;J)\times_L{{\mathscr M}}_{j+1,\beta_2}(M,L;J) @>>> {{\mathscr M}}_{j+1,\beta_2}(M,L;J) \\
@VVV @VV\operatorname{ev}_l V\\
{{\mathscr M}}_{i+1,\beta_1}(M,L;J) @>\operatorname{ev}_0>> L.
\end{CD}$$ Here $1\leq l\leq j$, $i+j=k+1$, and $\beta_1+\beta_2=\beta$. Indeed using this description of the boundary stratas the $A_\infty$ axiom for structure maps $m_k$ is an immediate consequence of Stokes formula.
We should emphasize that a mathematically rigorous realization of the above ideas involves lots of delicate constructions carried out by authors of [@FOOO]. Indeed the moduli spaces ${{\mathscr M}}_{k+1,\beta}(M,L;J)$ are not smooth manifolds, but Kuranishi orbifolds with corners, which causes trouble to define an integration theory. Even if this regularity problem is taken care of there are still transversality issues to define maps $m_{k,\beta}$ to have the expected dimension. Moreover it is not enough to take care of each individual moduli space since the $A_\infty$ relations for $m_k$ follows from analyzing the boundary stratas in ${{\mathscr M}}_{k+1,\beta}(M,L;J)$. Thus one needs to prove transversality of evaluation maps that are compatible for all $k$ and $\beta$. Furthermore one also need to deal with not only disk bubbles, but also sphere bubbles and regularity and transversality issues therein. We refer to the original constructions of [@FOOO] and [@Fukaya] for solutions of these problems.
#### Main properties of the $A_\infty$ algebra $\Omega^*(L,\Lambda_0)$.
Let us summarize some of the main properties of $\Omega^*(L,\Lambda_0)$ proved in [@FOOO] and [@Fukaya].
- (Invariants of symplectic geometry) The homotopy type of this $A_\infty$ structure on $\Omega(L,\Lambda_0)$ is independent of $J$, moreover it is invariant under symplectomorphism;
- (Deformation property) This $A_\infty$ structure is a deformation of the classical differential graded algebra structure on the de Rham complex with coefficients in $\Lambda_0$ [^8];
- (Algebraic property) The $A_\infty$ structure can be constructed to be strict unital and cyclic so that constant function ${{\mathbf 1}}$ is the strict unit and structure maps $m_k$ are cyclic with respect to the Poincaré pairing $<\alpha,\beta>=\int_L \alpha\wedge\beta$.
We describe one more important property of this $A_\infty$ which is crucial for applications in this paper. This is analogous to the divisor equation in Gromov-Witten theory of closed Riemann surfaces. Such a generalization was first observed by C.-H. Cho [@Cho] in the case of Fano toric manifolds. Later in [@Fukaya] Cho’s result was generalized by K. Fukaya to general symplectic manifolds.
\[lem:fukaya\] Let $b\in H^1(L,\Lambda_0)$ and consider any lift of it to an element of $\Omega^1(L,\Lambda_0)$ which we still denote by $b$. Then for any $k\geq 0$ and $l\geq 0$ we have $$\sum_{l_0+\cdots+l_k=l} m_{k+l,\beta}(b^{\otimes l_0},\alpha_1,\cdots,\alpha_k,b^{\otimes l_k}) = \frac{1}{l!}<b,\partial\beta>^l m_{k,\beta}(\alpha_1,\cdots,\alpha_k).$$ Here $\partial\beta \in H^1(L,{\mathbb{Z}})$ is the boundary of $\beta$.
[**Remark:** ]{}The main construction of [@Fukaya] was devoted to constructing compatible Kuranishi structure and continuous multi-section perturbations that are compatible with forgetful maps between Kuranishi spaces ${{\mathscr M}}_{k+l+1,\beta}(M,L;J){\rightarrow}{{\mathscr M}}_{k+1,\beta}(M,L;J)$ which forget the last $l$ marked points for various $k$ and $l$. Indeed with this structure being taken care of the above lemma is a simple exercise on iterated integrals.
#### Local family of $A_\infty$ algebras.
If $\pi:M(U){\rightarrow}U$ is a local smooth family of Lagrangian torus, we get an $A_\infty$ algebra for each point $u\in U$. We would like to consider this as giving us a family of $A_\infty$ algebra over $U$. However there is a delicate point involved here: in general this fiber-wise construction does not produce an $A_\infty$ algebra over $U$ on the relative de Rham complex of $\pi$ even for a generic almost complex structure $J$ due to wall-crossing discontinuity. In [@Fukaya] Section $13$ K. Fukaya constructed such a family by allowing almost complex structures to depend on the Lagrangians. Let us recall Fukaya’s construction here.
Let $u\in U$ be a fixed point inside the smooth locus $B_0\subset B$. We consider a neighborhood $U$ of $u$ in $B_0$ such that there is a symplectomorphism $s: \pi^{-1}(U){\rightarrow}{{\mathscr N}}$ identifying $\pi^{-1}(U)\subset M$ with a tubular neighborhood ${{\mathscr N}}$ of the zero section of $T^*L_u$. Using the affine coordinates $x_1,\cdots,x_n$ and $y_1^{{\scriptscriptstyle\vee}},\cdots,y_n^{{\scriptscriptstyle\vee}}$ to trivialize $T^*L_u$ we get an identification $\pi^{-1}(U)\cong U\times L_u$ where $U$ is considered as an open subset of $H^1(L_u,{\mathbb{R}})={\mathbb{R}}dy^{{\scriptscriptstyle\vee}}_1\oplus\cdots\oplus {\mathbb{R}}dy^{{\scriptscriptstyle\vee}}_n$. Equivalently this is to say that near-by Lagrangian fibers of $L_u$ maybe viewed as the graphs of one-forms on $L_u$ through the symplectomorphism $s$.
Let $p\in U$ be any point in $U$ corresponding to the one-form $p_1dy^{{\scriptscriptstyle\vee}}_1+\cdots+p_ndy^{{\scriptscriptstyle\vee}}_n$ on $L_u$. We define a *diffeomorphism* ${\varphi}_{u,p}$ of $M$ by formulas $$\begin{aligned}
V_{u,p} &:= p_1\partial/\partial x_1+\cdots+ p_n\partial/\partial x_n,\\
T_{u,p} &:= \epsilon_{U} \cdot (s)^{-1} V_{u,p},\\
{\varphi}_{u,p} &:= \mbox{time one flow of $T_{u,p}$}.\end{aligned}$$ Here $\epsilon_{U}$ is a cut-off function supported in a neighborhood $V$ of $U$, and is constant $1$ on $U$. The almost complex structure we shall use to form the Fukaya algebra on the fiber $L_p$ is $({\varphi}_{u,p})_*J$ where $J$ is the almost complex structure we use on the fiber $L_u$. By shrinking $U$ if necessary we can assume that $({\varphi}_{u,p})_*J$ is $\omega$-tamed for all $p\in U$.
The main advantage of this choice of almost complex structures depending on Lagrangians is that it induces identification of various moduli spaces: $$({\varphi}_{u,p})_*: {{\mathscr M}}_{k,\beta}(M,L_u;J) \cong {{\mathscr M}}_{k,({\varphi}_{u,p})_*\beta}(M,L_p;({\varphi}_{u,p})_*J).$$ Thus maps $m_{k,\beta}$ involved in the definition of the $A_\infty$ algebra associated a Lagrangian submanifold $L_p\; (p\in U)$ does not depend on the base parameter $p$, which implies that the structure maps $m_k:=\sum_\beta m_{k,\beta} T^{\int_\beta \omega}$ depend on the $u$-parameter only via symplectic area $\int_\beta \omega$. The follow lemma makes this dependence explicit.
\[lem:area\] Let $L_p$ be a near-by fiber of $L_u$ such that $L_p$ is defined as the graph of the one form $\alpha_p:=\sum_{i=1}^n p_i e_i \in H^1(L_u,{\mathbb{R}})$. Then for each $\beta\in \pi_2(M,L_u)$ we have $$\int_{({\varphi}_{u,p})_*\beta} \omega-\int_\beta \omega = <\alpha_p,\partial \beta>$$ where $\partial\beta\in \pi_1(L_u)$ is the boundary of $\beta$.
[**Proof.** ]{}This is an exercise in Stokes’ formula.
#### Relative Novikov ring.
To have a sheaf of $A_\infty$ algebras over $U$ we need to introduce another Novikov type coefficient ring. Let $\pi: M(U){\rightarrow}U$ be a small smooth local piece of Lagrangian torus fibration as above, and let $u\in U$ be the base point in $U$. The family of monoids $G(L_p) \; (p\in U)$ defines a bundle of monoids over $U$. This bundle is trivialized over $U$ to $U\times G(L_u)$ by the diffeomorphisms ${\varphi}_{u,p}$. In other words we have a local system of monoids over $U$. We denote it by $G$.
\[def:ring\] The relative Novikov ring $\Lambda^\pi$ associated to the family of Lagrangians $\pi:M(U){\rightarrow}U$ is defined by $$\Lambda^\pi:=\left\{\sum_{i=1}^\infty a_i T^{\beta_i} \mid a_i\in {\mathbb{C}}, \beta_i\in G; \sharp\left\{a_i\mid a_i\neq 0, \int_{\beta_i}\omega|_u\leq E\right\}<\infty, \forall E\in {\mathbb{R}}\right\}.$$
The ring $\Lambda^\pi$ is a ${\mathbb{Z}}$-graded ring with $T^\beta$ of degree $\mu(\beta)$. Note that the Maslov index map $\mu$ is well-defined on $G$ since it is preserved under isotopies. This grading makes the operator $m_{k,\beta} T^\beta$ homogeneous of degree $2-k$.
The ring $\Lambda^\pi$ can also be endowed with a valuation by evaluating symplectic area at the base point $u\in U$, i.e. $${{\mathsf{val}}}(\sum_{i=1}^\infty a_i T^\beta):=\inf_{a_i\neq 0} { \int_{\beta_i}\omega|_u}.$$ Using this valuation map we can define a decreasing filtration on $\Lambda^\pi$ by setting $F^{\leq E}:= {{\mathsf{val}}}^{-1}([E,\infty))$. This filtration is called energy filtration. The ring $\Lambda^\pi$ is complete with respect to this filtration. The energy filtration is a useful tool since it induces a spectral sequence to compute Floer homology, see Chapter $6$ of [@FOOO].
#### A sheaf of $A_\infty$ algebras.
Lagrangian Floer theory developed in [@FOOO] and [@Fukaya] can be formulated over the ring $\Lambda^\pi$. Its relationship to the ring $\Lambda_0$ is that there is a ring homomorphism $ \Lambda^\pi {\rightarrow}\Lambda_0$ for each point $p\in U$ defined by $$\sum_{i=1}^\infty a_i T^{\beta_i} \mapsto \sum_{i=1}^\infty a_i T^{\int_{\beta_i}\omega}$$ where we consider $\beta_i$ as an element of $G(L_p)$.
Using this Novikov ring we can define an $A_\infty$ algebra structure on $\Omega_\pi(\Lambda^\pi)$, the relative de Rham complex with coefficients in $\Lambda^\pi$. Explicitly an element of $\Omega_\pi(\Lambda^\pi)$ is of the form $\sum_{i=1}^\infty \alpha_i T^{\beta_i}$ satisfying the same finiteness condition as in the definition of $\Lambda^\pi$. Here $\alpha_i\in \Omega_\pi({\mathbb{C}})$ are ${\mathbb{C}}$-valued relative differential forms.
The structure maps of this $A_\infty$ algebra $\Omega_\pi(\Lambda^\pi)$ are defined by $$m_k(\alpha_1,\cdots,\alpha_k):=\sum_{\beta\in G} m_{k,\beta}(\alpha_1,\cdots,\alpha_k)T^\beta,$$ and we extend these maps $\Lambda^\pi$-linearly to all elements of $\Omega_\pi(\Lambda^\pi)$. Note that since the maps $m_{k,\beta}$ does not depend on the base parameter, they are in particular smooth, which make the above definition valid.
#### D-module structure on $\Omega_\pi(\Lambda^\pi)$.
Observe that the sheaf $\Omega_{\pi}({\mathbb{C}})$, being the relative de Rham complex with complex coefficients, has a D-module structure over $U$. We extend this D-module structure to $\Omega_\pi(\Lambda^\pi)$ by $$\label{eq:area}
\nabla (T^\beta):= -\nabla(\int_{(F_u)_*\beta} \omega)T^\beta$$ and Leibniz rule [^9]. We still denote this derivation by $\nabla$, and call it the Gauss-Manin connection.
#### Variational structure on $\Omega_\pi(\Lambda^\pi)$.
Our next goal is to study the variational structure of the $A_\infty$ structure $m_k$ using the Gauss-Manin connection $\nabla$. For this we consider the de Rham complex of the D-module $\Omega_\pi(\Lambda^\pi)$ over $U$. As a sheaf over $U$ this is the same as $\Omega_\pi(\Lambda^\pi)\otimes_{C^\infty_U} \Omega^*_U$. We wish to extend $m_k$ on $\Omega_\pi(\Lambda^\pi)$ to this tensor product. For this purpose it is more convenient to use a different sign convention which is better to form tensor products. We refer to the new sign convention as the $\epsilon$ sign convention. Structure maps in this sign convention will be denoted by $m_k^\epsilon$. It is related to the previous sign convention by formula $$m_k(a_1\otimes\cdots\otimes a_k)=(-1)^{\epsilon_k} m_k^{\epsilon}(a_1\otimes\cdots\otimes a_k)$$ where $\epsilon_k:= \sum_{i=1}^{k-1} (k-i)|a_i|$. We extend the maps $m_k^\epsilon$ on $\Omega_\pi(\Lambda^\pi)$ to its de Rham complex to get maps $m_k^\epsilon: (\Omega_\pi(\Lambda^\pi)\otimes \Omega^*_U)^k{\rightarrow}\Omega_\pi(\Lambda^\pi)\otimes \Omega^*_U$ which are defined by $$\begin{aligned}
m_0^\epsilon&:= m_0\otimes {{\mathbf 1}};\\
m_1^\epsilon &(f\otimes \alpha):=m_1^\epsilon(f)\otimes \alpha+(-1)^{|f|}f\otimes d_{{{\mathsf{dR}}}}\alpha;\\
m^\epsilon_k&((f_1\otimes \alpha_1)\otimes\cdots\otimes(f_k\otimes \alpha_k)):=(-1)^{\eta_k} m^\epsilon_k(f_1,\cdots,f_k)\otimes (\alpha_1\wedge\cdots\wedge\alpha_k)\end{aligned}$$ where the sign is given by $\eta_k=\sum_{i=1}^{k-1} |\alpha_i|(|f_{i+1}|+\cdots+|f_k|)$. Here we have abused the notation $m_k^\epsilon$, but no confusion should arise. It is straightforward to check that the maps $m_k^\epsilon$ defines an $A_\infty$ algebra structure on the tensor product $\Omega_\pi(\Lambda^\pi)\otimes \Omega^*_U$.
\[lem:diffeo\] For all $k\geq 0$ we have the following compatibility between the $A_\infty$ structure and the D-module structure on $\Omega_\pi(\Lambda^\pi)$: $$\label{equ:diffeo}
[\nabla, m^\epsilon_k]=\sum_{i=1}^{k+1} (-1)^{i-1} m^\epsilon_{k+1}(\operatorname{id}^{i-1}\otimes \omega\otimes \operatorname{id}^{k-i+1}).$$ Here $\omega$ is the symplectic form of $M$ restricted to $M(U)$, and it is viewed as an element of $\Omega_\pi(\Lambda^\pi)\otimes_{C^\infty_U} \Omega^*_U$. Locally in action-angle coordinates $\omega= \sum_{i=1}^n -dy_i\otimes dx_i^{{\scriptscriptstyle\vee}}=\sum_{i=1}^n -e_i\otimes dx_i^{{\scriptscriptstyle\vee}}$.
[**Proof.** ]{}Up to signs this lemma is a direct consequence of Lemma \[lem:area\] and Lemma \[lem:fukaya\]. We include the proof here to illustrate our sign conventions. It is enough to prove the lemma for flat sections $f_1,\cdots,f_k\in \Omega_\pi(\Lambda^\pi)$. The left hand side operator applied to $f_1\otimes\cdots\otimes f_k$ gives $$\begin{aligned}
&\nabla m_{k}^\epsilon (f_1\otimes\cdots\otimes f_k) = (-1)^{|f_1|+\cdots+|f_k|+2-k} \sum_\beta m_{k,\beta}^\epsilon (f_1\otimes\cdots\otimes f_k) \nabla (T^\beta)\\
&=(-1)^{|f_1|+\cdots+|f_k|+2-k} (-1)^{\epsilon_k} \sum_\beta m_{k,\beta} (f_1\otimes\cdots\otimes f_k) T^\beta \otimes \nabla(-\int_\beta \omega)\\
&= (-1)^{|f_1|+\cdots+|f_k|+1-k} (-1)^{\epsilon_k} \cdot\\
&\cdot \sum_\beta m_{k,\beta} (f_1\otimes\cdots\otimes f_k)<\partial\beta,e_i>T^\beta\otimes dx_i^{{\scriptscriptstyle\vee}}\mbox{\;\;(by Lemma~\ref{lem:area})}\\
&= \sum_{\beta, 1\leq j\leq k+1}(-1)^{|f_1|+\cdots+|f_k|+1-k} (-1)^{\epsilon_k}\cdot\\
&\cdot (-1)^{|f_1|k+\cdots+|f_{j-1}|(k-j+2)+(k-j+1)+\cdots |f_{k-1}|} \cdot m^\epsilon_{k+1,\beta} (f_1\cdots e_i \cdots f_k)T^\beta\otimes dx_i^{{\scriptscriptstyle\vee}}.\\\end{aligned}$$ The sign above can be simplified to $(-1)^{1-j+|f_1|+\cdots+|f_{j-1}|}$ which proves the lemma.
A differential $A_\infty$ algebra over a manifold $U$ is given by a triple $(E,\nabla,\omega)$ such that
- $E$ is a ${\mathbb{Z}}/2{\mathbb{Z}}$-graded D-module over $U$;
- $E$ is a sheaf of ${\mathbb{Z}}/2{\mathbb{Z}}$-graded $A_\infty$ algebras over $U$;
- $\omega$ is an even element in the de Rham complex of $E$.
Moreover these structures are compatible in the sense that equations \[equ:diffeo\] hold.
\[thm:daa\] Let $(E,\nabla,\omega)$ be a differential $A_\infty$ algebra over a smooth manifold $U$. Then its de Rham complex $\Omega_U^*(E)$ also has an $A_\infty$ algebra structure. Explicitly its structure maps are given by (in the $\epsilon$ sign convention)
- $\hat{m}^\epsilon_0:=m^\epsilon_0-\omega$;
- $\hat{m}^\epsilon_1:=m^\epsilon_1+\nabla$;
- $\hat{m}^\epsilon_k:=m^\epsilon_k$ $\;\;\;$ for $k\geq 2$.
[**Proof.** ]{}This is a direct computation keeping track of the signs involved. Indeed the left hand side of equation \[equ:diffeo\] is the additional terms resulting from adding $\nabla$ to $m_1$ while the right hand side is exactly the terms we get by adding $\omega$ to the curvature term. The theorem is proved.
Lemma \[lem:diffeo\] asserts that the triple $(\Omega_\pi(\Lambda^\pi),\nabla,\omega)$ forms a differential $A_\infty$ algebra over $U$. Theorem \[thm:daa\] implies that there is an $A_\infty$ algebra structure on the de Rham complex $\Omega_\pi(\Lambda^\pi)\otimes_{C^\infty_U} \Omega^*_U$. We denote this sheaf of $A_\infty$ algebras by ${{\mathscr O}}^\omega_{M(U)}$ (or simply ${{\mathscr O}}^\omega$), and refer to it as the sheaf of symplectic functions.
#### Deformation property of ${{\mathscr O}}^\omega$.
If $(X,\omega)$ is a symplectic manifold, then the triple $(C^\infty_X,d_{dR},\omega)$ forms a differential $A_\infty$ algebra over $X$ whose associated curved $A_\infty$ algebra is the de Rham algebra $\Omega^*_X$ endowed with a curvature term given by the symplectic form $-\omega$. This curved algebra may be thought of as “classical" symplectic functions, and the algebra ${{\mathscr O}}^\omega$ is a deformation of this classical algebra. The previous assertion follows from the deformation property of Fukaya algebras.
#### A variant construction.
In the end of this section we mention a variant of the sheaf ${{\mathscr O}}^\omega$ that will be used in Section \[sec:koszul\]. Namely in the above constructions we could have used the canonical model which is certain minimal model $H^*(L_u,\Lambda^\pi)$ of the full de Rham complex model $\Omega^*(L_u,\Lambda^\pi)$ for each Lagrangian torus fibers. All the previous constructions go through in this case as well.
More explicitly let $R\pi_*\Lambda^\pi$ be the push-forward of the constant sheaf $\Lambda^\pi$ via the map $\pi:M(U){\rightarrow}U$, and denote by ${{\mathscr H}}$ the sheaf $R\pi_*\Lambda^\pi\otimes_{\mathbb{C}}C^\infty_U$. Then ${{\mathscr H}}$ has a canonical Gauss-Manin connection $\nabla$ acting on it where we extend the action of $\nabla$ to $T^\beta$ by the same formula \[eq:area\]. Moreover the symplectic form $\omega$ can be viewed as an element in the de Rham complex of ${{\mathscr H}}$. Again locally in action-angle coordinates $\omega= \sum_{i=1}^n -dy_i\otimes dx_i^{{\scriptscriptstyle\vee}}=\sum_{i=1}^n -e_i\otimes dx_i^{{\scriptscriptstyle\vee}}$.
The triple $({{\mathscr H}},\nabla,\omega)$ forms a differential $A_\infty$ algebra over $U$. We shall denote by the resulting sheaf of $A_\infty$ algebras by ${{\mathscr O}}_{M(U)}^{\omega,{{\mathsf{can}}}}$ (or simply ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$).
[**Proof.** ]{}The proof is the same as for the triple $(\Omega_\pi(\Lambda^\pi),\nabla,\omega)$. We shall not repeat it here.
The main advantage of ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ over ${{\mathscr O}}^\omega$ is that the former is of finite rank over the differential graded algebra $\Omega^*_U(\Lambda^\pi)$; while its disadvantage is that its “mirror" gives the tangent bundle of $U$ rather than the dual torus bundle, which we will discuss in Sections \[sec:koszul\] and \[sec:fourier\]. In view of results in [@NZ] and [@FLTZ] it is likely that ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ is related to certain equivariant symplectic geometry.
From Lagrangians to modules {#sec:lag}
===========================
Let $\pi:M(U){\rightarrow}U$ as in the previous section, and we continue to use notations therein. In this section we construct $A_\infty$ modules over ${{\mathscr O}}^\omega$ from Lagrangian branes in $M$. We only consider Lagrangian branes of the form $(L_p,\alpha)$ for a Lagrangian torus fiber $L_p$ ($p\in U$) endowed with a purely imaginary torus invariant one form $\alpha$ on $L_p$. The case of general Lagrangian branes is left for future work.
Denote by ${\mathsf{Fuk}}^\pi(M)$ the full $A_\infty$ subcategory of ${\mathsf{Fuk}}(M)$ consisting of these objects. The main result of this section is the following theorem.
\[thm:lag\] There exists a linear $A_\infty$ functor $P:{\mathsf{Fuk}}^\pi(M){\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^\omega)$ which is a weak homotopy equivalence onto its image.
Here ${{\mathsf{tw}}}({{\mathscr O}}^\omega)$ is the $A_\infty$ category of twisted complexes over ${{\mathscr O}}^\omega$ possibly with internal curvatures. We refer to the Appendix \[app:modules\] for its definition. We also remark that the theorem remains true if we replace ${{\mathscr O}}^\omega$ by ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$. This version is used in the next section to interpret mirror symmetry as Koszul duality.
#### Weak unobstructedness and potential function.
We begin to recall the notion of weak unobstructedness from [@FOOO] Section $3.6$. Consider the Fukaya algebra $\Omega(L_p,\Lambda^\pi)$, and denote by ${{\mathbf 1}}$ its strict unit. An element $b\in \Omega^1(L_p,\Lambda^\pi)$ is called a weak Maurer-Cartan element if we have the equation $$\sum_{k=0}^\infty m_k(b^{\otimes k}) \equiv 0 \pmod {{\mathbf 1}}.$$ They are important to define Lagrangian Floer homology because they give rise to deformations of the $A_\infty$ structure on $\Omega(L_p,\Lambda^\pi)$ with square-zero differential.
In this paper we consider these elements from a more algebraic perspective: weak Maurer-Cartan elements give rise to $A_\infty$ modules with *internal curvatures*. If the left hand side of the above equation were equal to zero (i.e. if $b$ is a Maurer-Cartan element), such a $b$ by definition is an $A_\infty$ module over $\Omega(L_p,\Lambda^\pi)$ (in fact this is a twisted complex). For the case of weak Maurer-Cartan elements we include relevant homological algebras in Appendix \[app:modules\]. What we get in this case is an $A_\infty$ module with an *internal curvature*. In the following we shall freely use this notion.
On the set of weak Maurer-Cartan elements we define a function called potential function by formula $$\label{eq:potential}
W(p,b):= \mbox{the coefficient of ${{\mathbf 1}}$ of the sum}\;\; \sum_{k=0}^\infty m_k(b^{\otimes k}).$$
\[ass:wua\] For any $p\in U$, and any $b\in \Omega^1(L_p,\Lambda^\pi)$, the potential function $W(p,b)$ is a scalar multiple of ${{\mathbf 1}}$, the strict unit of $\Omega(L_p,\Lambda^\pi)$.
#### Torus fibers.
Let $L_p$ be a Lagrangian torus fiber for some point $p\in U$. By the weak unobstructedness assumption \[ass:wua\] the pair $(L_p,0)$ is a Lagrangian brane in ${\mathsf{Fuk}}(M)$. We will construct an $A_\infty$ module ${{\mathcal L}}_p$ over ${{\mathscr O}}^\omega$ with an internal curvature in $C^\infty_U(\Lambda^\pi)$. Since the element $b=0$ is weakly unobstructed, it defines an $A_\infty$ module structure on $\Omega(L_p,\Lambda^\pi)$ over itself with internal curvature $W(p,0)$. The question is how to “propagate" this structure to other points of $U$. For this we “propagate" the weak Maurer-Cartan element $b=0$ by a differential equation using the symplectic form $\omega$. More precisely we define $\theta\in{{\mathscr O}}^\omega$ over $U$ by the following differential equation with initial condition: $$\nabla \theta = \omega \mbox{\;\;\; and \;\;\;} \theta(p)=0.$$ Explicitly if $\theta$ is equal to $(x_1-p_1)e_1+\cdots+(x_n-p_n)e_n$ in local coordinates. Let us show that the element $\theta$ defines a weak Maurer-Cartan element of ${{\mathscr O}}^\omega$.
\[lem:constant\] We have $\nabla (\sum_{k=0}^\infty (-1)^{k(k-1)/2} m^\epsilon_k(\theta^k)=0$.
[**Proof.** ]{}This is a direct computation: $$\begin{aligned}
\nabla &(\sum_{k=0}^\infty (-1)^{k(k-1)/2} m^\epsilon_k(\theta^k) = \sum_{k=0}^\infty [\sum_{i=1}^{k+1} (-1)^{k(k-1)/2+i-1} m^\epsilon_{k+1} (\theta^{i-1},\omega,\theta^{k-i+1}) +\\
&+\sum_{j=1}^k (-1)^{k(k-1)/2+k+j-1} m^\epsilon_k (\theta^{j-1},\nabla\theta,\theta^{k-j}) ] \mbox{\;\;\; (by Lemma~\ref{lem:fukaya})} \\
&=\sum_{k=1}^\infty\sum_{i=1}^{k} (-1)^{(k-1)(k-2)/2+i-1} m_{k} (\theta^{i-1},\omega,\theta^{k-i}) - \\
&-\sum_{k=1}^\infty\sum_{j=1}^k (-1)^{k(k-1)/2+k+j-1} m_k (\theta^{j-1},\omega,\theta^{k-j}) \mbox{\;\;\; (by the equation $\nabla\theta=\omega$)} \\
&=0.\end{aligned}$$ For the last equality we observe that the sum $[(k-1)(k-2)/2+i-1]+[k(k-1)/2+k+i-1]$ from the signs is always odd, hence the two summations cancel out each other. The lemma is proved.
Thus the sum $\sum_{k=0}^\infty (-1)^{k(k-1)/2} m^\epsilon_k(\theta^k)$ is a constant with respect to $\nabla$. We note that this “constant" *is not* a constant function on $U$, but rather a flat section of the D-module $\Lambda^\pi$ over $U$. Let us denote this element in $C^\infty_U(\Lambda^\pi)$ by ${{\mathscr W}}_{(p,0)}$. By its definition we have $${{\mathscr W}}_{(p,0)}|_{p}=W(p,0).$$ Applying the construction in Appendix \[app:modules\] we get a sheaf of $A_\infty$ modules over ${{\mathscr O}}^{\omega}$ with internal curvature ${{\mathscr W}}_{(p,0)}$. If we assume convergence of relevant power series after evaluating $T=e^{-1}$, the function ${{\mathscr W}}_{(p,0)}$ is simply the constant function $W(p,0)$ over $U$.
#### Torus fibers with a purely imaginary closed one forms.
The above construction can also be generalized to the case $(L_p, \alpha)$ for some $\alpha\in H^1(L_p,{\mathbb{C}})$ that is purely imaginary [^10]. This time we define $\theta\in {{\mathscr O}}^\omega$ by $$\nabla\theta = \omega \mbox{\;\;\; and \;\;\;} \theta(p)= \alpha.$$ Assumption \[ass:wua\], together with the same computation as above, shows that $\theta$ defines a weak Maurer-Cartan element of ${{\mathscr O}}^\omega$ with internal curvature given by a $\nabla$-flat section ${{\mathscr W}}_{(p,\alpha)}\in C^\infty_U(\Lambda^\pi)$ such that ${{\mathscr W}}_{(p,\alpha)}|_p=W(p,\alpha)$. We denote the associated $A_\infty$ module by ${{\mathcal L}}_p(\alpha)$.
#### Proof of Theorem \[thm:lag\].
We begin to consider the endmorphism space of an object $(L_p,\alpha)$ in the Fukaya category. Recall by definition [@FOOO] Section $3.6$ the Endomorphism complex $\operatorname{Hom}_{{\mathsf{Fuk}}(M)}((L_p,\alpha),(L_p,\alpha))$ is just the $A_\infty$ algebra $\Omega(L_p,\Lambda^\pi)$ endowed with a differential twisted by the weak Maurer-Cartan element $\alpha$. Explicitly its differential is defined as $$m_1^\alpha(x):=\sum_{i,j=0}^{\infty} m_{i+j+1}(\alpha^i,x,\alpha^j).$$ Similarly the complex $\operatorname{Hom}_{{{\mathscr O}}^\omega}({{\mathcal L}}_p(\alpha),{{\mathcal L}}_p(\alpha))$ is the $A_\infty$ algebra ${{\mathscr O}}^\omega$ twisted by the weak Maurer-Cartan element $\theta$ associated to $(L_p,\alpha)$ as described in the previous paragraph.
For an element $\eta\in \Omega (L_p,\Lambda^\pi)$ we define an element $P(\eta)\in {{\mathscr O}}^\omega$ by propagating $\eta$ in a flat way. Namely $P(\eta)$ is such that $\nabla(P(\eta))=0$ and $P(\eta)|_p=\eta$. We shall show that the map $$P:\operatorname{Hom}_{{\mathsf{Fuk}}(M)}((L_p,\alpha),(L_p,\alpha)){\rightarrow}\operatorname{Hom}_{{{\mathscr O}}^{\omega}}({{\mathcal L}}_p(\alpha),{{\mathcal L}}_p(\alpha))$$ is a linear homomorphism of $A_\infty$ algebras and a quasi-isomorphism on the underlying complexes. The fact that $\theta$ has internal curvature ${{\mathscr W}}_{(p,\alpha)}$ implies that $P$ interchanges the curvature terms on both sides. To see that $P$ is compatible with all higher multiplications $m_k$ we need to show that $$P[m(e^\alpha,\eta_1,e^\alpha,\cdots,e^\alpha,\eta_k,e^\alpha)]=M(e^\theta,P(\eta_1),e^\theta,\cdots,e^\theta,P(\eta_k),e^\theta)$$ where $m$ and $M$ are $A_\infty$ structures on $\Omega(L_p,\Lambda^\pi)$ and ${{\mathscr O}}^{\omega}$ respectively, and $e^\alpha=1+\alpha+\alpha\otimes\alpha+\cdots$ is considered as an element of the bar complex of $\Omega(L_p,\Lambda^\pi)$, similarly for $e^\theta$. The two sides agree with each other at the point $p$ by definition. Hence it suffice to show that the right hand side is $\nabla$-flat. Computing $\nabla[M(e^\theta,P(\eta_1),e^\theta,\cdots,e^\theta,P(\eta_k),e^\theta)]$ using Lemma \[lem:diffeo\] and the condition $\nabla\theta=\omega$ shows that this is indeed the case. We omit this computation here as it is similar to the computation in Lemma \[lem:constant\].
It remains to prove that $P$ is a quasi-isomorphism. In fact we will show that $P$ is a homotopy equivalence. For this we need to use the assumption that $U$ is contractible. Let $H$ be a homotopy retraction between the one-point space $p$ and $U$. It induces a deformation retraction between functions on the point $p$ (one dimensional) and the de Rham complex of $U$ with coefficients in ${\mathbb{C}}$. We denote this algebraic homotopy also by $H$. To extend $H$ to $\Omega(L_p,\Lambda^\pi)$ observe that $$P(\sum\eta_i T^{\beta_i})=\sum P(\eta_i)P(T^\beta_i)=\sum P(\eta_i) e^{\sum_{j=1}^d<\partial\beta_i, e_j>(x_j-p_j)} T^{\beta_i}$$ where the factor $I(\beta_i):=e^{\sum_{j=1}^d<\partial\beta_i, e_j>(x_j-p_j)}$ is by Lemma \[lem:area\]. Thus for each sector $T^\beta$ the operator $I(\beta)\circ H\circ I(\beta)^{-1}$ is a contracting homotopy, yielding a contracting homotopy between $$(\Omega(L_p,\Lambda^\pi),0) \simeq(\Omega_\pi(\Lambda^\pi), \nabla).$$ Here the left hand side is endowed with a zero differential while the right hand side is endowed with only $\nabla$ as its differential.
Next we consider the operator $M_1^\theta(-):=M(e^\theta,-,e^\theta)$ on $\Omega_\pi(\Lambda^\pi)$ as a deformation of $\nabla$ and use homological perturbation lemma to prove our theorem. If we denote by $$i: (\Omega(L_p,\Lambda^\pi),0) \hookrightarrow (\Omega_\pi(\Lambda^\pi), \nabla)$$ the inclusion, and $$p: (\Omega_\pi(\Lambda^\pi), \nabla) {\rightarrow}(\Omega(L_p,\Lambda^\pi),0)$$ the projection, then we have $pi=\operatorname{id}$ and $ip=\operatorname{id}+\nabla H+H \nabla$. By standard homological perturbation technique, the perturbed differential on $(\Omega(L_p,\Lambda^\pi))$ is given by formula $$p M_1^\theta i + p M_1^\theta H M_1^\theta i+\cdots.$$ Observe that $M_1^\theta\circ M_1^\theta=0$ since $\theta$ is a weak Maurer-Cartan element, and that $M_1^\theta$ commutes with the homotopy $H$. These two facts imply that the induced perturbed differential on agrees with $p M_1^\theta i(-) = m_1^\alpha(-):=m(e^\alpha,-,e^\alpha)$. By the same reasoning, the perturbed homotopy $$H+ H M_1^\theta H + H M_1^\theta H M_1^\theta H+\cdots$$ is the same as the old homotopy $H$, providing a homotopy equivalence between the perturbed complexes. Thus the case of endomorphisms is proved.
The case of $\operatorname{Hom}_{{\mathsf{Fuk}}(M)}((L_p,\alpha_1),(L_p,\alpha_2))$ is similar. Note that by its very definition in order that this $\operatorname{Hom}$ space is non-vanishing it is necessary to have $W(p,\alpha_1)=W(p,\alpha_2)$. Since we work over the same point $p$, the propagation map $P$ can still be defined. The previous proof carries over word-by-word to this case.
For the last case the $\operatorname{Hom}$ space $\operatorname{Hom}_{{\mathsf{Fuk}}(M)}((L_{p_1},\alpha_1),(L_{p_2},\alpha_2))$ for distinct $p_1, p_2\in U$ is zero by definition in ${\mathsf{Fuk}}(M)$. On the ${{\mathscr O}}^\omega$-module side, the complex $\operatorname{Hom}_{{{\mathscr O}}^\omega} ({{\mathcal L}}_{p_1}(\alpha_1),{{\mathcal L}}_{p_2}(\alpha_2))$ also has zero cohomology. This is due to the fact their internal curvatures are different, i.e. $W(p_1,\alpha_1)\neq W(p_2,\alpha_2)$. Thus the proof of Theorem \[thm:lag\] is finished.
[**Remark:** ]{}\[rmk:gq\] The propagation equation $\nabla\theta=\omega$ is local in nature. Indeed when the base $B$ (or the smooth part of $B$) has nontrivial topology, there might not exist a global solution for this equation: we do not expect to be able to write the symplectic form as an exact form.
Mirror symmetry and Koszul duality {#sec:koszul}
==================================
In this section we study the Koszul dual algebra of the $A_\infty$ algebra ${{\mathscr O}}_{M(U)}^{\omega,{{\mathsf{can}}}}$ associated to a local Lagrangian torus fibration $\pi:M(U){\rightarrow}U$ (see Section \[sec:symp\] for its construction). We show that this Koszul dual algebra ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$ is certain Dolbeault complex of the structure sheaf of the complex manifold $TU$ with values in the relative Novikov coefficient $\Lambda^\pi$ (possibly with a holomorphic function as curvature). Applying a well-known correspondence of modules over Koszul dual algebras gives a natural construction of an $A_\infty$ functor $$\Phi^\tau:{{\mathsf{tw}}}({{\mathscr O}}^{\omega,{{\mathsf{can}}}}){\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{TU}).$$ Pre-composing with the propagation functor $P$ defined in the previous section gives a (local) mirror functor $$\Phi^\tau\circ P: {\mathsf{Fuk}}^\pi(M){\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{TU})$$ which is also an $A_\infty$ functor. We prove that this local mirror functor is a quasi-equivalence onto its images. Under certain conditions we can also identify the images of this functor which turn out to be skyscraper sheaves of points in $TU$.
Throughout the section we continue to work with the unobstructedness assumption \[ass:wua\] introduced in the previous section.
#### Potential function restricted to $TU$.
Recall by equation \[eq:potential\] we defined a potential function on the set of weak Maurer-Cartan elements whose elements are pairs $(p,b)$ for a point $p\in U$ and an element $b\in H^1(L_p,\Lambda^\pi)$. We consider the restriction of this function to the set of pairs $(p,\alpha)$ where $\alpha$ is a purely imaginary element of $H^1(L_p,{\mathbb{C}})$. The set of such pairs is canonically isomorphic to the tangent bundle $TU$. Locally in coordinates this identification is given by $$(x_1,\cdots,x_n,y_1,\cdots,y_n)\mapsto (x_1,\cdots,x_d; \sum_{i=1}^n -\sqrt{-1}y_i e_i).$$ Consider the set of $\Lambda^\pi$ valued smooth functions on $TU$ which we shall denote by $C^\infty_{TU}(\Lambda^\pi)$. More precisely $C^\infty_{TU}(\Lambda^\pi)$ consists elements of the form $\sum_j f_j T^{\beta_j}$ for $f_j\in C^\infty_{TU}$, and for any energy bound $E$ there are only finitely many nonzero terms in the series. Moreover we consider $C^\infty_{TU}(\Lambda^\pi)$ as a D-module over $U$ by letting $\partial/\partial x_i$ act on it by ${{\overline{\partial}}}_i:=(\partial/\partial x_i+\sqrt{-1}\partial/\partial y_i)$. Recall that the operator $\partial/\partial x_i$ acts on $T^\beta$ via formula \[eq:area\]. Denote by $\Omega^*_U(C^\infty_{TU}(\Lambda^\pi)):=C^\infty_{TU}(\Lambda^\pi)\otimes \Omega^*_U$ the associated de Rham complex with coefficients in $C^\infty_{TU}(\Lambda^\pi)$. The differential on this de Rham complex will be denoted by ${{\overline{\partial}}}$.
The notation ${{\overline{\partial}}}$ has a geometric explanation. If we assume all convergence in Lagrangian Floer theory, we can evaluate $T$ at $e^{-1}$ and work over ${\mathbb{C}}$. Then $\Omega^*(C^\infty_{TU}(\Lambda^\pi))$ after evaluating $T$ at $e^{-1}$ can be identified with the classical Dolbeault differential ${{\overline{\partial}}}$ of the structure sheaf of $TU$.
\[lem:holo\] We have ${{\overline{\partial}}}W=0.$ In the geometric situation this implies $W$ is a holomorphic function on $TU$.
[**Proof.** ]{}By the identification $(y_1,\cdots,y_d)\mapsto \sum_{i=1}^n -\sqrt{-1}y_i e_i$ mentioned above, we evaluate $W$ at $b=\sum_{i=1}^n-\sqrt{-1}y_i e_i$ and differentiate. We have $$\begin{aligned}
\sqrt{-1}\partial_{y_i} (m_{k+1}(b,\cdots,b)) &=\sum_{j=1}^{k+1} m_{k+1}(b,\cdots,e_i,\cdots,b) \mbox{\small \;\;\; ($e_i$ in the $j$-th spot)}\\
&=-\partial_{x_i} m_k(b,\cdots,b) \mbox{\;\;\; (by Lemma~\ref{lem:diffeo}).}\end{aligned}$$ Summing over $k$ yields the result. Hence the lemma is proved.
#### Koszul dual of ${{\mathscr O}}_{M(U)}^{\omega,{{\mathsf{can}}}}$.
The Koszul dual algebra ${{\mathscr O}}^{{\mathsf{hol}}}_{TU}$ (or simply ${{\mathscr O}}^{{\mathsf{hol}}}$) of ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ is defined as follows. This is a sheaf of curved differential graded algebras over $U$. Its underlying differential graded algebra is $\Omega^*_U(C^\infty_{TU}(\Lambda^\pi))=C^\infty_{TU}(\Lambda^\pi)\otimes \Omega^*_U$ endowed with the differential ${{\overline{\partial}}}$ and the natural tensor product algebra structure. Its curvature term is $-W$ (the sign is due to Theorem \[thm:koszul\] below). By the lemma above this defines a curved differential graded algebra. Geometrically we think of it as the Dolbeault complex of $TU$ endowed with a curvature term $-W$.
The two sheaves of algebras ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ and ${{\mathscr O}}^{{\mathsf{hol}}}$ are Koszul dual to each other in the sense of the following theorem.
\[thm:koszul\] Define the element $\tau:= \sum_{i=1}^n -\sqrt{-1} y_i\otimes e_i $ in an affine coordinates of $U$ (note that since $e_i$ and $y_i$ are dual to each other, the element $\tau$ is independent of coordinates). Then $\tau$ is a Maurer-Cartan element of the tensor product $A_\infty$ algebra ${{\mathscr O}}^{{\mathsf{hol}}}\otimes_{\Omega_U^*(\Lambda^\pi)} {{\mathscr O}}^{\omega,{{\mathsf{can}}}}$.
[**Proof.** ]{}This is a straight forward computation except that we need to pay extra attention to the signs. Indeed to form the tensor product $A_\infty$ algebra we use the $\epsilon$ sign convention as explained in more detail in the appendix. Let us denote by $M_k^\epsilon$ the resulting structure constants. Then we have $$\begin{aligned}
M^\epsilon_0 &= m^\epsilon_0-\omega-W;\\
M^\epsilon_1(\tau) &= m^\epsilon_1(\tau) + \nabla(\tau) + {{\overline{\partial}}}(\tau);\\
M^\epsilon_k(\tau) &= m^\epsilon_k(\tau,\cdots,\tau) \;\;\;\mbox{ for $k\geq 2$ .}\end{aligned}$$ Observe that the sum $\sum_{k=0}^\infty (-1)^{k(k-1)/2} m_k(\tau,\cdots,\tau)$ is by definition $W$. Moreover we have $\nabla(\tau)=0$ and ${{\overline{\partial}}}(\tau)=\sum dx_i\otimes e_i=\omega$. Thus summing over these equalities we get $$\sum_{k=0}^\infty (-1)^{k(k-1)/2} M^\epsilon_k(\tau,\cdots,\tau)=0.$$ Thus the theorem is proved.
#### Local construction of mirror functor.
Let $A$ be an $A_\infty$ algebra and $B$ be a curved differential graded algebra over a base ring $R$. Assume both $A$ and $B$ are free $R$-modules, and that $A$ is of finite rank of $R$. Then associated to any Maurer-Cartan element $\tau\in B\otimes A$ we can form an $A_\infty$ functor $\Phi^\tau: {{\mathsf{tw}}}(A){\rightarrow}{{\mathsf{tw}}}(B)$. Intuitively speaking the Maurer-Cartan element $\tau$ can be used to twist the tensor product $B\otimes A$ to get a rank one twisted complex over $B\otimes A$. Viewing this $B\otimes A$-module as a kernel we get a functor from ${{\mathsf{tw}}}(A)$ to ${{\mathsf{tw}}}(B)$. This construction is straightforward for ordinary algebras $A$ and $B$, but requires more explanations in the $A_\infty$ setting. Moreover we also would like to include modules with internal curvatures into this construction. Details of these homological constructions are included in the Appendix \[app:fm\].
Let us apply this construction to the case $A={{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ and $B={{\mathscr O}}^{{\mathsf{hol}}}_{TU}$ over the base ring $R=\Omega_U^*(\Lambda^\pi)$, which yields an $A_\infty$ functor $$\Phi^\tau: {{\mathsf{tw}}}({{\mathscr O}}^{\omega,{{\mathsf{can}}}}) {\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}).$$ Next we explicitly describe this local mirror functor $\Phi^\tau$. For an $A_\infty$ module ${{\mathcal L}}$ over ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ (in particular ${{\mathcal L}}$ must be a $\Omega_U^*(\Lambda^\pi)$-module since this is our base ring), define an ${{\mathscr O}}^{{\mathsf{hol}}}$-module $\Phi^\tau({{\mathcal L}})$ as follows. As a sheaf over $U$ this is just ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes_{\Omega_U^*(\Lambda^\pi)}{{\mathcal L}}$. Its ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$-module structure is induced from the first tensor component. On the tensor product we put a twisted differential defined by formula $$d:={{\overline{\partial}}}\otimes\operatorname{id}+ \sum_{k=0}^\infty \hat{\rho}_k(\tau,\cdots,\tau).$$ If we denote by $\rho_k:({{\mathscr O}}^{\omega,{{\mathsf{can}}}})^{\otimes k}\otimes_{\Omega_U^*(\Lambda^\pi)} {{\mathcal L}}{\rightarrow}{{\mathcal L}}$ the structure constants of the $A_\infty$ module ${{\mathcal L}}$, then the maps $\hat{\rho}_k(\tau,\cdots,\tau):{{\mathscr O}}^{{\mathsf{hol}}}\otimes_{\Omega_U^*(\Lambda^\pi)}{{\mathcal L}}{\rightarrow}{{\mathscr O}}^{{\mathsf{hol}}}\otimes_{\Omega_U^*(\Lambda^\pi)}{{\mathcal L}}$ is defined by $$\hat{\rho}_k(\tau,\cdots,\tau)(f\otimes m):= \sum_{i_1,i_2,\cdots,i_k} y_{i_1}\cdots y_{i_k}\cdot f\otimes\rho_k(e_{i_1},\cdots,e_{i_k};m).$$ The $A_\infty$ functors on $\operatorname{Hom}$ spaces can also be described explicitly. We refer to Appendix \[app:fm\] for more details.
#### Mirror of torus fibers.
Let us identify the object $\Phi^\tau({{\mathcal L}}_p(\alpha))$ for the ${{\mathscr O}}^\omega$-module ${{\mathcal L}}_p$ associated to a torus fiber $L_p$ endowed with a purely imaginary one form $\alpha\in H^1(L_p,{\mathbb{C}})$ on it. We refer to the previous section \[sec:lag\] for the construction of ${{\mathcal L}}_p(\alpha)$.
Let $p$ be a point in $U$, and let $\alpha$ be a purely imaginary one form in $H^1(L_p,{\mathbb{C}})$. Define an ${{\mathscr O}}^{{\mathsf{hol}}}_{TU}$-module $\Lambda^\pi(p,\alpha)$ as follows. As a sheaf over $U$ it is simply the skyscraper sheaf $\Lambda^\pi$ over the point $p\in U$. The ${{\mathscr O}}^{{\mathsf{hol}}}_{TU}$- module structure is defined by letting an element $f\in {{\mathscr O}}^{{\mathsf{hol}}}$ acting on $\Lambda^\pi$ via multiplication by $f(p,\alpha)$.
\[prop:torus\] Assume that strictly negative Maslov index does not contribute to structure maps $m_k$ of the Fukaya algebra $H^*(L_p,\Lambda_0)$, and assume further that the potential function $W\equiv 0$ over $U$. Then the object $\Phi^\tau({{\mathcal L}}_p(\alpha))$ is quasi-isomorphic to $\Lambda^\pi(p,\alpha)[-n]$.
[**Proof.** ]{}Let $e_1,\cdots,e_n$ be a trivialization of $R^1\pi_*{\mathbb{Z}}$ which induces a trivialization of $R\pi_*{\mathbb{Z}}$ whose flat sections are $e_I:=e_{i_1}\wedge\cdots\wedge e_{i_j}$ for any subset $I\subset\left\{ 1,\cdots,n\right\}$. This trivialization defines an isomorphism of sheaves on $U$ $$\Phi^\tau({{\mathcal L}}_p(\alpha)) \cong \prod_{I\subset\left\{1,\cdots,n\right\}} {{\mathscr O}}^{{\mathsf{hol}}}_{TU}\otimes e_I.$$ In this trivialization we can explicitly write down the differential on $\Phi^\tau({{\mathcal L}}_p(\alpha))$. The “un-twisted differential" on this sheaf is simply ${{\overline{\partial}}}$. By Theorem \[thm:duality\] in Appendix \[app:modules\] the twisted part is given by $$Q (f\otimes e_I):=\sum_{k\geq 0,l\geq 0} m_{k+l+1}(\tau^l,f\otimes e_I,\theta^k).$$ Here $\tau$ is as in Theorem \[thm:koszul\], and recall that $\theta$ was defined in the previous section by the differential equation $\nabla\theta=\omega$ with initial condition $\theta|_p=\alpha$. Again by Theorem \[thm:duality\] we have $$({{\mathscr W}}_{(p,\alpha)}-W)\operatorname{id}+[{{\overline{\partial}}},Q]-Q^2=0.$$ Since $W$ is assumed to be zero this equation simplifies to $[{{\overline{\partial}}},Q]-Q^2=0$. Moreover by degree reason, since $[{{\overline{\partial}}},Q]$ has Dolbeault degree one and $Q^2$ has Dolbeault degree zero, we have $[{{\overline{\partial}}},Q]=0$ and $Q^2=0$. The equation $[{{\overline{\partial}}},Q]=0$ implies that $Q$ is a holomorphic operator.
To understand the cohomology of the differential ${{\overline{\partial}}}+Q$ on $\Phi^\tau({{\mathcal L}}_p(\alpha))$, we first kill the ${{\overline{\partial}}}$ part. For this define a morphism $$F_1: ({{\mathscr A}}^\pi\otimes \prod_{I\subset\left\{1,\cdots,n\right\}} e_I,Q){\rightarrow}\Phi^\tau({{\mathcal L}}_p(\alpha))=({{\mathscr O}}^{{\mathsf{hol}}}_{TU}\otimes \prod_{I\subset\left\{1,\cdots,n\right\}} e_I,{{\overline{\partial}}}+Q)$$ where ${{\mathscr A}}^\pi$ denotes kernel of the operator ${{\overline{\partial}}}: C^\infty_{TU}(\Lambda^\pi) {\rightarrow}C^\infty_{TU}(\Lambda^\pi)\otimes \Omega^1_U$, and the map $F_1$ is simply the embedding map ${{\mathscr A}}^\pi\otimes e_I\hookrightarrow {{\mathscr O}}^{{\mathsf{hol}}}_{TU}\otimes e_I$. The fact that $F_1$ is a quasi-isomorphism follows from the acyclicity of Dolbeault complex. We can also explicitly describe ${{\mathscr A}}^\pi$. This is the set of of formal sums of the form $$\sum_\beta e^{\sum_i (<\partial \beta, e_i> x_i)} \cdot f_\beta \cdot T^\beta$$ for holomorphic functions $f_\beta$ on $TU$. These formal series satisfies the same finiteness condition as in the definition of $\Lambda^\pi$. Note that here the extra term $e^{\sum_i (<\partial \beta, e_i> x_i)}$ appears due to the non-trivial action of ${{\overline{\partial}}}$ on $T^\beta$ by formula \[eq:area\].
The cohomology of $({{\mathscr A}}^\pi\otimes \prod_{I\subset\left\{1,\cdots,n\right\}} e_I,Q)$ can be related to $\Lambda^\pi(p,\alpha)[-n]$ by defining another map $$F_2: ({{\mathscr A}}^\pi\otimes \prod_{I\subset\left\{1,\cdots,d\right\}} e_I,Q) {\rightarrow}\Lambda^\pi(p,\alpha)[-n]$$ which is defined by $$F_2(f \otimes e_I)=\begin{cases} 0; \mbox{\;\; if } I\neq\left\{1,\cdots,n\right\}\\
f(p,\alpha); \mbox{\;\; if } I=\left\{1,\cdots,n\right\}.\end{cases}$$ It is clear from the definition that $F_2$ is compatible with the ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$-modules structures.
The ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$-morphism $F_2$ is a map of complexes.
[**Proof.** ]{}Below we refer to the wedge degree of $e_I$ the integer $|I|$ . Recall the structure maps $m_{k,\beta}$ are of wedge degree $2-k-\mu(\beta)$. Thus the operator $Q_{\beta}$ $$e_I\mapsto \sum_{k\geq 0,l\geq 0} m_{k+l+1,\beta}(\tau^l, e_I,\theta^k)$$ is of wedge degree $1-\mu(\beta)$. This is because locally $\tau=\sum_{i=1}^n -\sqrt{-1}y_i e_i$ and $\theta=\sum_{i=1}^n (x_i-p_i-\sqrt{-1}\alpha_i) e_i$ which are both of wedge degree one. By our assumption negative Maslov index disks do not contribute to $m_k$, it follows that $Q_\beta$ can only increase the wedge degree when $\beta=0$ corresponding to constant maps. By constructions in [@Fukaya] this zero energy part is precisely the exterior algebra on cohomology of torus [^11]. By formula of $\tau$ and $\theta$ we get this zero energy part is $$Q_0(e_I)=m_{2,0}(\tau,e_I)+m_{2,0}(e_I,\theta)= \sum_i (x_i+\sqrt{-1}y_i-p_i-\sqrt{-1}\alpha_i) e_i\wedge e_I$$ which vanishes at $x=p$ and $y=\alpha$. Thus we have shown that $F_2$ is a map of complexes. The lemma is proved.
To prove the proposition it remains to show that $F_2$ is a quasi-isomorphism. For this we make use of the spectral sequence associated the energy filtration on both sides which is valid since $F_2$ preserves this filtration (we refer to [@FOOO] Chapter $6$ for details of the construction of this spectral sequence). We claim that the first page of this spectral sequence is already an isomorphism. The first page is obtained as the cohomology of the operator $Q_0$. Since the factor $e^{\sum_i (<\partial \beta, e_i> x_i)}$ is non-vanishing, the cohomology for each $\beta\in G$ is the same. Thus it suffices to analyze the case when $\beta=0$. That is we would like to show that the map $$F_0: (\prod_{I\subset\left\{1,\cdots,n\right\}} {{\mathscr A}}(TU) \otimes e_I, Q_0) {\rightarrow}{\mathbb{C}}(p,\alpha) [-n]$$ defined by evaluating at $(p,\alpha)$ is a quasi-isomorphism where ${{\mathscr A}}(TU)$ is the sheaf of holomorphic functions on $TU$. To this end we observe that $Q_0$ acts on the cohomology of ${{\overline{\partial}}}$ as the classical Koszul differential. It is well-known that the cohomology of this Koszul complex is ${\mathbb{C}}(p,\alpha)$ concentrated in degree $n$. Note that it is important here that we deal with holomorphic functions since the acyclicity of Koszul complex fails for $C^\infty$ functions. The proposition is proved.
[**Remark:** ]{}If $M$ is a compact symplectic manifold with vanishing first Chern class (computed with a choice of almost complex structure), and $L_p$ a *special* Lagrangian submanifold, then the assumption on the Maslov index is automatically satisfied. In fact in this case all the map $\mu:G(L_p){\rightarrow}2{\mathbb{Z}}$ is identically zero.
[**Remark:** ]{}The assumption on Maslov index is more of a technical nature while the condition that $W=0$ is necessary to describe the object $\Phi^\tau({{\mathcal L}}_p(\alpha))$ as a skyscraper sheaf, since otherwise $\Phi^\tau({{\mathcal L}}_p(\alpha))$ is only a matrix factorization of $W-W(p,\alpha)$ which is not a complex itself. In general without these assumptions we have the following theorem.
\[thm:hms\] The composition of $A_\infty$ functors $$\Phi^\tau\circ P: {\mathsf{Fuk}}^\pi(M){\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}_{TU}^{{\mathsf{hol}}})$$ is a quasi-equivalence onto its image.
[**Proof.** ]{}By Theorem \[thm:lag\] the first functor is in fact a linear $A_\infty$ functor which is a quasi-equivalence onto its image. Thus it remains to show this for the second functor $\Phi^\tau$. That is we would like to show that $$\Phi^\tau: \operatorname{Hom}_{{{\mathscr O}}^{\omega,{{\mathsf{can}}}}}({{\mathcal L}}_{p_1}(\alpha_1), {{\mathcal L}}_{p_2}(\alpha_2)) {\rightarrow}\operatorname{Hom}_{{{\mathscr O}}^{{\mathsf{hol}}}}(\Phi^\tau({{\mathcal L}}_{p_1}(\alpha_1)),\Phi^\tau( {{\mathcal L}}_{p_2}(\alpha_2)))$$ is a quasi-isomorphism. For this we can argue in the way as in the proof of Proposition \[prop:torus\].
In the case when $p_1\neq p_2$ or $\alpha_1\neq \alpha_2$ it was shown in the previous section that the complex $\operatorname{Hom}_{{{\mathscr O}}^{\omega,{{\mathsf{can}}}}}({{\mathcal L}}_{p_1}(\alpha_1), {{\mathcal L}}_{p_2}(\alpha_2))$ has zero cohomology. On the other hand the complex $\operatorname{Hom}_{{{\mathscr O}}^{{\mathsf{hol}}}}(\Phi^\tau({{\mathcal L}}_{p_1}(\alpha_1)),\Phi^\tau( {{\mathcal L}}_{p_2}(\alpha_2)))$ also has vanishing cohomology. For this we can use the spectral sequence associated to the energy filtration to calculate its cohomology. The first page of this spectral sequence is already zero due to the vanishing of $\operatorname{Ext}^*_{TU}({{\mathscr O}}_{(p_1,\alpha_1)},{{\mathscr O}}_{(p_2,\alpha_2)})$.
Thus in the following we consider the case when $p_1=p_2=p$ and $\alpha_1=\alpha_2=\alpha$. We can kill the $\Omega_U^*$ part by observing the following commutative diagram. $$\begin{CD}
\operatorname{Hom}_{{\mathsf{Fuk}}^\pi(M)} ((L_p,\alpha),(L_p,\alpha)) @>{\varphi}^\tau>> {{\mathscr A}}^\pi\otimes \operatorname{End}_{\mathbb{C}}(H^*(L_p,{\mathbb{C}}))\\
@V P VV @VVV \\
\operatorname{Hom}_{{{\mathscr O}}^{\omega,{{\mathsf{can}}}}}({{\mathcal L}}_{u}(\alpha), {{\mathcal L}}_{u}(\alpha)) @>\Phi^\tau>> \operatorname{Hom}_{{{\mathscr O}}^{{\mathsf{hol}}}}(\Phi^\tau({{\mathcal L}}_{u}(\alpha)),\Phi^\tau( {{\mathcal L}}_{u}(\alpha)))
\end{CD}$$ Several explanations of this diagram are in order. First of all as in the proof of Proposition \[prop:torus\], ${{\mathscr A}}^\pi$ is holomorphic functions on $TU$ with values in $\Lambda^\pi$, and the right vertical map is the inclusion $${{\mathscr A}}^\pi\otimes \operatorname{End}_{\mathbb{C}}(H^*(L_p,{\mathbb{C}}))\hookrightarrow {{\mathscr O}}^{{\mathsf{hol}}}\otimes \operatorname{End}_{\mathbb{C}}(H^*(L_p,{\mathbb{C}}))$$ which is a quasi-isomorphism proved in Proposition \[prop:torus\].
Secondly the left vertical arrow is the propagation map defined in Theorem \[thm:lag\] which is also a quasi-isomorphism. Thus in order to show the bottom arrow $\Phi^\tau$ is a quasi-isomorphism it suffices to define the top arrow ${\varphi}^\tau$ and show that it is also a quasi-isomorphism.
We define the map ${\varphi}^\tau$by $${\varphi}^\tau(e_I):= 1\otimes e_J^{{\scriptscriptstyle\vee}}\otimes \sum_{l\geq 0,i_0\geq 0,i_1\geq 0} m_{l+2+i_0+i_1}(\tau^l,e_J,\theta^{i_0},e_I,\theta^{i_1})$$ where as before $\tau=\sum -\sqrt{-1}y_i e_i$ and $\theta=\sum (x_i-p_i-\sqrt{-1}\alpha) e_i$. This formula is the same as the formula to define $\Phi^\tau$, and hence the above diagram is commutative. See Appendix \[app:fm\] for the formula for $\Phi^\tau$ on $\operatorname{Hom}$-spaces.
It remains to show that ${\varphi}^\tau$ is a quasi-isomorphism. For this we consider the spectral sequences associated energy filtrations on both sides. The first page of this spectral sequence is already an isomorphism, again it suffices to prove that the zero energy part sector (corresponding to $\beta=0$) is an isomorphism, which reduces to show that the map of complexes $$\begin{aligned}
(H^*(L_p,{\mathbb{C}}),0)&{\rightarrow}(\operatorname{End}_{{{\mathscr A}}(TU)}({{\mathscr A}}(TU)\otimes H^*(L_p,{\mathbb{C}})), [Q_0,-])\\
e_I & \mapsto (f\otimes e_J \mapsto f\otimes e_I\wedge e_J)\end{aligned}$$ where ${{\mathscr A}}(TU)$ is holomorphic functions on $TU$ and $Q_0$ is the Koszul differential associated to the regular sequence $\left\{x_i+\sqrt{-1}y_i-p_i-\sqrt{-1}\alpha_i\right\}_{i=1}^n$, is a quasi-isomorphism. This an exercise in classical Koszul duality theory between exterior algebras and symmetric algebras. The theorem is proved.
[**Remark:** ]{}Without the assumptions in Proposition \[prop:torus\] it is not clear how to identify the image of $\Phi^\tau$ in ${{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}})$. In general these objects are matrix factorizations of $W$ minus a constant (the internal curvatures), and we expect them to be homotopy equivalent to stabilization matrix factorizations introduced in [@Dyck] Section $2.3$. The trouble to prove this claim is that the matrix factorization defined by the operator $Q (f\otimes e_I):=\sum_{k\geq 0,l\geq 0} m_{k+l+1}(\tau^l,f\otimes e_I,\theta^k)$ mixes various the wedge degrees of $e_I$ while the stabilizations introduced in [@Dyck] only involves operators of wedge degrees $1$ and $-1$.
Mirror symmetry and Fourier-Mukai transform {#sec:fourier}
===========================================
From the symplectic point of view the two sheaves of $A_\infty$ algebras ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ and ${{\mathscr O}}^\omega$ are not much different since ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ in some sense is the minimal model of ${{\mathscr O}}^\omega$. However from the point of view of mirror symmetry there is an important difference between the two. Indeed we have seen from the previous section that the mirror of ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ is $TU$ while in this section we show the mirror of ${{\mathscr O}}^\omega$ is the dual torus bundle $M^{{\scriptscriptstyle\vee}}(U)$. Analogous results such as Proposition \[prop:torus\] and Theorem \[thm:hms\] are obtained in this case as well.
Throughout the section we continue to work with the unobstructedness assumption \[ass:wua\].
#### The case without quantum corrections.
Let us first consider the case when there exists no non-trivial holomorphic disks in $M$ with boundary in $L_u$ for all $u\in U$. In this case we can work over ${\mathbb{C}}$. We shall see how relative (over U) Poincaré bundles appear here.
In Theorem \[thm:koszul\] we constructed a Maurer-Cartan element in the tensor product algebra $ {{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes {{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ which is of the form $\tau:=\sum_{i=1}^n -\sqrt{-1}y_i\otimes e_i$. Considering elements $e_i$ as translation invariant one forms in $\Omega^1(L_u)$, the element $\tau$ can be viewed as an element in ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes{{\mathscr O}}^\omega$. As is explained in Appendix \[app:fm\] Koszul duality can be considered as a special case of Fourier-Mukai transform. Indeed the kernel to construct the Koszul functor $\Phi^\tau:{{\mathsf{tw}}}({{\mathscr O}}^\omega){\rightarrow}\operatorname{Tw}({{\mathscr O}}_{TU}^{{\mathsf{hol}}})$ [^12] is simply the rank one twisted complex over ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes{{\mathscr O}}^\omega$ defined by the Maurer-Cartan element $\tau$.
We can explicitly describe this twisted complex in coordinates $x$, $y$ and $y^{{\scriptscriptstyle\vee}}$. Recall $x$ is coordinates on the base $U$; $y^{{\scriptscriptstyle\vee}}$ and $y$ are angle coordinates on Lagrangian torus and dual torus respectively. We also trivialize $M(U)$ to $T\times U$, and denote by $H_1(T,{\mathbb{R}})$ by $V$, its dual by $V^{{\scriptscriptstyle\vee}}$. Then the tensor product ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes{{\mathscr O}}^\omega$ is simply $C^\infty_{V^{{\scriptscriptstyle\vee}}}\otimes \Omega^*(T) \otimes \Omega^*_U$ [^13] endowed with the differential $d_T+{{\overline{\partial}}}$. Here $d_T$ is the de Rham differential on $\Omega^*(T)$. The operator ${{\overline{\partial}}}$ is defined in the previous section as $\sum_{i=1}^n(\partial/\partial x_i+\sqrt{-1}\partial/\partial y_i) dx_i$. Using the Maurer-Cartan element $\tau$ this differential is twisted to $d_T-\sqrt{-1}y_i e_i +{{\overline{\partial}}}$ where the part additional operator stands for multiplication by $-\sqrt{-1}y_ie_i$. Note that the square of this twisted operator is not zero but the symplectic form $\omega$. Indeed this should be a twisted complex over ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes{{\mathscr O}}^\omega$ which has curvature $-\omega$. We shall denote this twisted complex by $K^\tau$.
To pass from $TU=V^{{\scriptscriptstyle\vee}}\times U$ to $M^{{\scriptscriptstyle\vee}}(U)=T^{{\scriptscriptstyle\vee}}\times U$ it suffices to quotient out the dual lattice group $\Gamma:=(H_1(T,{\mathbb{Z}}))^{{\scriptscriptstyle\vee}}=H_1(T^{{\scriptscriptstyle\vee}},{\mathbb{Z}})$ in $V^{{\scriptscriptstyle\vee}}$. However we note that the kernel does not descend to this quotient in an obvious way since the twisted operator $d_T-\sqrt{-1}y_ie_i +{{\overline{\partial}}}$ is not $\Gamma$-equivariant under the natural translation action. This is where Poincaré bundle comes into play: we can define another $\Gamma$ action on $K^\tau$ so that the operator $d_T-\sqrt{-1}y_ie_i +{{\overline{\partial}}}$ becomes equivariant. This “twisted action" is given by $$\label{eq:action}
\gamma[f(y^{{\scriptscriptstyle\vee}},y)]:= e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}} f(y^{{\scriptscriptstyle\vee}},y-\gamma)$$ where $\gamma\cdot y$ is the natural pairing between $V^{{\scriptscriptstyle\vee}}$ and $V$. It is well-known that if we take the above action and consider invariants in the function part $C^\infty_{V^{{\scriptscriptstyle\vee}}}\otimes C^\infty_T \otimes C^\infty_U$ of $K^\tau$ we get $C^\infty$-sections of the relative Poincaré bundle ${{\mathscr P}}$ on $M(U)\times_U M^{{\scriptscriptstyle\vee}}(U)$. A direct computation verifies the following commutative diagram $$\begin{CD}
K^\tau @>d_T-\sqrt{-1}y_ie_i +{{\overline{\partial}}}>> K^\tau \\
@V\gamma VV @VV\gamma V \\
K^\tau @>d_T-\sqrt{-1}y_ie_i +{{\overline{\partial}}}>> K^\tau.
\end{CD}$$ Thus the operator $d_T-\sqrt{-1}y_ie_i +{{\overline{\partial}}}$ descends to an operator on invariants $(K^\tau)^\Gamma$. Moreover the ${{\mathscr O}}^{{\mathsf{hol}}}_{TU}$-module structure $${{\mathscr O}}^{{\mathsf{hol}}}_{TU}\otimes K^\tau {\rightarrow}K^\tau$$ is $\Gamma$-equivariant if we put the ordinary translation action on ${{\mathscr O}}^{{\mathsf{hol}}}_{TU}$ and the twisted action on $K^\tau$. Taking invariants yields an action $${{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)} \otimes (K^\tau)^\Gamma {\rightarrow}(K^\tau)^\Gamma.$$ where ${{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}}$ is the $\Gamma$-invariants of ${{\mathscr O}}^{{\mathsf{hol}}}_{TU}$ under translation action, i.e. it is the Dolbeault complex of the structure sheaf of the complex manifold $M^{{\scriptscriptstyle\vee}}(U)$. We can then consider $(K^\tau)^\Gamma$ as a “twisted complex" over ${{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)}\otimes {{\mathscr O}}^\omega$. Here we put twisted complex in quote since the two objects $(K^\tau)^\Gamma$ and ${{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)}\otimes {{\mathscr O}}^\omega$ are not isomorphic even as sheaves on $U$ [^14]. As in the previous section we would like to use $(K^\tau)^\Gamma$ as a Kernel to define a local mirror symmetry functor. We shall perform this construction in the general case when quantum corrections are presented.
#### Fourier transform for families.
We briefly recall Fourier transform for families which may be thought of as mirror symmetry in the case when torus fibers do not bound any nontrivial holomorphic disk. Our main references are articles [@AP],[@LYZ] and [@BMP].
For a real torus $T$, and its dual torus $T^{{\scriptscriptstyle\vee}}$, there exists a Poincar' e bundle ${{\mathscr P}}$ over $T\times T^{{\scriptscriptstyle\vee}}$. The bundle ${{\mathscr P}}$ has a natural connection $\nabla$ whose curvature is equal to the canonical symplectic form on $T\times T^{{\scriptscriptstyle\vee}}$. If we split the connection $\nabla$ into $\nabla^{1,0} +\nabla^{0,1}$ corresponding to the splitting $\Omega_{T\times T^{{\scriptscriptstyle\vee}}}^1=p_1^*\Omega_T^1\otimes p_2^*\Omega_{T^{{\scriptscriptstyle\vee}}}^1$, then the partial connection $\nabla^{1,0}$ has zero curvature. Thus we may form the partial de Rham complex $p_1^*\Omega_T\otimes {{\mathscr P}}$ of the connection $\nabla^{1,0}$. Using $p_1^*\Omega_T\otimes {{\mathscr P}}$ as the kernel of Fourier-Mukai transform, one can show that there is an equivalence between the category of local systems on $T$ and the category of skyscraper sheaves on $T^{{\scriptscriptstyle\vee}}$. See for example Propositions $2.6$, $2.7$ and $2.8$ in [@BMP].
In the relative case, one considers a family of Lagrangian torus $X{\rightarrow}B$, and its dual family $X^{{\scriptscriptstyle\vee}}{\rightarrow}B$. Again we main form ${{\mathscr P}}$ over $X\times_B X^{{\scriptscriptstyle\vee}}$, and a partial flat connection $\nabla^{1,0}$. Using the partial de Rham complex of $({{\mathscr P}},\nabla^{1,0})$ as the integral kernel, one deduces an equivalence between the category of local systems supported on fibers of $X{\rightarrow}B$ and the category of skyscraper sheaves on $X^{{\scriptscriptstyle\vee}}$. See Section $3.2$ in [@BMP].
#### Quantum Fourier-Mukai transform.
Let us return to the general case to allow non-trivial holomorphic disks to enter the picture. The previous discussion motivates us to perform the construction in two steps:
- Replace the sheaf ${{\mathscr O}}^{\omega,{{\mathsf{can}}}}$ by ${{\mathscr O}}^\omega$, and perform the same construction as in the previous section;
- Descend to $\Gamma$-invariants by action \[eq:action\].
We begin with Step ${\expandafter\@slowromancap\romannumeral 1@}$ which is almost word by word as in the previous section. Consider the ${{\mathscr O}}^\omega$-module ${{\mathcal L}}_p(\alpha)$ associated to a Lagrangian $L_p$ endowed with a purely imaginary closed one form $\alpha$ (see Section \[sec:lag\] for the construction of ${{\mathscr O}}^\omega$). This is a twisted complex of rank one over ${{\mathscr O}}^\omega$ defined by a Maurer-Cartan element $\theta$ such that $\nabla\theta=\omega$ and $\theta|_p=\alpha$. In local coordinates $\theta=\sum_i (x_i-p_i+\alpha) e_i$, and let $\tau:=\sum_i -\sqrt{-1} y_i e_i \in {{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes {{\mathscr O}}^\omega$ be as before. By constructions in the previous section we get a twisted complex $\Phi^\tau({{\mathcal L}}_p(\alpha))$ over ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$ which, as a ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}$-module, is simply ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes {{\mathscr O}}^\omega$. It is endowed with a twisted differential $d:={{\overline{\partial}}}+Q$ where the operator $Q$ is $$Q(?)=\sum_{k\geq 0,l\geq 0} m_{k+l+1}(\tau^l,?,\theta^k)$$ expressed using structure maps on the tensor product $A_\infty$ algebra ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes {{\mathscr O}}^\omega$.
For the step ${\expandafter\@slowromancap\romannumeral 2@}$ we first observe that it follows from Lemma \[lem:fukaya\] the potential function $W$, *A priori* defined on $TU$, is in fact a function on $M^{{\scriptscriptstyle\vee}}(U)$ with values in $\Lambda^\pi$. This enables us to define the sheaf of curved algebras ${{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)}$ in general. Our next goal is to show that the operator $d:={{\overline{\partial}}}+Q$ on ${{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes {{\mathscr O}}^\omega$ intertwines with action \[eq:action\].
For this, we shall choose an almost complex structure $J$ on $M$ so that when restricted to $\pi^{-1}(U)$ it is given by a standard one in action-angle coordinates. Namely, over $\pi^{-1}(U)$, in action-angle coordinates $x_1,\cdots, x_n$, $y_1^{{\scriptscriptstyle\vee}},\cdots, y_n^{{\scriptscriptstyle\vee}}$, the almost complex structure is $$J: \partial/\partial x_i \mapsto \partial /\partial y_n^{{\scriptscriptstyle\vee}}.$$ The reason we need this is because the boundary of a $J$-holomorphic disk is geodesic in the metric $\omega(-,J-)$, and with this choice of $J$, geodesics are straight lines. Thus if we consider the evaluation maps $$\begin{aligned}
\operatorname{ev}_1& : {{\mathscr M}}_{1+1,\beta} {\rightarrow}L,\\
\operatorname{ev}_0&: {{\mathscr M}}_{0+1,\beta}{\rightarrow}L,\end{aligned}$$ and the forgetful map $${{\mathscr M}}_{1+1,\beta}{\rightarrow}{{\mathscr M}}_{0+1,\beta},$$ we have the identity $$~\label{geodesic:eq}
\operatorname{ev}_1([u,t])=\operatorname{ev}_0([u])+t\cdot \partial\beta$$ where $t$ is coordinate on the fiber of the forgetful map. We use this identification in the proof of the following lemma.
\[lem:intertwine\] For each $\beta\in G$, and $f\in {{\mathscr O}}_{TU}^{{\mathsf{hol}}}\otimes {{\mathscr O}}^\omega$ we have $$\sum_{l\geq 0} m_{l+1+k,\beta}(\tau^l,\gamma(f),\theta^k)= \gamma [\sum_{l\geq 0} m_{l+1+k,\beta}(\tau^l,f,\theta^k)].$$
[**Proof.** ]{}We denote by $T_\gamma$ the translation action $y\mapsto y-\gamma$. Thus $\gamma(f)=e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}}T_\gamma^*(f)=e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}} f(y^{{\scriptscriptstyle\vee}},y-\gamma)$. We have $$\begin{aligned}
& m_{l+1+k,\beta} (\tau^l,\gamma(f),\theta^k) = (\operatorname{ev}_0)_! [\int_{0\leq t_1\leq t_2\cdots\leq t_{k+l+1}\leq 1} \operatorname{ev}^*(\tau^l) \cdot \operatorname{ev}_{l+1}^*(\gamma(f))\cdot \operatorname{ev}^*(\theta^k)] \\
&=(\operatorname{ev}_0)_! [ \frac{1}{l!} <\partial\beta,\tau>^l\int_{0\leq t_{l+1}\cdots\leq t_{k+l+1}\leq 1} t_{l+1}^l\operatorname{ev}_{l+1}^*(\gamma(f))\cdot \operatorname{ev}^*(\theta^k)]\\
&=(\operatorname{ev}_0)_! [\frac{1}{l!}<\partial\beta,\tau>^l \int_{0\leq t_1\cdots\leq t_{k+1}\leq 1}
t_1^l\operatorname{ev}_1^*(e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}})\operatorname{ev}_1^*(T_\gamma f)\cdot\operatorname{ev}^*(\theta^k)]\\
&=(\operatorname{ev}_0)_! [\operatorname{ev}_0^*(e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}}) \frac{<\partial\beta,\tau>^l}{l!} \int_{0\leq t_1\cdots\leq t_{k+1}\leq 1}
t_1^l e^{\sqrt{-1}\gamma\cdot (t_1\partial\beta)}\operatorname{ev}_1^*(T_\gamma f)\cdot\operatorname{ev}^*(\theta^k)].\end{aligned}$$ The last equality follows from Equation \[geodesic:eq\]. Apply projection formula $f_!(f^*a\cdot b)=a \cdot f_! b$ to the last summation yields $$e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}} (\operatorname{ev}_0)_! [\frac{1}{l!}<\partial\beta,\tau>^l \int_{0\leq t_1\cdots\leq t_{k+1}\leq 1}
t_1^l e^{\sqrt{-1}\gamma\cdot (t_1\partial\beta)}\operatorname{ev}_1^*(T_\gamma f)\cdot\operatorname{ev}^*(\theta^k)].$$ Summing over $l$ yields $$\begin{aligned}
&\sum_{l\geq 0} m_{l+1+k,\beta} (\tau^l,\gamma(f),\theta^k)=\\
&= e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}} (\operatorname{ev}_0)_! [\int_{0\leq t_1\cdots\leq t_{k+1}\leq 1}
e^{(t_1\partial\beta)\cdot \tau} e^{\sqrt{-1}\gamma\cdot (t_1\partial\beta)}\operatorname{ev}_1^*(T_\gamma f)\cdot\operatorname{ev}^*(\theta^k)]\\
&= e^{\sqrt{-1}\gamma\cdot y^{{\scriptscriptstyle\vee}}} (\operatorname{ev}_0)_! [\int_{0\leq t_1\cdots\leq t_{k+1}\leq 1}
e^{(t_1\partial\beta)\cdot T_\gamma\tau}\operatorname{ev}_1^*(T_\gamma f)\cdot\operatorname{ev}^*(\theta^k)]\\
&= \gamma\left\{ (\operatorname{ev}_0)_! [\int_{0\leq t_1\cdots\leq t_{k+1}\leq 1}
e^{(t_1\partial\beta)\cdot\tau}\operatorname{ev}_1^*(f)\cdot\operatorname{ev}^*(\theta^k)]\right\}\\
&=\gamma\left\{ \sum_{l\geq 0} (\operatorname{ev}_0)_! [\int_{0\leq t_1\cdots\leq t_{k+1}\leq 1}
\frac{1}{l!} <\partial\beta,\tau>^l t_1^l\cdot \operatorname{ev}_1^*(f)\cdot\operatorname{ev}^*(\theta^k)]\right\}\\
&=\gamma [\sum_{l\geq 0} m_{l+1+k,\beta}(\tau^l,f,\theta^k)].\end{aligned}$$ Thus the lemma is proved.
[**Remark:** ]{}It follows from the proof that in the above formula the part $\theta^k$ can be replaced by any elements. Also, the proof we gave above assumes transversality for moduli spaces involved. At present there are several different approaches to deal with transversality issues. In any case, we believe that geometric argument as in the proof of the lemma should work in full generality with any successful approach towards solving the transversality problems.
Returning to the discussion of the twisted complex $\Phi^\tau({{\mathcal L}}_p(\alpha))$ endowed with differential ${{\overline{\partial}}}+Q$, the above lemma implies that the operator $Q$ is $\Gamma$-equivariant. Since ${{\overline{\partial}}}$ is also equivariant the sum operator ${{\overline{\partial}}}+Q$ is also $\Gamma$-equivariant. So we can take $\Gamma$-invariants to get a ${{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)}$-module structure on the complex $[\Phi^\tau({{\mathcal L}}_p(\alpha))]^\Gamma$. The generalization of this construction to general objects of ${{\mathsf{tw}}}({{\mathscr O}}^\omega)$ is straight-forward. Thus we have describe a functor from ${{\mathsf{tw}}}({{\mathscr O}}^\omega)$ to $\operatorname{Tw}({{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)})$ on the level of objects. The capital $\operatorname{Tw}$ stands for twisted complexes of possibly infinite rank. This is necessary for us here. We denote this functor by $\Phi^{{\mathscr P}}$ where we think of ${{\mathscr P}}:=[K^\tau]^\Gamma$ as a certain quantized relative Poincaré bundle.
We continue to define $\Phi^{{\mathscr P}}$ on morphisms. In fact the functor $\Phi^{{\mathscr P}}$ can be constructed as an $A_\infty$ functor ${{\mathsf{tw}}}({{\mathscr O}}^\omega){\rightarrow}\operatorname{Tw}({{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)})$. Indeed by Appendix \[app:fm\] the $A_\infty$ functor ${{\mathsf{tw}}}({{\mathscr O}}^\omega) {\rightarrow}\operatorname{Tw}({{\mathscr O}}^{{\mathsf{hol}}}_{TU})$ associated to the twisting cochain $\tau$ has the form $$\Phi^\tau(a_1,\cdots,a_k)(x):= \sum_{l\geq 0,i_0\geq 0,\cdots,i_k\geq 0} m_{l+k+1+i_0+\cdots+i_k}(\tau^l,x,\theta^{i_0},a_1,\theta^{i_1},\cdots,a_k,\theta^{i_k})$$ which by Lemma \[lem:intertwine\] (and its following remark) is also $\Gamma$-equivariant. Hence this functor descends to $\Gamma$-invariants to give the desired $A_\infty$ functor $\Phi^{{\mathscr P}}$.
#### Mirror dual of torus fibers.
Since the image of $\Phi^{{\mathscr P}}$ are twisted complexes of infinite rank over ${{\mathscr O}}^{{\mathsf{hol}}}_{M^{{\scriptscriptstyle\vee}}(U)}$, it is *A priori* unclear whether these objects are quasi-isomorphic to an object in the bounded derived category of coherent sheaves on $M^{{\scriptscriptstyle\vee}}(U)$ (with coefficients in $\Lambda^\pi$). We show this is the case for Lagrangian torus fibers endowed with a unitary line bundle $[\alpha]$ on it. Here the $\alpha$ is a connection one form corresponding to the line bundle $[\alpha]$ which is determined up to translation by lattice points.
\[prop:image\] Assume that strictly negative Maslov index does not contribute to structure maps $m_k$, and assume further that the potential function $W\equiv0$ over $U$. Then the object $\Phi^{{\mathscr P}}({{\mathcal L}}_p(\alpha))$ is quasi-isomorphic to the skyscraper sheaf $\Lambda^\pi(p,\alpha)[-n]$ over the point $p\in U$. As in Proposition \[prop:torus\] we let an element $f\in {{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}}$ act on $\Lambda^\pi$ via multiplication by $f(p,\alpha)$.
[**Proof.** ]{}The proof is analogous to that of Proposition \[prop:torus\]. Let us trivialize $M(U)=T\times U$ which induces an identification $$\Phi^{{\mathscr P}}({{\mathcal L}}_p(\alpha))\cong [(C^\infty_{V^{{\scriptscriptstyle\vee}}}\otimes \Omega^*_U) \otimes \Omega^*(T)]^\Gamma.$$ This complex is endowed with the $\Gamma$-equivariant differential ${{\overline{\partial}}}+Q$ as describe above. Moreover we have $[{{\overline{\partial}}},Q]=Q^2=0$ as before. To calculate the cohomology of this complex we first observe a quasi-isomorphism $$F_1: ( [{{\mathscr A}}(TU)\otimes \Omega^*(T)]^\Gamma, Q) {\rightarrow}([(C^\infty_{V^{{\scriptscriptstyle\vee}}}\otimes \Omega^*_U) \otimes \Omega^*(T)]^\Gamma,{{\overline{\partial}}}+Q)$$ where recall that ${{\mathscr A}}(TU)$ is $\Lambda^\pi$ valued holomorphic function on $TU$. The map $F_1$ is defined by $$[(f\cdot T^\beta )\otimes \zeta] \mapsto (e^{\sum_i (<\partial \beta, e_i> x_i)} \cdot f \cdot T^\beta)\otimes \zeta$$ where again the extra term $e^{\sum_i (<\partial \beta, e_i> x_i)}$ appears due to the non-trivial action of ${{\overline{\partial}}}$ on $T^\beta$ \[eq:area\]. That $F_1$ is a quasi-isomorphism follows from the exactness of Dolbeault complex.
Next we define a morphism of ${{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}}$-modules $$F_2: ( [{{\mathscr A}}(TU)\otimes \Omega^*(T)]^\Gamma, Q) {\rightarrow}\Lambda^\pi(p,\alpha)[-n]$$ by formula $F(f\otimes \zeta):=f(p,\alpha)\int_T \zeta$. It is clear that $F_2$ respects the action of ${{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}}$ on both sides. Let us check that $F_2$ is a morphism of complexes, i.e. we would like to show that $F\circ Q=0$. For this observe that integration on $T$ kills all elements of form degree strictly less than the dimension $n$ of $T$. Moreover by the Maslov index assumption the only operator that increases this degree is $Q_0$ corresponding to trivial holomorphic disks. As in Proposition \[prop:torus\] this operator is explicitly given by $$\begin{aligned}
Q_0(f\otimes \zeta) &=f\otimes d_{{{\mathsf{dR}}}} \zeta+ m_{2,0}(\tau,f\otimes \zeta)+m_{2,0}(f\otimes\zeta,\theta)\\
&= f\otimes d_{{{\mathsf{dR}}}} \zeta + \sum_i (x_i+\sqrt{-1}y_i-p_i-\sqrt{-1}\alpha_i)\cdot f \otimes e_i\wedge \zeta.\end{aligned}$$ Applying $F_2$ to this sum, which by definition is evaluation at $(p,\alpha)$ and integrate over $T$, yields zero. Thus we have shown that $F_2$ is a map of complexes. It remains to prove that it is also a quasi-isomorphism. For this we consider the spectral sequences associated to energy filtration on both sides. As in the proof of Proposition \[prop:torus\] it suffices to analyze the case for $\beta=0$. This is done in the following lemma, which finishes the proof the proposition.
Denote by ${{\mathscr A}}(TU,{\mathbb{C}})$ the sheaf of holomorphic functions on $TU$ with values in ${\mathbb{C}}$, then the map $${\varphi}: ([{{\mathscr A}}(TU,{\mathbb{C}}) \otimes \Omega^*(T)]^\Gamma,Q_0) {\rightarrow}{\mathbb{C}}[-n]$$ defined by ${\varphi}(f\otimes \zeta):=f(p,\alpha)\cdot \int_T \zeta$ is a quasi-isomorphism.
[**Proof.** ]{}This is a classical result in Fourier transform for families. We include a proof here for completeness. The proof is similar to that of Proposition $2.6$, $2.7$ and $2.8$ in [@BMP]. We only do this for the case when the dimension of $T$ is one, i.e. $T$ is a circle. The general case follows from Kunneth type argument.
We work in the universal cover of $T$ with affine coordinate $y^{{\scriptscriptstyle\vee}}$. Coordinates on $TU$ are $x$ and $y$. The operator $Q_0$ acts by $[\partial/\partial y^{{\scriptscriptstyle\vee}}+ (z-p-\sqrt{-1}\alpha)]dy^{{\scriptscriptstyle\vee}}$ where $z=x+\sqrt{-1}y$. To analyze the cohomology of $Q_0$ it is better to work in another “gauge", i.e. we conjugate the operator $Q_0$ with an automorphism which is given by multiplication by $e^{(z-p-\sqrt{-1}\alpha)\cdot y^{{\scriptscriptstyle\vee}}}$. Under this conjugation the operator $Q_0$ is identified with $\partial/\partial y^{{\scriptscriptstyle\vee}}\circ dy^{{\scriptscriptstyle\vee}}$, the de Rham differential in $y^{{\scriptscriptstyle\vee}}$-direction. Moreover the conjugation also changes the lattice group action on the variables $y^{{\scriptscriptstyle\vee}}$ and $y$. In the $y$-direction $\Gamma\subset {\mathbb{R}}^{{\scriptscriptstyle\vee}}$ acts simply by translation, while in the $y^{{\scriptscriptstyle\vee}}$-direction $m\in {\mathbb{Z}}\subset {\mathbb{R}}$ acts on $s\in C^\infty({\mathbb{R}}\times {\mathbb{R}}^{{\scriptscriptstyle\vee}})$ by $$s(y^{{\scriptscriptstyle\vee}},y) \mapsto e^{m\cdot(z-p-\sqrt{-1}\alpha)} s(y^{{\scriptscriptstyle\vee}}-m,y).$$ To prove the lemma it suffices to show that an element $f(z)\otimes s(y^{{\scriptscriptstyle\vee}}) dy^{{\scriptscriptstyle\vee}}$ is exact if and only if $f(p,\alpha)\int_0^1 s(v)dv=0$. The element $f\otimes s dy^{{\scriptscriptstyle\vee}}$ is exact if there exists an anti-derivative of the form $$f(z)\otimes t(y^{{\scriptscriptstyle\vee}}):= f(z) \otimes \int_0^{y^{{\scriptscriptstyle\vee}}} s(v) dv + h(z)$$ which is periodic in $y$-direction and in $y^{{\scriptscriptstyle\vee}}$-direction we have $$f(z) e^{m\cdot(z-p-\sqrt{-1}\alpha)}\otimes t(y^{{\scriptscriptstyle\vee}}-m)= f(z)\otimes t (y^{{\scriptscriptstyle\vee}}).$$ Using the fact that $f\otimes s$ is lattice group invariant we find that such an anti-derivative exists if and only if $$(e^{m\cdot(z-p-\sqrt{-1}\alpha)}-1) h= f\int_0^ms(v)dv$$ for all $m$. Denote by $q:=e^{z-p-\sqrt{-1}\alpha}$, then using the lattice group invariance of $s(v)$ we get $\int_0^m s(v)dv=(1+q+\cdots+q^{m-1})\int_0^1 s(v)dv$.
If $f(p,\alpha)\int_0^1 s(v)dv=0$, then either $f(p,\alpha)=0$ or $\int_0^1 s(v) dv=0$. In the first case the fraction $$\frac{f\int_0^ms(v)dv}{q^m-1}= \frac{f\int_0^1 s(v)dv}{q-1}$$ which is independent of $m$ extends to the point $(p,\alpha)$ since $(e^{m\cdot(z-p-\sqrt{-1}\alpha)}-1)$ vanishes in first order at $(p,\alpha)$. If $\int_0^1s(v)dv=0$, then $\int_0^m s(v)dv=0$. Hence we can take $h$ to be simply zero. So in either case the form $f\otimes s dy^{{\scriptscriptstyle\vee}}$ is exact.
Conversely if $f\otimes s dy^{{\scriptscriptstyle\vee}}$ is exact then $f(p,\alpha)\int_0^m s(v)dv=0$ of all $m$. In particular we have $f(p,\alpha)\int_0^1s(v)dv=0$. The lemma is proved.
Theorem \[thm:hms\] in the previous section can also be generalized to this situation for ${{\mathscr O}}^\omega$ and ${{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}}$. This result is summarized in the following theorem. Its proof is again to use spectral sequences associated to energy filtrations to reduce to classical results. In this case instead of using classical Koszul duality we use classical Fourier transform for families, see for instance [@BMP]. We shall not repeat the proof here.
\[thm:hms2\] The composition of $A_\infty$ functors $${\mathsf{Fuk}}^\pi(M)\stackrel{P}{{\rightarrow}} {{\mathsf{tw}}}({{\mathscr O}}^\omega) \stackrel{\Phi^{{\mathscr P}}}{{\rightarrow}} \operatorname{Tw}({{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}})$$ is a quasi-equivalence onto its image. Here the first functor $P$ was defined in Theorem \[thm:lag\].
[**Remark:** ]{}Here we need to use $\operatorname{Tw}({{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}})$ to include infinite rank objects over ${{\mathscr O}}_{M^{{\scriptscriptstyle\vee}}(U)}^{{\mathsf{hol}}}$. It is an interesting question to do homological perturbation on $\Phi^{{\mathscr P}}\circ P({{\mathcal L}}_p(\alpha))$ to reduce to an object of finite rank. This problem might be related to the appearance of theta functions in mirror symmetry [^15].
Homological mirror symmetry on toric manifolds {#sec:functor}
==============================================
As an immediate application of our general theory we prove a version of homological mirror symmetry between a toric symplectic manifold and its Landau-Ginzburg mirror.
\[thm:toric\] Let $M$ be a compact smooth toric symplectic manifold, and denote by $\pi:M(\Delta^{{\mathsf{int}}}){\rightarrow}\Delta^{{\mathsf{int}}}$ the Lagrangian torus fibration over the interior of the polytope of $M$. Then there exists an $A_\infty$ functor $\Psi: {\mathsf{Fuk}}^\pi(M) {\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{T\Delta^{{\mathsf{int}}}})$ which is a quasi-equivalence onto its image.
[**Proof.** ]{}We take $J$ to be the canonical integrable complex structure on $M$. Since biholomorphisms of $M$ acts transitively on torus fibers over $\Delta^{{\mathsf{int}}}$, we get a sheaf of $A_\infty$ algebras over $\Delta^{{\mathsf{int}}}$ by constructions in Section \[sec:symp\]. Moreover it is known after [@FOOO] that the weak unobstructedness assumption \[ass:wua\] holds in this context. Thus all results in this paper applies to this situation, and the theorem is simply an example of Theorem \[thm:hms\].
[**Remark:** ]{}The functor $\Psi$ in the above theorem is simply the composition $\Phi^\tau\circ P$ where $P$ is the propagation functor used in Theorem \[thm:lag\], and $\Phi^\tau$ is the Koszul duality functor $\Phi^\tau$ defined in Section \[sec:koszul\]. The image of $\Psi$ consists of matrix factorizations of $W$ on $T\Delta^{{\mathsf{int}}}$ which can be calculated by the formula $$Q (f\otimes e_I):=\sum_{k\geq 0,l\geq 0} m_{k+l+1}(\tau^l,f\otimes e_I,\theta^k).$$ It is plausible that these objects split generate ${{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{T\Delta^{{\mathsf{int}}}})$. But we do not know how to prove this generation result in general. Some of the difficulties are
- working over Novikov ring instead of ${\mathbb{C}}$;
- the map $Q$ mixes various wedge degrees.
In the Fano case we can specialize to $T=e^{-1}$ and work over ${\mathbb{C}}$, which enables us to get around the first issue. When the dimension is less than or equal to two the inhomogeneity does not appear, which allows us to prove the following.
\[thm:toricfano\] Let $M$ be a compact smooth toric Fano symplectic manifold of dimension less or equal to two. In this case we can work over ${\mathbb{C}}$ by evaluating the parameter $T$ at $e^{-1}$. Then there is a functor $\Psi^{\mathbb{C}}:{\mathsf{Fuk}}^\pi(M,{\mathbb{C}}) {\rightarrow}{{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{T\Delta^{{\mathsf{int}}}}\otimes{\mathbb{C}})$ which is a quasi-equivalence of $A_\infty$ categories.
[**Proof.** ]{}The functor $\Psi^{\mathbb{C}}$ is simply the reduction of $\Psi$ at the evaluation $T=e^{-1}$, which is valid under the Fano condition. By the previous theorem it suffices to show that the image of $\Psi^{\mathbb{C}}$ split generates the target category ${{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{T\Delta^{{\mathsf{int}}}}\otimes{\mathbb{C}})$. This generation follows from explicitly computing the operator $Q$ and using the generation result of T. Dyckerhoff [@Dyck] Section $4$. Note that this generation result requires $W$ to have isolated singularities which was proved in [@FOOOtoric] Theorem $10.4$.
Next we compute the operator $Q$. In the following the degree $|I|$ of $e_I$ is referred to as wedge degree. Recall the operator $Q_\beta$ is of wedge degree $1-\mu(\beta)$. Since for toric manifolds there are no negative Maslov index holomorphic disks, the operator $Q=\sum_{\beta\in G} Q_\beta$ has only one part $Q_0$ that increases the wedge degree. As we saw in the proof of Proposition \[prop:torus\] $Q_0$ is the Koszul differential associated to the regular sequence $z_i -p_i-\sqrt{-1}\alpha_i$.
If the dimension is less than or equal to two, then the operator $Q_\beta$ is necessary of wedge degree $-1$ corresponding to $\mu(\beta)=2$. Such type of matrix factorizations is shown to split generate ${{\mathsf{tw}}}({{\mathscr O}}^{{\mathsf{hol}}}_{T\Delta^{{\mathsf{int}}}}\otimes{\mathbb{C}})$ by [@Dyck] Section $4$. The theorem is proved.
#### An example: ${\mathbb{C}}P^1$.
A particularly simple example is the case $M={\mathbb{C}}P^1$. Since it is Fano we shall work over ${\mathbb{C}}$. With appropriate choice of its symplectic form we assume $U=(0,1)\subset {\mathbb{R}}$ as is in [@FOOOtoric] Section $5$. Let $e$ be a trivialization of $R^1\pi_*{\mathbb{Z}}$, and let $x$, $y^{{\scriptscriptstyle\vee}}$, $y$ be associated affine coordinates. It is known that the $A_\infty$ algebra associated to the Lagrangian torus fiber $L_{x}$ for $x\in (0,1)$ is a two dimensional vector space generated by ${{\mathbf 1}}, e$ with ${{\mathbf 1}}$ a strict unit. All the rest $A_\infty$ products are $$\begin{aligned}
m_0 &= \exp(-x)+ \exp(x-1);\\
m_1 (e) &= \exp(-x)-\exp(x-1); \\
&\cdots;\\
m_k (e^{\otimes k}) &=\frac{1}{k!}[ \exp(-x)+ (-1)^k \exp(x-1)];\\
&\cdots.\end{aligned}$$ The potential function is equal to $$\begin{aligned}
W(x,-\sqrt{-1}y)&= \sum_{i=0}^\infty m_k( (-\sqrt{-1}y e)^{\otimes k})\\
&= \sum_{i=0}^\infty \frac{1}{k!} (-\sqrt{-1}y)^k [ \exp(-x)+ (-1)^k \exp(x-1)]\\
&=\exp(-z)+\exp(z-1).\end{aligned}$$ Thus $W$ is indeed a holomorphic function. Let $a\in {\mathbb{R}}$ be a real number, and let $u\in (0,1)$ be a point. Consider the Lagrangian brane $(L_u,-\sqrt{-1}a)$. From this data we get define an $A_\infty$ module ${{\mathcal L}}_u(\sqrt{-1}a)$ over ${{\mathscr O}}_{M(U)}^{\omega,{{\mathsf{can}}}}$ with internal curvature $W(u,a)$ by constructions in Section \[sec:lag\]. Let us describe its image under the Koszul functor $\Phi^\tau$. It suffice to compute the operator $Q$ on generators ${{\mathbf 1}}$ and $e$. For this we have $$\begin{aligned}
Q(1)&=\sum_{k,l} m_{k+l+1}(\tau^l,{{\mathbf 1}},\theta^k)\\
&=\sum_{k,l} (x-u-\sqrt{-1}a)^k(-\sqrt{-1}y)^l m_{k+l+1}(e^{\otimes l},{{\mathbf 1}},e^{\otimes k})\\
&= e\otimes [(x-u-\sqrt{-1}a)-(-\sqrt{-1}y)] \mbox{\;\;\; (by our sign convention)}\\
&=e\otimes [z-u-\sqrt{-1}a].\end{aligned}$$ A more technical computation of $Q(e)$ by formula gives $\frac{W-W(u,a)}{u+\sqrt{-1}a-z}$, and hence $Q^2+[W-W(u,a)]\operatorname{id}=0$ as is expected from the general theory. Thus we see that $\Phi^\tau({{\mathcal L}}_u(\sqrt{-1}a))$ is a matrix factorization of $-[W-W(u,a)]$.
Modules with internal curvature. {#app:modules}
================================
Let $A$ be an $A_\infty$ algebra. In this section we define $A_\infty$ modules over $A$ possibly with an internal curvature. We show how weak Maurer-Cartan elements of $A$ give rise to such structures. These algebraic constructions naturally occur in Lagrangian Floer theory.
Throughout the construction we work over a base ring $R$, and all modules considered here are free $R$-modules. We follow the sign convention used in [@FOOO]. We refer to [@Keller] Section $3$ and $4$ for basics of $A_\infty$ algebras, homomorphisms and modules.
#### $A_\infty$ modules.
An $A_\infty$ module $M$ over an $A_\infty$ algebra $A$ is defined by a collection of maps $\rho_k(-;-): (A^{\otimes k})\otimes M{\rightarrow}M$ of degree $1-k$ such that $$\label{eq:module}
\sum_{i+j=N} \rho_i(\operatorname{id}^i;\rho_j(\operatorname{id}^j;-))+ \sum_{r+s+t=N} \rho_{r+t+1} (\operatorname{id}^r,m_s(\operatorname{id}^s),\operatorname{id}^t;-)=0$$ for all $N\geq 0$. When applied to elements $(a_1\otimes\cdots\otimes a_N\otimes x)\in A^{\otimes N}\otimes M$ extra signs come out by Koszul sign rule, for example when $N=0,1$ the above relation reads
$$\begin{aligned}
\rho_0(\rho_0(x))&+\rho_1(m_0;x)=0;\\
\rho_0(\rho_1(a;x))&+(-1)^{|a|-1}\rho_1(a;\rho_0(x))+\\
+&\rho_{1}(m_1(a);x)+\rho_{2}(m_0,a;x)+(-1)^{|a|-1}\rho_2(a,m_0;x)=0.\end{aligned}$$
Using the Bar construction we can interpret an $A_\infty$ module structure on a $R$-module $M$ as an $A_\infty$ homomorphism $\rho:A{\rightarrow}\operatorname{End}(M)$ [^16]. Recall an $A_\infty$ homomorphism between This correspondence is explicitly given by $$\left\{\rho_k\right\}_{k=0}^{\infty} \mapsto \rho:=\prod_{k=0}^\infty \rho_k\in \operatorname{Hom}_{A_\infty}(A,\operatorname{End}(M))$$
#### $A_\infty$-modules with internal curvature.
For applications in this paper we need to introduce a weaker notion of modules: those endowed with “internal curvatures". For its definition we will fix $\lambda\in R$ an even element in the ground ring.
\[def:module\] An $A_\infty$ module $M$ over $A$ with internal curvature $\lambda$ is defined by structure maps $\rho_k(-;-): (A^{\otimes k})\otimes M{\rightarrow}M$ of degree $1-k$ which satisfies the same axioms as in equation \[eq:module\] except for $N=0$ in which case we require that $$\rho_0(\rho_0(x))+\rho_1(m_0;x)=\lambda\operatorname{id}_M.$$
From the point of view of $A_\infty$ homomorphisms, we can add the element $\lambda\operatorname{id}_M$ as a curvature element for the matrix algebra $\operatorname{End}(M)$, and we denote the resulting curved algebra by $\operatorname{End}^\lambda(M)$. Then straight-forward computation shows that an $A_\infty$ module $M$ over $A$ with internal curvature $\lambda$ is the same as an $A_\infty$ homomorphism $\rho:A{\rightarrow}\operatorname{End}^\lambda(M)$.
#### From weak Maurer-Cartan elements to modules with internal curvature.
Let us see how $A_\infty$ modules with internal curvature can arise from a weak Maurer-Cartan element of $A$. Recall if $A$ is an $A_\infty$ algebra with a strict unit ${{\mathbf 1}}$, an odd element $b\in A^1$ is a weak Maurer-Cartan element if we have $$\sum_{k=0}^\infty m_k(b,\cdots,b)=\lambda{{\mathbf 1}}$$ for some even element $\lambda\in R$. Using such an element we can define an $A_\infty$ module structure on the same underlying space of $A$ which has internal curvature $\lambda$. We denote this $A_\infty$ module by $A^b$. Its structure maps are defined by
$$\rho_k^b(a_1,\cdots,a_k;x) :=\sum_{i=0}^{\infty} m_{i+k+1}(a_1,\cdots,a_k,x,b^{\otimes i})$$
Let us check the first axiom, i.e. $\rho^b_0(\rho^b_0(x))+\rho^b_1(m_0;x)=\lambda\operatorname{id}_{A^b}$.
$$\begin{aligned}
&\rho^b_0(\rho^b_0(x)) =\sum_{i,j} m_{i+j+1}(m_{i+1}(x,b^{\otimes i}),b^{\otimes j}) \\
&=-\sum_{r\geq 0, s\geq 0, t\geq 0} (-1)^{|x|-1} m_{r+t+2} (x,b^{\otimes r},m_s(b^{\otimes s}),b^{\otimes t}) -\sum_{k\geq 0} m_{k+2}(m_0,x,b^{\otimes k}) \\
&=(-1)^{|x|}m(x,\lambda{{\mathbf 1}})-\rho_1^b(m_0;x) \mbox{\;\;\; (by the weak Mauer-Cartan equation)}\\
&=\lambda\operatorname{id}-\rho_1^b(m_0;x) \mbox{\;\;\; (by our sign convection of strict unit).}\end{aligned}$$
The rest identities can be checked similarly using the fact that $\lambda{{\mathbf 1}}$ is a multiple of strict unit, and hence does not contribute to higher products. We denote by $\rho^b:A{\rightarrow}\operatorname{End}^\lambda (A^b)$ the corresponding $A_\infty$ homomorphism.
#### Twisted complexes.
The category of $A_\infty$ modules are usually defined as a differential graded category. However for purposes of the current paper it is better to use the category of twisted complexes of $A$ which is an $A_\infty$ category. We refer to the paper of B. Keller [@Keller] Section $7$ for details of these categorical constructions. The category of twisted complexes over an $A_\infty$ algebra will be denoted by ${{\mathsf{tw}}}(A)$. Intuitively this can be thought of as the $A_\infty$ analogue of differential graded modules over an algebra $A$ that are free of finite rank.
To include modules with internal curvatures we need to modify slightly the definition of ${{\mathsf{tw}}}(A)$. In the following we explain this modification. This modified version of ${{\mathsf{tw}}}(A)$ is a direct sum of $R$-linear categories $${{\mathsf{tw}}}(A):=\coprod_{\lambda\in R^{{\mathsf{even}}}} {{\mathsf{tw}}}^\lambda(A).$$ For each $\lambda\in R^{{\mathsf{even}}}$ the category ${{\mathsf{tw}}}^\lambda(A)$ consists of twisted complexes over $A$ with internal curvature $\lambda$. Thus the conventional definition of ${{\mathsf{tw}}}(A)$ corresponds to ${{\mathsf{tw}}}^0(A)$ in our notation. Let us explain in more detail the construction of ${{\mathsf{tw}}}^\lambda(A)$.
For each $\lambda$ the objects of ${{\mathsf{tw}}}^\lambda(A)$ are pairs $(V,b)$ where $V$ is a finite rank ${\mathbb{Z}}/2{\mathbb{Z}}$-graded free $R$-module, and $b$ is a weak Maurer-Cartan element of the tensor product $A\otimes \operatorname{End}_R(V)$ with internal curvature $\lambda$. By constructions in the previous paragraph these data give rise to an $A_\infty$ module over $A\otimes V$ with internal curvature $\lambda$. Strictly speaking in the previous paragraph we only dealt with the case when $V$ is of rank one over $R$, but the general case only requires more index.
Let us illustrate the morphism space between two pairs $(V,b)$ and $(W,\delta)$ when both $V$ and $W$ is one dimensional. In this case the $\operatorname{Hom}$ space, as a graded $R$-module, is simply $A$ itself. It is endowed with a differential $d$ twisted by $b$ and $\delta$ which is explicitly given by formula $$a\mapsto \sum_{k,l} m_{k+l+1}(b^k,a,\delta^l).$$ Using $A_\infty$ relations and Maurer-Cartan equations one shows that $d^2=[F(b)-F(\delta)]\operatorname{id}=[\lambda-\lambda]\operatorname{id}=0$ where $F(b)$ and $F(\delta)$ are internal curvatures of $b$ and $\delta$. This explains the reason why twisted complexes with different internal curvatures do not interact with each other. For the general case when $V$ and $W$ are of any finite rank, the definition is similar using matrix compositions. We refer the details to [@Keller] Section $8$.
#### Upper-triangular condition.
Finally we end this appendix with an important technical point involved in the construction of twisted complexes. In conventional definitions one usually assumes that the (weak) Maurer-Cartan element $b\in A\otimes \operatorname{End}_R(V)$ to satisfy strict upper-triangular condition. This has two important implications. Namely this assumption implies convergence of Maurer-Cartan equation and also the convergence of twisted differential. Secondly it also implies that the homotopy category of ${{\mathsf{tw}}}(A)$ embeds fully into the derived category of $A$; moreover the image of this embedding is simply the triangulated closure of $A$ as an $A_\infty$ module over itself (this works when $m_0$ of $A$ is trivial).
While working with such a condition has nice homological implications, it is too restrictive for applications in mirror symmetry. Indeed it follows from the upper-triangularity that the only rank one twisted complex is $A$ itself if $m_0$ vanishes. But as is shown in Section \[sec:lag\] we would like to associate to each Lagrangian torus fiber a non-trivial rank one twisted complex. Thus we would like to work with twisted complexes which might not satisfy the upper-triangular condition. In this case convergence of relevant series is not automatic, and needs to be taken care of separately. For Lagrangian Floer theory as needed in this paper this convergence follows from results in [@Fukaya]. Secondly the homotopy category of ${{\mathsf{tw}}}(A)$ in our definition might not admit a fully faithfully embedding into the derived category of $A$.
Koszul duality as Fourier-Mukai transform {#app:fm}
=========================================
In this appendix we construct a Koszul duality functor as a type of affine version of Fourier-Mukai transform. We also define such a functor on modules with internal curvatures. It follows from our definition the Koszul functors preserve internal curvatures of modules.
We will need to use another sign convection [@Keller] since the previous sign convention is not convenient to deal with tensor product of algebras. To avoid possible confusions from using two different signs, we first clarify the relationship between them.
#### Sign conventions.
In the sign convention used in the previous appendix, the maps $m_k$ are considered as degree one maps between suspensions $(A[1])^{\otimes k}{\rightarrow}A[1]$. The advantage of doing so is that there is no signs in the $A_\infty$ algebra axioms, i.e. for each $n\geq 0$ we have $$\sum_{r+s+t=n} m_{r+1+t}(\operatorname{id}^r\otimes m_s \otimes \operatorname{id}^t)=0.$$ When applying to elements we get signs by the Koszul rule. Using maps $m_k$ we can define its corresponding linear maps $m_k^\epsilon:A^{\otimes k}{\rightarrow}A$ by requiring the following diagram to be commutative: $$\begin{CD}
A^{\otimes k} @>m_k^\epsilon>> A \\
@VV [1]^{\otimes k} V @VV [1] V \\
(A[1])^{\otimes k} @>m_k>> A[1].
\end{CD}$$ Here the map $[1]: A{\rightarrow}A[1]$ defined by $a\mapsto a[1]$ is the identity map on the underlying $R$-module, but since it is a degree one map it yields signs when applied to tensor products by Koszul sign rule. Explicitly we have $$m_k^\epsilon(a_1,\cdots,a_k)[1]=(-1)^{\epsilon_k} m_k(a_1[1],\cdots,a_k[1])$$ where $\epsilon_k=\sum_{i=1}^k |a_i|(k-i)$. The above identity applied to the $A_\infty$ axioms of $m_k$ yields $$(-1)^{\epsilon_n}\sum_{r+s+t=n} (-1)^{r+st} m^\epsilon_{r+1+t}(\operatorname{id}^r\otimes m^\epsilon_s \otimes \operatorname{id}^t)=0.$$ Dividing the sign $(-1)^{\epsilon_n}$ gives the $A_\infty$ axioms for the maps $m^\epsilon_k$. Using this relationship between $m_k$ and $m_k^\epsilon$ we can freely pass from one to the other. For instance the weak ${\mathsf{MC}}$ equation expressed using $m_k^\epsilon$ reads $$\sum_{k=0}^\infty (-1)^{\frac{k(k-1)}{2}}m_k^\epsilon(b^{\otimes k}) =\lambda{{\mathbf 1}}.$$ A strict unit ${{\mathbf 1}}$ in the $\epsilon$-sign convention becomes $$\begin{aligned}
m_2^\epsilon(1,x)=m_2^\epsilon(x,1)&=x\mbox{\;\;\; and} \\
m_k^\epsilon(a_1,\cdots,a_i,{{\mathbf 1}},\cdots,a_{k-1})&=0 \mbox{\;\;\; for all $k\neq 2$.}\end{aligned}$$
#### Tensor product.
Let $A$ be a strict unital $A_\infty$ algebra, and let $B$ be a curved differential graded algebra. Form their tensor product $B\otimes A$ which is an $A_\infty$ algebra with structure maps defined by $$\begin{aligned}
m_0^\epsilon&:={{\mathbf 1}}\otimes m_0^\epsilon +W\otimes{{\mathbf 1}};\\
m_1^\epsilon(b\otimes a) &:=db\otimes a+(-1)^{|b|} b\otimes m_1^\epsilon (a);\\
m_k^\epsilon(b_1\otimes a_1,\cdots,b_k\otimes a_k)&:=(-1)^{\eta_k}(b_1\cdots b_k)\otimes m_k^\epsilon(a_1,\cdots,a_k) \mbox{\;\;\; for $k\geq 2$.}\end{aligned}$$ Here $W$ is the curvature term of $B$, the two ${{\mathbf 1}}$’s are units, and the sign in the last equation is $\eta_k=\sum_{i=1}^{k-1} |a_i|(|b_{i+1}|+\cdots+|b_k|)$. We have abused the notation $m_k^\epsilon$ for both structure maps on $A$ and $B\otimes A$. Since they are applied to different types of elements, no confusion can arise by doing so.
#### Koszul duality as Fourier-Mukai transform.
Next we describe a construction of a functor $\Phi^\tau:{{\mathsf{tw}}}(A){\rightarrow}{{\mathsf{tw}}}(B)$ associated to a given Maurer-Cartan element $\tau\in B\otimes A$. For simplicity we will assume that $A$ is of finite rank over $R$. This is for the purpose that $\Phi^\tau$ lands inside ${{\mathsf{tw}}}(B)$, i.e. it is of finite rank over $B$. If we replace the target category by $\operatorname{Tw}(B)$ consisting of twisted complexes over $B$ of possibly infinite rank then all constructions below still go through.
The intuitive idea to construct such a functor is that the element $\tau$ determines an $A_\infty$ module $(B\otimes A)^\tau$ over $B\otimes A$ which can be viewed as a kernel for an integral transform from ${{\mathsf{tw}}}(A)$ to ${{\mathsf{tw}}}(B)$. To realize this idea we proceed as follows.
Given an $A_\infty$ module $M$ with internal curvature $\lambda$, we denote by $\rho^M$ the corresponding $A_\infty$ homomorphism $A{\rightarrow}\operatorname{End}^\lambda(M)$. For $M\in{{\mathsf{tw}}}(A)$ by our assumption that $A$ is of finite rank, it follows that $M$ is also of finite rank. For general $M$ we assume $M$ is of finite rank below.
The map $\rho^M$ induces another $A_\infty$ homomorphism $\rho^M_B: B\otimes A{\rightarrow}B\otimes\operatorname{End}^\lambda(M)$ by scalar extension to $B$. Using $\rho^M_B$ we can push forward the given Maurer-Cartan element $\tau$ to get a Maurer-Cartan element of $\operatorname{End}^\lambda(M)\otimes B$. Such a Maurer-Cartan element by definition is a twisted complex structure on $B\otimes_R M$ with internal curvature $\lambda$. The following Theorem gives a more explicit description of this construction with formulas.
\[thm:duality\] The maps $(\rho^M_B)_k:(B\otimes A)^k {\rightarrow}B\otimes\operatorname{End}^\lambda(M)$ defined by $$\begin{aligned}
(\rho^M_B)_0&:={{\mathbf 1}}\otimes\rho^M_0;\\
(\rho^M_B)_1(b_1\otimes a_1)&:=b_1\otimes\rho^M_1(a_1);\\
(\rho^M_B)_k(b_1\otimes a_1,\cdots,b_k\otimes a_k) &:=(-1)^{\eta_k} (b_1\cdots b_k)\otimes \rho^M_k(a_1,\cdots,a_k)\end{aligned}$$ form an $A_\infty$ homomorphism $\rho^M_B: B\otimes A {\rightarrow}B\otimes\operatorname{End}^\lambda(M)$. Moreover if $\tau\in B\otimes A$ is a Maurer-Cartan element, its push-forward $Q:=(\rho_B^M)_*\tau=\sum_{k=0}^\infty (-1)^{\frac{k(k-1)}{2}} (\rho^M_B)_k(\tau^{\otimes k})$ is a Maurer-Cartan element of $B\otimes \operatorname{End}^\lambda(M)$, i.e. we have $$(\lambda-W)\operatorname{id}+[d,Q]-Q^2=0.$$ Here $-W$ is the curvature of $B$, and $d$ is its differential.
[**Proof.** ]{}The proof is straightforward verifications of formulas and keeping track of signs. We omit it here.
By the above theorem we described what $\Phi^\tau$ does on the level of objects. Namely we define $\Phi^\tau(M)$ to be the twisted complex on $B\otimes_R M$ defined by the weak Maurer-Cartan element $(\rho_B^M)_*\tau$. Note that $\Phi^\tau(M)$ and $M$ have the same internal curvature, i.e. we have a map $$\Phi^\tau: {{\mathsf{tw}}}^\lambda(A) {\rightarrow}{{\mathsf{tw}}}^\lambda(B)$$ for each $\lambda\in R^{{\mathsf{even}}}$. Next we describe $\Phi^\tau$ on the level of morphisms. We can define $\Phi^\tau$ as an $A_\infty$ functor from ${{\mathsf{tw}}}(A)$ to ${{\mathsf{tw}}}(B)$. Let us illustrate this for a rank one twisted complex $A^b$ over $A$ with internal curvature $\lambda$. The space $\operatorname{End}(A^b)$ is an $A_\infty$ algebra with structure maps $m_k^b$ defined by $$m_k^b(a_1,\cdots,a_k):=\sum_{i_0\geq 0,\cdots,i_k\geq 0} m_{i_0+i_1+\cdots+i_k+k} (b^{i_0},a_1,b^{i_1},\cdots,b^{i_{k-1}},a_k,b^{i_k}).$$ We need to define an $A_\infty$ homomorphism from $\operatorname{End}(A^b)$ to the differential graded algebra $\operatorname{End}(\Phi^\tau(A^b))$. Since $\Phi^\tau(A^b)=B\otimes_R A$ as a $R$-module, we use structure maps $m_k$ on the tensor product $B\otimes A$ to describe this homomorphism. Explicitly it is given by $$\Phi^\tau(a_1,\cdots,a_k)(x):= \sum_{l\geq 0,i_0\geq 0,\cdots,i_k\geq 0} m_{l+k+1+i_0+\cdots+i_k}(\tau^l,x,b^{i_0},a_1,b^{i_1},\cdots,a_k,b^{i_k}).$$ The case of higher rank twisted complexes requires no more than putting more index into the above equation. We refer to Section $7.3$ of K. Lefèvre-Hasegawa’s thesis [@LH] for a more detailed discussion of this $A_\infty$ homomorphism.
[GP]{}
M. Abouzaid. [On homological mirror symmetry for toric varieties.]{} Thesis (Ph.D.)–The University of Chicago. 2007. 92 pp.
D. Arinkin; A. Polishchuk. [Fukaya category and Fourier transform.]{} Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 261–274, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI, 2001.
D. Auroux; L. Katzarkov; D. Orlov. [Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves.]{} Invent. Math. 166 (2006), no. 3, 537–582.
U. Bruzzo; G. Marelli; F. Pioli. [A Fourier transform for sheaves on Lagrangian families of real tori]{} J. Geom. Phys. 39 (2001) 174-182; J. Geom. Phys. 41 (2002) 312-329.
K. Chan; N. C. Leung. [Mirror symmetry for toric Fano manifolds via SYZ transformations.]{} Adv. Math. 223 (2010), no. 3, 797–839.
K. Chan; S.-C. Lau; N. C. Leung. [SYZ mirror symmetry for toric Calabi-Yau manifolds.]{} J. Differential Geom. 90 (2012), no. 2, 177–250.
C.-H. Cho. [Products of Floer cohomology of torus fibers in toric Fano manifolds.]{}
C.-H. Cho; Y.-G. Oh. [Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds]{} Asian J. Math. Vol. 10, No. 4 (2006), 773-814.
T. Dyckerhoff. [Compact generators in categories of matrix factorizations]{} Duke Math. J. Volume 159, Number 2 (2011), 223-27.
K. Fukaya. [Cyclic symmetry and adic convergence in LagrangianFloer theory.]{} [arXiv:0907.4219.]{}
K. Fukaya. [Floer homology of Lagrangian foliation and noncommutative mirror symmetry [slowromancap1@]{}.]{} Preprint, available on the author’s website.
K. Fukaya; Y.-G. Oh; H. Ohta and K. Ono. [Lagrangian intersection Floer theory- anomaly and obstruction]{} AMS/IP Studies in Advanced Mathematics, vol 46, in press, AMS/International Press.
K. Fukaya; Y.-G. Oh; H. Ohta and K. Ono. [Canonical models of filtered $A_\infty$-algebras and Morse complexes]{} [arXiv:0812.1963]{}
K. Fukaya; Y.-G. Oh; H. Ohta and K. Ono. [Lagrangian Floer theory on compact toric manifolds [slowromancap1@]{}.]{} Duke. Math. J. 151 (2010), 23-174.
K. Fukaya; Y.-G. Oh; H. Ohta and K. Ono. [Lagranaign Floer theory on compact toric manifolds [slowromancap2@]{}; Bulk deformations.]{} Selecta Math. New Series, no. 3, (2011), 609-711.
K. Fukaya; Y.-G. Oh; H. Ohta and K. Ono. [Floer theory and Mirror symmetry on toric manifolds.]{} [arXiv:1009.1648.]{}
B. Fang; C.-C. M. Liu; D. Treumann; E. Zaslow. [The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties.]{} page 3-37, Geometry and Analysis Vol. 2, Adv. Lect. Math (ALM) 18, Higher Education Press, 2010.
M. Gross; B. Siebert. [Mirror symmetry via logarithmic degeneration data, [slowromancap1@]{}.]{} J. Differential Geom. 72 (2006), no. 2, 169–338.
M. Gross; B. Siebert. [Mirror symmetry via logarithmic degeneration data, [slowromancap2@]{}.]{} J. Algebraic Geom. 19 (2010), no. 4, 679–780.
B. Keller. [Introduction to A-infinity algebras and modules.]{} Homology Homotopy Appl. 3 (2001), no. 1, 1–35.
M. Kontsevich. [Homological Algebra of Mirror Symmetry.]{} [arXiv:alg-geom/9411018.]{}
M. Kontsevich; Y. Soibelman. [Homological mirror symmetry and torus fibrations.]{} Symplectic geometry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publ., River Edge, NJ, 2001.
K. Lefevre-Hasegawa. [Sur les A-infini catégories]{} [arXiv.org:math/0310337]{}
N. C. Leung; S.-T. Yau; E. Zaslow [From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform.]{} Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 209–225, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI, 2001.
D. Nadler; E. Zaslow. [Constructible sheaves and the Fukaya category.]{} J. Amer. Math. Soc. 22 (2009), no. 1, 233–286.
A. Polishchuk; E. Zaslow. [Categorical mirror symmetry: the elliptic curve.]{} Adv. Theor. Math. Phys. 2 (1998), no. 2, 443–470.
P. Seidel. [Homological mirror symmetry for the quartic surface.]{} [arXiv:math/0310414]{}
P. Seidel. [Homological mirror symmetry for the genus two curve.]{} J. Algebraic Geom. 20 (2011), no. 4, 727–769.
P. Seidel. [Some speculations on pairs-of-pants decompositions and Fukaya categories.]{} [arXiv:1004.0906]{}
N. Sheridan. [Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space.]{} [arXiv:1111.0632]{}
A. Strominger; S-T. Yau; E. Zaslow. [Mirror symmetry is T-duality.]{} Nuclear Phys. B 479 (1996), no. 1-2, 243–259.
J. Tu. [Algebraic structures of the family version of Lagrangian Floer theory. (In preparation)]{}
J. Tu. [On the reconstruction problem in mirror symmetry.]{} [arXiv:1208.5912]{}
[^1]: Mathematics Department, University of Oregon, Eugene OR 97403, USA, [*e-mail:*]{}[junwut@uoregon.edu]{}
[^2]: It seems to the author impossible to make a complete list, so he apologize for this.
[^3]: The original SYZ explanation for this was from the so-called T-duality in physics.
[^4]: The notion is due to the usage of the sheaf ${{\mathscr H}}$ which is fiber-wise cohomology ring which was referred to as the canonical model in [@FOOO2].
[^5]: This assertion is almost correct except we are only using polynomial function in the $y$ variable as opposed to smooth functions. In the main body of the paper we shall use smooth functions.
[^6]: Here several words are in quote since we need to work with Novikov ring, and hence these notions need to be interpreted correctly.
[^7]: Strictly speaking this version of Floer theory was developed by K. Fukaya in [@Fukaya] heavily based on the work of himself, Y.-G. Oh, H. Ohta and K. Ono [@FOOO].
[^8]: More precisely the construction in [@FOOO] yields an $A_\infty$ algebra structure whose zero energy part is only homotopic to the de Rham algebra of $L$; while in [@Fukaya] the zero energy part is exactly the same.
[^9]: This definition is due to the fact that in the convergent case we specialize $T$ to be $e^{-1}$, as is done in [@CO] Section $13$.
[^10]: In fact we can take any element in $H^1(L_p,\Lambda_0)$. We restricted to purely imaginary ones for the purpose of mirror symmetry. See Sections \[sec:koszul\] and \[sec:fourier\].
[^11]: It is essential for this proof that the zero energy part of $H^*(L_p,\Lambda^\pi)$ is given by the exterior product as opposed to an $A_\infty$ algebra homotopy equivalent to it. The latter was proved for constructions in [@FOOO] Section $7.5$.
[^12]: Here we need to use the capital $\operatorname{Tw}$ since ${{\mathscr O}}^\omega$ is of infinite rank over $\Omega_U^*(\Lambda^\pi)$.
[^13]: Here and in the following $\otimes$ means completed tensor product.
[^14]: Indeed the Poincaré bundle ${{\mathscr P}}$ is not a topologically trivial vector bundle.
[^15]: The assertion is motivated from the case of elliptic curves.
[^16]: Note that the product on the graded matrix algebra $\operatorname{End}(M)$ is defined by $({\varphi}\otimes\psi)\mapsto(-1)^{|{\varphi}|}{\varphi}\circ\psi$ where the sign appears due to our sign convention.
|
---
abstract: 'The argument shift method is a well-known method for generating commutative families of functions in Poisson algebras from central elements and a vector field, verifying a special condition with respect to the Poisson bracket. In this notice we give an analogous construction, which gives one a way to create commutative subalgebras of a deformed algebra from its center (which is as it is well known describable in the terms of the center of the Poisson algebra) and an $L_\infty$-differentiation of the algebra of Hochschild cochains, verifying some additional conditions with respect to the Poisson structure.'
---
<span style="font-variant:small-caps;">$L_\infty$-derivations and the argument shift method for deformation quantization algebras</span>
G.Sharygin
Introduction: classical argument shift method
=============================================
History and motivation
----------------------
In the study of integrable systems one naturally and inevitably encounters the question, whether there exists a sufficiently large set of first integrals of the given Hamiltonian equation. To answer it, it is often convenient to have a large collection of commutative subalgebras in the Poisson algebra of functions on the phase space. Thus, constructing and classifying such algebras is an important indispensable part of the integrable systems theory.
Among other methods of constructing commutative families of functions, the argument shift method is one of the simplest and relatively universal. It was first observed in the papers by Manakov in a particular case of Euler equation (see [@Manakov1976]), and later it was formulated in full generality (in the case, where the Poisson manifold is equal to the coadjoint representation of a Lie algebra equipped with the standard Poisson structure) by Mischenko and Fomenko, [@MiFo78].
Since that time, the method has been the subject of minute discussions and numerous generalizations (one of which we explain in this paper). In particular, it was shown that under mild conditions the commutative algebras yielded by it are maximal (if the direction of the shift is accurately chosen) and complete; see the papers of Bolsinov, Sadetov, Zhang, Izosimov and others, [@Bol1; @Bol2; @Sad; @BolZh; @Izo].
On the other hand, according to Kontsevich (see [@Kon97]) and many others one can apply a quantization procedure to any Poisson manifold, thus obtaining the “quantum observable” algebra: the associative noncommutative algebra, linearly isomorphic to the space of (usually smooth, or polynomial) functions on the phase space (often with a formal parameter $\hbar$ added to the picture), such that the product in it is a deformation of the usual commutative product of functions, and the linear part of the deformation is given by the Poisson structure (see the discussion at the end of the section \[sec:secintro2\] below). One can regard this algebra as the suitable domain for the investigation of the quantum mechanical problems; similarly to the classical Hamiltonian mechanics, solving such system involves finding a suitable system of mutually commuting elements in the quantized algebra (this time the commutation is understood in the usual algebraic sense). It is natural to assume that such algebras should be somehow related with the commutative Poisson subalgebras on the same space.
This idea, however simple in seems at first sight, is pretty hard to put into practice; the search for the corresponding commutative quantum algebras involves many nontrivial constructions and derives inspiration from most variagated sources. For example, in case when the phase space is given by the dual space of a Lie algebra, this involves the study of universal enveloping algebras and their generalizations, like Yangians and quantum groups, see [@FeiginFrenkel; @Molev1].
In this paper we are discussing a possible way to construct quantum counterpart of commutative algebras, yielded by the argument shift method. It turns out that in spite of being very algebraic in its nature, and allowing numerous interpretations on classical level, this method does not allow a straightforward interpretation in quantum case. The known constructions, which give “quantum integrable systems”, related to this method, involve hard results about the structure of universal enveloping algebras, Yangians, or the properties of the universal enveloping algebras of affine Lie algebras at the critical level, see [@Molev2; @Molev3; @Tarasov; @Rybnikov; @Talalaev].
In this paper we suggest an algebraic construction, which generalizes the argument shift method to the deformed algebra. It is based on the theory of $L_\infty$-algebras and $L_\infty$-morphisms. One can say, that it allows us to obtain an analogue of the procedure, that generates the commutative subalgebras, rather than constructs the commutative subalgebras in the quantized algebra directly from the algebras, generated by the shift at the classical level. A considerable drawback of this construction is that to this moment I have no example, where this procedure actually works, the conditions, that should be satisfied for it to be defined, being too hard to observe. I hope to amend this in the papers to come.
The remaining part of this paper is organised as follows: in section \[sec:secintro2\] we recall the classical version of the argument shift method, which we phrase in a greater generality than it is usually done: it turns out, that the method is based on a purely algebraic consideration and does not depend on the actual geometric nature of the ingredients, it involves. In section 2 we recall the definitions and basic properties of $L_\infty$ algebras and $L_\infty$-morphisms. We also recall the role they play in Kontsevich’s deformation quantization construction. Then in section 3 we define $L_\infty$ derivations of a DG Lie algebra and give the definition of weak Nijenhuis property. We then use this notion in the particular case of the Lie algebra of local Hochschild cochains (polydifferential operators) to show how one can get the first nontrivial commutation relation, analogous to $\{\xi(f),\xi(g)\}=0$ in the proposition \[prop:ashiftcl\]. In order to move forward we need to replace the weak Nijenhuis condition by the strong Nijenhuis condition, which we do in section 4; then we show (theorem \[theo:om\]), how in this case the $L_\infty$-derivation gives rise to the analogue of the argument shift method. Finally, in the last section we make few remarks on the possible further developments and applications of our ideas.
**Acknowledgements.** During the work on this paper, the author was supported by the Russian Science Foundartion grant 16-11-10069, RFFI grant 18-01-0398 and the Simons foundation. The major part of the paper was written during the visit to PKU; the author expresses his deepest gratitude to this university for wonderful working conditions.
The classical construction {#sec:secintro2}
--------------------------
In what follows we let $A$ be a Poisson algebra, i.e. a commutative unital algebra over a ground field $\Bbbk,\,char(\Bbbk)=0$ (usually $\Bbbk={\mathbb{R}}$ or ${\mathbb{C}}$) with a bracket $\{,\}:A\otimes A\to A$, verifying the following set of relations:
[(*i*)]{}
: $\{,\}$ is bilinear over $\Bbbk$;
[(*ii*)]{}
: $\{,\}$ is antisymmetric: $\{f,g\}=-\{g,f\}$;
[(*iii*)]{}
: $\{,\}$ verifies the Leibniz rule: $$\{f,gh\}=\{f,g\}h+\{f,h\}g.$$
[(*iv*)]{}
: $\{,\}$ verifies the Jacobi identity: $$\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0.$$
Center $Z_{\pi}(A)$ of Poisson algebra $A$ is the subalgebra in $A$, spanned by the elements $x\in A$ such that $\{x,y\}=0$ for all $y\in A$; elements of $Z_\pi(A)\setminus\Bbbk$ are sometimes called *Casimir elements* or simply *Casimirs*.
If $A=C^\infty(M)$ is the algebra of smooth functions on a manifold, then conditions (*i*)-(*iii*) mean that the bracket $\{,\}$ is determined by a bivector field $\pi\in\wedge^2TM$; in this case, as it is well-known (see e.g. [@Ciccoli]) the condition (*iv*) is equivalent to the equation $$\label{eq:Poi1}
[\pi,\pi]=0.$$ Here $[,]$ is the Nijenhuis-Schouten bracket of polyvector fields. In this situation the bracket $\{f,g\}$ of any two functions is given by the formula: $$\{f,g\}=\pi(df,dg).$$ One calls the bracket $\{,\}$ *Poisson bracket* and the bivector $\pi$ verifying *Poisson bivector*.
An important particular case is when $\pi$ has maximal rank; in this case its inverse differential $2$-form $\omega=\pi^{-1}$ satisfies the equation $d\omega=0$ and one calls it a *symplectic structure* on $M$. Remark, that in symplectic case there are no Casimirs in $C^\infty(M)$, i.e. $Z_\pi(A)={\mathbb{C}}$ or ${\mathbb{R}}$ in this case.
Let $\xi$ be a vector field on $M$; recall that one says that $\xi$ is *Poisson* field, if the Lie derivative of $\pi$ with respect to it vanishes; this condition is equivalent to the equation $$\label{eq:Poivf}
\xi(\{f,g\})=\{\xi(f),g\}+\{f,\xi(g)\}.$$ An important particular case of Poisson fields are the fields of the form $X_f=\pi^\sharp(df)$; here $\pi^\sharp$ is the “index raising” operator induced by $\pi$; in coordinates: $$\pi^\sharp(\alpha)^k=\pi^{kl}\alpha_l$$ for any $1$-form $\alpha$. The fields $X_f$ are called *Hamiltonian*, they are characterized by the equation $X_f(g)=-\{f,g\}$; the equality ${\ensuremath{\mathcal L}}_{X_f}\pi=0$ now follows from the Jacobi identity.
Another important class of vector fields consists of *Nijehuis* fields: *we shall say, that a field $\xi$ is Nijenhuis, if the second Lie derivative with respect to $\xi$ kills the Poisson bivector $\pi$, i.e. $$\label{eq:defNj}
{\ensuremath{\mathcal L}}_\xi^2\pi=0.$$* A good example of Nijenhuis field is any constant (in linear coordinates) field on a vector space, in case $\pi$ is linear; for example, one can take $M={\ensuremath{\mathfrak{g}}}^*$ with the standard Poisson-Lie structure.
The purpose of considering Nijenhuis fields follows from the next observation (here $A(M)$ is the Poisson algebra of smooth functions on a manifold with respect to a bivector $\pi$):
\[prop:ashiftcl\] For any $f,\,g\in Z_\pi(A(M))$, any Nijenhuis vector field $\xi$ and any natural $k,\,l$ the following relation holds: $\{{\ensuremath{\mathcal L}}_\xi^kf,{\ensuremath{\mathcal L}}_\xi^lg\}=0$.
Observe, that if $\xi$ is in fact Poisson field, then ${\ensuremath{\mathcal L}}_\xi^kf\in Z_\pi(A(M))$ for all $k\in\mathbb N,\,f\in Z_\pi(A(M))$, so the statement holds trivially; however, in the case of a “genuine” Nijenhuis field, i.e. if ${\ensuremath{\mathcal L}}_\xi\pi\neq0,\,{\ensuremath{\mathcal L}}_\xi^2\pi=0$, the functions ${\ensuremath{\mathcal L}}_\xi^kf$ are no more central. Let us now sketch the proof of this proposition:
We shall prove by induction in $N\ge0$ that ${\ensuremath{\mathcal L}}^k_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^lf,d{\ensuremath{\mathcal L}}_\xi^mg)=0$ for all $k+l+m=N$ and all $f,g\in Z_\pi(A(M))$ (this statement is a bit more than what the proposition needs, since $\pi(d{\ensuremath{\mathcal L}}_\xi^kf,d{\ensuremath{\mathcal L}}_\xi^lg)=\{{\ensuremath{\mathcal L}}_\xi^kf,{\ensuremath{\mathcal L}}_\xi^lg\}$). The base of this statement ($N=0$) is trivially true, since $f,g$ are central.
Next, assuming that this statement holds for all small $N$, and differentiating the corresponding equations by $\xi$ for all $k,\,l$ and $m$ we obtain the following system of linear equations on the values of ${\ensuremath{\mathcal L}}_\xi^k\pi(d{\ensuremath{\mathcal L}}_\xi^lf,d{\ensuremath{\mathcal L}}_\xi^mg)$: $${\ensuremath{\mathcal L}}^{k+1}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^lf,d{\ensuremath{\mathcal L}}_\xi^mg)+{\ensuremath{\mathcal L}}^k_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{l+1}f,d{\ensuremath{\mathcal L}}_\xi^mg)+{\ensuremath{\mathcal L}}^k_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^lf,d{\ensuremath{\mathcal L}}_\xi^{m+1}g)=0,\ \forall k+l+m=N.$$ Observe next that due to the conditions on $\xi$ and $\pi$ only $k=0$ and $k=1$ can appear, so this system is reduced to $$\begin{cases}
&{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^lf,d{\ensuremath{\mathcal L}}_\xi^mg)+\pi(d{\ensuremath{\mathcal L}}_\xi^{l+1}f,d{\ensuremath{\mathcal L}}_\xi^mg)+\pi(d{\ensuremath{\mathcal L}}_\xi^lf,d{\ensuremath{\mathcal L}}_\xi^{m+1}g)=0,\ l+m=N-1\\
&{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{l+1}f,d{\ensuremath{\mathcal L}}_\xi^mg)+{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^lf,d{\ensuremath{\mathcal L}}_\xi^{m+1}g)=0,\ l+m=N-1.
\end{cases}$$ We are going to show that this system under the conditions of our proposition can have only trivial solutions. To this end consider the second equation: varying $l$ from $0$ to $N-1$, we get inductively: $${\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1},dg)=-{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-2}f,d{\ensuremath{\mathcal L}}_\xi g)={\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-3}f,d{\ensuremath{\mathcal L}}_\xi^2g)=\dots=(-1)^{N-1}{\ensuremath{\mathcal L}}_\xi\pi(df,{\ensuremath{\mathcal L}}_\xi^{N-1}g),$$ i.e. ${\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-k-1}f,d{\ensuremath{\mathcal L}}_\xi^kg)=(-1)^k{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1},dg)$.
Now we turn to the first series of equations: if $m=0$, it is reduced to $${\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1}f,dg)+\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1}f,d{\ensuremath{\mathcal L}}_\xi g)=0,$$ since $g$ is central. So $$\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1}f,d{\ensuremath{\mathcal L}}_\xi g)=-{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1}f,dg)={\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-2}f,d{\ensuremath{\mathcal L}}_\xi g).$$ Now from the the equation of the first series with $m=1$ we get: $${\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-2}f,d{\ensuremath{\mathcal L}}_\xi g)+\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1}f,d{\ensuremath{\mathcal L}}_\xi g)+\pi(d{\ensuremath{\mathcal L}}_\xi^{N-2}f,d{\ensuremath{\mathcal L}}_\xi^{2}g)=0,$$ or $$\pi(d{\ensuremath{\mathcal L}}_\xi^{N-2}f,d{\ensuremath{\mathcal L}}_\xi^{2}g)=-2{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-2}f,d{\ensuremath{\mathcal L}}_\xi g)=2{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1}f,dg)=2{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-3}f,d{\ensuremath{\mathcal L}}_\xi^2g).$$ We plug this into the equation for $m=2$ in the first series: $${\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-3}f,d{\ensuremath{\mathcal L}}_\xi^2g)+\pi(d{\ensuremath{\mathcal L}}_\xi^{N-2}f,d{\ensuremath{\mathcal L}}_\xi^2g)+\pi(d{\ensuremath{\mathcal L}}_\xi^{N-3}f,d{\ensuremath{\mathcal L}}_\xi^{3}g)=0$$ we get $$\pi(d{\ensuremath{\mathcal L}}_\xi^{N-3}f,d{\ensuremath{\mathcal L}}_\xi^{3}g)=-3{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1},dg).$$ By induction we get $$\pi(d{\ensuremath{\mathcal L}}_\xi^{N-k}f,d{\ensuremath{\mathcal L}}_\xi^{k}g)=(-1)^kk{\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1},dg).$$ But when $k=N$ we see that $\pi(df,d{\ensuremath{\mathcal L}}_\xi^Ng)=0$, since $f$ is central. So ${\ensuremath{\mathcal L}}_\xi\pi(d{\ensuremath{\mathcal L}}_\xi^{N-1},dg)=0$ and all the other expressions vanish automatically.
In fact, this statement and its proof we gave here are applicable to binary operations of any kind $m:V\otimes V\to V$ on any vector space $V$ and any linear operator $\xi:V\to V$; we just put: $$\xi(m)(a,b)=\xi(m(a,b))-m(\xi(a),b)-m(a,\xi(b)),\ a,b\in V$$ for any two operators of this sort. Then *if $\xi^2(m)=\xi(\xi(m))=0$, and $$a,b\in Z_m(V)=\{x\in V\mid m(x,y)=0=m(y,x),\,\forall y\in V\}$$ then for all $k,\,l\in\mathbb N_0$ we have $m(\xi^k(a),\xi^l(b))=0$*.
As one knows from Kontsevich’s theorem, for every Poisson structure $\pi$ on a manifold $M$ there exists an associative star-product $\star=\star_\pi$ on the space of formal power series with coefficiewnts in functions on $M$: $$\label{eq:star1}
f\star g=fg+\frac{\hbar}{2}\{f,g\}+\sum_{k=2}^\infty\hbar^kB_k(f,g),$$ where $B_k(-,-)$ are suitable bidifferential operators; to make our notation more uniform we shall usually replace the term $\frac12\{f,g\}$ in this formula by $B_1(f,g)$. Below we shall give a brief description of Kontsevich’s construction, proving the existence of this formula. It is known, that many properties of the original algebra of functions on $M$ can be easily transferred to the algebra ${\ensuremath{\mathcal A}}(M)=(C^\infty(M)[[\hbar]],\star)$, called the deformation quantization of $M$. In particular, one can show that for every element in $Z_\pi(A(M))$ there will be a well-defined element $\tilde f\in Z({\ensuremath{\mathcal A}}(M))$ (where on the right we denote by $Z({\ensuremath{\mathcal A}}(M))$ the center of ${\ensuremath{\mathcal A}}(M)$, i.e. the subalgebra of elements, commuting with every other element in ${\ensuremath{\mathcal A}}(M)$).
One can ask, *if the argument shift method can be transferred to the quantized algebra as well*? The proof of proposition \[prop:ashiftcl\] being purely algebraic and suitable for any type of binary operation, this conjecture seems quite plausible. However the answer to this question still remains unknown, although in some important particular cases it is known to be positive. For example, this is the case when $M={\ensuremath{\mathfrak{g}}}^*$ for a semisimple Lie algebra ${\ensuremath{\mathfrak{g}}}$, but the construction of commutative families in that situation is quite complicated and is based on the subtle properties of affine Lie agebras and Yangians, i.e. on the study of infinite-dimensional Lie theory.
The purpose of the remaining part of this paper is to describe a construction analogous to the argument shift method in which the notion of Nijenhuis vector field is replaced with a suitable construction from $L_\infty$ algebras. We hope, that this notion being somewhat less restrictive than the usual Nijenhuis condition , it will be possible to find instances of this structure in a wider set of examples (and hence to find large commutative subalgebras in quantized algebras).
$L_\infty$ structures and morphisms
===================================
$L_\infty$-algebras
-------------------
We begin with the classical definition:
\[def:linf\] One says that a graded space $V=\bigoplus_{n\ge0} V_n$ is given the structure of $L_\infty$-algebra, if its graded exterior algebra $\Lambda_V=\Lambda^*(V[1])$ is equipped with operator $D:\Lambda_V\to\Lambda_V$ of degree $1$ such that
[(*i*)]{}
: $D^2=0$;
[(*ii*)]{}
: $D$ is differentiation with respect to the free cocommutative coalgebra structure $\Delta_V$ on $\Lambda_V$, i.e. $\Delta_V:\Lambda_V\to\Lambda_V\otimes\Lambda_V$ and $$\Delta_VD_V=(D_V\otimes 1+1\otimes D_V)\Delta_V.$$
It follows from condition (*ii*) that the map $D$ is determined by its “Taylor series” coefficients: $D=\{D_n\}_{n\ge 1}$ where $D_n:\Lambda^n(V[1])\to V[1]$ is a degree $1$ map (or, if we restore the original grading $D_n:\Lambda^n(V)\to V$ has degree $2-n$); now we can write the condition (*i*) “in coordinates”, i.e. in terms of $D_n$. For instance (we use the same symbols for homogeneous elements of $V$ and for their degrees, i.e. $a\in V_a,\,b\in V_b$ etc.): $$\begin{aligned}
&D_1(D_1(v))=0,\\
&D_1(D_2(a,b))-D_2(D_1(a),b)-(-1)^{a}D_2(a,D_1(b))=0.
\end{aligned}$$ Thus, the operator $D_1$ plays the role of differential in $V$ and is usually denoted by $d$; similarly $D_2$ determines a skew symmetric binary operation $[,]$ on $V$ of degree $0$, commuting with $d$. now the relation for $D_3$ in this notation can be written as $$(dD_3)(a,b,c)=[[a,b],c]+(-1)^{a(b+c)}[[b,c],a]+(-1)^{b(a+c)}[[c,a],b],$$ where $dD_3$ is the differential of a homogeneous map: $df=d\circ f-(-1)^ff\circ d$. In other words, $[,]$ verifies the (graded) Jacobi identity up to a homotopy. All the other relations are generalized Jacobi identities; they have the form: $$(dD_{n})(a_1,\dots,a_{n})=\sum_{i=2}^n\sum_{\sigma\in UnSh(i,n-i)}(-1)^{\epsilon(\sigma,a_1,\dots,a_n)}D_{n-i+1}(D_i(a_{\sigma(1)},\dots,a_{\sigma(i)}),a_{\sigma(i+1)},\dots,a_{\sigma(n)}).$$ Here $UnSh(i,j)$ is the set of all *unshuffles* of size $(i,j)$, i.e. of all permutations, whose inverse maps are $(i,j)$-shuffles (recall, that $\sigma\in S_{i+j}$ is called $(i,j)$-shuffle, if $\sigma(1)<\sigma(2)<\dots<\sigma(i),\,\sigma(i+1)<\sigma(i+2)<\dots<\sigma(i+j)$) and sign $\epsilon(\sigma,a_1,\dots,a_n)$ is determined by the Koszul rules (below we shall usually abbreviate this notation just to $\epsilon$ or $\epsilon(\sigma)$).
From these formulas it is easy to see that the usual DG Lie algebra structure on a space $\mathfrak g$ determines a particular kind of $L_\infty$-structure: we just put $D_1=d,\,D_2=[,]$ (the usual differential and the Lie bracket on $\mathfrak g$) and put $D_3=D_4=\dots=0$. In fact, $\Lambda_\mathfrak g$ with structure map $D$ in this case is just the Chevalley-Eilenberg complex with its standard differential.
Morphisms and homotopies in $L_\infty$ category
-----------------------------------------------
Given two $L_\infty$-algebras $V,\,W$ one can ask, what is the proper notion of morphisms between them. This question is answered by the following definition:
\[def:linfmo\] A degree $0$ map $F:\Lambda_V\to\Lambda_W$ for any two $L_\infty$-algebras $V,\,W$ is called an $L_\infty$-morphism, if
[(*i*)]{}
: $D_W\circ F=F\circ D_V$;
[(*ii*)]{}
: $F$ is a homomorphism with respect to the free cocommutative coalgebra structures on $\Lambda_V,\,\Lambda_W$.
As before this condition can be expressed in terms of “Taylor coefficients”: we represent $F$ as a collection of maps $F_n:\Lambda^n(V)\to W$ of degree $1-n$, then the condition (*ii*) from definition \[def:linfmo\] turns into the following equalities: $$\begin{aligned}
&F_1d_V=d_WF_1,\\
&F_1([a,b])-[F_1(a),F_1(b)]=d_W(F_2(a,b))+F_2(d_Va,b)+(-1)^aF_2(a,d_Vb),
\end{aligned}$$ and so on; for arbitrary $n$ this turns into the following somewhat cumbersome equation $$\label{eq:Linfmor}
\begin{aligned}
&\sum_{i=1}^n\sum_\sigma (-1)^{\epsilon_1}F_{n-i+1}(D^V_i(a_{\sigma(1)},\dots,a_{\sigma(i)}),a_{\sigma(i+1)},\dots,a_{\sigma(n)})\\
&=\sum_{k=1}^n\frac{1}{k!}\sum_{i_1+\dots+i_k=n}\sum_{\tau}(-1)^{\epsilon_2}D^W_k(F_{i_1}(a_{\tau(1)},\dots,a_{\tau(i_1)}),\dots,F_{i_k}(a_{\tau(i_1+\dots+i_{k-1}+1)},\dots,a_{\tau(n)})).
\end{aligned}$$ Here, as before, the second sum on the left is taken over all $(i,n-i)$ unshuffles, while the last sum on the right is over $(i_1,i_2,,\dots,i_k)$ unshuffles and the signs $\epsilon_1,\,\epsilon_2$ are chosen in accordance with the Koszul rules. Since the morphisms of $L_\infty$-algebras are determined by these “Taylor coefficients”, below we shall sometimes write $F=\{F_n\}:V\to W$.
As before, the usual homomorphism of DG Lie algebras is an example of $L_\infty$-morphism: we set $F_k=0$ for $k\ge2$. However, even in case of Lie algebras one can consider “genuine” $L_\infty$ maps, i.e. maps which do not coincide with the usual Lie-algebraic morphisms. In fact, equation in this case turns (with previous notation) into: $$\label{eq:Linfmog}
\begin{aligned}
&(dF_n)(a_1,\dots,a_n)-\sum_{i<j}(-1)^{\epsilon(i,j)}F_{n-1}([a_i,a_j]_V,a_1,\dots,\widehat{a_i},\dots,\widehat{a_j},\dots,a_n)\\
&=\sum_{i+j=n}\frac12\sum_{\sigma}(-1)^{\epsilon(\sigma)}[F_i(a_{\sigma(1)},\dots,a_{\sigma(i)}),F_j(a_{\sigma(i+1)},\dots,a_{\sigma(n)})]_W.
\end{aligned}$$ Finally, we give the following definition of homotopy between two $L_\infty$-morphisms:
\[def:linfho\] Two $L_\infty$-maps $F,\,G:\Lambda_V\to\Lambda_W$ are called homotopic, if there exist a map $H:\Lambda_V\to\Lambda_W$ of degree $-1$, such that
[(*i*)]{}
: $F-G=D_WH+HD_V$;
[(*ii*)]{}
: $H$ is a derivation with respect to the free cocommutative coalgebra structures on $\Lambda_V,\,\Lambda_W$, i.e. $$\Delta_W H=(H\otimes 1+1\otimes H)\Delta_V.$$
Once again, due to the condition (*ii*) of this definition, the first condition can be expressed in terms of the “Taylor coefficients” of $H$, i.e. $H=\{H_n\}_{n\ge1}$, where $H_n:\Lambda^nV\to W$ is a map of degree $-n$. Then the equation from (*i*) takes the form of the following sequence of equalities: $$\begin{aligned}
F_1(a)-G_1(a)&=d_WH_1(a)+H_1d_V(a),\\
F_2(a,b)-G_2(a,b)&=dH_2(a,b)+\left(H_1([a,b]_V)-[H_1(a),b]_W-(-1)^{ab}[H_1(b),a]_W\right)
\end{aligned}$$ and so on. In particular the map $H_1$ is just a chain homotopy between $F$ and $G$. The general formula looks rather intimidating: $$\label{eq:Linfhom}
\begin{aligned}
F_n&(a_1,\dots,a_n)-G_n(a_1,\dots,a_n)\\
&=\sum_{i+j=n}\sum_{\sigma\in UnSh(i,j)}(-1)^{\epsilon(\sigma)}\Bigl(D_{W,j+1}(H_i(a_{\sigma(1)},\dots,a_{\sigma(i)}),a_{\sigma(i+1)},\dots,a_{\sigma(n)})\\
&\quad\qquad\qquad\qquad\qquad\qquad+H_{j+1}(D_{V,i}(a_{\sigma(1)},\dots,a_{\sigma(i)}),a_{\sigma(i+1)},\dots,a_{\sigma(n)})\Bigr)
\end{aligned}$$ In what follows we shall deal with $L_\infty$-algebras, corresponding to DG Lie algebras; in this case equation will look as follows: $$\label{eq:Linfhodg}
\begin{aligned}
F_n&(a_1,\dots,a_n)-G_n(a_1,\dots,a_n)\\
&=dH_n(a_1,\dots,a_n)+\sum_{i=1}^n(-1)^{\epsilon(i)}[H_{n-1}(a_1,\dots,\widehat{a_i},\dots,a_n),a_i]_W\\
&\qquad\qquad\qquad\qquad+\sum_{1\le i<j\le n}(-1)^{\epsilon(i,j)}H_{n-1}([a_i,a_j]_V,a_1,\dots,\widehat{a_i},\dots,\widehat{a_j},\dots,a_n).
\end{aligned}$$ One of the main advantages of $L_\infty$-algebras and morphisms is the fact that in this context homotopy equivalence is equivalent to the equivalence in $L_\infty$ sense.
More accurately, recall that *quasi-isomorphism* of DG Lie algebras is a homomorphism $f:\mathfrak g\to\mathfrak h$, which induces isomorphism on the level of homology (observe that it is enough to speak about the isomorphism on the level of vector spaces in this case). It turns out that not every quasi-isomorphism of DG Lie algebras $f$ has homotopy inverse, i.e. not for every $f$ one can find a homomorphism $g:\mathfrak h\to\mathfrak g$, whose composditions with $f$ are homotopic to identity. But this is not so in the wider category of $L_\infty$-algebras.
Namely, one calls an $L_\infty$-morphism $F:\Lambda_V\to \Lambda_W$ *quasi-isomorphism* if its first “Taylor coefficient” $F_1:V\to W$ induces isomorphism in cohomology. Then the following statement is true (see [@Kon97]):
\[prop:Linfqua\] Every quasi-isomorphism $F:\Lambda_V\to\Lambda_W$ of $L_\infty$-algebras is homotopy-invertible, i.e. there exists an $L_\infty$-morphism $G:\Lambda_W\to\Lambda_V$ such that $F\circ G$ and $G\circ F$ are homotopic to identity.
In particular, by the virtue of this statement and due to the observations made in the previous paragraph every quasi-isomorphism of DG Lie algebras obtains its inverse in the framework of $L_\infty$ category.
Quasi-isomorphisms and $\star$-products {#sec:MCel}
---------------------------------------
The role of the $L_\infty$ algebras and morphisms in deformation theory is based on the following observations: let a star-product be given. Then the formal sum $B=\sum_{n\ge1}\hbar^n B_n$ determines a $2$-cochain in the ($\hbar$-linear) Hochschild complex of $A[[\hbar]]$; the associativity condition then is transcribed as the formula: $$\label{eq:MC1}
\delta(B)+\frac12[B,B]=0$$ where $\delta$ is the Hochschild cohomology differential and $[,]$ is the Gerstenhaber bracket, which introduces the DG Lie algebra structure on the Hochschild cohomology complex with shifted dimension (in fact, Gerstenhaber bracket is a morphism of degree $-1$, so in order to turn the Hochschild complex with $[,]$ into a DG Lie algebra we must shift all dimensions by $1$).
The equation exists in the case of an arbitrary DG Lie algebra $\mathfrak g$: just replace $\delta$ with proper differential in $\mathfrak g$ and use the given Lie bracket. This equation is called *Maurer-Cartan equation* and the elements $\Pi$ of degree $1$ in $\mathfrak g$ are called *Maurer-Cartan elements*. It turns out that the Maurer-Cartan elements behave very well with respect to the $L_\infty$-morphisms between the DG Lie algebras. Namely:
\[prop:MC1\] Let $\Pi$ be a Maurer-Cartan element in $\mathfrak g$ and let $F:\Lambda_\mathfrak g\to \Lambda_\mathfrak h$ be an $L_\infty$-morphism between these algebras. Than the following formula (assuming the convergence of the series in the right) determines a Maurer-Cartan element in $\mathfrak h$: $$\label{eq:MC2}
F(\Pi)=\sum_{n\ge1}\frac{1}{n!}F_n(\Pi,\dots,\Pi).$$
The proof of this statement is by direct computation with the help of equation and we omit it.
In fact, the notion of Maurer-Cartan element can be defined for arbitrary $L_\infty$ algebra: it is such an element $\Pi\in V$, that $D_V(\exp(\Pi))=0$, where $$\exp(\Pi)=\sum_{n\ge0}\frac{1}{n!}\underbrace{\Pi\wedge\dots\wedge\Pi}_{n\ \mbox{times}}.$$ Observe that in a generic case $\exp(\Pi)\in\widehat\Lambda_V$, i.e. it is not an element of the exterior algebra, but only an element in its suitable completion. If we rewrite the equation $D_V(\exp(\Pi))=0$ in “coordinate” terms, we obtain the following equation: $$d\Pi+\sum_{n\ge2}\frac{1}{n!}D_n(\Pi,\dots,\Pi)=0.$$ If $V=\mathfrak g$ is a DG Lie algebra, then this equation is reduced to , since $D_n=0\,n\ge3$. It turns out that in this context the statement of proposition \[prop:MC1\] remains true, i.e. the formula determines a Maurer-Cartan element in $W$. The proof in this case is even easier: just observe that $F(\exp(\Pi))=\exp(F(\Pi))$ under the assumption of general convergence of all series.
The observation, made in proposition \[prop:MC1\] helps to relate the $L_\infty$ theory with the deformation problem: first of all, if $A=C^\infty(M)$ for some manifold $M$, one can consider a subcomplex of the Hochschild cohomology complex of $A[[\hbar]]$, spanned by the $\hbar$-linear *local* cochains, i.e. by cochains, determined by polydifferential operators on $M$. Clearly, this subcomplex is closed with respect to the Gerstenhaber brackets, so we have a DG Lie algebra $\mathfrak g^\cdot=\mathcal D_{poly}^\cdot[1](M)[[\hbar]]$ of polydifferential operators on $M$ with Hochschild differential. It is our purpose to find in this DG Lie algebra a Maurer-Cartan element $B\in\mathfrak g^1=\mathcal D^1_{poly}[1](M)[[\hbar]]$, such that $B(f,g)=\frac{\hbar}{2}\{f,g\}+o(\hbar)$.
To this end we observe that the cohomology of local Hochschild complex $\mathcal D_{poly}^\cdot(M)$ on $M$ is isomorphic to the space of polyvector fields $\mathcal T_{poly}^\cdot(M)$ on $M$; the isomorphism is induced by the *Hoschild-Kostant-Rosenberg map* $\chi:\mathcal T_{poly}^\cdot(M)\to\mathcal D_{poly}^\cdot(M)$, determined by the formula: $$\chi(\Psi^p)(f_1,\dots,f_p)=\langle df_1\wedge\dots\wedge df_p,\Psi\rangle,$$ where $\langle,\rangle$ denotes the natural pairing between differential forms and polyvector fields. One can show that this map in fact on the level of cohomology induces not only an isomorphism of vector spaces, but also isomorphism of graded Lie algebras, if we use Gerstenhaber bracket to induce the Lie algebra structure on Hochschild cohomology (in effect it even induces the isomorphism of graded Poisson algebras, if we consider the wedge product on polyvectors and the standard product of Hochschild cochains). However, the map $\chi$ is not a homomorphism of DG Lie algebras (although it commutes with the differentials, if we allow zero differential on $\mathcal T_{poly}^\cdot(M)$). The same remarks of course are true with respect to the $\hbar$-linear complexes: one can define the map $\chi_\hbar:\mathcal T_{poly}^\cdot[1](M)[[\hbar]]\to \mathcal D_{poly}^\cdot[1](M)[[\hbar]]$, which will commute with differentials and induce the isomorphism of Lie algebras on the level of cohomology; however this map will not be a homomorphism of Lie algebras.
On the other hand, the bivector $\frac12\hbar\pi\in\mathcal T_{poly}^1[1](M)[[\hbar]]$ is a solution of the Maurer-Cartan equation if the differential in $\mathcal T_{poly}^1[1](M)[[\hbar]]$ is trivial, and hence as we have seen earlier, if the Hochschild-Kostant-Rosenberg map could be extended to a quasi-isomorphism of $L_\infty$ algebras, the formula applied to $\Pi=\frac12\hbar\pi$ would give us a Maurer-Cartan element in the complex $\mathcal D_{poly}^\cdot[1](M)[[\hbar]]$, beginning with $\chi_\hbar(\frac12\hbar\pi)$ as prescribed in the deformation problem.
The construction of such quasi-isomorphism for algebras of functions on $M={\mathbb{R}}^n$ with arbitrary Poisson structure was eventually given by Kontsevich. We shall denote this quasi-isomorphism by ${\ensuremath{\mathcal U}}=\{U_n\}:\mathcal T_{poly}^1[1](M)\to\mathcal D_{poly}^1[1](M)$, so that the $\star$-product associated with it is given by $$B=\sum_{n\ge 1}\frac{\hbar^n}{n!}U_n(\underbrace{\pi,\dots,\pi}_{n\ \mbox{times}}).$$ Observe that the convergence is guaranteed in the context of formal power series. We are not going to discuss the details of this construction now; for our purposes it is important to know that *when the $\star$-product is induced by an $L_\infty$-map, in particular, by Kontsevich’s quasi-isomorphism, the same construction as above induces the map from $Z_\pi(A(M))$ to $Z({\ensuremath{\mathcal A}}(M))$: for any $f\in Z_\pi(A(M))$ we put $$\hat f=\sum_{n\ge0}\frac{\hbar}{n!}U_{n+1}(f,\underbrace{\pi,\dots,\pi}_{n\ \mbox{times}});$$ an easy computation shows that when $f\in Z_\pi(A(M))$, the inner derivation of ${\ensuremath{\mathcal A}}(M)$, induced by $\hat f$ is equal to $0$*.
In effect, this claim follows from the next simple and rather well known observation: if $V$ is an $L_\infty$-algebra and $\Pi$ is a Maurer-Cartan element in $V$, then the formula $$D^\Pi_{V,n}(v_1,\dots,v_n)=\sum_{k\ge0}\frac{1}{k!}D_{n+k}(v_1,\dots,v_n,\underbrace{\Pi,\dots,\Pi}_{k\ \mbox{times}})$$ (under the general assumption of convergence in all such formulas) determines a new $L_\infty$-structure in $V$; we shall denote $V$ equipped with this structure by $(V,D_V^\Pi)$. Similarly if $F=\{F_n\}:V\to W$ is an $L_\infty$-morphism, then the formula $$F^\Pi_n(v_1,\dots,v_n)=\sum_{k\ge0}\frac{1}{k!}F_{n+k}(v_1,\dots,v_n,\underbrace{\Pi,\dots,\Pi}_{k\ \mbox{times}})$$ determines an $L_\infty$-morphism $F^\Pi=\{F_n^\Pi\}:(V,D_V^\Pi)\to (W,D_W^{F(\Pi)})$.
Now, the claim concerning the algebras’ centres follows from the fact that in the context of Poisson algebra $f\in Z_\pi(A(M))$ if and only if $D^\pi_{A(M),1}(f)=0$ and $\hat f\in{\ensuremath{\mathcal A}}(M)$ is in the center of ${\ensuremath{\mathcal A}}(M)$ if and only if $D^B_{{\ensuremath{\mathcal A}}(M),1}(\hat f)=0$.
Thus for all elements in $Z_\pi(A(M))$ we have their counterparts in the center of ${\ensuremath{\mathcal A}}(M)$. In the next section we shall discuss the analog of the Nijenhuis equation in the context of $L_\infty$-algebras and its meaning for constructing commutative subalgebras in ${\ensuremath{\mathcal A}}(M)$.
$L_\infty$-derivations and Nijenhuis property
=============================================
Properties of $\mathcal T_{poly}^\cdot[1](M)$ and $\mathcal D_{poly}^\cdot[1](M)$ in low degerees {#sec:introLinfder}
-------------------------------------------------------------------------------------------------
In the remaining part of this note, we are going to deal only with DG Lie algebras, related with the deformation theory, i.e. $\mathcal T_{poly}^\cdot[1](M)$ and $\mathcal D_{poly}^\cdot[1](M)$ (although we shall often omit the shift from our notation). Thus it is worth beginning this section with a short list of properties, that we shall need.
First of all, the grading in both algebras begins with $-1$ (after shift), or from $0$; the differential vanishes on the lowest degree in both algebras and the lowest degree elements in both algebras commute.
If we go further, we see that Hochschild-Kostant-Rosenberg map, although not a homomorphism of DG Lie algebras, intertwines the commutators in the lowest degrees (i.e $-1$ and $0$ in the shifted case): on the level $-1$ this is evident since on both sides commutator vanishes; for two vector fields $\xi,\,\eta\in\mathcal T_{poly}^0[1](M)$, their commutator is equal to the difference of their compositions: $$[\xi,\eta](f)=\xi(\eta(f))-\eta(\xi(f)),$$ but the same is true for $\chi(\xi)=\xi$ and $\chi(\eta)=\eta$. Similarly, the commutator of $f\in\mathcal T_{poly}^{-1}[1](M)$ with $\xi$ is equal to $\xi(f)\in\mathcal T_{poly}^{-1}[1](M)$; and similarly Gerstenhaber bracket of $f$ and $\xi=\chi(\xi)$ gives the same result.
Also let us recall that the Poisson bracket of two functions can be written as the following combination of elements in $\mathcal T_{poly}^\cdot[1](M)$: $$\{f,g\}=[f,[\pi,g]]$$ (where $[,]$ stand for the Schouten brackets). If we use the Lichnerowicz-Poisson differential $d_\pi(\psi)=[\pi,\psi]$ on $\mathcal T_{poly}^\cdot[1](M)$, we can rewrite this formula as $\{f,g\}=[f,d_\pi g]$. The skew-symmetry of this operation is then ensured by the Jacobi identity and the fact of commutativity of $\mathcal T_{poly}^{-1}[1](M)$, mentioned above.
Similarly, if $\Pi$ is a Maurer-Cartan element in $\mathcal D_{poly}^\cdot[1](M)$, then we have the following equation $$f\star g-g\star f=[f,[\Pi,g]],$$ for all $f,g\in A(M)$, where $\star$ is the deformed multiplication, determined by $\Pi$ and $[,]$ is the Gerstenhaber bracket. In other words, the commutator in ${\ensuremath{\mathcal A}}(M)$ is defined by the same formula as the Poisson algebra structure on $A(M)$, therefore we are going to denote this commutator by the same symbol $f\star g-g\star f=\{f,g\}$, when it can cause no ambiguity. As before, the skew-commutativity of this operation follows from Jacobi identity and the commutativity of the Gerstenhaber brackets on functions. Observe that in this context the Jacobi identity for both braces follows from the Maurer-Cartan equation and the fact that differentials vanish on degree $-1$ elements.
$L_\infty$-derivations, Maurer-Cartan elements and Nijenhuis conditions
-----------------------------------------------------------------------
We begin with the following definition, very similar to the definitions, given in previous section:
Let $V$ be an $L_\infty$-algebra, in particular we can take $V={\ensuremath{\mathfrak{g}}}$, where ${\ensuremath{\mathfrak{g}}}$ is a DG Lie algebra. A map $\mathscr X:\Lambda_V\to\Lambda_V$ is called $L_\infty$-derivation, if
[(*i*)]{}
: $\mathscr X$ is a coderivation of the free coalgebra;
[(*ii*)]{}
: $\mathscr X$ commutes with the structure map $D$.
In particular, as before the map $\mathscr X$ is determined by its “Taylor coefficients” and one can write down the condition (*ii*) in terms of these coefficients and the structure maps $D_{V,i}$. In what follows we shall only consider the $L_\infty$-derivations for DG Lie algebras, so let $\mathscr X=\{X_n\}$ be an $L_\infty$-derivation of a DGLa [$\mathfrak{g}$]{}; then the maps $X_n:\wedge^n{\ensuremath{\mathfrak{g}}}\to{\ensuremath{\mathfrak{g}}}$ verify the equalities $$\label{eq:xxxder2}
\begin{aligned}
dX_{n+1}(a_0,\dots,a_n)&=\sum_{0\le i<j\le n}(-1)^{\epsilon_{i,j}}X_n([a_i,a_j],a_0,\dots,\widehat{a}_i,\dots,\widehat{a}_j,\dots,a_n)\\
&+\sum_{i=0}^n(-1)^{\epsilon_i}[a_i,X_n(a_0,\dots,\widehat a_i,\dots,a_n)].
\end{aligned}$$ Here as earlier $\epsilon_{i,j},\ \epsilon_i$ are the signs, determined by the permutations, which place the elements $a_i,a_j$ in front of the others and $\widehat{}$ denotes the omission of an element.
Now for a MC element $\pi\in{\ensuremath{\mathfrak{g}}}$ consider the element $$\mathscr X(\pi)=\sum_{n=1}^\infty \frac{1}{n!}X_n(\pi,\dots,\pi).$$ As before we assume that all the formulas of this sort enjoy the convergence property. More generally, for arbitrary $a\in{\ensuremath{\mathfrak{g}}}$ we put: $$\mathscr X_\pi(a)=\sum_{n=1}^\infty\frac{1}{(n-1)!}X_n(a,\pi,\dots,\pi).$$ Then
\[prop:xscr1\] The element $\mathscr X(\pi)$ is closed with respect to the $\pi$-twisted differential in [$\mathfrak{g}$]{}: $d_\pi(a)=da+[\pi,a]$, i.e. $d\mathscr X(\pi)=-[\pi,\mathscr X(\pi)]$. If in addition we suppose that $\mathscr X_\pi(\mathscr X(\pi))=0$, i.e. $$\sum_{p\ge 1}\frac{1}{(p-1)!}X_p(\pi,\dots,\pi,\mathscr X(\pi,\dots,\pi))=0,$$ then it also verifies the equation $[\mathscr X(\pi),\mathscr X(\pi)]=0$.
We compute with the help of the equation (the signs do not appear, since the degree of $\pi$ is $0$ in $\wedge^*{\ensuremath{\mathfrak{g}}}$): $$\begin{aligned}
d(\mathscr X(\pi))&=\sum_{n=1}^\infty \frac{1}{n!}\Bigl(dX_n(\pi,\dots,\pi)+\sum_{i=1}^nX_n(\pi,\dots,\underset{i}{d\pi},\dots,\pi)\Bigr)\\
&=\sum_{n=1}^\infty\frac{1}{n!}\Bigl(\sum_{1\le i<j\le n}X_{n-1}([\pi,\pi],\pi,\dots,\underset{i}{\widehat{\pi}},\dots,\underset{j}{\widehat{\pi}},\dots,\pi)\\
&\quad+\sum_{i=1}^n(-1)^{\epsilon_i}[\pi,X_{n-1}(\pi,\dots,\underset{i}{\widehat\pi},\dots,\pi)]+\sum_{i=1}^nX_n(\pi,\dots,\underset{i}{d\pi},\dots,\pi)\Bigr)\\
&=\sum_{n=1}^\infty\frac{1}{n!}\Bigl(\frac{n(n-1)}{2}X_{n-1}([\pi,\pi],\pi,\dots,\pi)+n[\pi,X_{n-1}(\pi,\dots,\pi)]\\
&\quad+nX_n(d\pi,\pi,\dots,\pi)\Bigr)\\
&=\sum_{n=1}^\infty\Bigl(\frac{1}{(n-1)!}X_n(d\pi,\pi,\dots,\pi)+\frac{1}{2(n-2)!}X_{n-1}([\pi,\pi],\pi,\dots,\pi)\\
&\quad+\frac{1}{(n-1)!}[\pi,X_{n-1}(\pi,\dots,\pi)]\Bigl)\\
&=\sum_{n=2}^\infty\frac{1}{(n-1)!}X_n(d\pi+\frac12[\pi,\pi],\pi,\dots,\pi)-\left[\pi,\sum_{n=1}^\infty\frac{1}{n!}X_n(\pi,\dots,\pi)\right]\\
&=-[\pi,\mathscr X(\pi)].
\end{aligned}$$ Here we used the Maurer-Cartan equation $d\pi+\frac12[\pi,\pi]=0$. In a similar way, one can prove the equality $$\label{eq:xpi1}
d\left(\mathscr X_\pi(a)\right)=[\pi,\mathscr X_\pi(a)]+[\mathscr X(\pi),a]+\mathscr X_\pi(d_\pi a).$$ for arbitrary $a\in{\ensuremath{\mathfrak{g}}}$. Now, from the latter formula, the equality $d_\pi\mathscr X(\pi)=0$ and the equation $$\mathscr X_\pi(\mathscr X(\pi))=\sum_{p,q=1}^\infty \frac{1}{(p-1)!q!}X_{p}(\pi,\dots,\pi,X_q(\pi,\dots,\pi))=0,$$ we see $$0=d(\mathscr X_\pi(\mathscr X(\pi)))=[\pi,\mathscr X_\pi(\mathscr X(\pi))]+[\mathscr X(\pi),\mathscr X(\pi)]+\mathscr X_\pi(d_\pi\mathscr X(\pi))=[\mathscr X(\pi),\mathscr X(\pi)].
$$
We shall say, that the $L_\infty$-derivarion $\mathscr X$ of [$\mathfrak{g}$]{} verifies weak $L_\infty$-Nijenhuis property with respect to a Maurer-Cartan element $\pi$ in ${\ensuremath{\mathfrak{g}}}$, if $\mathscr X_\pi(\mathscr X(\pi))=0$.
The proposition \[prop:xscr1\] means that $\pi+\mathscr X(\pi)$ is a Maurer-Cartan elementin [$\mathfrak{g}$]{}: $$\begin{aligned}
d(\pi+\mathscr X(\pi))&+\frac12[\pi+\mathscr X(\pi),\pi+\mathscr X(\pi)]=\\
&=d\pi+\frac12[\pi,\pi]+d\mathscr X(\pi)+\frac12([\pi,\mathscr X(\pi)]+[\mathscr X(\pi),\pi])+\frac12[\mathscr X(\pi),\mathscr X(\pi)]=0.
\end{aligned}$$ Also observe that we can rewrite equation as $$\label{eq:xpi2}
(d_\pi\mathscr X_\pi)(a)=d_{\pi}(\mathscr X_\pi(a))-\mathscr X_\pi(d_\pi a)=[\mathscr X(\pi),a].$$
$L_\infty$-Nijenhuis property and commutation relations
-------------------------------------------------------
We are going to find the relation between the map $\mathscr X_\pi$ and the brackets in [$\mathfrak{g}$]{}. To this end we compute: $$\begin{aligned}
d(X_{p+2}&(x,y,\pi,\dots,\pi))=X_{p+2}(dx,y,\pi,\dots,\pi)+(-1)^{|x|}\Bigl(X_{p+2}(x,dy,\pi,\dots,\pi)\\
&+\sum_{i=3}^{p+2}(-1)^{|y|}X_{p+2}(x,y,\pi,\dots,\underset{i}{d\pi},\dots,\pi)\Bigr)+X_{p+1}([x,y],\pi,\dots,\pi)\\
&+\sum_{i=3}^{p+2}\left(X_{p+1}([x,\pi],y,\pi,\dots,\underset{i}{\widehat\pi},\dots,\pi)+(-1)^{|x|}X_{p+1}(x,[y,\pi],\pi,\dots,\underset{i}{\widehat\pi},\dots,\pi)\right)\\
&+\sum_{3\le i<j\le p+1}(-1)^{|x|+|y|}X_{p+1}(x,y,[\pi,\pi],\pi,\dots,\underset{i}{\widehat\pi}\dots,\underset{j}{\widehat\pi},\dots,\pi)\\
&+[x,X_{p+1}(y,\pi,\dots,\pi)]+(-1)^{|x||y|}[y,X_{p+1}(x,\pi,\dots,\pi)]\\
&+\sum_{i=3}^{p+1}[\pi,X_{p+1}(x,y,\pi,\dots,\underset{i}{\widehat\pi},\dots,\pi)].
\end{aligned}$$ Here $|x|,\,|y|$ are the degrees of $x$ and $y$ in the exterior powers of [$\mathfrak{g}$]{}. Multiplying this expression by $\frac{1}{p!}$ and summing up for $p=0,1,2,\dots$ we obtain the formula: $$d_\pi\mathscr X_\pi(x,y)=\mathscr X_\pi([x,y])-[x,\mathscr X_\pi(y)]-(-1)^{|x||y|}[y,\mathscr X_\pi(x)],$$ where we use the notation $\mathscr X_\pi(x,y)=\sum_{p=0}^\infty\frac{1}{p!}X_{p+2}(x,y,\pi,\dots,\pi)$ and $d_\pi$ on the left hand side denotes the usual differential of a map. Let us now assume that ${\ensuremath{\mathfrak{g}}}=\mathcal T_{poly}^\cdot[1](M)$ or ${\ensuremath{\mathfrak{g}}}=\mathcal D_{poly}^\cdot[1](M)$ and consider $y=d_\pi g,\,x=f$, where $|f|=|g|=-1$. In this case $\mathscr X_\pi(x,y)=0$ (because of dimension restrictions) and we have: $$\mathscr X_\pi(d_\pi f,d_\pi g)=\mathscr X_\pi(\{f,g\})-[f,\mathscr X_\pi(d_{\pi}g)]+\{g,\mathscr X_\pi(f)\}$$ where we use the observation that $[f,d_\pi g]=\{f,g\}$, the commutator of $f$ and $g$ with respect to $\pi$, see section \[sec:introLinfder\]. Similarly, from we see $[f,\mathscr X_\pi(d_{\pi}g)]=\{f,\mathscr X_\pi(g)\}+[f,[\mathscr X(\pi),g]]$ and we have: $$\label{eq:xder1}
\mathscr X_\pi(d_\pi f,d_\pi g)=\mathscr X_\pi(\{f,g\})-\{\mathscr X_\pi(f),g\}-\{f,\mathscr X_\pi(g)\}-[f,[\mathscr X(\pi),g]].$$ Another important example is $x=f,\,y=[\mathscr X(\pi),g]$: first we compute $$d_\pi\mathscr X_\pi(\mathscr X(\pi),g)=\mathscr X_\pi([\mathscr X(\pi),g])-[\mathscr X(\pi),\mathscr X_\pi(g)].$$ Here we used the equation $d_\pi\mathscr X(\pi)=0$. Next, using this equation we compute $$\label{eq:xder2}
\begin{aligned}[m]
d_\pi&\mathscr X_\pi(f,[\mathscr X(\pi),g])=\mathscr X_\pi([f,[\mathscr X(\pi),g]])-[f,\mathscr X_\pi([\mathscr X(\pi),g])]-[[\mathscr X(\pi),g],\mathscr X_\pi(f)]\\
&=\mathscr X_\pi([f,[\mathscr X(\pi),g]])-[f,d_\pi\mathscr X_\pi(\mathscr X(\pi),g)]-[f,[\mathscr X(\pi),\mathscr X_\pi(g)]]-[\mathscr X_\pi(f),[\mathscr X(\pi),g]]
\end{aligned}$$ Let now $f,\,g$ belong to the “center of $\pi$-deformed product”, i.e. let $d_\pi f=d_\pi g=0$ (see section \[sec:introLinfder\]). Then by we have: $$\mathscr X_\pi(\{f,g\})=\{\mathscr X_\pi(f),g\}+\{f,\mathscr X_\pi(g)\}+[f,[\mathscr X(\pi),g]],$$ and so $\mathscr X_\pi(\{f,g\})=\{\mathscr X_\pi(f),g\}=\{f,\mathscr X_\pi(g)\}=[f,[\mathscr X(\pi),g]]=0$. Next we apply to $f,\,\mathscr X_\pi(g)$ where $d_\pi f=0$: $$\mathscr X_\pi(\{f,\mathscr X_\pi(g)\})=\{\mathscr X_\pi(f),\mathscr X_\pi(g)\}+\{f,\mathscr X_\pi(\mathscr X_\pi(g))\}+[f,[\mathscr X(\pi),\mathscr X_\pi(g)]].$$ So: $$\label{eq:xder3}
\{\mathscr X_\pi(f),\mathscr X_\pi(g)\}+[f,[\mathscr X(\pi),\mathscr X_\pi(g)]]=0.$$ Similarly, taking the pair $\mathscr X_\pi(f),\,g$ with $d_\pi g=0$ we get: $$\label{eq:xder4}
\{\mathscr X_\pi(f),\mathscr X_\pi(g)\}+[g,[\mathscr X(\pi),\mathscr X_\pi(f)]]=0$$ (here we used the identity $[f,g]=0$ for all functions). Next, we apply equation : since $d_\pi f=d_\pi g=d_\pi\mathscr X(\pi)=0$ the left hand side of this equation vanishes. The first term on the right is equal to $0$ since $[f,[\mathscr X(\pi),g]]=0$, and the second term is equal to $\{f,\mathscr X_\pi(\mathscr X(\pi),g)\}=0$ since $f$ is in center. Thus this equation amounts to $$\label{eq:xder5}
[f,[\mathscr X(\pi),\mathscr X_\pi(g)]]+[g,[\mathscr X(\pi),\mathscr X_\pi(f)]]=0.$$ Now, summing up the equations and and subtracting we get: $$2\{\mathscr X_\pi(f),\mathscr X_\pi(g)\}=0.$$ Thus, $\{\mathscr X_\pi(f),\mathscr X_\pi(g)\}=0$.
Argument shift in deformed algebras
===================================
Strong Nijenhuis property
-------------------------
We saw, that in the case when $\mathscr X$ verifies the weak Nijenhuis condition, the elements $\mathscr X_\pi(f),\,\mathscr X_\pi(g)$ commute, although they are not in general central: one sees from equation that in this case $$d_\pi\mathscr X_\pi(f)=[\mathscr X(\pi),f],$$ which need not vanish. However, weak Nijenhuis property is not enough to prove the commutativity of $\mathscr X_\pi^k(f),\,\mathscr X_\pi^l(g)$ for all $k$ and $l$. In order to prove this alongside the reasoning from the first section, we shall need a stronger condition. To this end we observe that the formal exponentiation of an $L_\infty$-derivation as a map from $\Lambda_V$ to itself gives an $L_\infty$-automorphism of $V$. Indeed, the commutation with $D_V$ follows from the definitions, and the fact that exponent of a coderivation is a homomorphism of coalgebras is trivial. As we have explained earlier in section \[sec:MCel\], any Maurer-Cartan element $\Pi$ in $V$ determines a deformation of the differential $D_{V,1}=d$ in $V$ and any $L_\infty$-morphism can be applied to Maurer-Cartan elements and can be extended to an $L_\infty$-morphism between the $L_\infty$-algebras with differentials, deformed by the Maurer-Cartan elements.
Summing up, we have the following a bit cumbersome proposition:
Let $V$ be an $L_\infty$-algebra, $\mathscr X=\{X_n\}_{n\ge1}:\Lambda_V\to\Lambda_V$ an $L_\infty$-derivation of $V$ and $\Pi\in V$ a Maurer-Cartan element. Let $\exp(\mathscr X)$ be the map with “Taylor coefficients” $$\begin{aligned}
&\exp(\mathscr X)_n(a_1,\dots,a_n)=\\
&=\sum_k\frac{1}{k!}\sum_{i_1+i_2+\dots+i_p-p=n+1}\frac{1}{i_1!(i_2-1)!\dots(i_p-1)!}\\
&\,\sum_{\sigma\in S_n}X_{i_p}(\dots(X_{i_2}(X_{i_1}(a_{\sigma(1)},\dots,a_{\sigma(i_1)}),a_{\sigma(i_1+1)},\dots,a_{\sigma(i_1+i_2-1)}),\dots),a_{\sigma(i_1+\dots+i_{p-1}+3-p)},\dots,a_{\sigma(n)}).
\end{aligned}$$ Then $\exp(\mathscr X)$ is an $L_\infty$-morphism. Moreover, $\exp(\mathscr X)(\Pi)$ is a Maurer-Cartan element in $V$ and if $\mathscr X_\Pi$ is given by the formula $$\label{eq:pider}
\mathscr X_{\Pi,n}(a_1,\dots,a_n)=\sum_{p\ge0}\frac{1}{p!}X_{n+p}(a_1,\dots,a_n,\underbrace{\Pi,\dots,\Pi}_{p\ \mbox{times}})$$ then $\mathscr X_\Pi$ is an $L_\infty$-derivative between $(V,D_{V,\Pi})$ (the $L_\infty$-algebra $V$ with $\Pi$-deformed differential $D_{V,\Pi}$) and $(V,D_{V,\exp(\mathscr X)(\Pi)})$, which means that the following equality holds: $$\mathscr X_\Pi\circ D_{V,\Pi}=D_{V,\exp(\mathscr X)(\Pi)}\circ\mathscr X_\Pi.$$ In particular, $\exp(\mathscr X_\Pi)$ is an $L_\infty$-morphism between $(V,D_{V,\Pi})$ and $(V,D_{V,\exp(\mathscr X)(\Pi)})$.
In the particular case $V={\ensuremath{\mathfrak{g}}}$ and $\Pi=\pi$ (the usual Maurer-Cartan element in DG Lie algebra) the map $\mathscr X_\pi=\{\mathscr X_{\pi,n}\}_{n\ge0}:\Lambda_{\ensuremath{\mathfrak{g}}}\to\Lambda_{\ensuremath{\mathfrak{g}}}$, with “Taylor coefficients” given by equation will verify the following identities, similar to $$\begin{aligned}
(d\mathscr X_{\pi,n+1})(a_0,\dots,a_n)&=\sum_{0\le i<j\le n}(-1)^{\epsilon_{i,j}}\mathscr X_{\pi,n}([a_i,a_j],a_0,\dots,\widehat{a}_i,\dots,\widehat{a}_j,\dots,a_n)\\
&+\sum_{i=0}^n(-1)^{\epsilon_i}[a_i,\mathscr X_{\pi,n}(a_0,\dots,\widehat a_i,\dots,a_n)].
\end{aligned}$$ However, here $d\mathscr X_{\pi,n+1}$ stands for the commutator $$[d,\mathscr X_{\pi,n+1}]=d_{\exp(\mathscr X)(\pi)}\circ\mathscr X_{\pi,n+1}+(-1)^n\mathscr X_{\pi,n+1}\circ d_\pi.$$ We shall call maps with this property *twisted $L_\infty$-derivations*.
Of course, all the statements of this section are still true for the maps $\exp(t\mathscr X),\,\exp(t\mathscr X_\pi)$ etc. We shall use this observation below.
It is clear, that the maps we discussed above are closely related with this construction: $\mathscr X_\pi(a),\,\mathscr X_\pi(x,y)$ are just the $1$-st and the $2$-nd “Taylor coefficients” of this twisted $L_\infty$-derivation. This brings forth the following definition:
We shall say, that $\mathscr X$ verifies the (strong) Nijenhuis condition with respect to $\pi$, if $$\mathscr X_{\pi,n}(a_1,\dots,a_n)=0,\ \mbox{when $a_i=\mathscr X(\pi)$ for some $i=1,\dots,n$}.$$
Of course, every strong Nijenhuis derivation verifies the weak Nijenhuis property, so all the statements from the previous section remain valid. Below we shall show that strong Nijenhuis property of an $L_\infty$-derivation of DG Lie algebras allows one perform the trick from the proposition \[prop:ashiftcl\] and thus obtain commutative subalgebras in quantized algebras. One of the problems of the theory that we develop here is that so far we have no example of a “genuine” (strong) Nijenhuis derivations, which would not come from the considerations of the usual Nijenhuis fields on a Poisson manifold.
Argument shift for strong Nijenhuis derivations
-----------------------------------------------
Let now [$\mathfrak{g}$]{} be a deformation the DG Lie algebra, i.e. ${\ensuremath{\mathfrak{g}}}=\mathcal T_{poly}^{-1}[1](M)$ or ${\ensuremath{\mathfrak{g}}}=\mathcal D_{poly}^{-1}[1](M)$. Now we claim that the following is true:
\[prop:shiftLinf1\] Let $\mathscr X$ be an $L_\infty$-derivation of a deformation DG Lie algebra [$\mathfrak{g}$]{}, which verifies the strong Nijenhuis condition with respect to a Maurer-Cartan element $\pi$, then $$\mathscr X_{\pi,1}(\{x,y\})=\{\mathscr X_{\pi,1}(x),y\}+\{x,\mathscr X_{\pi,1}(y)\}+[x,[\mathscr X(\pi),y]]$$ and $$\mathscr X_{\pi,1}([x,[\mathscr X(\pi),y]])=[\mathscr X_{\pi,1}(x),[\mathscr X(\pi),y]]+[x,[\mathscr X(\pi),\mathscr X_{\pi,1}(y)]]$$ for all $x,y\in{\ensuremath{\mathfrak{g}}}^{-1}$ of the form $x=\mathscr X_{\pi,1}^k(f),\,y=\mathscr X_{\pi,1}^l(g),\, k,l=0,1,2,\dots$ with $f,\,g$ in the center of $\pi$, i.e. if $d_\pi f=0=d_\pi g$.
First of all, observe that under the conditions we have $\exp(t\mathscr X)(\pi)=\pi+t\mathscr X(\pi)$. Now we have the following lemma
\[lem:lemsh1\] Let $x_k$ be of the form $x_k=\mathscr X_{\pi,1}^k(f),\ d_\pi f=0$, as prescribed in proposition \[prop:shiftLinf1\]. Then $d_\pi(x_k)=[\mathscr X(\pi),x_{k-1}]$.
The map $\exp(t\mathscr X_{\pi,1})=\sum_{k\ge0}\frac{t^k}{k!}\mathscr X_{\pi,1}^k$ is the $1$-st “Taylor coefficient” of $\exp(t\mathscr X_\pi),\,t\in{\mathbb{R}}$. Thus, it verifies the condition: $$\exp(t\mathscr X_{\pi,1})\circ d_\pi=d_{\exp(t\mathscr X)(\pi)}\circ\exp(t\mathscr X_{\pi,1}).$$ This is true for all $t\in{\mathbb{R}}$, so for every $k=1,2,\dots$ and every $f\in{\ensuremath{\mathfrak{g}}}$ we have $$d_\pi\circ \mathscr X_{\pi,1}^k(f)+(-1)^{|f|}[\mathscr X(\pi),\mathscr X_{\pi,1}^{k-1}(f)]=-\mathscr X_{\pi,1}^k\circ d_\pi(f).$$ Since $d_\pi f=0$, the statement follows.
Now, using the equation we see that the first equation from proposition \[prop:shiftLinf1\] would follow if we prove that $\mathscr X_{\pi,2}(d_\pi x,d_\pi y)=0$ for $x=x_k,\,y=y_l$ and for all $k,l=1,2,\dots$ as in the conditions of proposition \[prop:shiftLinf1\]. To this end consider $\mathscr X_{\pi,3}(\mathscr X(\pi),x,d_\pi y)$; we may assume that $k,l>0$ since otherwise the statement is trivially true. Due to the dimensional restrictions $$\mathscr X_{\pi,3}(\mathscr X(\pi),x,d_\pi y)=0=\mathscr X_{\pi,2}(x,d_\pi y).$$ On the other hand, since $\mathscr X_\pi$ is a twisted $L_\infty$-derivation and verifies the strong Nijenhuis property, $\mathscr X(\pi)$ is $d_\pi$-closed and we have: $$0=(d\mathscr X_{\pi,3})(\mathscr X(\pi),x_{k-1},d_\pi y)=\mathscr X_{\pi,2}([\mathscr X(\pi),x_{k-1}],d_\pi y)=\mathscr X_{\pi,2}(d_\pi x_k,d_\pi y).$$ We used the statement of lemma \[lem:lemsh1\] in the last equality.
Similarly, we have $(d\mathscr X_{\pi,2})(x,[\mathscr X(\pi),y])=0$ due to the dimension restrictions, result of lemma \[lem:lemsh1\] and the equation we just proved; on the other hand $$(d\mathscr X_{\pi,2})(x,[\mathscr X(\pi),y])=-[x,\mathscr X_{\pi,1}([\mathscr X(\pi),y])]-[\mathscr X_{\pi,1}(x),[\mathscr X(\pi),y]]+\mathscr X_{\pi,1}([x_k,[\mathscr X(\pi),y]])$$ Finally, consider $\mathscr X_{\pi,2}(\mathscr X(\pi),y)$: due to the strong Nijenhuis condition $(d\mathscr X_{\pi,2})(\mathscr X(\pi),y)=0$, on the other hand, due to the same condition we have $$(d\mathscr X_{\pi,2})(\mathscr X(\pi),y)=\mathscr X_{\pi,1}([\mathscr X(\pi),y])-[\mathscr X(\pi),\mathscr X_{\pi,1}(y)].$$ Comparing the last two equalities we obtain the second formula from proposition \[prop:shiftLinf1\].
Now the following statement is a direct consequence of proposition \[prop:shiftLinf1\] and the method we used in the proof of the proposition \[prop:ashiftcl\]:
\[theo:om\] Let $\mathscr X$ be a strong Nijenhuis derivation of a deformation DG Lie algebra with respect to the Maurer-Cartan element $\pi$. Then for any $f,\,g\in{\ensuremath{\mathfrak{g}}}^{-1}$, such that $d_\pi f=0=d_\pi g$, the elements $\mathscr X_{\pi,1}^k(f),\,\mathscr X_{\pi,1}^l(g),\,k,l=0,1,2,\dots$ commute with respect to $\pi$: $$\{\mathscr X_{\pi,1}^k(f),\mathscr X_{\pi,1}^l(g)\}=[\mathscr X_{\pi,1}^k(f),[\pi,\mathscr X_{\pi,1}^l(g)]]=0.$$
Conclusions and remarks
=======================
As we have just seen, the (strong) Nijenhuis property of an $L_\infty$ derivation allows one to reproduce in a word for word manner the proof of the proposition \[prop:ashiftcl\], thus yielding the commutative algebras in the deformation quantization of Poisson manifolds (in case the quantization is done according to the Kontsevich’s recipe). Let us now briefly discuss the possible directions of future research in relation with this our construction and the other topics, arising from it.
The first and most acute problem of our result is that so far we know of no other examples of $L_\infty$-derivations, verifying the (strong) Nijenhuis property except for the classical ones, i.e. those which arise in the study of usual Poisson algebras, for instance the linear Poisson structures on Euclidean spaces (in particular on the dual spaces of Lie algebras). Finding such a nontrivial example (or disproving its existence) is the most important question to be addressed in any paper, dedicated to the elaboration of the methods, considered above.
Next, the manner of our proof is not the most economical one. In fact, we just showed, that the map $\mathscr X_{\pi,1}$ in this case verifies the equations, similar to a Nijenhuis vector field $\xi$, see proposition \[prop:shiftLinf1\]. Moreover, as we saw in section 3, one can get some intermediate result with a much a much less restrictive assumptions. Thus the question, which should also be addressed in a future investigation is whether the strong Nijenhuis property can in some way be relaxed. Some evident improvements in this side can be made right now; for instance, since our proof of proposition \[prop:shiftLinf1\] only involved manipulations with the maps $\mathscr X_{k,\pi}$ for $k=1,2,3$, we could have freely removed the conditions, involving all other maps $\mathscr X_{k,\pi},\,k\ge4$ from our considerations. However, this is too small an improvement to make this point at present. And of course, this question is closely related with the previous one: one might suppose that relaxing the strong Nijenuis property would make the quest for the corresponding examples easier.
Another consideration, which can be helpful in the search of applications of this construction, is that in the case of semisimple Lie algebras, the quantum counterparts of the commutative subalgebras rendered by the argument shift method, are known; we imply at the results of Tarasov, Rybnikov, Molev and others (see [@Tarasov; @Rybnikov; @Molev2]). These algebras are certain commutative subalgebras inside the universal enveloping algebra, which coincide with the argument shift results modulo the terms, linear in deformation parameter; however, the methods in which they are obtained, are totally different from each other and from anything, resembling the argument shifting. Thus, one of the first questions, one could ask about these algebras, is whether there is any shifting construction, that would underlie these results. Another interesting observation is that all these results are about the semisimple Lie algebras, whereas the usual method is applicable to any Lie algebra, and even to any Poisson structure with any Nijenhuis field associated to it. The role of the semisimplicity assumption is very far from being clearly understood, as well as the degree to which it can be dispensed of.
Last, but not least, is the question about the homological meaning of the Nijenhuis property: as one knows, the major step towards the construction of Kontsevich’s quantization is the observation, that solutions of Maurer-Cartan equations can be “moved around” by $L_\infty$-morphisms. Now, consider the pair $(\pi,\xi)$, where $\pi$ is a solution of the Maurer-Cartan equation in a DG Lie algebra [$\mathfrak{g}$]{} and $\xi$ is a Nijenhuis vector field for $\pi$ (regarded as a derivation of [$\mathfrak{g}$]{} or more generally as a linear operator on [$\mathfrak{g}$]{}). Let $F=\{F_n\}:{\ensuremath{\mathfrak{g}}}\to\mathfrak h$ be an $L_\infty$-morphism of DG Lie algebras. What kind of structure can we induce on $\mathfrak h$ from $(\pi,\xi)$ with the help of $F$? In particular, if $F$ is an $L_\infty$-quasi-isomorphism, then can one use $F$ to obtain a Maurer-Cartan element $F(\pi)$ in $\mathfrak h$ with an $L_\infty$-derivation, verifying the weak or strong Nijenhuis condition, associated with $F(\pi)$?
[99]{} S.V. Manakov. *Note on the integration of Euler’s equations of the dynamics of an $n$-dimensional rigid body.* Functional Analysis and Its Applications, [**10**]{}:328-329, 1976. A.S. Mishchenko and A.T. Fomenko. *Euler equations on finite-dimensional Lie groups*. Mathematics of the USSR-Izvestiya, [**12**]{}(2):371-389, 1978. A.V. Bolsinov. *Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution.* Mathematics of the USSR-Izvestiya, [**38**]{}(1):69-90, 1992. A.V. Bolsinov, K.M. Zuev. *A formal Frobenius theorem and argument shift.* Mathematical Notes, [**86**]{}(1-2):10-18, 2009. S.T. Sadetov. *A proof of the Mishchenko-Fomenko conjecture.* Doklady Math., [**70**]{}(1):634-638, 2004. A. Bolsinov, P. Zhang. *Jordan-Kronecker invariants of finite-dimensional Lie algebras.* arXiv:1211.0579, 2012. A. Izosimov. *Generalized argument shift method and complete commutative subalgebras in polynomial Poisson algebras.* arXiv:1406.3777, 2014. M. Kontsevich. *Deformation quantization of Poisson manifolds, I.* Lett. Math.Phys., [**66**]{}(3):157-216, 2003. B. Feigin, E. Frenkel. *Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras.* Int. Jour. Mod. Phys., [**A7**]{} Supplement 1A:197-215, 1992. A. I. Molev. *Yangians and their applications.* in: Handbook of algebra, vol. [**3**]{}, North-Holland, Amsterdam, 2003, 907-959. A. Futorny, A. Molev. *Quantization of the shift of argument subalgebras in type A.* Adv. Math. [**285**]{}:1358-1375, 2015. A. Molev, O. Yakimova. *Quantisation and nilpotent limits of Mishchenko-Fomenko subalgebras.* Represent. Theory [**23**]{}:350-378, 2019. A.A. Tarasov. *On some commutative subalgebras of the universal enveloping algebra of the Lie algebra $\mathfrak{gl}(n,\mathbb C)$.* Math. Sbornik [**191**]{}(9):115-122, 2000 L.G. Rybnikov. *The Argument Shift Method and the Gaudin Model.* Funktsional. Anal. i Prilozhen. [**40**]{}(3):30-43, 2006 D. Talalaev. *Quantization of the Gaudin System.* arXiv:hep-th/0404153, 2004. N. Ciccoli, P. Witkowski. *From Poisson to Quantum Geometry.* Warsaw, 2006
|
---
abstract: 'The skin effect is analyzed to provide the numerous measurements of the penetration depth of the electromagnetic field in superconducting materials with a theoretical basis. Both the normal and anomalous skin effects are accounted for within a single framework. The emphasis is laid on the conditions required for the penetration depth to be equal to London’s length, which enables us to validate an assumption widely used in the interpretation of all current experimental results.'
author:
- Jacob Szeftel$^1$
- Nicolas Sandeau$^2$
- Antoine Khater$^3$
title: Study of the skin effect in superconducting materials
---
introduction
============
Superconductivity is characterized by two prominent properties[@par; @ash; @gen; @sch; @tin], namely persistent currents in vanishing electric field and the Meissner effect[@mei], which latter highlights the rapid decay of an applied magnetic field in bulk matter in a superconductor. Early insight into the Meissner effect was achieved thanks to London’s equation[@lon] $$B+\mu_0\lambda^2_L\textrm{curl} j=0\quad,$$ where $\mu_0,j,\lambda_L$ stand for the magnetic permeability of vacuum, the persistent current, induced by the magnetic induction $B$ and London’s length, respectively. London’s equation, combined with those of Newton and Maxwell, entails[@lon; @par; @ash; @gen; @sch; @tin] that the penetration depth of the magnetic field is equal to $\lambda_L$ $$\lambda_L=\sqrt{\frac{m}{\mu_0\rho e^2}}\quad,$$ where $e,\quad m,\quad\rho$ stand for the charge, effective mass and concentration of superconducting electrons. Thus the measurement of $\lambda_L$ is all the more important, since there is no other experimental access to $\rho$.
Pippard[@pip1; @pip; @pip2] carried out the first measurements of electromagnetic energy absorption at a frequency $\omega\approx 10GHz$ in superconducting $Sn$, containing impurities, and interpreted his results within the framework of the anomalous skin effect. In normal conductors, the real part of the dielectric constant $\epsilon_R(\omega)$ being negative for $\omega<\omega_p$, where $\omega_p\approx 10^{16}Hz$ stands for the plasma frequency, causes the electromagnetic field to remain confined within a thin layer of frequency dependent thickness $\delta(\omega)$, called the skin depth, and located at the outer edge of the conductor. $\delta$ is well known[@jac; @bor] to behave like $\omega^{-1/2}$ at low frequency and to reach, in very pure metals at low temperature, a $\omega$-independent, lower bound $\delta_a$, characteristic of the anomalous skin effect[@reu; @cha]. Actually, in the wake of Pippard’s work, all current determinations of $\lambda_L$, made in superconducting materials[@h3; @h4; @h2; @h1; @gor; @har], including high $T_c$ compounds ($T_c$ stands for the critical temperature), consist of measuring the penetration depth of the electromagnetic field at frequencies $\omega\in\left[10MHz,100GHz\right]$, while assuming $\lambda_L=\delta_a$.
As the latter assumption has hardly been questioned, the main purpose of this work is to ascertain its validity by working out a comprehensive analysis of the skin effect, including both the usual and anomalous cases. The treatment of the electrical conductivity at finite frequency in a superconducting material runs into further difficulty, because, according to the mainstream model[@ash; @par; @gen; @tin; @sch], the conduction electrons make up, for $T< T_c$, a two-component fluid, comprising normal and superconducting electrons. This work, which is intended at deriving the respective contributions of the two kinds of electrons, is based solely on Newton and Maxwell’s equations.
The outline is as follows: Sections II deals with the skin effect; the results are used to work out the conduction properties of the two-fluid model in Section III, while contact is made with the experimental results in Section IV. The conclusions are given in Section V.
![Cross-section of the superconducting sample (dotted) and the coil (hatched); $E_\theta$ and $j_\theta$ are both normal to the unit vectors along the $r$ and $z$ coordinates; vertical arrows illustrate the $r$ dependence of $B_z(r)$; $r_c$ has been magnified for the reader’s convenience; Eq.(\[coil\]) has been integrated from $A$ ($B_z(r_0+2r_c)=0$) to $B$[]{data-label="Bzr"}](Bz.eps){height="5" width="7.5"}
skin effect
===========
Consider as in Fig.1 a superconducting material of cylindrical shape, characterized by its symmetry axis $z$ and radius $r_0$ in a cylindrical frame with coordinates ($r,\theta,z$). The material is taken to contain conduction electrons of charge $e$, effective mass $m$, and total concentration $\rho$. It is subjected to an oscillating electric field $E(t,r)=E_\theta(r)e^{i\omega t}$, with $t$ referring to time. As $E_\theta(r)$ is normal to the unit vectors along the $r$ and $z$ coordinates, there is $\textrm{div} E=0$. $E$ induces a current $j(t,r)=j_\theta(r)e^{i\omega t}$ along the field direction, as given by Newton’s law $$\label{newt}
\frac{dj}{dt}=\frac{\rho e^2}{m}E-\frac{j}{\tau}\quad,$$ where $\frac{\rho e^2}{m}E$ and $-\frac{j}{\tau}$ are respectively proportional to the driving force accelerating the conduction electrons and a generalized friction term, which is non zero in any superconducting material provided $\frac{dj}{dt}\neq0$.
The existence of a *friction force* in superconductors, carrying an *ac current*, is known experimentally (see[@sch] p.4, $2^{nd}$ paragraph, line $9$). For example, the measured ac conductivity for the superconducting phase of $BaFe_2(As_{1-x}P_x)_2$ has been found (see[@h1] p.1555, $3^{rd}$ column, $2^{nd}$ paragraph, line $11$) to be $\approx .03\sigma_n$, where $\sigma_n$ stands for the normal conductivity[@foo] measured just above the critical temperature $T_c$. However because the current is carried by electrons, making up either a BCS state[@bar] or a Fermi gas[@ash] in a superconducting and normal metal, respectively, the physical sense of $\tau$ in Eq.(\[newt\]) for superconductors may be different from that given by the Drude model[@ash] for a normal metal. To understand this difference and to model the new $\tau$, we shall next work out the equivalent of Ohm’s law for a superconducting material when submitted to an electric field.
The superconducting state, carrying no current in the absence of external fields, is assumed to comprise two subsets of equal concentration $\rho/2$, moving in opposite directions with respective mass center velocity $v,-v$, which ensures $j=0,\quad p=0$, where $p$ refers to the average electron momentum. Under a driving field $E$, an ensemble $\delta\rho/2$ of electrons is transferred from one subset to the other, so as to give rise to a finite current $j=\delta\rho ev=e\delta p/m$, where $\delta p$ stands for the electron momentum variation. The generalized friction force is responsible for the reverse mechanism, whereby electrons are transferred from the majority subset of concentration $\frac{\rho+\delta\rho}{2}$ back to the minority one ($\frac{\rho-\delta\rho}{2}$). It follows from flux quantization and the Josephson’s effect[@par; @ash; @gen; @sch; @tin; @jos] that the elementary transfer process involves a pair[@coo] rather than a single electron. Hence if $\tau^{-1}$ is defined as the transfer probability per unit time of one electron pair, the net electron transfer rate is equal to $\frac{\rho+\delta\rho-(\rho-\delta\rho)}{2\tau}=\frac{\delta\rho}{\tau}$. By virtue of Newton’s law, the resulting generalized friction term is $mv\delta\rho/\tau=\delta p/\tau\propto j/\tau$, which validates Eq.(\[newt\]). Furthermore for $\omega\tau<<1$, the inertial term $\propto\frac{dj}{dt}$ in Eq.(\[newt\]) is negligible, so that we can write the equivalent of Ohm’s law for the superconducting material as $$j=\sigma E\quad,\quad\sigma=\frac{\rho e^2\tau}{m}\quad.$$ Thus both Ohm’s law and $\sigma$ are seen to display[@ash] the same form in normal and superconducting metals, as well.
$E$ induces a magnetic induction $B(r,t)=B_z(r)e^{i\omega t}$, parallel to the $z$ axis. $B$ is given by the Faraday-Maxwell equation as $$\label{Bz}
-\frac{\partial B}{\partial t}=\textrm{curl}E=\frac{E}{r}+\frac{\partial E}{\partial r}\quad.$$ The displacement vector $D$, is parallel to $E$ and is defined as $$D=\epsilon_0 E+\rho e u\quad,$$ where $\epsilon_0,\quad u$ refer to the electric permittivity of vacuum and displacement coordinate of the conduction electron center of mass, parallel to $E$. The term $\rho e u$ represents the polarization of conduction electrons[@fo2]. Because $\textrm{div} E=0$ entails that $\textrm{div} D=0$, Poisson’s law warrants the lack of charge fluctuation around $\rho e$. Thence since there is by definition $j=\rho e\frac{du}{dt}$, the displacement current reads $$\frac{\partial D}{\partial t}=j+\epsilon_0\frac{\partial E}{\partial t}\quad.$$ Finally the magnetic field $H(t,r)=H_z(r)e^{i\omega t}$, parallel to the $z$ axis, is given by the Ampère-Maxwell equation as $$\label{Hz}
\textrm{curl}H=-\frac{\partial H}{\partial r}=j+\frac{\partial D}{\partial t}=2j+\epsilon_0\frac{\partial E}{\partial t} \quad.$$ Replacing $E(t,r),j(t,r),B(t,r),H(t,r)$ in Eqs.(\[newt\],\[Bz\],\[Hz\]) by their time-Fourier transforms $E_\theta(\omega,r),j_\theta(\omega,r),B_z(\omega,r),H_z(\omega,r)$, while taking into account $$B_z\left(\omega,r\right)=\mu\left(\omega\right)H_z\left(\omega,r\right)\quad,$$ where $\mu\left(\omega\right)=\mu_0\left(1+\chi_s\left(\omega\right)\right)$ ($\chi_s\left(\omega\right)$ is the magnetic susceptibility of superconducting electrons at frequency $\omega$) yields $$\label{fou2}
\begin{array}{l}
E_\theta\left(\omega,r\right)=\frac{1+i\omega\tau}{\sigma}j_\theta\left(\omega,r\right)\\
i\omega B_z\left(\omega,r\right)=-\left(\frac{E_\theta\left(\omega,r\right)}{r}+\frac{\partial E_\theta\left(\omega,r\right)}{\partial r}\right)\\
\frac{\partial B_z\left(\omega,r\right)}{\partial r}=-\mu\left(\omega\right)\left(2j_\theta\left(\omega,r\right)+i\omega\epsilon_0E_\theta\left(\omega,r\right)\right)
\end{array}$$ Eliminating $E_\theta\left(\omega,r\right),j_\theta\left(\omega,r\right)$ from Eqs.(\[fou2\]) gives $$\label{skin}
\frac{\partial^2 B_z\left(\omega,r\right)}{\partial r^2}=\frac{B_z\left(\omega,r\right)}{\delta^2(\omega)}-\frac{\partial B_z\left(\omega,r\right)}{r\partial r}\quad.$$ The skin depth $\delta$ and plasma frequency $\omega_p$ are defined[@ash; @jac; @bor] as $$\begin{array}{l} \delta(\omega)=\frac{\lambda_L}{\sqrt{\left(1+\chi_s\left(\omega\right)\right)\left(\frac{2i\omega\tau}{1+i\omega\tau}-\frac{\omega^2}{\omega^2_p}\right)}}\quad,\\
\omega_p=\sqrt{\frac{\rho e^2}{\epsilon_0m}}
\end{array}$$ Note that the above formula of $\delta(\omega)$ retrieves indeed both, the usual[@jac; @bor] expression $|\delta|=\frac{1}{\sqrt{2\mu_0\sigma\omega}}$, valid for $\omega\tau<<1$, and the $\omega\tau>>1$ limit $\delta_a=\frac{\lambda_L}{\sqrt{2}}$, typical of the anomalous skin effect[@reu; @cha] and widely used in the interpretation of the experimental work[@pip1; @pip; @pip2; @h3; @h4; @h2; @h1; @gor; @har].
![Semi-logarithmic plots of $B_z(u),e^u$.[]{data-label="f2"}](f2.eps){height="6" width="6"}
As Eqs.(\[fou2\]) make up a system of $3$ linear equations in terms of $3$ unknowns $j_\theta,E_\theta,B_z$, there is a single solution, embodied by Eq.(\[skin\]). The solution of Eq.(\[skin\]), which has been integrated over $r\in \left[0,r_0\right]$ with the initial condition $\dfrac{dB_z}{dr}\left(r=0\right)=0$, is a Bessel function, having the property $B_z(r)\approx e^{r/\delta(\omega)}$ if $r>>|\delta(\omega)|$, as illustrated in Fig.\[f2\].
the two-fluid model
===================
The total current $j_t$ reads as $$j_t=j_n+j_s=(\sigma_n+\sigma_s)E\quad,$$ where $j_n,\sigma_n=\frac{\rho_n e^2\tau_n}{m_n}$ ($j_s,\sigma_s=\frac{\rho_s e^2\tau_s}{m_s}$) designate the normal (superconducting) current and conductivity, with $\rho_n,\tau_n,m_n$ ($\rho_s,\tau_s,m_s$) being the concentration, decay time of the kinetic energy, associated with $j_n$ ($j_s$), and effective mass of normal (superconducting) electrons. Replacing $j_\theta$ in Eqs.(\[fou2\]) by the expression $j_t$ hereabove leads to the following expression of the skin depth for the practical case $\omega<<\omega_p$ and $\chi_s<<1$ $$\label{skin3}
\delta^{-2}=2i\mu_0\omega\left(\frac{\sigma_n}{1+i\omega\tau_n}+\frac{\sigma_s}{1+i\omega\tau_s}\right)\quad.$$ The inequalities $\rho_n<<\rho_s,\tau_n<<\tau_s,\sigma_n<<\sigma_s$ can be inferred from observation[@h3; @h4; @h2; @h1; @gor; @har] to hold at $T$ well below $T_c$, so that $\delta$ in Eq.(\[skin3\]) is recast finally as $$\begin{array}{l}
|\delta\left(\omega<<\tau_s^{-1}\right)|=\frac{1}{\sqrt{2\mu_0\sigma_s\omega}}=\frac{\lambda_L}{\sqrt{2\omega\tau}}\quad,\\
\delta\left(\omega>>\tau_s^{-1}\right)=\sqrt{\frac{m_s}{2\mu_0\rho_s e^2}}=\frac{\lambda_L}{\sqrt{2}}\quad,
\end{array}$$ where both limits $\omega\tau_s<<1,\quad \omega\tau_s>>1$ are seen to correspond to the usual and anomalous skin effect, respectively. Besides it is concluded that the conduction properties, at $\omega\neq 0$ and $T$ well below $T_c$, are assessed solely by the superconducting electrons, while the normal ones play hardly any role.
comparison with experiment
==========================
The experiments, performed at $\omega\approx 10GHz$, give access to the imaginary part of the complex impedance of the resonant cavity, containing the superconducting sample, equal to $\mu_0\omega l_p$, where $l_p$ designates the penetration depth of the electromagnetic field, i.e. $l_p=\delta$. Moreover at $\omega\approx 10MHz$, the cavity is to be replaced by a resonant circuit, combining a capacitor and a cylindrical coil of radius $r_0$. The sample is inserted into the coil, which is flown through by an oscillating current $I_0(\omega)e^{i\omega t}$. The coil is made up of a wire of length $l$ and radius $r_c$ (see Fig.\[Bzr\]). Since the relationship between the penetration depth $l_p$ and the observed self inductance of the coil $L$ is seemingly lacking, we first set out to work out an expression for $L$.
Applying Ohm’s law yields $$\label{ohm}\begin{array}{l}
-l\left(E_a(\omega)+E_\theta(\omega,r_0)\right)=RI_0(\omega)\Rightarrow\\
E_\theta(\omega,r_0)=\frac{U(\omega)-RI_0(\omega)}{l}
\end{array}\quad,$$ where $E_a(\omega)e^{i\omega t},E_\theta(\omega,r)e^{i\omega t},Ue^{i\omega t}=-lE_a(\omega)e^{i\omega t},R$ are the applied and induced electric fields, both normal to the $r,z$ axes, the voltage drop throughout the coil and its resistance, respectively $\left(E_a(\omega),E_\theta(\omega,r),U(\omega)\in\mathds{C}\right)$. Besides $E_\theta(\omega,r_0)$ is obtained from Eq.(\[fou2\]) as $$\label{eth}E_\theta(\omega,r_0)=-i\omega\delta(\omega)B_z(\omega,r_0)\quad.$$ where $E_\theta(\omega,r\rightarrow r_0)\approx E_\theta(\omega,r_0)e^{\frac{r-r_0}{\delta(\omega)}}$. Working out $B_z(\omega,r_0)$ in Eq.(\[eth\]) requires to solve the Ampère-Maxwell equation for $B_z(\omega,r)e^{i\omega t}$ inside a cross-section of the coil wire $$\label{coil}
\frac{\partial B_z}{\partial r}=-\mu_0\left(2j_c+i\epsilon_0\omega E_c\right) \quad,$$ where $j_c(\omega)=\frac{I_0(\omega)}{\pi r_c^2}$ and $E_c(\omega)=E_a(\omega)+E_\theta(\omega,r_0)$ are both assumed to be $r$-independent. Moreover integrating Eq.(\[coil\]) for $r\in\left[r_0+2r_c,r_0\right]$ with the boundary condition $B_z(\omega,r_0+2r_c)=0,\forall t$ (see Fig.\[Bzr\]), while taking advantage of Eq.(\[ohm\]), yields $$\label{Bzt}
B_z(\omega,r_0)=2\mu_0\left(\frac{2}{\pi r_c}-\frac{i\epsilon_0\omega r_cR}{l}\right)I_0(\omega)\quad.$$ Finally it ensues from the definition of $L$ and Eq.(\[Bzt\]) that $$LI_0=2\pi Re\left(\int_0^{r_0}B_z(\omega,r_0)e^{\frac{r-r_0}{\delta}}rdr\right)\Rightarrow L\approx 2^{\frac{5}{2}}\mu_0|\delta|\frac{r_0}{r_c},$$ where $Re$ means real part and $\delta=\lambda_L/\sqrt{2i\omega\tau}$ for $\omega\tau<<1$.
Both imaginary parts of the coil and cavity impedance are thus found to be equal to $C(\omega)\omega|\delta(\omega)|$, where $C(\omega)$ is an unknown coefficient depending on the experimental conditions. As $C(\omega)$ is likely to be $T$ independent at least inside the limited range $0<T<T_c$, this hurdle can be overcome by plotting the dimensionless ratio $\left|\frac{\delta(T)}{\delta(T_c)}\right|$ versus $T\in\left[0,T_c\right]$, where $\delta(T)$ is the skin depth measured at two frequencies far apart from each other, e.g. $\omega_1=10GHz,\quad\omega_2=10MHz$. The reason for suggesting such an experimental procedure is that the value of $\delta(T_c)$ is well known, because $\delta(T_c)=\frac{1}{\sqrt{2\mu_0\sigma_n(T_c)\omega}}$ and the normal conductivity $\sigma_n(T_c)$ can be measured with great accuracy. Consequently two cases should be considered :\
*i)* the two curves representing $\left|\frac{\delta(T\in\left[0,T_c\right])}{\delta(T_c)}\right|$, corresponding to $\omega_1,\quad\omega_2$ cannot be distinguished from each other, which entails that $\delta(T<T_c)\propto1/\sqrt{\omega_{i=1,2}}$ even for $T\rightarrow T_c^-$. As this conclusion implies also that $\omega_{i=1,2}\tau_s<<1$, this case is likely to be observed in high-$T_c$ superconducting alloys, wherein $\tau_s$ is bound to be relatively short because of numerous impurities;\
*ii)* conversely the two curves turn out to be conspicuously different, which points toward the limit $\omega_1\tau_s>>1,\quad\omega_2\tau_s<<1$, characterized by $\delta(\omega_1,T<T_c)=\frac{\lambda_L(T)}{\sqrt{2}},\quad \delta(\omega_2,T<T_c)=\frac{\lambda_L(T)}{\sqrt{2\omega_2\tau_s}}$. Since a very long $\tau_s$ is required, the good candidates are likely to be chosen among the elementary superconducting metals of very high purity[@pip1; @pip; @pip2]. However, because of $\rho_s(T\rightarrow T_c^-)\rightarrow0$, the $\omega$ dependence of $\delta(T<T_c)$ will switch from $\delta(T)$ independent from $\omega$ for $T$ well below $T_c$ to $\delta\propto\omega^{-1/2}$ for $T\rightarrow T_c^-$.
Because $\tau_s$ is not known, $\lambda_L(T)$ cannot be measured in the limit $\omega\tau_s<<1$. Therefore it is suggested to work at higher frequencies, such that $\omega>>1/\tau_s,\omega<<\omega_p$, because $\delta(\omega)=\lambda_L/\sqrt{2}$ is independent from $\tau_s$ in that range. Typical values $\tau_s\approx 10^{-11}s,\omega_p\approx 10^{16}Hz$ would imply to measure light absorption in the IR range. Then for an incoming beam being shone at normal incidence on a superconductor of refractive index $\tilde{n}\in\mathds{C}$, the absorption and reflection coefficients $A,R$ read[@jac; @bor] $$A=1-R=1-\left|\frac{1-\tilde{n}}{1+\tilde{n}}\right|^2\quad.$$ The refractive index $\tilde{n}$ and the complex dielectric constant $\epsilon=\epsilon_R+i\epsilon_I$, conveying the contribution of conduction electrons, are related[@jac; @bor] by $$\tilde{n}^2=\frac{\epsilon}{\epsilon_0}=1-\frac{\left(\omega_p/\omega\right)^2}{1-i/(\omega\tau_s)}\quad.$$ At last we get $$\lambda_L=\frac{2}{\mu_0c\sigma_s A}\quad,\quad\tau_s=\mu_0\sigma_s\lambda_L^2\quad,$$ where $c$ refers to light velocity in vacuum.
conclusion
==========
The skin effect has been analyzed in a single framework, to account for both the cases of the usual and anomalous skin effect. The calculation of the skin depth has then been applied to the study of electromagnetic wave propagation in superconducting materials and to the interpretation of skin depth measurements. An experiment has been proposed to measure London’s length in dirty superconductors.
[00]{} R.D. Parks, Superconductivity, ed. CRC Press (1969) N.W. Ashcroft and N. D. Mermin, Solid State Physics, ed. Saunders College (1976) P.G. de Gennes, Superconductivity of Metals and Alloys, ed. Addison-Wesley, Reading, MA (1989) J.R. Schrieffer, Theory of Superconductivity, ed. Addison-Wesley (1993) M. Tinkham, Introduction to Superconductivity, ed. Dover Books (2004) W. Meissner and R. Ochsenfeld, Naturwiss., 21, 787 (1933) F. London, Superfluids, ed. Wiley, vol.1 (1950) A.B. Pippard, Proc.Roy.Soc., A203, 98 (1950) A.B. Pippard, Proc.Roy.Soc., A216, 547 (1953) A.B. Pippard, Phil.Mag., 41, 243 (1950) J.D. Jackson, Classical Electrodynamics, ed. John Wiley (1998) M. Born and E. Wolf, Principles of Optics, ed. Cambridge University Press (1999) G. E. H. Reuter and E. H. Sondheimer, Proc.Roy.Soc., A195, 336 (1948) R. G. Chambers, Proc.Roy.Soc., A215, 481 (I952) K. Hashimoto et al., Phys. Rev. Lett., 102, 017002 (2009) K. Hashimoto et al., Phys. Rev. Lett., 102, 207001 (2009) K. Hashimoto et al., Phys. Rev.B, 81, 220501R (2010) K. Hashimoto et al., Science, 336, 1554 (2012) R. T. Gordon et al., Phys.Rev.B, 82, 054507 (2010) W. N. Hardy et al., Phys. Rev. Lett., 70, 3999 (1993)
the finite conductivity, measured at $\omega\neq 0$ in superconductors, is consistent with the observation of persistent currents at *vanishing* electric field, even though a compelling explanation is still lacking[@par; @ash; @gen; @sch; @tin] J. Bardeen, L.N. Cooper and J.R.Schrieffer, Phys. Rev., 108, 1175 (1957) B. D. Josephson, Phys. Letters, 1, 251 (1962) L.N. Cooper, Phys. Rev., 104, 1189 (1956) the contribution of electrons bound to the nucleus can be taken into account, by replacing hereafter $\epsilon_0$ by $\epsilon_0+\alpha$, where $\alpha$ refers to their polarisability
|
---
author:
- 'L. Inno $^,$'
- 'N. Matsunaga'
- 'M. Romaniello'
- 'G. Bono$^,$'
- 'A. Monson'
- 'I. Ferraro'
- 'G. Iannicola'
- 'E. Persson'
- 'R. Buonanno$^,$'
- 'W. Freedman'
- 'W. Gieren'
- 'M.A.T. Groenewegen'
- 'Y. Ita'
- 'C.D. Laney$^,$'
- 'B. Lemasle'
- 'B.F. Madore'
- 'T. Nagayama$^,$'
- 'Y. Nakada'
- 'M. Nonino'
- 'G. Pietrzy[ń]{}ski$^,$'
- 'F. Primas'
- 'V. Scowcroft'
- 'I. Soszy[ń]{}ski'
- 'T. Tanabé'
- 'A. Udalski'
date: 'drafted / Received / Accepted 10/12/14 '
title: 'New NIR light-curve templates for classical Cepheids'
---
[The mean NIR magnitudes based on the new templates are up to 80% more accurate than single–epoch NIR measurements and up to 50% more accurate than the mean magnitudes based on previous NIR templates, with typical associated uncertainties ranging from 0.015 mag ($J$ band) to 0.019 mag ($K_{\rm{S}}$ band). Moreover, we find that errors on individual distance estimates for Small Magellanic Cloud Cepheids derived from NIR PW relations are essentially reduced to the intrinsic scatter of the adopted relations.]{} [Thus, the new templates are the ultimate tool for estimating precise Cepheid distances from NIR single-epoch observations, which can be safely adopted for future interesting applications, including deriving the 3D structure of the Magellanic Clouds. ]{}
Introduction
============
Radially pulsating variables, and in particular RR Lyrae and Classical Cepheids, play a key role in modern astrophysics because they are robust primary distance indicators and solid tracers of old (t$\sim$10–12 Gyr) and young (t$\sim$10–300 Myr) stellar populations, respectively. The radially pulsating variables when compared with canonical stellar tracers have the key advantage of being easily recognized by their characteristic light-curves and to provide firm constraints on the metallicity gradient and the kinematics of both the thin disk [@pedicelli09; @luck11; @luck11a; @genovali13] and the halo [@kinman12]. The above evidence applies not only to the Galaxy, but also to nearby resolved stellar systems [@minniti03].
The main drawback in using classical Cepheids is that they have periods ranging from one day to several tens of days. This means that identifying and characterizing them is demanding from the observational point of view. An unprecedented improvement on the number of known radially pulsating variables was indeed provided by microlensing experiments (MACHO, EROS, OGLE) as a byproduct of their large-area surveys. In particular, the ongoing OGLE IV project became a large-scale, long-term, sky-variability survey, and will further increase the variable star identification in the Galactic Bulge and in the Magellanic Clouds [OGLE IV, @sos2012]. These surveys have been deeply complemented by large experiments aimed at detecting variable phenomena covering a significant fraction of the Southern Sky, either in the optical, such as the ASAS Survey [@pojmanski02], the QUEST Survey [@vivas04], the NSVS survey [@kinemuchi06], the LONEOS Survey [@miceli08], the Catalina Real-time Transient survey [@drake09], the SEKBO survey [@akhter12] and the LINEAR Survey [@palaversa13]; or in the near-infrared (NIR), such as the IRSF [@ita04], the VVV [@minniti10]; or in the mid-infrared (MIR) such as the Carnegie RRL Program [CRRLP; @freedman12].
The intrinsic feature of current surveys is that both the identification and the characterization is performed in optical bands, since the pulsation amplitude is typically larger in the B–band than in the NIR bands. However, recent theoretical [@bono10] and empirical [@storm11a; @storm11b; @inno13; @groe13] evidence indicates that NIR and MIR photometry has several indisputable advantages when compared with optical photometry: a) it is minimally affected by metallicity dependence [@bono10; @madore10]; b) it is minimally affected by reddening uncertainties; and c) the luminosity amplitude is a factor of 3–5 smaller than in optical bands. This means that NIR observations are not very efficient in identify new variables, but they play a crucial role in heavily reddened regions [@matsunaga11; @matsunaga13]. Moreover, accurate mean NIR and MIR magnitudes can be provided even with a limited number of phase points, because of their reduced luminosity amplitudes in this wavelength regime. However, NIR and MIR ground–based observations are even more time-consuming than optical observations, because of sky subtraction. This is the reason why during the past 20 years NIR light-curve templates have been developed for RR Lyrae [@jones96] and classical Cepheids [@soszynski05; @pejcha12]. The key advantage of this approach is that variables for which the pulsation period, the epoch of maximum, and the $B$- and/or the $V$–band amplitude are available, a single-epoch NIR measurement is enough to provide accurate estimates of their mean NIR magnitudes, which can then be used to compute their distances.
The most recent Cepheid NIR light-curve template was published by @soszynski05 [hereinafter S05]. They used 30 Galactic and 31 Large Magellanic Cloud (LMC) calibrating Cepheids and provided analytical Fourier fits in the $J$, $H$, and $K$ bands by using two period bins for Galactic (0.5$< \log P \le$ 1.3 and $\log P > 1.3$) and LMC (0.5$<\log P \le 1.1$ and $\log P > 1.1$) Cepheids.
We derived new sets of NIR light-curve templates for classical Cepheids covering the entire period range (0.0 $< \log P \le$ 1.8). The advantages of the current approach compared with previous NIR templates available in the literature are the following:
a\) Statistics – We collected optical and NIR accurate photometry for more than 180 Galactic and 500 Magellanic Cloud Cepheids. Among these data, we selected the light curves characterized by full phase coverage and high photometric quality in the $V$,$J$,$H$ and $K_{\rm{S}}$ bands. We ended up with a sample of more than 200 calibrating Cepheids. This sample is a three times larger than the sample adopted by S05. The sample size enabled us to split the calibrating Cepheids into ten period bins ranging from one day to approximately 100 days;
b\) Hertzsprung progression – The sample size allowed us to properly trace the change in light-curve morphology across the Hertzsprung Progression (HP). Cepheids in the period range 6$<$ P$<$16 days show a bump along the light-curves. The HP indicates the relation between this secondary feature and the pulsation period: the bump crosses the light-curve from the decreasing branch to the maximum for periods close to the center of the HP and moves to earlier phases for longer periods. To properly trace the change in the shape of the light curve, we adopted a new anchor for the phase zero–point. The classical approach was to use the phase of maximum light of optical light curves to phase the NIR light-curves. The use of the phase of maximum light as zero–point ($\phi$=0) was justified by the fact that the photometry was more accurate along the brighter pulsation phases. However, this anchor has an intrinsic limit in dealing with bump Cepheids. At the center of the HP the optical light-curves are either flat topped or show a double peak. This means that from an empirical point of view it is quite difficult to identify the phase of maximum light. Moreover, the center of the HP is metallicity dependent (see Sect. 2.1). To overcome this problem, we decided to use the phase of the mean magnitude along the rising branch. This phase zero–point can be easily estimated even if the light-curve is not uniformly sampled;
c\) First overtones – We derived for the first time the template for first overtone Cepheids in the $J$ band;
d\) Analytical fit – Together with the classical analytical fits based on Fourier series we also provide a new analytical fit based on periodic Gaussian functions. The key advantage in using the latter functions is that the precision is quite similar to the canonical fit, but the number of parameters decreases from fifteen to nine.
The structure of the paper is the following: In Sect. 2 we discuss in detail the different samples of calibrating light curves we adopted to derive the template for fundamental (FU) and first overtone (FO) Cepheids. In particular, in Sect. 2.1 we describe the new technique adopted for phasing and merge the light-curves. The preliminary analysis of the calibrating NIR light-curves and the development of the template is described in Sect. 3, while the analytical formula are given in Sect. 4. In Sect. 5, we discuss in detail the NIR–to–optical amplitude ratios that we adopted to apply the new templates. Sect. 6 describes the application of the templates and with the error budget associated with the new templates. Finally, in Sect. 7 we summarize the results of this investigation and briefly outline possible future developments.
Optical and NIR data sets for calibrating Cepheids
==================================================
Our analysis is based on the largest available sample of fundamental–mode Cepheids with well-covered light-curves in the NIR and in optical ($V$,$I$) bands. This sample covers a very broad period range (1–100 days). We collected $J$, $H$, and $K_{\rm{S}}$band observations from four different data sets: @laney92 [51] and @monson11 [131] for Galactic Cepheids, @persson04 [92] for LMC Cepheids, and the IRSF survey catalog for $\sim$500 Small Magellanic Cloud (SMC) Cepheids.\
The optical ($V$,$I$) light-curves for the galactic Cepheids were collected from the literature [@laney92; @berdnikov04 and references therein][^1], while for Magellanic Cloud Cepheids we adopted the data from the OGLE III Catalog of Variable Stars [@sos2008; @sos2010][^2]. When compared with the other microlensing surveys mentioned in the previous section, the OGLE III Catalog provides very accurate $V$– and $I$–band light-curves, with the typical photometric error associated with individual measurements $\le$0.008 mag and $\le$ 0.006 mag for $I$ and $V$ bands, respectively. Moreover, the sky coverage of the OGLE III survey fully matches the coverage by the IRSF survey.\
[*i) Calibrating Cepheids in the IRSF/OGLE Sample (SMC)*]{}\
During the past few years, the IRSF Survey (Ita et al., 2014, in prep.) collected more than $\sim$500 complete NIR light-curves (571 $J$, 434 $H$, 219 $K_{\rm{S}}$) for SMC Cepheids. These Cepheids have optical ($V$,$I$) mean magnitudes, periods, amplitudes, and positions from the OGLE III Catalog of Variable Stars [@sos2010].Typically we have more than 1,000 measurements in the optical and at least 100 in NIR bands for SMC Cepheids. This means that phasing optical and NIR light-curves is relatively simple. The photometric accuracy of the data in the IRSF catalog is $\pm$0.02 mag for the brightest ($J \approx 13$) and $\pm$0.06 mag for the faintest ($J \approx 17$) Cepheids. To improve the quality of the calibrating sample, we performed a selection based on the root mean square ($rms$) between the individual data points and the analytical fit of the individual light-curves. We adopted a seventh-order Fourier series and the selection criterion $rms \le \frac{1}{20} A_J$, where $A_J$ is the pulsation amplitude in the $J$ band ($A_J$=$J_{max}-J_{min}$). The $rms/A_J$ ratio is an indication of how strongly the photometric errors of the individual observations affect the shape of the light-curve. The threshold we chose allows us to select the most accurate light-curve while keeping a statistically significant sample for each bin. However, small changes of the adopted values do not significantly affect our results. The selection criterion was relaxed to $rms \le \frac{1}{15} A_{H,K}$ for the $H$– and $K_{\rm{S}}$–band light-curves, because the amplitude decreases with increasing wavelength. For Cepheids with shortest period Cepheids (1–3 days) we selected the light-curves with $rms \le \frac{1}{10} A_{J,H,K}$. The data for these fainter Cepheids are characterized by larger photometric errors. The *J,H,K$_{\rm S}$* measurements were transformed into the 2MASS NIR photometric system following @kato07. However, the corrections adopted for the transformations between different NIR photometric system transformations are smaller than a few hundredths of magnitude and affect neither the shape nor the amplitude of the light curves.\
![Period distribution for the calibrating Cepheids in our sample for the three different bands $J$ (top), $H$ (middle), and $K_{\rm{S}}$ (bottom). The color coding indicates different data sets (red: MP sample; blue: LS sample; green: SMC sample; orange: P04 sample). The total number of calibrating Cepheids per data set are also labeled. See text for more details.[]{data-label="f1"}](f1.pdf){width="0.90\columnwidth"}
[*ii) Calibrating Cepheids in the Persson et al. 2004 Sample (P04)*]{}\
This sample is based on the $J$,$H$, and $K$ light-curves for 92 Cepheids in the LMC published by @persson04 [P04] and the $V$ and $I$ photometric data available in the OGLE Catalog for 60 of them. The photometric precision of the data in the P04 catalog is $\pm$0.02 mag for the brightest ($J \approx 12$) and $\pm$0.06 mag for the faintest ($J \approx 14$) Cepheids. We included Cepheids with complete coverage of the light-curve in the three bands (more than 20 phase points) and with low $rms$ (i.e. $\le$ 0.4 mag in the J band), for a total of $\sim$30 selected Cepheids in the P04 sample. These Cepheids have periods between 6 and 50 days, thus increasing the number of calibrating Cepheids ($\gtrsim$50%) in the long-period regime (see Fig. \[f1\]). To transform the NIR measurements from the original LCO photometric system into the 2MASS photometric system, we adopted the relations given by @carpenter01.\
[*iii) Calibrating Cepheids in the Laney & Stobie Sample (LS)*]{}\
This sample includes 51 Galactic Cepheids with optical and NIR light curves with periods ranging from 3 to 69 days [@laney92; @laney94]. This is the most accurate photometric sample for Classical Cepheids available in literature, with intrinsic errors ranging from about $\pm$0.004 mag for the brightest ($J \approx 3$) to $\pm$0.011 mag for the faintest ($J \approx 8$). The optical light curves are also highly accurate, with typical values of the $rms \le \frac{1}{100} A_{V}$. However, we still performed a selection based on the number of available phase points (we need at least 15 measurements for the Fourier fit) in the optical and the NIR light-curves, for a total of $\sim$30 selected Cepheids in the LS sample. The *J,H,K$_{\rm S}$* measurements in @laney92 are in the SAAO photometric system. We have converted them into the 2MASS photometric system by applying the transformation equations given by @carpenter01.\
[*iv) Calibrating Cepheids in the Monson & Pierce Sample (MP)*]{}\
This sample is based on NIR photometric measurements for 131 Northern Galactic Cepheids [@monson11]. These Cepheid light-curves are sampled with an average of 22 measurements per star with an associated photometric error of $\pm$0.015 mag. However, we selected only the light curves with more than 17 measurements in each NIR band, for a total of 64, 72, and 93 light-curves in the $J$,$H$, and $K_{\rm{S}}$ bands, respectively. The original *J,H,K$_{\rm S}$* data were taken in the BIRCAM system and transformed into the 2MASS photometric system by applying the equations given by @monson11. For all the Cepheids in this sample, $V$ and $I$ light curves were collected from the literature [@berdnikov04 and references therein]. We did not perform any selection on the optical light-curves, because no high accuracy in the $V$-band data is required by our method. However, the $rms$ is always better than $\frac{1}{15} A_{V}$ for all the Cepheids in this sample.\
In total, we collected a sample of light-curves that includes more than 200 calibrating Cepheids and is three times larger than the sample adopted by S05 (60 Cepheids) to derive the NIR light-curve templates for classical Cepheids.
To further improve the sampling of the light curve over the entire period range and to reduce the $rms$ of the light-curve templates, the sample of Galactic and Magellanic calibrating Cepheids was split into ten period bins.
Note that the approach we adopted is completely reddening independent. In particular, the period is the safest diagnostic to bin the calibrating sample, because it can be easily measured with high accuracy, it does not depend on the wavelength, and it is not affected by reddening. This means that the binning in period will not introduce any systematic effect when combining optical and NIR photometric data from different instruments. Moreover, theoretical predictions [@marconi05] clearly show that the light-curve shape changes with the mass at fixed chemical composition and luminosity and that the period is the best observable to account for this trend.
The adopted period ranges and the number of calibrating Cepheids per bin are listed in Table \[tab\_bins\].
Bin Period range \[days\] N$_J$ N$_H$ N$_{Ks}$
----- ----------------------- ------- ------- ----------
1 1–3 12 3 3
2 3–5 40 35 44
3 5–7 26 29 29
4 7–9.5 18 23 29
5 9.5–10.5 11 13 13
6 10.5–13.5 19 21 25
7 13.5–15.5 24 22 23
8 15.5–20 17 19 18
9 20–30 14 17 19
10 30–100 16 19 17
1–100 197 201 220
: Adopted period bins.\[tab\_bins\]
There are typically twenty Cepheids per period bin with two exceptions: bin 1 (P$\le$3 days), for which we have fewer than a dozen objects, and bin 5 (9.5$\ge$ P $<$10), for which the number of Cepheids ranges from 11 ($J$–band) to 13 ($K_{\rm{S}}$ band). Fig. \[f1\] shows the histograms of the calibrating Cepheids in $J$ (top), $H$ (middle) and in the $K_{\rm{S}}$ (bottom) band. Cepheids belonging to different data sets are plotted with different colors.
A similar selection was also performed for FO Cepheids. The IRSF monitoring survey collected $\sim$231 complete NIR light-curves for FO Cepheids (231 $J$, 85 $H$, 10 $K_{\rm{S}}$) with periods ranging from 0.8 to 4 days. We selected from those the calibrating light curves by adopting the following criterion: $rms_J \le \frac{1}{10} A_J$. Again, the threshold was chosen to guarantee the good photometric quality of the calibrating light curves. However, because of the limited photometric accuracy of individual measurements compared with FU light-curves, the final sample of calibrating FO Cepheids only includes ten $J$-band light curves, with periods ranging from 1.4 to 3.5 days. We did not apply any binning in period for FO Cepheids, because the shape of the light-curve in this period range is almost exactly sinusoidal.
Phasing the light-curves
------------------------
Precise period determinations are required to derive correct phase shifts between optical and NIR light curves. The constraint is less severe if optical and NIR time-series data are collected in the same time interval. The $V$ and $I$ photometric data for Galactic Cepheids cover a time interval that ranges from several years to more than 20 years. Thus, we adopted the new period estimate published by @groe13 [G13] for all the Cepheids in the LS sample. The G13 sample includes $\sim$130 Galactic Cepheids, and 50% of them are in common with the MP sample. The light curves from MP were phased by adopting the period given in the General Catalog of Variable Stars (GCVS). To check the consistency of the period listed in GCVS, we compared them with periods estimated by using either the Lomb-Scargle algorithm [@press89] or the PDM2 [@stell11]. The difference between the two sets of periods are about of $10^{-3}$ days, thus we adopted the GCVS for all the MP Cepheids not included in the G13 catalog.
The phase of the light-curve is usually defined by $$\phi^{V}_{obs} = \bmod \left( \frac{JD^{V}_{obs}- JD^V_{max}}{P} \right),$$ where $JD^{V}_{obs}$ is the epoch of the observation and $JD^V_{max}$ is the epoch of the maximum in the $V$ band. The epoch of maximum in the $V$ and $I$ bands for the LMC and SMC Cepheids is available from the OGLE III catalog, while for the Galactic Cepheids in LS and the 50% in MP we used the epoch of the maximum estimated by G13. To estimate the maximum brightness for the Cepheids for which the epoch of maximum was not available, we fitted the V–band light-curves with a seventh-order Fourier series.
![$V$–band (left panels) and $J$–band (right panels) normalized light-curves for Galactic bump Cepheids in the LS sample. The period increases from bottom to top. For all these light-curves the phase $\phi$=0 was fixed according to the maximum brightness in the $V$ band and was marked with the arrow and the red circle. However, the $J$–band maximum brightness, marked by the arrow, moves across the light-curves, and occurs at later phases than the optical maximum. The drift in phase arises because the secondary bump can be brighter than the true maximum, that corresponds to the phase of minimum radius along the pulsation cycle. The overplotted orange dots show the position of the mean magnitude along the rising branch, which we adopt as the new phase zero–point ($\phi$=0).[]{data-label="f2"}](f2.pdf){width="0.90\columnwidth"}
The identification of the epoch of maximum in our optical light-curves is straightforward, because of the very accurate time sampling of $V$- and $I$-band light-curves of calibrating Cepheids. However, light-curves of extragalactic Cepheids are covered with a limited number of phase points, typically fewer than two dozen [@sandage06; @bono10; @madore10]. The problem becomes even more severe for bump Cepheids, because the bump moves from the decreasing to the rising branch in a narrow period range. As mentioned above, the light-curve at the center of HP becomes either flat topped or double peaked with the real maximum and the bump located at close phases. Fig. \[f2\] shows the normalized optical ($V$; left) and NIR ($J$; right) light-curves for three Galactic Cepheids located across the center of the HP and with periods ranging from 7.02 (U Aql, bottom) to 9.84 days($\beta$ Dor, top) . If we apply the strict definition of epoch of maximum the phase shift between the optical and the NIR light-curve ranges from $\sim$0.1 ($\beta$ Dor) to $\sim$0.8 (U Aql). The red circles and the red arrows show that the identification of the luminosity maximum is hampered by photometric errors and by the fact that the bump can be brighter than the true maximum that corresponds to the phases of maximum contraction (minimum radius).
The scenario is further complicated because theory [@bono00; @marconi05] and observations [@moskalik92; @welch97; @beaulieu98; @moskalik00] indicate that the center of the HP is anticorrelated with the metal content. It moves from 9.5 days for Galactic Cepheids to 10.5 and to 11.0 days for LMC and SMC Cepheids. For a more quantitative analysis of the physical mechanism(s) driving the HP refer to @bono02 [@marconi05] and refer to @sos2008 [@sos2010] for a thorough analysis of the observed light-curves.
To overcome this problem, we decided to use a different zero–point to phase optical and NIR light-curves. Our phase zero–point is defined as the phase of the mean magnitude along the rising branch of the $V$–band light-curve
$$\phi_{obs}^V = \bmod \left( \frac{JD^V_{obs}- JD^V_{mean}}{P}\right),$$
We selected this phase point, because the mean magnitudes are more precise than the maximum brightness in Cepheids with modest phase coverage. The new phase zero–point allows us to highly improve the precision of the light-curve template in the period bins located across the bump (bin 4 to 6). A more detailed discussion of the impact of the new phase zero–point is given in Sect. 3.1. The top panel of Fig. \[f3\] shows the comparison between the phase lags of $V$ and $J$ light curves for a sample of SMC Cepheids based on the epoch of the maximum (Eq. 1, black open circles) and on the epoch of the mean (Eq. 2, orange dots). The same comparison for the $H$ and $K_{\rm{S}}$–bands are shown in the middle and bottom panels of the same figure. Data plotted in this figure clearly show the advantages in using the epoch of the mean magnitude as the phase zero–point.
![Top: Phase lag between the $V$– and the $J$–band light-curve of SMC Cepheids. The black open circles are estimated by adopting the epoch of maximum brightness as phase zero–point, while the orange dots by adopting the new phasing, i.e. the epoch of the mean–magnitude along the rising branch. The red lines show the linear fit of the orange data set. The $rms$ (dashed lines) are also overplotted. The values of the medians and of the $rms$ are labeled in the top of the panel. Middle: Same as the top, but for the $H$ band. Bottom: Same as the top, but for the $K_{\rm{S}}$ band.[]{data-label="f3"}](f3.pdf){width="0.90\columnwidth"}
i\) The phase lag anchored to the epoch of the mean can be approximated by linear relations on a broad period range. The intercept values of the phase lag are almost zero for all the bands and it slightly increases from $\sim$0 for the $J$and $H$ bands to 0.011 for the $K_{\rm{S}}$ band. The slope also systematically increases from 0.05 for the $J$ band, to 0.08 for the $H$ and 0.09 for the $K_{\rm{S}}$ bands. Moreover, the standard deviations based on the epoch of the mean magnitude are at least a factor of two smaller than the standard deviations of the phase lags based on the epoch of maximum light (0.02 vs 0.11 in the $J$ band). Thus, using the epoch of the maximum introduces a spurious shift in the epoch of the maximum of the NIR light-curves of bump Cepheids, and in turn a systematic error in the estimate of their mean NIR magnitudes. ii) The zero–point and the slopes of the linear relations to predict the phase lag between optical and NIR light-curves are similar for Magellanic and Galactic Cepheids.
iii\) The slope of the light-curve’s rising branch is steeper than that of the decreasing branch. This means that the error on the estimate of the mean magnitudes propagates into a smaller error in the phase determination. For this reason, the epoch of the mean along the rising branch provides a more accurate phase zero–point than the phase along the decreasing branch. However, the shape of the light curves changes once again for periods longer than the HP. The rising branch of Cepheids with periods longer than 15.5 days is shallower than the decreasing branch. This means that the latter provides a more solid phase zero–point. However, this problem only affects a minor fraction of our sample ($\approx$10%), therefore we adopted the phase of the mean magnitude along the rising branch.
The phase lags between $V$- and $J$-band FO Cepheid light-curves are similar to those of the FU Cepheid, with a median value of 0.05.
{width="80.00000%" height="0.4\textheight"}
To estimate the phase of the mean magnitude for the entire sample of calibrating Cepheids, we fit the V–band light-curves transformed into intensity with a seventh-order Fourier series. The mean in intensity was estimated as the constant term of the analytical fit and eventually transformed into magnitude. The comparison with the $V$–band mean magnitudes provided by @sos2010 by using the same definition indicates that the difference is at most of the order of a few hundredths of magnitude. The luminosity minimum and the luminosity maximum were estimated as the mean of the three closest observed point located across the luminosity maximum and the luminosity minimum of the analytical fit. We adopted this approach because the analytical fit in the period ranges in which the light-curves show secondary features (3 $\le$ P $\le$ 5 days; 7 $\le$ P $\le$ 15 days) slightly underestimate the luminosity amplitude. Moreover, the error associated with the amplitude estimated by adopting this approach does not depend on the analytical fit, but is given by the propagation of the photometric error of the observations.
After estimating the epoch of the mean V-band magnitude ($\phi_{mean}^V$) for FU and FO calibrating Cepheids, the epoch of the mean magnitude in the NIR bands is given by
$$JD_{mean}^{J,H,Ks}=JD_{mean}^{V}+ P \times \phi_{Lag}^{J,H,Ks},$$
where $\phi_{Lag}^{J,H,Ks}$ is a constant for the different bands. Its value for FU Cepheids with periods shorter than 20 days is 0.03 ($J$), 0.07 ($H$), and 0.13($K_{\rm{S}}$), while for longer periods it is: 0.06 ($J$), 0.14 ($H$), and 0.16($K_{\rm{S}}$). The phase lag in the $J$ band for FO Cepheids is 0.05. This equation can also be written in terms of $JD_{max}^{V}$ and $\phi_{mean}^{V}$: $$JD_{mean}^{J,H,Ks}=JD_{max}^{V}+P \times (\phi_{mean}^{V}+\phi_{Lag}^{J,H,Ks})$$. Thus, the pulsation phase can also be defined as
$$\phi_{obs}^{V,J,H,Ks} = \bmod \left( \frac{JD_{obs}^{V,J,H,Ks}-JD_{mean}^{V,J,H,Ks}}{P}\right),$$
where the symbols have their usual meanings. The name, the period, the $V$, $J$, $H$, $K_S$ mean magnitudes, the amplitude pulsations, and the epoch of the mean magnitude along the rising branch for the entire sample of calibrating FU Cepheids are listed in Table \[tab\_cat\_fu\]. The same parameters for FO calibrating Cepheids are listed in Table \[tab\_cat\_fo\].
Merged NIR light-curves of calibrating Cepheids
===============================================
To compute the light-curve template for FU Cepheids in the different period bins, we performed a fit with a seventh-order Fourier series of the $V$,$J$,$H$,$K_{\rm{S}}$ light curves of the calibrating Cepheids. The analytical fit provides several pulsation parameters: mean magnitude[^3], pulsation amplitude, and the phase of the mean along the rising branch. The fit with a seventh-order Fourier series is the most often used for Classical Cepheid light-curves [@laney92; @sos2008; @sos2010]. Analytical fits with lower order Fourier series cause an underestimate of FU Cepheid luminosity amplitudes. On the other hand, higher order analytical fits cause minimal changes (a few thousandths of magnitude) in the luminosity amplitudes and in the mean magnitudes. Following the same approach as adopted by S05, we normalized the light curves in such a way that the mean magnitude is equal to zero and the total luminosity amplitude is equal to one. In particular, for the $J$ band, the normalized light-curve is defined as
$$T_{J,l} = (J_{i,l} - <J>_l)/A_{J,l},$$
where $J_i$ are the individual measurements in the $J$ band, $<J>$ is the mean magnitude and $A_J$ is the luminosity amplitude of the variable in the $J$ band for the $l$-th light-curve.
This approach allowed us to compute the merged light-curve for each period bin ($T_J^{bin}$). The merged light-curves for the ten period bins in the $J$ (left), $H$ (middle), and $K_{\rm{S}}$ (right) are shown in Fig. \[f4\]. Data plotted in this figure clearly show that current NIR data set properly cover the entire pulsation cycle in the three different bands in the short and the long period range. Moreover, the intrinsic scatter at fixed pulsation phase is quite small and ranges from $\sim$0.03 to $\sim$0.05 over the entire data set. This evidence underlines the photometric precision and homogeneity of the adopted data sets together with the selection of calibrating Cepheids.
We adopted the same approach for the FO calibrating Cepheids. To estimate the main physical parameters, we fit the FO light-curves with a third-order Fourier series, because they have a sinusoidal shape in the adopted period range (see Fig. \[f5\]).
![Merged light-curve for FO Cepheids. The F3 (red line) and G2 (green line) templates are also shown.[]{data-label="f5"}](f5.pdf){width="0.90\columnwidth" height="0.8\columnwidth"}
Analytical fits of the merged NIR light-curves of calibrating Cepheids
======================================================================
The precision of the mean NIR magnitudes based on light-curve templates depends on the accuracy of the analytical fits in reproducing the shape of the individual light-curves in the different period bins. The fit of the merged light-curves was performed by using seventh-order Fourier series
$${\rm F7(\phi)}= A_0 + \sum_{i=1}^7 A_i \cos (2 \pi i \phi + \Phi_i).$$
Fitting the light curves with Fourier series is a very popular approach for both regular and irregular variables. They have many advantages, but also several limits. In particular, the F7 templates (red lines) for the bin 1 in the $J$,$H$, and $K_{\rm{S}}$ bands –see top panels of Fig. 4– show several spurious ripples along the decreasing branch of the light-curve. Moreover, the F7 templates for the period bins located across the bump display a wiggle close to the phases of maximum brightness (bin 5) and a stiff trend close to the phases of minimum brightness (bin 7).
To provide an independent approach for the analytical modeling of the light curves, we adopted a fit with multiple periodic Gaussian functions
$${\rm G3(\phi)}= \sum_{i=1}^3 G_i \exp \left[ \frac{- \sin \pi (\phi- \Gamma_i)}{\tau_i}\right]^2.$$
We called these analytical functions PEGASUS, because they provide PEriodic GAuSsian Uniform and Smooth fits. The key advantage of these functions is that they follow the features of the light-curves, but the wings remain stiff. The fits of the calibrating light-curves with the linear combination of three PEGASUS functions are plotted as green lines in Fig. \[f4\] and are very accurate over the entire period range. The fits based on PEGASUS show two indisputable advantages over the Fourier series: a) the PEGASUS fit (G3) only requires nine parameters, while the Fourier fit (F7) needs 15 parameters, and b) the G3 fit does not show the ripples in either the short period bins (bins 1 and 2; see Fig. \[f4\]) or across the Hertzsprung progression (bins 4, 6, and 7; see Fig. \[f6\])
![From left to right: residuals of the $J$–band merged light-curves (silver dots) obtained with the new templates: F7 (red line; top) and G3 (green line; bottom) for the period bin 4 (7–9.5 days), bin 6 (10.5–13.5 days) and bin 7 (13.5–15.5 days). The residuals attain vanishing mean values for the F7 and the G3 templates.[]{data-label="f6"}](f6.pdf){width="0.90\columnwidth"}
However, the standard deviations of the individual fits and the errors on the coefficients in G3 and in F7 fits attain similar values. The standard deviations are on average on the order of a few hundredths of magnitude, wile the errors of the coefficients are smaller than one thousandth of magnitude. This is the main reason why we decided to provide the analytical fits for both of them. The coefficients $A_i$ and $\Phi_i$ for the F7 fits and the $G_i$, $\Gamma_i$ and $\tau_i$ coefficients for the G3 templates are given in Table \[tab\_c\].
For the FO light-curve template, we adopted a third-order Fourier (F3) series and a second-order PEGASUS function (G2) to fit the merged light-curves. Fig. \[f5\] shows the $J$–band template for FO Cepheids together with the F3 and the G2 best fits. The $rms$ for the merged light-curve is $\sim$0.07 (F3) and $\sim$0.08 (G2) mag. The coefficients $A_i$ and $\Phi_i$ for the F3 templates and the exponents $G_i$, $\Gamma_i$, $\tau_i$ for the G2 templates are listed in Table \[tab\_c\]
The *IDL* procedure for estimating the mean NIR magnitudes by using the templates is available on the webpage: <http://www.laurainno.com/#!idl/c5wp>.
Validation of the new phase zero–point: bump Cepheids
-----------------------------------------------------
![Comparison between the merged light-curve T$^{bin~4}_J$ with phasing anchored on the phase of maximum light (Eq. 1; left) and with the new phasing anchored on the phase of mean magnitude along the rising branch (Eq. 2; right) for bin 4 Cepheids.[]{data-label="f7"}](f7.pdf){width="0.80\columnwidth"}
To improve the mean NIR magnitude of bump Cepheids, we adopted (see Sect. 2.1) a new phase zero–point anchored on the phase of the mean magnitude along the rising branch. Fig. \[f7\] shows the comparison between the merged $J$–band light curve for the period bin 4 computed by using as phase zero–point both the phase of maximum light (Eq. 1, left panel) and the phase of mean magnitude along the rising branch (Eq. 5, right panel). A glance at the data plotted in this figure show that the $rms$ significantly decreases in the merged light-curve that was computed using the new phase zero–point, and the $rms$ decreases by roughly a factor of two (0.06 vs 0.13).
To further constrain the precision of current templates, we also performed a comparison with the light-curve template provided by S05[^4]. The left panel of Fig. \[f8\] shows that the $J$–band S05 template predicts a shape of the normalized light-curve (left panel, blue line) for Cepheids with period approaching the center of HP (U Aql, P=7.024) that differs from the observed shape. The difference is quite clear not only close to the phase of the maximum ($\phi \sim$0.05), but also close to the phase of the bump ($\phi \sim$0.15), and in particular, along the decreasing branch. The middle and right panels show the comparison between our templates (F7, middle; G3, right) and the observed data.
![From left to right: comparison between the normalized $J$–band light-curve (black dots) of U Aql and the S05 (blue line; left), F7 (red line; center) and the G3 (green line; right) templates. The typical error associated with observations ($\pm$0.01 mag) and rescaled in normalized units is about $\sim$0.001 and is shown in the top left corner of the plot. The residuals between the data and the templates are plotted in the bottom panels. The dashed lines indicates the $rms$ of the residuals; it decreases from 0.10 (S05) to 0.04 (F7, G3).[]{data-label="f8"}](f8.pdf){width="0.80\columnwidth"}
The residuals for the S05 template plotted in the bottom left panel of the same figure display a phase delay between the data and the light-curve template along the rising and the decreasing branch. On the other hand, the residuals plotted in the middle and right panels show that the F7 and G3 templates provide a good approximation of the observed light-curve. The residuals have an $rms$ (dashed lines) of 0.04 mag, that is a factor of two smaller than the $rms$ of the S05 template (0.10).
NIR–to-optical amplitude ratios
===============================
{width="80.00000%" height="0.4\textheight"}
The light-curve template allows us to estimate the NIR mean magnitudes from single-epoch observations if the amplitude in that specific band is already known. Indeed, Eq. 5 gives
$$<J>_l= J_{i,l}-A_{J,l} \times T_J$$
and similar equations for the other NIR bands (see also Eq. 4 in S05). To estimate the NIR mean magnitudes, the luminosity amplitudes can be estimated by using the luminosity amplitudes in the optical bands. We derived new amplitude ratios between optical and NIR bands by using our calibrating Cepheids. The results are shown in Fig. \[f9\] from top to bottom: $A_J$/$A_V$, $A_H$/$A_V$, and $A_{Ks}$/$A_V$. We estimated the mean value (black solid line) over two different period ranges: P $\le$ 20 days and P$>$20 days for the MW+LMC (left panels) and the SMC calibrating Cepheids (right panels).
The data plotted in this figure disclose several interesting features that need to be addressed in more detail, because these ratios are prone to systematic uncertainties. Classical Cepheids are young objects and a significant fraction of them are still members of binary systems [@szabados12b]. Their companions are mainly young low–mass stars, which meant that they mainly affect the $V$–band amplitude. Moreover, recent accurate optical and NIR interferometric [@kervella06], mid-infrared [@marengo10; @barmby11], and radio [@matthews12] measurements indicate the presence of a circumstellar envelope around several Galactic Cepheids. This evidence implies that the NIR amplitudes might also be affected by systematic uncertainties and it accounts for a significant fraction of the dispersion around the mean values because the photometric errors are significantly smaller (see the typical error bars in the top left corners).
Moreover, current theoretical [@bono00] and preliminary empirical evidence [@paczy00; @szabados12a] indicates that the luminosity amplitudes depend on the metal content. The $V$–band amplitudes of SMC Cepheids in the short–period range (P$\le$ 11 days) are, at fixed period, larger than those of Galactic and LMC Cepheids. The difference is caused by the HP dependence on the metallicity (see Sect. 2.1). The trend is opposite in the long–period range. The difference in the optical amplitude is also clear in the Fourier amplitude of Magellanic and Galactic Cepheids [see Fig. 5 and Fig. 2 in @sos2008; @sos2010; @matsunaga13 Fig. 2].
This evidence indicates that solid empirical constraints on the dependence of the luminosity amplitudes on metallicity requires accurate information on individual metal abundances. However, @genovali13 did not find, within the errors, any significant dependence on iron abundance, by adopting a homogeneous sample of 350 Galactic and 77 Magellanic Cepheids with precise and homogeneous iron abundances. However, this finding is far from being definitive, because the number of SMC Cepheids for which accurate iron abundances are available is quite limited (19).
The data plotted in the top panel of Fig. \[f9\] indicate that the NIR–to–optical amplitude ratios of SMC Cepheids (right panel) are smaller over the entire period range than the amplitude ratios of MW plus LMC Cepheids (left panel). This is the reason why we decided to adopt independent values for the NIR–to–optical amplitude ratios of SMC and MW plus LMC Cepheids. The high dispersion in the amplitude ratios is mainly due to the $V$–band amplitude distribution in the Bailey diagram (amplitude vs period), while the NIR amplitudes show a similar distribution, but tighter. Current theoretical and empirical evidence indicates that the amplitude distribution in the Bailey diagram for Galactic Cepheids has the typical V shape [@vangenderen74; @bono00; @szabados12a; @genovali14], with the largest luminosity amplitudes attained in the short– ($\log P\le$ 1.0 days) and in the long–period ($\log P \ge$ 1.2 days) ranges, while the minimum, at fixed chemical composition, is reached at the center of the HP. This peculiar behavior does not allow a straightforward prediction of the NIR amplitude on the basis of the period. On the other hand, the NIR–to–optical amplitude ratios are almost constant for a broad range of periods, as shown in Fig. \[f9\]. In particular, we find that the mean $A(J)/A(V)$, and $A(H)/A(V)$ amplitude ratios are smaller for periods shorter than P$\le$20 days and larger for periods longer than P$>$20 days. For the SMC mean $A(K)/A(V)$ amplitude ratios we chose a different cut in period: P=15.5 days. The estimated NIR–to–V amplitude ratios for MW+LMC and SMC calibrating Cepheids are listed in Table \[tab\_amp\].
We also note that the typical dispersion for the NIR–to–optical amplitude ratios of the MP sample is almost twice as high as that of the LS sample (MP: $\sigma$=0.05; LS: $\sigma$= 0.03; $J$-band). The main reason for this is the photometric quality of $V$-band light-curves in the MP sample. As already mentioned in Sect. 2, the optical photometry for these Cepheids was collected from the literature, with the source data spanning a long time and coming from different instruments. Indeed, the typical $rms$ for the MP calibrating $V$-band light-curve is $\sim$0.05 mag, ten times higher than the typical $rms$ for the LS $V$-band light-curves ($\sim$0.005 mag, see also Sect. 2). We also performed a test by adopting different mean amplitude ratios for each period bin and we did not find any significant improvement in the final results.
Similar considerations apply to the use of linear regression to fit the amplitude ratios as a function of the period. A linear relation on the entire period range provides a good approximation for the short-period range, where the ratios are almost constant, but it underestimates the values in the long-period range. On the other hand, by adopting two different relations in the two period ranges, the number of parameters will double without significantly improving the template accuracy compared with the two horizontal lines.
The pulsational amplitudes and Fourier parameters of FO Cepheids also show a sudden jump for periods close to P=3 days [@kienzle99]. This behavior is associated with the possible presence of a short-period bump along the light-curves of FO Cepheids with periods between 2 and 3.8 days [@bono00]. Thus, we adopted two different $A(J)/A(V)$ mean values for the two different regimes of the amplitude before and after the appearance of this short-period bump. Fig. \[f10\] shows the phase lags (top panel) and the amplitude ratios (bottom panel) computed for the FO calibrating Cepheids. The mean values (solid lines) are $A_J$/$A_V$= 0.40 (P$\le$2.8 days) and $A_J$/$A_V$= 0.30 (P$>$2.8 days).
![Top: Phase lags in $J$ band for FO Cepheids based on the epoch of the maximum (black circles) and on the epoch of the mean magnitude along the rising branch (orange dots). The median (solid lines) and $rms$ (dashed lines) of the ten calibrating Cepheids are overplotted. The black labels refer to the epoch of maximum, orange labels to the epoch of the mean magnitude. Bottom: the amplitude ratio $A(J)/A(V)$ for FO Cepheids. The mean values for the two adopted bins in period (P$\le$2.8, and P$>$2.8 days) are also labeled (black solid lines).[]{data-label="f10"}](f10.pdf){width="0.90\columnwidth" height="0.8\columnwidth"}
Validation of the light-curve templates
=======================================
To further evaluate the accuracy of current templates, we performed a new test by using the complete light-curves of the calibrating Cepheids. For each period bin, we have several calibrating light-curves (see Fig. \[f1\]) for which all the parameters –mean magnitudes, NIR luminosity amplitudes, period, phase zero–points– have already been estimated. Therefore, we randomly selected a phase point from the calibrating light curve to simulate a single-epoch observation and applied the new templates to estimate the mean magnitude, which we then compared with the true one. We define $\delta J$ as the difference in $J$ band between the true mean magnitude –estimated as the mean along the light-curve– and the mean magnitude computed by using the new $J$–band templates. A similar approach was also adopted for the $H$ and $K_{\rm{S}}$–bands. Fig. \[f11\] shows the $\delta J$ for two period bins: bin 3 (top) and bin 4 (bottom) by adopting the F7 (red dots, left panels), the G3 (green dots, middle panels) and the S05 (blue dots, right panels) light-curve templates. The $\delta J$ based on both F7 and G3 templates give a vanishing mean ($\le 10^{-3}$ mag) and a small standard deviation $\sim 0.03$ mag. The S05 templates also give a mean close to zero ($\sim 10^{-3}$ mag) and a slightly larger standard deviation ($\sim 0.04$ mag). The data plotted in Fig. \[f11\] show that the residuals of the S05 template are not symmetric, therefore we estimated the interquartile range and found that the difference becomes about 40% ($\sim$0.06 vs $\sim$0.04 mag). We also divided the data into ten phase bins and estimated the mean and standard deviation for each bin. The values are overplotted in Fig. \[f11\] (red dots, F7; green dots, G3; blue dots, S05). The horizontal error bars display the range in phase covered by individual bins, while the vertical error bars display their standard deviations.
![Top: $\delta J$ for period bin 3 (5–7 days) by adopting the F7 (red dots, left panel), the G3 (green dots, central panel), and the S05 templates (blue dots, right panel). The gray dots on the background are the difference between the mean magnitude estimated by applying the templates and the true mean magnitude estimate by the individual fits. By binning the phase in ten different bins, we estimated the mean of the residuals (dots) and the standard deviation (error bar). The solid black lines show the mean values, which are $<$10$^{-3}$ mag for the three templates. Bottom: Same as top, but for period bin 4 (7–9.5 days).[]{data-label="f11"}](f11.pdf){width="0.99\columnwidth"}
The data plotted in this figure indicate that the residuals of the F7 and G3 are independent of the phase, while the residuals of the S05 template show a clear phase dependence for $\phi\sim$0.5 and $\phi\sim$0.8. In particular, the mean $J$ magnitudes based on the S05 template are 2$\sigma$ fainter close to $\phi\sim$0.5 and brighter close to $\phi\sim$0.8. Most of this discrepancy is due to bump Cepheids for which the phase zero–point anchored on the epoch of maximum brightness introduces systematic phase shifts in using the template. Similar trends were also found when estimating the $\delta H$ and $\delta K_{\rm{S}}$ by applying the F7, G3, and S05 templates. We also performed the same test for FO Cepheids, and the results are plotted in Fig. \[f12\]. Once again, the residuals for the F7 and the G3 templates attain vanishing values ($\le 10^{-3}$) and the standard deviations are smaller than 0.04 mag. Note that above standard deviations account for the entire error budget, because they include the photometric error (measurements, absolute calibration) and the standard deviations of the analytical fits.
![Same as Fig. 11, but for FO calibrating Cepheids. The $\delta J$ are based on the F7 (top) and the G2 (bottom) templates. The red (F3) and green (G2) dots indicate the mean for each of the 20 bins in phase and the standard deviation (error bars). The mean of the residuals is $\le 0.01$, with a mean $rms$ of 0.04 mag.[]{data-label="f12"}](f12.pdf){width="0.90\columnwidth" height="0.8\columnwidth"}
Test based on single-epoch measurements
---------------------------------------
The validation of the light-curve templates performed in the last section has a limited use, bceause it relies on Cepheids with a good coverage of the pulsation cycle, and in turn on accurate luminosity amplitudes. Therefore, we performed an independent validation by using the amplitude ratios discussed in Sect. 5 and by randomly extracting phase points from the $J$, $H$, and $K_{\rm{S}}$–band light-curves of the calibrating SMC Cepheids. This is an acid test, because we are mimicking the typical use of the light-curve template.
The difference in mean magnitude between the true mean magnitudes and the mean magnitudes estimated using the new NIR templates are plotted in Fig. \[f13\] as red triangles (F7, left panels), green triangles (G3, middle panels), and blue triangles (S05, right panels). The red, green and light blue shadowed areas indicate the standard deviation $\pm \sigma$, for the $\delta J$ (top), the $\delta H$ (middle) and the $\delta K_{\rm{S}}$ (bottom) estimated with the three different templates. The mean values for the three bands ($\delta J$, $\delta H$ and $\delta K_{\rm{S}}$) are lower than a few thousandths in all the cases. However, the $\sigma$ for the F7 and the G3 $J$ and $H$–band templates are at least 40% lower than for the S05 template (0.03 vs 0.05 mag). The difference for the $K_{\rm{S}}$ band is lower and of about $\sim$20% (0.04 vs 0.05 mag). Moreover, the residuals of the S05 template show a phase dependence that is not present in the residuals of the F7 and the G3 templates[^5].
The evidence that the new NIR templates do not significantly reduce the scatter in the $K_{\rm{S}}$ band is a consequence of the fact that the pulsation amplitude in this band is smaller than in the $J$ and in the $H$ bands. Moreover, the photometric errors on individual measurements become larger.
We performed the same test for the Galactic calibrating Cepheids, and the results are given in Fig. \[f14\]. The difference we found for Galactic calibrating Cepheids from using the new NIR templates is between two ($K_{\rm{S}}$) to three ($J,H$) times smaller than for the S05 templates. The mean in the three different bands approaches zero ($\sim10^{-3}$ mag), but the residuals of the S05 template show a clear phase dependence. Moreover, data plotted in the right panels show a systematic overestimate of the mean magnitude for phases close to the rising branch ($\phi\sim$1). The main reason for this discrepancy is once again the adopted phase zero–point. The use of the maximum brightness to anchor the phase causes loose constraints along the rising branch, i.e. close to phases in which the luminosity rapidly increases. The F7 and the G3 templates do not show evidence of similar systematic effects. However, the former exhibits a mild phase dependence in the $K_{\rm{S}}$ bands –and probably in the $H$ bands– close to $\phi$=0.8.
The difference between the residuals of MW+LMC and SMC calibrating Cepheids is a consequence of the fact the former sample is characterized by a better photometric precision over the entire period range. The mild phase dependence is mainly due to the smaller photometric errors, and in turn, to the smaller standard deviations (see labeled values). The key points to explain the above trends are a) the definition of the phase zero–point: the use of the mean magnitude along the rising branch (Sect.2.1) instead of the decreasing branch reduces the precision of the template for periods longer than $13.5$ days; b) the NIR–to–optical amplitude ratios adopted for the $H$ and $K_{\rm{S}}$: the $V$–band amplitude of the Galactic calibrating Cepheids shows, at fixed period, a high dispersion, that propagates into the NIR–to–optical amplitude ratios.
The same test was also applied to the FO calibrating Cepheids. The residuals plotted in Fig. \[f15\] clearly show that the standard deviation of the $J$–band template decreases by $\sim$50% when compared with single-epoch measurements extracted along the light-curves. Note that the typical pulsation amplitude in the $J$ band for FO pulsators is $\sim$0.15 mag. This means that the use of single-epoch measurement as a mean magnitude introduces an error of about $\sim$0.07 mag. Thus, the new FO NIR templates allow us to reduce the error budget for FO Cepheids by almost a factor of two. Moreover, the new templates do not show evidence of a phase dependence.
To fully exploit the impact of the new NIR templates on FU Cepheids, we also performed a test with the mean Wesenheit magnitudes. The NIR Wesenheit magnitudes are closely related to apparent magnitudes, but they are minimally affected by uncertainties on reddening and are defined as
$$W(JK_{\rm{S}})=K_{\rm{S}}-0.69 \times (J- K_{\rm{S}}),$$ $$W(HK_{\rm{S}})=K_{\rm{S}}-1.92 \times (H- K_{\rm{S}}),$$ $$W(JH)=H-1.63 \times (J- H),$$ where the coefficients of the color terms are based on the reddening law provided by @cardelli89, and for the SMC selective absorption coefficient we adopted $R_V$=3.23. This is an acid test concerning the NIR templates, since the color coefficients of the Wesenheit relations attain values higher than one for bands with limited difference in central wavelength. This means that uncertainties affecting the mean colors are magnified in using Wesenheit relations.
To simulate the effect of non-simultaneous NIR observations, we randomly extracted for the entire set of SMC calibrating light-curves three different ($J$,$H$,$K_{\rm{S}}$) measurements. The NIR templates were applied to each of them and we obtained three independent estimates of the mean NIR magnitudes. Then we computed the three mean Wesenheit magnitudes –$W(JH)$,$W(HK_{\rm{S}})$, $W(JK_{\rm{S}})$– by using these relations and estimated the difference in magnitude with the true mean Wesenheit magnitudes. To properly sample the luminosity variations along the entire pulsation cycle, the procedure was repeated ten times per light-curve. This test was performed with the new (F7, G3) and the S05 templates. The residuals are plotted in Fig. \[f16\], $W(JK_{\rm{S}})$ (top), $W(HK_{\rm{S}})$ (middle) and $W(JH)$ (bottom). A Gaussian fit to the histograms performed to evaluate the mean and the standard deviation is also overplotted. We found that the means once again vanished. The $\sigma$ of the mean Wesenheit magnitudes based on the F7 \[a) panels\] and on the G3 \[b) panels\] templates are between 15–30% lower than the $\sigma$ of the residuals based on the S05 \[c) panels\] template (see labeled values).
Finally, we also compared the difference between the mean Wesenheit magnitudes based on the new NIR templates with single-epoch measurements randomly extracted from the light-curve of the SMC calibrating Cepheids. The gray shaded areas plotted in the d) panels of Fig. \[f16\] show that the standard deviations of the new NIR templates are a factor of two smaller than those of the single-epoch measurements.
![Random-phase extraction test for all the period bins in the different NIR bands (from top to bottom: $J$, $H$ and $K_{\rm{S}}$) in the SMC sample. The red triangles show the difference between the mean magnitude predicted by applying the F7 template to the single-epoch NIR observation (left panel). In the middle panel the green triangles were evaluated by applying the G3 template, while the blue triangles in the left panel are evaluated by applying the S05 template. The shadowed areas shows the 2$\sigma$ of the results (F7: light red; G3: light green; S05: light blue), while the black lines indicate the zero. The residuals of F7 and G3 template show no dependence on the phase, and the dispersion is from $\sim$20% ($K_{\rm{S}}$) to 40%($J$) lower than the residuals of the S05 template.[]{data-label="f13"}](f13.pdf){width="0.99\columnwidth" height="0.8\columnwidth"}
Error budget of the analytical fits
-----------------------------------
These results clearly show that the application of NIR light curve templates increases the accuracy on the mean magnitude compared with single-epoch measurements. However, the templates are affected by several uncertainties that contribute to the total error budget. The test discussed in Sect. 6.1 and shown in Fig. \[f13\] was also applied to constrain the impact of the individual uncertainties on the total dispersion of the $\delta J$, $\delta H$, and $\delta K_{\rm{S}}$ residuals.
[*i) Photometric error*]{} – The photometric error is the main source of error, and it affects the precision of the template and its application. However, only the former source should be taken into account when estimating the precision of current templates. To artificially remove the photometric error on the measured NIR magnitudes, we extracted the individual measurements from the Fourier fits of the light-curves. The result of this numerical experiment shows that 60% of the total dispersion is due to the photometric error on the observed magnitudes. This accounts for 0.02 mag in the total error budget.
[*ii) Use of the template*]{} – The use of a template plus a single-epoch measurement to estimate the mean magnitude accounts for 12% of the total dispersion in Fig. \[f13\].
[*iii) Merging of the light-curves in period bins*]{} – Our approach assumes that all the light-curves inside the same period bin are identical within the errors. This assumption is verified inside the confidence level given by the $rms$ of the merged light-curves, typically $\sim$0.03 mag. However, small differences in shape may occur between the true light-curve of the Cepheid and the given template. This is a simple consequence of the merging process of the Cepheid light-curves in a limited number of period bins. We tested the impact of this approach by using synthetic light-curve based on analytical fits (F7, G3). We found that 15% of the total dispersion is due to the process of merging the light-curves in a modest number of period bins.
[*iv) $V$-NIR amplitude ratio*]{} – The prediction of the pulsation amplitudes in the $J$, $H$, and $ K_{\rm{S}}$ bands based on the optical amplitude introduces an uncertainty on the mean magnitude provided by the templates. However, we can quantify this effect by adopting the true amplitudes measured for the calibrating Cepheids. The error associated with the use of the $V$-NIR amplitude ratios given in Sect. 5 accounts for 10% of the total budget.
[*v) $V$-NIR phase lags*]{} – We estimated the epoch of the mean magnitude along the rising branch by adopting $V$-NIR phase lags (Sect. 2.1). The comparison between the $\delta J$, $\delta H$, and $\delta K_{\rm{S}}$ residuals estimated with the adopted and the measured epoch of the mean magnitudes indicates that this assumption accounts for $\sim$3% of the error budget. The error associated with the NIR mean magnitudes estimated by applying the new NIR templates is $\sim$0.015 mag for the $J$ band, 0.017 mag for the $H$ band and 0.019 mag for the $K_{\rm{S}}$ band. These errors have to be added in quadrature to the photometric error on the single-epoch measurements to which the template is applied. The use of two or more measurements for the same Cepheid and the weighted average of the independent mean magnitudes implies a better precision on the final mean magnitude.
![Same as Fig. \[f13\], but for Galactic Cepheids (LS, MP). The red triangles show the difference between the mean magnitude predicted by applying the F7 template (left panel), the G3 template (central panel, green triangles), and the S05 template (right panel, blue triangles). The shadowed areas shows the 2$\sigma$ of the results (F7: light red; G3: light green; S05: light blue), while the black lines indicate the zero for each data set.[]{data-label="f14"}](f14.pdf){width="0.99\columnwidth" height="0.8\columnwidth"}
![Random-phase extraction for FO calibrating Cepheids in the $J$–band. The red triangles show the differences between the mean magnitude based on the F3 template and on single–epoch NIR observations (top panel). The middle panel shows the residuals (green triangles) based on the G2 template, while the bottom panel shows the residuals (black triangles) of the single–epoch measurements. The shadowed areas shows the 2$\sigma$ of the results (F7: light red; G2: light green; single-epoch: light gray), while the black lines display the mean. []{data-label="f15"}](f15.pdf){width="0.90\columnwidth" height="0.8\columnwidth"}
Summary and final remarks
=========================
We developed new NIR $J$, $H$, and $K_{\rm{S}}$ light-curve templates for FU and FO Cepheids. The new templates compared with those already available in the literature have several advantages:
[*i)*]{} [**Period binning**]{} – We divided the entire period range (1–100 days) into ten different period bins. The binning was performed to a) reduce the $rms$ of the merged light-curves, b) properly trace the change in the shape of the light-curve across the HP, and to c) minimize the discrepancy in the amplitude ratio and in the phase difference ($R_{l,m}$, $\phi_{l,m}$; Fourier parameters) between templates and individual light curves. The adopted binning in period allowed us to limit this difference to less than 15% of the total error associated with the estimate of the NIR mean magnitudes.
[*ii)*]{} [**Phase zero–point**]{} – The phase zero–point of the mean magnitude was fixed along the rising branch of the light-curve. The main advantage of this new definition is that it allow us to estimate the phase lag between optical and NIR light-curves. Moreover, the identification of the new phase zero–point is straightforward even for bump cepheids and thus overcomes possible systematic errors introduced by the secondary bumps along the light curves.
[*iii)*]{} [**NIR–to–optical amplitude ratios**]{} – To apply the new templates, we need to know the luminosity amplitude in an optical band ($V$, $B$) in advance together with the ratio between optical and NIR bands. The optical amplitudes come from the OGLE data set. We provided a new estimate of the $V$–NIR amplitude ratios for the calibrating Cepheids and found that a) the ratios for SMC Cepheids are, at fixed period, systematically lower than the ratios of Galactic and LMC Cepheids; b) the difference between SMC and MW$+$LMC decreases for periods longer than the center of the HP. The optical amplitudes of Galactic Cepheids are smaller than the amplitudes of SMC Cepheids for periods shorter than the center of the HP. Therefore, we adopted two different ratios for the short– and the long–period regime.
[*iv)*]{} [**Analytical Functions**]{} – Together with the popular seventh-order Fourier series fitting, we also provided a template based on multi-Gaussian periodic functions. The main advantage in using this new template is that it provides a solid fit of the light-curves by using fewer parameters than the Fourier fit. Moreover, it is less sensitive to spurious features that can be introduced in the light-curves by photometric errors and to secondary features (bumps, dips) that can appear along the light-curves.
[*v)*]{} [**FO pulsators**]{} – We provide for the first time the $J$-band template for FO Cepheids, following the same approach as we adopted for FU Cepheids. The new template reduces the uncertainty on the mean $J$–band magnitude of FO Cepheids by a factor of two.
The application of the new NIR templates when compared with single-epoch NIR data provides mean magnitudes that are 80% more accurate, and their typical error is smaller than $0.02$ mag. Cepheids mean magnitudes can be used to estimate their distances by adopting Period–Luminosity (PL) and Period–Wesenheit (PW) relations. The error associated with these distances includes both the error on the observed mean magnitude and the uncertainty on the absolute magnitude estimated by the adopted relation. This uncertainty is formally derived by the dispersion of the relation, which is produced by three different error sources: the photometric errors associated with the measured mean magnitudes from which the adopted PL or the PW relation is derived, the line-of-sight depth of the galaxy, and the intrinsic scatter. This last term is a consequence of the fact that PL and PW relations do not account for all the physical parameters that contribute to the stellar luminosity, such as temperature, metallicity, helium-content. Recent theoretical predictions [@bono00; @marconi05; @fiorentino07] and empirical results [@bono10; @inno13] indicate that the intrinsic dispersion decreases for NIR PW relations. In particular, @fiorentino07 predicted a dispersion lower than 0.05 mag for PW$(J,K)$ relation. This uncertainty of $\sim$ 3% on individual distances is thus the precision that intrinsically limits the method we adopted. The main advantage of the new templates is that they reach the precision limit, even with single-epoch NIR observations. Indeed, for single-epoch measurements with 1% accuracy or better, the error on the mean magnitude is lower than 2%. Computing the sum in quadrature of all these error sources, the dominant term is then the intrinsic scatter of the PW relation.
{width="80.00000%" height="0.4\textheight"}
This means that Cepheid distances can be determined with the highest possible accuracy by using the new templates and one single-epoch NIR observation.
Compared with the S05 templates, F7 and G3 templates have the advantage to be minimally affected by problems in phase dependences, and they provide new NIR mean magnitudes that are more accurate by 30%($K_{\rm{S}}$) to 50% ($J$). This means that if single-epoch measurements are available with photometric precision better than 0.03 mag, the new templates already reduce the total uncertainty on distances by 20% with respect to the S05 templates. For instance, by applying the new templates to the NIR single-epoch data presented in @inno13 for the SMC Cepheids, the total dispersion of the optical–NIR PW relations decreases by up to 30% (i.e. 0.15 mag vs 0.26 mag, PW$(VJ)$) when compared to single-epoch data, and up to 5% (i.e. 0.15 mag vs 0.16 mag PW$(VJ)$) when compared with the S05 template. Moreover,the total dispersion for PW$(VJ)$ is 0.15 mag, indicating that the scatter due to spatial effects is still significantly larger then the intrinsic dispersion ($\sim$3 times larger). This means that Cepheid relative distances can be safely used to derive the three-dimensional structure of the SMC, with an accuracy limited by the total error estimated above that is $\le$5%, which corresponds to the physical limit of the method itself.
If we instead consider all of the Cepheids as a statistical ensemble representing the stellar distribution in the galaxy, the mean distances to the SMC as derived from different PW relations can be determined with a precision of up to 0.1% (0.002 magnitudes, PW$(VJ)$), with the precision scaling as the square root of the number of stars in the ensemble itself ($\gtrsim$2,200 Cepheids).
A more detailed discussion on the application of the new templates to derive new MC Cepheids relative distances will be given in a forthcoming paper (Inno et al., in preparation).
Our findings rely on a panoply of Galactic and MC Cepheid light-curves. The new templates and reddening-free optical–NIR PW relations will provide accurate absolute and relative distances. The latter appear very promising, because the intrinsic error is on the order of 1–2 percent. This gives the opportunity to derive 3D structure of nearby stellar systems by using single-epoch NIR observations.
In spite of the substantial improvement in the intrinsic accuracy of the NIR light-curve templates, the observational scenario is far from being complete. Current NIR light-curves did not allow us to derive accurate $H$- and $K_{\rm{S}}$-band templates for FO Cepheids, because of the limited photometric accuracy in the short-period regime (faintest Cepheids). Moreover, we found evidence that the NIR–to–optical amplitude ratios of SMC Cepheids are lower when compared with MW+LMC Cepheids. Current data did not allow us to constrain whether a similar difference is present in addition between MW and LMC Cepheids, because of the limited sample of LMC Cepheids, The new NIR time series data that are being collected by IRSF for MC Cepheids appear a very good viaticum to address these open problems.
This work was partially supported by PRIN–INAF 2011 “Tracing the formation and evolution of the Galactic halo with VST" (P.I.: M. Marconi) and by PRIN–MIUR (2010LY5N2T) “Chemical and dynamical evolution of the Milky Way and Local Group galaxies" (P.I.: F. Matteucci). One of us (G.B.) thanks The Carnegie Observatories visitor program for support as science visitor. N.M. acknowledges the support by Grants-in-Aid for Scientific Research (Nos. 23684005 and 26287028) from the Japan Society for the Promotion of Science (JSPS). Support from the Polish National Science Center grant MAESTRO 2012/06/A/ST9/00269 is also acknowledged. WG gratefully acknowledges support for this work from the BASAL Centro de Astrofísica y Tecnologías Afines (CATA) PFB-06/2007, and from the Chilean Ministry of Economy, Development and Tourism’s Millenium Science Iniciative through grant IC120009 awarded to the Millenium Institute of Astrophysics (MAS).\
We also acknowledge G. Fiorentino for many useful discussions concerning the theoretical predictions on NIR Period Wesenheit relations. It is also a pleasure to thank an anonymous referee for his/her supportive attitude and insightful suggestions that helped us to improve the readability of the paper.
Akhter, S., Da Costa, G. S., Keller, S. C., & Schmidt, B. P. 2012, , 756, 23 Barmby, P., Marengo, M., Evans, N. R., et al. 2011, , 141, 42 Beaulieu, J. P. 1998, , 69, 21 Berdnikov, L. N., & Turner, D. G. 2004, Astronomical and Astrophysical Transactions, 23, 253 Bono, G., Caputo, F., Marconi, M., & Musella, I. 2010, , 715, 277 Bono, G., Castellani, V., & Marconi, M. 2002, , 565, L83 Bono, G., Marconi, M., & Stellingwerf, R. F. 2000, , 360, 245 Bono, G., Castellani, V., & Marconi, M. 2000, , 529, 293 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, , 345, 245 Carpenter, J. M. 2001, , 121, 2851 Drake, A. J., Djorgovski, S. G., Mahabal, A., et al. 2009, , 696, 870 Fiorentino, G., Marconi, M., Musella, I., & Caputo, F. 2007, , 476, 863 Freedman, W. L., & Madore, B. F. 2010a, , 48, 673 Freedman, W., Scowcroft, V., Madore, B., et al. 2012, Spitzer Proposal, 90002 Genovali, K., Lemasle, B., Bono, G., et al. 2013, , 554, A132 Genovali, K., Lemasle, B., Bono, G., et al. 2014, arXiv:1403.6128 Groenewegen, M. A. T. 2013, , 550, A70 (G13) Groenewegen, M. A. T. 2000, , 363, 901 Inno, L., Matsunaga, N., Bono, G., et al. 2013, , 764, 84 Ita, Y., Tanab[é]{}, T., Matsunaga, N., et al. 2004, , 347, 720 Kato, D., Nagashima, C., Nagayama, T., et al. 2007, , 59, 615 Kervella, P., M[é]{}rand, A., Perrin, G., & Coud[é]{} du Foresto, V. 2006, , 448, 623 Kienzle, F., Moskalik, P., Bersier, D., & Pont, F. 1999, , 341, 818 Kinemuchi, K., Smith, H. A., Wo[ź]{}niak, P. R., McKay, T. A., & ROTSE Collaboration 2006, , 132, 1202 Kinman, T. D., Cacciari, C., Bragaglia, A., Smart, R., & Spagna, A. 2012, , 422, 2116 Jones, R. V., Carney, B. W., & Fulbright, J. P. 1996, , 108, 877 Laney, C. D., & Stobie, R. S. 1992, , 93, 93 (LS) Laney, C. D., & Stobie, R. S. 1994, , 266, 441 Luck, R. E., Andrievsky, S. M., Kovtyukh, V. V., Gieren, W., & Graczyk, D. 2011, , 142, 51 Luck, R. E., & Lambert, D. L. 2011, , 142, 136 Marconi, M., Molinaro, R., Ripepi, V., Musella, I., & Brocato, E. 2013, , 428, 2185 Marconi, M., Musella, I., & Fiorentino, G. 2005, , 632, 590 Marengo, M., Evans, N. R., Barmby, P., et al. 2010, , 709, 120 Matsunaga, N., Feast, M. W., & Soszy[ń]{}ski, I. 2011, , 413, 223 Matsunaga, N., Feast, M. W., Kawadu, T., et al. 2013, , 429, 385 Matthews, L. D., Marengo, M., Evans, N. R., & Bono, G. 2012, , 744, 53 Miceli, A., Rest, A., Stubbs, C. W., et al. 2008, , 678, 865 Minniti, D., Borissova, J., Rejkuba, M., et al. 2003, Science, 301, 1508 Minniti, D., Lucas, P. W., Emerson, J. P., et al. 2010, New Astronomy, 15, 433 Monson, A. J., & Pierce, M. J. 2011, , 193, 12 (MP) Moskalik, P., Buchler, J. R., & Marom, A. 1992, , 385, 685 Moskalik, P., Krzyt, T., Gorynya, N. A., & Samus, N. N. 2000, IAU Colloq. 176: The Impact of Large-Scale Surveys on Pulsating Star Research, 203, 233 Paczy[ń]{}ski, B., & Pindor, B. 2000, , 533, L103 Palaversa, L., Ivezi[ć]{}, [Ž]{}., Eyer, L., et al. 2013, , 146, 101 Pedicelli, S., Bono, G., Lemasle, B., et al. 2009, , 504, 81 Pejcha, O., & Kochanek, C. S. 2012, , 748, 107 Persson, S. E., Madore, B. F., Krzemi[ń]{}ski, W., et al. 2004, , 128, 2239 (P04) Pojmanski, G. 2002, ACTA, 52, 397 Press, W. H., Flannery, B. P., Teukolsky, S. A., & Vetterling, W. T. Numerical recipes in C. The art of scientific computing1989, Cambridge: University Press, 1989, Romaniello, M., Primas, F., Mottini, M., et al. 2008, , 488, 731 Sandage, A., Tammann, G. A., Saha, A., et al. 2006, , 653, 843 Szabados, L., & Klagyivik, P. 2012, , 537, A81 Szabados, L., & Neh[é]{}z, D. 2012, , 426, 3148 Soszy[ń]{}ski, I., Gieren, W., & Pietrzy[ń]{}ski, G. 2005, , 117, 823 (S05) Soszy[ń]{}nski, I., Poleski, R., Udalski, A., et al. 2008, AcA, 58, 163 Soszy[ń]{}ski, I., Poleski, R., Udalski, A., et al. 2010, AcA, 60, 17 Soszy[ń]{}ski, I., Udalski, A., Poleski, R., et al. 2012, AcA, 62, 219 Stellingwerf, R. F. 2011, RR Lyrae Stars, Metal-Poor Stars, and the Galaxy, 47 Storm, J., Gieren, W., Fouqu[é]{}, P., et al. 2011, , 534, A94 (S11a) Storm, J., Gieren, W., Fouqu[é]{}, P., et al. 2011, , 534, A95 (S11b) van Genderen, A. M. 1974, , 34, 279 Vivas, A. K., Zinn, R., Abad, C., et al. 2004, , 127, 1158 Welch, D. L., Alcock, C., Allsman, R. A., et al. 1997, Variables Stars and the Astrophysical Returns of the Microlensing Surveys, 205 Zinn, R., Horowitz, B., Vivas, A. K., et al. 2014, , 781, 22
[^1]: see also [www.astronet.ru/db/varstars](www.astronet.ru/db/varstars) and [www.astro.utoronto.ca/DDO/research/cepheids](www.astro.utoronto.ca/DDO/research/cepheids)
[^2]: [www.ogledb.astrouw.edu.pl](www.ogledb.astrouw.edu.pl)
[^3]: Throughout the paper, mean magnitude means a mean in intensity transformed into magnitude.
[^4]: We applied the S05 templates by using their phase zero–point –the epoch of the maximum– and their NIR–to–optical amplitude ratios.
[^5]: Note that for this test we only adopted calibrating Cepheids with $\log P \geqslant$0.5, because the S05 template does not include shorter period Cepheids.
|
---
abstract: 'We investigate a lattice-fluid model of water, defined on a 3-dimensional body-centered cubic lattice. Model molecules possess a tetrahedral symmetry, with four equivalent bonding arms. The model is similar to the one proposed by Roberts and Debenedetti \[J. Chem. Phys. [**105**]{}, 658 (1996)\], simplified by removing distinction between “donors” and “acceptors”. We focus on solvation properties, mainly as far as an ideally inert (hydrophobic) solute is concerned. As in our previous analysis, devoted to neat water \[J. Chem. Phys. [**121**]{}, 11856 (2004)\], we make use of a generalized first order approximation on a tetrahedral cluster. We show that the model exhibits quite a coherent picture of water thermodynamics, reproducing qualitatively several anomalous properties observed both in pure water and in solutions of hydrophobic solutes. As far as supercooled liquid water is concerned, the model is consistent with the second critical point scenario.'
author:
- 'M. Pretti and C. Buzano'
title: ' Thermodynamic anomalies in a lattice model of water: Solvation properties '
---
Introduction
============
From the experimental point of view, water is known to exhibit several thermodynamic anomalies [@EisenbergKauzmann1969; @Franks1982; @Stanley2003]. Contrary to most fluids, at ordinary pressure the solid phase (ice) is less dense than the liquid phase. The latter displays a temperature of maximum density at constant pressure, while both isothermal compressibility and isobaric heat capacity display a minimum as a function of temperature. Moreover, heat capacity is on average much larger than usual. Anomalous properties of neat water have been studied for long, but much interest has been devoted also to unusual properties of water as a solvent, in particular for nonpolar (hydrophobic) chemical species [@FrankEvans1945; @BenNaim1980; @DillScience1990; @SouthallDillHaymet2002]. Insertion of a nonpolar solute molecule in water is characterized by positive solvation Gibbs free energy (it is thermodynamically unfavored), negative solvation enthalpy (it is energetically [*favored*]{}), negative solvation entropy (it has an ordering effect), and large positive solvation heat capacity [@CrovettoFernandez-PriniJapas1982; @BenNaim1987]. More precisely, for prototype hydrophobic species (that is, for instance, noble gases), solvation entropies and enthalpies, which are negative at room temperature, increase upon increasing temperature, and eventually become positive. These properties define the so-called hydrophobic effect, whose importance in biological processes, such as protein folding, has been emphasized in the latest years [@Dill1990].
From the theoretical point of view, one can relate the anomalous properties of neat water to the formation of a large amount of hydrogen bonds, because of peculiar features of water molecules [@Stanley1998; @Poole1994]. The same physics is believed to underly the hydrophobic effect [@FrankEvans1945; @BenNaim1980; @Stillinger1980], but a comprehensive theory which explains all of these phenomena has not been developed yet. A possible way of investigation consists of computer simulations [@MahoneyJorgensen2000; @Stanley2002; @Paschek2004], based on very detailed (but still phenomenological) interaction potentials. In this way, quite a high level of accuracy in describing water thermodynamics has been achieved. Nevertheless, simulations are generally limited by the large computational effort required, while microscopic physical mechanisms are sometimes hidden by a large number of model details. A complementary approach involves investigations of simplified models [@SouthallDillHaymet2002; @AshbaughTruskettDebenedetti2002; @WidomBhimalapuramKoga2003; @BruscoliniCasetti2001pre]. Although quantitative accuracy is sometimes poor, this approach usually allows more detailed analysis, in a wide range of thermodynamic conditions, with relatively low computational cost. One of these attempts is based on the application of scaled-particle theory [@ReissFrischLebowitz1965] to hydrophobic hydration [@Lee1991]. A recent descendant of scaled-particle theory is the information theory approach by Pratt and coworkers [@Hummer_et_al1996], based on previous knowledge of water properties, such as the pair correlation function, which can be obtained by experiments or by simulations. According to the cited studies, the hydrophobic effect would result mostly from small size of water molecules, and not from water structuring by the solute, as in the classical view [@FrankEvans1945]. Such effect, though existing, would be scarcely relevant for a description of the hydrophobic effect. The simplified molecular thermodynamic theory of Ref. is basically in agreement with this conclusion. Different theories stress that the large positive heat capacity variation, observed upon insertion of apolar solutes into water, can only arise from cooperativity, that is, from induced ordering of water molecules, so that a theory of the hydrophobic effect should be based on a description of this phenomenon. This position is supported by the results of the 2-dimensional “Mercedes Benz” model, first introduced by Ben-Naim in 1971 [@BenNaim1971], and extensively investigated by Dill and coworkers in the latest years [@SouthallDillHaymet2002]. Contrary to the previously mentioned approaches, the Mercedes Benz model, though involving high simplifications, is based on well defined microscopic interactions, that is, on an energy function, without previous knowledge of water properties. One important reason to do so is the need of modelling water in a computationally convenient way, in order to investigations on complex systems such as biomolecules, for which water plays a key role. An even more simplified approach [@BesselingLyklema1997; @SharmaKumar1998; @WidomBhimalapuramKoga2003], which nevertheless can in principle satisfy this criterion, relies on the long standing tradition of lattice fluid models. As far as neat water is concerned, several different models, both in 2 [@BellLavisI1970; @BellLavisII1970; @Lavis1973; @LavisChristou1979; @HuckabyHanna1987; @BuzanoDestefanisPelizzolaPretti2004; @BalladaresBarbosa2004; @DeoliveiraBarbosa2005] and 3 dimensions [@Bell1972; @BellSalt1976; @LavisChristou1977; @MeijerKikuchiVanRoyen1982; @SastrySciortinoStanley1993jcp; @BesselingLyklema1994; @RobertsDebenedetti1996; @PrettiBuzano2004] have been investigated, some of which are variations of the early model proposed by Bell in 1972 [@Bell1972]. One of them is the 3-dimensional model by Roberts and Debenedetti [@RobertsDebenedetti1996; @RobertsPanagiotopoulosDebenedetti1996; @RobertsKarayiannakisDebenedetti1998], defined on the body-centered cubic lattice. In this model, water molecules possess four bonding arms (2 donors and 2 acceptors) arranged in a tetrahedral symmetry. Hydrogen (H) bond formation requires that 2 nearest neighbor molecules point respectively a donor and an acceptor towards each other. A number of nonbonding configurations is allowed, to account for H bond directionality. Bond weakening occurs (the bond energy is increased of some fraction) whenever a third molecule is placed near a formed bond. The latter feature basically mimics the fact that too closely packed water molecules disfavor H bonding. Let us notice that, while bonding properties are equivalent to those of the Bell model [@Bell1972], the weakening criterion is different. The Roberts-Debenedetti model is quite appealing in that it has been shown to predict not only some of real water thermodynamic anomalies, such as the temperature of maximum density, but also a liquid-liquid phase separation in the supercooled region, and a second critical point. Nevertheless, the distinction between donors and acceptors is likely to be not so crucial to the physics of water. Therefore, in a previous paper [@PrettiBuzano2004], we have investigated a simplified version of the model (without donor/acceptor distinction), showing that the same basic properties could be reproduced. Here we extend the simplified model to deal with aqueous solutions, working out solvation thermodynamics for an inert (apolar) solute. Our purpose is to verify whether the model is able to reproduce also the main features of hydrophobicity. This analysis might be interesting also in view of investigations on mixtures of water with more complex chemical species, such as polymers. We shall carry out the analysis by means of a generalized first-order approximation on a tetrahedral cluster, which has been verified to be quite accurate for the neat water model [@PrettiBuzano2004].
The model
=========
Let us introduce the model. Molecules are placed on the sites of a body-centered cubic lattice, whose structure is sketched in Fig. \[fig:reticolo\]. A site may be empty or occupied by a water molecule (${\textsf{w}}$) or by a solute molecule (${\textsf{s}}$). An attractive potential energy $-\epsilon_{{\textsf{x}} {\textsf{y}}}<0$ is assigned to any pair of nearest neighbor (NN) sites occupied by molecules of species ${\textsf{x}},{\textsf{y}}$, where ${\textsf{x}}$ and ${\textsf{y}}$ can take the values ${\textsf{w}},{\textsf{s}}$. This is the ordinary Van der Waals contribution. Water molecules possess four equivalent arms that can form H bonds, arranged in a tetrahedral symmetry, so that they can point towards 4 out of 8 NNs of a given site. There is no distinction between donors and acceptors, so that a H bond is formed whenever two NN molecules have a bonding arm pointing to each other, yielding an energy $-\eta<0$. It is easily seen that water molecules have only 2 different configurations in which they can form H bonds (see Fig. \[fig:molecole\]). We assume that $w$ more configurations are allowed, in which water molecules cannot form bonds ($w$ is related to the bond-breaking entropy). Moreover, we assign an energy increase $\eta c_{\textsf{x}}/6$, with $c_{\textsf{x}}\in[0,1]$, for each of the $6$ sites closest to a formed bond (i.e., 3 out of 6 second neighbors of each bonded molecule), occupied by an ${\textsf{x}}$ molecule. A bond surrounded by $6$ molecules of species ${\textsf{x}}$ contributes an energy $-\eta (1-c_{\textsf{x}})$. As far as water molecule are concerned, the weakening parameter mainly accounts for the fact that H bonds are most favorably formed when water molecules are located at a certain distance, larger than the optimal Van der Waals distance. Therefore, if too many molecules are present, the average distance among them is decreased, and hydrogen bonds are (on average) weakened. Moreover, the presence of an external molecule may perturb the electronic density, resulting in a lowered H bond strength as well. A weakening parameter for the solute ($c_{\textsf{s}}$) takes into account possible perturbation effects for a generic chemical species, even if in the following we shall mainly consider an ideally inert solute with $c_{\textsf{s}} = 0$.
The hamiltonian of the system can be written as a sum over irregular tetrahedra, whose vertices lie on 4 different face-centered cubic sublattices, shown in Fig. \[fig:reticolo\]. One of such tetrahedra is shown in Fig. \[fig:cactustetraedro\](a). We have $${\mathcal{H}}=
\frac{1}{6} \sum_{\langle \alpha,\beta,\gamma,\delta \rangle}
{\mathcal{H}}_{i_\alpha i_\beta i_\gamma i_\delta}
,
\label{eq:ham}$$ where ${\mathcal{H}}_{ijkl}$ is a contribution which will be referred to as tetrahedron hamiltonian, and the subscripts $i_\alpha,i_\beta,i_\gamma,i_\delta$ label site configurations for the 4 vertices $\alpha,\beta,\gamma,\delta$, respectively. Possible site configurations are: “empty” ($i=0$), “bonding water” (site occupied by a water molecule in one of the $2$ orientations which can form bonds: $i=1,2$; see Fig. \[fig:molecole\]), “nonbonding water” (site occupied by a water molecule in one of the $w$ orientations which cannot form bonds: $i=3$), “solute” (site occupied by a solute molecule: $i=4$). Assuming that $(i,j)$, $(j,k)$, $(k,l)$, and $(l,i)$ refer to NN pair configurations, the tetrahedron hamiltonian reads $${\mathcal{H}}_{ijkl} = H_{ijkl} + H_{jkli} + H_{klij} + H_{lijk}
,
\label{eq:tetraham}$$ where $$H_{ijkl} =
- \epsilon_{\textsf{xy}} {n_{{\textsf{x}},{i}}} {n_{{\textsf{y}},{j}}}
- \eta h_{ij} \left( 1 - c_{\textsf{x}} \frac{{n_{{\textsf{x}},{k}}} + {n_{{\textsf{x}},{l}}}}{2} \right)
,
\label{eq:tetraham2}$$ ${n_{{\textsf{x}},{i}}}$ is an occupation variable for the ${\textsf{x}}$ species, defined as in Tab. \[tab:configurazioni\], while $h_{ij}$ are bond variables, defined as $h_{ij}=1$ if the pair configuration $(i,j)$ forms a H bond, and $h_{ij}=0$ otherwise. Here and in the following, we understand that repeated ${\textsf{x}},{\textsf{y}}$ indices are to be summed over their possible values ${\textsf{w}},{\textsf{s}}$.
[l|ccccc]{} $i$ & $0$ & $1$ & $2$ & $3$ & $4$ $w_i$ & $1$ & $1$ & $1$ & $w$ & $1$ ${n_{{\textsf{w}},{i}}}$ & $0$ & $1$ & $1$ & $1$ & $0$ ${n_{{\textsf{s}},{i}}}$ & $0$ & $0$ & $0$ & $0$ & $1$
\[tab:configurazioni\]
Let us also assume that $i,j,k,l$ (in this order) denote configurations of sites placed on, say, $A,B,C,D$ sublattices respectively. If $A,B,C,D$ are defined as in Fig. \[fig:reticolo\], we can define $h_{ij}=1$ if $i=1$ and $j=2$, and $h_{ij}=0$ otherwise. With the above assumptions, the tetrahedron hamiltonian is independent of the orientation, that is of the arrangement of $A,B,C,D$ on its vertices. Let us notice that both Van der Waals ($-\epsilon_{\textsf{xy}} {n_{{\textsf{x}},{i}}}
{n_{{\textsf{y}},{j}}}$) and H bond energies ($-\eta h_{ij}$), which are 2-body terms, are split among $6$ tetrahedra, whence the $1/6$ prefactor in Eq. . On the contrary, the 3-body weakening terms ($\eta c_{\textsf{x}} h_{ij} {n_{{\textsf{x}},{k}}} /6$) are split between $2$ tetrahedra, thus the $1/6$ factor is absorbed in the prefactor, while a $1/2$ factor is left in the tetrahedron hamiltonian.
First order approximation
=========================
We perform the investigation by means of a generalized first order approximation on a tetrahedral cluster. In the previous paper [@PrettiBuzano2004], we have introduced the approximation in the variational approach [@An1988], as a particular choice of the largest clusters left in the entropy expansion (basic clusters). Such a choice, sometimes denoted as cluster-site approximation [@Oates1999] (the only clusters to be taken into account in the expansion are basic clusters and single sites), has not only the advantage of high simplicity, but also of relative accuracy, which has been recognized for different models [@Oates1999; @BuzanoDestefanisPelizzolaPretti2004]. The basic clusters are a number of irregular tetrahedra, namely, 4 out of 24 tetrahedra sharing a given site, as sketched in Fig. \[fig:cactustetraedro\]. This choice turns out to coincide with the (generalized) first order approximation (on a tetrahedron), also equivalent to the exact calculation on a Husimi lattice [@Pretti2003], whose (tetrahedral) building blocks are just arranged as in Fig. \[fig:cactustetraedro\](b). In the present paper, we employ the latter approach, which, not having to treat cluster probability distributions explicitly, is numerically more convenient.
Husimi lattice thermodynamics can be studied exactly, in a numerical way, by solving a suitable recursion relation [@Pretti2003], since the system is locally treelike. First of all, we have to choose, as a Husimi lattice hamiltonian, the following expression $${\mathcal{H}}' =
\sum_{\langle \alpha,\beta,\gamma,\delta \rangle}
{\mathcal{H}}_{i_\alpha i_\beta i_\gamma i_\delta}
,
\label{eq:hamprime}$$ obtained from Eq. by understanding that the sum runs over tetrahedra in the treelike system only, and removing the $1/6$ prefactor. Let us notice that the latter change is required, in order to obtain equal internal energy densities (internal energies per site) for the Husimi lattice and the ordinary lattice system. If we denote the tetrahedron configuration probability by $p_{ijkl}$, and assume that it is equal on every tetrahedron, the internal energy density can be written as $$u = \sum_{i=0}^4 w_i \sum_{j=0}^4 w_j \sum_{k=0}^4 w_k
\sum_{l=0}^4 w_l p_{ijkl} {\mathcal{H}}_{ijkl}
,
\label{eq:intenergy}$$ where $w_i$ is the multiplicity of the $i$-th site configuration, equal to $w$ for the nonbonding configuration of water ($i=3$) and equal to $1$ otherwise (see Tab. \[tab:configurazioni\]). One then has to define partial partition functions (PPFs) in the following way. Let us consider a single branch of a Husimi tree, made up of tetrahedral blocks, and a corresponding partial hamiltonian, obtained by Eq. with the sum restricted to tetrahedra in the branch. The PPF $Q_i$ is defined as a sum of the Boltzmann weights of the partial hamiltonian over the configurations of the branch minus the base site (that is why the PPF depends on a site configuration variable $i$). Working in the grand-canonical ensemble, as we are interested in, we also have to take into account chemical potential contributions, which is done by replacing the tetrahedron hamiltonian ${\mathcal{H}}_{ijkl}$ with $$\tilde{{\mathcal{H}}}_{ijkl} = {\mathcal{H}}_{ijkl}
- \mu_{\textsf{x}} \frac{{n_{{\textsf{x}},{i}}} + {n_{{\textsf{x}},{j}}} + {n_{{\textsf{x}},{k}}} + {n_{{\textsf{x}},{l}}}}{4}
\label{eq:tetrahamtilde}
,$$ where the usual convention on repeated indices holds. Let us notice that of course the PPF tends to infinity in the thermodynamic limit, that is for an “infinite generation branch”, therefore it is convenient to define a normalized PPF $q_i \propto Q_i$, for instance in such a way that $$\sum_{i=0}^4 w_i q_i = 1
.
\label{eq:norm}$$ In this way, $q_i$ represents the $i$-th configuration probability that the base site would have if it were not attached to any other branch.
Let us now consider again the single branch of our Husimi tree. In the infinite generation limit, and in the hypothesis of a homogeneous system, the subbranches attached to the first tetrahedral block should be equivalent to the main one, so that one can write the recursion relation $$q_i = y^{-1}
\sum_{j=0}^4 w_j
\sum_{k=0}^4 w_k
\sum_{l=0}^4 w_l
e^{-\tilde{{\mathcal{H}}}_{ijkl}/T}
\left( q_j q_k q_l \right)^3
,
\label{eq:rec}$$ where the sums run over configuration variables in the tetrahedron except $i$, $T$ is the temperature expressed in energy units (whence entropy will be expressed in natural units), and $y$ is a normalization constant, imposed by Eq. (\[eq:norm\]). The recursion relation can be iterated numerically to determine a fixed point, representing the PPF of a branch whose base site lies in the bulk of the Husimi tree (generally denoted as Husimi lattice). Husimi lattice properties are equivalent to those obtained by the cluster-site approximation [@Pretti2003]. We can compute the site probability distribution $p_i$, by considering the operation of attaching $4$ equivalent branches to the given site. We obtain $$p_i = z^{-1} q_i^4 ,
\label{eq:pjoint}$$ where $$z = \sum_{i=0}^4 w_i q_i^4
\label{eq:zjoint}$$ provides normalization. We can also compute the tetrahedron probability distribution, by considering the operation of attaching $3$ equivalent branches to each site of a given tetrahedron, yielding $$p_{ijkl} \propto
e^{-\tilde{{\mathcal{H}}}_{ijkl}/T}
\left( q_i q_j q_k q_l \right)^3
,$$ where of course the proportionality constant is determined by normalization. From the knowledge of the tetrahedron probability distribution $\{p_{ijkl}\}$ one can compute the thermal average of every observable in the first order approximation, the internal energy density by Eq. , and the grand-canonical free energy by Eq. (10) in Ref. . According to Eq. (31) in Ref. , the latter can be also related to normalization constants as $$\omega = - T \left( \ln y - 2 \ln z \right)
,$$ where $y$ is the normalization constant of the recursion relation (\[eq:rec\]) and $z$ is given by Eq. (\[eq:zjoint\]). Finally, the entropy density can be computed as $$s = \frac{u - \mu_{\textsf{x}} \rho_{\textsf{x}} - \omega}{T}
,$$ where $u-\mu_{\textsf{x}}\rho_{\textsf{x}}$ has formally the same expression as $u$ in Eq. , with the tetrahedron hamiltonian ${\mathcal{H}}_{ijkl}$ replaced by $\tilde{{\mathcal{H}}}_{ijkl}$.
Results
=======
In order to investigate the model properties, we fix a set of parameters, as a result of several attempts. First of all, we take $\epsilon_{\textsf{ww}}/\eta = 0.3$. This value of water-water Van der Waals interaction is equal to the one employed for the very detailed analysis by Roberts et al. [@RobertsKarayiannakisDebenedetti1998], though a bit larger than previously employed by us [@PrettiBuzano2004]. Anyway, this choice accounts for the greater binding energy of hydrogen bonds with respect to Van der Waals interactions. As far as the multiplicity of nonbonding water configurations is concerned, we set $w = 20$ (as in the previous work), which mimics the high directionality of hydrogen bonds. For neat water, it is necessary to set this parameter large enough to let anomalous properties appear, but further increase does not change qualitatively the phase diagram and the thermodynamic properties. As far as the weakening parameters are concerned, for water we choose $c_{\textsf{w}}=0.5$, which, given the other parameter values, corresponds to a situation without a reentrant spinodal, in agreement with most recent molecular dynamics results [@Stanley2003].
Pure water properties
---------------------
Since the parameter choice is slightly different from the former work [@PrettiBuzano2004], we first reconsider neat water properties, taking the limit $\mu_{\textsf{s}} \to -\infty$. This analysis is mainly meant to show that the model behavior still remains consistent with a “realistic” phase diagram, that is, with several experimental evidences as well as simulation predictions. In fact, we have observed that relatively small variations in parameter values may give rise to quite dramatic changes in the phase behavior [@PrettiBuzano2004], as observed also in other models of water [@TruskettDebenedettiSastryTorquato1999]. The temperature-pressure phase diagram is reported in Fig. \[fig:tp\]. Imposing homogeneity, our analysis includes both thermodynamically stable and metastable (supercooled) phases, even if stability is not investigated. Let us notice that pressure can be determined as $P = -\omega$, where $\omega$ is the grand-canonical potential per site, introduced in the previous section. We have assumed the volume per site equal to unit, so that pressure is expressed in energy units. The appropriate order parameter is the density, i.e., the probability $\rho_{\textsf{w}}$ that a site is occupied by a water molecule, which can be evaluated from the formula $$\rho_{\textsf{x}} = \sum_{i=0}^4 w_i p_i {n_{{\textsf{x}},{i}}}
.
\label{eq:density}$$
We find two different first order transition lines, terminating in two different critical points. The positively sloped line, at lower pressures, corresponds to the coexistence of a very low density phase and a high density phase, and represents the ordinary vapor-liquid transition. The other one, negatively sloped and placed at higher pressures, corresponds to coexistence between the high density liquid and and a lower density one. The related critical point may reasonably represent the so-called second critical point, which has been conjectured and observed in simulations, and of which also some experimental evidences have been given. As one could expect, the low density liquid turns out to be more hydrogen bonded than the high density one. We find a density maximum as a function of temperature for liquid coexisting with vapor (and at constant pressure as well), and the temperature of maximum density slightly decreases upon increasing pressure, as observed in experiments.
We also report the liquid phase spinodals and the Kauzmann line. Details about the (semi-analytical) calculation of spinodals, which allows to determine density response functions as well, are given in the previous paper [@PrettiBuzano2004]. The limit of stability of the liquid phase (spinodal) is the locus at which the metastable liquid ceases to be a minimum of (a variational form of) the free energy, and becomes a saddle point. On the contrary, the Kauzmann line is the locus at which the liquid phase entropy vanishes, and corresponds to the ideal glass transition. It can be easily determined numerically. As previously mentioned, we do not observe a reentrance of the liquid-vapor spinodal in the positive pressure half-plane, which is actually a possibility of our model, for a different parameter choice, namely, for higher values of the weakening parameter. The reentrant spinodal scenario was one of the conjectures invoked to explain thermodynamic anomalies of liquid water, and in particular of the divergent-like behavior of response functions in the supercooled regime [@Speedy1982I; @ZhengDurbenWolfAngell1991], even if at the moment the second critical point scenario is believed to be more realistic [@Stanley2003].
For the present parameter choice, in our model the critical point lies just above the Kauzmann line, while most of the high-low density liquid transition lies in the negative entropy region. This is consistent with the experimental observation of two different forms of amorfous ice in this region, while the “underlying” liquid phase is just an extrapolation of the equation of state for the liquid. The Kauzmann line displays a cusp (actually a slight discontinuity), while intersecting the high-low density liquid transition. It is noticeable that a similar feature has been predicted also by a recent analysis of the potential energy landscape of simulated water, performed by Sciortino and coworkers [@SciortinoLaNaveTartaglia2003], on the basis of the inherent structure theory.
Let us also report the density response functions and the specific heat of the liquid at constant pressure $P/\eta=0.015,0.030$, roughly corresponding to $1/5,2/5$ of the liquid-vapor critical pressure. Also for these calculations, details are reported in Ref. . We find anomalous behavior, qualitatively similar to that of real liquid water. The first response function we consider is the thermal expansion coefficient $\alpha_P = (-\partial\ln\rho/\partial T)_P$, which, from statistical mechanics, is known to be proportional to the entropy-volume cross-correlation. For ordinary fluids, $\alpha_P$ is always positive, i.e., the local entropy and the local specific volume are positively correlated. On the contrary, for our model $\alpha_P$ (Fig. \[fig:pcost\], top panel) is anomalous. As temperature is lowered, the expansion coefficient vanishes (at the temperature of maximum density), and then becomes negative. Of course, we do not observe a really divergent behavior of this coefficient, but a pronounced peak instead. Let us notice that, upon increasing pressure, the peak is observed to become broader, indicating that the liquid is becoming more normal, in agreement with experiments. The trend of the isothermal compressibility $\kappa_T = (\partial\ln\rho/\partial P)_T$ is also anomalous (Fig. \[fig:pcost\], middle panel). For a typical liquid, $\kappa_T$ decreases as one lowers temperature, because it is proportional to density fluctuations, whose magnitude decreases upon decreasing temperature. On the contrary, we observe that $\kappa_T$, once reached a minimum, begins to increase upon decreasing temperature. The constant pressure specific heat $c_P =
(T\partial s/\partial T)_P$ (Fig. \[fig:pcost\], bottom panel) displays a completely analogous behavior, with the minimum occurring at a higher temperature.
Solution properties
-------------------
Let us now consider an ideal inert solute molecule, with no Van der Waals interaction with water ($\epsilon_{\textsf{ws}}=0$), nor with other solute molecules ($\epsilon_{\textsf{ss}}=0$), and no weakening effect on H bonds ($c_{\textsf{s}}=0$). As a first analysis of the model mixture of water with this kind of (hydrophobic) solute, let us investigate phase diagrams at constant temperature and constant pressure, corresponding to the “cuts” reported in Fig. \[fig:tp\]. We take into account, as a composition variable, the solute molar fraction, defined as $$x_{\textsf{s}} = \frac{\rho_{\textsf{s}}}{\rho_{\textsf{w}} + \rho_{\textsf{s}}}
,$$ where $\rho_{\textsf{w}}$ and $\rho_{\textsf{s}}$ are determined by Eq. . In Fig. \[fig:xptx\] (top panel), we report a constant temperature phase diagram, computed at $T/\eta=0.34$. We can observe a first order transition between a water-rich and a solute-rich phase, which arises continuously from the vapor-liquid transition of neat water, as solute concentration is increased. To determine this transition, we have fixed several different values of water chemical potential $\mu_{\textsf{w}}$, then we have determined numerically the value of solute chemical potential $\mu_{\textsf{s}}$, for which both phases had the same pressure $P=-\omega$. The phase-separated (coexistence) region occupies large part of the diagram, that is, the molar fraction of solute which can be dissolved into water, without giving rise to phase separation, is very small. The behavior is very similar to that of an ordinary solution of two disaffine chemical species (at a temperature higher than the critical one for the solute), with no peculiar anomaly related to the hydrophobic effect. Notice that actually we cannot observe any phase transition for neat solute ($x_{\textsf{s}}=1$), because we have described it as a perfect gas, with $\epsilon_{\textsf{ss}}=0$. We have also computed a constant pressure phase diagram at $P/\eta=0.015$, which we have reported in Fig. \[fig:xptx\] (bottom panel). Here, to compute the transition, we have adjusted numerically both chemical potentials $\mu_{\textsf{w}}$ and $\mu_{\textsf{s}}$ to impose equality between pressures of both phases and the reference one. As expected, also in this case we observe no phase transition for pure solute, whereas a clearly anomalous behavior is observed for the water-rich phase boundary. Upon decreasing temperature, near $T/\eta \approx 0.31$, the slope of this curve begins to change rapidly. Such a behavior gives rise to an absorption coefficient $x_{\textsf{s}}^{w}/x_{\textsf{s}}^{s}$ (the superscripts denoting the water-rich and the solute-rich phase, respectively) with a minimum around $T/\eta \approx 0.34$, well above the temperature of maximum density at that pressure (see Fig. \[fig:pcost\]). This is typical signature of hydrophobic effect [@WidomBhimalapuramKoga2003].
Let us now turn to the transfer properties of a single molecule in water, that is, to a dilute solution. Large amounts of experimental data are available for this case [@BenNaim1987]. According to the Ben-Naim standard [@BenNaim1987], a transfer process (for a chemical species ${\textsf{x}}$) can be characterized by means of the pseudo-chemical potential $\mu^{*}_{\textsf{x}}$ , related to the ordinary chemical potential $\mu_{\textsf{x}}$ by the formula $$\mu_{\textsf{x}} = \mu^{*}_{\textsf{x}} + T\log\rho_{\textsf{x}}
.$$ The use of pseudo-chemical potentials is meant to remove translational entropy contributions, which are not directly related to the solvation process. The solvation free energy per molecule $\Delta g^*_{\textsf{x}}$, that is, the free energy of transfer for a molecule ${\textsf{x}}$ from the gas phase to the liquid phase, can then be defined as $$\Delta g^*_{\textsf{x}} = \mu^{*l}_{\textsf{x}} - \mu^{*g}_{\textsf{x}}
,
\label{eq:mudg}$$ where $\mu^{*l}_{\textsf{x}}$ and $\mu^{*g}_{\textsf{x}}$ denote pseudo-chemical potentials in the liquid and gas phase, respectively. If the gas and liquid phases coexist in equilibrium, the ordinary chemical potentials for the given species must be equal, so that we obtain $$\Delta g^*_{\textsf{x}} = -T \ln
\frac{\rho_{\textsf{x}}^l}{\rho_{\textsf{x}}^g}
, \label{eq:realdg}$$ where $\rho_{\textsf{x}}^l$ and $\rho_{\textsf{x}}^g$ denote densities for the ${\textsf{x}}$ species in the two phases. Derived quantities, of interest in experiments, are the solvation entropy $$\Delta s^*_{\textsf{x}} = - \frac{\partial \Delta g^*_{\textsf{x}}}
{\partial T}\biggl\lvert_P
,
\label{eq:constpentropy}$$ the solvation enthalpy $$\Delta h^*_{\textsf{x}} = \Delta g^*_{\textsf{x}} + T \Delta s^*_{\textsf{x}}
,$$ and the solvation heat-capacity $$\Delta {c_P}^*_{\textsf{x}} = \frac{\partial\Delta h^*_{\textsf{x}}}
{\partial T}\biggl\lvert_P
.$$ In principle, we should distinguish between derivatives taken at constant pressure (as stated by definition) or along the liquid-vapor equilibrium curve. In particular, we could not even use Eq. , because we would move out of the equilibrium curve. Nevertheless, we have verified that the difference between the two sets of results is negligible, in agreement with experimental observations [@BenNaim1987], and one can usually take the “equilibrium” derivative without further care.
Let us start studying solvation properties for the ideal inert solute, in the framework of our model. Water parameters are fixed as in the previous case. The temperature trends of the free energy, entropy, and enthalpy of transfer are given in Fig. \[fig:tdxs\]a; the transfer heat capacity in Fig. \[fig:tdxs\]c. In order to compare with experimental data [@CrovettoFernandez-PriniJapas1982; @BenNaim1987], also reported in Fig. \[fig:tdxs\]b and \[fig:tdxs\]d, respectively, all quantities are evaluated at liquid-vapor coexistence, and for very low solute density with respect to water density (dilute solution limit). In practical calculations, we have adjusted numerically water and solute chemical potentials, in order to impose the equilibrium condition (equal pressure) between liquid and vapor, and to fix the solute molar fraction. Nevertheless, we have also verified that, only for the perfectly inert solute, concentration does not affect the results at all, so that we could also set an arbitrary value for the solute chemical potential. As shown in the previous section, our parameter set for pure water corresponds to a liquid-vapor critical temperature $T/\eta \approx 0.52$, and to a temperature of maximum density for the liquid phase around $T/\eta \approx 0.32$, at low pressure. Therefore, in order to represent roughly the experimental temperature range (between $0^\circ\,\mathrm{C}$ and $300^\circ\,\mathrm{C}$) we report model results between $T/\eta =
0.31$ (just below the temperature of maximum density for pure liquid water) and $T/\eta = 0.40$ (about half way between the previous temperature and the critical temperature). Remarkably, it turns out that the model displays the defining features of hydrophobic solvation. The solvation free energy is positive and large, while the solvation entropy and enthalpy are negative at low temperatures and become positive upon increasing temperature. The solvation heat capacity is positive and large, and also the decreasing trend with temperature is basically reproduced. A slightly increasing trend at high temperature is related to the the fact that we are approaching the liquid-vapor critical point. Negative solvation entropy at low (room) temperature is a clear indication that solute insertion into the mixture orders the system. The corresponding positive (unfavorable) contribution to free energy compensates a negative (favorable) enthalpic contribution, giving rise to a positive solvation free energy. At higher temperature, enthalpic and entropic contributions change sign, but they still have the same compensating trend. The model also predicts two different temperatures at which the transfer enthalpy and entropy vanish (see Fig. \[fig:tdxs\]a), as observed in experiments (Fig. \[fig:tdxs\]b).
The whole observed behavior is to be ascribed to the thermodynamics of H bonding and, in order to rationalize this fact in the model framework, we have also analysed transfer quantities, upon removing H bond interactions (see Fig. \[fig:tdxs\]a,\[fig:tdxs\]c). As reasonable, the results are quite similar in the high temperature regime, where there is a high probability that H bonds are broken by thermal fluctuations, whereas they change more and more dramatically upon decreasing temperature, and, in particular, the regions of negative transfer entropy and enthalpy completely disappear. This facts confirm that H bonding is the key element for system ordering, upon insertion of an inert molecule. Accordingly, also the increasing trend of the heat capacity upon decreasing temperature is suppressed. The process is now dominated by enthalpy, with a large and positive transfer free energy (but without a maximum), and a positive transfer entropy. Transfer quantities now behave qualitatively as observed in solvation experiments of noble gas molecules in ordinary liquids [@DaviesDuncan1967; @BenNaim1987], and are relatively independent of temperature. Actually, a water molecule for which H bond formation has been “turned off”, can be viewed as a nonpolar molecule with only Van der Waals interaction energy $\epsilon_{{\textsf{ww}}}$.
Let us now consider also the solvation of water in its own pure liquid. Corresponding transfer quantities obtained by the model are displayed in Fig. \[fig:tdxw\]a, where we have reduced the temperature interval, in order to compare with available experimental results [@BenNaim1987], reported in Fig. \[fig:tdxw\]b. With respect to the inert molecule case, here absolute values of solvation free energy and entropy are considerably smaller. Enthalpy, rather than entropy, dominates the solvation process, while all quantities are relatively independent of temperature. These features characterize a regular transfer process, like the solvation of an ordinary fluid molecule from a gas phase into its pure liquid phase. In this case, upon removing H bond interactions (thin lines in Fig. \[fig:tdxw\]a), very little changes are observed. Let us discuss two issues about these results. First, the fact that so little changes are caused by turning on or off H bonds can be rationalized on the basis of the microscopic model interactions. Insertion of a water molecule into pure liquid water should imply in principle the formation of new H bonds, but the model is such that insertion of a new water molecule also weakens other H bonds in its neighborhood, and the two effects nearly compensate each other. Second, let us notice that solvation enthalpy decreases upon increasing temperature, that is, the solvation heat capacity is negative, in contrast with experiments. We do not have an explanation for this fact, but we can observe that it is basically unchanged when H bonds are turned off, that is, when the model is reduced to describe a “regular” solvation process. This suggest that there is probably a limitation of the lattice description, that anyway has nothing to do with peculiarities of water. Indeed, the effect is quantitatively small, so that it is hidden by other large (enthalpic and entropic) effects, observed in the case of hydrophobic solvation.
“Nonideal” solute and entropy convergence
-----------------------------------------
So far, we have always turned off all interactions involving solute molecules, except excluded volume. Here we report some results concerning the role of nonzero solute-water interaction parameters ($\epsilon_{\textsf{ws}}$, $c_{\textsf{s}}$), still assuming that solute molecules have no relevant interaction with one another ($\epsilon_{\textsf{ss}} = 0$). In order to have a single parameter to be varied, we have performed our investigation for increasing values of the solute weakening parameter $c_{\textsf{s}}$, and defined solute-water interaction $\epsilon_{\textsf{ws}}$ according to the following proportionality condition $$\epsilon_{\textsf{ws}}/\epsilon_{\textsf{ww}} = c_{\textsf{s}}/c_{\textsf{w}}
.
\label{eq:proporz}$$ At the beginning, we took this assumption as a simple trial, but the results we obtained were quite interesting, so that we have carried on with the analysis. As far as transfer free energies are concerned, we have observed a qualitatively unchanged behavior, with a broad maximum at some temperature, and free energy values getting smaller and smaller, upon increasing $\epsilon_{\textsf{ws}}$ and $c_{\textsf{s}}$. On the contrary, we have observed peculiar features concerning entropies, actually related to one another, which are displayed in Fig. \[fig:entconv\]a. Still upon increasing $\epsilon_{\textsf{ws}}$ and $c_{\textsf{s}}$, the temperature of zero entropy is progressively shifted towards higher values, while different entropy curves converge in a very narrow temperature range, relatively close to zero entropy temperatures. Moreover, entropy values at convergence are negative and relatively small. As one can observe in Fig. \[fig:entconv\]b, all of these features correspond remarkably well to phenomenology observed for the series of noble gases [@BenNaim1987], in particular the entropy convergence, which has attracted some interest, due to the fact that a similar effect has been observed for the entropies of protein unfolding [@Lee1991pnas]. Because of this unexpected result, we have tried and justified a posteriori the working hypothesis , and actually a naive explanation may be the following one. As a first approximation, different hydrophobic species, such as noble gases, may be viewed as hard spheres distinguished by their diameter, that is their volume, only. In this way, the proportionality condition can be conceived as a trick that, in the framework of the lattice model, mimics the fact that “a fraction” $c_{\textsf{s}}/c_{\textsf{w}}$ of the site occupied by the solute actually “behaves like water”. As a consequence, higher values of the ratio $c_{\textsf{s}}/c_{\textsf{w}}$ should correspond to smaller solute molecules, which turns out to be consistent, comparing Figs. \[fig:entconv\]a and \[fig:entconv\]b. Let us notice also that, for parameter values as in Fig. \[fig:entconv\]a, the “fractions that behaves like solute” ($1-c_{\textsf{s}}/c_{\textsf{w}}$) correlate well with the squares of the hard sphere diameters of the corresponding substances [@Garde1996], which is consistent with the common assumption that hydrophobicity is proportional to exposed surface.
Discussion and conclusions
==========================
In this paper we have considered a 3-dimensional lattice fluid model of water, which we had previously shown to exhibit realistic thermodynamic anomalies [@PrettiBuzano2004], and extended the model to describe aqueous solutions. The motivation for this work resides mainly in the interest for the hydrophobic effect, whose relevance for biological processes such as protein folding, taking place in aqueous solutions, has been more and more recognized in the latest years. Moreover, a simplified but accurate modelling of water is an appealing issue, in view of investigations on such processes, because detailed water models may be extremely time consuming [@SouthallDillHaymet2002].
As far as water is concerned, our model is a simplified version of a previous model proposed by Roberts and Debenedetti [@RobertsDebenedetti1996], without a distinction between hydrogen bond donors and acceptors. In the framework of this model, the microscopic description of water anomalies, is essentially based on the competition between an isotropic (Van der Waals like) interaction and an highly directional (H bonding) interaction, and on the difference between the respective optimal interaction distances. In the lattice environment, the latter is taken into account by a trick, that is, the weakening effect of a water molecule near a formed bond. The same assumptions also accounts for possible perturbations of the electronic density, due to interaction with other water molecules. We have extended the weakening trick to take into account the presence of a different chemical species (solute), with no internal degrees of freedom (bonding arms). Calculations have been performed in a generalized first-order approximation on a tetrahedral cluster, which requires small computational effort, and had been shown to be quite accurate for the pure water model [@PrettiBuzano2004].
Due to the fact that we have chosen slightly different interaction parameters for the present investigation, we have first analysed again pure water behavior. At constant pressure, the typical thermodynamic anomalies are reproduced, with a density maximum, and a minimum of isothermal compressibility and specific heat. In the ordinary temperature and pressure region, the temperature of maximum density decreases upon increasing pressure, as observed in experiments. Moreover, as far as the supercooled regime is concerned, there is still evidence of a second (metastable) critical point, which terminates a line of coexistence between two liquid phases at different densities. Let us recall that the pure water model could predict, for different values of the weakening parameter $c_{\textsf{w}}$, two different scenarios, that is, with or without a reentrant spinodal. The present parameter choice predicts a nonreentrant spinodal. The reentrant spinodal scenario was the first conjecture put forth to justify water anomalies, whereas the most recent and accurate molecular dynamics simulations of water suggest a scenario with a nonreentrant spinodal and a metastable liquid-liquid critical point [@Stanley2003]. As far as the critical point is concerned, in our calculation, it lies at some temperature just above the Kauzmann line, at which the configurational entropy vanishes, while the Kauzmann line displays a cusp upon crossing the metastable coexistence line. All of these features turn out to be in a remarkably good agreement with the inherent state analysis of the potential energy landscape of simulated water, recently performed by Sciortino and coworkers [@SciortinoLaNaveTartaglia2003].
As far as the solution model is concerned, we have mainly considered solutions of inert, that is, hydrophobic solutes. First of all, we have investigated phase diagrams for arbitrary solute concentration, pointing out a minimum of solubility as a function of temperature, as typical for hydrophobic solutions. We have investigated in more detail the dilute solution limit, at liquid-vapor equilibrium, for which many experimental data are available. Solvation quantities turn out to exhibit peculiar features that are believed to be the fingerprints of hydrophobicity. The solvation free energy is positive (unfavorable solvation), while entropy and enthalpy are negative at low temperatures and positive at high temperatures. The solvation heat capacity is large and decreases upon increasing temperature. These results compare qualitatively well with solvation experiments for noble gases in water. Let us notice that a previous lattice model by Besseling and Lyklema [@BesselingLyklema1997], based on a different description of water interactions, was also able to account for the qualitative behavior of free energy, enthalphy, and entropy of transfer. Nevertheless, it failed in reproducing the correct temperature trend of the transfer heat capacity, which, according to some authors [@SouthallDillHaymet2002] is a key feature, revealing the cooperative nature of the hydrophobic effect. Let us recall, by the way, that the same difficulty about heat capacity is encountered by the information theory approach [@ArthurHaymet1999].
We have investigated explicitly on the effect of H bonding, in the framework of our model, performing calculations also when this interaction is completely turned off. In this case, we have obtained transfer quantities that approach the ones computed [*with*]{} H bonds at high temperatures, but that largely deviates from them upon decreasing temperature, that is, in the region were H bonding begins to dominate. In particular, we have observed that, while disaffinity between solute and solvent is left (the solvation free energy is still positive), this is mainly of enthalpic nature. Both the enthalpy and entropy of solvation remain positive at all temperatures, so that also the typical strong temperature dependence of hydrophobic solvation, disappears.
We have also taken into account the solvation process of water into its own pure liquid, for which experimental data are available. We have found qualitative agreement, as far as the values of transfer free energy, entropy and enthalpy are concerned, but we have observed some discrepancy in the temperature dependence of enthalpy, indicating a negative solvation heat capacity, in disagreement with experiments. We have verified that the same kind of discrepancy can be observed if H bonding is turned off, that is, for an ordinary lattice gas. Therefore, we suggest that the discrepancy is to be related to an intrinsic limitation of the lattice environment, that has nothing to do with peculiarities of the water model. The effect is relatively small, so that it is completely invisible, when the dominant effect of H bonds is introduced.
Let us notice that the results concerning transfer quantities are qualitatively similar to those observed for a 2-dimensional lattice model recently investigated by us [@BuzanoDestefanisPretti2005], which in turn is a simplified “lattice version” of the Mercedes Benz model investigated by Dill and coworkers. In spite of similarities in the experimental temperature range, there is one major difference between the two models. The 2-dimensional model is not able to account for the metastable critical point of water, probably due to impossibility of reproducing a high density ordered packing of water molecules. As a consequence, thermodynamic anomalies are entirely due to the presence of a reentrant spinodal, which is no longer believed to be a realistic scenario for real water [@Stanley2003]. In this sense, the present 3-dimensional model seems to provide a more coherent view of water thermodynamics, relating each other neat water anomalies and hydrophobic effect.
Moreover, we have shown that, with quite a reasonable assumption for the solute interaction parameters $\epsilon_{\textsf{ws}}$,$c_{\textsf{s}}$ (attempting to describe solutes of different volume in the lattice framework), the model is able to reproduce also the entropy convergence phenomenon, in a qualitatively correct way. Such a phenomenon has been theoretically described, for instance, by the simplified molecular theory of Debenedetti and coworkers [@AshbaughTruskettDebenedetti2002]. Moreover, an almost quantitative explanation has been proposed by Pratt and coworkers [@Garde1996], on the basis of the information theory approach, which we mentioned in the Introduction. In the cited work, the authors argue that entropy convergence, and in particular the negative entropy at convergence, are related to the weak temperature dependence of free volume fluctuations in liquid water, that is, isothermal compressibility. Although we cannot provide a clear explanation of why our lattice model, with the proportionality assumption , exhibits qualitatively correct behavior, let us notice that such result is somehow consistent with the previous explanation. In fact, entropy convergence occurs very close to the minimum of isothermal compressibility, as one can argue from Fig. \[fig:pcost\], that is, in a region where compressibility is nearly constant, as a function of temperature. Nonetheless, the proportionality assumption still remains not well justified.
Let us finally recall that our model, at least in the present treatment, is not able to provide microscopic structural details as simulations do, but its most appealing feature is simplicity. In the present paper we have shown that, in spite of this, the model yields a qualitatively coherent description of peculiar thermodynamics of water, not only as a pure substance but also as a solvent, and is consistent with predictions based on much more sophisticated models and simulations. Moreover, thanks to the 3-dimensional embedding, it may be suitable for quite a realistic analysis of more complex, for instance polymeric, solutes. We are going to report about such investigations in a forthcoming article.
|
---
abstract: 'We give a positive solution to a conjecture of Faith stating that a self-injective semiprimary ring is QF, for algebras which are at most countable dimensional modulo their Jacobson radical. As a consequence of the method used, we also give short proofs of several other known positive answers to this conjecture.'
address:
- |
University of Southern California\
3620 S Vermont Ave, KAP 108\
Los Angeles, CA 90089, USA
- |
University of Bucharest, Facultatea de Matematica\
Str. Academiei 14, Bucharest 1, RO-010014, Romania
- 'e-mail: iovanov@usc.edu, yovanov@gmail.com'
author:
- 'M.C. Iovanov'
title: Semiprimary selfinjective algebras with at most countable dimensional Jacobson quotient are QF
---
Introduction and Preliminaries
==============================
In classical ring theory, among the rings of interest and intensively studied in literature are the left or right selfinjective rings. Left selfinjective rings which are also artinian form another important class of rings called quasi-Frobenius (QF) rings. There are many equivalent definitions of these rings, and they have an intrinsic symmetry: a ring is QF if it is right selfinjective and right semiartinian, or equivalently, noetherian or artinian on one side and injective on one side. Classical results also include those of Faith and Walker stating that such rings are characterized by the fact that all right (equivalently, all left) injective (equivalently, projective) modules are projective (injective). These rings are important generalizations of Frobenius algebras, retaining the categorical properties of these; examples include group algebras of finite groups, Hopf algebras, certain cohomology rings. Moreover, such rings are important in many fields of mathematics, from representation theory, category theory, homological algebra and topology to coding theory.
Perhaps one of the most interesting questions regarding QF rings, and also in ring theory in general, is the following question, known in literature as Faith’s QF conjecture:
[**Conjecture \[Faith\]**]{}\
*A left selfinjective semiprimary ring is QF.*
Much work has been dedicated to this problem over the years [@ANY; @CH; @CS; @NY; @NY2; @NY3; @O; @X]; we also refer to the recent survey [@FH] which contains a comprehensive account of the history and known results on QF rings and related topics.
In this note we present a positive answer for algebras $A$ over a field $K$, which are at most countable dimensional modulo their Jacobson radical, i.e. $A/Jac(A)$ is at most $\aleph_0$. This includes, for example, the important situation when $A/J$ is not only semisimple but finite dimensional. As consequence of our method, we also give short straightforward proofs of two other results of [@L] and [@Ko], stating that Faith’s Conjecture is true for countable dimensional algebras, or for rings $A$ for which $|A/J|\leq \aleph_0$ or $|A/J|<|A|$.
For sake of completeness, we recall a few facts most of which are fairly easy to see and well known in literature. Let $A$ be a ring and $J$ its Jacobson radical. If $A$ is semilocal, i.e. $A/J$ is semisimple, then an $A$-module is semisimple if and only if it is canceled by $J$. Indeed, every simple is canceled by $J$, and if $JM=0$ then $M$ has an $A/J$-module structure which is semisimple; therefore $M$ is semisimple as the lattice of $A$-submodules and $A/J$-submodules of $M$ coincide in this case. If $A$ is semiprimary, and $n$ is such that $J^n=0\neq J^{n-1}$ then $A$ is semiartinian with a Loewy series of length $n-1$ since $J^k/J^{k+1}$ is semisimple for all $k$. Write $A\bigoplus\limits_eAe$ a sum of indecomposable $A$-modules; such a decomposition obviously exists because $A/J$ has finite length, and each $Ae$ is obtained for some indecomposable idempotent $e$. Note that if $A$ is left self-injective, then each indecomposable $Ae$ has simple socle: indeed, if we have a nontrivial decomposition of the socle $s(Ae)=M\oplus N$, then we can find $E(M),E(N)$ injective hulls of $M,N$ contained in $Ae$, and we obtain $Ae=E(M)\oplus E(N)$ a nontrivial decomposition. This is a contradiction.
Note also that if $A$ is left self-injective semiprimary, for each simple left $A$-module, the right $A$-module ${{\rm Hom}}(S,A)$ is simple. First, note that it is nonzero. For this, we look at the isomorphism types of indecomposable modules $Ae$; these are projective and local, and are the cover of some simple $A$-module. They are isomorphic if and only if their respective “tops” are isomorphic. The number of isomorphism types of such modules equals the number of isomorphism types of simple modules $t$. Moreover, since the indecomposable $Ae$’s are also injective with simple socle, we see that they are isomorphic if and only if their socle is isomorphic. This shows that the distinct types of isomorphism of simples occurring as socle of some $Ae$ is also $t$, and so each simple $S$ must appear as socle of some $Ae$ (i.e. it embeds in $A$). This shows that ${{\rm Hom}}(S,A)\neq 0$ for each simple $A$-module $S$. If $f,g\in{{\rm Hom}}(S,A)$, and $f\neq 0$, then $f:S\rightarrow A$ is a mono and since ${}_AA$ is injective there is some $h:A\rightarrow A$ such that $h\circ f=g$. If $f(x)=xc,\,\forall x\in A$, we get $f(x)c=g(x)$ i.e. $f\cdot c=g$ in ${{\rm Hom}}(S,A)$. This shows that ${{\rm Hom}}(S,A)$ is generated by any $f\neq 0$, so it is simple. In particular, since each simple module embeds in $A$ which is left injective, it follows that $A$ is an injective cogenerator of the category of left $A$-modules, i.e. it is a left PF (pseudo-Frobenius) ring. It is easy to see that the same conclusions follow in case $A$ is semilocal, left semiartinian and left selfinjective.
The Main Result {#s1}
===============
Let ${\mathcal{S}}$ be a set of representatives for the simple left $A$-modules, $t=|{\mathcal{S}}|$ and $A/J=\bigoplus\limits_{S\in{\mathcal{S}}}S^{n_S}$. Let $\Sigma=s({}_AA)$ be the left socle of $A$. It is easy to see that this is an $A$-sub-bimodule of $A$. Note that since each indecomposable module $Ae$ has simple socle, we have that $length(\Sigma)$ equals the number of terms in the indecomposable decomposition $A=\bigoplus\limits_eAe$, which equals $length({}_AA/J)$ since each indecomposable $Ae$ is local. Let $\Sigma=\bigoplus\limits_{S\in{\mathcal{S}}}S^{p_s}$. We have $\sum\limits_{S\in{\mathcal{S}}}p_S=\sum\limits_{S\in{\mathcal{S}}}n_S$.
Let $A$ be left self-injective and semiprimary. Then the set $\{{{\rm Hom}}(S,A)| S\in {\mathcal{S}}\}$ is a set of representatives for the simple right $A$-modules. In particular, ${{\rm Hom}}(S,A),{{\rm Hom}}(T,A)$ are non-isomorphic for non-isomorphic $S,L\in{\mathcal{S}}$.
Since $A$ is left injective, the monomorphism $0\rightarrow \Sigma\rightarrow A$ gives rise to the epimorphism of right $A$-modules ${{\rm Hom}}(A,A)\rightarrow {{\rm Hom}}(\Sigma,A)\rightarrow 0$. Note that ${{\rm Hom}}(\Sigma,A)=\bigoplus\limits_{S\in{\mathcal{S}}}{{\rm Hom}}(S,A)$. Since ${{\rm Hom}}(S,A)\neq 0$ for each $S\in{\mathcal{S}}$ we have ${{\rm Hom}}(\Sigma,A)=\bigoplus\limits_{S\in{\mathcal{S}}}{{\rm Hom}}(S,A)^{p_S}$ has length equal to $length(\Sigma)=\sum\limits_{S\in{\mathcal{S}}}{p_S}=length({}_AA/J)$. But by the classical Wedderburn-Artin theorem, $length{}_A(A/J)=length(A/J)_A$. Since ${{\rm Hom}}(\Sigma,A)$ is semisimple, the kernel of $A\rightarrow {{\rm Hom}}(\Sigma,A)$ contains $J$, and furthermore since $length(A/J)=length({{\rm Hom}}(\Sigma,A))$, we obtain $A/J\cong {{\rm Hom}}(\Sigma,A)$ as right $A$-modules. This shows that all types of isomorphism of right $A$-modules are found among components of ${{\rm Hom}}(\Sigma,A)$, and so the statement is proved.
We note that the above proof further shows that there is an exact sequence of right $A$-modules $$0\longrightarrow J\longrightarrow A\longrightarrow {{\rm Hom}}(\Sigma,A)\longrightarrow 0$$ But it is immediate to see that this means that $\{a\in A|\Sigma\cdot a=0\}=J$, i.e. $ann(\Sigma_A)=J$. In particular, this shows that $\Sigma$ is also semisimple as a right $A$-module, i.e. the left socle of $A$ is contained in the right socle. In fact, it is known that the left and right socles of a left PF-ring coincide [@Ka Theorem 6], and if $A$ is semiprimary with same left and right socle, then it is easy to show that the left and right Loewy series of $A$ coincide [@AP Proposition 2.1] (see also [@Ko Lemma 3.7]).
For a left $A$-module $M$, let us denote for short $M^*={{\rm Hom}}(M,A)$; this is a right $A$-module.
\[p.1\] Let $A$ be a left self-injective ring and let $M$ be a left $A$-module such that there is an exact sequence $0\rightarrow S\rightarrow M\rightarrow L^{(\alpha)}\rightarrow 0$, with $S,L$ simple modules, and assume $S=s(M)$ the socle of $M$, and $L^{(\alpha)}$ denoting the coproduct of $\alpha$ copies of $L$. Then $M^*$ is a local right module with unique maximal ideal $S^\perp=\{f\in {{\rm Hom}}(M,A)|f_{\vert S}\neq 0\}$ which is semisimple isomorphic to $(L^*)^\alpha$.
We have an exact sequence $0\rightarrow (L^*)^\alpha\rightarrow M^*\rightarrow S^*\rightarrow 0$; it is easy to see that the kernel of the morphism $M^*={{\rm Hom}}(M,A)\rightarrow S^*={{\rm Hom}}(S,A)$ is $S^\perp$. Hence $S^\perp\cong (L^*)^\alpha$ which is right semisimple since it is canceled by $J$. Now since $M$ has simple socle, and its socle embeds in $A$ which is injective, it follows that $M$ embeds in $A$. We now note that $M^*$ is generated by any $f\not\in S^\perp$, which will show that $M^*$ is . Indeed, such an $f$ must be a monomorphism, and given any other $h:M\rightarrow A$, by the injectivity of ${}_AA$ there is $g\in{{\rm Hom}}(A,A)$ such that $g\circ f=h$. If $g(x)=xc$ for $c\in A$ then we have $h=f\cdot c$ in $M^*$. This shows that $f\cdot A=M^*$. This obviously shows that $S^\perp$ is the only maximal submodule of the cyclic right $A$-module $M^*$.
Note that the fact that $M^*$ is local can also be proved by embedding $M$ in some indecomposable $Ae$ for an indecomposable idempotent $e$, and then, by applying the exact functor ${{\rm Hom}}(-,A)$, one obtains an epimorphism ${{\rm Hom}}(Ae,A)=eA\rightarrow M^*$, and so $M^*$ is local because $eA$ is.
Let $\alpha$ be the largest cardinality for which there is a left module $M$ with simple socle and such that $M/s(M)\cong L^{\alpha}$ for some simple module $L$. Such a cardinality obviously exists, since any such module is contained in $A$ because $A$ is injective. In fact, if $\Sigma_1$ is the second socle of $A$, then $\alpha\leq length(\Sigma_2/\Sigma)$. We note that if $\alpha$ is infinite, this is an equality. Indeed, if for each simple modules $S,L$ we denote by $\alpha_{S,L}=[E(S)/S):L]$ - the multiplicity of $L$ in the second socle of the injective hull $E(S)$ of $S$, then $\alpha=\max_{S,L\in{\mathcal{S}}}\alpha_{S,L}$. Therefore, $\alpha\leq\sum\limits_{S,L\in{\mathcal{S}}}\alpha_{S,L}\leq n\alpha=\alpha$ if $\alpha$ is infinite. We note also that if $\Sigma_k$ is the $k$’th socle, then $length(\Sigma_k/\Sigma_{k-1})\leq \alpha$; this follows by induction on $k$: if this is true for $k$, then there is an embedding $\Sigma_k/\Sigma_{k-1}\hookrightarrow A^{(\alpha)}$, and therefore we have $length(\Sigma_{k+1}/\Sigma_k)\leq length(\Sigma_1/\Sigma_0)^{(\alpha)}=\alpha\times \alpha=\alpha$ since $\alpha$ is an infinite cardinal. We therefore have
\[t.1\] Let $A$ be a self-injective semiprimary algebra such that the dimension of each simple $A$-module is at most countable (equivalently, the dimension of $A/J$ is at most countable).
With the above notations, assume $\alpha$ is infinite. The length of each $\Sigma_k/\Sigma_{k-1}$ is at most $\alpha$, so since the dimension of each simple is at most $\aleph_0$, its dimension is at most $\aleph_0\times \alpha=\alpha$ (since $\alpha$ is infinite). Thus, the dimension of $A$ is at most $\alpha$, and so it equals $\alpha$ (since $length(\Sigma_1/\Sigma_0)=\alpha$). On the other hand, by Proposition \[p.1\], there is a local right $A$-module $M^*$, with socle $L^\alpha$ for some simple $L$. Note that $\dim(L^\alpha)\geq 2^\alpha$, and that there is an epimorphism $A\rightarrow M^*$, so $\dim(A)\geq 2^\alpha$. This is obviously a contradiction.
We note that can also prove this by using the exact sequences $0\rightarrow \Sigma_k/\Sigma_{k-1}\rightarrow A/\Sigma_{k-1}\rightarrow A/\Sigma_k\rightarrow 0$, which by the left injectivity of $A$ yield the exact sequences of right $A$-modules $0\rightarrow (\Sigma_k/\Sigma_{k-1})^*\rightarrow (A/\Sigma_{k-1})^*\rightarrow (A/\Sigma_k)^*\rightarrow 0$ so $\dim(A/\Sigma_{k-1})^*-\dim(A/\Sigma_k)^*=\dim(\Sigma_k/\Sigma_{k-1})^*$, which, by summing for $k$ yields $$\dim(A)=\sum\limits_k\dim(\Sigma_k/\Sigma_{k-1})^*.$$ Then one can proceed as above to note that in the situation when $\alpha$ is infinite and $\dim(A/J)$ is at most countable, then one of the dimensions on the right of the above equalities is at least $2^\alpha$, while $\dim(A)=\alpha$, a contradiction.
We note several other corollaries that can be obtained applying the above method. The following can also be obtained from the results of [@L], which shows that a left self-injective at most countable dimensional algebra is QF; nevertheless, the proofs of [@L] use some further assumptions on $A$, such that the cardinality of $A$ is regular, and also makes use of the generalized continuum hypothesis (see also MR512077, Erratum to: \[*A countable self-injective ring is quasi-Frobenius*, Proc. Amer. Math. Soc. 65 (1977), no. 2, 217–-220\]; Proc. Amer. Math. Soc. 73 (1979), no. 1, 140).
A semiprimary left self-injective algebra of countable dimension then is QF.
The following is known from [@Ko Corollary 3.10]. We also give a very short (and straightforward) proof of this using the method above.
\(1) A left self-injective semiprimary ring $A$ with $|A/J|\leq \aleph_0$ is QF.\
(2) A left self-injective semiprimary ring $A$ with $|A/J|<|A|$ is QF.
\(1) We proceed as in Theorem \[t.1\], and keep the notations above. The length of each $\Sigma_k/\Sigma_{k-1}$ is at most $\alpha$, and since each simple module has cardinality at most $\aleph_0$, $|\Sigma_{k}/\Sigma_{k-1}|\leq \aleph_0\times \alpha=\alpha$. As in Theorem \[t.1\], using Proposition \[p.1\] we find the right module $M^*$ with socle $L^\alpha$, which has cardinality at least $2^{\alpha}$, and is a quotient of $A$. This yields a contradiction.\
(2) Let $c$ be the largest cardinality of a simple left $A$-module; we have $c<|A|$. Again, as above, if $\alpha$ is infinite, we obtain that the cardinality of $A$ has to be at least $2^{\alpha}$. On the other hand, the cardinality of the modules $\Sigma_k/\Sigma_{k-1}$ is less than $c\times \alpha={\rm max}(c,\alpha)$. Since $\Sigma_n=A$ for some $n$, this shows that $|A|\leq \max{c,\alpha}$. But $\alpha<2^\alpha\leq |A|$ and $c<|A|$, yielding a contradiction.
We now note another interesting fact about the general situation of Faith’s QF conjecture.
Let $A$ be a left self-injective semiprimary ring. Then\
(i) each right $A$-module $eA$ has simple socle. Consequently, the right socle of $A$ (which coincides with $\Sigma$) is finitely generated, and has the same left and right lengths.\
(ii) ${{\rm Hom}}(T,A)\neq 0$ for each right simple $A$-module $T$.
\(i) We have $eA={{\rm Hom}}(Ae,A)$. Let $M$ be the unique maximal submodule of $Ae$. We show that $M^\perp=\{f:Ae\rightarrow A| \,f\vert_M=0\}\subset eA$ is essential in $A$. Let $0\neq h:Ae\rightarrow A$. Then, $\ker(h)\neq Ae$, so $\ker(h)\subseteq M$, and thus we have the following commutative diagram $$\xymatrix{
& Ae \ar[d]_p \ar[dr]^h & \\
0 \ar[r] & \frac{Ae}{\ker(h)}\ar[d]_\pi \ar[r]^i & A \ar@{..>}[ddl]^g \\
& \frac{A}{M}\ar[d]_u &\\
& A &
}$$ Here, $p$ and $\pi$ are the canonical projections, $h=i\circ p$ is the canonical decomposition, and $u$ is a nonzero morphism from $A/M$ to $A$, which exists since we know all isomorphism types of simple modules embed in $A$. Since $A$ is injective, then the above diagram is completed commutatively by a $g:A\rightarrow A$, $g(x)=xc$ for $x\in A$. Let $f=u\circ\pi\circ p$; then obviously $f\neq 0$, $f\in M^\perp$ and $g\circ h=f$, i.e. $g\cdot c=f$ in ${{\rm Hom}}(Ae,A)$. This shows that $M^\perp \cap hA\neq 0$ whenever $h\neq 0$. This shows that $M^\perp$ is essential in $eA$. Moreover, it is easy to see that $M^\perp\cong (Ae/M)^*$ by dualizing the exact sequence $0\rightarrow M\rightarrow Ae\rightarrow Ae/M\rightarrow 0$, so $M^\perp$ is simple. Thus, $eA={{\rm Hom}}(Ae,A)$ has simple (essential) socle.\
(ii) We have already noticed that each simple right $A$-module $T$ is of the form ${{\rm Hom}}(S,A)$ for a simple left $A$-module. But since there is an epimorphism $A\rightarrow S$, by duality we get a monomorphism of right $A$-modules $0\rightarrow T={{\rm Hom}}(S,A)\rightarrow A$.
We note that a possible procedure for proving this conjecture for other cases, would be the following. For a semiprimary left self-injective ring $A$, consider the Loewy series $0\subset \Sigma_0\subset \dots \Sigma_k\dots \Sigma_{n-1}=A)$ of $A$ - this is the same to the left and to the right. The first term has the same left and right length, as shown before. If this is true for all the factors in the Loewy series, that is, if the length of $\Sigma_{k}/\Sigma_{k-1}$ would be the same as left and right modules, one could apply the above procedure of Proposition \[p.1\] and Theorem \[p.1\] to obtain a positive answer to Faith’s Conjecture (in fact it is enough to show that $\Sigma_1/\Sigma_0$ has the same left and right length). Specifically, let $M$ be a module like in Proposition \[p.1\] of maximal (infinite) length $\alpha$ modulo its socle; one then sees that $M^*/M^*J$ has $A$-length greater than $\alpha$ (in fact, it is semisimple of length $2^\alpha$), by regarding everything as vector spaces over some division algebra and using methods similar to those of vector spaces. This would again be a contradiction to the fact that the left and right lengths of $\Sigma_{n-1}/\Sigma_{n-2}$ coincide. On the other hand, this also shows that if a counterexample to this conjecture exists, then some $\Sigma_{k}/\Sigma_{k-1}$ would have different left and right lengths.
One way one could try to compare the left and right lengths of $\Sigma_{k}/\Sigma_{k-1}$ is to decompose it into a direct sum of bimodules $\bigoplus\limits_{S,T}M_{S,T}$, with each $M_{S,T}$ being left and right semisimple and iso-typical (isomorphic to powers of some $S$ as a left module and some $T$ as a right module). Then one essentially needs to compare left and right lengths of certain $\Delta-\Delta'$-bimodules for some division algebras $\Delta$ and $\Delta'$.
[99]{}
D. Anderson, K. Fuller, *Rings and Categories of Modules*, Grad. Texts in Math., Springer, Berlin-Heidelberg-New York, 1974.
P. Ara, W.K. Nicholson, M.F. Yousif, *A look at the Faith-Conjecture*, Glasgow Math. J. 42 (2001), 391–404.
P. Ara, J.K.Park, *On continuous semiprimary rings*, Comm. Algebra 19, no.7 (1991), 1945–1957.
J. Clark, D.V. Huynh, *A note on perfect left self-injective rings*, Quart. J. Math. Oxford 45 (1994), 13–17.
J. Clark, D.V. Huynh, *On semiartinian modules and injective modules*, Proc. Edinburgh Math. Soc. 39 (1996), 263–270.
C. Faith, *Algebra II: Ring Theory*. Vol 191, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
C. Faith, D. van Huynh, *When self-injective rings are QF: a report on a problem*, J. Alg. Appl. Vol. 1, No.1 (2002), 75–105.
T. Kato, *Self-injective rings*, Tohoku Math. J. 19, No. 4 (1967).
K. Koike, *On self-injective semiprimary rings*, Comm. Algebra 28 (2000), 4303–4319.
J. Lawrence, *A countable self-injective ring is Quasi-Frobenius*, Proc. Amer. Math. Soc. 65 (1977), 217–220.
T-Y. Lam, *Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Berlin - New York, Springer-Verlag (1999).
W.K. Nicholson, M.F. Yousif, *Quasi-Frobenius rings*, Cambridge University Press (2003).
W.K. Nicholson, M.F. Yousif, *Miniinjective rings*, J. Algebra (1997), 548–578.
W.K. Nicholson, M.F. Yousif, *On finitely embedded rings*, Comm. Algebra 28 (2000), 5311–5315.
B. Osofsky, *A generalization of Quasi-Frobenius rings*, J. Algebra 4 (1966), 373–387.
W. Xue, *A note on perfect self-injective rings*, Comm. Algebra 24 (1996), 749–755.
|
---
abstract: 'We review a method that we recently introduced to calculate the finite-temperature distribution of the axial quadrupole operator in the laboratory frame using the auxiliary-field Monte Carlo technique in the framework of the configuration-interaction shell model. We also discuss recent work to determine the probability distribution of the quadrupole shape tensor as a function of intrinsic deformation $\beta,\gamma$ by expanding its logarithm in quadrupole invariants. We demonstrate our method for an isotope chain of samarium nuclei whose ground states describe a crossover from spherical to deformed shapes.'
address: |
$^{1}$Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520, USA\
$^{2}$Institute of Nuclear Theory, Box 351550, University of Washington, Seattle, WA 98915\
$^{3}$Department of Physics, Box 351560, University of Washington, Seattle, WA 98195
author:
- 'Y. Alhassid,$^{1}$ G.F. Bertsch$^{2,3}$ C.N. Gilbreth,$^{2}$ and M.T. Mustonen$^{1}$'
title: 'Nuclear deformation in the configuration-interaction shell model'
---
Introduction
============
Deformation is a central concept in understanding the physics of heavy nuclei [@BM75]. However, since intrinsic deformation is introduced by invoking a mean-field approximation that breaks rotational symmetry, it is a challenge to determine the probability density of the intrinsic deformation in the configuration-interaction (CI) shell model, a framework that preserves rotational symmetry.
Here we review a recent technique we introduced to calculate the axial quadrupole distribution in the laboratory frame using the auxiliary-field Monte Carlo (AFMC) method [@al14; @gi17]. We found that this lab-frame distribution exhibits a model-independent signature of deformation. We then discuss recent work in which we used quadrupole invariants [@ku72; @cl86] to model the quadrupole shape distribution in the intrinsic frame [@mu17]. We demonstrate our method for an isotope chain of samarium nuclei, using the model space and interaction of Refs. [@al08; @oz13]. Quadrupole invariants were used to extract the effective intrinsic deformation within the framework of the CI shell model in lighter nuclei; see the recent examples in Refs. [@ha16; @sc17] and references therein.
Auxiliary-field Monte Carlo method
==================================
AFMC, also known in the context of the nuclear shell model as the shell model Monte Carlo (SMMC) method [@la93; @al94; @al17], is based on the Hubbard-Stratonovich (HS) transformation [@hu59]. The Gibbs operator $e^{-\hat H/T}$ of a nucleus described by the Hamiltonian $\hat H$ at temperature $T$ is represented as a superposition of non-interacting propagators $\hat U_\sigma$ of nucleons moving in auxiliary fields $\sigma=\sigma(\tau)$ that depend on imaginary time $\tau$ \[HS\] e\^[-H/T]{} = D \[\] G\_U\_, where $G_\sigma$ is a Gaussian weight. The thermal expectation value of an observable $\hat O$ can then be written as \[observable\] O = [ [Tr]{}(O e\^[-H/T]{}) e\^[-H/T]{}]{}=[[\[\] G\_O \_U\_\[\] G\_U\_]{}]{} , where $\langle \hat O \rangle_\sigma\equiv {\rm Tr} \,( \hat O \hat U_\sigma)/ {\rm Tr}\,\hat U_\sigma$. The integrands in Eq. (\[observable\]) can be calculated using matrix algebra in the single-particle space, and the integration over the large number of auxiliary fields $\sigma(\tau)$ is carried out by Monte Carlo methods. Canonical expectation values at fixed number of protons and neutrons are calculated using a discrete Fourier representation of the particle-number projection [@or94; @al99].
Quadrupole distribution in the laboratory frame
===============================================
The lab-frame distribution $P(q)$ of the axial quadrupole $\hat Q_{20}$ at temperature $T$ is defined by \[Pq\] P(q )= [ [Tr]{} / [Tr]{} e\^[-H/T]{}]{} . Using compete sets of many-particle eigenstates $| e_m\rangle$ and $| q_n \rangle$ of $\hat H$ and $\hat Q_{20}$, respectively, we have (note that $[H,\hat Q_{20}] \ne 0$) $$\label{prob1}
P(q) = \sum_n \delta(q - q_n) \sum_m \langle q_n |e_m \rangle^2 e^{-e_m/T} \;.$$ In the CI shell model, the spectrum of $\hat Q_{20}$ is discrete, but for a heavy nucleus it becomes a quasi-continuum.
Projection on the axial quadrupole
----------------------------------
To carry out the projection in AFMC, we represent the $\delta$ function as a Fourier integral $$\label{delta-q}
\delta(\hat Q_{20} - q) = {1 \over 2 \pi} \int_{-\infty}^\infty d \varphi \, e^{-i \varphi q }\, e^{i \varphi \hat Q_{20}} \,,$$ and use this in (\[Pq\]) together with the HS transformation (\[HS\]) for $e^{-\hat H/T}$. For each configuration $\sigma$ of the auxiliary fields, we replace the Fourier integral by a discrete Fourier transform. Choosing an interval $[-q_{\rm max}, q_{\rm max}]$, dividing it into $2M+1$ intervals of equal length $\Delta q = 2q_{\rm max} /(2M+1)$, and defining $q_m = m\Delta q$, we have $$\label{fourier-q}
\Tr\left[\delta(\hat Q_{20} - q_m) \hat U_\sigma \right] \approx {1\over 2 q_{\rm max}} \sum_{k=-M}^M
\!\! e^{-i \varphi_k q_m} \Tr(e^{i \varphi_k \hat Q_{20}} \hat U_\sigma) \;,$$ where $\varphi_k = \pi k/q_{\rm max}$ ($k=-M,\ldots, M$). Since $\hat Q_{20}$ is a one-body operator, we can calculate the grand-canonical traces on the r.h.s. of (\[fourier-q\]) in terms of the matrices ${\bf Q}_{20}$ and ${\bf U}_\sigma$ representing, respectively, $\hat Q_{20}$ and $\hat U_\sigma$ in the single-particle space, i.e., $\Tr\left(e^{i\varphi_k \hat Q_{20}} \hat U_\sigma \right) =
\det \left( 1+ e^{i \varphi_k {\bf Q}_{20}} {\bf U}_\sigma\right)$.
Angle averaging
---------------
When using the usual Metropolis algorithm, we find that for a deformed nucleus, the distribution $P(q)$ and its moments are slow to thermalize and have a large decorrelation length. We resolved this problem by averaging over a specific set of rotation angles $\Omega_j$ e\^[i Q\_[20]{}]{} \_ \_[j=1]{}\^[N\_]{} e\^[i Q\_[20]{}]{} \_[,\_j]{} , where $\langle e^{i \varphi \hat Q_{20}} \rangle_{\sigma,\Omega} = {\Tr \left( e^{i \varphi \hat Q_{20}} \, \hat R \hat U_\sigma \hat R^\dagger \right)}/{\Tr \left (\hat R \hat U_\sigma \hat R^\dagger \right )}$, with $\hat R=\hat R(\Omega)$ being the rotation operator with angles $\Omega$. We note that any rotation of $\hat Q_{20}$ in (\[delta-q\]) does not affect the distribution $P(q)$ since the Hamiltonian $\hat H$ is invariant under rotations. The angles $\Omega_j$ are chosen such that $\hat Q_{20}^m$ is proportional to the invariant of order $m$ up to a given order $n$. We have determined a set of 6 angles for $n=2$ and a set of 21 angles for $n=3$ [@gi17]. All calculations shown here are based on a 21-angle average. We used a time slice of $\Delta \beta=1/64$ MeV$^{-1}$ in a discretized version of the HS transformation (\[HS\]) and $\sim 5000$ auxiliary-field configurations for each temperature.
Application to samarium isotopes
--------------------------------
In Fig. 1 we show AFMC distributions $P(q)$ for an isotope chain of samarium nuclei $^{148-154}$Sm at low, intermediate and high temperatures. We observed that the low-temperature distribution for $^{154}$Sm, whose Hartree-Fock-Bogoliubov (HFB) ground state is deformed, is skewed and in qualitative agreement with the distribution for a prolate rigid rotor (dashed line). In contrast, the low-temperature distribution for the spherical nucleus $^{148}$Sm is close to a Gaussian. We conclude that the axial quadrupole distribution in the lab frame is a model-independent signature of deformation. At low temperatures, we observe a crossover from a spherical to a prolate shape as we increase the number of neutrons. In the isotopes that are deformed at low temperature ($^{150-154}$Sm), we observe a crossover to a spherical shape as we increase $T$.
\[Sm\_Pq\] ![ AFMC distributions $P(q)$ vs. $q$ (blue circles) for an isotope chain of samarium nuclei at high, intermediate and low temperatures. The dashed lines are rigid-rotor distributions. The red solid lines are the distributions obtained from (\[landau\]) (see Sec. \[landau\_expansion\]). Adapted from Ref. [@gi17].](Sm_zvq_therm.eps "fig:"){width="95.00000%"}
Quadrupole distribution in the intrinsic frame
==============================================
In physical applications, we are interested in the intrinsic deformation of the nucleus. Information on intrinsic deformation can be extracted without invoking a mean-field approximation by using quadrupole invariants which are frame-independent.
Quadrupole invariants and their relation to moments of $\hat Q_{20}$.
---------------------------------------------------------------------
A quadrupole invariant is a linear combination of products of the components $\hat Q_{2\mu}$ that is invariant under rotations. These invariants can be constructed from tensor products of the second-rank quadrupole tensor $\hat Q_{2\mu}$ [@ku72; @cl86]. For any given order $2\le n \le 4$, these invariants are unique and their expectation values can be expressed in terms of the corresponding moments of $\hat Q_{20}$: \[q\_invariants\] = 5 \_[20]{}\^2 , ( )\^[(2)]{} = - 5 \_[20]{}\^3 , ( )\^2 = \_[20]{}\^4 . For given values $q_{2\mu}$ of the quadrupole tensor, we define dimensionless quadrupole deformation parameters $\alpha_{2\mu}$ as in the liquid drop model, i.e., $q_{2\mu} = \frac{3}{\sqrt{5 \pi}} 3 r_0^2 A^{5/3} \alpha_{2\mu}$, where $r_0 = 1.2$ fm and $A$ is the mass number of the nucleus. For each set of deformation parameters $\alpha_{2\mu}$, we define an intrinsic frame whose orientation is characterized by Euler angles $\Omega$, and in which the deformation parameters $\tilde \alpha_{2\mu}$ are given by \_[20]{} = , \_[21]{} = \_[2,-1]{} = 0, \_[22]{} = \_[2,-2]{} = [real]{}= . The parameters $\beta,\gamma$ are known as the Hill-Wheeler parameters. The metric of the transformation from the lab-frame variables $\alpha_{2\mu}$ to the intrinsic-frame variables $\beta,\gamma,\Omega$ is given by \[metric\] \_d [\_[2]{}]{}= \^4 |(3)| d d d . Quadrupole invariants can also be constructed from $\alpha_{2\mu}$, and up to fourth order, they are given by \[alpha\_invariants\] = \^2 , \[\]\_2 = - \^3 (3) , ()\^2 = \^4 .
Landau-like expansion {#landau_expansion}
---------------------
The distribution $P(T,\alpha_{2\mu})$ of the quadrupole deformation $\alpha_{2\mu}$ at temperature $T$ is a rotational invariant and therefore it depends only on the intrinsic parameters $\beta,\gamma$. Using a Landau-like expansion [@al86], we expand the logarithm of $P$ in the quadrupole invariants up to fourth order \[landau\] P(T, , ) = (T) e\^[ -a(T) \^2 - b(T) \^3 (3) - c(T) \^4 ]{}, where $a,b,c$ are temperature-dependent coefficients and $\mathcal{N}$ is a normalization constant determined from $4\pi^2 \int \,d {\beta} \,d {\gamma} \, \beta^4 |\sin (3\gamma)| P(T,\beta,\gamma) =1$. The parameters $a,b,c$ are determined by matching the expectation values \[calculated with the distribution (\[landau\])\] of the three quadrupole invariants as expressed in (\[alpha\_invariants\]) with their AFMC values, which can be computed from the corresponding moments of $P(q)$ using Eqs. (\[q\_invariants\]).
Validation of the Landau-like expansion
---------------------------------------
To test the validity of (\[landau\]), we construct the lab-frame distribution $P(T,\alpha_{2\mu})$ by expressing the quadrupole invariants in terms of the lab-frame deformation $\alpha_{2\mu}$ \[see Eq. (\[alpha\_invariants\])\]. We then integrate over all $\alpha_{2\mu}$ with $\mu \ne 0$ to find the lab-frame distribution of $\alpha_{20}$, or equivalently $P(q)$, and compare it with the AFMC distribution. The distributions $P(q)$ calculated from the model (\[landau\]) are shown by the solid red lines in Fig. 1 and are in excellent agreement with the AFMC distributions (open blue circles).
Application to samarium isotopes
--------------------------------
![\[probability\_distribution\] Intrinsic shape distributions $P(T,\beta,\gamma)$ at low, intermediate and high temperatures for the even-mass samarium isotopes $^{148-154}$Sm. Adapted from Ref. [@mu17].](probability_distribution.pdf){width="\textwidth"}
Figure \[probability\_distribution\] shows the calculated shape distributions $P(T,\beta,\gamma)$ defined in (\[landau\]) vs. $\beta,\gamma$ for the samarium isotopes at the same temperatures as in Fig. 1. The maxima of the distributions (\[landau\]) mimic the shape transition observed in the HFB mean-field approximation but in the framework of the CI shell model [@note]. As a function of neutron number we observe a transition from a spherical to prolate shape, while nuclei that are deformed in their ground state make a transition from deformed to spherical shape as a function of temperature.
![\[illustration\_shapes\] Definition of a spherical, prolate and oblate shape regions in the $\beta-\gamma$ plane. These regions are used in presenting the results of Fig. \[pshapes\]. Taken from Ref. [@mu17].](illustration_shapes.pdf){width="65.00000%"}
To simplify the presentation of our results we divide the $\beta$-$\gamma$ plane into the three regions as shown in Fig. \[illustration\_shapes\], which we choose to represent spherical, prolate and oblate shapes. For each region, we then define $P_{\rm shape}(T)$ to be the probability to find the nucleus in the corresponding region, i.e., $P_{\rm shape}(T) = 4\pi^2 \int_{\rm shape} \,d\beta\,d\gamma \, \beta^4|\sin 3\gamma| P(T,\beta,\gamma)$. In Fig. \[pshapes\] we show these probabilities as a function of temperature $T$ for the four even-mass samarium isotopes. In the spherical $^{148}$Sm, the spherical region dominates at all temperatures, while in the deformed $^{152,154}$Sm isotopes, the prolate region has a probability close to $1$ at low temperatures and the spherical region becomes the most probable above a certain temperature. The transitional nucleus$^{150}$Sm exhibits an intermediate behavior.
![\[pshapes\] Probabilities of spherical (red open circles), prolate (green solid circles), and oblate (blue pluses) regions as a function of $T$ for $^{148-154}$Sm isotopes. Adapted from Ref. [@mu17].](pshapes.pdf){width="\textwidth"}
Conclusion and outlook
======================
We discussed a method we recently introduced to calculate lab-frame and intrinsic shape distributions within the CI shell model without invoking a mean-field approximation. Using the saddle-point approximation, it is also possible to convert the finite-temperature intrinsic shape distribution (\[landau\]) to level densities $\rho(E_x,\beta,\gamma)$ as a function of excitation energy $E_x$ and intrinsic deformation $\beta,\gamma$ [@mu17]. Deformation-dependent level densities are useful in the modeling of nuclear shape dynamics, such as fission.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the U.S. DOE grant Nos. DE-FG02-91ER40608 and DE-FG02-00ER41132. The research presented here used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work was also supported by the HPC facilities operated by, and the staff of, the Yale Center for Research Computing.
References {#references .unnumbered}
==========
Bohr A and Mottelson B R 1975 [*Nuclear Structure*]{} vol II (Reading, MA: Benjamin) Alhassid Y, Gilbreth C N and Bertsch G F 2014 [*Phys. Rev. Lett.*]{} [**113**]{} 262503 Gilbreth C N, Alhassid Y and Bertsch G F arXiv:1710.00072 Kumar K 1972 [*Phys. Rev. Lett.*]{} [**28**]{} 249 Cline D 1986 [*Ann. Rev. Nucl. Part. Sci.*]{} [**36**]{} 683 Mustonen M T, Gilbreth C N, Alhassid Y and Bertsch G F, to be published. Alhassid Y, Fang L and Nakada H 2008 [*Phys. Rev. Lett.*]{} [**101**]{} 082501 Özen C, Alhassid Y and Nakada H 2013 [*Phys. Rev. Lett.*]{} [**110**]{} 042502 Hadyńska-Klek K [*et al.*]{} 2016 [*Phys. Rev. Lett.*]{} [**117**]{} 062501 Schmidt T, Heyde K L G, Blazhev A and Jolie J 2017 [*Phys. Rev.*]{} C [**96**]{} 014302 Lang G H, Johnson C W, Koonin S E and Ormand W E 1993 [*Phys. Rev.*]{} C [**48**]{} 1518 Alhassid Y, Dean D J, Koonin S E, Lang G, and Ormand W E 1994 [*Phys. Rev. Lett.*]{} [**72**]{} 613 For a recent review, see Alhassid Y in 2017 [*Emergent Phenomena in Atomic Nuclei from Large-Scale Modeling: a Symmetry-Guided Perspective*]{}, ed K D Launey (Singapore: World Scientific) Hubbard J 1959 [*Phys. Rev. Lett.*]{} [**3**]{} 77; Stratonovich R L 1957 [*Sov. Phys. - Dokl.*]{} [**115**]{} 1097 Ormand W E, Dean D J, Johnson C W, Lang G H and Koonin S E 1994 [*Phys. Rev.*]{} C [**49**]{} 1422 Alhassid Y, Liu S and Nakada H 1999 [*Phys. Rev. Lett.*]{} [**83**]{} 4265 Alhassid Y, Levit S and Zingman J 1986 [*Phys. Rev. Lett.*]{} [**57**]{} 539 Note, however, that the maxima of the probability density distribution in $\beta,\gamma$, given by $4\pi^2 \beta^4|\sin 3\gamma| P(T,\beta,\gamma)$, do not exhibit sharp shape transitions.
|
---
author:
- 'Nicolau C. Saldanha'
title: |
The homotopy and cohomology of spaces\
of locally convex curves in the sphere — II
---
|
---
abstract: 'An action for supersymmetric D0-branes in curved backgrounds is obtained by dimensional reduction of N=1 ten-dimensional supergravity coupled to super Yang-Mills system to 0+1 dimensions. The resultant action exhibits the coset-space symmetry $\frac{SO(9,9+n)}{SO(9)\times SO(9+n)}\times U(1)$ where $n=N^{2}-1$ is the dimension of the SU(N) gauge group.'
---
CAMS/99-03\
**Ali H. Chamseddine**\
0.5cm *Center for Advanced Mathematical Sciences\
and\
Physics Department\
American University of Beirut\
Beirut, Lebanon*\
chams@aub.edu.lb 0.2cm
Introduction
============
Very little is known about D-branes [@polchinski], [@A.; @Sen], [@susskind], in curved backgrounds [@Douglas] and what restrictions, if any, should be imposed on such backgrounds . Supersymmetry imposes constraints on the background metric. In the case of the superstring with world sheet supersymmetry, there is a direct relation between the number of supersymmetries and the background metric [@Frohlich]. In flat background geometry the D0-brane action has 16 space-time supersymmetries, and it is natural to ask for the type of curved backgrounds compatible with this symmetry. The action with 8 space-time supersymmetries ($N=2$ in 4 dimensions ) was shown to correspond to Kähler backgrounds [@Kato].
In general it is difficult to construct such actions without having determined the underlying symmetry. There are few possible routes to handle this problem, the most obvious one is to quantize the D-brane action in the presence of a general superspace metric, thus keeping all supersymmetries, and to determine what are the required constraints on such a metric [@Sezgin], [@hoppe]. This is expected to be extremely complicated and only recently some work has been done in this direction [@wati]. The other possibility is to determine the necessary fields to make a supersymmetric multiplet and to find the corresponding invariant action under such transformations. We shall follow a simpler approach, which is straightforward but which the drawback that the underlying symmetry is not manifest. The idea is based on the observation that the supersymmetric D-0 brane action was obtained by dimensionally reducing the super Yang-Mills action from ten to 0+1 dimensions [@hoppe]. The most general supersymmetric interaction in ten dimensions with $N=1$ supersymmetry is that of supergravity coupled to super Yang-Mills with an arbitrary gauge group [@chams]. Dimensional reduction keeps maximal supresymmetry and rearranges the scalar fields to have the action of a non-linear sigma model on a coset space. Reducing to 0+1 dimensions have the peculiarity that there is no gravitational part and the only fields coming from the gravity sector in ten-dimensions yields scalar fields. Similarly the vector fields coming from the Yang-Mills part give scalar fields taking values in the adjoint representation of the gauge group. As with compactification of supergravity theory it is expected that the coset space to be of the form [@coset] $$\frac{SO(9,9+n)}{SO(9)\times SO(9+n)}\times U(1)$$ where $n=N^{2}-1$ is the dimension of $SU(N)$ gauge group. The absence of the gravitational sector in 0+1 dimensions also enables us to identify the gravitational coupling in higher dimensions with the string tension to insure that all rescaled scalar fields have the same dimensions and could serve as coordinates of the D-brane.
The aim of this letter is to derive the D0-brane action with the most general curved background compatible with maximal space-time supersymmetry. This is done by compactifying the ten-dimensional theory and grouping all the resultant fields. The result obtained does not have manifest symmetry. Nonetheless, this suggests that a direct derivation in terms of supermultiplets, where some auxiliary fields as well as constrained variables are used, might drastically simplify the answer. This is the situation encountered in the derivation of the four-dimensional $N=4$ supersymmetric action where superconformal methods were used to simplify the analysis [@thesis]. This more systematic approach will be left for the future and our study here will be limited to the action obtained by dimensional reduction. The plan of this paper is as follows. In section 2 we derive the bosonic dimensionally reduced action and in section 3 we give the fermionic part. Section four includes comments on the results.
Bosonic action of curved D0-branes.
===================================
Our starting point is the $N=1$ supergravity Lagrangian in ten dimensions coupled to super Yang-Mills system. This is given by (up to quartic fermionic terms) [@chams] $$\begin{aligned}
\det \left( e_{M}^{A}\right) ^{-1}L &=&-\frac{1}{4\kappa ^{2}}R(\omega )-%
\frac{i}{2}\overline{\psi }_{M}\Gamma ^{MNP}D_{N}\psi _{P}+\frac{1}{2\kappa
^{2}}\partial _{M}\phi \partial _{N}\phi g^{MN} \\
&&+\frac{i}{2}\overline{\chi }\Gamma ^{M}D_{M}\chi +\frac{1}{\sqrt{2}}%
\partial _{M}\phi \overline{\psi }_{M}\chi +e^{-2\phi }F_{MNP}^{\prime
}F_{QRS}^{\prime }g^{MQ}g^{NR}g^{PS} \\
&&+\frac{i\kappa }{24}e^{-\phi }\overline{\psi }_{M}\left( \Gamma
^{MNPQR}-6g^{MP}\Gamma ^{Q}g^{RN}\right) \psi _{N}F_{PQR}^{\prime } \\
&&-\frac{1}{4}e^{-\phi }Tr\left( G_{MN}G_{PQ}\right) g^{MP}g^{NQ}+\frac{i}{2}%
Tr\left( \overline{\lambda }\Gamma ^{M}D_{M}\lambda \right) \\
&&-\frac{i\kappa }{2\sqrt{2}}Tr\left( \overline{\lambda }\Gamma ^{M}\Gamma
^{NP}\psi _{M}G_{NP}\right)\end{aligned}$$ where $F_{MNP\ }^{\prime }$is the field strength of the antisymmetric tensor $B_{MN}$ modified by the gauge Chern-Simons three form. $$F_{MNP}^{\prime }=F_{MNP}+\omega _{MNP}^{\left( CS\right) }$$ and $F_{MNP}=\frac{3}{\kappa }\partial _{\left[ M\right. }B_{NP\left.
{}\right] }$, while $$\omega _{MNP}^{\left( CS\right) }=6\kappa Tr\left( A_{\left[ M\right.
}\partial _{N}A_{P\left. {}\right] }+\frac{2}{3}A_{\left[ M\right.
}A_{N}A_{\left. P\right] }\right)$$ We can reduce this action from $10$ to $d$ dimensions with the following distribution of fields. The metric $g_{MN}$ gives a metric $g_{\mu \nu }$, $%
m $ vectors and $\frac{1}{2}m\left( m+1\right) $ scalars in $d$ dimensions, where $m=10-d$. The antisymmetric tensor $B_{MN}$ gives $B_{\mu \nu }$, $m$ vectors and $\frac{1}{2}m\left( m-1\right) $ scalars. The gauge fields $%
A_{M}^{i}$ give $A_{\mu }^{i}$, and $nm$ scalars $A_{m}^{i}$, where $n$ is the dimension of the gauge group. All in all we will have $m\left(
n+m\right) $ scalars which will span the coset space [@coset] $$\frac{SO\left( m,n+m\right) }{SO\left( m\right) \times SO\left( n+m\right) }$$ The case when $d=4$ (i.e. $m=6$) is well established [@chams]. The case we are interested in have $\ m=9$.
To reduce this action to 0+1 dimensions we decompose: $$e_{M}^{A}=\left(
\begin{array}{cc}
e_{\stackrel{.}{0}}^{0} & B_{\stackrel{.}{0}}^{a} \\
0 & e_{m}^{a}
\end{array}
\right)$$ The inverse metric is $$e_{A}^{M}=\left(
\begin{array}{cc}
e_{0}^{\stackrel{.}{0}} & e_{0}^{m} \\
0 & e_{a}^{m}
\end{array}
\right)$$ where $e_{0}^{\stackrel{.}{0}}=\left( e_{\stackrel{.}{0}}^{0}\right) ^{-1}$, $e_{a}^{m}e_{m}^{b}=\delta _{a}^{b}$, $e_{0}^{m}=-e_{0}^{\stackrel{.}{0}}B_{%
\stackrel{.}{0}}^{a}e_{a}^{m}.$
To evaluate $-\frac{1}{4}\det \left( e_{M}^{A}\right) R(\omega )$ we have [@cremmer] $$\frac{1}{4}\det \left( e_{M}^{A}\right) R(\omega )=\frac{1}{16}e\left(
\Omega _{ABC}\Omega ^{ABC}-2\Omega _{ABC}\Omega ^{CAB}-4\Omega _{CA}^{\quad
A}\Omega _{\quad B}^{CB}\right)$$ where $\Omega _{ABC}=-e_{A}^{M}e_{B}^{N}(\partial _{M}e_{N}^{C}-\partial
_{N}e_{M}^{C})$. The only non-vanishing $\Omega _{ABC}$ is $$\Omega _{0bc}=-e_{0}^{\stackrel{.}{0}}e_{b}^{n}\partial _{\stackrel{.}{0}%
}e_{nc}$$ Substituting this gives $$-\frac{1}{4}\det \left( e_{M}^{A}\right) R(\omega )=\frac{e}{8}\left( -\frac{%
1}{2}\partial _{\stackrel{.}{0}}g_{mn}\partial _{\stackrel{.}{0}%
}g^{mn}-2\left( \partial _{\stackrel{.}{0}}\ln e\right) ^{2}\right)$$ where $e=\det (e_{m}^{a})$. The antisymmetric tensor piece gives $$\frac{e}{12}e_{\stackrel{.}{0}}^{0}e^{-2\phi }\left( 3F_{ab0}^{\prime
}F_{ab0}^{\prime }-F_{abc}^{\prime }F_{abc}^{\prime }\right)$$ where we are raising and lowering tangent space indices with the euclidean metric $\delta _{ab}$, and $$F_{abc}^{\prime }=4\kappa e_{a}^{m}e_{b}^{n}e_{c}^{p}Tr\left( A_{\left[
m\right. }A_{n}A_{\left. p\right] }\right)$$
$$F_{ab0}^{\prime }=\frac{1}{\kappa }e_{a}^{m}e_{b}^{n}\left( \partial _{%
\stackrel{.}{0}}B_{mn}-\kappa ^{2}Tr\left( A_{m}D_{\stackrel{.}{0}%
}A_{n}-A_{n}D_{\stackrel{.}{0}}A_{m}\right) \right)$$ where $D_{\stackrel{.}{0}}A_{m}=\partial _{\stackrel{.}{0}}A_{m}+\left[ A_{_{%
\stackrel{.}{0}}}^{\prime },A_{m}\right] $, and $A_{_{\stackrel{.}{0}%
}}^{\prime }=A_{_{\stackrel{.}{0}}}-B_{_{\stackrel{.}{0}}}^{m}A_{m}$. The redefinition of $A_{_{\stackrel{.}{0}}}$ will insure that the field $B_{_{%
\stackrel{.}{0}}}^{m}$ will not appear in the action and therefore is irrelevant.
The Yang-Mills part is $$-\frac{1}{4}\det \left( e_{M}^{A}\right) e^{-\phi }Tr\left(
G_{MN}G_{PQ}g^{MN}g^{PQ}\right) =-\frac{e}{4}e_{\stackrel{.}{0}}^{0}e^{-\phi
}Tr\left( G_{ab}G_{ab}-2G_{a0}G_{a0}\right)$$ where $$G_{ab}=e_{a}^{m}e_{b}^{n}\left[ A_{m},A_{n}\right]$$ $$G_{a0}=-e_{a}^{m}e_{0}^{\stackrel{.}{0}}D_{\stackrel{.}{0}}A_{m}$$ Grouping all terms together we obtain the bosonic part of the D0-brane action: $$\begin{aligned}
e^{-1}L_{b} &=&\frac{1}{\kappa ^{2}}\left( e_{0}^{\stackrel{.}{0}}\left( -%
\frac{1}{16}\partial _{\stackrel{.}{0}}g_{mn}\partial _{\stackrel{.}{0}%
}g^{mn}-\frac{1}{4}\left( \partial _{\stackrel{.}{0}}\ln e\right) ^{2}+\frac{%
1}{2}\partial _{\stackrel{.}{0}}\phi \partial _{\stackrel{.}{0}}\phi \right.
\right. \\
&&\qquad \qquad \left. +\frac{1}{4}e^{-2\phi }g^{mp}g^{nq}D_{\stackrel{.}{0}%
}B_{mn}D_{\stackrel{.}{0}}B_{pq}+\frac{\kappa ^{2}}{2}e^{-\phi
}g^{mn}Tr\left( D_{\stackrel{.}{0}}A_{m}D_{\stackrel{.}{0}}A_{n}\right)
\right) \\
&&\qquad +e_{\stackrel{.}{0}}^{0}\left( -\frac{4\kappa ^{2}}{3}e^{-2\phi
}g^{mq}g^{nr}g^{ps}Tr\left( A_{\left[ m\right. }A_{n}A_{p\left. {}\right]
}\right) Tr\left( A_{\left[ q\right. }A_{r}A_{s\left. {}\right] }\right)
\right. \\
&&\qquad \qquad \qquad \left. \left. -\frac{1}{4}e^{-\phi
}g^{mq}g^{nr}Tr\left( \left[ A_{m,}A_{n}\right] \right) Tr\left( \left[
A_{q,}A_{r}\right] \right) \right) \right)\end{aligned}$$ where $D_{\stackrel{.}{0}}B_{mn}=\partial _{\stackrel{.}{0}}B_{mn}-\kappa
^{2}Tr\left( A_{m}D_{\stackrel{.}{0}}A_{n}-A_{n}D_{\stackrel{.}{0}%
}A_{m}\right) $. To get the correct dimensions we identify the gravitational coupling $\kappa $ with the string tension $\alpha ^{\prime }$ and redefine the gauge fields $A_{m}^{i}=\frac{1}{\alpha ^{\prime }}X_{m}^{i},$thus identifying them with the D0-brane coordinates. Multiplying the Lagrangian with an overall factor of $\left( \alpha ^{\prime }\right) ^{2}$ gives the bosonic part of the D0-brane action. The 82 fields $g_{mn}$, $B_{mn}$ and $%
\phi $ are needed, beside the coordinates $X_{m}^{i}$, to provide coordinates for a D0-brane action with a curved background. The rescaled bosonic Lagrangian becomes $$\begin{aligned}
e^{-1}L_{b} &=&\left( e_{0}^{\stackrel{.}{0}}\left( -\frac{1}{16}\partial _{%
\stackrel{.}{0}}g_{mn}\partial _{\stackrel{.}{0}}g^{mn}-\frac{1}{4}\left(
\partial _{\stackrel{.}{0}}\ln e\right) ^{2}+\frac{1}{2}\partial _{\stackrel{%
.}{0}}\phi \partial _{\stackrel{.}{0}}\phi \right. \right. \\
&&\qquad \qquad \left. +\frac{1}{4}e^{-2\phi }g^{mp}g^{nq}D_{\stackrel{.}{0}%
}B_{mn}D_{\stackrel{.}{0}}B_{pq}+\frac{\kappa ^{2}}{2}e^{-\phi
}g^{mn}Tr\left( D_{\stackrel{.}{0}}X_{m}D_{\stackrel{.}{0}}X_{n}\right)
\right) \\
&&\qquad +e_{\stackrel{.}{0}}^{0}\left( -\frac{4}{3\left( \alpha ^{\prime
}\right) ^{2}}e^{-2\phi }g^{mq}g^{nr}g^{ps}Tr\left( X_{\left[ m\right.
}X_{n}X_{p\left. {}\right] }\right) Tr\left( X_{\left[ q\right.
}X_{r}X_{s\left. {}\right] }\right) \right. \\
&&\qquad \qquad \qquad \left. \left. -\frac{1}{4\left( \alpha ^{\prime
}\right) ^{2}}e^{-\phi }g^{mq}g^{nr}Tr\left( \left[ X_{m,}X_{n}\right]
\right) Tr\left( \left[ X_{q,}X_{r}\right] \right) \right) \right)\end{aligned}$$
The scalar fields can be regrouped into a set $X_{m}^{R}$ where $R=1,\cdots
,9+n$ plus an additional scalar field as a combination of the fields $g_{mn}$, $B_{mn}$ , $\phi $ and $X_{m}^{i}.$Can one take the limit where $%
g_{mn}=\delta _{mn}$ $,$ $B_{mn}=0$ and $\phi =0$ ? This gives the flat background D-0 brane action plus the order 6 terms in $X_{m}^{i}.$This is usually incompatible with supersymmetry as we shall show later. The proper limit to flat backgrounds can be obtained by keeping the couplings $\kappa $ and $\alpha ^{\prime }$ independent, then taking the limit $\kappa
\rightarrow 0$.
The transformation law for $e_{\stackrel{.}{0}}^{0}$ with respect to time transformation is given by $$\delta e_{\stackrel{.}{0}}^{0}=\partial _{_{\stackrel{.}{0}}}\left( \xi
^{^{_{\stackrel{.}{0}}}}e_{\stackrel{.}{0}}^{0}\right)$$ which would allow us to set $e_{\stackrel{.}{0}}^{0}=1$.
In this action the coset space symmetry is not manifest. The coset space metric is a non-polynomial function of $g_{mn}$, $B_{mn}$, $X_{m}^{i}$ and $%
\phi .$ To obtain manifest symmetry, one method would be to start with the symmetry $SO(9,9+n)$ using supersymmetric multiplets, and then gauge the $%
SO(9)\times SO(9+n)$ subgroup. This will be the topic of a forthcoming project where a systematic analysis of all possible background symmetries would be carried out.
The fermionic action
====================
The Rarita-Schwinger term $$-\frac{i}{2}\det (e_{M}^{A})\overline{\psi }_{A}\Gamma ^{ABC}D_{B}\psi _{C}$$ where $\psi _{A}=e_{A}^{M}\psi _{M}$, and $D_{M}\psi _{N}=\left( \partial
_{M}+\frac{1}{4}\omega _{M}^{\quad AB}\Gamma _{AB}\right) \psi _{N{\ ,}}$gives upon compactification $$\begin{aligned}
&&-\frac{i}{8}e\left( \overline{\psi }_{a}\Gamma _{b}\psi _{d}\left(
e_{b}^{n}\partial _{\stackrel{.}{0}}e_{nd}+e_{d}^{n}\partial _{\stackrel{.}{0%
}}e_{nb}\right) -2\overline{\psi }_{0}\Gamma ^{c}\psi _{c}\left(
e_{d}^{n}\partial _{\stackrel{.}{0}}e_{nd}\right) \right) \\
&&+\frac{i}{2}e\left( \overline{\psi }_{a}\Gamma ^{ac}\Gamma _{0}\partial
_{_{\stackrel{.}{0}}}\psi _{c}+\frac{1}{4}\overline{\psi }_{a}\Gamma
^{ac}\Gamma _{de}\Gamma _{0}\psi _{c}\left( e_{d}^{n}\partial _{\stackrel{.}{%
0}}e_{ne}\right) \right) \\
&&+\frac{i}{4}ee_{0}^{\stackrel{.}{0}}\left( \overline{\psi }_{a}\Gamma
^{a}\psi _{0}\left( e_{b}^{n}\partial _{\stackrel{.}{0}}e_{nb}\right) +\frac{%
1}{2}\overline{\psi }_{a}\Gamma _{b}\psi _{0}\left( e_{b}^{n}\partial _{%
\stackrel{.}{0}}e_{na}+e_{a}^{n}\partial _{\stackrel{.}{0}}e_{nb}\right)
\right)\end{aligned}$$ where we have used $\psi _{0}=e_{0}^{_{_{\stackrel{.}{0}}}}\left( \psi _{_{_{%
\stackrel{.}{0}}}}-B_{_{_{\stackrel{.}{0}}}}^{a}e_{a}^{m}\psi _{m}\right) $ and $\psi _{a}=e_{a}^{m}\psi _{m}$. Again, the definition of $\psi _{0}$ insures that $B_{_{_{\stackrel{.}{0}}}}^{a}$ does not appear in the action. The nonvanishing components of the spin-connection are $$\begin{aligned}
\omega _{0bc} &=&\frac{1}{2}e_{0}^{\stackrel{.}{0}}\left( e_{b}^{n}\partial
_{\stackrel{.}{0}}e_{nc}-e_{c}^{n}\partial _{\stackrel{.}{0}}e_{nb}\right) \\
\omega _{ab0} &=&-\frac{1}{2}e_{0}^{\stackrel{.}{0}}\left( e_{b}^{n}\partial
_{\stackrel{.}{0}}e_{na}+e_{a}^{n}\partial _{\stackrel{.}{0}}e_{nb}\right)\end{aligned}$$ The term $\frac{i}{2}\det \left( e_{M}^{A}\right) \overline{\chi }\Gamma
^{A}D_{A}\chi $ reduces to $$-\frac{i}{4}e\left( e_{\stackrel{.}{0}}^{0}\overline{\chi }\Gamma ^{0}\chi
\left( e_{a}^{n}\partial _{\stackrel{.}{0}}e_{na}\right) -2\overline{\chi }%
\Gamma ^{0}\left( \partial _{\stackrel{.}{0}}+\frac{1}{4}e_{b}^{n}\partial _{%
\stackrel{.}{0}}e_{nc}\Gamma _{bc}\right) \chi \right)$$ Similarly the gaugino kinetic term gives upon reduction $$-\frac{i}{4}eTr\left( e_{\stackrel{.}{0}}^{0}\overline{\lambda }\Gamma
^{0}\lambda \left( e_{a}^{n}\partial _{\stackrel{.}{0}}e_{na}\right) -2%
\overline{\lambda }\Gamma ^{0}\left( D_{\stackrel{.}{0}}+\frac{1}{4}%
e_{b}^{n}\partial _{\stackrel{.}{0}}e_{nc}\Gamma _{bc}\right) \lambda
\right)$$
Next, the fermi-bose interaction $\overline{\psi }\psi F^{\prime }$ gives $$\begin{aligned}
&&\frac{i\kappa ^{2}}{24}e^{-\phi }ee_{\stackrel{.}{0}}^{0}\left( 4\left(
\overline{\psi }_{a}\Gamma ^{abcde}\psi _{b}+6\overline{\psi }_{0}\Gamma
^{0bcde}\psi _{b}-6\overline{\psi }_{c}\Gamma _{d}\psi _{e}\right)
TrA_{\left[ c\right. }A_{d}A_{\left. e\right] }\right. \\
&&\qquad \qquad \quad +\left( 3\overline{\psi }_{a}\Gamma ^{abcd0}\psi _{b}-2%
\overline{\psi }_{0}\Gamma _{c}\psi _{d}+2\overline{\psi }_{c}\Gamma
_{0}\psi _{d}-2\overline{\psi }_{c}\Gamma _{d}\psi _{0}\right) \\
&&\ \qquad \qquad \quad \left. \times e_{c}^{m}e_{d}^{n}e_{0}^{\stackrel{.}{0%
}}\left( \frac{1}{\kappa }\partial _{\stackrel{.}{0}}B_{mn}-\kappa Tr\left(
A_{m}D_{\stackrel{.}{0}}A_{n}-A_{n}D_{\stackrel{.}{0}}A_{m}\right) \right)
\right)\end{aligned}$$ Finally the $\overline{\psi }\chi \partial \phi $ coupling gives $$\frac{1}{2}e\partial _{_{\stackrel{.}{0}}}\phi \overline{\psi }^{0}\chi .$$
The supersymmetry transformations in ten dimensions are $$\begin{aligned}
\delta e_{M}^{A} &=&-i\kappa \overline{\epsilon \,}\Gamma ^{A}\psi _{M} \\
\delta \phi &=&-\frac{\kappa }{\sqrt{2}}\overline{\epsilon \,}\chi \\
\delta B_{MN} &=&\kappa e^{\phi }\left( i\overline{\epsilon }\,\Gamma
_{\left[ M\right. }\psi _{\left. N\right] }+\frac{1}{2}\overline{\epsilon }%
\,\Gamma _{MN}\chi \right) \\
\delta \psi _{M} &=&\frac{1}{\kappa }D_{M}\epsilon +\frac{1}{48}e^{-\phi
}\left( \Gamma _{\qquad M}^{NPQ}+9\delta _{M}^{N}\Gamma ^{PQ}\right)
\epsilon \,F_{NPQ}^{\prime } \\
\delta \chi &=&\frac{i}{2\kappa }\Gamma ^{M}\epsilon \,\partial _{M}\phi \\
\delta A_{M} &=&\frac{i}{\sqrt{2}}e^{\phi }\overline{\epsilon \,}\Gamma
_{M}\lambda \\
\delta \lambda &=&e^{-\frac{1}{2}\phi }\Gamma ^{MN}\epsilon \,G_{MN}\end{aligned}$$
The compactified supersymmetry transformations become after rescaling $$\begin{aligned}
\delta e_{\stackrel{.}{0}}^{0} &=&-i\kappa \overline{\epsilon }\,\Gamma
^{0}\psi _{\stackrel{.}{0}} \\
\delta e_{\stackrel{.}{0}}^{a} &=&-i\kappa \overline{\epsilon }\,\Gamma
^{a}\psi _{\stackrel{.}{0}} \\
\delta e_{m}^{a} &=&-i\overline{\epsilon }\,\Gamma ^{a}\psi _{m} \\
\delta \phi &=&-\frac{\kappa }{\sqrt{2}}\overline{\epsilon }\chi \\
\delta B_{mn} &=&\kappa e^{\phi }\left( i\overline{\epsilon }\,\Gamma
_{\left[ m\right. }\psi _{\left. n\right] }+\frac{1}{2\sqrt{2}}\overline{%
\epsilon }\,\Gamma _{mn}\chi \right) \\
\delta \psi _{0} &=&\frac{1}{\kappa }e_{0}^{\stackrel{.}{0}}\left( \partial
_{\stackrel{.}{0}}+\frac{1}{4}e_{a}^{m}\partial _{\stackrel{.}{0}%
}e_{mb}\Gamma ^{ab}\right) \epsilon \\
&&+\frac{1}{12}e^{-\phi }\Gamma ^{cd}\epsilon \,e_{c}^{m}e_{d}^{n}\left(
\frac{1}{\kappa }\partial _{\stackrel{.}{0}}B_{mn}-\frac{\kappa }{\left(
\alpha ^{\prime }\right) ^{2}}Tr\left( X_{m}D_{\stackrel{.}{0}}X_{n}-X_{n}D_{%
\stackrel{.}{0}}X_{m}\right) \right) \epsilon \\
&&+\frac{\kappa }{12\left( \alpha ^{\prime }\right) ^{3}}e^{-\phi }\Gamma
^{bcd}\Gamma _{0}\epsilon \,e_{b}^{m}e_{c}^{n}e_{d}^{p}Tr\left( X_{\left[
m\right. }X_{n}X_{\left. p\right] }\right) \\
\delta \psi _{a} &=&\frac{1}{4\kappa }e_{0}^{\stackrel{.}{0}}\left(
e_{a}^{n}\partial _{\stackrel{.}{0}}e_{nb}+e_{b}^{n}\partial _{\stackrel{.}{0%
}}e_{na}\right) \Gamma _{b0}\epsilon \\
&&-\frac{\kappa }{12\left( \alpha ^{\prime }\right) ^{3}}e^{-\phi }\left(
\Gamma _{abcd}-4\delta _{a}^{b}\Gamma _{cd}\right) \epsilon
\,e_{b}^{m}e_{c}^{n}e_{d}^{p}Tr\left( X_{\left[ m\right. }X_{n}X_{\left.
p\right] }\right) \\
&&+\frac{1}{16}e^{-\phi }\left( \Gamma _{0cda}+\frac{4}{3}\Gamma _{0c}\delta
_{d}^{a}\right) \epsilon \,e_{c}^{m}e_{d}^{n}e_{0}^{_{\stackrel{.}{0}%
}}\left( \frac{1}{\kappa }\partial _{\stackrel{.}{0}}B_{mn}-\frac{\kappa }{%
\left( \alpha ^{\prime }\right) ^{3}}Tr\left( X_{m}D_{\stackrel{.}{0}%
}X_{n}-X_{n}D_{\stackrel{.}{0}}X_{m}\right) \right) \\
\delta \chi &=&\frac{i}{\kappa \sqrt{2}}\Gamma ^{0}\epsilon \,e_{0}^{%
\stackrel{.}{0}}\partial _{\stackrel{.}{0}}\phi +\frac{i\kappa }{3\sqrt{2}%
\left( \alpha ^{\prime }\right) ^{3}}e^{-\phi }\Gamma ^{abc}\epsilon
\,e_{a}^{m}e_{b}^{n}e_{c}^{p}Tr\left( X_{\left[ m\right. }X_{n}X_{\left.
p\right] }\right) \\
&&+\frac{i}{4\sqrt{2}}\Gamma ^{ab0}\epsilon \,e_{a}^{m}e_{b}^{n}e_{0}^{_{%
\stackrel{.}{0}}}\left( \frac{1}{\kappa }\partial _{\stackrel{.}{0}}B_{mn}-%
\frac{\kappa }{\left( \alpha ^{\prime }\right) ^{3}}Tr\left( X_{m}D_{%
\stackrel{.}{0}}X_{n}-X_{n}D_{\stackrel{.}{0}}X_{m}\right) \right) \\
\delta A_{\stackrel{.}{0}} &=&\frac{i}{\alpha ^{\prime }\sqrt{2}}e^{\frac{1}{%
2}\phi }\overline{\epsilon }\,\Gamma _{\stackrel{.}{0}}\lambda \\
\delta X_{m} &=&\frac{i\alpha ^{\prime }}{\sqrt{2}}e^{\frac{1}{2}\phi }%
\overline{\epsilon }\,\Gamma _{m}\lambda \\
\delta \lambda &=&\frac{1}{\alpha ^{\prime }\sqrt{2}}e^{-\frac{1}{2}\phi
}\Gamma ^{a0}\epsilon \,e_{a}^{m}e_{0}^{\stackrel{.}{0}}D_{\stackrel{.}{0}%
}X_{m}\end{aligned}$$
From these transformations it should be clear that the truncation $%
g_{mn}=\delta _{mn}$, $B_{mn}=0$, $\phi =0$ is not consistent with supersymmetry because the fields $X_{m}^{i}$, $g_{mn}$, $B_{mn}$ and $\phi $ are now mixed to form $9(9+n)+1$ coordinates for the D-0 brane. A proper way of going to the flat background limit is to keep $\kappa $ and $\alpha
^{\prime }$ distinct, and then take the limit $\kappa \rightarrow 0$.
Comments
========
The D0-brane action with maximal $N=16$ space-time supersymmetry, derived here have the coset symmetry $\frac{SO(9,9+n)}{SO(9)\times
SO(9+n)}$ which, however, is not manifest. The 81 fields which are not related to the $SU(N)$ gauge group are essential to provide curvature for the background. We can say that curved backgrounds are only possible once $n$ gauge fields are embedded into the above coset structure. One way to improve on this solution is to start with the light-cone formulation of the supermembrane in arbitrary background and find under what conditions the quantized action simplifies in such a manner as not to involve a square root. This is a difficult problem and the recent work of weakly coupling D0-branes to curved backgrounds may help to clarify the situation [@wati]. The action constructed in [@wati] contains higher order time derivatives indicating that higher derivative terms in ten-dimensional supergravity should also be included. It would also be very interesting to find out how the 82 scalar fields arise in this formulation.
Another possibility is to study supersymmetry representations in 0+1 dimensions and form multiplets in complete analogy with the one obtained by superconformal methods in four dimensions. This would have the advantage of getting the coset space symmetry in a manifestly invariant way. In addition this would allow to investigate the general problem of finding the relation between the required degree of space-time supersymmetry and the nature of the curved background.
[99]{} J. Polchinski, *Phys.Rev. Lett.* **75** (1995) 4724.
A. Sen, *Adv. Theor. Math. Phys.* **2** (1998) 51.
T. Banks, W. Fischler, S. H. Shenker and L. Susskind, *Phys. Rev*. **D55** (1997) 5112.
M. R. Douglas, *Nucl. Phys. Proc. Suppl.* **68** (1998) 381.
J. Frohlich, O. Grandjean and A. Recknagel, *Comm. Math. Phys.* **193** (1998) 527.
M. R. Douglas, A. Kato, H. Ooguri, *Adv. Theor. Math. Phys.* **1** (1998) 237.
E. Bergshoeff, E. Sezgin and P. K. Townsend, *Phys. Lett.* **B189** (1987) 95.
B. de Wit, J. Hoppe and H. Nicolai, *Nucl. Phys.* **B305** (1988) 545.
W. Taylor and M. Van Raamsdonk, `hep-th/9904095`. A. H. Chamseddine, *Nucl. Phys.* **B185** (1981) 403; *Phys. Rev.* **D29** (1981) 3065;
E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, *Nucl. Phys.* **B195** (1982) 97:
G. Chapline and N. Manton, *Phys. Lett.* **120B** (1983) 105.
J.-P. Derendinger and S. Ferrara, in *Supersymmetry and Supergravity 84* editors B. de Wit, P. Fayet and P. van Nieuwenhuizen, World Scientific p. 159, 1984.
M. de Roo and P. Wagemans, *Nucl. Phys.* **B262** (1995) 644.
E. Cremmer and B. Julia, *Nucl. Phys.* **B159** (1979) 141.
|
---
author:
- |
Jiun-Jie Wang\
Email: jiunjiew@buffalo.edu
title: Layouts for Plane Graphs on Constant Number of Tracks
---
Introduction
============
A *track* layout of a graph consists of a vertex $k$ colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. A *queue* layout of a graph consists of a total order of the vertices, and a partition of the edges into sets (called queues) such that no two edges that are in the same set are nested with respect to the vertex ordering. The minimum number of queues in a queue layout of a graph is its *queue number*. Track layouts have been extensively studied in [@D15; @DFJW12; @DMW05; @DMW13; @DPW04; @DW04; @DW05; @FLW02]. Queue layouts have been introduced by Heath, Leighton, and Rosenberg [@HLR92; @HR92] and have been extensively studied from [@BFP10; @DMW05; @DMW13; @DPW04; @DW05; @EI71; @H04; @HLR92; @HR92; @P92; @RM95; @SS00; @T72; @W05; @W08]. Both track and queue layouts have applications in parallel process scheduling, fault-tolerant processing, matrix computations, and sorting networks (see [@P92] for a survey). Queue layouts of directed acyclic graphs [@BCLR96; @HP99; @HPT99; @P92] and posets [@HP97; @P92] have also been investigated.
The question in Heath et al. [@HLR92; @HR92], whether the queue number of a planar graphs is constant-bound (it also leads to constant-bound track number), remains open. Heath et al. [@HLR92; @HR92] conjectured that the question has an affirmative answer. However, Pemmaraju [@P92] conjectured that every planar graph has $O(\log n)$ queue number. Also, he conjectured that this is the correct lower bound. Up to now, the best known lower bound is still constant-bound. On the other hand, the well-known upper bound for the queue number of planar graphs had remained stagnant as $O(\sqrt{n})$ roughly two decades. This upper bound utilizes the fact that planar graphs have path width at most $O(\sqrt{n})$. Recently, the upper bounds of queue and track numbers for planar graphs were reduced to $O(\log^2 n)$, by Di Battista, Frati and Pach [@BFP10] and $O(\log n)$, by Vida Djumovic [@D15], respectively.
In this paper, we provide a layout on constant number of tracks for a plane graph. Our result attempts to break Pemmaraju’s conjecture in a positive direction. The proof that a plane graph has constant-bound track number is simple. It utilizes a novel graph partition technique. In particular, our main result states that every $n$-vertex plane graph has such a graph partition and it leads to $O(1)$-track layouts for plane graphs.
One of the most important motivations for studying queue layouts is 3D crossing-free straight-line grid drawing in a small volume. Particularly, a 3D crossing-free straight-line grid drawing of a graph is a placement of the vertices at distinct points in a 3D grid, and the straight-line representing the edges are pairwise non-crossing. One of the most important open problems that Felsner et al. [@FLW02] present in graph drawing questions is whether planar graphs have 3D crossing-free straight-line grid drawings in a linear volume. A 3D crossing-free straight-line grid drawing with volume $X \times Y \times Z$ is an $X \times Y \times Z$ drawing that fits in an axis-aligned box with side lengths $X-1$, $Y-1$, and $Z-1$. The following theorem has been established in [@DMW05; @DPW04].
An $n$-vertex graph $G$ has a 3D crossing-free straight grid drawing in an $O(1) \times O(1) \times O(n)$ volume, if and only if $G$ has a constant-bound queue number. (constant-bound track number.)
The road map for this paper is as follows: the first half part from Sections \[sec:prelim\] to \[sec:cons-tracks\] explain the basic framework and ideas for this article. The second half part explain more details in the first half part of this article.
Preliminaries {#sec:prelim}
=============
In this section, Some definitions and important preliminary results are given. Definitions not mentioned here are standard. A graph $G=(V,E)$ is called [*planar*]{} if it can be drawn on the plane with no edge crossings. Such a drawing is called a [*plane embedding*]{} of $G$. A [*plane graph*]{} is a planar graph with a fixed plane embedding.
A *layerlike* graph $\Pi$ is a graph whose vertices are partitioned and placed on contiguous layers such that no edge is placed between any two non-contiguous layers and no edges are crossing. Given a layerlike graph $\Pi$, a *down-pointing* triangle $\triangledown$ is a a cycle $(l, \cdots, r, m)$ that vertices on the cycle $(l, \cdots, r, m)$ are on the two contiguous layers where the path from $l$ to $r$ are on the upper layer and the vertex $m$ is on the lower layer. A *bowl* $\heartsuit$ is a cycle $(l, \cdots, r)$ that the cycle are on the same layer where each vertex of the cycle is on the same layer. In Fig. \[fig:perfect-layer-graph\], vertices $(b_8, b_9, b_{10}, b_{11}, b_{12}, b_{13})$ form a bowl in the composite-layerlike graph ${\mathcal{G}}$.
A *composite-layerlike* graph ${\mathcal{G}}$ can be recursively defined as follows: ${\mathcal{G}}$ consists of a layerlike graph $\Pi$ such that each bowl $\heartsuit$ of ${\mathcal{G}}$ has a smaller composite-layerlike graph ${\mathcal{G}}_1$ where ${\mathcal{G}}_1$’s first layer is the bowl $\heartsuit$, and each down-pointing triangle $\triangledown$ has a composite-layerlike graph ${\mathcal{G}}_2$ where the first layer of ${\mathcal{G}}_2$ is the upper layer of $\triangledown$.
An edge $e=(u, v)$ is called a *chord* if both end-vertices $u$ and $v$ are on the same layer in a composite-layerlike graph ${\mathcal{G}}$. A *region* ${\mathcal{W}}$, rooted at a vertex $r$ in a composite-layerlike graph ${\mathcal{G}}$, consists of a left boundary ${\mathcal{B}}^L$ and a right boundary ${\mathcal{B}}^R$ such that ${\mathcal{W}}$ satisfies (1): $B^L$ and ${\mathcal{B}}^R$ are two paths walking along contiguous layers from the vertex $r$ to two different vertices on lower layers in ${\mathcal{G}}$, and (2) ${\mathcal{W}}$ is a separator of the composite-layerlike graph ${\mathcal{G}}$. Also, we denote the left and right boundaries of a region ${\mathcal{W}}$ as ${\mathcal{B}}^L({\mathcal{W}})$ and ${\mathcal{B}}^R({\mathcal{W}})$, respectively. Consider a region ${\mathcal{W}}$ rooted at a vertex $r$ in a composite-layerlike graph ${\mathcal{G}}$. A composite-layerlike graph ${\mathcal{G}}({\mathcal{W}})$ is *induced* by ${\mathcal{W}}$ if ${\mathcal{G}}({\mathcal{W}})$ is a subgraph of ${\mathcal{G}}$ inside by the two boundaries ${\mathcal{B}}^L({\mathcal{W}})$ and ${\mathcal{B}}^R({\mathcal{W}})$. Also, we denote ${\mathcal{W}}^{{\mathcal{M}}}$ as the maximum region bounded by the leftmost and rightmost boundaries of ${\mathcal{G}}$. Obviously, a composite-layerlike graph ${\mathcal{G}}$ is a maximum composite-layerlike graph ${\mathcal{G}}({\mathcal{W}}^{{\mathcal{M}}})$ induced by the maximum region ${\mathcal{W}}^{{\mathcal{M}}}$.
A *ladder* ${\mathcal{H}}$ is defined to consist of contiguous tracks. A *layout* of a composite-layerlike graph ${\mathcal{G}}$ in a ladder ${\mathcal{H}}$ is defined to be an arbitrary vertices’s partition of ${\mathcal{G}}$ on tracks of ${\mathcal{H}}$. For a layout of a composite-layerlike graph ${\mathcal{G}}$ in a ladder ${\mathcal{H}}$, a set of chords $\{e_1=(u_1, v_1), e_2=(u_2, v_2), \cdots, e_q=(u_q, v_q)\}$ are called *nest* if $\{e_1, e_2, \cdots, e_q\}$ are placed on a track in ${\mathcal{H}}$ as the order: $(u_1,$ $u_2,$ $\cdots,$ $u_q,$ $v_q,$ $\cdots,$ $v_2,$ $v_1)$. A set of edges $\{e_1=(u_1, v_1),$ $e_2=(u_2, v_2),$ $\cdots,$ $e_q=(u_q, v_q)\}$ are called *$X$-cross* if $(u_1,$ $u_2,$ $\cdots,$ $u_q)$ are orderly placed as $(u_1,$ $u_2,$ $\cdots,$ $u_q)$ on a track and $(v_1,$ $v_2,$ $\cdots,$ $v_q)$ are reversely placed as $(v_q,$ $\cdots,$ $v_2,$ $v_1)$ on another track in ${\mathcal{H}}$.
Given an edge $e=(u, v)$ a layout in ${\mathcal{H}}$, let ${\mathcal{L}}_{{\mathcal{H}}}(u)$ and ${\mathcal{L}}_{{\mathcal{H}}}(v)$ be the track numbers where the vertices $u$ and $v$ placed in ${\mathcal{H}}({\mathcal{G}})$, respectively. The *gap* of an edge $e=(u, v)$ is the absolute difference $|{\mathcal{L}}_{{\mathcal{H}}}(u)-{\mathcal{L}}_{{\mathcal{H}}}(v)|$. Also, the *queue number* on a track is defined as the maximum size of edges nest on the track, and the $X$-*crossing number* for any two tracks in ${\mathcal{H}}$ is defined as the maximum size of edges $X$-cross between the two track. The *distance-number* of a layout in a ladder ${\mathcal{H}}$ is defined as the maximum gaps among all edges.
\[def:well-placed\] A layout of a composite-layerlike graph ${\mathcal{G}}$ in a ladder ${\mathcal{H}}$ is called $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-*well-placed* if they can be placed in ${\mathcal{H}}$ such that
- each track’s queue number is less than ${\mathcal{Q}}$,
- the $X$-crossing number between any two tracks is less than ${\mathcal{X}}$,
- the distance-number in ${\mathcal{H}}$ is less than ${\mathcal{D}}$, and
- ${\mathcal{G}}$ can be placed as sequential regions in ${\mathcal{H}}$; The sequential regions are denoted as $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{G}})$.
\[thm:wrap\] If a composite-layerlike graph ${\mathcal{G}}$ is an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout in ${\mathcal{H}}'$, then ${\mathcal{G}}$ can be placed as an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout on $2{\mathcal{D}}$ tracks in ${\mathcal{H}}$.
Assume that a composite-layerlike graph ${\mathcal{G}}$ can be placed in a ladder ${\mathcal{H}}'$ such that
1. the queue number of each track in ${\mathcal{H}}'$ is less than or equal to ${\mathcal{Q}}$,
2. the $X$-crossing number of between any two tracks in ${\mathcal{H}}'$ is less than or equal to ${\mathcal{X}}$, and
3. the difference $|{\mathcal{L}}_{{\mathcal{H}}}(u)-{\mathcal{L}}_{{\mathcal{H}}}(v)|$ is less than or equal to ${\mathcal{D}}$ for any edge $e=(u, v)$ in ${\mathcal{H}}'$.
Since the total tracks in ${\mathcal{H}}'$ could grow beyond constant bound, we need to wrap ${\mathcal{H}}'$ as follows: move vertices on $(i\times 2{\mathcal{D}}+ j)$-th track to right of vertices on $((i-1)\times 2{\mathcal{D}}+ j)$-th track on the wrapped ${\mathcal{H}}$’s $j$-th track.
Now each track $i$ in the wrapped ladder ${\mathcal{H}}$, vertices are from tracks $(0\times 2{\mathcal{D}}+ j)$, $(1\times 2{\mathcal{D}}+ i)$, $(2\times 2{\mathcal{D}}+ j)$, $\cdots$ in the unwrapped ladder ${\mathcal{H}}'$. And, for each edge $e=(u, v)$ in the wrapped ladder ${\mathcal{H}}$, the edge $e$ comes from pair of tracks $(0\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), 0\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$, $(1\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), 1\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$, $(2\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), 2\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$, $\cdots$ in the unwrapped ladder ${\mathcal{H}}'$.
Because for any edge $e=(u, v)$ in the unwrapped ladder ${\mathcal{H}}'$, the difference $|{\mathcal{L}}_{{\mathcal{H}}}(u)-{\mathcal{L}}_{{\mathcal{H}}}(v)|$ is at most ${\mathcal{D}}$, only edges on pair tracks $(0\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), 0\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$, $(1\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), 1\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$, $(2\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), 2\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$, $\cdots$ in the unwrapped ladder ${\mathcal{H}}'$ can be placed on the pair tracks $({\mathcal{L}}_{{\mathcal{H}}}(u), {\mathcal{L}}_{{\mathcal{H}}}(v))$ in the wrapped ladder ${\mathcal{H}}$. Also, for a track $({\mathcal{L}}_{{\mathcal{H}}}(u))$ (${\mathcal{L}}_{{\mathcal{H}}}(v)$, respectively) on the wrapped ladder ${\mathcal{H}}$, we know that vertices on a track $i\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u)$ ($i\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v)$, respectively) from the unwrapped ladder ${\mathcal{H}}'$ are placed at left of vertices on a track $(i+1) \times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v)$ ($(i+1)\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u)$, respectively) from the unwrapped ladder ${\mathcal{H}}'$.
Hence there is no any $X$-crossing edge between edges from pair tracks $(i\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), i\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$ and pair tracks $((i+1)\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(u), (i+1)\times 2{\mathcal{D}}+ {\mathcal{L}}_{{\mathcal{H}}}(v))$.
Finally, We can conclude that a composite-layerlike graph ${\mathcal{G}}$ can be placed in the wrapped laddder graph ${\mathcal{H}}$ such that
1. the queue number of each track in the wrapped ladder ${\mathcal{H}}$ is less than or equal to ${\mathcal{Q}}$,
2. the $X$-crossing number of between any two track in the wrapped ladder ${\mathcal{H}}$ is less than or equal to ${\mathcal{X}}$, and
3. the number of tracks in the wrapped ladder ${\mathcal{H}}$ is $2{\mathcal{D}}$.
A Framework to Construct an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-Well-Placed Layout on Constant Number of Tracks for a Composite-Layerlike Graph ${\mathcal{G}}$ {#sec:perfect-layer-graph-layout}
===========================================================================================================================================================================
Place the ${\mathcal{W}}^{{\mathcal{M}}}$’s root on the first track in ${\mathcal{H}}$
Place the contiguous layers $(2, 3, \cdots)$ of ${\mathcal{W}}^{{\mathcal{M}}}$ at right of ${\mathcal{W}}^{{\mathcal{M}}}$’s root on the contiguous $({\mathcal{Z}}+2, {\mathcal{Z}}+3, \cdots)$ tracks in ${\mathcal{H}}$
Add the maximum region ${\mathcal{W}}^{{\mathcal{M}}}$ into the empty first-in-first-out queue $\tilde{{\mathcal{Y}}}$
Wrap ${\mathcal{H}}$
In this section, we provide a framework in Algorithm \[alg:framework\] to place a composite-layerlike graph ${\mathcal{G}}$ as $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout in ${\mathcal{H}}$ on constant number of tracks. Before we describe the framework, we introduce a structure *skeleton* as follows:
\[def:skeleton\] Consider a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$ rooted at a vertex $r$. A subgraph $\Psi$ of a composite-layerlike graph ${\mathcal{G}}({\mathcal{W}})$ is called a *skeleton* of ${\mathcal{W}}$ if $\Psi$ consists of
1. the ${\mathcal{W}}$’s root $r$,
2. sequential regions $\tilde{{\mathcal{W}}}(\Psi)$ such that there are subsequential regions $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots,$ ${\mathcal{W}}^M_1,$ $\cdots,$ ${\mathcal{W}}^M_m,$ ${\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots)$ $\subseteq \tilde{{\mathcal{W}}}(\Psi)$ where (1) for each vertex $u\in {\mathcal{B}}^L$, each $u$’s child is inside a region in $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots)$, (2) for each vertex $v\in {\mathcal{B}}^R$, each $v$’s child is inside a region in $({\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots)$, and (3) each $r$’s child is inside a region in $({\mathcal{W}}^M_1, {\mathcal{W}}^M_2, \cdots, {\mathcal{W}}^M_m)$.
and the subgraph induced by ${\mathcal{W}}$; (2) shows a ${\mathcal{W}}$’s skeleton that consists of sequential regions ${\mathcal{W}}_1=((b_1, c_1, \cdots), (b_1, c_3, \cdots))$, ${\mathcal{W}}_2=((b_1, c_3, \cdots), (b_1, c_5, \cdots))$, ${\mathcal{W}}_3=((a_1, b_1, c_5, \cdots), (a_1, b_2, c_6, \cdots))$, ${\mathcal{W}}_4=((a_1, b_2, c_6), (a_1, b_2, c_6))$, ${\mathcal{W}}_5=((a_1, b_4, c_7), (a_1, b_5, c_7))$, ${\mathcal{W}}_6=((a_1, b_5, c_7, \cdots), (a_1, b_6, c_9, \cdots))$, ${\mathcal{W}}_7=((b_6, c_9, \cdots), (b_6, c_{12}, \cdots))$. In addition, the region ${\mathcal{W}}_1$ consists of two subregions ${\mathcal{W}}'_1=((b_1, c_1, \cdots), (b_1, c_2, \cdots))$ and ${\mathcal{W}}'_2=((b_1, c_2, \cdots), (b_1, c_3, \cdots))$. The region ${\mathcal{W}}_2$ consists of two subregions ${\mathcal{W}}'_3=((b_1, c_3, \cdots), (b_1, c_4, \cdots))$ and ${\mathcal{W}}'_4=((b_1, c_4, \cdots), (b_1, c_5, \cdots))$. the region ${\mathcal{W}}_7$ consists of three subregions ${\mathcal{W}}'_5=((b_6, c_9, \cdots), (b_6, c_{10}, \cdots))$, ${\mathcal{W}}'_6=((b_6, c_{10}, \cdots), (b_6, c_{11}, \cdots))$ and ${\mathcal{W}}'_7=((b_6, c_{11}, \cdots), (b_6, c_{12}, \cdots))$; (3): the sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, {\mathcal{W}}_3, {\mathcal{W}}_4, {\mathcal{W}}_5, {\mathcal{W}}_6, {\mathcal{W}}_7)$ are placed as: $({\mathcal{W}}_1, {\mathcal{W}}_2, {\mathcal{W}}_3, {\mathcal{W}}_4, {\mathcal{W}}_7, {\mathcal{W}}_6, {\mathcal{W}}_5)$ in a ladder ${\mathcal{H}}$; \[fig:skeleton\]
Firstly, this framework place ${\mathcal{W}}^M$ on the contiguous tracks from the first track in ${\mathcal{H}}$. Next, this framework also consists of a loop and each iteration of the loop in Algorithm \[alg:framework\] starts from the first region ${\mathcal{W}}$ rooted at $r$ of the first-in-first-out queue $\tilde{{\mathcal{Y}}}$ and executes the following steps: find sequential skeletons $(\Psi_1, \Psi_2, \cdots)$ such that each skeleton $\Psi_i, i\geq 1,$ roots at the vertex $r({\mathcal{W}})$ and each $r({\mathcal{W}})$’s child inside ${\mathcal{W}}$ is a vertex in a skeleton of $(\Psi_1, \Psi_2, \cdots)$. Also, the sequential skeletons $(\Psi_1, \Psi_2, \cdots)$ are placed orderly at the rightmost part in ${\mathcal{H}}$ and starts from the track $({\mathcal{L}}_{{\mathcal{H}}}(r({\mathcal{W}}))+2{\mathcal{Z}})$ in ${\mathcal{H}}$.
Before we prove the correctness of Algorithm \[alg:framework\] in Subsection \[subsec:correctness-framework\], we assume the following conjecture in advance. This conjecture is proved in Section \[sec:skeleton\].
\[conj:skeleton\] Given a region ${\mathcal{W}}$ rooted at a vertex $r$, we have a skeleton $\Psi$ that $\Psi$ can have sequential regions $\tilde{{\mathcal{W}}}(\Psi)$. And, the skeleton $\Psi$ can be placed in ${\mathcal{H}}$ as $\Psi_{{\mathcal{H}}}$ such that $\Psi_{{\mathcal{H}}}$ is $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{J}})$-well-placed in ${\mathcal{H}}$.
Sequential Skeletons $(\Psi_1, \Psi_2, \cdots)$ inside a Region ${\mathcal{W}}$ Rooted at a Vertex $r$ {#subsec:correctness-framework}
------------------------------------------------------------------------------------------------------
In this subsection, we start to show two lemmas, the first one shows how to find sequential skeletons $(\Psi_1, \Psi_2, \cdots)$ inside a region ${\mathcal{W}}$ rooted at $r$ such that each $r$’s child in ${\mathcal{W}}$ is a vertex of a skeleton of $(\Psi_1, \Psi_2, \cdots)$, And the second one shows how to place the sequential skeletons $(\Psi_1, \Psi_2, \cdots)$ in ${\mathcal{H}}$ and $(\Psi_1, \Psi_2, \cdots)$ can partition the region ${\mathcal{W}}$ into sequential regions such that $(\Psi_1, \Psi_2, \cdots)$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{J}})$-well-placed in ${\mathcal{H}}$.
In the following lemma, we give a constructive proof to find sequential skeletons $(\Psi_1, \Psi_2, \cdots)$ consisting of all $r$’s children inside a region ${\mathcal{W}}$.
\[lem:partition\] For a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$ rooted at a vertex $r$, sequential skeletons $\tilde{\Psi}({\mathcal{W}})=$ $(\Psi_1, \Psi_2, \cdots)$ can be constructed such that each $r$’s child is a vertex of some skeleton $\Psi$ of $\tilde{\Psi}({\mathcal{W}})$.
From Theorem \[thm:skeleton\], there exists a skeleton $\Psi$ for the region ${\mathcal{W}}$ and $\Psi$ can partition the region ${\mathcal{W}}$ into sequential regions $\tilde{{\mathcal{W}}}(\Psi)=({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$. From Definition \[def:skeleton\], each $r$’s children not included into $\Psi$ is inside in a region of $\tilde{{\mathcal{W}}}(\Psi)$. Let $({\mathcal{W}}^M_1, {\mathcal{W}}^M_2, \cdots, {\mathcal{W}}^M_m)$ be the maximal subsequential regions of $\tilde{{\mathcal{W}}}(\Psi)$ such that each $r$’s child not included in $\Psi$ is inside a region of $({\mathcal{W}}^M_1, {\mathcal{W}}^M_2, \cdots, {\mathcal{W}}^M_m)$ rooted at the vertex $r$.
Now for each region ${\mathcal{W}}^M_i\in ({\mathcal{W}}^M_1, {\mathcal{W}}^M_2, \cdots, {\mathcal{W}}^M_m)$, a skeleton $\Psi_i$ for the region ${\mathcal{W}}_i$ can be found. Then by the above discussion, for each skeleton $\Psi_i, 1\leq i\leq m$, the skeleton $\Psi_i$ can partition the region ${\mathcal{W}}^M_i$ into sequential regions $\tilde{{\mathcal{W}}}(\Psi_i)$. And, each $r$’s children inside ${\mathcal{W}}^M_i$ not included into $\Psi_i$ is inside a region of $\tilde{{\mathcal{W}}}(\Psi_i)$. the same partition can be repeatedly executed till each $r$’s child is included into a skeleton. Hence we can conclude that given a region ${\mathcal{W}}$ rooted at a vertex $r$, we can have sequential skeletons $\tilde{\Psi}({\mathcal{W}})$ such that each $r$’s child is a vertex of some skeleton $\Psi$ in $\tilde{\Psi}({\mathcal{W}})$.
Consider sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$ in ${\mathcal{H}}$. The sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$ and the sequential subgraphs $(\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_1), \tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_2), \cdots)$ are called an *ordered* layout in ${\mathcal{H}}$ if the sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$ are at left of the sequential subgraphs $(\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_1), \tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_2), \cdots)$ in ${\mathcal{H}}$, and for any two regions ${\mathcal{W}}_i$ and ${\mathcal{W}}_j$ in $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$, the region ${\mathcal{W}}_i$ is at left of the region ${\mathcal{W}}_j$ if and only if the subgraph $\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_i)$ is at left of the subgraph $\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_j)$ in ${\mathcal{H}}$. The task of this subsection is to show that if the sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$ and the sequential subgraphs $(\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_1), \tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_2), \cdots)$ in ${\mathcal{H}}$ are an ordered layout in ${\mathcal{H}}$, then the layout is also $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$. Now consider sequential edges $({\mathcal{E}}_1, {\mathcal{E}}_2, \cdots)$ where each ${\mathcal{E}}_i, i\geq 1,$ are edges connected between the region ${\mathcal{W}}_i$ and the subgraph $\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}}_i)$.
Now we prove that the sequential skeletons $\tilde{\Psi}({\mathcal{W}})$ can be $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{J}})$-well-placed on contiguous tracks in ${\mathcal{H}}$. From Theorem \[thm:skeleton\], we can have a skeleton $\Psi$ for the region ${\mathcal{W}}$ such that $\Psi$ can be $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{J}})$-well-placed in ${\mathcal{H}}$ as $\Psi_{{\mathcal{H}}}$ and have sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi)$ in ${\mathcal{H}}$. Then we can pick the maximum subsequential regions $({\mathcal{W}}^M_1, {\mathcal{W}}^M_2, \cdots, {\mathcal{W}}^M_m)$ of $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi)$ such that each $r$’s child is inside a region of $({\mathcal{W}}^M_1, {\mathcal{W}}^M_2, \cdots, {\mathcal{W}}^M_m)$. And, for the sequential regions $({\mathcal{W}}^M_1, {\mathcal{W}}^M_2, \cdots, {\mathcal{W}}^M_m)$, we can have corresponding sequential skeletons $(\Psi_1, \Psi_2, \cdots, \Psi_m)$ such that $(\Psi_1, \Psi_2, \cdots, \Psi_m)$ can be $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{J}})$-well-placed as $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_1), \tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_2), \cdots, \tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_m))$ in ${\mathcal{H}}$. Now we have placed $(\Psi, \Psi_1, \Psi_2, \cdots, \Psi_m)$ in ${\mathcal{H}}$ as sequential regions $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_m))$. Since regions in $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_1),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_m))$ are mutually disjoint and $({\mathcal{W}}^M_1,$ ${\mathcal{W}}^M_2,$ $\cdots,$ ${\mathcal{W}}^M_m,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_1),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_m))$ are an ordered layout, edges ${\mathcal{E}}_i$ between ${\mathcal{W}}^M_i$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_i)$ and edges ${\mathcal{E}}_j$ between ${\mathcal{W}}^M_j$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\Psi_j)$ don’t nest in ${\mathcal{H}}$. Hence the layout in ${\mathcal{H}}$ is $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{J}})$-well-placed. We can repeat the above steps until each $r$’s child inside the region ${\mathcal{W}}$ is a vertex of some skeleton of the sequential skeletons $\tilde{\Psi}({\mathcal{W}})$ and have sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}))$ in ${\mathcal{H}}$. From the above discussion, we immediately have the following lemma.
\[lem:skeleton-wrap\] For a region ${\mathcal{W}}$ rooted at a vertex $r$, sequential skeletons $\tilde{\Psi}({\mathcal{W}})$ can be placed as the new order $\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}})$ on contiguous tracks in ${\mathcal{H}}$ such that $\tilde{\Psi}_{{\mathcal{H}}}({\mathcal{W}})$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{J}})$-well-placed in ${\mathcal{H}}$ where the three numbers ${\mathcal{Q}}$, ${\mathcal{X}}$ and ${\mathcal{J}}$ are constant-bound.
In the next lemma, we prove that when a sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}))$ are placed in ${\mathcal{H}}$ in Algorithm \[alg:framework\], all edges connecting between ${\mathcal{W}}$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}))$ cannot make $X$-crossing with the existing layout in ${\mathcal{H}}$.
\[lem:noncrossing\] Given sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$ in ${\mathcal{H}}$ where each region ${\mathcal{W}}_i, i\geq 1,$ roots at a vertex $r_i$. If each sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_i)), i\geq 1,$ is placed at right of $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_1)),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_{i-1}))$ and from the track ${\mathcal{L}}_{{\mathcal{H}}}(r_i)+2{\mathcal{Z}}$ in ${\mathcal{H}}$, then (1): the edges set ${\mathcal{E}}_i$ connecting between ${\mathcal{W}}_i$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_i))$ don’t have $X$-crossing edges with the layout $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_1)),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_i))$ in ${\mathcal{H}}$.
Because the region ${\mathcal{W}}_i$ is placed at right of the sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots, {\mathcal{W}}_{i-1})$ and $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots, {\mathcal{W}}_i)$ are mutually disjoint, the edges sets ${\mathcal{E}}_i$ and ${\mathcal{E}}_j, 1\leq i-1$ don’t have any $X$-crossing edge in ${\mathcal{H}}$ where ${\mathcal{E}}_j$ are all edges connecting between ${\mathcal{W}}_j$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_j))$.
From Lemma \[lem:skeleton-wrap\], we can assume that each edge’s gap number on each region ${\mathcal{W}}_i$ is less than or equal to ${\mathcal{J}}$. Because the gap numbers of edges in each edges set ${\mathcal{E}}_i$ is greater than or equal to ${\mathcal{Z}}$ and ${\mathcal{Z}}$ is greater than ${\mathcal{J}}$, each region of $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_1)),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{\Psi}({\mathcal{W}}_{i-1}))$ and ${\mathcal{E}}_j$ cannot make $X$-crossing in ${\mathcal{H}}$.
\[lem:gap\_number\] For each edge $e$ placed in Algorithm \[alg:framework\], $e$’s gap number is at most $2{\mathcal{Z}}$.
Initially, Each edge of the maximum region ${\mathcal{W}}^{{\mathcal{M}}}$ except the edges connecting to ${\mathcal{W}}^{{\mathcal{M}}}$’s root can be placed in ${\mathcal{H}}$ with gap number = 1. Each Edge connecting to ${\mathcal{W}}^{{\mathcal{M}}}$’s root has gap number = ${\mathcal{Z}}$. Each edge $e$ connecting to ${\mathcal{W}}^{{\mathcal{M}}}$’s root inside ${\mathcal{W}}^{{\mathcal{M}}}$ is an edge of a skeleton of $\tilde{\Psi}({\mathcal{W}}^{{\mathcal{M}}})$ and $\tilde{\Psi}({\mathcal{W}}^{{\mathcal{M}}})$ is placed in ${\mathcal{H}}$ from the track $2{\mathcal{Z}}+1$ in ${\mathcal{H}}$. It leads to $e$’s gap number = $2{\mathcal{Z}}$ in ${\mathcal{H}}$.
For a region ${\mathcal{W}}$ in ${\mathcal{H}}$, the region ${\mathcal{W}}$ is called *processed* if ${\mathcal{W}}$ has been partitioned by sequential skeletons $\tilde{\Psi}({\mathcal{W}})$ and each edge $e$ connecting to a boundary of ${\mathcal{W}}$, either $e$ is an edge of a skeleton $\Psi$ for ${\mathcal{W}}^{{\mathcal{M}}}$ or $e$ is inside a region in $\tilde{{\mathcal{W}}}(\Psi)$ in ${\mathcal{H}}$.
In each iteration of Algorithm \[alg:framework\], we pick the leftmost unprocessed region ${\mathcal{W}}$ in ${\mathcal{H}}$ and place its sequential skeletons $\tilde{\Psi}({\mathcal{W}})$ into ${\mathcal{H}}$. From Lemma \[lem:skeleton-wrap\], for each region ${\mathcal{W}}' \in \tilde{\Psi}({\mathcal{W}})$, each edge of ${\mathcal{W}}'$ except connecting to ${\mathcal{W}}'$’s root has gap number = ${\mathcal{J}}$. Now for each vertex $v$ on a boundary of the leftmost unprocessed region ${\mathcal{W}}$, all edges connecting to the vertex $v$ inside the region ${\mathcal{W}}$ are processed by two consecutive steps: (1): some edges connecting to the vertex $v$ are edges of a skeleton of $\tilde{\Psi}({\mathcal{W}})$ and have gap number ${\mathcal{Z}}$ in ${\mathcal{H}}$. And, (2): each remaining edge connecting to the vertex $v$ is inside a region ${\mathcal{W}}'$ rooted at the vertex $v$ of $\tilde{{\mathcal{W}}}(\tilde{\Psi}({\mathcal{W}}))$. Then the region ${\mathcal{W}}'$ has sequential skeletons $\tilde{\Psi}({\mathcal{W}}')$ such that each remaining edge $e'$ is an edge of some skeleton of $\tilde{\Psi}({\mathcal{W}}')$ and has gap number $2{\mathcal{Z}}$. Hence we can conclude that each edge placed in Algorithm \[alg:framework\] has gap number at most $2{\mathcal{Z}}$.
From Lemmas \[lem:partition\], \[lem:skeleton-wrap\] and \[lem:noncrossing\], we know that each current step in Algorithm \[alg:framework\], the placement of sequential skeletons for a region ${\mathcal{W}}$ in ${\mathcal{H}}$ cannot make ${\mathcal{X}}$-crossing with the placement of previous steps in ${\mathcal{H}}$ because each edge’s gap number in each skeleton is at most ${\mathcal{J}}$. Also, from Lemma \[lem:gap\_number\], we know that the distance number of the layout in Algorithm \[alg:framework\] is at most $2{\mathcal{Z}}$.
Hence we can immediately prove that the framework in Algorithm \[alg:framework\] can place a composite-layerlike graph ${\mathcal{G}}$ as an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}}=2{\mathcal{Z}})$-well-placed layout on $2{\mathcal{D}}$ tracks of a ladder ${\mathcal{H}}$ in Theorem \[thm:cons-track\].
\[thm:cons-track\] If Conjecture \[conj:skeleton\] can be proven, then by Algorithm \[alg:framework\], every composite-layerlike graph ${\mathcal{G}}$ can be an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout on $2{\mathcal{D}}$ tracks in a ladder ${\mathcal{H}}$.
From a Plane Graph $G$ To a Composite-Layerlike Graph ${\mathcal{G}}$ {#sec:transformation}
=====================================================================
In this section, we show how to reform a plane graph $G$ to a composite-layerlike graph ${\mathcal{G}}$. Let $G$ be a plane graph and ${{\cal O}}(G)$ be its outer boundary. Each layer in a composite-layerlike graph ${\mathcal{G}}$ can be recursively defined as follows: the first layer is ${{\cal O}}(G)$ and ${{\cal O}}_{G}$ is placed as clockwise order $(m, u_1, u_2, \cdots)$. Then $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$ are the sequential maximal inner cycles inside ${{\cal O}}$ such that for each maximal inner cycle ${{\cal O}}_i, 1\leq i\leq p$, there are some vertices on ${{\cal O}}_i$ connecting to the vertex $m$. Now for each maximal inner cycle ${{\cal O}}_i, 1\leq i\leq p$, we can walk around the cycle ${{\cal O}}_i$ by clockwise order to get two contiguous vertices $(L^U({{\cal O}}_i)=(v^i_1=u^i_y, v^i_2, \cdots, v^i_x=u^i_1), L^B({{\cal O}}_i)=(u^i_1=v^i_x, u^i_2, \cdots, u^i_y=v^i_1) )$ where each vertex of $L^U({{\cal O}}_i)$ don’t connect to the vertex $m$ except the first and last vertices $\{v^i_1, v^i_y\}$ of $L^U({{\cal O}}_i)$ and each vertex of $L^B({{\cal O}}_i)$ connects to the vertex $m$. All maximal inner cycles $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$ can be placed on the second layer as the order: $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$. Moreover, for each maximal inner cycle ${{\cal O}}_i$, ${{\cal O}}_i$ is placed on the second layer from the vertex $u^i_1$ by clockwise order as: $(v^i_1=u^i_y, v^i_2, \cdots, v^i_x=u^i_y, u^i_{y-1}, \cdots, u^i_2)$. By the above placement, we can have sequential induced subgraphs $(G|\triangledown_1, G|\triangledown_2, \cdots, G|\triangledown_q)$ that each induced subgraph $G|\triangledown_i, 1\leq i\leq q$, is a subgraph induced by a maximal down-pointing triangle $(l_i, \cdots, r_i, m_i)$ where the path from vertices $l_i$ and $r_i$ are on the outer boundary ${{\cal O}}(G)$ and $m_i$ is a vertex on some inner cycle ${{\cal O}}_i \in ({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$. In the followings, we discuss how to place each induced subgraphs $G|\triangledown_i, 1\leq i\leq q,$ on the subsequent layers in a composite-layerlike graph ${\mathcal{G}}$.
For a subgraph induced by a maximal down-pointing triangle $\triangledown=(l, \cdots, r, m)$. Sequential maximal inner cycles $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_q)$ inside $\triangledown=(l, \cdots, r, m)$ can be found such that each maximal inner cycle ${{\cal O}}_i, 1\leq i\leq q$ inside $\triangledown$ connects to the vertex $m$ and each maximal inner cycle ${{\cal O}}_i$ can be partitioned into two contiguous vertices $(L^U({{\cal O}}_i), L^B({{\cal O}}_i))$ by clockwise order where each vertex of $L^U({{\cal O}}'_i)$ doesn’t connect the vertex $m$ except the first and last vertices of $L^U({{\cal O}}_i)$ and each vertex of $L^B({{\cal O}}'_i)$ connects to the vertex $m$. Also, (1) the sequential cycles $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_q)$ is placed orderly, (2) the contiguous vertices ${\mathcal{L}}^U({{\cal O}}_i)$ are placed orderly and the contiguous vertices ${\mathcal{L}}^B({{\cal O}}_i)$ are placed reversely, and (3) ${\mathcal{L}}^U({{\cal O}}_i)$ is placed at left of ${\mathcal{L}}^B({{\cal O}}_i)$ on the subsequent layer of a new frame $\Pi$ in a composite-layerlike graph ${\mathcal{G}}$ where the $\Pi$’s first layer is the same as the down-pointing triangle $\triangledown$’s upper layer. From the above description, we assume there is no any chord $(u, v)$ on each cycle ${{\cal O}}$. Next we start to explain how to eliminate each chord on each cycle.
Now we plan to remove edges from the above construction such that there is no any $X$-crossing edge in a composite-layerlike graph. For each maximal inner cycle ${{\cal O}}$ that the cycle ${{\cal O}}$ is placed as the clockwise order: $(m, u_1, \cdots)$ on a layerlike graph $\Pi$, and the sequential maximal inner cycles $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$ inside ${{\cal O}}$ such that each maximal inner cycle ${{\cal O}}_i, 1\leq i\leq q,$ connects to the vertex $u_1$. we remove edges between the vertex $m$ and all vertices on the $(L^B({{\cal O}}'_1), L^B({{\cal O}}'_2), \cdots, L^B({{\cal O}}'_q))$ from the frame $\Pi$. Moreover, let $v_i$ be the vertex on the cycle ${{\cal O}}$ such that $v_i$ connects to the both cycles ${{\cal O}}_i$ and ${{\cal O}}_{i+1}$. Then, we add each sequential edges between $v_i$ and $L^B({{\cal O}}_i), 1\leq i\leq q,$ in the frame $\Pi$. Similarly, for each maximal down-pointing triangle $\triangledown=(l, \cdots, r, m)$, we execute the above procedure for the vertex $m$ of the down-pointing triangle $\triangledown$.
During the above transformation, how to process that if there exists a chord on a cycle ${{\cal O}}$ is neglected to discuss. The reason is explained below. A $d$-subdivision of a graph $G$ is a graph obtained by replacing each edge of $G$ with a path having at most $2 + d$ vertices. For each chord $(u, v)$ on a cycle ${{\cal O}}$, a 1-subdivision plane graph $G^1$ without any chord on a cycle ${{\cal O}}$ can be constructed by the following steps: (1): find another vertex $w\notin {{\cal O}}$ such that the triple vertices $(u, v, w)$ form a triangle face in a plane graph $G$, (2): a vertex $w'$ can be added inside the face $(u, v, w)$, (3): the chord $(u, v)$ can be replaced with two edges $\{(u, w'), (w', v)\}$, and (4): a dummy edge $(w, w')$ can be added to satisfy triangulation property. Hence how to process a chord on a cycle ${{\cal O}}$ found during the above transformation can be neglected by replaced a chord with two edges. Now we immediately have the following theorem.
\[thm:reform\] For each plane graph $G$, a $1$-subdivision $G^1$ of $G$ can be reformed into a composite-layerlike graph ${\mathcal{G}}$.
A Track Layout for a Plane Graph on Constant Number of Tracks {#sec:cons-tracks}
=============================================================
We explain how a plane graph $G$ can be placed as a track layout on constant number of tracks.
\[thm:subdivision\]*[@DW05]* Suppose a graph $G$ has a $d$-subdivision $k$-track layout. If the two numbers $k$ and $d$ are constant-bound, $G$ also have a track layout on constant number of tracks.
Every plane graph $G$ has a track layout on constant number of tracks.
From Theorem \[thm:G1-well-placed\], a $1$-subdivision plane graph $G^1$ can have a track layout on constant number of tracks ${\mathcal{H}}$. Because $G^1$’s track number is constant-bound, by Theorem \[thm:subdivision\], $G$ can also have a track layout on constant number of tracks.
An $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-Well-Placed Layout for a Raising Fan $\tilde{{\mathcal{F}}}$ in a Ladder ${\mathcal{H}}$ {#sec:layout-fan-path}
===========================================================================================================================================
In this section, we present an approach to have a specified type of sequential fans $\tilde{{\mathcal{F}}}$ in a composite-layerlike graph ${\mathcal{G}}({\mathcal{W}}^{{\mathcal{M}}})$ called a *raising-fan* path and defined later and show hot to place it as an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout in a ladder ${\mathcal{H}}$. Also, an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout for a *raising-fan* path $\tilde{{\mathcal{F}}}$ can partition the region ${\mathcal{W}}^{{\mathcal{M}}}$ into sequential regions in ${\mathcal{H}}$.
A *fan* ${\mathcal{F}}$ consists of sequential vertices $(u_1, \cdots, u_a, m)$ such that $(u_1, \cdots, u_a)$ and $w$ are on two contiguous layers $i$ and $i+1$ of a layerlike graph, and $w$ connects each vertex in $(u_1, \cdots, u_a$. The sequential vertices $(u_1, \cdots, u_a)$ are called upper vertices of the fan ${\mathcal{F}}$. Also, the vertex $m$ are called the lower vertex of the fan ${\mathcal{F}}$. A *raising-fan* path $\tilde{{\mathcal{F}}}$ consists of sequential fans $({\mathcal{F}}_1, {\mathcal{F}}_2, \cdots)$ such that for each fan ${\mathcal{F}}_i, i\geq 1$, there is a down-pointing triangle in ${\mathcal{F}}_{i+1}$ bounds ${\mathcal{F}}_i$. Also, the lower vertices of the raising fan $\tilde{{\mathcal{F}}}$ form sequential vertices ${\mathcal{M}}=(m_1, m_2, \cdots)$ and is called a *middle path* where each vertex $m_i, i\geq 1$, is the lower vertex of the fan ${\mathcal{F}}_i$. For a fan ${\mathcal{F}}\in \tilde{{\mathcal{F}}}$ with the middle path ${\mathcal{M}}$, edges of the fan ${\mathcal{F}}$ at left and right of the middle path ${\mathcal{M}}$ are called a *left arm* and *right arm* of the fan ${\mathcal{F}}$, respectively
A path $\Re(v)$ is called a *raising-path* from a vertex $v$ in a composite-layerlike graph ${\mathcal{G}}$ if the path $\Re(v)$ starts from the vertex $v$ along edges from the lower layer to the upper layer till the first layer in ${\mathcal{G}}$,. Any two raising-paths $\Re(v_1)$ and $\Re(v_2)$, $\Re(v_1)$ and $\Re(v_2)$ are called *upward-merging* if either the two raising-paths $\Re(v_1)$ and $\Re(v_2)$ are vertex-disjoint or the intersection of the two raising-paths $\Re(v_1) \cap \Re(v_2)$ is a subpath from some vertex to a vertex on the first layer in ${\mathcal{G}}$. A set of raising-paths $\tilde{\Re}=\{\Re(v)|v\in V({\mathcal{G}})\}$ in a composite-layerlike graph ${\mathcal{G}}$ are called *upward-merging* to ${\mathcal{G}}$ if for any two raising-paths $\Re(v_1)$ and $\Re(v_2)$ in $\tilde{\Re}$ are upward-merging and each vertex in ${\mathcal{G}}$ is in a raising-path $\Re(v)$ in $\tilde{\Re}$. From now on, when mention a raising-path $\Re(v)$, it always means that the path $\Re(v)$ is a raising-path in some specific upward-merging raising-path set $\tilde{\Re}$ for a composite-layerlike graph ${\mathcal{G}}$.
Assume that $\triangledown$ is a down-pointing triangle inside a fan ${\mathcal{F}}$ of a raising fan $\tilde{{\mathcal{F}}}$ with the middle path ${\mathcal{M}}$.
- A *left spine* of $\triangledown$ with respect to the $\tilde{{\mathcal{F}}}$’s middle path ${\mathcal{M}}$ consists of sequential maximal cycles $LS(\triangledown)=({{\cal O}}^L_1, {{\cal O}}^L_2, \cdots, {{\cal O}}^L_p)$ at left of the middle path ${\mathcal{M}}$ such that for each cycle ${{\cal O}}^L_i, 1\leq i\leq p$, there is an edge connecting between the upper vertices of $\triangledown$ and the boundary of ${{\cal O}}^L_i$.
- A *right spine* of $\triangledown$ with respect to the $\tilde{{\mathcal{F}}}$’s middle path ${\mathcal{M}}$ consists of sequential maximal cycles $RS(\triangledown)=({{\cal O}}^R_1, {{\cal O}}^R_2, \cdots, {{\cal O}}^R_q)$ at right of the middle path ${\mathcal{M}}$ such that for each cycle ${{\cal O}}^R_i, 1\leq i\leq q$, there is an edge connecting between the upper vertices of $\triangledown$ and the boundary of ${{\cal O}}^R_i$.
- the $i$-th *joint* of a spine inside ${\mathcal{F}}'$ consists of the $i$-th cycle’s the leftmost and rightmost vertices in the spine of $\triangledown$.
If a down-pointing triangle $\triangledown\in {\mathcal{F}}$ which is not passed through the middle path ${\mathcal{M}}$, then the sequential maximal cycles $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_t)$ is called a *spine* in $\triangledown$ because the spine is not partitioned by the middle path ${\mathcal{M}}$.
Let ${\mathcal{F}}$ be a fan in a raising fan $\tilde{{\mathcal{F}}}$ with the middle path ${\mathcal{M}}$ that Then the fan ${\mathcal{F}}$ are partitioned by the middle path ${\mathcal{M}}$ into the following sequential down-pointing triangles $(\triangledown^L_1,$ $\triangledown^L_2,$ $\cdots,$ $\triangledown^L_a),$ $\triangledown^M,$ $(\triangledown^R_1,$ $\triangledown^R_2,$ $\cdots,$ $\triangledown^R_b)$ where each down-pointing triangle $\triangledown^L_i, 1\leq i\leq a,$ is at left of the middle path ${\mathcal{M}}$, the down-pointing triangle $\triangledown^M$ consists of the middle path ${\mathcal{M}}$ and each down-pointing triangle $\triangledown^R_i, 1\leq i\leq b,$ is at right of the middle path ${\mathcal{M}}$.
- A *left wing* ${\mathcal{L}}({\mathcal{F}})$ of a fan ${\mathcal{F}}$ is the union of raising-paths consisting of all joints in the sequential left down-pointing triangles $\triangledown^L_i, 1\leq i\leq a$ of the fan ${\mathcal{F}}$, and the down-pointing triangle $\triangledown^M$’s left joints with respect to the middle path ${\mathcal{M}}$. And
- A *right wing* ${\mathcal{R}}({\mathcal{F}})$ of a fan ${\mathcal{F}}$ is the union of raising-paths consisting of all joints in the sequential right down-pointing triangles $\triangledown^R_i, 1\leq i\leq b$ of the fan ${\mathcal{F}}$, and the down-pointing triangle $\triangledown^M$’s right joints with respect to the middle path ${\mathcal{M}}$.
From the definition of a raising-path, two contiguous raising-paths $(\Re(u), \Re(u')) \in {\mathcal{L}}$, $\Re(u)$ and $\Re(u')$ form a disjoint region ${\mathcal{W}}^L$, and $\Re(u)$ and $\Re(u')$ are the left and right boundaries of ${\mathcal{W}}^L$, respectively. Similarly, two raising-paths $(\Re(u), \Re(u')) \in {\mathcal{R}}$ form a region ${\mathcal{W}}^R$, and $\Re(u)$ and $\Re(u')$ are the left and right boundaries of ${\mathcal{W}}^R$, respectively. Hence we have two sequential regions for a fan ${\mathcal{F}}$: $\tilde{{\mathcal{W}}}^L({\mathcal{F}})$ and $\tilde{{\mathcal{W}}}^R({\mathcal{F}})$ that are at left and right of the middle path ${\mathcal{M}}$, respectively.
From the definitions of a raising fan $\tilde{{\mathcal{F}}}$ and a composite-layerlike graph ${\mathcal{G}}$, we can give another representation for a raising fan $\tilde{{\mathcal{F}}}$ as $({\mathcal{F}}_{1, 1}, {\mathcal{F}}_{1, 2}, \cdots, {\mathcal{F}}_{1, a_1},$ ${\mathcal{F}}_{2, 1}, {\mathcal{F}}_{2, 2}, \cdots, {\mathcal{F}}_{2, a_2},$ $\cdots)$ where for each subsequential fans $({\mathcal{F}}_{i, 1}, {\mathcal{F}}_{i, 2}, \cdots, {\mathcal{F}}_{i, a_i}), i\geq 1$, all upper vertices of each fan ${\mathcal{F}}_{i, j}, 1\leq j\leq a_i,$ are on the $i$-th layer of the layerlike graph $\Pi$ of ${\mathcal{G}}$. Also, subsequential fans $\tilde{{\mathcal{F}}}^C=({\mathcal{F}}_{1, 1}, {\mathcal{F}}_{1, 2}, \cdots,$ ${\mathcal{F}}_{1, a_1}, {\mathcal{F}}_{2, 1}, {\mathcal{F}}_{2, 2},$ $\cdots, {\mathcal{F}}_{2, a_2}, \cdots)$ are called *characteristic-fans* of $\tilde{{\mathcal{F}}}$ if for each $i\geq 1$, all upper vertices of $({\mathcal{F}}_{i, 1}, {\mathcal{F}}_{i, 2}, \cdots, {\mathcal{F}}_{i, a_i})$ are on the $i$-th layer of the ${\mathcal{G}}$’s layerlike graph $\Pi$. Also, the union of left wings and right wings of all fans in a characteristic-raising fan on the $i$-th layer are denoted as ${\mathcal{L}}_i(\tilde{{\mathcal{F}}}^C)$ and ${\mathcal{R}}_i(\tilde{{\mathcal{F}}}^C)$, respectively. And, the sequential lower vertices $(m_{i, 1}, m_{i, 2}, \cdots, m_{i, a_i})$ of the sequential fans $({\mathcal{F}}_{i, 1}, {\mathcal{F}}_{i, 2}, \cdots, {\mathcal{F}}_{i, a_i})$ are denoted as ${\mathcal{M}}_i(\tilde{{\mathcal{F}}}^C)$.
Let $\tilde{{\mathcal{F}}}^C$ $=({\mathcal{F}}_{1, 1}, {\mathcal{F}}_{1, 2},$ $\cdots,$ ${\mathcal{F}}_{1, a_1},$ ${\mathcal{F}}_{2, 1},$ ${\mathcal{F}}_{2, 2},$ $\cdots,$ ${\mathcal{F}}_{2, a_2},$ $\cdots)$ be the characteristic-raising fan of $\tilde{{\mathcal{F}}}$. The characteristic-raising fan $\tilde{{\mathcal{F}}}^C$ can form sequential regions as follows: initially, let the two term $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C)$ be empty sequence. Then repeatedly find the last fan ${\mathcal{F}}\in \tilde{{\mathcal{F}}}^C$ (the last fan ${\mathcal{F}}\in \tilde{{\mathcal{F}}}^C$ is inside a region between the rightmost and leftmost boundaries of $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C)$, respectively), remove the fan ${\mathcal{F}}$ from $\tilde{{\mathcal{F}}}^C$, and add two sequential regions $\tilde{{\mathcal{W}}}^L({\mathcal{F}})$ and $\tilde{{\mathcal{W}}}^R({\mathcal{F}})$ into $\tilde{{\mathcal{F}}}^L(\tilde{{\mathcal{F}}}^C)$ and $\tilde{{\mathcal{F}}}^L(\tilde{{\mathcal{F}}}^C)$, respectively where the two sequential regions $\tilde{{\mathcal{W}}}^L({\mathcal{F}})$ and $\tilde{{\mathcal{W}}}^R({\mathcal{F}})$ are formed by the fan ${\mathcal{F}}$ that are at left and at right of the middle path ${\mathcal{M}}$, respectively. After all fan are removed from the characteristic-raising fan $\tilde{{\mathcal{F}}}^C$, we can have two sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}})$ that are at left and right of the middle path ${\mathcal{M}}$, respectively.
For each subsequential fans $\tilde{{\mathcal{F}}}_{i, j}$ from ${\mathcal{F}}_{(i, j)+1}$ to ${\mathcal{F}}_{(i, j+1)-1}, i, j\geq1$, we know that the unions of left wings ${\mathcal{L}}_{i, j}$ and right wings ${\mathcal{R}}_{i, j}$ for all fans ${\mathcal{F}}$ from ${\mathcal{F}}_{(i, j)+1}$ to ${\mathcal{F}}_{(i, j+1)-1}$ are inside the rightmost region ${\mathcal{W}}^L_{i, j}$ and leftmost regions ${\mathcal{W}}^R_{i, j}$ of $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)_{i, j}$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C)_{i, j}$, respectively. Hence we can recursively partition the regions ${\mathcal{W}}^L_{i, j}$ and ${\mathcal{W}}^R_{i, j}$ by the two left wings ${\mathcal{L}}_{i, j}$ and right wings ${\mathcal{R}}_{i, j}$ and replace the two regions $\tilde{{\mathcal{W}}}^L_{i, j}$ and $\tilde{{\mathcal{W}}}^R_{i, j}$ by the two sequential regions $\tilde{{\mathcal{W}}}^L_{i, j}$ and $\tilde{{\mathcal{W}}}^R_{i, j}$, respectively. Now we can conclude that the union of the left wings and right wings of the raising fan $\tilde{{\mathcal{F}}}$ form two sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}})=({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p)$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}})=({\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots, {\mathcal{W}}^R_q)$.
We can conclude that a raising fan $\tilde{{\mathcal{F}}}=({\mathcal{F}}_1, {\mathcal{F}}_2, \cdots)$ can form sequential regions as $(\cdots,$ $\tilde{{\mathcal{W}}}^L_2,$ $\tilde{{\mathcal{W}}}^L_1,$ $\tilde{{\mathcal{W}}}^R_1,$ $\tilde{{\mathcal{W}}}^R_2,$ $\cdots)$ where $\tilde{{\mathcal{W}}}^L_i$ and $\tilde{{\mathcal{W}}}^R_i$ are sequential regions at left and right of the middle path ${\mathcal{M}}$ for a fan ${\mathcal{F}}_i$ of a raising fan $\tilde{{\mathcal{F}}}$, respectively. From the above discussion, we immediately have the following lemma.
\[lemma:fan-path-partition\] Given a raising fan $\tilde{{\mathcal{F}}}$ and its middle path ${\mathcal{M}}$, the unions of $\tilde{{\mathcal{F}}}$’s left wings and right wings can form sequential regions $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p,$ ${\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots, {\mathcal{W}}^R_q)$ where the two subsequential regions $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2,$ $\cdots,$ ${\mathcal{W}}^L_p)$ and $({\mathcal{W}}^R_1, {\mathcal{W}}^R_2,$ $\cdots,$ ${\mathcal{W}}^R_q)$ are at left and right of the middle path ${\mathcal{M}}$, respectively. Moreover, $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p)$ and $({\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots, {\mathcal{W}}^R_q)$ are called left and right sequential regions of $\tilde{{\mathcal{F}}}$, and denoted to $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}})$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}})$, respectively.
Now we can consider how to have an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout for a raising fan $\tilde{{\mathcal{F}}}=({\mathcal{F}}_1, {\mathcal{F}}_2, \cdots)$ in a ladder ${\mathcal{H}}$. When we have a raising fan $\tilde{{\mathcal{F}}}$, it implies that we have two sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}})$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}})$ from Lemma \[lemma:fan-path-partition\]. The basic idea to have an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout in ${\mathcal{H}}$ is that we orderly place the sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}})$ and reversely place the sequential regions ${\mathcal{W}}^R(\tilde{{\mathcal{F}}})$ in ${\mathcal{H}}$. To place the left wing and right wing of a raising fan $\tilde{{\mathcal{F}}}$, we can start to place the $\tilde{{\mathcal{F}}}$’s characteristic-fans $({\mathcal{F}}_{1, 1}, {\mathcal{F}}_{1, 2}, \cdots, {\mathcal{F}}_{1, a_1},$ $\cdots,$ ${\mathcal{F}}_{2, 1}, {\mathcal{F}}_{2, 2}, \cdots, {\mathcal{F}}_{2, a_2}, \cdots)$ and their union of left wings ${\mathcal{L}}_i$, union of right wings ${\mathcal{R}}_i$ and their sequential lower vertices ${\mathcal{M}}_i=(m_{i, 1}, m_{i, 2}, \cdots, m_{i, a_i})$ as the order: for each $i\geq 1$, we place ${\mathcal{L}}_i$ orderly, ${\mathcal{M}}_i$ reversely and ${\mathcal{R}}_i$ reversely on the $i$-th layer of ${\mathcal{H}}$.
Since the sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C_i)$ are placed orderly in ${\mathcal{H}}$, we must place ${\mathcal{M}}_i(\tilde{{\mathcal{F}}}^C)=(m_{i, 1},$ $m_{i, 2},$ $\cdots,$ $m_{i, a_i})$ reversely in ${\mathcal{H}}$ as $(m_{i, a_i},$ $\cdots,$ $m_{i, 2},$ $m_{i, 1})$ to avoid $X$-crossing edges in ${\mathcal{H}}$. For each raising fan $\tilde{{\mathcal{F}}}^C_i=({\mathcal{F}}^C_{i, 1}, {\mathcal{F}}^C_{i, 2}, \cdots, {\mathcal{F}}^C_{i, a_i})$, the sequential lower vertices ${\mathcal{M}}_i=(m_{i, 1}, m_{i, 2}, \cdots, m_{i, a_i})$ are reversely placed as $(m_{i, a_i},$ $\cdots,$ $m_{i, 2},$ $m_{i, 1})$ in ${\mathcal{H}}$, the sequential regions $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C_i)$ need to be placed reversely in ${\mathcal{H}}$ to avoid that $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C_i)$ make $X$-crossing edges in ${\mathcal{H}}$.
\[lem:fan-path-region-order\] Given the characteristic-raising fan $\tilde{{\mathcal{F}}}^C$ of a raising fan $\tilde{{\mathcal{F}}}$, the unions of left wings and right wings of $\tilde{{\mathcal{F}}}^C$ form two sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C)$, the sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)$ are orderly placed as $\tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C)$ and the sequential regions $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C)$ are reversely placed as $\tilde{{\mathcal{W}}}^R_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C)$. Also, the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C)=(\tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C),
\tilde{{\mathcal{W}}}^R_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
In the next theorem, we prove that a raising fan $\tilde{{\mathcal{F}}}$ can be $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in a ladder ${\mathcal{H}}$.
\[thm:fan-path-region-order\] Given a raising fan $\tilde{{\mathcal{F}}}$, the left wings and right wings of $\tilde{{\mathcal{F}}}$ form two sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}})=({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p)$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}})=({\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots, {\mathcal{W}}^R_q)$. If the sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}})=({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p)$ are sequentially placed in ${\mathcal{H}}$ as the order: $\tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})=({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p)$ and the sequential regions $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}})=({\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots, {\mathcal{W}}^R_q)$ are reversely placed as the order in ${\mathcal{H}}$: $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})=$ $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2,$ $\cdots, {\mathcal{W}}^R_p,$ ${\mathcal{W}}^R_q, \cdots,$ ${\mathcal{W}}^R_2, {\mathcal{W}}^R_1)$, then the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})=(\tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}), \tilde{{\mathcal{W}}}^R_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
From Lemma \[lemma:fan-path-partition\], we know that a raising fan $\tilde{{\mathcal{F}}}^C$ can form two sequential regions $(\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}),$ $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}))$ that are at left and right of the middle path ${\mathcal{M}}$, respectively.
For a characteristic-raising fan $\tilde{{\mathcal{F}}}^C$, we know that $(\tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C), \tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in a ladder ${\mathcal{H}}$ by Lemma \[lem:fan-path-region-order\]. Moreover, the sequential regions $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)$ are orderly placed in ${\mathcal{H}}$ as$\tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C)$ and the sequential regions $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C)$ are reversely placed in ${\mathcal{H}}$ as $\tilde{{\mathcal{W}}}^R_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^C)$. For each subsequential fans $\tilde{{\mathcal{F}}}_{i, j}$ from ${\mathcal{F}}_{(i, j)+1}$ to ${\mathcal{F}}_{(i, j+1)-1}, i, j\geq1$, we know that the unions of left wings ${\mathcal{L}}_{i, j}$ and right wings ${\mathcal{R}}_{i, j}$ of all fans from ${\mathcal{F}}_{(i, j)+1}$ to ${\mathcal{F}}_{(i, j+1)-1}$ are inside the rightmost region ${\mathcal{W}}^L_{i, j}$ and leftmost regions ${\mathcal{W}}^R_{i, j}$ of $\tilde{{\mathcal{W}}}^L(\tilde{{\mathcal{F}}}^C)_{i, j}$ and $\tilde{{\mathcal{W}}}^R(\tilde{{\mathcal{F}}}^C)_{i, j}$, respectively. Hence we can recursively partition the regions ${\mathcal{W}}^L_{i, j}$ and ${\mathcal{W}}^R_{i, j}$ by the two left wings ${\mathcal{L}}_{i, j}$ and right wings ${\mathcal{R}}_{i, j}$ to have two sequential regions $\tilde{{\mathcal{W}}}({\mathcal{L}}_{i, j})$ and $\tilde{{\mathcal{W}}}({\mathcal{R}}_{i, j})$ that can be $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed as $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{L}}_{i, j})$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{R}}_{i, j})$ in ${\mathcal{H}}$. Because the regions ${\mathcal{W}}^L_{i, j}$ and ${\mathcal{W}}^R_{i, j}$ are orderly and reversely placed in ${\mathcal{H}}$, respectively, we can place the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{L}}_{i, j})$ orderly and the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{R}}_{i, j})$ reversely inside the two regions $\tilde{{\mathcal{W}}}^L_{i, j}$ and $\tilde{{\mathcal{W}}}^R_{i, j}$, respectively to have an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout in ${\mathcal{H}}$. Moreover, the middle path from ${\mathcal{F}}_{(i, j)+1}$ to ${\mathcal{F}}_{(i, j+1)-1}$ is placed reversely between the two vertices $m_{(i, j)}$ and $m_{(i, j+1)-1}$.
Because (1) ${\mathcal{L}}_{i, j}$ are placed between ${\mathcal{F}}_{i, j}$ and ${\mathcal{F}}_{i, j+1}$, and (2) the middle path from ${\mathcal{F}}_{(i, j)+1}$ to ${\mathcal{F}}_{(i, j+1)-1}$ is placed reversely between the two vertices $m_{(i, j)}$ and $m_{(i, j+1)-1}$, each left arm from ${\mathcal{F}}_{(i, j)+1}$ to ${\mathcal{F}}_{i, j+1}-1$ is placed between tracks $i$ and $i+1$ cannot make $X$-crossing with left arms of the characteristic-raising fan $\tilde{{\mathcal{F}}}^C$ between tracks $i$ and $i+1$in ${\mathcal{H}}$. Similarly, each right arm of ${\mathcal{R}}_{i, j}$ is placed between tracks $i$ and $i+1$ cannot make $X$-crossing with right arms of the characteristic-raising fan $\tilde{{\mathcal{F}}}^C$ between tracks $i$ and $i+1$ in ${\mathcal{H}}$.
Finally, we conclude that the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})=(\tilde{{\mathcal{W}}}^L_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}), \tilde{{\mathcal{W}}}^R_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
How to Find a Skeleton $\Psi$ in a Region ${\mathcal{W}}$ to Have an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-Well-Placed Layout $\Psi_{{\mathcal{H}}}$ in a Ladder ${\mathcal{H}}$? {#sec:skeleton}
===========================================================================================================================================================================================
In this section, we prove that a skeleton $\Psi$ of a regions ${\mathcal{W}}$ can be obtained from all *rightward-outer* and *leftward-outer* fans in ${\mathcal{W}}$ and can have an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout in ${\mathcal{H}}$.
A Forest-Like Representation $\clubsuit_{{\mathcal{B}}}$ of Raising Fans from a Boundary ${\mathcal{B}}$ {#sec:forest-like}
--------------------------------------------------------------------------------------------------------
In this subsection, we show how to build a forest-like representation $\clubsuit_{{\mathcal{B}}}$ from all *rightward-outer* fans of all vertices on a boundary ${\mathcal{B}}$.
Given a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$, let $v$ be a vertex on the boundary ${\mathcal{B}}^L$ and $(v_1, v_2, \cdots)$ be the sequential children of $v$ inside the region ${\mathcal{W}}$. A fan ${\mathcal{F}}=(u_1, u_2, \cdots, m)$ is called a *rightward* for $v$’ children on a left boundary of a region ${\mathcal{W}}'$ if ${\mathcal{F}}$’s upper vertices can be partitioned into two contiguous subsequences that the first contiguous subsequence is among $v$’s children and the second contiguous subsequence is not among $v$’s children.
A fan ${\mathcal{F}}=(u_1, u_2, \cdots, m)$ is called a *rightward* fan for a vertex $v$ on a left boundary of a region ${\mathcal{W}}'$ if ${\mathcal{F}}$’s the first upper vertex $u_1$ is the vertex $v$.
Given a subregion ${\mathcal{W}}'$ inside a region ${\mathcal{W}}$ and the vertex $v$ is on the lowest layer of the intersection of the ${\mathcal{W}}$’s left boundary ${\mathcal{B}}^L$ and ${\mathcal{W}}'$’s left boundary, a *leftmost-raising-fan* path $\tilde{{\mathcal{F}}}$ inside the subregion ${\mathcal{W}}'$ of ${\mathcal{W}}$ is a maximal raising fan consisting of all $v$’s rightward-outer fans. Intuitively, the layer of any rightward-outer fan not included in $\tilde{{\mathcal{F}}}$ can not be lower than the lowest layer of a rightward-outer fan included in $\tilde{{\mathcal{F}}}$.
Similarly, a *leftward-outer* fan and a *leftward* fan of a vertex $v$ in a region ${\mathcal{W}}'$ also can be defined symmetrically to a rightward fan and a rightward-outer fan of a vertex $v\in {\mathcal{B}}^R$, respectively. And, a *rightmost-raising-fan* path inside a region ${\mathcal{W}}'$ also can be defined symmetrically to a leftmost raising fan inside a region ${\mathcal{W}}'$.
Without loss of generality, assume that ${\mathcal{B}}$ is ${\mathcal{W}}$’s the left boundary, we plan to find a set of raising fans from ${\mathcal{B}}$’s all rightward-outer fans and represent them as a forest-like representation $\clubsuit_{{\mathcal{B}}}$ where each vertex of $\clubsuit_{{\mathcal{B}}}$ represents a raising fan. Initially $\clubsuit_{{\mathcal{B}}}$’s root is a raising fan $\tilde{{\mathcal{F}}}$ consisting the only region ${\mathcal{W}}$ and add the region ${\mathcal{W}}$ into $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}})$. Then $\clubsuit_{{\mathcal{B}}}$ can be constructed as follows: repeatedly find a region ${\mathcal{W}}'$ from the sequential regions having uncolored rightward-outer fans inside the region ${\mathcal{W}}'$ and ${\mathcal{W}}'$ is a region of a raising fan $\tilde{{\mathcal{F}}}$ , to
1. find a maximal raising fan $\tilde{{\mathcal{F}}}'$ consisting of all leftmost and uncolored fans inside the region ${\mathcal{W}}'$,
2. assign the raising fan $\tilde{{\mathcal{F}}}'$ to a child of $\tilde{{\mathcal{F}}}$ consisting of the region ${\mathcal{W}}'$ in $\clubsuit_{{\mathcal{B}}}$ and color the raising fan $\tilde{{\mathcal{F}}}'$,
3. partition the region ${\mathcal{W}}'$ into sequential regions $\tilde{{\mathcal{W}}}'$ by the raising fan $\tilde{{\mathcal{F}}}'$ and replace the region ${\mathcal{W}}'$ by the sequential regions $\tilde{{\mathcal{W}}}'$ in $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}})$, and
till no any such region can be found. Now we immediately have the following two lemmas:
Given a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$ with the left and right boundaries ${\mathcal{B}}^L$ and ${\mathcal{B}}^R$, the collection of all rightward-outer and leftward-outer fans of the boundary ${\mathcal{B}}^L$ and ${\mathcal{B}}^R$ form sequential regions $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p)$ and $({\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots, {\mathcal{W}}^R_q)$, respectively. Also, the two sequential regions partition the region ${\mathcal{W}}$ into sequential regions $({\mathcal{W}}^L_1, {\mathcal{W}}^L_2, \cdots, {\mathcal{W}}^L_p,$ ${\mathcal{W}}^M,$ ${\mathcal{W}}^R_1, {\mathcal{W}}^R_2, \cdots, {\mathcal{W}}^R_q)$ where ${\mathcal{W}}^M$ is a region bounded by the right boundary of ${\mathcal{W}}^L_p$ and the left boundary of ${\mathcal{W}}^R_1$.
\[lem:forest-like\] Given a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$, we can have two forest-like representations $\clubsuit_{{\mathcal{B}}^L}$ and $\clubsuit_{{\mathcal{B}}^R}$ from the collection of all rightward-outer and leftward-outer fans of ${\mathcal{B}}^L$ and ${\mathcal{B}}^R$, respectively.
A forest-like structure $\clubsuit_{{\mathcal{B}}}$ is called *left-forest-like* structure, if ${\mathcal{B}}$ is the left boundary of a region ${\mathcal{W}}$.
Recall that a boundary ${\mathcal{B}}$’s forest-like structure $\clubsuit_{{\mathcal{B}}}$ is the union of all rightward-outer fans of $(u_1, u_2, \cdots, u_l)$ where each $u_i, i\geq 1,$ is a vertex on the boundary ${\mathcal{B}}$. Observe that for each vertex $u\in {\mathcal{B}}$, every $u$’s child $u'$ is inside a region of $u$’s rightward-outer fan ${\mathcal{F}}$ and ${\mathcal{F}}$ is a fan in a raising fan $\tilde{{\mathcal{F}}}\in (\tilde{{\mathcal{F}}}_1, \tilde{{\mathcal{F}}}_2, \cdots)$. Hence $u'$ is inside a region of $\tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}})$ and $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}})$. From the above discussion, we have the following lemma:
\[lem:region-partition\] Given a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$ and a boundary ${\mathcal{B}}\in ({\mathcal{B}}^L, {\mathcal{B}}^R)$, for each vertex $u\in {\mathcal{B}}$, every $u$’s child is inside a region of sequential regions $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}})$.
From the above lemma, we know that the union of the two forest-like structures $\clubsuit_{{\mathcal{B}}^L}$ and $\clubsuit_{{\mathcal{B}}^R}$ form a skeleton of a region ${\mathcal{W}}$. $\clubsuit_{{\mathcal{B}}^L}$ and $\clubsuit_{{\mathcal{B}}^R}$ are called *left skeleton* $\Psi^L({\mathcal{W}})$ and *right skeleton* $\Psi^R({\mathcal{W}})$, respectively.
\[thm:skeleton\] Given a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$, the union of the left skeleton and right skeleton $\Psi^L({\mathcal{W}})$ and $\Psi^R({\mathcal{W}})$ is a skeleton of ${\mathcal{W}}$.
An $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-Well-Placed Layout in a Ladder ${\mathcal{H}}$ for a Forest-Like Structure $\clubsuit_{{\mathcal{B}}}$ {#sec:placement-forest-like}
---------------------------------------------------------------------------------------------------------------------------------------------------------
Let $\tilde{{\mathcal{F}}}$ be the raising fan at the $\clubsuit_{{\mathcal{B}}}$’s root $r$
Orderly place the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})$ in ${\mathcal{H}}$
Orderly add the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})$ into the first-in-first-out queue $\tilde{{\mathcal{W}}}$
Before describing Algorithm \[alg:framework-boundary\], we need to roughly define the term $\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i)$ as a subset of raising fans in $\clubsuit_{{\mathcal{B}}}$ which consist of all raising fans inside $\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i)$ meet at a vertex on the right boundary of ${\mathcal{W}}_i$. Moreover, the subset of raising fans $\clubsuit({\mathcal{W}}_i)$ partition ${\mathcal{W}}_i$ into sequential regions and can be a $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout in ${\mathcal{H}}$. (They are proven in Lemmas \[lem:partition-boundary\] and \[lem:layout-boundary\], respectively.) Initially, in Algorithm \[alg:framework-boundary\], the root $r$’s raising fan $\tilde{{\mathcal{F}}}$ in a forest-like structure $\clubsuit_{{\mathcal{B}}}$ form sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})$ in ${\mathcal{H}}$ and we put them into the first-in-first-out queue orderly. Next each iteration $i$ of Algorithm \[alg:framework-boundary\] picks up the first region ${\mathcal{W}}_i$ in the first-in-first-out queue $\tilde{{\mathcal{W}}}$ (it means that the region ${\mathcal{W}}_i$ at the leftmost region such that the region ${\mathcal{W}}_i$ consists of raising fans in $\clubsuit_{{\mathcal{B}}}$ in ${\mathcal{H}}$; $|\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i)|>0$) to place the sequential regions $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ (it can be proven in Lemma \[lem:partition-boundary\]) at the rightmost side in ${\mathcal{H}}$ as $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$. (It can be proven in Lemma \[lem:layout-boundary\].) It means that all regions ${\mathcal{W}}_j$ at left of ${\mathcal{W}}_i$ in ${\mathcal{H}}$ doesn’t consist any raising fan in $\clubsuit_{{\mathcal{B}}}$ ($|\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j)|=0$) and the chords between ${\mathcal{W}}_j$ and $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ don’t nest with the chords between ${\mathcal{W}}_i$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ because ${\mathcal{W}}_j$ is at left of ${\mathcal{W}}_i$ in ${\mathcal{H}}$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ are at left of $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ in ${\mathcal{H}}$. It means that the left end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ are placed at left of the left end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$, and the right end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ are placed at left of the right end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ in ${\mathcal{H}}$.
Next, we give a precise definition of $\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i)$ as follows:
Given a region ${\mathcal{W}}_i=({\mathcal{B}}^L, {\mathcal{B}}^R)$ and a left forest-like structure $\clubsuit_{{\mathcal{B}}}$, $\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i)$ is defined to consist of two sequential raising fans $\clubsuit^L=(\tilde{{\mathcal{F}}}^L_1, \tilde{{\mathcal{F}}}^L_2, \cdots, \tilde{{\mathcal{F}}}^L_p)$ and $\clubsuit^R=(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$ where
- $\clubsuit^L=(\tilde{{\mathcal{F}}}^L_1, \tilde{{\mathcal{F}}}^L_2, \cdots, \tilde{{\mathcal{F}}}^L_p)$ are the maximal subsequential raising fans in $\clubsuit_{{\mathcal{B}}}$ such that for each raising fan $\tilde{{\mathcal{F}}}^L_i, 1\leq i\leq p$, (1) $\tilde{{\mathcal{F}}}^L_i$ does not touch any vertex on the right boundary ${\mathcal{B}}^R$, (2) $\tilde{{\mathcal{F}}}^L_i$ is not a descendant of any raising fan $\tilde{{\mathcal{F}}}^R_j, 1\leq j\leq q$, and (3) $\tilde{{\mathcal{F}}}^L_i$ is a right sibling of $\tilde{{\mathcal{F}}}^L_{i-1}$ in $\clubsuit_{{\mathcal{B}}}$. And,
- $\clubsuit^R=(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$ are the maximal subsequential raising fans in $\clubsuit_{{\mathcal{B}}}$ such that for each raising fan $\tilde{{\mathcal{F}}}^R_i, 1\leq i\leq q$, $\tilde{{\mathcal{F}}}^R_i$’s right wings touch the right boundary ${\mathcal{B}}^R$.
In the followings, we prove three properties: the first one proves that sequential raising fans $\clubsuit^R$ form a contiguous path in $\clubsuit_{{\mathcal{B}}}$. Properties \[prop:left-contiguous\] and \[prop:right-contiguous\] state that for each vertex $v$, all rightward-outer fans for children of a vertex $v\notin {\mathcal{B}}$ form a contiguous subsequence in $\clubsuit^R$. Also, all rightward fans for a vertex $v\notin {\mathcal{B}}$ also form contiguous subsequence in $\clubsuit^R$.
For any region ${\mathcal{W}}_i\in \tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}^R_i})$, ${\mathcal{W}}_i$ has three different types: the type one is the ${\mathcal{W}}_i$’s root is at a vertex on the left boundary of ${\mathcal{W}}_{i-1}$. The type two is that ${\mathcal{B}}^L({\mathcal{W}}_i)\cap {\mathcal{B}}^L({\mathcal{W}}_{i-1})$ and ${\mathcal{B}}^R({\mathcal{W}}_i)\cap {\mathcal{B}}^R({\mathcal{W}}_{i-1})$ are sub-paths of ${\mathcal{B}}^L({\mathcal{W}}_{i-1})$ and ${\mathcal{B}}^R({\mathcal{W}}_{i-1})$ from the ${\mathcal{W}}_{i-1}$’s root, respectively. The type three is that the ${\mathcal{W}}_i$’s is at a vertex on the right boundary of ${\mathcal{W}}_{i-1}$. In the sequential regions $\tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^R_i)$, types one, two and three are orderly appeared from left to right. Notes that the third type of region can bound contiguous raising fans of $\clubsuit^R$.
Given a region ${\mathcal{W}}_i\in \tilde{{\mathcal{F}}}^R_i$, the region ${\mathcal{W}}_i$ is called a *black-hole* if ${\mathcal{W}}_i$ is the type two. Intuitively, a black-hole is a region such that it can bound a contiguous subsequence $(\tilde{{\mathcal{F}}}^R_{i+1}, \tilde{{\mathcal{F}}}^R_{i+2}, \cdots, \tilde{{\mathcal{F}}}^R_q)$ in $\clubsuit^R$. The following observation states that for each raising fan in $\clubsuit^R$, there is the only one region which can be a black-hole. Note that for a region ${\mathcal{W}}_i$ in a raising fan $\tilde{{\mathcal{F}}}^R_i$. If the ${\mathcal{W}}_i$’s root is at a vertex on the left boundary of ${\mathcal{W}}_{i-1}$ or on the right boundary of ${\mathcal{W}}_{i-1}$, then ${\mathcal{W}}_i$ cannot consist of any raising fan in $\clubsuit^R$. Hence the only region in $\tilde{{\mathcal{F}}}_i$ which can consist of contiguous raising fans $(\tilde{{\mathcal{F}}}^R_{i+1}, \tilde{{\mathcal{F}}}^R_{i+2}, \cdots, \tilde{{\mathcal{F}}}^R_q)$ in $\clubsuit^R$ is the $\tilde{{\mathcal{F}}}$’s black-hole.
\[obs:black-hole\] If a region ${\mathcal{W}}_i\in \tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^R_i)$ is a black-hole, then the region ${\mathcal{W}}_i$ bounds subsequential raising fans $(\tilde{{\mathcal{F}}}^R_{i+1}, \tilde{{\mathcal{F}}}^R_{i+2}, \cdots, \tilde{{\mathcal{F}}}^R_q)\subseteq \clubsuit^R$ where $(\tilde{{\mathcal{F}}}^R_{i+1}, \tilde{{\mathcal{F}}}^R_{i+2}, \cdots, \tilde{{\mathcal{F}}}^R_q)\subseteq \clubsuit^R$ form a contiguous subpath in $\clubsuit^R$ and $\clubsuit_{{\mathcal{B}}}$. And, ${\mathcal{W}}_i$ is the only one black-hole in the raising fan $\tilde{{\mathcal{F}}}^R_i$.
Sequential raising fans $\clubsuit^R=(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$ are an ancestor-descendant path in $\clubsuit_{{\mathcal{B}}}$ such that for each $1\leq i\leq q-1$, a raising fan $\tilde{{\mathcal{F}}}^R_i$ is the parent of $\tilde{{\mathcal{F}}}^R_{i+1}$ in $\clubsuit_{{\mathcal{B}}}$. ($\tilde{{\mathcal{F}}}^R_i$ bounds $\tilde{{\mathcal{F}}}^R_{i+1}$.)
From Observation \[obs:black-hole\], we know that for each raising fan $\tilde{{\mathcal{F}}}^R_i, 1\leq i\leq q$, there is the only one black-hole in the raising fan $\tilde{{\mathcal{F}}}^R_i$. Then for each raising fan $\tilde{{\mathcal{F}}}^R_i, 1\leq i\leq q-1$, $\tilde{{\mathcal{F}}}^R_i$ bounds the raising fan $\tilde{{\mathcal{F}}}^R_{i+1}$ in $\clubsuit^R$ and $\tilde{{\mathcal{F}}}^R_i$ is the parent of $\tilde{{\mathcal{F}}}^R_{i+1}$ in $\clubsuit_{{\mathcal{B}}}$. Hence we can prove that $\clubsuit^R=(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)
$ is a path in $\clubsuit_{{\mathcal{B}}}$.
\[prop:left-contiguous\] Sequential raising fans $\clubsuit^R$ have the following property: for each vertex $v \notin {\mathcal{B}}$ in the right wing of a raising fan $\tilde{{\mathcal{F}}} \in \clubsuit^R$,
1. all rightward fans for the vertex $v$ form at most one raising fan in $\clubsuit^R$ and
2. all rightward fans for $v$’s children form at most one raising fan in $\clubsuit^R$.
Let $v$ be a vertex at the right boundary of $\tilde{{\mathcal{F}}}$’s right wing. All rightward fans for the vertex $v$ and rightward-outer fans for $v$’s children are consisted in at most one raising fan $\tilde{{\mathcal{F}}}' \in \clubsuit^R$. We know that no any raising fan in $\clubsuit^R$ can be inside a region formed by $\tilde{{\mathcal{F}}}'$’s left wing and it leads that the right boundary of $\tilde{{\mathcal{F}}}$’s left wing cannot overlap $\tilde{{\mathcal{F}}}'$’s right wing.
Hence we can conclude that for each vertex $v\notin {\mathcal{B}}$ that $v$ is a vertex of the right wing of $\tilde{{\mathcal{F}}} \in \clubsuit^R$, all rightward fans for the vertex $v$ and all rightward-outer fans for $v$’s children form at most one raising fan $\tilde{{\mathcal{F}}}'$ in $\clubsuit^R$.
\[prop:right-contiguous\] Sequential raising fans $\clubsuit^R=(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$ have the following property: for each vertex $v\notin {\mathcal{B}}$ in the right wing of a raising fan $\tilde{{\mathcal{F}}}^R_i\in \clubsuit^R$, all $u$’s leftward-outer fans form contiguous raising fans $(\tilde{{\mathcal{F}}}^R_i, \tilde{{\mathcal{F}}}^R_{i+1}, \cdots, \tilde{{\mathcal{F}}}^R_a) \in \clubsuit^R$.
Let ${\mathcal{W}}'$ be a black-hole passes through a vertex $u$ where the vertex $u$ is on the right boundary ${\mathcal{B}}^R({\mathcal{W}}')$ of ${\mathcal{W}}'$. Let $\tilde{{\mathcal{F}}}^R_{a_1}$ be the first raising fan in $\clubsuit^R$ such that the black-hole ${\mathcal{W}}_{a_1} \in \tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^R_{a_1})$ passes through the vertex $u$. Observe that all $u$’s leftward-outer fans are shared by a maximal contiguous black-holes $({\mathcal{W}}_{a_1}, {\mathcal{W}}_{a_1+2}, \cdots, {\mathcal{W}}_{a_2})$ where each black-hole ${\mathcal{W}}_j, a_1\leq j\leq a_2$, passes through the vertex $u$ till the black-hole ${\mathcal{W}}_{a_2+1}\in \tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^R_{a_2+1})$ doesn’t pass through the vertex $u$. Now we know that there are sequential raising fans $(\tilde{{\mathcal{F}}}^R_{a_1}, \tilde{{\mathcal{F}}}^R_{a_1+1}, \cdots, \tilde{{\mathcal{F}}}^R_{a_2}) \subseteq \clubsuit^R$ such that each raising fan $\tilde{{\mathcal{F}}}^R_i, a_1\leq i\leq a_2,$ consists of a black-hole ${\mathcal{W}}_i$. When the black-hole in $\tilde{{\mathcal{F}}}^R_{a_2+1}$ doesn’t passes through the vertex $u$, each raising fan $\tilde{{\mathcal{F}}}^R_j, a_2+1\leq j\leq q$, cannot consists of any $u$’s leftward-outer fan. Hence we can prove that $u$’s leftward-outer fans are shared by contiguous raising fans $(\tilde{{\mathcal{F}}}^R_{a_1}, \tilde{{\mathcal{F}}}^R_{a_1+1}, \cdots, \tilde{{\mathcal{F}}}^R_{a_2}) \subseteq \clubsuit^R$.
Now we can construct sequential regions $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))$ from the sequential raising fans $(\clubsuit^L, \clubsuit^R)$ in a region ${\mathcal{W}}$ as follows: firstly, because the sequential raising fans $(\tilde{{\mathcal{F}}}^L_1, \tilde{{\mathcal{F}}}^L_2, \cdots, \tilde{{\mathcal{F}}}^L_p, \tilde{{\mathcal{F}}}^R_1)$ are mutually disjoint (they don’t have any ancestor-descendant relation in $\clubsuit_{{\mathcal{B}}}$), the sequential raising fans $(\tilde{{\mathcal{F}}}^L_1, \tilde{{\mathcal{F}}}^L_2, \cdots, \tilde{{\mathcal{F}}}^L_p, \tilde{{\mathcal{F}}}^R_1)$ partition ${\mathcal{W}}$ into sequential disjoint regions $(\tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^L_1),$ $ \tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^L_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^L_p),$ $\tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^R_1))$. Secondly, for each raising fan $\tilde{{\mathcal{F}}}^R_i, 2\leq i\leq q$, process the following steps: let ${\mathcal{W}}_i$ be a region in $\tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^R_i)$ bounds a raising fan $\tilde{{\mathcal{F}}}^R_{i+1}$. Then replace ${\mathcal{W}}_i$ by $\tilde{{\mathcal{W}}}(\tilde{{\mathcal{F}}}^R_{i+1})$. After the last raising fan $\tilde{{\mathcal{F}}}^R_q$ is processed, we get sequential disjoint regions $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))$.
Now we show how to place sequential regions $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))$ in ${\mathcal{H}}$ as $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))$ and utilize it to place sequential regions $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}})$ in ${\mathcal{H}}$ as $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}})$ such that $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}})$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
1. Firstly, we place the sequential regions $(\tilde{{\mathcal{W}}}({\mathcal{F}}^L_1), \tilde{{\mathcal{W}}}({\mathcal{F}}^L_2), \cdots, \tilde{{\mathcal{W}}}({\mathcal{F}}^L_p), \tilde{{\mathcal{W}}}({\mathcal{F}}^R_1))$ as $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{F}}^L_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{F}}^L_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{F}}^L_p),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}({\mathcal{F}}^R_1))$ in ${\mathcal{H}}$.
2. Because each raising fan $\tilde{{\mathcal{F}}}^R_i, 2\leq i\leq q$, is inside a region ${\mathcal{W}}_i$ of $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_{i-1})$, we place the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_i)$ at right of the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_{i-1})$ in ${\mathcal{H}}$.
Now we have new sequential regions in ${\mathcal{H}}$ as follows: $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))=$ $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_p),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_q))$. And, we immediately have the following lemma.
\[lem:partition-boundary\] Regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))$ are sequential in ${\mathcal{H}}$.
The next lemma proves that the sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))=$ $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_p),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_q))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
\[lem:layout-boundary\] Sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
Recall that $\clubsuit^R=(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$ are sequentially placed in ${\mathcal{H}}$ as $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_1)$, $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_q))$ such that each raising fan $\tilde{{\mathcal{F}}}^R_i, 2\leq i\leq q$, is at right of $\tilde{{\mathcal{F}}}^R_{i-1}$ in ${\mathcal{H}}$.
In the followings, we utilize Property \[prop:left-contiguous\] to prove that for each vertex $v\notin {\mathcal{B}}$, edges between $v$ and $v$’s children have constant $X$-crossing edges with other edges in $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_1)$, $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_q))$. Because all $v$’ rightward-outer fans form contiguous raising fans $(\tilde{{\mathcal{F}}}_i, \tilde{{\mathcal{F}}}_{i+1})$ with length at most two in $(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$. The edges between $v$ and $v$’s children make $X$-crossing edges with the only raising fan $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}_{i+1})$ in ${\mathcal{H}}$. Hence the edges between $v$ and $v$’s children have $X$-crossing number at most one.
As we place $\clubsuit^R=(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$ as $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_q))$ in ${\mathcal{H}}$, we have sequential edges $(\tilde{e}_1, \tilde{e}_2, \cdots, \tilde{e}_{q-1})$ that each edge $e\in \tilde{e}_i, 1\leq i\leq q-1$, connects between two raising fans $\tilde{{\mathcal{F}}}^R_i$ and $\tilde{{\mathcal{F}}}^R_{i+1}$ except for edges between $v$ and $v$’s children, Observe that $(\tilde{e}_1, \tilde{e}_2, \cdots, \tilde{e}_{q-1})$ are orderly placed in ${\mathcal{H}}$. So, there is no any $X$-crossing edge among $(\tilde{e}_1, \tilde{e}_2, \cdots, \tilde{e}_{q-1})$.
Since any two raising fans in $(\tilde{{\mathcal{F}}}^L_1,$ $\tilde{{\mathcal{F}}}^L_2,$ $\cdots,$ $\tilde{{\mathcal{F}}}^L_p)$ are siblings in $\clubsuit_{{\mathcal{B}}}$ (two raising fans $\tilde{{\mathcal{F}}}$ and $\tilde{{\mathcal{F}}}'$ are siblings in $\clubsuit_{{\mathcal{B}}}$, $\tilde{{\mathcal{F}}}$ and $\tilde{{\mathcal{F}}}'$ are not bounded to each other), the placement: $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_p))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$. Also, because $(\tilde{{\mathcal{F}}}^L_1, \tilde{{\mathcal{F}}}^L_2, \cdots, \tilde{{\mathcal{F}}}^L_p)$ and $(\tilde{{\mathcal{F}}}^R_1, \tilde{{\mathcal{F}}}^R_2, \cdots, \tilde{{\mathcal{F}}}^R_q)$ are not bounded to each other, except for the edges between a vertex on a black-hole, the placement: $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_1), \tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_2), \cdots, \tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_p))$ and $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_q))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
In the followings, we utilize Property \[prop:right-contiguous\] to prove that for each vertex $v\notin {\mathcal{B}}$, all $v$’s leftward-outer fans have constant number of $X$-crossing edges in ${\mathcal{H}}$. Let $v$ be a vertex on the right boundary of a black-hole. By Property \[prop:right-contiguous\], there exists contiguous raising fans $(\tilde{{\mathcal{F}}}^R_i, \tilde{{\mathcal{F}}}^R_{i+1}, \cdots, \tilde{{\mathcal{F}}}^R_a) \subseteq \clubsuit^R$ which consists of $v$’s leftward-outer fans. All edges between $v$ and $v$’s children cross from the sequential raising fans $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_i),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_{i+1}),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_a))$ in ${\mathcal{H}}$.
Let the sequential vertices $(v_1, v_2, \cdots, v_h)$ be orderly on a track in ${\mathcal{H}}$ such that each vertex $v_j, j\geq 1,$ consists of some leftward-outer fans. Let $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots, {\mathcal{W}}_h)$ be sequential black-holes in ${\mathcal{H}}$ such that each vertex $v_j, 1\leq j\leq h$, is on the right boundary of ${\mathcal{W}}_j$.
We know that for each vertex $v_j, 1\leq j\leq h$, there are contiguous raising fans $(\tilde{{\mathcal{F}}}_{a_j},$ $\tilde{{\mathcal{F}}}_{a_j+1},$ $\cdots$ $\tilde{{\mathcal{F}}}_{b_j})$ passing through $v_j$. Also, since the sequential black-holes $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots, {\mathcal{W}}_h)$ are disjoint, we have the following ordered relation $(a_1 < a_2 \cdots < a_h)$. Now, the sequential vertices $(v_1, v_2, \cdots, v_h)$ are orderly placed on a track in ${\mathcal{H}}$ and the sequential children $({\mathcal{C}}_{v_1}, {\mathcal{C}}_{v_2}, \cdots, {\mathcal{C}}_{v_h})$ of $(v_1, v_2, \cdots, v_h)$ are also orderly placed on other track in ${\mathcal{H}}$ because the order relation $(a_1\leq b_1 < a_2 \leq b_2 \cdots < a_h \leq b_h)$.
Hence for all sequential edges $(\tilde{e}(v_1),$ $\tilde{e}(v_2),$ $\cdots,$ $\tilde{e}(v_h))$ where each edges $\tilde{e}(v_j), 1\leq j\leq h$, are edges between the vertex $v_j$ and and $v_j$’s children ${\mathcal{C}}_{v_j}$, $(\tilde{e}(v_1),$ $\tilde{e}(v_2),$ $\cdots,$ $\tilde{e}(v_h))$ are not $X$-crossing in ${\mathcal{H}}$.
Finally, when we add the sequential edges $(\tilde{e}(v_1), \tilde{e}(v_2), \cdots, \tilde{e}(v_h))$ into ${\mathcal{H}}$, the $X$-crossing number in ${\mathcal{H}}$ increase one in ${\mathcal{H}}$ because the sequential edges $(\tilde{e}(v_1), \tilde{e}(v_2), \cdots, \tilde{e}(v_h))$ make $X$-crossing edges with the sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots, {\mathcal{W}}_h)$ in ${\mathcal{H}}$. and the layout: $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_2),$ $cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^L_p),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_1),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_2),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}}^R_q))$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
\[lem:full-layout-boundary\] Sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}})$ are $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
Initially, the root $r$’s raising fan $\tilde{{\mathcal{F}}}$ in a forest-like structure $\clubsuit_{{\mathcal{B}}}$ form sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\tilde{{\mathcal{F}}})$ in ${\mathcal{H}}$. Next each iteration of Algorithm \[alg:framework-boundary\] picks up the first region ${\mathcal{W}}_i$ in the first-in-first-out queue $\tilde{{\mathcal{W}}}$ to place the sequential regions $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ at the rightmost side in ${\mathcal{H}}$; The region ${\mathcal{W}}_i$ at the leftmost region consists of raising fans in $\clubsuit_{{\mathcal{B}}}$ in ${\mathcal{H}}$. ($|\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i)|>0$.)
All regions ${\mathcal{W}}_j$ at left of ${\mathcal{W}}_i$ in ${\mathcal{H}}$ doesn’t consist of any raising fan in $\clubsuit_{{\mathcal{B}}}$ ($|\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j)|=0$) and the chords between ${\mathcal{W}}_j$ and $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ don’t nest with the chords between ${\mathcal{W}}_i$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ because ${\mathcal{W}}_j$ is at left of ${\mathcal{W}}_i$ in ${\mathcal{H}}$ and $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ are at left of $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ in ${\mathcal{H}}$. The above description also implies that the left end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ are placed at left of the left end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$, and the right end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_j))$ are placed at left of the right end-vertices of the chords $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ in ${\mathcal{H}}$.
Observe that if sequential regions $(\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_1)),$ $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_2)),$ $\cdots)$ are orderly placed in ${\mathcal{H}}$ as $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_1)),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_2)),$ $\cdots)$ in Algorithm \[alg:framework-boundary\], then the sequential regions $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$ are also orderly placed in ${\mathcal{H}}$. Let $({\mathcal{W}}_1, {\mathcal{W}}_2, \cdots)$ be sequential regions in ${\mathcal{H}}$ and $(\tilde{e}_1, \tilde{e}_2, \cdots)$ be sequential chords in ${\mathcal{H}}$ where $\tilde{e}_i, i\geq 1,$ are edges between ${\mathcal{W}}_i$ and $\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$. When we place a new sequential regions $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i))$ at right of the sequential regions $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_1)),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_2)),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_{i-1})))$, the sequential chords $(\tilde{e}_1,$ $\tilde{e}_2,$ $\cdots,$ $\tilde{e}_i)$ don’t nest to each other in ${\mathcal{H}}$ because the order of $({\mathcal{W}}_1,$ ${\mathcal{W}}_2,$ $\cdots,$ ${\mathcal{W}}_i)$ in ${\mathcal{H}}$ is the same as the order of $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_1)),$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_2)),$ $\cdots,$ $\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}}({\mathcal{W}}_i)))$ in ${\mathcal{H}}$. Hence the layout in Algorithm \[alg:framework-boundary\] is $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$.
In Algorithm \[alg:framework-boundary\], we place a forest-like raising fans $\clubsuit_{{\mathcal{B}}}$ of a boundary ${\mathcal{B}}$ in a ladder ${\mathcal{H}}$ where the input $\clubsuit_{{\mathcal{B}}}$ consists of the only one root. However, $\clubsuit_{{\mathcal{B}}}$ would be a forest with sequential roots $(r_1, r_2, \cdots, r_s)$, we can slightly modified Algorithm \[alg:framework-boundary\] as follows: if $\clubsuit_{{\mathcal{B}}}$ is a forest with the sequential roots $(r_1, r_2, \cdots, r_s)$, we can orderly place the sequential regions $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(r_1), \tilde{{\mathcal{W}}}_{{\mathcal{H}}}(r_2), \cdots, \tilde{{\mathcal{W}}}_{{\mathcal{H}}}(r_s))$ in ${\mathcal{H}}$.
Given a region ${\mathcal{W}}=({\mathcal{B}}^L, {\mathcal{B}}^R)$, the collection of all rightward-outer and leftward-outer fans from the boundaries ${\mathcal{B}}^L$ and ${\mathcal{B}}^R$ can be $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in ${\mathcal{H}}$ and there are sequential regions $(\tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}^L}), {\mathcal{W}}^M, \tilde{{\mathcal{W}}}_{{\mathcal{H}}}(\clubsuit_{{\mathcal{B}}^R}))$ in ${\mathcal{H}}$ where ${\mathcal{W}}^M$ is a region between the rightmost boundary of $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}^L})$ and the leftmost boundary of $\tilde{{\mathcal{W}}}(\clubsuit_{{\mathcal{B}}^R})$. Moreover, the union of the two forest-like structures $\clubsuit_{{\mathcal{B}}^L}$ and $\clubsuit_{{\mathcal{B}}^R}$ is a skeleton $\Psi({\mathcal{W}})$ of ${\mathcal{W}}$.
Deleted Edges Increase $X$-Crossing, Queue and Gap Numbers Sightly
==================================================================
In this section, we explain why our layout is still $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed after deleted edges are re-added into our layout.
Recall that in the Section \[sec:transformation\], for a cycle ${{\cal O}}$, the cycle ${{\cal O}}$ is clockwisely placed from the vertex $m$ of ${{\cal O}}$ in a composite-layerlike graph ${\mathcal{G}}$. And, for a down-pointing triangle $\triangledown$, the down-pointing triagle $\triangledown$ is clockwisely placed from the lower vertex $m$ of $\triangledown$ where all vertices except the lower vertex $m$ of $\triangledown$ are placed at a upper layer and the lower vertex $m$ is placed at a lower layer of a composite-layerlike graph ${\mathcal{G}}$. Also, there exist sequential maximal inner cycles $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$ inside ${{\cal O}}$ or $\triangledown$; The spines of the cycle ${{\cal O}}$ or the down-pointing triangle $\triangledown$. And, the leftmost and rightmost vertices of the $i$-th cycle ${{\cal O}}_i \in ({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$ are the $i$-th joint of the spine. Moreover, for the $i$-th cycle ${{\cal O}}_i \in ({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$, the $i$-th *hoop* $(u_i, u'_i)$ on the cycle ${{\cal O}}$ is defined that the vertices $u_i$ and $u'_i$ are the parents of the $i$-th joint of the spine of the cycle ${{\cal O}}$ or the down-pointing triangle $\triangledown$. The vertex $m$ is called the *bad* vertex of a cycle ${{\cal O}}$ or a down-pointing triangle $\triangledown$.
For a cycle ${{\cal O}}$, a deleted edge on the bad vertex $m$ of the cycle ${{\cal O}}$ are called a *wire* of $m$ inside ${{\cal O}}$ denoted as $\curlywedge(m)$. Also, a removed edge $e_i, 1\leq i\leq p-1,$ which connects between two contiguous maximal inner cycles ${{\cal O}}_i$ and ${{\cal O}}_{i+1}$ of the spine is called a *bridge*.
Recall that in Algorithm \[alg:framework\], we simultaneously pick all joints of ${{\cal O}}_i, 1\leq i\leq p$ and their hoops $\{(u_1, u'_1), (u_2, u'_2), \cdots, (u_p, u'_p)\}$ on the two contiguous tracks in ${\mathcal{H}}$. Also, we order all hoops $\{(u_1, u'_1),$ $(u_2, u'_2),$ $\cdots,$ $(u_p, u'_p)\}$ as ordering the lower boundary $\{L^B({{\cal O}}_1), L^B({{\cal O}}_2), \cdots, L^B({{\cal O}}_p)\}$ of the ${{\cal O}}$’s spine. Moreover, for all hoops $((u_1, u'_1),$ $(u_2, u'_2),$ $\cdots,$ $(u_p, u'_p))$ and the lower boundary $\{L^B({{\cal O}}_1),$ $L^B({{\cal O}}_2),$ $\cdots,$ $L^B({{\cal O}}_p)\}$ of the spine, we place the two sequential vertices contiguously on any track in ${\mathcal{H}}$. Hence we have the following observations that state the key reasons why wires cannot make $X$-crossing in our layout.
Given a bad vertex $m$ on a cycle ${{\cal O}}$, let $\{{{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p\}$ be ${{\cal O}}$’s spine and $\{(u_1, u'_1),$ $(u_2, u'_2),$ $\cdots,$ $(u_p, u'_p)\}$ be corresponding hoops on ${{\cal O}}$, the layout in ${\mathcal{H}}$ has the following properties:
1. all hoops are placed contiguously on the same track in ${\mathcal{H}}$,
2. all joints are placed contiguously on a track in ${\mathcal{H}}$,
3. the order of all joints on a track is the same as the order of all hoops on a track in ${\mathcal{H}}$. And,
4. all hoops and all joints are placed at two contiguous tracks in ${\mathcal{H}}$.
Let $m$ and $m'$ be bad vertices on cycles ${{\cal O}}$ and ${{\cal O}}'$, respectively. The bad vertex $m$ is placed at left of the bad vertex $m'$ on a track in ${\mathcal{H}}$ if and only if the spine of ${{\cal O}}$ is placed at left of the spine of ${{\cal O}}'$ at any track in ${\mathcal{H}}$.
For a cycle ${{\cal O}}$ with the bad vertex $m$,
1. the gap number between the bad vertex $m$ and any vertex on the lower boundary $\{L^B({{\cal O}}_1),$ $L^B({{\cal O}}_2),$ $\cdots,$ $L^B({{\cal O}}_p)\}$ of the ${{\cal O}}$’s spine is at most $2{\mathcal{J}}$, and
2. each lower boundary $L^B({{\cal O}}_i),$ $1\leq i\leq p$ except its joint is placed contiguously on a track in ${\mathcal{H}}$.
For any two bad vertices $m_1$ and $m_2$ that $m_1$ is at left of $m_2$ on a track in ${\mathcal{H}}$, the wires $\curlywedge(m_1)$ are placed at left of the wires $\curlywedge(m_2)$ in ${\mathcal{H}}$. So, there is no any $X$-crossing edge between $\curlywedge(m_1)$ and $\curlywedge(m_2)$. From the above observations, we have the following lemma:
\[lem:vertical-X-crossing\] Suppose sequential bad vertices $(m_1, m_2, \cdots, m_p)$ are placed from left to right on a track in ${\mathcal{H}}$, the sequential wires $(\curlywedge(m_1),$ $\curlywedge(m_2),$ $\cdots,$ $\curlywedge(m_p))$ are not $X$-crossing in ${\mathcal{H}}$.
Suppose (1) $\triangledown$ is a down-pointing triangle with the bad vertex $m$ and (2) the down-pointing triangle $\triangledown$ is in a fan ${\mathcal{F}}$ of a raising fan $\tilde{{\mathcal{F}}}$ with the middle path ${\mathcal{M}}$,
1. a left (right, respectively) *pile* of the bad vertex $m$ is defined as an edge connecting between the bad vertex $m$ and a vertex in the lower boundary $\{L^B({{\cal O}}_1), L^B({{\cal O}}_2), \cdots, L^B({{\cal O}}_p)\}$ of the spine of $\triangledown$ that is at left (right, respectively) of the middle path ${\mathcal{M}}$. A left (right, respectively) pile with respect to the middle path ${\mathcal{M}}$ is denoted to $\curlyvee^L(m)$ ($\curlyvee^R(m)$, respectively).
2. A left (right, respectively) spine of the down-pointing triangle $\triangledown$ with respect to the middle path ${\mathcal{M}}$ is the subsequential spine of the down-pointing triangle $\triangledown$ at left (right, respectively) of the middle path ${\mathcal{M}}$.
3. A left (right, respectively) hoop with respect to the middle path ${\mathcal{M}}$ is a hoop of the down-pointing triangle $\triangledown$ at left (right, respectively) of the middle path ${\mathcal{M}}$.
Suppose $\triangledown'$ is a down-pointing triangle with the bad vertex $m'$ inside the down-pointing triangle $\triangledown$. Then the sequential regions $\tilde{{\mathcal{W}}}(m)$ consisting of all $m$’s left joints are placed at left of the sequential regions $\tilde{{\mathcal{W}}}(m')$ consisting of all $m'$’s left joints. Because we place the bad vertex $m$ at left of the bad vertex $m'$ on a track in ${\mathcal{H}}$ and all $m$’s joints at left of $m'$’s hoops on a track in ${\mathcal{H}}$, we can have that the left piles $\curlyvee^L(m)$ are not nested with the left piles $\curlyvee^L(m')$ on any track in ${\mathcal{H}}$. Similarly, the sequential regions $\tilde{{\mathcal{W}}}(m)$ consisting of all $m$’s right joints are at left of the sequential regions $\tilde{{\mathcal{W}}}(m')$ consisting of all $m'$’s right joints. Because we place the bad vertex $m$ at left of bad vertex $m'$ on a track in ${\mathcal{H}}$ and place all $m$’s right joints at left of all $m'$’s right joints on a track in ${\mathcal{H}}$, there is no any nested edge between all right piles $\curlyvee^R(m)$ of the bad vertex $m$ and all right piles $\curlyvee^R(m')$ of the bad vertex $m'$. Now we can the following observations:
Suppose raising down-pointing triangles $(\triangledown_1, \triangledown_2, \cdots, \triangledown_p)$ and their sequential bad vertices ${\mathcal{M}}=(m_1, m_2, \cdots, m_p)$ are placed on the same track in ${\mathcal{H}}$,
1. the sequential left joints of the sequential down-pointing triangles $(\triangledown_1, \triangledown_2, \cdots, \triangledown_p)$ are orderly placed at a track in ${\mathcal{H}}$. And,
2. the sequential right joints of the sequential down-pointing triangles $(\triangledown_1, \triangledown_2, \cdots, \triangledown_p)$ are orderly placed at a track in ${\mathcal{H}}$.
For a down-pointing triangle $\triangledown$ with the bad vertex $m$,
1. the gap number between the bad vertex $m$ and any vertex on the lower boundary $\{L^B({{\cal O}}_1),$ $L^B({{\cal O}}_2),$ $\cdots,$ $L^B({{\cal O}}_p)\}$ of the $\triangledown$’s spine is at most $2{\mathcal{J}}$, and
2. each lower boundary $L^B({{\cal O}}_i), 1\leq i\leq p$ except its joint is placed contiguously on a track in ${\mathcal{H}}$.
From the above observations, we can have that the right piles $\curlyvee^R(m_i)$ are not nested with the left piles $\curlyvee^R(m_j)$ on any track in ${\mathcal{H}}$ in the following lemma:
\[lem:pile-nested\] Given sequential bad vertices $(m_1, m_2, \cdots, m_p)$ orderly placed on a track in ${\mathcal{H}}$, their sequential left and right piles $(\curlyvee^L(m_1), \curlyvee^L(m_2), \cdots, \curlyvee^L(m_p))$ and $(\curlyvee^R(m_1), \curlyvee^R(m_2), \cdots, \curlyvee^R(m_p))$ are not nested on any track in ${\mathcal{H}}$.
For a cycle ${{\cal O}}$, all joints of the spine of the cycle are orderly placed on any track in ${\mathcal{H}}$, all bridges of the spine cannot have nested chords on any track in ${\mathcal{H}}$. Similarly, for a down-pointing triangle $\triangledown$, all left and right joints of the left and right spines of the cycle are orderly placed on any track in ${\mathcal{H}}$, respectively, all bridges of the spine cannot have nested chords on any track in ${\mathcal{H}}$. From the above fact, we can have the following lemma:
\[lem:bridge-nested\] Given a cycle ${{\cal O}}$ or a down-pointing triangle $\triangledown$ with their spine $({{\cal O}}_1, {{\cal O}}_2, \cdots, {{\cal O}}_p)$, their sequential bridges $(e_1, e_2, \cdots, e_{p-1})$ are not nested on any track in ${\mathcal{H}}$ where $e_i, 1\leq i\leq p-1,$ is the bridge between the cycles ${{\cal O}}_i$ and ${{\cal O}}_{i+1}$.
\[thm:G1-well-placed\] Every $1$-subdivision plane graph $G^1$ can have an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout on constant number of tracks.
From Theorem \[thm:reform\], we know a plane graph $G$ can be reformed into a composite-layerlike graph ${\mathcal{G}}$. From Lemmas \[lem:vertical-X-crossing\], \[lem:pile-nested\] and \[lem:bridge-nested\], deleted edges slightly increase $X$-crossing number in any two tracks and queue number in any track in ${\mathcal{H}}$. Hence we conclude that a plane graph $G$ can be $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed in a ladder ${\mathcal{H}}$. Also, from Theorem \[thm:cons-track\], an $({\mathcal{Q}}, {\mathcal{X}}, {\mathcal{D}})$-well-placed layout can be wrapped into a ladder ${\mathcal{H}}$ on constant number of tracks.
[5]{}
Giuseppe Di Battista, Fabrizio Frati and Janos Pach, On the queue number of planar graphs, in Foundations of Computer Science, 2010, pp. 365-374.
Sandeep N. Bhatt, Fan R. K. Chung, F. Thomson Leighton, and Arnold L. Rosenberg, Scheduling tree-dags using FIFO queues: A control-memory trade-off, , 33, 1996, pp. 55-68.
Robin Blankenship. Book Embeddings of Graphs. Ph.D. thesis, Department of Mathematics, Louisiana State University, U.S.A., 2003.
Robin Blankenship and Bogdan Oporowski, Book embeddings of graphs and minor closed classes, in Proceedings of 32nd Southeastern International Conference on Combinatorics, Graph Theory and Computing. Department of Mathematics, Louisiana State University, 2001.
Vida Dujmović, Graph layouts via layered separators, , 110, 2015, pp. 79-89.
Vida Dujmović, Fabrizio Frati, Gwena'’[e]{}l Joret, and David R. Wood ˝ , Nonrepetitive colourings of planar graphs with $O(\log n)$ colours, 2012. http://arxiv.org/abs/1202.1569.
Vida Dujmović, Pat Morin, and David R. Wood, Layout of graphs with bounded tree-width, , 34(3), 2005, pp. 553-579.
Vida Dujmović, Pat Morin, and David R. Wood, Layered separators in minor-closed families with applications, 2013. arXiv: 1306.1595.
Vida Dujmović, Attila Pór, and David R. Wood, Track layouts of graphs, , 6(2), 2004, pp. 497-522.
Vida Dujmović and David R. Wood, Three-dimensional grid drawings with subquadratic volume. In Janos Pach , ed., Towards a Theory of Geometric Graphs, vol. 342 of Contemporary Mathematics, pp. 55-66. Amer. Math. Soc., 2004.
Vida Dujmović and David R. Wood, Stacks, queues and tracks: Layouts of graph subdivisions, , 7, 2005, pp. 155-202.
Shimon Even and A. Itai, Queues, stacks, and graphs, in Zvi Kohavi and Azaria Paz, eds., Proc. International Symp. on Theory of Machines and Computations, pp. 71-86. Academic Press, 1971.
Stefan Felsner, Giussepe Liotta, and Stephen K. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions, in Proc. 9th International Symp. on Graph Drawing (GD ’01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328-342. Springer, 2002.
Stefan Felsner, Giussepe Liotta, and Stephen K. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions, , 7(4), 2003, pp. 363-398.
John R. Gilbert, Joan P. Hutchinson, and Robert E. Tarjan, A separator theorem for graphs of bounded genus, , 5(3), 1984, pp. 391-407.
Toru Hasunuma, Laying out iterated line digraphs using queues, in Giuseppe Liotta, ed., Proc. 11th International Symp. on Graph Drawing (GD ’03), vol. 2912 of Lecture Notes in Comput. Sci., pp. 202-213. Springer, 2004.
Lenwood S. Heath, F. Thomson Leighton, and Arnold L. Rosenberg, Comparing queues and stacks as mechanisms for laying out graphs, , 5(3), 1992, pp. 398-412.
Lenwood S. Heath and Sriram V. Pemmaraju, Stack and queue layouts of posets, , 10(4), 1997, pp. 599-625.
Lenwood S. Heath, Sriram V. Pemmaraju, and Ann N. Trenk, Stack and queue layouts of directed acyclic graphs. I, , 28(4), 1999, pp. 1510-1539.
Lenwood S. Heath and Sriram V. Pemmaraju, Stack and queue layouts of directed acyclic graphs. II, , 28(5), 1999, pp. 1588-1626.
Lenwood S. Heath and Arnold L. Rosenberg, Laying out graphs using queues, , 21(5), 1992, pp. 927-958.
Richard J. Lipton and Robert E. Tarjan, A separator theorem for planar graphs, , 36(2), 1979, pp. 177-189.
Seth M. Malitz, Genus g graphs have pagenumber $O(
\sqrt{g})$, , 17(1), 1994, pp. 85-109.
Sriram V. Pemmaraju, Exploring the Powers of Stacks and Queues via Graph Layouts, Ph.D. thesis, Virginia Polytechnic Institute and State University, U.S.A., 1992.
S. Rengarajan and C. E. Veni Madhavan, Stack and queue number of 2-trees, in Ding-Zhu Du and Ming Li, eds., Proc. 1st Annual International Conf. on Computing and Combinatorics (COCOON ’95), vol. 959 of Lecture Notes in Comput. Sci., pp. 203-212. Springer, 1995.
Farhad Shahrokhi and Weiping Shi, On crossing sets, disjoint sets, and pagenumber, , 34(1), 2000, pp. 40-53.
Robert E. Tarjan, Sorting using networks of queues and stacks, , 19, 1972, pp. 341-346.
Jiun-Jie Wang, Layouts of Chordal Trees on Constant Number of Tracks, , 2016.
David R. Wood, Queue layouts of graph products and powers, , 7(1), 2005, pp. 255-268.
David R. Wood, Bounded-degree graphs have arbitrarily large queue-number, , 10(1), 2008, pp. 27-34.
Mihalis Yannakakis, Embedding planar graphs in four pages, , 38(1), 1989, pp. 36-67.
|
---
abstract: |
Topic models are Bayesian models that are frequently used to capture the latent structure of certain corpora of documents or images. Each data element in such a corpus (for instance each item in a collection of scientific articles) is regarded as a convex combination of a small number of vectors corresponding to ‘topics’ or ‘components’. The weights are assumed to have a Dirichlet prior distribution. The standard approach towards approximating the posterior is to use variational inference algorithms, and in particular a mean field approximation.
We show that this approach suffers from an instability that can produce misleading conclusions. Namely, for certain regimes of the model parameters, variational inference outputs a non-trivial decomposition into topics. However –for the same parameter values– the data contain no actual information about the true decomposition, and hence the output of the algorithm is uncorrelated with the true topic decomposition. Among other consequences, the estimated posterior mean is significantly wrong, and estimated Bayesian credible regions do not achieve the nominal coverage. We discuss how this instability is remedied by more accurate mean field approximations.
author:
- 'Behrooz Ghorbani, Hamid Javadi[^1], Andrea Montanari[^2]'
title: An Instability in Variational Inference for Topic Models
---
Introduction {#sec:Intro}
============
Topic modeling [@blei2012probabilistic] aims at extracting the latent structure from a corpus of documents (either images or texts), that are represented as vectors $\bx_1,\bx_2,\dots,\bx_n\in\reals^d$. The key assumption is that the $n$ documents are (approximately) convex combinations of a small number $k$ of topics $\tbh_1,\dots,\tbh_k\in\reals^d$. Conditional on the topics, documents are generated independently by letting $$\begin{aligned}
\bx_a = \frac{\sqrt{\beta}}{d}\sum_{\ell=1}^kw_{a,\ell}\tbh_{\ell}+\bz_a\, ,
$$ where the weights $\bw_a= (w_{a,\ell})_{1\le \ell\le k}$ and noise vectors $\bz_a$ are i.i.d. across $a\in\{1,\dots,n\}$. The scaling factor $\sqrt{\beta}/d$ is introduced for mathematical convenience (an equivalent parametrization would have been to scale $\bZ$ by a noise-level parameter $\sigma$), and $\beta>0$ can be interpreted as a signal-to-noise ratio. It is also useful to introduce the matrix $\bX\in \reals^{n\times d}$ whose $i$-th row is $\bx_i$, and therefore $$\begin{aligned}
\bX = \frac{\sqrt{\beta}}{d}\,\bW\bH^{\sT} +\bZ\, ,\label{eq:LDAModel}
$$ where $\bW\in\reals^{n\times k}$ and $\bH\in\reals^{d\times k}$. The $a$-th row of $\bW$, is the vector of weights $\bw_a$, while the rows of $\bH$ will be denoted by $\bh_i\in\reals^k$.
Note that $\bw_a$ belongs to the simplex $\sP_1(k) = \{\bw\in\reals^k_{\ge 0}\; :\;\;\<\bw,\bfone_k\> =1\}$. It is common to assume that its prior is Dirichlet: this class of models is known as *Latent Dirichlet Allocations*, or LDA [@blei2003latent]. Here we will take a particularly simple example of this type, and assume that the prior is Dirichlet in $k$ dimensions with all parameters equal to $\nu$ (which we will denote by $\Dir(\nu;k)$). As for the topics $\bH$, their prior distribution depends on the specific application. For instance, when applied to text corpora, the $\tbh_i$ are typically non-negative and represent normalized word count vectors. Here we will make the simplifying assumption that they are standard Gaussian $(\tbh_{i})_{i\le d}\sim_{iid}\normal(0,\id_k)$. Finally, $\bZ$ will be a noise matrix with entries $(Z_{ij})_{i\in
[n], j\in [d]}\sim_{iid}\normal(0,1/d)$.
In fully Bayesian topic models, the parameters of the Dirichlet distribution, as well as the topic distributions are themselves unknown and to be learned from data. Here we will work in an idealized setting in which they are known. We will also assume that data are in fact distributed according to the postulated generative model. Since we are interested in studying some limitations of current approaches, our main point is only reinforced by assuming this idealized scenario.
As is common with Bayesian approaches, computing the posterior distribution of the factors $\bH$, $\bW$ given the data $\bX$ is computationally challenging. Since the seminal work of Blei, Ng and Jordan [@blei2003latent], variational inference is the method of choice for addressing this problem within topic models. The term ‘variational inference’ refers to a broad class of methods that aim at approximating the posterior computation by solving an optimization problem, see [@jordan1999introduction; @wainwright2008graphical; @blei2017variational] for background. A popular starting point is the Gibbs variational principle, namely the fact that the posterior solves the following convex optimization problem: $$\begin{aligned}
p_{\bW,\bH|\bX}(\,\cdot\,,\cdot,|\bX) & = \arg\min_{q\in\cP_{n,d,k}} \KL(q\| p_{\bW,\bH|\bX}) \label{eq:Gibbs}\\
&= \arg\min_{q\in\cP_{n,d,k}}\Big\{-\E_{q}\log p_{\bX|\bW,\bH}(\bX|\bH,\bW) +\KL(q\| p_{\bW}\times p_{\bH})\Big\}\,, \label{eq:Gibbs2}
$$ where $\KL(\, \cdot\,\|\,\cdot\, )$ denotes the Kullback-Leibler divergence. The variational expression in Eq. (\[eq:Gibbs2\]) is also known as the Gibbs free energy. Optimization is within the space $\cP_{n,d,k}$ of probability measures on $\bH,\bW$. To be precise, we always assume that a dominating measure $\nu_0$ over $\reals^{n\times k}\times \reals^{d\times k}$ is given for $\bW,\bH$, and both $p_{\bW,\bH|\bX}$ and $q$ have densities with respect to $\nu_0$: we hence identify the measure with its density. Throughout the paper (with the exception of the example in Section \[sec:Toy\]) $\nu_0$ can be taken to be the Lebesgue measure.
Even for $\bW,\bH$ discrete, the Gibbs principle has exponentially many decision variables. Variational methods differ in the way the problem (\[eq:Gibbs\]) is approximated. The main approach within topic modeling is *naive mean field*, which restricts the optimization problem to the space of probability measures that factorize over the rows of $\bW,\bH$: $$\begin{aligned}
\hq\left(\bW,\bH\right) = q\left(\bH\right)\tilde q\left(\bW\right) = \prod_{i=1}^d q_i\left(\bh_i\right)\prod_{a=1}^n\tq_a\left(\bw_a\right)\, . \label{eq:ProductForm}
$$ By a suitable parametrization of the marginals $q_i$, $\tq_a$, this leads to an optimization problem of dimension $O((n+d)k)$, cf. Section \[sec:Inst\]. Despite being non-convex, this problem is separately convex in the $(q_i)_{i\le d}$ and $(\tq_a)_{a\le n}$, which naturally suggests the use of an alternating minimization algorithm which has been successfully deployed in a broad range of applications ranging from computer vision to genetics [@fei2005bayesian; @wang2011collaborative; @raj2014faststructure]. We will refer to this as to the *naive mean field iteration*. Following a common use in the topics models literature, we will use the terms ‘variational inference’ and ‘naive mean field’ interchangeably.
The main result of this paper is that naive mean field presents an instability for learning Latent Dirichlet Allocations. We will focus on the limit $n,d\to\infty$ with $n/d=\delta$ fixed. Hence, an LDA distribution is determined by the parameters $(k,\delta,\nu,\beta)$. We will show that there are regions in this parameter space such that the following two findings hold simultaneously:
No non-trivial estimator.
: Any estimator $\hbH$, $\hbW$ of the topic or weight matrices is asymptotically uncorrelated with the real model parameters $\bH, \bW$. In other words, the data do not contain enough signal to perform any strong inference.
Variational inference is randomly biased.
: Given the above, one would hope the Bayesian posterior to be centered on an unbiased estimate. In particular, $p(\bw_a|\bX)$ (the posterior distribution over weights of document $a$) should be centered around the uniform distribution $\bw_a= (1/k,\dots,1/k)$. In contrast, we will show that the posterior produced by naive mean field is centered around a random distribution that is uncorrelated with the actual weights. Similarly, the posterior over topic vectors is centered around random vectors uncorrelated with the true topics.
One key argument in support of Bayesian methods is the hope that they provide a measure of uncertainty of the estimated variables. In view of this, the failure just described is particularly dangerous because it suggests some measure of certainty, although the estimates are essentially random.
Is there a way to eliminate this instability by using a better mean field approximation? We show that a promising approach is provided by a classical idea in statistical physics, the Thouless-Anderson-Palmer (TAP) free energy [@thouless1977solution; @opper2001adaptive]. This suggests a variational principle that is analogous in form to naive mean field, but provides a more accurate approximation of the Gibbs principle:
Variational inference via the TAP free energy.
: We show that the instability of naive mean field is remedied by using the TAP free energy instead of the naive mean field free energy. The latter can be optimized using an iterative scheme that is analogous to the naive mean field iteration and is known as approximate message passing (AMP).
While the TAP approach is promising –at least for synthetic data– we believe that further work is needed to develop a reliable inference scheme.
The rest of the paper is organized as follows. Section \[sec:Toy\] discusses a simpler example, $\integers_2$-synchronization, which shares important features with latent Dirichlet allocations. Since calculations are fairly straightforward, this example allows to explain the main mathematical points in a simple context. We then present our main results about instability of naive mean field in Section \[sec:Inst\], and discuss the use of TAP free energy to overcome the instability in Section \[sec:Fixing\].
Related literature
------------------
Over the last fifteen years, topic models have been generalized to cover an impressive number of applications. A short list includes mixed membership models [@erosheva2004mixed; @airoldi2008mixed], dynamic topic models [@blei2006dynamic], correlated topic models [@lafferty2006correlated; @blei2007correlated], spatial LDA [@wang2008spatial], relational topic models [@chang2009relational], Bayesian tensor models [@zhou2015bayesian]. While other approaches have been used (e.g. Gibbs sampling), variational algorithms are among the most popular methods for Bayesian inference in these models. Variational methods provide a fairly complete and interpretable description of the posterior, while allowing to leverage advances in optimization algorithms and architectures towards this goal (see [@hoffman2010online; @broderick2013streaming]).
Despite this broad empirical success, little is rigorously known about the accuracy of variational inference in concrete statistical problems. Wang and Titterington [@wang2004convergence; @wang2006convergence] prove local convergence of naive mean field estimate to the true parameters for exponential families with missing data and Gaussian mixture models. In the context of Gaussian mixtures, the same authors prove that the covariance of the variational posterior is asymptotically smaller (in the positive semidefinite order) than the inverse of the Fisher information matrix [@wang2005inadequacy]. All of these results are established in the classical large sample asymptotics $n\to\infty$ with $d$ fixed. In the present paper we focus instead on the high-dimensional limit $n = \Theta(d)$ and prove that also the mode (or mean) of the variational posterior is incorrect. Notice that the high-dimensional regime is particularly relevant for the analysis of Bayesian methods. Indeed, in the classical low-dimensional asymptotics Bayesian approaches do not outperform maximum likelihood.
In order to correct for the underestimation of covariances, [@wang2005inadequacy] suggest replacing its variational estimate by the inverse Fisher information matrix. A different approach is developed in [@giordano2015linear], building on linear response theory.
Naive mean field variational inference was used in [@celisse2012consistency; @bickel2013asymptotic] to estimate the parameters of the stochastic block model. These works establish consistency and asymptotic normality of the variational estimates in a large signal-to-noise ratio regime. Our work focuses on estimating the latent factors: it would be interesting to consider implications on parameter estimation as well.
The recent paper [@zhang2017theoretical] also studies variational inference in the context of the stochastic block model, but focuses on reconstructing the latent vertex labels. The authors prove that naive mean field achieves minimax optimal statistical rates. Let us emphasize that this problem is closely related to topic models: both are models for approximately low-rank matrices, with a probabilistic prior on the factors. The results of [@zhang2017theoretical] are complementary to ours, in the sense that [@zhang2017theoretical] establishes positive results at large signal-to-noise ratio (albeit for a different model), while we prove inconsistency at low signal-to-noise ratio. General conditions for consistency of variational Bayes methods are proposed in [@pati2017statistical].
Our work also builds on recent theoretical advances in high-dimensional low-rank models, that were mainly driven by techniques from mathematical statistical physics (more specifically, spin glass theory). An incomplete list of relevant references includes [@korada2009exact; @deshpande2014information; @deshpande2017asymptotic; @krzakala2016mutual; @barbier2016mutual; @lelarge2016fundamental; @miolane2017fundamental; @lesieur2017constrained; @alaoui2018estimation]. These papers prove asymptotically exact characterizations of the Bayes optimal estimation error in low-rank models, to an increasing degree of generality, under the high-dimensional scaling $n,d\to\infty$ with $n/d\to\delta\in (0,\infty)$.
Related ideas also suggest an iterative algorithm for Bayesian estimation, namely Bayes Approximate Message Passing [@DMM09; @DMM_ITW_I]. As mentioned above, Bayes AMP can be regarded as minimizing a different variational approximation known as the TAP free energy. An important advantage over naive mean field is that AMP can be rigorously analyzed using a method known as state evolution [@BM-MPCS-2011; @javanmard2013state; @berthier2017state].
Let us finally mention that a parallel line of work develops polynomial-time algorithms to construct non-negative matrix factorizations under certain structural assumptions on the data matrix $\bX$, such as separability [@arora2012learning; @arora2012computing; @recht2012factoring]. It should be emphasized that the objective of these algorithms is different from the one of Bayesian methods: they return a factorization that is guaranteed to be unique under separability. In contrast, variational methods attempt to approximate the posterior distribution, when the data are generated according to the LDA model.
Notations
---------
We denote by $\id_m$ the identity matrix, and by $\bJ_m$ the all-ones matrix in $m$ dimensions (subscripts will be dropped when the number of dimensions is clear from the context). We use $\bfone_k\in\reals^k$ for the all-ones vector.
We will use $\otimes$ for the tensor (outer) product. In particular, given vectors expressed in the canonical basis as $\bu = \sum_{i=1}^{d_1} u_i\be_i\in\reals^{d_1}$ and $\bv = \sum_{i=j}^{d_2} v_j\be_j\in\reals^{d_2}$, $\bu\otimes\bv\in\reals^{d_1}\otimes \reals^{d_2}$ is the tensor with coordinates $(\bu\otimes\bv)_{ij} = u_iv_j$ in the basis $\be_i\otimes \be_j$. We will identify the space of matrices $\reals^{d_1\times d_2}$ with the tensor product $\reals^{d_1}\otimes \reals^{d_2}$. In particular, for $\bu\in\reals^{d_1}$, $\bv\in\reals^{d_2}$, we identify $\bu\otimes \bv$ with the matrix $\bu\bv^{\sT}$.
Given a symmetric matrix $\bM\in\reals^{n\times n}$, we denote by $\lambda_1(\bM)\ge \lambda_2(\bM)\ge \dots\ge \lambda_n(\bM)$ its eigenvalues in decreasing order. For a matrix (or vector) $\bA \in \reals^{d\times n}$ we denote the orthogonal projector operator onto the subspace spanned by the columns of $\bA$ by $\bP_\bA\in\reals^{d\times d}$, and its orthogonal complement by $\bP_\bA^{\perp} = \id_d-\bP_{\bA}$. When the subscript is omitted, this is understood to be the projector onto the space spanned by the all-ones vector: $\bP=\bfone_d\bfone_d/d$ and $\bP_{\perp}=\id_d-\bP$.
A simple example: $\integers_2$-synchronization {#sec:Toy}
===============================================
In $\integers_2$ synchronization we are interested in estimating a vector $\bsigma\in\{+1,-1\}^n$ from observations $\bX\in\reals^{n\times n}$, generated according to $$\begin{aligned}
\bX = \frac{\lambda}{n}\bsigma\bsigma^{\sT}+\bZ\, ,\label{eq:Z2-synch}
$$ where $\bZ=\bZ^{\sT}\in\reals^{n\times n}$ is distributed according to the Gaussian Orthogonal Ensemble $\GOE(n)$, namely $(Z_{ij})_{i<j\le n}\sim_{iid}\normal(0,1/n)$ are independent of $(Z_{ii})_{i\le n}\sim_{iid}\normal(0,2/n)$. The parameter $\lambda\ge 0$ corresponds to the signal-to-noise ratio.
It is known that for $\lambda\le 1$ no algorithm can estimate $\bsigma$ from data $\bX$ with positive correlation in the limit $n\to\infty$. The following is an immediate consequence of [@korada2009exact; @deshpande2017asymptotic], see Appendix \[app:LemmaITZ2\].
\[lemma:IT-Threshold-Z2\] Under model (\[eq:Z2-synch\]), for $\lambda\le 1$ and any estimator $\hbsigma:\reals^{n\times n}\to\reals^n\setminus \{\bzero\}$, the following limit holds in probability: $$\begin{aligned}
\lim\sup_{n\to\infty}\frac{|\<\hbsigma(\bX),\bsigma\>|}{\|\hbsigma(\bX)\|_2\|\bsigma\|_2} = 0\, .
$$
How does variational inference perform on this problem? Any product probability distribution $\hq(\bsigma) = \prod_{i=1}^n q_i(\sigma_i)$ can be parametrized by the means $m_i= \sum_{\sigma_i\in\{+1,-1\}} q_i(\sigma_i)\,\sigma_i$, and it is immediate to get $$\begin{aligned}
\KL(\hq\|p_{\bsigma|\bX}) &= \cF(\bm) + {\rm const.}\, ,\\
\cF(\bm) & \equiv -\frac{\lambda}{2}\<\bm,\bX_0\bm\> -\sum_{i=1}^n\entro(m_i)\, .\label{eq:Z2_FreeEnergy}
$$ Here $\bX_0$ is obtained from $\bX$ by setting the diagonal entries to $0$, and $\entro(x) = -\frac{(1+x)}{2}\log\frac{(1+x)}{2} -\frac{(1-x)}{2}\log\frac{(1-x)}{2}$ is the binary entropy function. In view of Lemma \[lemma:IT-Threshold-Z2\], the correct posterior distribution should be essentially uniform, resulting in $\bm$ vanishing. Indeed, $\bm_* = 0$ is a stationary point of the mean field free energy $\cF(\bm)$: $\left.\nabla\cF(\bm) \right|_{\bm = \bm_*}=0$. We refer to this as the ‘uninformative fixed point’.
*Is $\bm_*$ a local minimum?* Computing the Hessian at the uninformative fixed point yields $$\begin{aligned}
\left.\nabla^2\cF(\bm) \right|_{\bm = \bm_*} = -\lambda\bX_0 +\id\, .
$$ The matrix $\bX_0$ is a rank-one deformation of a Wigner matrix and its spectrum is well understood [@baik2005phase; @feral2007largest; @benaych2011eigenvalues]. For $\lambda\le 1$, its eigenvalues are contained with high probability in the interval $[-2,2]$, with $\lambda_{\min}(\bX)\to -2$, $\lambda_{\max}(\bX)\to 2$ as $n\to\infty$. For $\lambda>1$, $\lambda_{\max}(\bX)\to \lambda+\lambda^{-1}$, while the other eigenvalues are contained in $[-2,2]$. This implies $$\begin{aligned}
\lim_{n\to\infty}\lambda_{\rm min}(\left.\nabla^2\cF\right|_{\bm_*}) = \begin{cases}
1-2\lambda& \;\; \mbox{if $\lambda\le 1$,}\\
-\lambda^2 & \;\; \mbox{if $\lambda > 1$.}\\
\end{cases}
$$ In other words, $\bm_* = 0$ is a local minimum for $\lambda<1/2$, but becomes a saddle point for $\lambda>1/2$. In particular, for $\lambda\in (1/2,1)$, variational inference will produce an estimate $\hbm\neq 0$, although the posterior should be essentially uniform. In fact, it is possible to make this conclusion more quantitative.
\[propo:Toy\] Let $\hbm \in [-1,1]^n$ be any local minimum of the mean field free energy $\cF(\bm)$, under the $\integers_2$-synchronization model (\[eq:Z2-synch\]). Then there exists a numerical constant $c_0>0$ such that, with high probability, for $\lambda>1/2$, $$\begin{aligned}
\frac{1}{n}\|\hbm\|_2^2 \ge c_0\, \min\big((2\lambda-1)^2,1\big)\, .
$$
In other words, although no estimator is positively correlated with the true signal $\bsigma$, variational inference outputs biases $\hm_i$ that are non-zero (and indeed of order one, for a positive fraction of them).
The last statement immediately implies that naive mean field leads to incorrect inferential statements for $\lambda\in (1/2,1)$. In order to formalize this point, given any estimators $\{\hq_i(\,\cdot\, )\}_{i\le n}$ of the posterior marginals, we define the per-coordinate expected coverage as $$\begin{aligned}
\cuQ(\hq) = \frac{1}{n}\sum_{i=1}^n \prob\big(\sigma_i=\arg\max_{\tau_i\in\{+1,-1\}} \hq_i(\tau_i)\big) \, .
$$ This is the expected fraction of coordinates that are estimated correctly by choosing $\bsigma$ according to the estimated posterior. Since the prior is assumed to be correct, it can be interpreted either as the expectation (with respect to the parameters) of the frequentist coverage, or as the expectation (with respect to the data) of the Bayesian coverage. On the other hand, if the $\hq_i$ were accurate, Bayesian theory would suggest claiming the coverage $$\begin{aligned}
\widehat{\cuQ}(\hq) \equiv \frac{1}{n}\sum_{i\le n}\max_{\tau_i}\hq_i(\tau_i)\, .
$$ The following corollary is a direct consequence of Proposition \[propo:Toy\], and formalizes the claim that naive mean field leads to incorrect inferential statements. More precisely, it overestimates the coverage achieved.
Let $\hbm \in [-1,1]^n$ be any local minimum of the mean field free energy $\cF(\bm)$, under the $\integers_2$-synchronization model (\[eq:Z2-synch\]), and consider the corresponding posterior marginal estimates $\hq_i(\sigma_i) = (1+\hm_i\sigma_i)/2$. Then, there exists a numerical constant $c_0>0$ such that, with high probability, for $\lambda\in (1/2,1)$, $$\begin{aligned}
\cuQ(\hq)\le \frac{1}{2}+o_n(1)\, ,\;\;\;\;\;\;
\widehat{\cuQ}(\hq) \ge \frac{1}{2}+c_0\, \min\big((2\lambda-1),1\big)\, .
$$
While similar formal coverage statements can be obtained also for the more complex case of topic models, we will not make them explicit, since they are relatively straightforward consequences of our analysis.
Instability of variational inference for topic models {#sec:Inst}
=====================================================
Information-theoretic limit {#sec:IT-main}
---------------------------
As in the case of $\integers_2$ synchronization discussed in Section \[sec:Toy\], we expect it to be impossible to estimate the factors $\bW,\bH$ with strictly positive correlation for small enough signal-to-noise ratio $\beta$ (or small enough sample size $\delta$). The exact threshold was characterized recently in [@miolane2017fundamental] (but see also [@deshpande2014information; @barbier2016mutual; @lelarge2016fundamental; @lesieur2017constrained] for closely related results). The characterization in [@miolane2017fundamental] is given in terms of a variational principle over $k\times k$ matrices.
\[thm:IT\_Limit\] Let $\Info_n(\bX;\bW,\bH)$ denote the mutual information between the data $\bX$ and the factors $\bH,\bW$ under the LDA model (\[eq:LDAModel\]). Then, the following limit holds almost surely $$\begin{aligned}
\lim_{n,d\to\infty}\frac{1}{d}\Info_n(\bX;\bW,\bH) = \inf_{\bM\in\bbS_k} \RS(\bM;k,\delta,\nu)\, ,\label{eq:InfimumFreeEnergy}
$$ where $\bbS_k$ is the cone of $k\times k$ positive semidefinite matrices and $\RS(\,\cdots\,) $ is a function given explicitly in Appendix \[app:ProofBayes\].
It is also shown in Appendix \[app:ProofBayes\] that $\bM^* = (\delta\beta/k^2)\bJ_k$ is a stationary point of the free energy $\RS(\bM;k,\delta,\nu)$. We shall refer to $\bM^*$ as the uninformative point. Let $\beta_{\sBayes} = \beta_{\sBayes}(k,\delta,\nu)$ be the supremum value of $\beta$ such that the infimum in Eq. (\[eq:InfimumFreeEnergy\]) is uniquely achieved at $\bM^*$: $$\begin{aligned}
\beta_{\sBayes}(k,\delta,\nu) = \sup\Big\{\beta\ge 0:\;\; \RS(\bM;k,\delta,\nu)>\RS(\bM_*;k,\delta,\nu) \mbox{\;\; for all\;\;\;} \bM\neq\bM_*\Big\}\, .\end{aligned}$$ As formalized below, for $\beta<\beta_{\sBayes}$ the data $\bX$ do not contain sufficient information for estimating $\bH$, $\bW$ in a non-trivial manner.
\[propo:Bayes\] Let $\bM_* = \delta\beta\bJ_k/k^2$. Then $\bM^*$ is a stationary point of the function $\bM\mapsto \RS(\bM;\beta,k,\delta,\nu)$. Further, it is a local minimum provided $\beta<\beta_{\sp}(k,\delta,\nu)$ where the spectral threshold is given by $$\begin{aligned}
\beta_{\sp} \equiv \frac{k(k\nu+1)}{\sqrt{\delta}}.
$$ Finally, if $\beta<\beta_{\sBayes}(k,\delta,\nu)$, for any estimator $\bX\mapsto \hbF_n(\bX)$, we have $$\begin{aligned}
\lim\inf_{n\to \infty}\E\big\{\left\|\bW\bH^{\sT}-\hbF_n(\bX)\right\|_F^2\big\} \ge \lim_{n\to\infty}\E\left\{\left\|\bW\bH^{\sT}-
c\bfone_n(\bX^{\sT}\bfone_n)^{\sT}\right\|_F^2\right\} \,,\label{eq:TrivialEst}
$$ for $c\equiv\sqrt{\beta}/(k+\beta\delta)$ a constant.
We refer to Appendix \[app:IT\] for a proof of this statement.
Note that Eq. (\[eq:TrivialEst\]) compares the mean square error of an arbitrary estimator $\hbF_n$, to the mean square error of the trivial estimator that replaces each column of $\bX$ by its average. This is equivalent to estimating all the weights $\bw_i$ by the uniform distribution $\bfone_k/k$. Of course, $\beta_{\sBayes}\le \beta_{\sp}$. However, this upper bound appears to be tight for small $k$.
Solving numerically the $k(k+1)/2$-dimensional problem (\[eq:InfimumFreeEnergy\]) indicates that $\beta_{\sBayes}(k,\nu,\delta) = \beta_{\sp}(k,\nu,\delta)$ for $k\in\{2,3\}$ and $\nu=1$.
Naive mean field free energy
----------------------------
We consider a trial joint distribution that factorizes according to rows of $\bW$ and $\bH$ according to Eq. (\[eq:ProductForm\]). It turns out (see Appendix \[app:NMF\_ansatz\]) that, for any stationary point of $\KL(\hq\|p_{\bH,\bW|\bX})$ over such product distributions, the marginals take the form $$\begin{aligned}
\label{eq:densityforms_main}
\begin{split}
&q_i(\bh) = \exp\left\{\left\langle\bm_i,\bh\right\rangle-\frac{1}{2}\left\langle\bh, \bQ_i\bh\right\rangle-\phi(\bm_i,\bQ_i)\right\}q_0\left(\bh\right)\, ,\\
&\tq_a(\bw) = \exp\left\{\left\langle\tbm_a,\bw\right\rangle-\frac{1}{2}\left\langle\bw, \tbQ_a\bw\right\rangle-\tphi(\tbm_a,\tbQ_a)\right\}\tq_0\left(\bw\right)\, ,
\end{split}\end{aligned}$$ where $q_0(\,\cdot\,)$ is the density of $\normal(0,\id_k)$, and $\tq_0(\,\cdot\,)$ is the density of $\Dir(\nu;k)$, and $\phi,\tphi:\reals^k\times \reals^{k\times k}\to \reals$ are defined implicitly by the normalization condition $\int q_i(\de\bh_i) = \int \tq_a(\de\bw_a) = 1$. In the following we let $\bm = (\bm_i)_{i\le d}$, $\tbm = (\tbm_a)_{a\le n}$ denote the set of parameters in these distributions; these can also be viewed as matrices $\bm\in\reals^{d\times k}$ and $\tbm\in\reals^{d\times k}$ whose $i$-th row is $\bm_i$ (in the former case) or $\tbm_i$ (in the latter).
It is useful to define the functions $\sF, \tsF :\reals^k\times \reals^{k\times k}\to\reals^k$ and $\sG,\tsG :\reals^k\times \reals^{k\times k}\to\reals^{k\times k}$ as (proportional to) expectations with respect to the approximate posteriors (\[eq:densityforms\_main\]) $$\begin{aligned}
\sF(\bm_i; \bQ) &\equiv\sqrt{\beta}\, \int \bh\,\, q_i(\de \bh) \, ,\;\;\;\;\;
\tsF(\tbm_a; \tbQ) \equiv \sqrt{\beta}\, \int \bw \, \, \tq_a(\de \bw)\, ,\label{eq:defF_main}\\
\sG(\bm_i; \bQ) &\equiv \beta\, \int \bh^{\otimes 2} \,\,q_i(\de \bh) \, ,\;\;\;\;\;
\tsG(\tbm_a; \tbQ) \equiv \beta\, \int \bw^{\otimes 2} \, \, \tq_a(\de \bw)\, .
$$ For $\bm\in\reals^{d\times k}$, we overload the notation and denote by $\sF(\bm;\bQ)\in\reals^{d\times k}$ the matrix whose $i$-th row is $\sF(\bm_i;\bQ)$ (and similarly for $\tsF(\tbm;\tbQ)$).
When restricted to a product-form ansatz with parametrization (\[eq:densityforms\_main\]), the mean field free energy takes the form (see Appendix \[app:NMF\_Free\_Energy\]) $$\begin{aligned}
\KL(\hq\|p_{\bW,\bH|\bX}) = \cF(\br,\tbr,\bOmega,\tbOmega) +\frac{d}{2}\|\bX\|_{F}^2+\log p_{\bX}(\bX)\, ,
$$ where $$\begin{aligned}
\label{eq:FreeEnergy_main}
\cF(\br,\tbr,\bOmega,\tbOmega) = &
\sum_{i=1}^d\psi_*(\br_i,\bOmega_i)+\sum_{a=1}^n\tpsi_*(\tbr_a,\tbOmega) -\sqrt{\beta}\Tr\left(\bX\br\tbr^{\sT}\right)+
\frac{\beta}{2d}\sum_{i=1}^d \sum_{a=1}^n\<\bOmega_i,\tbOmega_a\> \, ,\\
\psi_*(\br,\bOmega) \equiv \sup_{\bm, \bQ}&\left\{\< \br, \bm\> -\frac{1}{2}\<\bOmega,\bQ\>- \phi(\bm, \bQ)\right\} \, ,\;\;\;\;
\tpsi_*(\tbr,\tbOmega) \equiv \sup_{\tbm,\tbQ}\left\{\< \tbr, \tbm\> -\frac{1}{2}\<\tbOmega,\tbQ\>- \tphi(\tbm, \tbQ)\right\} \, ,\label{eq:LegendrePhi}
$$ Note that Eq. (\[eq:LegendrePhi\]) implies the following convex duality relation between $(\br,\tbr,\bOmega,\tbOmega)$ and $(\bm,\tbm,\bQ,\tbQ)$ $$\begin{aligned}
\br_i &\equiv \frac{1}{\sqrt{\beta}}\sF(\bm_i;\bQ)\,,\;\;\;\;\;\;\;\;\tbr_a \equiv \frac{1}{\sqrt{\beta}}\tsF(\tbm_a;\tbQ)\,,\label{eq:r_def}\\
\bOmega_i &\equiv \frac{1}{\beta} \sG(\bm_i;\bQ)\,,\;\;\;\;\;\;\;\;\tbOmega_a \equiv \frac{1}{\beta}\tsG(\tbm_a;\tbQ)\, .\label{eq:Omega_def}
$$ By strict convexity of $\phi(\bm,\bQ)$, $\tphi(\tbm,\tbQ)$ (the latter is strongly convex on the hyperplane $\<\bfone,\tbm\>=0$, $\<\bfone,\tbQ\bfone\>=0$) we can view $\cF(\cdots )$ as a function of $(\br,\tbr,\bOmega,\tbOmega)$ or $(\bm,\tbm,\bQ,\tbQ)$. With an abuse of notation, we will write $\cF(\br,\tbr,\bOmega,\tbOmega)$ or $\cF(\bm,\tbm,\bQ,\tbQ)$ interchangeably.
A critical (stationary) point of the free energy (\[eq:FreeEnergy\_main\]) is a point at which $\nabla\cF(\bm,\tbm,\bQ,\tbQ) =\bzero$. It turns out that the mean field free energy always admits a point that does not distinguish between the $k$ latent factors, and in particular $\bm = \bv\bfone_k^{\sT}$, $\tbm = \tbv\bfone_k^{\sT}$, as stated in detail below. We will refer to this as the *uninformative critical point* (or *uninformative fixed point*).
\[lemma:Uninf\] Define $\sE(q;\nu) \equiv (\int w_1^2 e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw))/(\int e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw))$ and let $q_1^*$ be any solution of the following equation in $[0,\infty)$ $$\begin{aligned}
q_1^* = \frac{k\beta\delta}{k-1}\, \left\{\sE\left(\frac{\beta}{1+q_1^*};\nu\right) - \frac{1}{k^2}\right\}\, . \label{eq:qs_1_main}
$$ (Such a solution always exists.) Further define $$\begin{aligned}
q_2^* &= \frac{\beta\delta-kq_1^*}{k^2}\, ,\;\;\;\;\;\tq_1^* = \frac{\beta}{1+q_1^*}\, ,\label{eq:qs_2_main}\\
\tq_2^* &= \beta\left(\frac{\|\bX^{\sT}\bfone_n\|_2^2}{d(1+q_1^*+kq_2^*)^2} - \frac{q_2^*}{(1+q_1^*)(1+q_1^*+kq_2^*)}\right)\, . \label{eq:qs_3_main}
$$ Then the naive mean field free energy of Eq. (\[eq:FreeEnergy\_main\]) admits a stationary point whereby, for all $i\in [d]$, $a\in [n]$, $$\begin{aligned}
\bm_i^* &= \frac{\sqrt{\beta}}{k}\, (\bX^{\sT}\bfone_n)_i \, \bfone_k\, ,\\
\tbm_a^* &= \frac{\beta}{k(1+q_1^*+kq_2^*)}\, (\bX\bX^{\sT}\bfone_n)_a\, \bfone_k\, ,\\
\bQ_i^* &= q_1^*\id_k + q_2^*\bJ_k\, ,\;\;\;\; \tbQ_a^* = \tq_1^*\id_k + \tq_2^*\bJ_k\,.
$$
The proof of this lemma is deferred to Appendix \[app:Uninformative\]. We note that Eq. (\[eq:qs\_1\_main\]) appears to always have a unique solution. Although we do not have a proof of uniqueness, in Appendix \[app:Uniqueness\] we prove that the solution is unique conditional on a certain inequality that can be easily checked numerically.
Naive mean field iteration
--------------------------
As mentioned in the introduction, the variational approximation of the free energy is often minimized by alternating minimization over the marginals $(q_i)_{i\le d}$, $(\tq_a)_{a\le n}$ of Eq. (\[eq:ProductForm\]). Using the parametrization (\[eq:densityforms\_main\]), we obtain the following naive mean field iteration for $\bm^t, \tbm^t, \bQ^t,\tbQ^t$ (see Appendix \[app:NMF\_ansatz\]): $$\begin{aligned}
\bm^{t+1}&= \bX^{\sT}\,\tsF(\tbm^t;\tbQ^t)\, ,\;\;\;\;\;\bQ^{t+1} = \frac{1}{d}\sum_{a=1}^n \tsG(\tbm^t_{a};\tbQ^t)\, ,\label{eq:NMF1_Main}\\
\tbm^t &= \bX\,\sF(\bm^t;\bQ^t)\, , \;\;\;\;\;\tbQ^{t} = \frac{1}{d}\sum_{i=1}^d \sG(\bm^t_{i};\bQ^t)\, .\label{eq:NMF2_Main}
$$ Note that, while the free energy naturally depends on the $(\bQ_i)_{i\le d}$, $(\tbQ_a)_{a\le n}$, the iteration sets $\bQ^t_i = \bQ^t$, $\tbQ^t_a = \tbQ^t$, independent of the indices $i,a$. In fact, any stationary point of $\cF(\bm,\tbm,\bQ,\tbQ)$ can be shown to be of this form.
The state of the iteration in Eqs. (\[eq:NMF1\_Main\]), (\[eq:NMF2\_Main\]) is given by the pair $(\bm^t,\bQ^t)\in\reals^{d\times k}\times\reals^{k\times k}$, and $(\tbm^t,\tbQ^t)$ can be viewed as derived variables. The iteration hence defines a mapping $\NMF_{\bX}:\reals^{d\times k}\times\reals^{k\times k}\to \reals^{d\times k}\times\reals^{k\times k}$, and we can write it in the form $$\begin{aligned}
(\bm^{t+1},\bQ^{t+1}) = \NMF_{\bX}(\bm^{t},\bQ^{t})\, .
$$ Any critical point of the free energy (\[eq:FreeEnergy\_main\]) is a fixed point of the naive mean field iteration and vice-versa, as follows from Appendix \[app:NMF\_Free\_Energy\]. In particular, the uninformative critical point $(\bm^*,\tbm^*,\bQ^*,\tbQ^*)$ is a fixed point of the naive mean field iteration.
Instability
-----------
In view of Section \[sec:IT-main\], for $\beta<\beta_{\sBayes}(k,\delta,\nu)$, the real posterior should be centered around a point symmetric under permutations of the topics. In particular, the posterior $\tq(\bw_a)$ over the weights of document $a$ should be centered around the symmetric distribution $\bw_a = (1/k,\dots,1/k)$. In other words, the uninformative fixed point should be a good approximation of the posterior for $\beta \leq \beta_{\sBayes}$.
A minimum consistency condition for variational inference is that the uninformative stationary point is a local minimum of the posterior for $\beta<\beta_{\sBayes}$. The next theorem provides a necessary condition for stability of the uninformative point, which we expect to be tight. As discussed below, it implies that this point is a saddle in an interval of $\beta$ below $\beta_{\sBayes}$. We recall that the index of a smooth function $f$ at stationary point $\bx_*$ is the number of the negative eigenvalues of the Hessian $\nabla^2f(\bx_*)$.
\[thm:Main\] Define $q_1^*$, $q_2^*$ as in Eqs. (\[eq:qs\_1\_main\]), (\[eq:qs\_2\_main\]), and let $$\begin{aligned}
L(\beta,k, \delta,\nu) \equiv \frac{\beta(1+\sqrt{\delta})^2 }{1+q_1^*}\left(\frac{q_1^*}{\delta\beta} + k\left[\frac{q_2^*}{1+q_1^*+kq_2^*}\left(\frac{1}{\delta\beta}+\frac{1}{k}\right)-\frac{1}{k^2}\right]_+\right)\, .
$$ If $L(\beta,k,\delta,\nu)>1$, then there exists $\eps_1,\eps_2>0$ such that the uninformative critical point of Lemma \[lemma:Uninf\], $(\bm^*,\tbm^*,\bQ^*,\tbQ^*)$ is, with high probability, a saddle point, with index at least $n\eps_1$ and $\lambda_{\min}(\cF|_{\bm^*,\tbm^*,\bQ^*,\tbQ^*})\le -\eps_2$.
Correspondingly $(\bm^*,\bQ^*)$ is an unstable critical point of the mapping $\NMF_{\bX}$ in the sense that the Jacobian $\bD\NMF_{\bX}$ has spectral radius larger than one at $(\bm^*,\bQ^*)$.
In the following, we will say that a fixed point $(\bm^*,\bQ^*)$ is stable if the linearization of $\NMF_{\bX}(\, \cdot\,)$ at $(\bm^*,\bQ^*)$ (i.e. the Jacobian matrix $\bD \NMF_{\bX}(\bm^*,\bQ^*)$) has spectral radius smaller than one. By the Hartman-Grobman linearization theorem [@perko2013differential], this implies that $(\bm^*,\bQ^*)$ is an attractive fixed point. Namely, there exists a neighborhood $\cO$ of $(\bm^*,\bQ^*)$ such that, initializing the naive mean field iteration within that neighborhood, results in $(\bm^t,\bQ^t)\to (\bm^*,\bQ^*)$ as $t\to\infty$. Vice-versa, we say that $(\bm^*,\bQ^*)$ is unstable if the Jacobian $\bD \NMF_{\bX}(\bm^*,\bQ^*)$ has spectral radius larger than one. In this case, for any neighborhood of $(\bm^*,\bQ^*)$, and a generic initialization in that neighborhood, $(\bm^t,\bQ^t)$ does not converge to the fixed point.
Motivated by Theorem \[thm:Main\], we define the instability threshold $\beta_{\inst} = \beta_{\inst}(k,\delta,\nu)$ by $$\begin{aligned}
\beta_{\inst}(k,\delta,\nu) \equiv \inf\Big\{\beta\ge 0\, :\;\; L(\beta,k,\delta,\nu)>1\, \Big\}\, .\end{aligned}$$ Let us emphasize that, while we discuss the consequences of the instability at $\beta_{\inst}$ on the naive mean field iteration, this is a problem of the variational free energy (\[eq:FreeEnergy\_main\]) and not of the specific optimization algorithm.
Numerical results for naive mean field {#sec:NMF_numerical}
--------------------------------------
In order to investigate the impact of the instability described above, we carried out extensive numerical simulations with the variational algorithm (\[eq:NMF1\_Main\]), (\[eq:NMF2\_Main\]). After any number of iterations $t$, estimates of the factors $\bH$, $\bW$ are obtained by computing expectations with respect to the marginals (\[eq:densityforms\_main\]). This results in $$\begin{aligned}
\hbH^t = \br^t= \frac{1}{\sqrt{\beta}}\sF(\bm^t;\bQ_t)\, ,\;\;\;\;\;\;\hbW^t = \tbr^t=\frac{1}{\sqrt{\beta}}\tsF(\tbm^t;\tbQ_t)\, .\label{eq:Estimates}
$$ Note that $(\hbH^t, \hbQ^t)$ can be used as the state of the naive mean-field iteration instead of $(\bm^t,\bQ^t)$.
We select a two-dimensional grid of $(\delta, \beta)$’s and generate $400$ different instances according to the LDA model for each grid point. We report various statistics of the estimates aggregated over the $400$ instances. We have performed the simulations for $\nu =1 $ and $k\in\{2,3\}$. For space considerations, we focus here on the case $\nu = 1$, $k=2$, and discuss other results in Appendix \[app:Numerical\_MF\]. (Simulations for other values of $\nu$ also yield similar results.)
We initialize both the naive mean field iteration near the uninformative fixed-point as follows: $$\begin{aligned}
\hbH^0 &=(1 - \epsilon) \,\bH_* + \epsilon \frac{\bG }{\|\bG\|_F}\|\bH_* \|_F, \\
\bQ_0 &= \bQ_*\,.
$$ Here $\bG$ has entries $(G_{ij})_{i\le d,j\le k}\sim_{iid}\normal(0,1)$ and $\epsilon=0.01$ and $\bH_* = \sF(\bm_*,\bQ_*)/\sqrt{\beta}$ is the estimate at the uninformative fixed point. We run a maximum of $300$ and a minimum of $40$ iterations, and assess convergence at iteration $t$ by evaluating $$\label{eq:convergence_criteria_main}
\Delta_t = \min_{\bPi\in \Sym_k} \big\| \hbW^{t-1}\bPi - \hbW^{t} \big\|_\infty\, ,
$$ where the minimum is over the set $\Sym_k$ of $k\times k$ permutation matrices. We declare convergence when $\Delta_t<0.005$. We denote by $\hbH$, $\hbW$ the estimates obtained at convergence.
![Normalized distances $\Norm(\hbH)$, $\Norm(\hbW)$ of the naive mean field estimates from the uninformative fixed point. Here $k=2$, $d = 1000$ and $n= d\delta$: each data point corresponds to an average over $400$ random realizations.[]{data-label="fig:H_norm_k_2"}](k_2_norm-eps-converted-to.pdf){height="5.5in"}
![Empirical fraction of instances such that $\Norm(\hbW)\ge \eps_0=10^{-4}$ (left frame) or $\Norm(\hbH)\ge \eps_0$ (right frame), where $\hbW,\hbH$ are the naive mean field estimate. Here $k=2$, $d=1000$ and, for each $(\delta,\beta)$ point on a grid, we used $400$ random realizations to estimate the probability of $\Norm(\hbW)\ge \eps_0$.[]{data-label="fig:H_norm_k_2_HM"}](k_2_norm_heatmap-eps-converted-to.pdf){height="2.66in"}
Recall the definition $\bPp=\id_k-\bfone_k\bfone_k^{\sT}/k$. In order to investigate the instability of Theorem \[thm:Main\], we define the quantities $$\begin{aligned}
\Norm(\hbW)\equiv \frac{1}{\sqrt{n}}\,\|\hbW\bPp\|_F\, ,\;\;\;\;\;\;\Norm(\hbH)\equiv \frac{1}{\sqrt{d}}\,\|\hbH\bPp\|_F
$$ In Figure \[fig:H\_norm\_k\_2\] we plot empirical results for the average $\Norm(\hbW)$, $\Norm(\hbH)$ for $k=2$, $\nu=1$ and four values of $\delta$. In Figure \[fig:H\_norm\_k\_2\_HM\], we plot the empirical probability that variational inference does not converge to the uninformative fixed point or, more precisely, $\hprob(\Norm(\hbW)\ge \eps_0)$ with $\eps_0= 10^{-4}$, evaluated on a grid of $(\beta,\delta)$ values. We also plot the Bayes threshold $\beta_{\sBayes}$ (which we find numerically that it coincides with the spectral threshold $\beta_{\sp}$) and the instability threshold $\beta_{\inst}$.
It is clear from Figures \[fig:H\_norm\_k\_2\], \[fig:H\_norm\_k\_2\_HM\], that variational inference stops converging to the uninformative fixed point (although we initialize close to it) when $\beta$ is still significantly smaller than the Bayes threshold $\beta_{\sBayes}$ (i.e. in a regime in which the uninformative fixed point would a reasonable output). The data are consistent with the hypothesis that variational inference becomes unstable at $\beta_{\inst}$, as predicted by Theorem \[thm:Main\].
![Binder cumulant for the correlation between the naive mean field estimates $\hbH$ and the true topics $\bH$, see Eq. (\[eq:Binder\_Def\]). Here we report results for $k=2$, $d\in \{500,2000,4000\}$ and $n=d\delta$, obtained by averaging over $400$ realizations. Note that for $\beta<\beta_{\sBayes}(k,\nu,\delta)$, $\Bind_{\bH}$ decreases with increasing dimensions, suggesting asymptotically vanishing correlations.[]{data-label="fig:Binder_k_2"}](k_2_corrs_binder_asymptotic_BH_v2-eps-converted-to.pdf){height="5.5in"}
![Binder cumulant for the correlation between the naive mean field estimates $\hbW$, $\hbH$ and the true weights and topics $\bW$, $\bH$. Here $k=2$, $d=1000$ and $n=d\delta$, and we averaged over $400$ realizations.[]{data-label="fig:Binder_k_2_HM"}](k_2_correlation_heatmap_binder-eps-converted-to.pdf){height="2.66in"}
Because of Proposition \[propo:Bayes\], we expect the estimates $\hbH,\hbW$ produced by variational inference to be asymptotically uncorrelated with the true factors for $\beta_{\inst}<\beta<\beta_{\sBayes}$. In order to test this hypothesis, we borrow a technique that has been developed in the study of phase transitions in statistical physics, and is known as the Binder cumulant [@binder1981finite]. For the sake of simplicity, we focus here –again– on the case $k=2$, deferring the general case to Appendix \[app:Numerical\_MF\]. Since in this case $\hbH,\bH\in \reals^{d\times 2}$, $\hbW,\bW\in \reals^{n\times 2}$, we can encode the informative component of these matrices by taking the difference between their columns. For instance, we define $\hbh_{\perp} \equiv \hbH(\be_1-\be_2)$, and analogously $\bh_{\perp}$, $\hbw_{\perp}$, $\bw_{\perp}$. We then define $$\begin{aligned}
\Corr_{\eta}(\bH,\hbH) &\equiv \<\hbh_{\perp}+\eta \bg,\bh_{\perp}\>\, ,\;\;\;\;\;\;\;\; \Bind_{\bH} \equiv\frac{3}{2}-\frac{\hE\{\Corr_{\eta}(\bH,\hbH)^4\}}{2
\hE\{\Corr_{\eta}(\bH,\hbH)^2\}^2} \, .
\label{eq:Binder_Def}
$$ Here $\hE$ denotes empirical average with respect to the sample, $\bg\sim\normal(0,\id_d)$, and we set $\eta=10^{-4}$. An analogous definition holds for $\Corr_{\eta}(\hbW)$, $\Bind_{\eta}(\hbW)$.
![Bayesian credible intervals as computed by variational inference at nominal coverage level $1-\alpha= 0.9$. Here $k=2$, $n=d=5000$, and we consider three values of $\beta$: $\beta\in\{2,4.1,6\}$ (for reference $\beta_{\inst}\approx 2.2, \beta_{\sBayes}=6$). Circles correspond to the posterior mean, and squares to the actual weights. We use red for the coordinates on which the credible interval does not cover the actual value of $w_{i,1}$.[]{data-label="fig:Uncertainty_delta1"}](Uncertainty_delta_1_new-eps-converted-to.pdf){height="3.in"}
The rationale for definition (\[eq:Binder\_Def\]) is easy to explain. At small signal-to-noise ratio $\beta$, we expect $\hbh_{\perp}$ to be essentially uncorrelated from $\bh_{\perp}$ and hence the correlation $\Corr_{\eta}(\bH,\hbH)$ to be roughly normal with mean zero and variance $\sigma^2_{\bH}$. In particular $\E\{\Corr_{\eta}(\bH,\hbH)^4\}\approx 3 \E\{\Corr_{\eta}(\bH,\hbH)^4\}$ and therefore $\Bind_{\bH}\approx 0$. (Note that the term $\eta\bg$ is added to avoid that empirical correlation vanishes, and hence $\Bind_{\bH}$ is not defined.)
In contrast, for large $\beta$, we expect $\hbh_{\perp}$ to be positively correlated with $\bh_{\perp}$, and $\Corr_{\eta}(\bH,\hbH)$ should concentrate around a non-random positive value. As a consequence, $\Bind_{\bH}\approx 1$.
In Figures \[fig:Binder\_k\_2\] we report our empirical results for $\Bind_{\bH}$ and $\Bind_{\bW}$ for four different values of $\delta$, and several values of $d$. As expected, these quantities grow from $0$ to $1$ as $\beta$ grows, and the transition is centered around $\beta_{\sBayes}$. Figure \[fig:Binder\_k\_2\_HM\] reports the results on a grid of $(\beta,\delta)$ values. Again, the transition is well predicted by the analytical curve $\beta_{\sBayes}$. These data support our claim that, for $\beta_{\inst}<\beta<\beta_{\sBayes}$, the output of variational inference is non-uniform but uncorrelated with the true signal.
Finally, in Figure \[fig:Uncertainty\_delta1\] we plot the estimates obtained for $100$ entries of the weights vector $w_{i,1}$ for three instances with $n=d=5000$ and $\beta=2<\beta_{\inst}$, $\beta= 4.1\in(\beta_{\inst},\beta_{\sBayes})$ and $\beta=6=\beta_{\sBayes}$. The interval for $w_{a,1}$ is the form $\{w_{a,1}\in [0,1]: \tq_a(w_{a,1})\ge t_{a}(\alpha)\}$ and are constructed to achieve nominal coverage level $1-\alpha=0.9$. It is visually clear that the claimed coverage level is not verified in these simulations for $\beta>\beta_{\inst}$, confirming our analytical results. Indeed, for the three simulations in Figure \[fig:Uncertainty\_delta1\] we achieve coverage $0.87$ (for $\beta=2<\beta_{\inst}$), $0.65$ (for $\beta= 4.1\in(\beta_{\inst},\beta_{\sBayes})$), and $0.51$ (for $\beta=6=\beta_{\sBayes}$). Further results of this type are reported in Appendix \[app:Numerical\_MF\].
Fixing the instability {#sec:Fixing}
======================
The fact that naive mean field is not accurate for certain classes of random high-dimensional probability distributions is well understood within statistical physics. In particular, in the context of mean field spin glasses [@SpinGlass], naive mean field is known to lead to an asymptotically incorrect expression for the free energy. We expect the same mechanism to be relevant in the context of topic models.
Namely, the product-form expression (\[eq:ProductForm\]) only holds asymptotically in the sense of finite-dimensional marginals. However, when computing the term $\E_{\hq}\log p_{\bX|\bW,\bH}(\bX|\bH,\bW)$ in the KL divergence (\[eq:Gibbs2\]), the error due to the product form approximation is non-negligible. Keeping track of this error leads to the so-called TAP free energy.
Revisiting $\integers_2$-synchronization
----------------------------------------
It is instructive to briefly discuss the $\integers_2$-synchronization example of Section \[sec:Toy\], as the basic concepts can be explained more easily in this example. For this problem, the TAP approximation replaces the free energy (\[eq:Z2\_FreeEnergy\]) with $$\begin{aligned}
\cF_{\sTAP}(\bm) & \equiv -\frac{\lambda}{2}\<\bm,\bX_0\bm\> -\sum_{i=1}^n\entro(m_i)
-\frac{n\lambda^2}{4}\big(1-Q(\bm)\big)^2\,,
$$ where $Q(\bm) \equiv \|\bm\|_2^2/n$.
We can now repeat the analysis of Section \[sec:Toy\] with this new free energy approximation. It is easy to see that $\bm_*=\bzero$ is again a stationary point. However, the Hessian is now $$\begin{aligned}
\left. \nabla^2\cF(\bm) \right|_{\bm = \bm_*} = -\lambda\bX_0 +\left(1+\lambda^2\right)\id\, .
$$ In particular, for $\lambda<1$, $\lambda_{\rm min}(\left.\nabla^2\cF\right|_{\bm = \bm_*})$ converges to $(1-\lambda)^2>0$: the uninformative stationary point is (with high probability) a local minimum.
The stationarity condition for the TAP free energy are known as TAP equations, and the algorithm that corresponds to the naive mean field iteration is Bayesian approximate message passing (AMP). For the $\integers_2$ synchronization problem, Bayes AMP is known to achieve the Bayes optimal estimation error [@deshpande2017asymptotic; @montanari2017estimation].
TAP free energy for topic models {#sec:TAP-Topic}
--------------------------------
We now turn to topic models. The TAP approach replaces the free energy (\[eq:FreeEnergy\_main\]) with the following (see Appendix \[app:TAP\_Derivation\] for a derivation) $$\begin{aligned}
\label{eq:FreeEnergy_TAP_TM}
\cF_{\sTAP}(\br,\tbr) = & \sum_{i=1}^d\psi\left(\br_i, \frac{\beta}{d}\sum_{a=1}^n\tbr_a^{\otimes 2}\right)+
\sum_{a=1}^n \tpsi\left(\tbr_a, \frac{\beta}{d}\sum_{i=1}^d\br_i^{\otimes 2}\right) -\sqrt{\beta}\Tr\left(\bX\br\tbr^{\sT}\right) -
\frac{\beta}{2d}\sum_{i=1}^d\sum_{a=1}^n\<\br_i,\tbr_a\>^2\, ,\end{aligned}$$ where $\tbr\bfone_k = \bfone_n$, and we defined the partial Legendre transforms $$\begin{aligned}
\psi(\br,\bQ) \equiv \sup_{\bm}\left\{\< \br, \bm\> - \phi(\bm, \bQ)\right\} \, ,\;\;\;\;
\tpsi(\tbr,\tbQ) \equiv \sup_{\tbm}\left\{\< \tbr, \tbm\>- \tphi(\tbm, \tbQ)\right\} \, .\label{eq:LegendrePhiPartial}
$$ Notice that $\tpsi(\tbr,\tbQ)$ is finite only if $\<\bfone_k,\tbr\>=1$.
When substituting in Eq. (\[eq:FreeEnergy\_TAP\_TM\]), the supremum of Eq. (\[eq:LegendrePhiPartial\]) is achieved at $$\begin{aligned}
\br& = \frac{1}{\sqrt{\beta}}\sF(\bm;\bQ)\, ,\;\;\;\;\;\tbr = \frac{1}{\sqrt{\beta}}\tsF(\tbm;\tbQ)\, ,\label{eq:MapM-R}\\
\bQ&=\frac{\beta}{d}\sum_{a=1}^n\tbr_a^{\otimes 2},\;\;\;\;\;\;\tbQ\equiv \frac{\beta}{d}\sum_{i=1}^d\br_i^{\otimes 2}\, .
$$
Calculus shows that stationary points of this free energy are in one-to-one correspondence (via Eq. (\[eq:MapM-R\])) with the fixed points of the following iteration: $$\begin{aligned}
\bm^{t+1}&= \bX^{\sT}\,\tsF(\tbm^t;\tbQ^t)-\sF(\bm^t;\bQ^t) \tbOmega_t\, ,\label{eq:AMP1}\\
\tbm^t &= \bX\,\sF(\bm^t;\bQ^t)-\tsF(\tbm^{t-1};\tbQ^{t-1}) \bOmega_t\, ,\label{eq:AMP2}\\
\bQ^{t+1} &= \frac{1}{d}\sum_{a=1}^n \tsF(\tbm^t_a;\tbQ^t)^{\otimes 2}\, ,\;\;\;\; \tbQ^t = \frac{1}{d}\sum_{i=1}^d \sF(\bm^t_i;\bQ^t)^{\otimes 2}\, .
\label{eq:Calibration}
$$ where $\bOmega_t$, $\tbOmega_t$ are defined as $$\begin{aligned}
\bOmega_t& =\frac{1}{d\sqrt{\beta}}\sum_{i=1}^d [\sG(\bm^t_i,\bQ^t)-\sF(\bm^t_i;\bQ^t)^{\otimes 2}]= \frac{1}{d}\sum_{i=1}^d\frac{\partial\sF}{\partial \bm_i}(\bm^t_i;\bQ^t)\, ,\label{eq:OmegaTAP1}\\
\tbOmega_t& =\frac{1}{d\sqrt{\beta}}\sum_{a=1}^n[\tsG(\tbm^t_a,\tbQ)- \tsF(\tbm^t_a;\tbQ^t)^{\otimes 2}]= \frac{1}{d}\sum_{a=1}^n\frac{\partial\tsF}{\partial \tbm_a}(\tbm^t_a;\tbQ^t)\, .\label{eq:OmegaTAP2}
$$ The stationarity conditions for the TAP free energy (\[eq:FreeEnergy\_TAP\_TM\]) are known as TAP equations, and the corresponding iterative algorithm (\[eq:AMP1\]), (\[eq:AMP2\]) is a special case of approximate message passing (AMP), with Bayesian updates. Note that the specific choice of time indices in Eqs. (\[eq:AMP1\]), (\[eq:AMP2\]) is instrumental for the analysis in the next section to hold. We also note that the general AMP analysis of [@BM-MPCS-2011; @javanmard2013state] allows for quite general choices of the sequence of matrices $\bQ_t, \tbQ_t$. However, stationarity of the TAP free energy (\[eq:FreeEnergy\_TAP\_TM\]) requires that at convergence the condition (\[eq:Calibration\]) holds at the fixed point
Estimates of the factors $\bW$, $\bH$ are computed following the same recipe as for naive mean field, cf. Eq. (\[eq:Estimates\]), namely $\hbH^t = \br^t = \sF(\bm^t;\bQ_t)/\sqrt{\beta}$, $\hbW^t = \tbr^t=\tsF(\tbm^t;\tbQ_t)/\sqrt{\beta}$.
It is not hard to see that the AMP iteration admits an uninformative fixed point, which is a stationary point of the TAP free energy, see proof in Appendix \[app:UninformativeTAP\].
\[lemma:Uninf\_TAP\] Define $q_0^* = \beta\delta/k^2$ and $\tq_0^* = \beta^2\|\bX^{\sT}\bfone_n\|_2^2/(dk^2(1+kq_0)^2)$. Then, AMP iteration admits the following fixed point $$\begin{aligned}
\bm^* & = \frac{\sqrt{\beta}}{k}(\bX^{\sT}\bfone_n)\otimes \bfone_k\, ,\label{eq:UninfTAP_1}\\
\tbm^* & = \frac{\beta}{k(1+kq_0)} (\bX\bX^{\sT}\bfone_n)\otimes\bfone_k - \frac{\beta}{k+\delta\beta} \, \bfone_n\otimes\bfone_k\, ,\label{eq:UninfTAP_2}\\
\bQ^*& = q_0^*\, \bJ_k\, ,\;\;\;\;\;\;\;\; \tbQ^* = \tq_0^* \, \bJ_k\, .
$$ This corresponds to a stationary point of the TAP free energy (\[eq:FreeEnergy\_TAP\_TM\]), via Eq. (\[eq:MapM-R\]): $$\begin{aligned}
\br_* = \frac{\sqrt{\beta}}{k(1+kq_0^*)} (\bX^{\sT}\bfone_n)\otimes \bfone_k\,,\;\;\;\;\;\;\; \tbr_* = \frac{1}{k}\bfone_n\otimes\bfone_k\, .
$$ Further, this is the only stationary point that is unchanged under permutations of the topics.
State evolution analysis {#sec:StateEvol}
------------------------
State evolution is a recursion over matrices $\bM_t$, $\tbM_t\in\reals^{k\times k}$, defined by $$\begin{aligned}
\bM_{t+1} & = \delta\, \E\Big\{\tsF(\tbM_t\bw+\tbM_t^{1/2}\bz;\tbM_t)^{\otimes 2}\Big\}\, ,\label{eq:FirstSE}\\
\tbM_{t} & = \E\Big\{\sF(\bM_t\bh+\bM_t^{1/2}\bz;\bM_t)^{\otimes 2}\Big\}\, , \label{eq:SecondSE}
$$ where expectation is with respect to $\bh\sim q_0(\,\cdot\,)$, $\bw\sim \tq_0(\,\cdot\,)$ and $\bz\sim \normal(0,\id_k)$ independent. Note that $\bM_t, \tbM_t$ are positive semidefinite symmetric matrices. Also, Eq. (\[eq:SecondSE\]) can be written explicitly as $$\begin{aligned}
\tbM_t = \beta(\id_k+\bM_t)^{-1}\bM_t\, .
$$ State evolution provides an asymptotically exact characterization of the behavior of AMP, as formalized by the next theorem (which is a direct application of [@javanmard2013state]).
\[thm:SE\] Consider the AMP algorithm of Eqs. (\[eq:AMP1\]), with deterministic initialization $\bm^0,\bQ^0$. Assume $\bG\in\reals^{d\times k}$ to be independent of data $\bX$, with entries $(G_{ij})_{i\le d,j\le k}\sim_{iid}\normal(0,1)$, and let $\bm^0 = \bH\bM_0+\bZ\bM_0^{1/2}$ for $\bM_0\in \reals^{k\times k}$ non-random, $\bM_0\succeq 0$. Let $\{\bM_t,\tbM_t\}_{t\ge 1}$ be defined by the state evolution recursion (\[eq:FirstSE\]), (\[eq:SecondSE\]). Then, for any pseudo-Lipschitz function $g:\reals^k\times\reals^k\to\reals$, we have, almost surely, $$\begin{aligned}
\lim_{n\to\infty}\frac{1}{d}\sum_{i=1}^d g(\bh_i,\bm^t_i) & =\E\Big\{g(\bh,\bM_t\bh+\bM_t^{1/2}\bz)\Big\} \, ,\\
\lim_{n\to\infty}\frac{1}{n}\sum_{a=1}^n g(\bw_a,\tbm^t_a) & =\E\Big\{g(\bw,\tbM_t\bw+\tbM_t^{1/2}\bz)\Big\} \, ,
$$ where it is understood that $n,d\to\infty$ with $n/d\to\delta$. In particular $$\begin{aligned}
\lim_{n\to\infty}\frac{1}{d}\bH^{\sT}\hbH^t & =\frac{1}{\sqrt{\beta}}\tbM_t\, ,\\
\lim_{n\to\infty}\frac{1}{n}\bW^{\sT}\hbW^t& =\frac{1}{\sqrt{\beta}}\bM_{t+1} \, .
$$ Further $\lim_{n\to \infty}\bQ^t = \bM_t$, $\lim_{n\to\infty}\tbQ^t = \tbM_t$.
Using state evolution, we can establish a stability result for AMP. First of all, notice that the state evolution iteration (\[eq:FirstSE\]), (\[eq:SecondSE\]) admits a fixed point of the form $\bM^* = (\delta\beta/k^2)\bJ_k$, $\tbM^* = \rho_0\bJ_k$, for $\rho_0 = \delta\beta^2/(k\delta\beta + k^2)$, see Appendix \[app:SE\_FP\]. This is an uninformative fixed point, in the sense that the $k$ topics are asymptotically identical. The next theorem is proved in Appendix \[sec:StabilitySE\].
\[thm:StateEvolStable\] If $\beta<\beta_{\sp}(k,\nu,\delta)$, then the uninformative fixed point is stable under the state evolution iteration (\[eq:FirstSE\]), (\[eq:SecondSE\]).
In particular, for $\beta<\beta_{\sp}(k,\nu,\delta)$, there exists $c_0=c_0(\beta,k\nu,\delta)$ such that, if we initialize AMP as in Theorem \[thm:SE\] with $\|\bM_0-\bM^*\|_F\le c_0$, then (recalling $\bPp = \id_k-\bfone_k\bfone_k/k$) $$\begin{aligned}
\lim_{t\to\infty}\lim_{n\to\infty}\frac{1}{n}\big\|\bm^t\bPp\|_F^2 = 0\, ,\;\;\;\;\;\;\lim_{t\to\infty}\lim_{n\to\infty}\frac{1}{n}\big\|\bm^t\bPp\|_F^2 = 0\, .
$$
Stability of the uninformative fixed point
------------------------------------------
The next theorem establishes that the uninformative fixed point of the TAP free energy is a local minimum for all $\beta$ below the spectral threshold $\beta_{\sp}(k,\nu,\delta)$. Since $\beta_{\sBayes}(k,\nu,\delta)\le \beta_{\sp}(k,\nu,\delta)$, this shows that the instability we discovered in the case of naive mean field is corrected by the TAP free energy.
\[thm:StabilityTAP\] Let $(\br_*,\tbr_*)$ be the uninformative stationary point of the TAP free energy, cf. Lemma \[lemma:Uninf\_TAP\]. If $\beta<\beta_{\sp}(k,\nu,\delta)$, then there exists $\eps>0$ such that, with high probability $$\begin{aligned}
\lambda_{\min}\left(\left.\nabla^2\cF_{\sTAP}\right|_{(\br_*,\tbr_*)}\right)\ge \eps\, .
$$
Let us emphasize that this result is not implied by the state evolution result of Theorem \[thm:StateEvolStable\], which only establishes stability in a certain asymptotic sense. Vice-versa, Theorem \[thm:StabilityTAP\] does not directly imply Theorem \[thm:StateEvolStable\].
Numerical results for TAP free energy {#sec:TAP_numerical}
-------------------------------------
![Normalized distances $\Norm(\hbH)$, $\Norm(\hbW)$ of the AMP estimates from the uninformative fixed point. Here, $k=2$, $d = 1000$ and $n= d\delta$: each data point corresponds to an average over $400$ random realizations.[]{data-label="fig:AMP_norm_k_2"}](new_k_2_amp_norm-eps-converted-to.pdf){height="5.5in"}
![Empirical fraction of instances such that $\Norm(\hbW)\ge
\eps_0=5\cdot 10^{-3}$, where $\hbW$ is the AMP estimate. Here $k=2$, $d=1000$, and for each $(\delta,\beta)$ point on the grid we ran AMP on $400$ random realizations.[]{data-label="fig:AMP_norm_k_2_HM"}](new_k_2_amp_norm_heatmap_fractions-eps-converted-to.pdf){height="2.66in"}
![Binder cumulant for the correlation between AMP estimates $\hbH$ and the true topics $\bH$, and between $\hbW$ and $\bW$, see Eq. (\[eq:Binder\_Def\]). Here $k=2$, $d=1000$, $n=d\delta$ and estimates are obtained by averaging over $400$ realizations.[]{data-label="fig:AMP_corrs_k_2"}](new_k_2_amp_binder-eps-converted-to.pdf){height="5.5in"}
![Binder cumulant for the correlation between AMP estimates $\hbW$, $\hbH$ and the true weights and topics $\bW, \bH$. Here $k=2$,$d=1000$ and estimates are obtained by averaging over $400$ realizations.[]{data-label="fig:AMP_corrs_k_2_HM"}](new_k_2_amp_Binder_heatmap-eps-converted-to.pdf){height="2.66in"}
In order to confirm the stability analysis at the previous section, we carried out numerical simulations analogous to the ones of Section \[sec:NMF\_numerical\]. We found that the AMP iteration of Eqs. (\[eq:AMP1\]), (\[eq:AMP2\]) is somewhat unstable when $\beta\approx \beta_{\sp}$. In order to remedy this problem, we used a damped version of the same iteration, see Appendix \[app:Damped\]. Notice that damping does not change the stability of a local minimum or saddle, it merely reduces oscillations due to aggressive step sizes.
We initialize the iteration as for naive mean field, and monitor the same quantities, as in Section \[sec:NMF\_numerical\]. In particular, here we report results on the distance from the uninformative subspace $\Norm(\hbH)$, $\Norm(\hbW)$, in Figures \[fig:AMP\_norm\_k\_2\] and \[fig:AMP\_norm\_k\_2\_HM\], and the Binder cumulants $\Bind_{\bH}$ and $\Bind_{\bW}$, measuring the correlation between AMP estimates and the true factors $\bW, \bH$, in Figures \[fig:AMP\_corrs\_k\_2\], \[fig:AMP\_corrs\_k\_2\_HM\]. We focus on the case $k=2$, deferring $k=3$ to the appendices.
In the intermediate regime $\beta\in (\beta_{\inst},\beta_{\sp})$, the behavior of AMP is strikingly different from the one of naive mean field. AMP remains close to the uninformative fixed point, confirming that this is a local minimum of the TAP free energy. The distance from the uninformative subspace starts growing only at the spectral threshold $\beta_{\sp}$ (which coincides, in the present cases, with the Bayes threshold $\beta_{\sBayes}$). At the same point, the correlation with the true factors $\bW$, $\bH$ also becomes strictly positive.
Discussion
==========
Bayesian methods are particularly attractive in unsupervised learning problems such as topic modeling. Faced with a collection of documents $\bx_1$,…$\bx_n$, it is not clear a priori whether they should be modeled as convex combinations of topics, or how many topics should be used. Even after a low-rank factorization $\bX\approx \bW\bH^{\sT}$ is computed, it is still unclear how to evaluate it, or to which extent it should be trusted.
Bayesian approaches provide estimates of the factors $\bW$, $\bH$, but also a probabilistic measure of how much these estimates should be trusted. To the extent that the posterior concentrates around its mean, this can be considered as a good estimate of a true underlying signal.
It is well understood that Bayesian estimates can be unreliable if the prior is not chosen carefully. Our work points at a second reason for caution. When variational inference is used for approximating the posterior, the result can be incorrect even if the data are generated according to the prior. More precisely, we showed that for a certain regime of parameters, naive mean field ‘believes’ that there is a signal, even if it is information-theoretically impossible to extract any non-trivial estimate from the data.
Given that naive mean field is the method of choice for inference with topic models [@blei2003latent], it would be of great interest to remedy this instability. We showed that the TAP free energy provides a better mean field approximation, and in particular does not have the same instability. However, this approximation is also based on the correctness of the generative model, and further investigation is warranted on its robustness.
Acknowledgements {#acknowledgements .unnumbered}
================
H.J. and A.M. were partially supported by grants NSF CCF-1714305 and NSF IIS-1741162. B.G. was supported by Stanford’s Caroline and Fabian Pease Graduate Fellowship.
[AGKM12]{}
Edoardo M Airoldi, David M Blei, Stephen E Fienberg, and Eric P Xing, *Mixed membership stochastic blockmodels*, Journal of Machine Learning Research **9** (2008), no. Sep, 1981–2014.
Sanjeev Arora, Rong Ge, Ravindran Kannan, and Ankur Moitra, *Computing a nonnegative matrix factorization–provably*, Proceedings of the forty-fourth annual ACM symposium on Theory of computing, ACM, 2012, pp. 145–162.
Sanjeev Arora, Rong Ge, and Ankur Moitra, *Learning topic models–going beyond svd*, Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on, IEEE, 2012, pp. 1–10.
Greg W. Anderson, Alice Guionnet, and Ofer Zeitouni, *[An introduction to random matrices]{}*, Cambridge University Press, 2009.
Ahmed El Alaoui and Florent Krzakala, *Estimation in the spiked wigner model: A short proof of the replica formula*, arXiv preprint arXiv:1801.01593 (2018).
Jinho Baik, G[é]{}rard Ben Arous, and Sandrine P[é]{}ch[é]{}, *Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices*, Annals of Probability (2005), 1643–1697.
Tamara Broderick, Nicholas Boyd, Andre Wibisono, Ashia C Wilson, and Michael I Jordan, *Streaming variational bayes*, Advances in Neural Information Processing Systems, 2013, pp. 1727–1735.
Peter Bickel, David Choi, Xiangyu Chang, and Hai Zhang, *Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels*, The Annals of Statistics (2013), 1922–1943.
Jean Barbier, Mohamad Dia, Nicolas Macris, Florent Krzakala, Thibault Lesieur, and Lenka Zdeborov[á]{}, *Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula*, Advances in Neural Information Processing Systems, 2016, pp. 424–432.
Florent Benaych-Georges and Raj Rao Nadakuditi, *The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices*, Advances in Mathematics **227** (2011), no. 1, 494–521.
[to3em]{}, *The singular values and vectors of low rank perturbations of large rectangular random matrices*, Journal of Multivariate Analysis **111** (2012), 120–135.
Kurt Binder, *Finite size scaling analysis of ising model block distribution functions*, Zeitschrift f[ü]{}r Physik B Condensed Matter **43** (1981), no. 2, 119–140.
David M Blei, Alp Kucukelbir, and Jon D McAuliffe, *Variational inference: A review for statisticians*, Journal of the American Statistical Association (2017), no. just-accepted.
David M Blei and John D Lafferty, *Dynamic topic models*, Proceedings of the 23rd international conference on Machine learning, ACM, 2006, pp. 113–120.
[to3em]{}, *A correlated topic model of science*, The Annals of Applied Statistics (2007), 17–35.
David M Blei, *Probabilistic topic models*, Communications of the ACM **55** (2012), no. 4, 77–84.
Mohsen Bayati and Andrea Montanari, *[The dynamics of message passing on dense graphs, with applications to compressed sensing]{}*, IEEE Trans. on Inform. Theory **57** (2011), 764–785.
Raphael Berthier, Andrea Montanari, and Phan-Minh Nguyen, *State evolution for approximate message passing with non-separable functions*, [ arXiv:1708.03950]{} (2017).
David M Blei, Andrew Y Ng, and Michael I Jordan, *Latent dirichlet allocation*, Journal of machine Learning research **3** (2003), no. Jan, 993–1022.
Z. Bai and J. Silverstein, *[Spectral Analysis of Large Dimensional Random Matrices]{}*, Springer, 2010.
Jonathan Chang and David Blei, *Relational topic models for document networks*, Artificial Intelligence and Statistics, 2009, pp. 81–88.
Alain Celisse, Jean-Jacques Daudin, Laurent Pierre, et al., *Consistency of maximum-likelihood and variational estimators in the stochastic block model*, Electronic Journal of Statistics **6** (2012), 1847–1899.
Yash Deshpande, Emmanuel Abbe, and Andrea Montanari, *Asymptotic mutual information for the balanced binary stochastic block model*, Information and Inference: A Journal of the IMA **6** (2017), no. 2, 125–170.
Yash Deshpande and Andrea Montanari, *Information-theoretically optimal sparse pca*, Information Theory (ISIT), 2014 IEEE International Symposium on, IEEE, 2014, pp. 2197–2201.
David L. Donoho, Arian Maleki, and Andrea Montanari, *[Message Passing Algorithms for Compressed Sensing]{}*, Proceedings of the National Academy of Sciences **106** (2009), 18914–18919.
[to3em]{}, *[Message Passing Algorithms for Compressed Sensing: I. Motivation and Construction]{}*, Proceedings of IEEE Inform. Theory Workshop (Cairo), 2010.
Elena Erosheva, Stephen Fienberg, and John Lafferty, *Mixed-membership models of scientific publications*, Proceedings of the National Academy of Sciences **101** (2004), no. suppl 1, 5220–5227.
Li Fei-Fei and Pietro Perona, *A bayesian hierarchical model for learning natural scene categories*, Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, vol. 2, IEEE, 2005, pp. 524–531.
Delphine F[é]{}ral and Sandrine P[é]{}ch[é]{}, *The largest eigenvalue of rank one deformation of large wigner matrices*, Communications in mathematical physics **272** (2007), no. 1, 185–228.
Ryan J Giordano, Tamara Broderick, and Michael I Jordan, *Linear response methods for accurate covariance estimates from mean field variational bayes*, Advances in Neural Information Processing Systems, 2015, pp. 1441–1449.
Matthew Hoffman, Francis R Bach, and David M Blei, *[Online learning for Latent Dirichlet Allocation]{}*, Advances in neural information processing systems, 2010, pp. 856–864.
Michael I Jordan, Zoubin Ghahramani, Tommi S Jaakkola, and Lawrence K Saul, *An introduction to variational methods for graphical models*, Machine learning **37** (1999), no. 2, 183–233.
Adel Javanmard and Andrea Montanari, *State evolution for general approximate message passing algorithms, with applications to spatial coupling*, Information and Inference: A Journal of the IMA **2** (2013), no. 2, 115–144.
Daphne Koller and Nir Friedman, *Probabilistic graphical models: principles and techniques*, MIT press, 2009.
Satish Babu Korada and Nicolas Macris, *Exact solution of the gauge symmetric p-spin glass model on a complete graph*, Journal of Statistical Physics **136** (2009), no. 2, 205–230.
Florent Krzakala, Jiaming Xu, and Lenka Zdeborov[á]{}, *Mutual information in rank-one matrix estimation*, IEEE Information Theory Workshop (ITW), 2016, pp. 71–75.
John D Lafferty and David M Blei, *Correlated topic models*, Advances in neural information processing systems, 2006, pp. 147–154.
Thibault Lesieur, Florent Krzakala, and Lenka Zdeborov[á]{}, *Constrained low-rank matrix estimation: Phase transitions, approximate message passing and applications*, [arXiv:1701.00858]{} (2017).
Marc Lelarge and L[é]{}o Miolane, *Fundamental limits of symmetric low-rank matrix estimation*, [arXiv:1611.03888]{} (2016).
L[é]{}o Miolane, *Fundamental limits of low-rank matrix estimation*, [ arXiv:1702.00473]{} (2017).
Marc M[é]{}zard and Andrea Montanari, *[Information, Physics and Computation]{}*, Oxford, 2009.
Marc Mézard, Giorgio Parisi, and Miguel A. Virasoro, *Spin glass theory and beyond*, World Scientific, 1987.
Andrea Montanari, Daniel Reichman, and Ofer Zeitouni, *On the limitation of spectral methods: From the gaussian hidden clique problem to rank one perturbations of gaussian tensors*, IEEE Transactions on Information Theory **63** (2017), no. 3, 1572–1579.
Andrea Montanari and Ramji Venkataramanan, *Estimation of low-rank matrices via approximate message passing*, [arXiv:1711.01682]{} (2017).
Manfred Opper and Ole Winther, *Adaptive and self-averaging thouless-anderson-palmer mean-field theory for probabilistic modeling*, Physical Review E **64** (2001), no. 5, 056131.
Debdeep Pati, Anirban Bhattacharya, and Yun Yang, *On statistical optimality of variational bayes*, [arXiv:1712.08983]{} (2017).
Lawrence Perko, *Differential equations and dynamical systems*, vol. 7, Springer Science & Business Media, 2013.
Ben Recht, Christopher Re, Joel Tropp, and Victor Bittorf, *Factoring nonnegative matrices with linear programs*, Advances in Neural Information Processing Systems, 2012, pp. 1214–1222.
Anil Raj, Matthew Stephens, and Jonathan K Pritchard, *faststructure: variational inference of population structure in large snp data sets*, Genetics **197** (2014), no. 2, 573–589.
David J. Thouless, Philip W. Anderson, and Richard G. Palmer, *Solution of’solvable model of a spin glass’*, Philosophical Magazine **35** (1977), no. 3, 593–601.
Chong Wang and David M Blei, *Collaborative topic modeling for recommending scientific articles*, Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, 2011, pp. 448–456.
Xiaogang Wang and Eric Grimson, *Spatial latent dirichlet allocation*, Advances in neural information processing systems, 2008, pp. 1577–1584.
Martin J Wainwright and Michael I Jordan, *Graphical models, exponential families, and variational inference*, Foundations and Trends in Machine Learning **1** (2008), no. 1-2, 1–305.
Bo Wang and D. Michael Titterington, *[Convergence and asymptotic normality of variational Bayesian approximations for exponential family models with missing values]{}*, Proceedings of the 20th conference on Uncertainty in artificial intelligence, AUAI Press, 2004, pp. 577–584.
[to3em]{}, *[Inadequacy of interval estimates corresponding to variational Bayesian approximations]{}*, AISTATS, 2005.
[to3em]{}, *[Convergence properties of a general algorithm for calculating variational Bayesian estimates for a normal mixture model]{}*, Bayesian Analysis **1** (2006), no. 3, 625–650.
Jing Zhou, Anirban Bhattacharya, Amy H Herring, and David B Dunson, *Bayesian factorizations of big sparse tensors*, Journal of the American Statistical Association **110** (2015), no. 512, 1562–1576.
Anderson Y Zhang and Harrison H Zhou, *Theoretical and computational guarantees of mean field variational inference for community detection*, [ arXiv:1710.11268]{} (2017).
Some remarks on alternating minimization
========================================
Let $f: \reals^{n}\times \reals^d \to \reals$ be twice continuously differentiable in an open neighborhood $\Omega_1\times \Omega_2\subseteq\reals^{n}\times\reals^d$ of a critical point $(\bx^*, \by^*)$ (i.e. a point for which $\nabla_{(\bx,\by)}f(\bx,\by) = \bzero$). Further assume that, fixing $\bx_0\in \Omega_1$, $f(\bx_0,\,\cdot\,)$ is strongly convex with a minimizer in $\Omega_2$, and fixing $\by_0\in\Omega_2$, $f(\,\cdot\,,\by_0)$ is strongly convex with a minimizer in $\Omega_1$. By taking $\Omega_1$ and $\Omega_2$ sufficiently small, these conditions follow by requiring that the partial Hessians satisfy $\nabla^2_{\bx}f(\bx^*,\by^*)\succ \bzero$ and $\nabla^2_{\by}f(\bx^*,\by^*)\succ \bzero$ (i.e. they are strictly positive definite).
By strong convexity, the minimizers of $f(\bx_0,\,\cdot\,)$ and $f(\,\cdot\,,\by_0)$ are unique, and we can define the functions $g:\reals^d\to\reals^n$ and $h:\reals^n\to\reals^d$ by $$\begin{aligned}
h(\bx_0) = \arg\min_{\by\in\Omega_2} f(\bx_0,\by)\, ,\\
g(\by_0) = \arg\min_{\bx\in\Omega_1} f(\bx,\by_0)\, .
$$ We then define the alternating minimization iteration $$\begin{aligned}
\label{eq:alternatemin}
\bx^{t+1} = h(\by^t),\;\;\;\;\;\;
\by^t = g(\bx^t)\, .
$$ If $d=n$ and $h:\Omega_1\to\Omega_2$, $g:\Omega_2\to\Omega_1$ are bijective, we also define the dual iteration $$\begin{aligned}
\label{eq:alternatemin_dual}
\obx^{t+1} = g^{-1}(\oby^t),\;\;\;\;\;\;
\oby^t = h^{-1}(\obx^t)\, .
$$
\[lemma:hessianstable\] Let $f: \reals^{n}\times \reals^d \to \reals$ by twice continuously differentiable in $\Omega_1\times \Omega_2$, satisfying the above assumptions. Then the following are equivalent:
- The Hessian $\bH = \nabla^2_{(\bx, \by)} f\big|_{(\bx, \by) = (\bx^*, \by^*)}$ is strictly positive definite.
- $(\bx^*,\by^*)$ is a stable fixed point of the alternate minimization algorithm (\[eq:alternatemin\]).
- $f_1(\bx) \equiv \min_{\by\in\Omega_2} f(\bx,\by)$ is strongly convex in a neighborhood of $\bx^*$ (and in particular, $\bx^*$ is a local minimum of $f_1$).
Further, if $n=d$ and the matrix $\left.\frac{\partial f}{\partial\bx\partial \by}\right|_{\bx^*,\by^*}$ is invertible, then the following are equivalent:
- $(\bx^*,\by^*)$ is a stable fixed point of the dual algorithm (\[eq:alternatemin\_dual\]).
- $f_1(\bx) \equiv \min_{\by\in\Omega_2} f(\bx,\by)$ is strongly concave in a neighborhood of $\bx^*$ (and in particular, $\bx^*$ is a local maximum).
Let $$\begin{aligned}
\bH = \begin{bmatrix}
\bH_{\bx\bx} & \bH_{\bx\by} \\
\bH_{\bx\by}^\sT & \bH_{\by\by}
\end{bmatrix} = \nabla^2_{(\bx, \by)} f\big|_{(\bx, \by) = (\bx^*, \by^*)}.\end{aligned}$$ [**(A1)**]{}$\equiv$[**(A2)**]{} We compute the linearization of the iterations in around the fixed point $(\bx^*, \by^*)$. Note that since $\bx^*$ is a minimizer of $f(\,\cdot\, ,\by^*)$, using the implicit function theorem for the Jacobian of the update rule for $\bx$ in we have $$\begin{aligned}
\frac {\partial^2 f}{\partial \bx \partial \by}\bigg|_{(\bx, \by) = (\bx^*, \by^*)} + \left[\frac{\partial^2 f}{\partial \bx^{2}}\bigg|_{(\bx, \by) = (\bx^*, \by^*)}\right]\left[\bD h(\by^*)\right] = 0.\end{aligned}$$ Hence, we get $$\begin{aligned}
\bD h(\by^*)= - \left[\left(\frac{\partial^2 f}{\partial \bx^{2}}\right)^{-1} \left(\frac {\partial^2 f}{\partial \bx \partial \by}\right)\right]_{(\bx, \by) = (\bx^*, \by^*)} = -\bH_{\bx\bx}^{-1}\bH_{\bx\by}.\end{aligned}$$ Similarly, for the Jacobian of the update rule for $\by$ in we have $$\begin{aligned}
\bD g(\bx^*)= - \left[\left(\frac{\partial^2 f}{\partial \by^{2}}\right)^{-1} \left(\frac {\partial^2 f}{\partial \by \partial \bx}\right)\right]_{(\bx, \by) = (\bx^*, \by^*)} = -\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT.
\label{eq:JacobianG}\end{aligned}$$ Hence, $(\bx^*, \by^*)$ is stable if and only if the operator $$\begin{aligned}
\bL = \bD h(\bx^*) \cdot \bD g(\by^*) = \bH_{\bx\bx}^{-1}\bH_{\bx\by} \bH_{\by\by}^{-1}\bH_{\bx\by}^\sT\, ,\end{aligned}$$ has spectral radius $$\begin{aligned}
\sigma(\bL) \equiv \max_{i}\left|\lambda_i\left(\bL\right)\right| < 1.\end{aligned}$$ Since $f(\,\cdot\,,\bx^*)$ is strongly convex, the matrices $\bH_{\bx\bx}, \bH_{\bx\bx}^{-1}$ are positive definite. Hence, the eigenvalues of $\bH_{\bx\bx}^{-1}\bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT$ are real and equal to the eigenvalues of the symmetric positive semi-definite matrix $\bH_{\bx\bx}^{-1/2}\bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT\bH_{\bx\bx}^{-1/2}$. Therefore, $\sigma(\bL)<1$ if and only if $$\begin{aligned}
\bH_{\bx\bx}^{-1/2}\bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT\bH_{\bx\bx}^{-1/2} \prec \id_n \iff
\bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT \prec \bH_{\bx\bx} \iff \bH_{\bx\bx} - \bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT\succ 0.\end{aligned}$$ Note that since $f(\bx^*,\,\cdot\,)$ is convex, $\bH_{\by\by} \succ 0$. Therefore, $\bH_{\bx\bx} - \bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT\succ 0$ if and only if $\bH \succ 0$. Hence, the fixed point is stable if and only if $\bH \succ 0$ and this completes the proof.
[**(A1)**]{}$\equiv$ [**(A3)**]{} By differentiating $f_1(\bz) = f(\bx,g(\bx))$, we obtain $$\begin{aligned}
\left .\frac{\partial^2f_1}{\partial\bx^2}\right|_{\bx^*} &= \left.\frac{\partial^2f}{\partial\bx^2}\right|_{\bx^*,\by^*} +
\left.\frac{\partial^2f}{\partial\bx\partial\by}\right|_{\bx^*,\by^*} \cdot\bD g(\bx^*)\\
& = \bH_{\bx\bx} -\bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^{\sT}\, ,
$$ where in the last line we used Eq. (\[eq:JacobianG\]). Hence $\left .\frac{\partial^2f_1}{\partial\bx^2}\right|_{\bx^*} \succ \bzero$ if and only if $\bH_{\bx\bx} \succ \bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT$ which, by Schur’s complement formula is equivalent to $\bH\succ \bzero$. Further, since $f\in C^2(\reals^{n+d})$, $\left .\frac{\partial^2f_1}{\partial\bx^2}\right|_{\bx^*} \succ \bzero$ if and only if $\frac{\partial^2f_1}{\partial\bx^2} \succ \bzero$ in a neighborhood of $\bx^*$.
[**(B1)**]{}$\equiv$ [**(B2)**]{} Linearizing the iteration (\[eq:alternatemin\_dual\]), we get that $(\bx^*,\by^*)$ is a stable fixed point if and only if the operator $$\begin{aligned}
\bL^{-1} = \bD g(\bx^*)^{-1} \bD h(\by^*)^{-1}=(\bH_{\bx\by}^\sT)^{-1}\bH_{\by\by}\bH_{\bx\by}^{-1}\bH_{\bx\bx}
$$ has spectral radius $$\begin{aligned}
\sigma(\bL^{-1}) \equiv \max_{i\le n}\left|\lambda_i\left(\bL^{-1}\right)\right| < 1.
$$ Using the fact that $\bH_{\bx\bx}\succ \bzero$, we have that $\sigma(\bL^{-1})<1$ if and only if $$\begin{aligned}
\bH_{\bx\bx}^{1/2}(\bH^\sT_{\bx\by})^{-1}\bH_{\by\by}\bH_{\bx\by}^{-1}\bH_{\bx\bx}^{1/2} \prec \id_n \iff
(\bH_{\bx\by}^{\sT})^{-1}\bH_{\by\by}\bH_{\bx\by}^{-1} \prec \bH^{-1}_{\bx\bx} \iff \bH_{\bx\bx} - \bH_{\bx\by}\bH_{\by\by}^{-1}\bH_{\bx\by}^\sT\prec \bzero.\end{aligned}$$ As shown above, the last condition is equivalent to $\left.\frac{\partial^2 f_1}{\partial\bx^2}\right|_{\bx^*}\prec \bzero$, and by continuity of the Hessian, this is equivalent to $f_1$ being strongly concave in a neighborhood of $\bx^*$.
Proof of Proposition \[propo:Toy\]
==================================
It is useful to first prove a simple random matrix theory remark.
\[lemma:Submatrix\] For $S\subseteq [n]$, let $\bX_{S,S}$ be the submatrix of $\bX$ with rows and columns with index in $S$. Then, for any $\eps\in [0,1)$, the following holds with high probability: $$\begin{aligned}
\min\big\{\lambda_{\max}(\bX_{S,S}):\, |S|\ge n(1-\eps)\big\} \ge 2\sqrt{1-\eps}-o_n(1)\, .
$$
Without loss of generality we can assume $\bX\sim\GOE(n)$ (because the rank-one deformation cannot decrease the maximum eigenvalue), and $|S|=n(1-\eps)$ (because $\lambda_{\max}(\bX_{S,S})$ is non-decreasing in $S$). Note that $\bX_{S,S}$ is distributed as $\sqrt{1-\eps}$ times a $\GOE(n(1-\eps))$ matrix. Large deviation bounds on the eigenvalues of $\GOE$ matrices imply that, for any $\delta>0$, there exists $c(\delta)>0$ such that $$\begin{aligned}
\prob\big(\lambda_{\max}(\bX_{S,S})\le 2\sqrt{1-\eps}-\delta\big)\le 2\, e^{-c(\delta)n^2} \, ,\end{aligned}$$ for all $n$ large enough. The claim follows by union bound since there is at most $2^n$ such sets $S$.
First notice that Lemma \[lemma:Submatrix\] continues to hold if $\bX$ is replaced by $\bX_0$ since $\|\bX_{S,S}-(\bX_0)_{S,S}\|_{\op}\le \max_{i\le n}|X_{ii}|\le 4\sqrt{\log n/n}$ (where the last bound holds with high probability since $(X_{ii})_{i\le n}\sim\normal(0,2/n)$.
Note that $\nabla\cF(\bm)_i=\pm \infty$ if $m_i= \pm 1$, whence any local minimum must be in the interior of $[-1,+1]^n$. Let $\bm\in (-1,-1)^n$ be a local minimum of $\cF(\,\cdot\,)$. By the second-order minimality conditions, we must have $$\begin{aligned}
\nabla^2\cF(\bm) = -\lambda\bX_0+ \diag\left((1-m_i^2)^{-1}_{i\le n}\right)\succeq \bzero\, .
$$ Denote by $m_{(1)}$, $m_{(2)}$, $\dots$ the entries of $\bm$ ordered by decreasing absolute value, and let $S_{\ell}$ be the set of indices corresponding to entries $m_{(\ell+1)},\dots, m_{(n)}$. Finally let $\bv^{(\ell)}\in\reals^n$ be the eigenvector corresponding to the largest eigenvalue of $(\bX_0)_{S_{\ell},S_{\ell}}$ (extended with zeros outside $S_{\ell}$). We then have, for $\ell=n\eps$ $$\begin{aligned}
0&\le \<\bv^{(\ell)},\nabla^2\cF(\bm) \bv^{(\ell)}\> \\
&= -\lambda\cdot\lambda_{\max}\big((\bX_0)_{S_{\ell},S_{\ell}}\big)+ \sum_{i\in S_{\ell}}\frac{(v^{(\ell)}_i)^2}{1-m_i^2}\\
&\le -2\lambda\sqrt{1-\eps} +\frac{1}{1-m_{(n\eps)}^2}+o_n(1)\, .
$$ The last inequality holds with high probability by Lemma \[lemma:Submatrix\]. Inverting it, we get $$\begin{aligned}
m^2_{(n\eps)}\ge 1-\frac{1}{2\lambda\sqrt{1-\eps}}-o_n(1)\, ,
$$ and therefore $$\begin{aligned}
\frac{1}{n}\|\bm\|_2^2\ge \eps\left(1-\frac{1}{2\lambda\sqrt{1-\eps}}\right)-o_n(1).
$$ The claim follows by taking $\eps=c_1$ a small constant (for which the right-hand side is lower bounded by $c_0$ for all $\lambda\ge 1$), or $\eps = c_2(2\lambda-1)$ (for which the right-hand side is lower bounded by $c_0(2\lambda-1)^2$).
Information-theoretic limits {#app:IT}
============================
Proof of Lemma \[lemma:IT-Threshold-Z2\] {#app:LemmaITZ2}
----------------------------------------
Let $\hbQ:\reals^{n\times n}\mapsto\reals^{n\times n}$, $\bX\mapsto \hbQ(\bX)$ be any estimator of $\bsigma\bsigma^{\sT}$. By [@deshpande2017asymptotic Theorem 1.6], for $\lambda\in [0,1]$, $$\begin{aligned}
\lim\inf_{n\to\infty}\frac{1}{n^2}\E\Big\{\big\|\bsigma\bsigma^{\sT}-\hbQ(\bX)\big\|_F^2\Big\} \ge 1\, .
$$ Given $\hbsigma:\reals^{n\times n}\to\reals^n\setminus \{\bzero\}$, set $$\begin{aligned}
\hbQ(\bX) = c\, \frac{\hbsigma(\bX)\hbsigma(\bX)^{\sT}}{\|\hbsigma(\bX)\|_2^2}
\, ,\;\;\;\;\; c = \E\left(\frac{\<\hbsigma(\bX),\bsigma\>^2}{\|\hbsigma(\bX)\|_2^2}\right)\, .\end{aligned}$$ By a simple calculation $$\begin{aligned}
1-o_n(1)\le \frac{1}{n^2}\E\Big\{\big\|\bsigma\bsigma^{\sT}-\hbQ(\bX)\big\|_F^2\Big\} = 1-\E\left(\frac{\<\hbsigma(\bX),\bsigma\>^2}{\|\hbsigma(\bX)\|_2^2}\right)^2
\, ,
$$ which obviously implies the claim.
Proof of Proposition \[propo:Bayes\] {#app:ProofBayes}
------------------------------------
We begin by providing the expression for the free energy functional $\RS(\bM;k,\delta,\nu)$ of Theorem \[thm:IT\_Limit\], which is obtained by specializing the expression in [@miolane2017fundamental]. Recall the functions $\phi(\,\cdots\, )$, $\tphi(\,\cdots\, )$, introduced in Eq. (\[eq:densityforms\_main\]). We then define a function $\RS_0(\,\cdot\,,\,\cdot\,;k,\delta,\nu):\bbS_k\times\bbS_k\to\reals$ by $$\begin{aligned}
\RS_0(\bM,\tbM;k,\delta,\nu)&= \frac{\beta\delta(\nu+1)}{k\nu+1}+
\frac{1}{2\beta}\<\bM,\tbM\>\label{eq:PsiGeneral}\\
&-\E\,\phi(\bM\bh+\bM^{1/2}\bz;\bM)-\delta\,\E\, \tphi(\tbM\bw+\tbM^{1/2}\bz;\tbM)\, ,\nonumber
$$ where expectations are with respect to $\bz\sim\normal(0,\id_k)$ independent of $\bh\sim\normal(0,\id_k)$ and $\bw\sim\Dir(\nu;k)$. We then have $$\begin{aligned}
\RS(\bM;k,\delta,\nu) = \sup_{\tbM\in\bbS_k}\RS_0(\bM,\tbM;k,\delta,\nu)\, .
$$ Further, the function $\RS_0(\bM,\tbM;k,\delta,\nu)$ on Eq. (\[eq:PsiGeneral\]) is separately strictly concave in $\bM$ and $\tbM$, and in particular the last supremum is uniquely achieved at a point $\tbM = \tbM(\bM)$.
A simple calculation shows that $$\begin{aligned}
\frac{\partial\RS_0}{\partial\bM}(\bM,\tbM;k,\delta,\nu) & = \frac{1}{2\beta}\left\{\tbM - \E\Big\{\sF(\bM\bh+\bM^{1/2}\bz;\bM)^{\otimes 2}\Big\}\right\}\, ,
\label{eq:PdPsi1}\\
\frac{\partial\RS_0}{\partial\tbM}(\bM,\tbM;k,\delta,\nu) & = \frac{1}{2\beta}\left\{\bM - \delta\E\Big\{\tsF(\tbM\bw+\tbM^{1/2}\bz;\tbM)^{\otimes 2}\Big\}\right\}\, .
\label{eq:PdPsi2}
$$ By Lemma \[lemma:UsefulFormulae\], for $\bM = a\bJ_k$, $\tbM = b\bJ_k$, we have $$\begin{aligned}
\frac{\partial\RS_0}{\partial\bM}(\bM,\tbM;k,\delta,\nu) & = \frac{1}{2\beta}\left\{b\bJ_k- \frac{\beta a}{1+ka}\bJ_k\right\}\, ,\\
\frac{\partial\RS_0}{\partial\tbM}(\bM,\tbM;k,\delta,\nu) & = \frac{1}{2\beta}\left\{a\bJ_k - \frac{\beta\delta}{k^2}\bJ_k\right\}\, .
$$ Therefore, this is a stationary point of $\RS_0$ provided $a = \beta\delta/k^2$ and $b=\beta^2\delta/(k(k+\beta\delta))$ (in particular, $\bM = \bM^*$). Since $\RS(\bM;k,\delta,\nu) = \RS_0(\bM,\tbM(\bM);k,\delta,\nu)$, for $\tbM(\,\cdot\, )$ a differentiable function, it also follows that $\bM_*$ is a stationary point of $\RS$.
In order to prove that $\bM^*$ is a local minimum of $\RS$ for $\beta<\beta_{\sp}$, we apply Lemma \[lemma:hessianstable\] to the function $f(\bx,\by)= -\RS_0(\bx ,\by;k,\delta,\nu)$, whence $f_1(\bx) = -\RS(\bx;k,\delta,\nu)$. It follows from Eqs. (\[eq:PdPsi1\]) and (\[eq:PdPsi2\]) that the dynamics (\[eq:alternatemin\_dual\]) then coincides with the state evolution dynamics discussed in Section \[sec:StateEvol\], namely $$\begin{aligned}
\bM_{t+1} & = \delta\, \E\Big\{\tsF(\tbM_t\bw+\tbM_t^{1/2}\bz;\tbM_t)^{\otimes 2}\Big\}\, ,\\
\tbM_{t} & = \E\Big\{\sF(\bM_t\bh+\bM_t^{1/2}\bz;\bM_t)^{\otimes 2}\Big\}\, .
$$ Hence, the claim follows immediately from Theorem \[thm:StateEvolStable\] and Lemma \[lemma:hessianstable\].
Finally, we prove that Eq. (\[eq:TrivialEst\]) holds for $\beta<\beta_{\sBayes}$. Note that the estimator $\hbF_n(\bX)$ that minimizes the left-hand side is $\hbF_n(\bX) = \E\{\bW\bH^{\sT}|\bX\}$. By [@miolane2017fundamental Proposition 29], for $\beta<\beta_{\sBayes}$, $$\begin{aligned}
\lim_{n\to\infty}\frac{1}{nd}\E\left\{\left\|\bW\bH^{\sT}-\E\{\bW\bH^{\sT}|\bX\}\right\|_F^2\right\} &=
\lim_{n\to\infty}\frac{1}{nd}\E\left\{\left\|\bW\bH^{\sT}\right\|_F^2\right\}-\frac{1}{\beta^2\delta}\Tr(\bM^*\tbM^*)\\
&= \lim_{n\to\infty}\frac{1}{nd}\E\left\{\left\|\bW\bH^{\sT}\right\|_F^2\right\}-\frac{\beta\delta}{k(\beta\delta+k)}\, .\label{eq:CBA}
$$ On the other hand, $$\begin{aligned}
\lim_{n\to\infty}\frac{1}{nd}\E\left\{\left\|\bW\bH^{\sT} -c\bfone_n(\bX^{\sT}\bfone_n)^\sT\right\|_F^2\right\}& = \lim_{n\to\infty}\frac{1}{nd}\E\left\{\left\|\bW\bH^{\sT}\right\|_F^2\right\}-2c\, A+c^2 B\, . \label{eq:ABC}
$$ Here, we defined $A$ via $$\begin{aligned}
A &\equiv\lim_{n\to\infty}\frac{1}{nd}\E\Tr\Big(\bH\bW^{\sT}\bfone_n(\bX^{\sT}\bfone_n)^\sT\Big)\\
& = \lim_{n\to\infty}\frac{\sqrt{\beta}}{nd^2}\E\Tr\Big(\bW^{\sT}\bfone_n\bfone_n^{\sT}\bW\bH^{\sT}\bH\Big)\\
& = \sqrt{\beta} \delta \Tr\Big(\frac{\bfone_k}{k}\frac{\bfone_k^{\sT}}{k}\id_k\Big) = \frac{\sqrt{\beta}\delta}{k}\, ,\
$$ (where we used $\bW^{\sT}\bfone_n/n\to \bfone_k/k$ and $\bH^{\sT}\bH/d\to \id_k$ by the law of large numbers) and $$\begin{aligned}
B &\equiv \lim_{n\to\infty} \frac{1}{nd}\E\Tr\big(\bfone_n(\bX^{\sT}\bfone_n)^{\sT}(\bX^{\sT}\bfone_n)\bfone_n\big)\\
&= \lim_{n\to\infty} \frac{1}{d}\E\big\<\bfone_n,\bX\bX^{\sT}\bfone_n\big\>\\
& = \lim_{n\to\infty} \frac{1}{d}\E\left\{\frac{\beta}{d^2}\Tr\big((\bW^{\sT}\bfone_n)^{\sT}\bH^{\sT}\bH(\bW^{\sT}\bfone_n)\big)+n\right\}\\
& = \beta\delta^2 \Tr\Big(\frac{\bfone_k^{\sT}}{k}\frac{\bfone_k}{k}\id_k\big)+\delta = \frac{\beta\delta^2}{k}+\delta\, .
$$ Setting $c=A/B$, and substituting in Eq. (\[eq:ABC\]), we obtain $$\begin{aligned}
\lim_{n\to\infty}\frac{1}{nd}\E\left\{\left\|\bW\bH^{\sT} -c\bfone_n(\bX^{\sT}\bfone_n)^\sT\right\|_F^2\right\}& =
\lim_{n\to\infty}\frac{1}{nd}\E\left\{\left\|\bW\bH^{\sT}\right\|_F^2\right\}-\frac{\beta\delta}{k(\beta\delta+k)}\, ,
$$ which coincides with Eq. (\[eq:CBA\]) as claimed.
Naive Mean Field: Analytical results
====================================
Preliminary definitions
-----------------------
The functions $\sF, \tsF:\reals^k\times \reals^{k\times k}\to\reals^k$ are defined in Eq. (\[eq:defF\_main\]). Explicitly $$\begin{aligned}
\label{eq:defF}
\sF(\by; \bQ) &\equiv\sqrt{\beta}\, \frac{\int \bh \, e^{\<\by,\bh\>-\<\bh,\bQ\bh\>/2} \, q_0(\de \bh)}{\int\, e^{\<\by,\bh\>-\<\bh,\bQ\bh\>/2} \,
q_0(\de \bh)}\, ,\\
\label{eq:deftF}
\tsF(\hby; \tbQ) &\equiv \sqrt{\beta}\, \frac{\int \bw \, e^{\<\hby,\bw\>-\<\bw,\tbQ\bw\>/2} \, \tq_0(\de \bw)}{\int \, e^{\<\hby,\bw\>-\<\bw,\tbQ\bw\>/2} \,
\tq_0(\de \bw)}\, ,
$$ where $q_0(\,\cdot\, )$ is the prior distribution of the rows of $\bH$, and $\tq_0(\,\cdot\,)$ is the prior distribution of the rows of $\bW$.
For $\bQ$ positive semidefinite and symmetric, $\sF(\by;\bQ)/\sqrt{\beta}$ can be interpreted as the posterior expectation of $\bh\sim q_0(\,\cdot\,)$, given observations $\by = \bQ\bh+\bQ^{1/2}\bz$, where $\bz\sim\normal(0,\id_k)$, and analogously for $\tsF(\hby;\tbQ)$. Explicitly $$\begin{aligned}
\sF(\by; \bQ) = \sqrt{\beta}\,\E\Big\{ \bh \Big|\; \bQ\bh +\bQ^{1/2}\bz = \by\Big\}\, ,\;\;\;\;\;\;\;\;
\tsF(\hby; \tbQ) = \sqrt{\beta}\,\E\Big\{ \bw \Big|\; \tbQ\bw +\tbQ^{1/2}\bz = \hby\Big\}\, .
$$ In our specific application $q_0(\,\cdot\,)$ is $\normal(0,\id_k)$, and $\tq_0(\,\cdot\,)$ is $\Dir(\nu;k)$, namely $$\begin{aligned}
\label{eq:initmeasures}
q_0(\de\bh) = \frac{1}{(2\pi)^{k/2}}\, e^{-\|\bh\|_2^2/2}\de\bh\, ,\;\;\;\;\;\;\;\;
\tq_0(\de\bw) = \frac{1}{Z(\nu;k)} \prod_{i=1}^kw_i^{\nu-1} \, \oq(\de\bw)\, ,
$$ where $\oq(\,\cdot\,)$ is the uniform measure over the simplex $\sP_1(k) = \{\bw\in\reals^k_{\ge 0}\; :\;\;\<\bw,\bfone_k\> =1\}$. In particular, $\sF(\by;\bQ)$ can be computed explicitly, yielding $$\begin{aligned}
\label{eq:sFexplicit}
\sF(\by;\bQ) = \sqrt{\beta}(\id_k+\bQ)^{-1}\by\, .
$$
We also define the second moment functions $\sG, \tsG:\reals^k\times \reals^{k\times k}\to \reals^{k\times k}$ by $$\begin{aligned}
\label{eq:defG}
\sG(\by; \bQ) &\equiv {\beta}\, \frac{\int \bh^{\otimes 2} \, e^{\<\by,\bh\>-\<\bh,\bQ\bh\>/2} \, q_0(\de \bh)}{\int \, e^{\<\by,\bh\>-\<\bh,\bQ\bh\>/2} \,
q_0(\de \bh)}\,,\\
\label{eq:deftG}
\tsG(\tby; \tbQ) &\equiv {\beta}\, \frac{\int \bw^{\otimes 2} \, e^{\<\tby,\bw\>-\<\bw,\tbQ\bw\>/2} \, \tq_0(\de \bw)}{\int \, e^{\<\tby,\bw\>-\<\bw,\tbQ\bw\>/2} \,
\tq_0(\de \bw)}\,.
$$ Again, $\sG(\,\cdots\,)$ can be written explicitly as $$\begin{aligned}
\label{eq:sGexplicit}
\sG(\by; \bQ) & = \beta\Big\{(\id_k+\bQ)^{-1}\by\by^{\sT}(\id_k+\bQ)^{-1}+(\id_k+\bQ)^{-1}\Big\}\, .
$$
Derivation of the iteration (\[eq:NMF1\_Main\]), (\[eq:NMF2\_Main\]) {#app:NMF_ansatz}
--------------------------------------------------------------------
Let $\mathcal D$, the set of joint distributions $\hat{q}\left(\bW,\bH\right)$ that factorize over the rows of $\bW, \bH$, namely $$\begin{aligned}
\hat{q}\left(\bW,\bH\right) = q\left(\bH\right)\tilde q\left(\bW\right) = \prod_{i=1}^d q_i\left(\bh_i\right)\prod_{a=1}^n\tilde q_a\left(\bw_a\right)\, .\end{aligned}$$ The goal in variational inference is to find the distribution in $\cD$ that minimizes the Kullback-Leibler (KL) divergence with respect to the actual posterior distribution of $\bX,\bW$ given $\bX$ $$\begin{aligned}
\hat{q}^*\left(\,\cdot\,,\,\cdot\,\right) = \arg\min_{\hq\in\cD}\KL\left(\hat{q}\left(\,\cdot\,,\,\cdot\,\right)||\;p\left(\, \cdot\, ,\,\cdot\,|\bX\right)\right)\end{aligned}$$ The KL divergence can also be written as (denoting by $\E_{\hq}$ expectation over $(\bW,\bH)\sim \hq(\,\cdot\,,\,\cdot\,)$) $$\begin{aligned}
\label{eq:KL}
\KL\left(\hat{q}\left(\,\cdot\,,\,\cdot\,\right)||\;p\left(\,\cdot\,,\,\cdot\,|\bX\right)\right) &= \E_{\hq}\left[\log \hat{q}\left(\bW,\bH\right) \right]
- \E_{\hq}\left[\log p\left(\bX,\bW,\bH\right)\right] + \log p\left(\bX\right)\\
& \equiv \cF(\hq)+ \log p\left(\bX\right)\, .
$$ The function $\cF(\hq)$ is known as Gibbs free energy or –within the topic models literature– as the opposite of the evidence lower bound $\cF(\hq) = -\ELBO(\hq)$ [@blei2017variational]. Since $\log p\left(\bX\right)$ does not depend on $\hq$, minimizing the KL divergence is equivalent to minimizing the Gibbs free energy.
In order to find $\hat{q}^*\left(\bW,\bH\right) = q^*\left(\bH\right)\tilde q^*\left(\bW\right)$, the naive mean field iteration minimizes the Gibbs free energy by alternating minimization: we minimize the Gibbs free energy over $q\left(\bH\right)$ (while keeping $\tilde q\left(\bW\right)$ fixed), then minimize over $\tilde q\left(\bW\right)$ (while keeping $q\left(\bH\right)$ fixed), and repeat. With a slight abuse of notation, we will write $\cF(\hq)=\cF(q,\tq)$. Note that if we keep $\tq\left(\bW\right)$ fixed, we have $$\begin{aligned}
\arg\min _{q}\cF(q,\tq) &=
\arg\min _{q}\left\{\E_{q(\bH)}\left[\log q\left(\bH\right)\right] - \E_{q(\bH)}\left[\E_{\tilde q\left(\bW\right)}\left[\log p\left(\bX,\bW,\bH\right)\right]\right]\right\}\nonumber\\
&= \arg\min _{q} {\KL}\left(q\left(\bH\right)||\, C\exp\left\{\E_{\tilde q(\bW)}\left[\log p\left(\bX,\bW,\bH\right)\right]\right\}\right)\nonumber\\
&\propto \exp\left\{\E_{\tilde q(\bW)}\left[\log p\left(\bX,\bW,\bH\right)\right]\right\}\, .\end{aligned}$$ Similarly, by taking $q\left(\bH\right)$ fixed, we have $$\begin{aligned}
\arg\min _{\tilde q}\cF(q,\tq) \propto \exp\left\{\E_{q\left(\bH\right)}\left[\log p\left(\bX,\bW,\bH\right)\right] \right\}.\end{aligned}$$ Therefore, the naive mean field iterations have the form $$\begin{aligned}
\label{eq:densityevol}
\begin{split}
&q^{t+1}\left(\bH\right) = \prod_{i=1}^d q_i^{t+1}\left(\bh_i\right) \propto \exp\left\{\E_{\tilde q^t\left(\bW\right)}\left[\log p\left(\bX,\bW,\bH\right)\right]\right\},\\
&\tilde q^{t}\left(\bW\right) = \prod_{a=1}^n \tilde q_a^{t}\left(\bw_a\right) \propto\exp\left\{ \E_{q^t\left(\bH\right)}\left[\log p\left(\bX,\bW,\bH\right)\right] \right\}.
\end{split}\end{aligned}$$ with initialization $$\begin{aligned}
q^0\left(\bH\right) = \prod_{i=1}^d q_0\left(\bh_i\right)\,,\;\;\;\;\;\tilde q^0\left(\bW\right) = \prod_{a=1}^n \tilde q_0\left(\bw_a\right)\end{aligned}$$ where $q_0\left(\bh_i\right)$, $\tilde q_0\left(\bw_a\right)$ are the prior distributions on the rows of $\bH$ and $\bW$, cf. Eq. . Note that the iterations in can be further simplified by noting that the densities $q_i^{t}$ and $\tilde q_i^t$ have the form $$\begin{aligned}
\label{eq:densityforms}
\begin{split}
&q_i^t\left(\bh\right) \propto \exp\left\{\left\langle\bm_i^t,\bh\right\rangle-\frac{1}{2}\left\langle\bh, \bQ^t\bh\right\rangle\right\}q_0\left(\bh\right),\\
&\tilde q_a^t\left(\bw\right) \propto \exp\left\{\left\langle\tbm_a^t,\bw\right\rangle-\frac{1}{2}\left\langle\bw, \tbQ^t\bw\right\rangle\right\}\tilde q_0\left(\bw\right).
\end{split}\end{aligned}$$ In order to see this, note that the initial densities $q_0\left(\bh\right)$, $\tilde q_0\left(\bw\right)$ are in the form . Further, if we assume that $q_i^t\left(\bh\right)$, $\tilde q_a^t\left(\bw\right)$ are in the form , using the update equations , we have $$\begin{aligned}
q^{t+1}\left(\bH\right) = \prod_{i=1}^d q_{i}^{t+1}\left(\bh_i\right) &\propto \exp\left\{\E_{\tilde q^t\left(\bW\right)}\log p\left(\bX,\bH, \bW\right)\right\}\\
& \propto \exp\left\{\E_{\tilde q^t\left(\bW\right)}\log p \left(\bH, \bX|\bW\right)\right\}\\
& \propto q_0\left(\bH\right) \exp\left\{\E_{\tilde q^t\left(\bW\right)}\log p\left(\bX|\bH, \bW\right)\right\}\\
& \propto q_0\left(\bH\right)\exp\left\{-\E_{\tilde q^t\left(\bW\right)}\left[\frac{d}{2}\left\|\bX - \frac{\sqrt{\beta}}{d}\bW\bH^\sT\right\|_F^2\right]\right\}\\
&\propto q_0\left(\bH\right)\exp\left\{\E_{\tilde q^t\left(\bW\right)}\Tr\left(\sqrt{\beta}\bX\bH\bW^\sT - \frac{\beta}{2d}\bW\bH^\sT\bH\bW^\sT\right)\right\}\\
&= q_0\left(\bH\right)\exp\left\{\E_{\tilde q^t\left(\bW\right)}\sum_{a=1}^n\left(\sqrt{\beta}\left\langle\bx_a,\bH\bw_a\right\rangle-\frac{\beta}{2d}\left\langle\bw_a,\bH^\sT\bH\bw_a\right\rangle\right)\right\}\\
&= q_0\left(\bH\right)\exp\left\{\sum_{a=1}^n\left\langle\bx_a, \bH\tsF\left(\tbm_a^t; \tbQ^t\right)\right\rangle - \frac{1}{2d}\left\langle\bH^\sT\bH, \sum_{a=1}^n\tsG\left(\tbm_a^t; \tbQ_t\right)\right\rangle\right\}\\
&= \prod_{i=1}^d \left(q_0\left(\bh_i\right)\exp\left\{\left\langle\bm_i^{t+1},\bh_i\right\rangle-\frac{1}{2}\left\langle\bh_i, \bQ^{t+1}\bh_i\right\rangle\right\}\right)\end{aligned}$$ where $\tsF(\,\cdot\, ;\,\cdot\, ), \tsG(\,\cdot\,;\,\cdot\,)$ are given in , and $$\begin{aligned}
\label{eq:densityevolH}
\begin{split}
&\bm^{t+1} = \bX^\sT\tsF\left(\tbm^t; \tbQ^t\right),\\
&\bQ^{t+1} = \frac{1}{d}\sum_{a=1}^n \tsG\left(\tbm_a^t; \tbQ^t\right).
\end{split}\end{aligned}$$ Therefore, $q_i^{t+1}\left(\bh\right)$ has the form in and the update formula for $\bm^{t+1}$, $\bQ^{t+1}$ are given in . Similarly, for $\tq^{t+1}\left(\bW\right)$ we have $$\begin{aligned}
\tq^{t+1}\left(\bW\right) = \prod_{a=1}^n \tq_{a}^{t+1}\left(\bw_a\right) &\propto \exp\left\{\E_{q^{t+1}\left(\bH\right)}\log p\left(\bX, \bH,\bW\right)\right\}\\
& \propto \exp\left\{\E_{ q^{t+1}\left(\bH\right)}\log p \left(\bW, \bX|\bH\right)\right\}\\
& = \tq_0\left(\bW\right) \exp\left\{\E_{q^{t+1}\left(\bH\right)}\log p\left(\bX|\bH, \bW\right)\right\}\\
& \propto \tq_0\left(\bW\right)\exp\left\{\E_{q^{t+1}\left(\bH\right)}\left[-\frac{d}{2}\left\|\bX - \frac{\sqrt{\beta}}{d}\bW\bH^\sT\right\|_F^2\right]\right\}\\
&\propto \tq_0\left(\bW\right)\exp\left\{\E_{q^{t+1}\left(\bH\right)}\Tr\left(\sqrt{\beta}\bW\bH^\sT\bX^\sT - \frac{\beta}{2d}\bW\bH^\sT\bH\bW^\sT\right)\right\}\end{aligned}$$ Hence, $$\begin{aligned}
\tq^{t+1}\left(\bW\right) &\propto \tq_0\left(\bW\right)\exp\left\{\E_{ q^{t+1}\left(\bH\right)}\sum_{a=1}^n\left(\sqrt{\beta}\left\langle\bw_a,\bx_a\bH\right\rangle-\frac{\beta}{2d}\left\langle\bw_a,\bH^\sT\bH\bw_a\right\rangle\right)\right\}\\
&= \tq_0\left(\bW\right)\exp\left\{\sum_{a=1}^n\left\langle\bw_a, \bx_a\sF\left(\bm^{t+1}; \bQ^{t+1}\right)\right\rangle - \frac{1}{2d}\left\langle\bw_a,\left(\sum_{i=1}^d \sG\left(\bm_i^{t+1}; \bQ^{t+1}\right)\right)\bw_a\right\rangle\right\}\\
&= \prod_{a=1}^n \left(\tq_0\left(\bw_a\right)\exp\left\{\left\langle\bw_a, \tbm_a^{t+1}\right\rangle-\frac{1}{2}\left\langle\bw_a, \tbQ^{t+1}\bw_a\right\rangle\right\}\right)\end{aligned}$$ where $\sF(\,\cdot\,;\,\cdot\,), \sG(\,\cdot\,;\,\cdot\,)$ are given in , and $$\begin{aligned}
\label{eq:densityevolW}
\begin{split}
&\tbm^{t+1} = \bX\sF\left(\bm^{t+1}; \bQ^{t+1}\right),\\
&\tbQ^{t+1} = \frac{1}{d}\sum_{i=1}^d \sG\left(\bm_i^{t+1}; \bQ^{t+1}\right).
\end{split}\end{aligned}$$ Therefore, $\tq_a^{t+1}\left(\bw\right)$ has the form in and the update formula for $\tbm^{t+1}$, $\tbQ^{t+1}$ are given in .
Derivation of the variational free energy (\[eq:FreeEnergy\_main\]) {#app:NMF_Free_Energy}
-------------------------------------------------------------------
As already mentioned, naive mean field minimizes the KL divergence between a factorized distribution $\hat q(\bW, \bH) = \prod_{a=1}^{n} \tq(\bw_a) \prod_{i = 1}^{d} q(\bh_i)$ and the real posterior $p(\bW,\bH|\bX)$. The KL divergence takes the form $$\begin{aligned}
\KL(\hat q(\,\cdot\,, \,\cdot\, )||p(\,\cdot\,,\,\cdot\,|\bX)) = \cF(\hq) +\log p(\bX) +\frac{d}{2}\|\bX\|_{F}^2\, ,\end{aligned}$$ where $\cF(\hq)$ is the Gibbs free energy. In this appendix we derive an explicit form for $\cF(\hq)$ when $\hq$ is factorized. We have $$\begin{aligned}
\cF(\hq) &= \E_{\hat q}[-\log p(\bW, \bH| \bX)] + \E_{\hat q}[\log \hat q(\bW,\bH)] -\frac{d}{2}\|\bX\|_{F}^2\\
&= \E_{\hat q}[-\log p(\bW, \bH, \bX)] + \E_{\hat q}[\log \hat q(\bW,\bH)] -\frac{d}{2}\|\bX\|_{F}^2\\
&= \E_{\hat q}[-\log p(\bX|\bW, \bH) -\log p(\bW, \bH)] + \E_{\hat q}[\log \hat q(\bW,\bH)] -\frac{d}{2}\|\bX\|_{F}^2\\
&= \E_{\hat q}\left[\frac{d\Vert \bX - \frac{\sqrt{\beta}}{d}\bW\bH^\sT \Vert_F^2}{2}-\frac{d}{2}\|\bX\|_{F}^2 -\log (p(\bW, \bH))\right] + \E_{\hat q}[\log \hat q(\bW,\bH)] \\
&= \frac{d}{2}\E_{\hat q}\left[\Vert \bX - \frac{\sqrt{\beta}}{d}\bW\bH^\sT \Vert_F^2\right] -\frac{d}{2}\|\bX\|_{F}^2+
\KL(\hat q(\,\cdot\,, \,\cdot\,)\| q_0(\,\cdot\,, \,\cdot\,))\, .
$$ (The last term is the KL divergence between $\hq$ and the prior.)
We can explicitly calculate each term. Let’s denote by $\br_i,\bOmega_i$ the first and second moments of $q_i$ and by $\tbr_a$, $\tbQ_a$ the first and second moments of $\tq_a$: $$\begin{aligned}
\br_i&= \int \bh\,\, q_i(\de \bh) \, ,\;\;\;\;\;
\tbr_a = \int \bw \, \, \tq_a(\de \bw)\, ,\label{eq:Moment1}\\
\bOmega_i&= \int \bh^{\otimes 2} \,\,q_i(\de \bh) \, ,\;\;\;\;\;
\tbOmega_a = \int \bw^{\otimes 2} \, \, \tq_a(\de \bw)\, . \label{eq:Moment2}
$$ We then have $$\begin{aligned}
\frac{d}{2}\E_{\hat q}\Vert \bX - \frac{\sqrt{\beta}}{d}\bW\bH^\sT \Vert_F^2 &-\frac{d}{2}\|\bX\|_{F}^2
= \frac{d}{2}\E_{\hat q}\left[\Tr\left(-\frac{2\sqrt{\beta}}{d}\bX^\sT\bW\bH^\sT\right) + \Tr\left(\frac{\beta}{d^2}\bH\bW^\sT\bW\bH^\sT\right)\right]
\\
&= -\sqrt{\beta} \Tr\left(\bX^\sT\E_{\hat q}[\bW\bH^\sT]\right) + \frac{\beta}{2d} \Tr\left(\E_{\hat q}[\bH\bW^\sT\bW\bH^\sT]\right) \\
&= -\sqrt{\beta} \Tr\left(\bX^\sT\br \tbr^\sT\right) +\frac{\beta}{2d} \sum_{i=1}^d \sum_{a=1}^n\<\bOmega_i,\tbOmega_a\>\,. \label{eq:expected_loglike} \end{aligned}$$ Since both $\hq$ and $q_0$ have product form, their KL divergence is just a sum of KL divergences for each row of $\bW$ and each row of $\bH$: $$\begin{aligned}
\label{kl_div_2}
\KL(\hat q(\,\cdot\,, \,\cdot\,)\|q_0(\,\cdot\,, \,\cdot\,))) &= \sum_{i = 1}^{d} \KL(q_i\| q_0) +\sum_{a = 1}^{n} \KL(\tq_a\| q_0)\, .
$$ Each of these terms is treated in the same manner: we minimize over $q_i$ or $\tq_a$ subject to the moment constraints (\[eq:Moment1\]), and define $$\begin{aligned}
\psi_*(\br_i,\bOmega_i) = \min\left\{ \KL(q_i\| q_0) :\;\;\; \int \bh\,\, q_i(\de \bh) =\br_i\, ,\;\;
\int \bh^{\otimes 2} \, \, q_i(\de \bh) =\bOmega_i\right\}\, ,\label{eqs:PsiKL1}\\
\tpsi_*(\tbr_a,\tbOmega_a) = \min\left\{ \KL(\tq_a\| \tq_0) :\;\;\; \int \bw\,\, \tq_a(\de \bw)=\tbr_a\, ,\;\;
\int \bw^{\otimes 2} \, \, \tq_a(\de \bw) =\tbOmega_a\right\}\, . \label{eqs:PsiKL2}
$$ Standard duality between entropy and moment generating functions yields that $\psi_*$, $\tpsi_*$ are defined as per Eq. (\[eq:LegendrePhi\]). We briefly recall the argument for the reader’s convenience. Considering for instance $\tpsi_*(\tbr,\tbOmega)$, we introduce the Lagrangian $$\begin{aligned}
\cL(\tq_a,\tbm_a,\tbQ_a) = \KL(\tq_a\| \tq_0) +\<\tbm_a,\tbr_a\>
-\frac{1}{2}\<\tbQ_a,\tbOmega_a\>-\int \Big\{\<\tbm_a,\bw\>-\frac{1}{2}\<\bw,\tbQ_a\bw\>\Big\}\, \tq_a(\de \bw)\, .
$$ This is minimized easily with respect to $\tq_a$. The minimum is achieved at the distribution (\[eq:densityforms\_main\]), with $$\begin{aligned}
\min_{\tq_a}\cL(\tq_a,\tbm_a,\tbQ_a) =
\< \tbr_a, \tbm_a\> -\frac{1}{2}\<\tbOmega_a,\tbQ_a\>- \tphi(\tbm_a, \tbQ_a)\, ,
$$ and the claim (\[eq:LegendrePhi\]) follows by strong duality.
Putting together Eqs. (\[eq:expected\_loglike\]), (\[kl\_div\_2\]), and (\[eqs:PsiKL1\])-(\[eqs:PsiKL2\]), we obtain the desired expression (\[eq:FreeEnergy\_main\]).
Using (\[eq:LegendrePhi\]), we get the following expressions for the gradients of $\psi_*$ $$\begin{aligned}
\frac{\partial \psi_*}{\partial \br}(\br,\bOmega) = \bm\, ,\;\;\;\;\;\frac{\partial \psi_*}{\partial \bOmega}(\br,\bOmega) = -\frac{1}{2}\bQ\,,
$$ and similarly for $\tpsi_*$ (where $\bm,\bQ$ are related to $\br,\bOmega$ via Eqs. (\[eq:r\_def\]), (\[eq:r\_def\])). Hence, the gradients of $\cF$ with respect to $\br_i$, $\bOmega_i$ read $$\begin{aligned}
\frac{\partial \cF}{\partial \br_i}(\br,\tbr,\bOmega,\tbOmega)& = -\sqrt{\beta}(\bX^{\sT}\tbr)_{i,\cdot}+\bm_i\, ,\;\;\;\;\;\;\;
\frac{\partial \cF}{\partial \tbr_a}(\br,\tbr,\bOmega,\tbOmega) = -\sqrt{\beta}(\bX\br)_{a,\cdot}+\tbm_a\,,\\
\frac{\partial \cF}{\partial \bOmega_i}(\br,\tbr,\bOmega,\tbOmega) & = -\frac{1}{2}\bQ_i+\frac{\beta}{2d}\sum_{a=1}^n\tbOmega_a\, ,\;\;\;\;\;\;\;
\frac{\partial \cF}{\partial \tbOmega_a}(\br,\tbr,\bOmega,\tbOmega) = -\frac{1}{2}\bQ_a+\frac{\beta}{2d}\sum_{i=1}^d\bOmega_i
\, .\label{eq:DF_Omega}
$$ Notice that at stationarity points, we have $\bQ_i = \bQ = (\beta/d)\sum_{a=1}^n\tbOmega_a$ independent of $i$.
Proof of Lemma \[lemma:Uninf\] {#app:Uninformative}
------------------------------
We start with some useful formulae.
\[lemma:UsefulFormulae\] For $q\in\reals$ define $\sE(q)$ by $$\begin{aligned}
\sE(q;\nu) = \frac{\int w_1^2 e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)}{\int e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)}\, . \label{eq:Edef}\end{aligned}$$ Then, we have $$\begin{aligned}
\sF(\by = y \bfone_k;\bQ =q_1\id_k+q_2\bJ_k) &= \frac{\sqrt{\beta}\, y}{1+q_1+kq_2} \, \bfone_k\, ,\label{eq:sfsimplifiedsymm}\\
\sG(\by = y \bfone_k;\bQ = q_1\id_k+q_2\bJ_k) &= \frac{\beta}{(1+q_1)} \,\id_k+\beta \left\{\frac{y^2}{(1+q_1+kq_2)^2}-\frac{q_1}{(1+q_1)(1+q_1+kq_2)}\right\} \, \bJ_k \, ,\label{eq:sgsimplifiedsymm}\\
\tsF(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k) &= \frac{\sqrt{\beta}}{k} \, \bfone_k\, ,\label{eq:tsfsimplifiedsymm}\\
\tsG(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k) &= \beta \, \frac{k^2\sE(\tq_1;\nu)-1}{k(k-1)}\, \id_k-\beta \, \frac{k\sE(\tq_1;\nu)-1}{k(k-1)}\, \bJ_k \, .\label{eq:tsgsimplifiedsymm}
$$ In particular $$\begin{aligned}
\sF(\by = y \bfone_k;\bQ =q\bJ_k) &= \frac{\sqrt{\beta}\, y}{1+kq} \, \bfone_k\, ,\\
\sG(\by = y \bfone_k;\bQ = q\bJ_k) &= \beta \,\id_k+ \beta \, \frac{y^2}{(1+kq)^2} \, \bJ_k \, ,\\
\tsF(\tby = \ty \bfone_k;\tbQ = \tq\bJ_k) &= \frac{\sqrt{\beta}}{k} \, \bfone_k\, ,\\
\tsG(\tby = \ty \bfone_k;\tbQ = \tq\bJ_k) &= \frac{\beta}{k(k\nu+1)}\, \left(\id_k+\nu\bJ_k\right)\, .
$$
First note that $$\begin{aligned}
\left[\left(1+q_1\right)\id_k + q_2\bJ_k\right]^{-1} = \frac{1}{1+q_1} \id_k - \frac{q_2}{(1+q_1)(1+q_1+kq_2)}\bJ_k.\end{aligned}$$ Hence, by we have $$\begin{aligned}
\sF(\by = y \bfone_k;\bQ =q_1\id_k+q_2\bJ_k) &= \sqrt{\beta}y\left[\left(1+q_1\right)\id_k + q_2\bJ_k\right]^{-1}\bfone_k\\
&=\sqrt{\beta}y \left(\frac{1}{1+q_1} \id_k - \frac{q_2}{(1+q_1)(1+q_1+kq_2)}\bJ_k\right) \bfone_k\\
&= \sqrt{\beta}y \left(\frac{1}{1+q_1} - \frac{kq_2}{(1+q_1)(1+q_1+kq_2)}\right)\bfone_k\\
&= \frac{\sqrt{\beta}\, y}{1+q_1+kq_2} \, \bfone_k\,.\end{aligned}$$ Thus, by $$\begin{aligned}
\sG(\by = y \bfone_k;\bQ =q_1\id_k+q_2\bJ_k) &= \frac{\beta\, y^2}{(1+q_1+kq_2)^2} \bJ_k\,
+ \beta\left(\frac{1}{1+q_1} \id_k - \frac{q_2}{(1+q_1)(1+q_1+kq_2)}\bJ_k\right)\\
&= \frac{\beta}{(1+q_1)} \,\id_k+\beta \left\{\frac{y^2}{(1+q_1+kq_2)^2}-\frac{q_1}{(1+q_1)(1+q_1+kq_2)}\right\} \, \bJ_k \,.\end{aligned}$$ In addition, using , by symmetry, all entries of $\tsF(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k)$ are equal. Further, $$\begin{aligned}
\left\langle \bfone_k\,, \tsF(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k)\right\rangle &=
\sqrt{\beta}\, \frac{\int \left\langle\bfone_k,\bw\right\rangle \, e^{\<\hby,\bw\>-\<\bw,\tbQ\bw\>/2} \, \tq_0(\de \bw)}{\int \, e^{\<\hby,\bw\>-\<\bw,\tbQ\bw\>/2} \,
\tq_0(\de \bw)}\\
&= \sqrt{\beta}\, \frac{\int e^{\<\hby,\bw\>-\<\bw,\tbQ\bw\>/2} \, \tq_0(\de \bw)}{\int \, e^{\<\hby,\bw\>-\<\bw,\tbQ\bw\>/2} \,
\tq_0(\de \bw)} = \sqrt{\beta}.\end{aligned}$$ Therefore, $$\begin{aligned}
\tsF(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k) &= \frac{\sqrt{\beta}}{k} \, \bfone_k.\end{aligned}$$ Finally, again by symmetry, $\tsG(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k)$ has the same diagonal entries. Further, the off-diagonal entries of this matrix are equal. Thus, we have $$\begin{aligned}
\tsG(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k) = \left(\tsG_{11}-\tsG_{12}\right)\id_k + \tsG_{12}\bJ_k.\label{eq:diagG}\end{aligned}$$ Note that by , $$\begin{aligned}
\tsG_{1,1} &=
\beta\,\frac{\int w_1^2 e^{\ty\left\langle \bw, \bfone_k\right\rangle-\tq_1\|\bw\|_2^2/2-\tq_2\left\langle \bw, \bfone_k\right\rangle^2/2}\, \tq_0(\de\bw)}{\int e^{\ty\left\langle \bw, \bfone_k\right\rangle-\tq_1\|\bw\|_2^2/2-\tq_2\left\langle \bw, \bfone_k\right\rangle^2/2}\tq_0(\de\bw)} \\
&= \beta\,\frac{e^{\ty - \tq_2/2}\int w_1^2 e^{-\tq_1\|\bw\|_2^2/2}\, \tq_0(\de\bw)}{e^{\ty - \tq_2/2}\int e^{-\tq_1\|\bw\|_2^2/2}\tq_0(\de\bw)} = \beta\,\sE(\tq_1;\nu).\end{aligned}$$ Further, by $$\begin{aligned}
k\tsG_{1,1} + k(k-1)\tsG_{1,2} &= \left\langle\tsG(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k), \bJ_k\right\rangle \\
&= {\beta}\, \frac{\int \left\langle\bw, \bfone_k\right\rangle^2 \, e^{\<\tby,\bw\>-\<\bw,\tbQ\bw\>/2} \, \tq_0(\de \bw)}{\int \, e^{\<\tby,\bw\>-\<\bw,\tbQ\bw\>/2} \,
\tq_0(\de \bw)}\,\\
&= {\beta}\, \frac{\int e^{\<\tby,\bw\>-\<\bw,\tbQ\bw\>/2} \, \tq_0(\de \bw)}{\int \, e^{\<\tby,\bw\>-\<\bw,\tbQ\bw\>/2} \,
\tq_0(\de \bw)}\, = \beta.\label{eq:sumallG}\end{aligned}$$ Therefore, by , , we get $$\begin{aligned}
\tsG_{1,1} = \beta\,\sE(\tq_1;\nu)\,, \;\;\;\;\; \tsG_{1,2} = -\beta \, \frac{k\sE(\tq_1;\nu)-1}{k(k-1)}.\end{aligned}$$ Hence, $$\begin{aligned}
\tsG(\tby = \ty \bfone_k;\tbQ = \tq_1\id_k+\tq_2\bJ_k) &= \beta \, \frac{k^2\sE(\tq_1;\nu)-1}{k(k-1)}\, \id_k-\beta \, \frac{k\sE(\tq_1;\nu)-1}{k(k-1)}\, \bJ_k. \end{aligned}$$ In addition, note that $$\begin{aligned}
\sE(0;\nu) = \int w_1^2 \tq_0(\de\bw) = \frac{\nu+1}{k(k\nu+1)}.\end{aligned}$$ Using this, and replacing $q_1, \tq_1 = 0$ in - will complete the proof.
Note that $q\geq 0$ $$\begin{aligned}
k^2\sE(q;\nu) = \frac{\int k^2w_1^2 e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)}{\int e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)} =
\frac{\int k\|\bw\|_2^2 e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)}{\int e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)} \geq
\frac{\int \|\bw\|_1^2 e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)}{\int e^{-q\|\bw\|_2^2}\, \tq_0(\de\bw)} = 1.\end{aligned}$$ In addition, we have $$\begin{aligned}
&\sE(0;\nu) = \int w_1^2 \tq_0(\de\bw) = \frac{\nu+1}{k(k\nu+1)}\, ,\\
&\lim_{q_1 \to \infty} \frac{k\beta\delta}{k-1}\, \left\{\sE\left(\frac{\beta}{1+q_1};\nu\right) - \frac{1}{k^2}\right\} = \frac{k\beta\delta}{k-1}\left\{\sE\left(0;\nu\right) - \frac{1}{k^2}\right\} = \frac{\beta\delta}{k(k\nu+1)} < \infty.
$$ Therefore, the right hand side of is non-negative, continuous, bounded for $q_1^* \in [0, \infty)$. Hence, using intermediate value theorem, has a solution in $[0,\infty)$.
Now we will check that equations (\[eq:NMF1\_Main\]) and (\[eq:NMF2\_Main\]) hold for $\bm^{t+1} = \bm^t = \bm^*$, $\tbm^t = \tbm^*$, $\bQ^t = \bQ^{t+1} = \bQ^*$, $\tbQ^t = \tbQ^*$. We start with the first equation in (\[eq:NMF1\_Main\]). Using Lemma \[lemma:UsefulFormulae\], we have $$\begin{aligned}
\tsF(\tbm^*_a; \tbQ^*) = \frac{\sqrt{\beta}}{k}\bfone_k.\end{aligned}$$ Therefore, $$\begin{aligned}
\tsF(\tbm^*; \tbQ^*) = \frac{\sqrt{\beta}}{k}\bfone_n\otimes \bfone_k\,, \;\;\;\;\; \bX^\sT\tsF(\tbm^*; \tbQ^*) = \frac{\sqrt{\beta}}{k}\, (\bX^{\sT}\bfone_n)\otimes \bfone_k = \bm^*.\end{aligned}$$ Now we consider the first equation in . Using Lemma \[lemma:UsefulFormulae\], we have $$\begin{aligned}
\sF(\bm_i^*; \bQ^*) = \frac{\beta}{k(1+q_1^*+kq_2^*)}\left\langle\bX_{.,i}, \bfone_n\right\rangle\bfone_k.\end{aligned}$$ Hence, $$\begin{aligned}
\sF(\bm^*; \bQ^*) &= \frac{\beta}{k(1+q_1^*+kq_2^*)}(\bX^\sT\bfone_n)\otimes \bfone_k\,,\\
\bX\sF(\bm^*; \bQ^*) &= \frac{\beta}{k(1+q_1^*+kq_2^*)}(\bX\bX^\sT\bfone_n)\otimes \bfone_k = \tbm^*.\end{aligned}$$ For the second equation in , note that using Lemma \[lemma:UsefulFormulae\], we have $$\begin{aligned}
\frac{1}{d}\sum_{a=1}^n \tsG(\tbm^*_{a};\tbQ^*) = \delta\beta \left(\, \frac{k^2\sE(\tq_1^*;\nu)-1}{k(k-1)}\, \id_k- \frac{k\sE(\tq_1^*;\nu)-1}{k(k-1)}\, \bJ_k\right).\end{aligned}$$ Note that using , $$\begin{aligned}
\frac{k^2\sE(\tq_1^*;\nu)-1}{k(k-1)} &= \frac{1}{k(k-1)}\left[k^2\sE\left(\frac{\beta}{1+q_1^*};\nu\right)-1\right] = \frac{q_1^*}{\delta\beta},\\
\frac{-k\sE(\tq_1^*;\nu)+1}{k(k-1)} &= \frac{-1}{k(k-1)}\left[k\sE\left(\frac{\beta}{1+q_1^*};\nu\right)-1\right] = \frac{-1}{k(k-1)}\left[\frac{k-1}{\delta\beta}q_1^*+\frac{1-k}{k}\right] \\
&= \frac{1}{\delta\beta}\left(\frac{\beta\delta-kq_1^*}{k^2}\right) = \frac{q_2^*}{\delta\beta}.\end{aligned}$$ Therefore, $$\begin{aligned}
\frac{1}{d}\sum_{a=1}^n \tsG(\tbm^*_{a};\tbQ^*) = q_1^* \id_k + q_2^*\bJ_k = \bQ^*.\end{aligned}$$ Finally, we check the second equation in . Using Lemma \[lemma:UsefulFormulae\], we have $$\begin{aligned}
\sG(\bm^*_{i};\bQ^*) = \frac{\beta}{(1+q_1^*)} \,\id_k+\beta \left\{\frac{\left\langle\bX_{.,i}, \bfone_n\right\rangle^2}{(1+q_1^*+kq_2^*)^2}-\frac{q_1^*}{(1+q_1^*)(1+q_1^*+kq_2^*)}\right\} \, \bJ_k.\end{aligned}$$ Hence, $$\begin{aligned}
\frac{1}{d}\sum_{i=1}^d \sG(\bm^*_{i};\bQ^*) &= \frac{\beta}{(1+q_1^*)} \,\id_k+\beta \left\{\frac{\|\bX^\sT\bfone_n\|_2^2}{d(1+q_1^*+kq_2^*)^2}-\frac{q_1^*}{(1+q_1^*)(1+q_1^*+kq_2^*)}\right\} \, \bJ_k\\
&= \tq_1^* \id_k + \tq_2^*\bJ_k = \tbQ^*,\end{aligned}$$ this completes the proof.
Proof of Theorem \[thm:Main\] {#app:ProofMain}
-----------------------------
We will first prove that, if $L(\beta,k,\delta,\nu)>1$, then the uninformative fixed point $(\br^*,\tbr^*,\bOmega^*,\tbOmega^*)$ (or equivalently, its conjugate $(\bm^*,\tbm^*,\bQ^*,\tbQ^*)$) is (with high probability) a saddle point of the naive mean field free energy (\[eq:FreeEnergy\_main\]). This implies immediately that the naive mean field iteration is unstable at that fixed point.
Note that the mapping $(\br,\tbr,\bOmega,\tbOmega)\to (\br,\tbr,\bQ,\tbQ)$ is a diffeomorphism (since the Jacobian is always invertible by strict convexity of $\phi$, $\tphi$). We define $\cF_*$ to be the restriction of $\cF$ to the submanifold defined by $\bQ=\bQ_*$, $\tbQ=\tbQ_*$. Explicitly, this can be written in terms of the partial Legendre transforms (we repeat the definition of Eq. (\[eq:LegendrePhiPartial\]) for the reader’s convenience): $$\begin{aligned}
\psi(\br,\bQ) \equiv \sup_{\bm}\left\{\< \br, \bm\> - \phi(\bm, \bQ)\right\} \, ,\;\;\;\;
\tpsi(\tbr,\tbQ) \equiv \sup_{\tbm}\left\{\< \tbr, \tbm\>- \tphi(\tbm, \tbQ)\right\} \, .
$$ We then have $$\begin{aligned}
\cF_*(\br,\tbr) = & \sum_{i=1}^d\psi(\br_i,\bQ_*)+\sum_{a=1}^n\tpsi(\tbr_a,\tbQ_*) -\sqrt{\beta}\Tr\left(\bX\br\tbr^{\sT}\right)\nonumber\\
&- \frac{d}{2}\<\bQ_*,\bOmega\>- \frac{n}{2}\<\tbQ_*,\tbOmega\>+\frac{\beta n}{2}\<\bOmega,\tbOmega\>\, ,\\
\bOmega &\equiv \frac{1}{d\beta} \sum_{i=1}^d\sG(\bm_i;\bQ^*)\,,\;\;\;\;\;\;\;\;\tbOmega \equiv \frac{1}{n\beta}\sum_{a=1}^n\tsG(\tbm_a;\tbQ^*)\, ,
\label{eq:OmegaM}\\
\br_i &\equiv \frac{1}{\sqrt{\beta}}\sF(\bm_i;\bQ^*)\,,\;\;\;\;\;\;\;\;\tbr_a \equiv \frac{1}{\sqrt{\beta}}\tsF(\tbm_a;\tbQ^*)\,,\label{eq:r_def_app}\, .
$$ In order to prove that $(\br_*,\tbr_*)$ is a saddle point of $\cF$, it is sufficient to show that it is a saddle along a submanifold, and hence that the Hessian of $\cF_*$ has a negative eigenvalue at $(\br_*,\tbr_*)$.
Next notice that $$\begin{aligned}
\cF_*(\br, \tbr) &= \cG_1(\br, \tbr) + \cG_2(\br, \tbr)\, ,\\
\cG_1(\br, \tbr) &\equiv \sum_{i=1}^d\psi(\br_i,\bQ_*)+\sum_{a=1}^n\tpsi(\tbr_a,\tbQ_*) -\sqrt{\beta}\Tr\left(\bX\br\tbr^{\sT}\right)\, ,\\
\cG_2(\br, \tbr)& \equiv - \frac{d}{2}\<\bQ_*,\bOmega\>- \frac{n}{2}\<\tbQ_*,\tbOmega\>+\frac{\beta n}{2}\<\bOmega,\tbOmega\>\, .
$$ Consider deviations from the stationary point $\br_i = \br_i^*+\bdelta_i$, $\tbr_a = \tbr_a^*+\tbdelta_a$. By Eqs. (\[eq:OmegaM\]) and (\[eq:r\_def\_app\]), we have (for some tensors $\bT,\tbT\in (\reals^k)^{\otimes 3})$) $$\begin{aligned}
\bOmega = \bOmega^*+\frac{1}{d}\sum_{i=1}^{d}\bT\bdelta_i +\bDelta\, ,\;\;\;\;\;\tbOmega = \tbOmega^*+\frac{1}{n}\sum_{a=1}^{n}\tbT\tbdelta_a +\tbDelta\, ,
$$ where $\bDelta$, $\tbDelta$ are of second order in $\bdelta,\tbdelta$. At the stationary point, by Eq. (\[eq:DF\_Omega\]), we have $\bQ^*= \beta\bOmega^*$, $\tbQ^*= \beta\delta\tbOmega^*$. Hence, substituting in $\cG_2$, and letting $M_{ij}= \sum_{s,t}T_{st,i}\tilde{T}_{st,j}$, we obtain $$\begin{aligned}
\cG_2(\br, \tbr)& = \cG_2(\br_*, \tbr_*) +\frac{\beta}{2d}\sum_{i=1}^d\sum_{a=1}^n\<\bdelta_i,\bM\tbdelta_a\> +o(\bdelta^2)
$$ Therefore, the Hessian $\nabla^2\bG_2(\br_*,\tbr_*)$ has rank at most $k$.
Since $\psi(\,\cdot\,,\bQ^*)$, $\tilde \psi(\,\cdot\,, \tbQ^*)$ are Legendre transforms of $\phi(\,\cdot\,, \bQ^*)$, $\tilde \phi(\,\cdot\,, \tbQ^*)$, respectively, we have $$\begin{aligned}
&\nabla_{\br\br}^2 \psi(\br, \bQ^*) = \left(\nabla_{\bm\bm}^2\phi(\bm, \bQ^*)\right)^{-1} = \id_k + \bQ^*,\\
&\nabla_{\tbr\tbr}^2 \tilde\psi(\tbr, \tbQ^*) = \left(\nabla_{\tbm\tbm}^2 \tilde\phi(\tbm, \tbQ^*)\right)^{-1} = \bD^{-1}\end{aligned}$$ where $\bD \in \reals^{k\times k}$ is as $$\begin{aligned}
D_{ij} = \frac{1}{\sqrt{\beta}}\frac{\partial \tsF_i\left(\tbm; \tbQ\right)}{\partial \tilde m_j}\Bigg|_{\tbm = 0, \tbQ = \tbQ^*}.\end{aligned}$$ Thus, $$\begin{aligned}
\bD &= \frac{\left(\int \bw^{\otimes 2}e^{-\tq_1^*\|\bw\|_2^2/2}\tq_0(\de \bw)\right)\left(\int e^{-\tq_1^*\|\bw\|_2^2/2}\tq_0(\de \bw)\right) - \left(\int \bw e^{-\tq_1^*\|\bw\|_2^2/2}\tq_0(\de \bw)\right)^{\otimes 2}}{\left(\int e^{-\tq_1^*\|\bw\|_2^2/2}\tq_0(\de \bw)\right)^2}\\
&= \frac{\bQ^*}{\delta\beta} - \frac{\bJ_k}{k^2}.\end{aligned}$$ Hence, $$\begin{aligned}
\nabla^2\mathcal G_1 =
\begin{bmatrix}
\id_d \otimes \left(\id_k + \tbQ^*\right) & -\sqrt{\beta}\bX^\sT \otimes \id_k \\
-\sqrt{\beta}\bX \otimes \id_k & \id_n \otimes \bD^{-1}
\end{bmatrix}.\end{aligned}$$ Since $\id_k + \tbQ^*$ is positive definite, $\nabla^2\mathcal G \succeq 0$ if and only if $$\begin{aligned}
&\id_n \otimes \bD^{-1} \succeq \beta\left(\bX\otimes \id_k\right)\left(\id_d\otimes\left(\id_k+\tbQ^*\right)\right)^{-1}\left(\bX^\sT\otimes \id_k\right) \iff \\
&\id_n\otimes \bD^{-1} \succeq \beta\left(\bX\bX^\sT\right) \otimes\left(\id_k+\tbQ^*\right)^{-1} \iff\\
&\id_n\otimes \id_k \succeq \beta\left(\bX\bX^\sT\right)\otimes\left(\id_k + \tbQ^*\right)^{-1}\bD.\end{aligned}$$ Hence, $\nabla^2\mathcal G_1$ has a negative eigenvalue if and only if $$\begin{aligned}
\beta \lambda_{\max}\left(\bX\bX^\sT \right) \lambda_{\max}\left(\left(\id_k + \tbQ^*\right)^{-1}\bD\right) > 1.\end{aligned}$$ Further, by the same argument, if $\beta\lambda_{\ell}(\bX\bX^\sT) \lambda_{\max}((\id_k + \tbQ^*)^{-1}\bD) > 1$, then $\nabla^2\cG_1$ has at least $\ell$ negative eigenvalues (recall that $\lambda_{\ell}(\bM)$ denotes the $\ell$-th eigenvalue of $\bM$ in decreasing order).
Note that $$\begin{aligned}
\left(\id_k + \bQ^*\right)^{-1}\bD &= \left(\frac{\id_k}{1+q_1^*}-\frac{q_2^*}{(1+q_1^*)(1+q_1^*+kq_2^*)}\bJ_k\right)\left(\frac{q_1^*}{\delta\beta}\id_k+\left(\frac{q_2^*}{\delta\beta}-\frac{1}{k^2}\right)\bJ_k\right)\\
&= \frac{1}{1+q_1^*}\left(\frac{q_1^*}{\delta\beta}\id_k + \left(\frac{q_2^*}{1+q_1^*+kq_2^*}\left(\frac{1}{\delta\beta}+\frac{1}{k}\right)-\frac{1}{k^2}\right)\bJ_k\right),\\
\mu(\beta,\delta)&\equiv \lambda_{\max}\left(\left(\id_k + \bQ^*\right)^{-1}\bD\right)\\
&= \frac{1}{1+q_1^*}\left(\frac{q_1^*}{\delta\beta} + k\left[\frac{q_2^*}{1+q_1^*+kq_2^*}\left(\frac{1}{\delta\beta}+\frac{1}{k}\right)-\frac{1}{k^2}\right]_+\right).\end{aligned}$$ where $q_1^*, \tq_1^*$, $q_2^*$ are given in , , . Further $\bX\bX^\sT$ is a low-rank deformation of a Wishart matrix. Hence, for any fixed $\ell$, we have, almost surely $$\begin{aligned}
\lim\inf_{n,d\to\infty}\lambda_{\ell}(\bX\bX^{\sT})\ge \left(1+\frac{1}{\sqrt{\delta}}\right)^2\, .
$$ Thus, if $$\begin{aligned}
L(\beta, \delta) = \beta\lambda_{\max}\left(1+\frac{1}{\sqrt{\delta}}\right)^2\mu(\beta,\delta) > 1,\end{aligned}$$ we have $\lambda_{n}(\nabla^2 \cG_1) \le \dots\le \lambda_{n-\ell}(\nabla^2 \cG_1)<0$ with high probability for any fixed $\ell$.
As explained above, $\nabla^2\cG_2$ has rank at most $k$. Therefore, by Cauchy’s interlacing inequality, if $L(\beta,k, \delta,\nu) > 1$, $$\begin{aligned}
\lambda_{\min}\left(\nabla^2\cF_*\right) \le \lambda_{n+k}\left(\nabla^2\cG_1 + \nabla^2\cG_2\right) < 0.\end{aligned}$$ Hence, for $L(\beta, \delta) > 1$, $\nabla^2 \cF_*$ has a negative eigenvalue.
Note that the mapping $(\br,\tbr,\bOmega,\tbOmega)\to (\bm,\tbm,\bQ,\tbQ)$ is a diffeomorphism, and therefore, uninformative fixed point $(\bm^*,\tbm^*,\bQ^*,\tbQ^*)$ is a saddle also when we consider the free energy as a function of the parameters $(\bm,\tbm,\bQ,\tbQ)$. The claim that $(\bm^*,\bQ^*)$ is unstable under the naive mean field iteration follows immediately from the above, by using Lemma \[lemma:hessianstable\], applied to $f(\bx,\by) = \cF(\bm,\tbm,\bQ,\tbQ)$, whereby $\bx = (\bm,\bQ)$, $\by=(\tbm,\tbQ)$.
Naive Mean Field: Further numerical results {#app:Numerical_MF}
===========================================
In this section we report on additional numerical simulations using the alternate minimization to minimize the naive mean field free energy. These results confirm the one presented in the main text in Section \[sec:NMF\_numerical\].
Credible intervals
------------------
![Bayesian credible intervals as computed by variational inference at nominal coverage level $1-\alpha= 0.9$. Here $k=2$, $d=5000$, $n=2500$ and we consider three values of $\beta$: $\beta\in\{2,5.7,8.5\}$ (for reference $\beta_{\inst}\approx 2.9$, $\beta_{\sBayes}\approx 8.5$. Circles correspond to the posterior mean, and squares to the actual weights. We use red for the coordinates on which the credible interval does not cover the actual value of $w_{i,1}$.[]{data-label="fig:Uncertainty_delta_p5"}](Uncertainty_delta_p5_new-eps-converted-to.pdf){height="3.in"}
![Bayesian credible intervals as computed by variational inference at nominal coverage level $1-\alpha= 0.9$. Here $k=2$, $d=5000$, $n=10000$ and we consider three values of $\beta$: $\beta\in\{1,3,4.2\}$ (for reference $\beta_{\inst}\approx 1.7, \beta_{\sBayes}\approx 4.2$. Circles correspond to the posterior mean, and squares to the actual weights. We use red for the coordinates on which the credible interval does not cover the actual value of $w_{i,1}$.[]{data-label="fig:Uncertainty_delta_2"}](Uncertainty_delta_2_new-eps-converted-to.pdf){height="3.in"}
In Figures \[fig:Uncertainty\_delta\_p5\] and \[fig:Uncertainty\_delta\_2\] we plot Bayesian credible intervals for the weights $w_{i,1}$ as computed within naive mean field, for $k=2$, $d=5000$. These simulations are analogous to the one reported in the main text in Figure \[fig:Uncertainty\_delta1\], but we use $n = 2500$ ($\delta=0.5$) in Figure \[fig:Uncertainty\_delta\_p5\] and $n = 10000$ ($\delta=2$) in Figure \[fig:Uncertainty\_delta\_p5\].
The nominal coverage of these intervals is $0.9$, but we obtain a smaller empirical coverage. For $\delta=0.5$, the empirical coverage was $0.87$ (for $\beta = 2<\beta_{\inst}$), $0.61$ (for $\beta=5.7\in(\beta_{\inst},\beta_{\sBayes})$), and $0.64$ (for $\beta=8.5\approx\beta_{\sBayes}$). For $\delta =2$, the empirical coverage was $0.89$ (for $\beta = 1<\beta_{\inst}$), $0.69$ (for $\beta=3\in(\beta_{\inst},\beta_{\sBayes})$), and $0.65$ (for $\beta=4.2\approx\beta_{\sBayes}$).
Results for $k=3$ topics {#sec:NMF_k3}
------------------------
![Normalized distances $\Norm(\hbH)$, $\Norm(\hbW)$ of the naive mean field estimates from the uninformative fixed point. Here $d = 1000$ and changed $n= d\delta$: each data point corresponds to an average over $400$ random realizations.[]{data-label="fig:H_norm_k_3"}](new_k_3_variational_norm-eps-converted-to.pdf){height="5.5in"}
![Empirical fraction of instances such that $\Norm(\hbW)\ge \eps_0=5\cdot 10^{-3}$ (left) or $\Norm(\hbH)\ge \eps_0$ (right), where $\hbW, \hbH$ are the naive mean field estimate. Here $k=3$, $d=1000$ and, for each $(\delta,\beta)$ point on a grid, we used $400$ random realizations to estimate the probability of $\Norm(\hbW)\ge \eps_0$.[]{data-label="fig:H_norm_k_3_HM"}](new_k_3_variational_norm_heatmap_fractions-eps-converted-to.pdf){height="2.66in"}
![Binder cumulant for the correlation between the naive mean field estimates $\hbH$ and the true topics $\bH$. Here we report results for $k=3$, $d =1000$ and $n=d\delta$, obtained by averaging over $400$ realizations. Note that for $\beta<\beta_{\sBayes}(k,\nu,\delta)$, $\Bind_{\bH}$ decreases with the dimensions, suggesting asymptotically vanishing correlations.[]{data-label="fig:Binder_k_3"}](new_k_3_variational_binder-eps-converted-to.pdf){height="5.5in"}
![Binder cumulant for the correlation between the naive mean field estimates $\hbW$, $\hbH$ and the true weights and topics $\bW$, $\bH$. Here $k=3$, $d=1000$ and $n=d\delta$, and we averaged over $400$ realizations.[]{data-label="fig:Binder_k_3_HM"}](new_k_3_variational_Binder_heatmap-eps-converted-to.pdf){height="2.66in"}
In Figures \[fig:H\_norm\_k\_3\] to \[fig:Binder\_k\_3\_HM\] we report our results using alternating minimization to minimize the naive mean field free energy for $k=3$.
In Figures \[fig:H\_norm\_k\_3\], \[fig:H\_norm\_k\_3\_HM\] we plot (respectively) the normalized distances $\Norm(\hbH)$, $\Norm(\hbW)$ from the uninformative subspaces $\{\bH = \bv\otimes\bfone_k:\; \bv\in\reals^d\}$ and $\{\bW = \bv\otimes\bfone_k:\; \bv\in\reals^d\}$. Data are consistent with the claim that this distance becomes significant when $\beta\ge \beta_{\inst}(k,\nu,\delta)$.
In Figures \[fig:Binder\_k\_3\], \[fig:Binder\_k\_3\_HM\] we consider the correlation between the estimates $\hbH, \hbW$ and the true factorization $\bH, \bW$, and define a Binder cumulant as follows for $k\ge 3$. Let $\Corr_{\eta}(\bH,\hbH)$ be the $k \times k$ matrix with entries $$\begin{aligned}
\Corr_{\eta}(\bH,\hbH)_{i, j} &=&\frac{\<(\hbH_{\perp})_i + \eta \bg,(\bH_{\perp})_j\>}{\Vert (\hbH_{\perp})_i + \eta \bg \Vert_2 \Vert (\bH_{\perp})_j \Vert_2}\end{aligned}$$ We then define $$\begin{aligned}
\label{eqn:Binder_General}
\hat{\bR} &\equiv& \frac{\hE\left\{\sum_{i, j \leq k} \Corr_{\eta}(\bH,\hbH)_{i, j}^4\right\}}{ \hE\left\{ \sum_{i, j \leq k} \Corr_{\eta}(\bH,\hbH)_{i, j}^2 \right\}^2} \\
\Bind_{\bH} &\equiv& \left\{ \begin{array}{cl}
6\bigg(\max\big\{\frac{2}{3} - \hat{\bR}\big\} - \frac{1}{3}\bigg) & \mbox{if } \hE\left\{ \sum_{i, j \leq k} \Corr_{\eta}(\bH,\hbH)_{i, j}^2 \right\} > 0.01 \, ,\\
0 & \mbox{otherwise.}
\end{array} \right.\end{aligned}$$ Here $\hE$ denotes empirical average with respect to the sample and $\bg\sim\normal(0,\id_d)$. We set $\eta=10^{-4}$. An analogous definition holds for $\Corr_{\eta}(\hbW)$, $\Bind_{\eta}(\hbW)$. In equation we introduced a max thresholding step and a threshold on the denominator. These are added to ensure the stability of the fraction below the phase transition region where the denominator of $\hat{\boldsymbol{R}}$ vanishes.
Figures \[fig:Binder\_k\_3\], \[fig:Binder\_k\_3\_HM\] are consistent with the prediction that the correlation between the AMP estimates and the true factors $\bW,\bH$ starts to be non-negligible at the Bayes threshold.
TAP free energy and approximate message passing {#app:TAP}
===============================================
Heuristic derivation of the TAP free energy {#app:TAP_Derivation}
-------------------------------------------
Several heuristic approaches exist to construct the TAP free energy. Here we will derive the expression (\[eq:FreeEnergy\_TAP\_TM\]) of the TAP free energy for topic models as an approximation of the Bethe free energy for the same problem: we refer to [@wainwright2008graphical; @MezardMontanari; @koller2009probabilistic] for background on the latter. Let us emphasize that our derivation will be only heuristic, since our rigorous results are obtained by analyzing the resulting expression $\cF_{\sTAP}(\br,\tbr)$ and do not require a rigorous justification of Eq. (\[eq:FreeEnergy\_TAP\_TM\]).
The posterior $p_{\bH,\bW|\bX}$ takes the form $$\begin{aligned}
p_{\bH,\bW|\bX}(\bH,\bW|\bX) = \frac{1}{Z(\bX)}\prod_{(a,i)\in [n]\times [d]} \exp\left\{\sqrt{\beta}X_{ai} \<\bw_a,\bh_i\> -\frac{\beta}{2d} \<\bw_a,\bh_i\>^2\right\}
\prod_{a=1}^d\tq_0(\bw_a)\prod_{i=1}^dq_0(\bh_i)\, .
$$ This can be regarded as a pairwise graphical model whose underlying graph is the complete bipartite graph over vertex sets $[n]$ (associated to variables $\bw_1$, …$\bw_n$) and $[d]$ (associated to variables $\bh_1$, …$\bh_d$). The Bethe free energy $\cF_{\sBethe}$ takes as input messages $\bq \equiv (q_{i\to a})_{i\in [d],a\in [n]}$, $\tbq= (\tq_{a\to i})_{i\in [d],a\in [n]}$. Messages are probability densities over the $\bh_i$’s (for $q_{i\to a}$) or the $\bw_a$’s (for $\tq_{a\to i}$), indexed by the directed edges in this graph (each pair $(a,i)$, $a\in [n]$, $i\in [d]$ gives rise to two directed edges). The free energy takes the form [@MezardMontanari] $$\begin{aligned}
\cF_{\sBethe}(\bq,\tbq)&= \sum_{a=1}^n\sum_{i=1}^d\log Z_{ai}-\sum_{i=1}^d\log Z_i-\sum_{a=1}^n\log \tZ_a \,,\\
Z_i & = \int \prod_{a=1}^n e^{\sqrt{\beta}X_{ai} \<\bw_a,\bh_i\> -\frac{\beta}{2d} \<\bw_a,\bh_i\>^2} \de q_0(\bh_i)\, \prod_{a=1}^n \de\tq_{a\to i}(\bw_a)\, ,\label{eq:Zi}\\
\tZ_a & = \int \prod_{i=1}^d e^{\sqrt{\beta}X_{ai} \<\bw_a,\bh_i\> -\frac{\beta}{2d} \<\bw_a,\bh_i\>^2} \de \tq_0(\bw_a)\, \prod_{i=1}^d \de q_{i\to a}(\bh_i)\, ,\label{eq:Za}\\
Z_{ai} & = \int e^{\sqrt{\beta}X_{ai} \<\bw_a,\bh_i\> -\frac{\beta}{2d} \<\bw_a,\bh_i\>^2} \, \, \de q_{i\to a}(\bh_i)\,\de \tq_{a\to i}(\bw_a)\, . \label{eq:Zai}
$$ The stationarity conditions for $\cF_{\sBethe}(\bq,\tbq)$ correspond to the belief propagation fixed point equations $$\begin{aligned}
q_{i\to b}(\bh_i) & = \frac{1}{C_{i\to b}}\, q_0(\bh_i) \, \prod_{a\in[n]\setminus b} \int e^{\sqrt{\beta}X_{ai} \<\bw_a,\bh_i\> -\frac{\beta}{2d} \<\bw_a,\bh_i\>^2} \de\tq_{a\to i}(\bw_a)\, ,\label{eq:BP_FP1}\\
\tq_{a\to j}(\bw_i) & = \frac{1}{\tilde{C}_{a\to j}}\, \tq_0(\bw_i) \, \prod_{i\in [d]\setminus j} \int e^{\sqrt{\beta}X_{ai} \<\bw_a,\bh_i\> -\frac{\beta}{2d} \<\bw_a,\bh_i\>^2} \de q_{i\to a}(\bh_i)\, .
\label{eq:BP_FP2}
$$ We define $\fb_{i\to a}= \int \bh_i \de q_{i\to a}(\bh_i)$, $\tfb_{a\to i}= \int \bw_a \de \tq_{a\to i}(\bw_a)$, and $\bg_{i\to a}= \int \bh^{\otimes 2}_i \de q_{i\to a}(\bh_i)$, $\tbg_{a\to i}= \int \bw^{\otimes 2}_a \de \tq_{a\to i}(\bw_a)$. Since $X_{ai} = O(1/\sqrt{n})$, we have $$\begin{aligned}
\prod_{i=1}^d&\int e^{\sqrt{\beta}X_{ai} \<\bw_a,\bh_i\> -\frac{\beta}{2d} \<\bw_a,\bh_i\>^2} \de q_{i\to a}(\bh_i) =\\
&=\prod_{i=1}^d
\exp\left\{\sqrt{\beta} X_{ai}\<\fb_{i\to a}, \bw_a\>-\frac{\beta}{2d}\<\fb_{i\to a},\bw_a\>^2+\frac{\beta}{2}\Big(X_{ai}^2-\frac{1}{d}\Big)
\<\bg_{i\to a}-\fb_{i\to a}^{\otimes 2},\bw_a^{\otimes 2}\>+O(n^{-3/2})\right\}\\
& =\exp\left\{\sum_{i=1}^d\sqrt{\beta} X_{ai}\<\fb_{i\to a}, \bw_a\>-\frac{\beta}{2d}\sum_{i=1}^d\<\fb_{i\to a},\bw_a\>^2+O(n^{-1/2})\right\}\, ,
\label{eq:ExpansionLargeDeg}
$$ where in the last step we used the fact that $\E\{X^2_{ai}-d^{-1}\}=O(n^{-3/2})$ and applied the central limit theorem.
Using the expression (\[eq:ExpansionLargeDeg\]) in Eq. (\[eq:Za\]), and repeating a similar calculation for (\[eq:Zi\]), we get $$\begin{aligned}
\log Z_i & =\phi\left(\sqrt{\beta}\sum_{a=1}^nX_{ai}\tfb_{a\to i},\frac{\beta}{d}\sum_{a=1}^n\tfb_{a\to i}^{\otimes 2}\right) +O(n^{-1/2})\, ,\label{eq:Zi_formula}\\
\log \tZ_a & = \tphi\left(\sqrt{\beta}\sum_{i=1}^dX_{ai}\fb_{i\to a},\frac{\beta}{d}\sum_{i=1}^d\fb_{i\to a}^{\otimes 2}\right) +O(n^{-1/2})\, ,\label{eq:Za_formula}
$$ where the functions $\phi$, $\tphi$ are defined implicitly in Eq. (\[eq:densityforms\_main\]).
We can similarly expand $Z_{ai}$ for large $n,d$: $$\begin{aligned}
Z_{ai}&= 1+\sqrt{\beta} X_{ai}\<\tfb_{a\to i},\fb_{i\to a}\> +\frac{\beta}{2}\Big(X_{ai}^2-\frac{1}{d}\Big) \<\tbg_{a\to i},\bg_{i\to a}\> +O(n^{-3/2})\\
& = \exp\left\{\sqrt{\beta} X_{ai}\<\tfb_{a\to i},\fb_{i\to a}\>-\frac{\beta}{2} X_{ai}^2\<\tfb_{a\to i},\fb_{i\to a}\>^2 +\frac{\beta}{2}\Big(X_{ai}^2-
\frac{1}{d}\Big) \<\tbg_{a\to i},\bg_{i\to a}\> +O(n^{-3/2})\right\}\, .
$$ Therefore, using again the central limit theorem, $$\begin{aligned}
\sum_{a\le n, i\le d}\log Z_{ai}& = \sqrt{\beta} \sum_{a\le n, i\le d} X_{ai}\<\tfb_{a\to i},\fb_{i\to a}\>-\frac{\beta}{2d} \sum_{a\le n, i\le d}\<\tfb_{a\to i},\fb_{i\to a}\>^2+
O(n^{1/2})\, . \label{eq:Zai_formula}
$$ Putting together Eqs. (\[eq:Zi\_formula\]), (\[eq:Za\_formula\]), and (\[eq:Zai\_formula\]), we obtain $$\begin{aligned}
\cF_{\sBethe}(\bq,\tbq)&= -\sum_{i=1}^d \phi\left(\sqrt{\beta}\sum_{a=1}^nX_{ai}\tfb_{a\to i},\frac{\beta}{d}\sum_{a=1}^n\tfb_{a\to i}^{\otimes 2}\right)
-\sum_{a=1}^n \tphi\left(\sqrt{\beta}\sum_{i=1}^dX_{ai}\fb_{i\to a},\frac{\beta}{d}\sum_{i=1}^d\fb_{i\to a}^{\otimes 2}\right)\nonumber\\
& +\sqrt{\beta}\sum_{a\le n, i\le d} X_{ai}\<\tfb_{a\to i},\fb_{i\to a}\>-\frac{\beta}{2d} \sum_{a\le n, i\le d}\<\tfb_{a\to i},\fb_{i\to a}\>^2 +O(n^{1/2})\,.
$$ Close to the solution of the stationarity conditions (\[eq:BP\_FP1\]), (\[eq:BP\_FP2\]), the message $\fb_{i\to a}$ should be roughly independent of $a\in [n]$ and $\tfb_{a\to i}$ should be roughly independent of $i\in [d]$. Hence, we can approximate $$\begin{aligned}
-\frac{\beta}{2d} \sum_{a\le n, i\le d}\<\tfb_{a\to i},\fb_{i\to a}\>^2= -\frac{\beta}{2nd^2} \sum_{a\le n, i\le d}\sum_{b\le n, j\le d}\<\tfb_{a\to j},\fb_{i\to b}\>^2+o(n)\, .
\label{eq:ApproximationMessages}
$$ In order to obtain the expression of Eq. (\[eq:FreeEnergy\_TAP\_TM\]) we add auxiliary variables $\bm_i,\tbm_a\in\reals^k$, and $\bQ_i,\tbQ_a\in\reals^{k\times k}$, alongside Lagrange multipliers $\br_i$, $\tbr_a$, $\bOmega_i$, $\tbOmega_a$ to enforce the constraints $$\begin{aligned}
\bm_i = \sqrt{\beta}\sum_{a=1}^nX_{ai}\tfb_{a\to i}\, ,\;\;\;\;\;\;\; \bQ_i = \frac{\beta}{d}\sum_{a=1}^n\tfb_{a\to i}^{\otimes 2}\, ,\\
\bm_a = \sqrt{\beta}\sum_{i=1}^dX_{ai}\fb_{i\to a}\, ,\;\;\;\;\;\;\; \tbQ_a = \frac{\beta}{d}\sum_{i=1}^d\fb_{i\to a}^{\otimes 2}\, .
$$ Denoting by $\bm\in\reals^{d\times k}$ the matrix whose $i$-th row is $\bm_i$ (and analogously for $\tbm$, $\fb$, $\tfb$ and the Lagrange multipliers $\br$, $\tbr$), and using Eq. (\[eq:ApproximationMessages\]) we obtain the Lagrangian (here all sums run over $a\in [n]$ and $i\in [d]$) $$\begin{aligned}
\cL = & \<\br,\bm\>-\sqrt{\beta}\sum_{a, i}X_{ai}\<\br_i,\tfb_{a\to i}\>+\<\tbr,\tbm\>-\sqrt{\beta}\sum_{a, i}X_{ai}\<\tbr_a,\fb_{i\to a}\>
+\sqrt{\beta}\sum_{a, i}X_{ai}\<\tfb_{a\to i},\fb_{i\to a}\>\nonumber \\
&+\frac{\sqrt{\beta}}{2n} \sum_{a, i}\<\tbOmega_a,\fb_{i\to a}^{\otimes 2}\>
-\frac{d}{2n\sqrt{\beta}} \sum_{a}\<\tbOmega_a,\tbQ_a\>+\frac{\sqrt{\beta}}{2d} \sum_{a, i}\<\tbOmega_i,\tfb_{a\to i}^{\otimes 2}\>
-\frac{d}{2d\sqrt{\beta}} \sum_{a}\<\bOmega_i,\tbQ_i\>\nonumber\\
&-\sum_{i}\phi(\bm_i,\bQ_i)-\sum_{a}\tphi(\tbm_i,\tbQ_i)-\frac{d}{2\beta dn}\sum_{a,i}\<\tbQ_a,\bQ_i\>\, . \label{eq:LagrangianForm}
$$ We next minimize with respect to the message variables $(\fb_{i\to a})$, $(\tfb_{a\to i})$. The first order stationarity conditions read $$\begin{aligned}
X_{ai} \tfb_{a\to i} & = X_{ai}\tbr_a-\frac{1}{n}\tbOmega_a \fb_{i\to a}\, ,\label{eq:Fstationarity}\\
X_{ai} \fb_{i\to a} & = X_{ai}\br_i -\frac{1}{d}\bOmega_i \tfb_{a\to i}\, . \label{eq:tFstationarity}
$$ In particular these imply that $\tfb_{a\to i} = \tbr_a+O(1/\sqrt{n})$ and $\tfb_{a\to i} = \tbr_a+O(1/\sqrt{n})$. Multiplying the first of these equations by $\fb_{i\to a}$ and the second by $\tfb_{a\to i}$, and summing over $i,a$ we obtain $$\begin{aligned}
\sum_{a,i}X_{ai} \<\tfb_{a\to i},\fb_{i\to a}\> =& \frac{1}{2} \sum_{i,a}X_{ai}\Big(\<\fb_{i\to a},\tbr_a\>+\<\tfb_{a\to i},\br_i\>\Big)
-\frac{1}{2n} \sum_{i,a}\<\tbOmega_a,\fb_{i\to a}^{\otimes 2}\>-\frac{1}{2d} \sum_{i,a}\<\bOmega_i,\tfb_{a\to i}^{\otimes 2}\>\nonumber\\
=&\frac{1}{2} \sum_{i,a}X_{ai}\Big(\<\fb_{i\to a},\tbr_a\>+\<\tfb_{a\to i},\br_i\>\Big)
-\frac{1}{2n} \sum_{i,a}\<\tbOmega_a,\br_i^{\otimes 2}\>-\frac{1}{2d} \sum_{i,a}\<\bOmega_i,\tbr_{a}^{\otimes 2}\>+O(n^{1/2})\, .
$$ Further, multiplying Eqs. (\[eq:Fstationarity\]), (\[eq:tFstationarity\]) respectively by $\br_i$ and $\tbr_a$, we get $$\begin{aligned}
\frac{1}{2} \sum_{i,a}X_{ai}\Big(\<\fb_{i\to a},\tbr_a\>&+\<\tfb_{a\to i},\br_i\>\Big) = \sum_{a,i} X_{ai}\<\tbr_a,\br_i\>-\frac{1}{2n}\sum_{a,i}\<\br_i,\tbOmega_a \fb_{i\to a}\>
-\frac{1}{2d}\sum_{a,i}\<\tbr_a,\bOmega_i \tfb_{a\to i}\>\nonumber\\
&= \sum_{a,i} X_{ai}\<\tbr_a,\br_i\>-\frac{1}{2n}\sum_{a,i}\<\tbOmega_a ,\br_i^{\otimes 2}\>
-\frac{1}{2d}\sum_{a,i}\<\bOmega_i ,\tbr_{a}^{\otimes 2}\>+O(n^{1/2})\, .
$$ Substituting the last two expressions in Eq. (\[eq:LagrangianForm\]), we obtain $$\begin{aligned}
\cL = & \; \<\br,\bm\>+\<\tbr,\tbm\>-\sqrt{\beta}\<\tbr,\bX\br\>+\frac{\sqrt{\beta}}{2n} \sum_{a, i}\<\tbOmega_a,\br_{i}^{\otimes 2}\>
+\frac{\sqrt{\beta}}{2d} \sum_{a, i}\<\bOmega_i,\tbr_{a}^{\otimes 2}\>
-\frac{d}{2n\sqrt{\beta}} \sum_{a}\<\tbOmega_a,\tbQ_a\>\nonumber\\
&-\frac{d}{2d\sqrt{\beta}} \sum_{i}\<\bOmega_i,\bQ_i\>
-\sum_{i}\phi(\bm_i,\bQ_i)-\sum_{a}\tphi(\tbm_i,\tbQ_i)-\frac{d}{2\beta dn}\sum_{a,i}\<\tbQ_a,\bQ_i\>+O(n^{1/2})\, . \label{eq:LagrangianForm2}
$$ Setting $\bQ_i = \bQ$ independent of $i$, $\tbQ_a = \tbQ$ independent of $a$, defining $\bOmega = d^{-1}\sum_{i=1}^d \bOmega_i$, $\tbOmega = n^{-1}\sum_{a=1}^n \tbOmega_a$, and neglecting $o(n)$ terms, we get $$\begin{aligned}
\begin{split}
\tilde{\cF}_{\sTAP} = & \frac{d}{2}\|\bX\|_{F} -\sqrt{\beta}\Tr\left(\bX\br\tbr^{\sT}\right) +\Tr(\br^{\sT}\bm) +\Tr(\tbr^{\sT}\tbm) -\frac{d}{2\sqrt{\beta}}\Tr(\bQ\bOmega) -\frac{d}{2\sqrt{\beta}}\Tr(\tbQ\tbOmega)\\
&-\sum_{a=1}^n \tphi(\tbm_a, \tbQ) - \sum_{i=1}^d\phi(\bm_i, \bQ)
+\frac{\sqrt{\beta}}{2}\sum_{i=1}^d\<\tbOmega,\br_i^{\otimes 2}\>+\frac{\sqrt{\beta}}{2}\sum_{a=1}^n\<\bOmega,\tbr_a^{\otimes 2}\>\\
&-\frac{d}{2\beta}\<\bQ,\tbQ\> \, .
\end{split}\end{aligned}$$ Finally, the expression (\[eq:FreeEnergy\_TAP\_TM\]) is recovered by using the stationarity conditions with respect to $\bOmega$ and $\tbOmega$, which imply $\bQ = (\sqrt{\beta}/d)\sum_{a=1}^n\tbr_a^{\otimes 2}$ and $\tbQ = (\sqrt{\beta}/d)\sum_{i=1}^d\br_i^{\otimes 2}$, and maximizing with respect to $\bm$, $\tbm$.
Gradient of the TAP free energy
-------------------------------
From the definition of the partial Legendre transforms $\psi(\br,\bQ)$, $\tpsi(\tbr,\tbQ)$, the following derivatives hold $$\begin{aligned}
\frac{\partial\psi}{\partial\br}(\br,\bQ) = \bm(\br,\bQ)\, ,\;\;\;\;\;\;\;
\frac{\partial\psi}{\partial\bQ}(\br,\bQ) = -\frac{1}{2\beta}\sG\big(\bm(\br,\bQ),\bQ\big)\, ,
$$ where $\bm(\br,\bQ)\in\reals^k$ is the unique solution of $$\begin{aligned}
\br = \frac{1}{\sqrt{\beta}}\,\sF(\bm;\bQ) \,.\label{eq:mrApp}
$$ Using these derivatives we can compute the gradient of the free energy $$\begin{aligned}
\frac{\partial \cF_{\sTAP}}{\partial \br_i}(\br,\tbr) & = -\sqrt{\beta} (\bX^{\sT}\tbr)_i+\bm_i -\frac{\beta}{d}\sum_{a=1}^n\<\tbr_a,\br_i\>\, \tbr_a
+\frac{1}{d}\sum_{a=1}^n\tsG(\tbm_a,\tbQ)\br_i\nonumber\\
& = -\sqrt{\beta} (\bX^{\sT}\tbr)_i+\bm_i + \sqrt{\beta}\tbOmega\br_i\, ,\label{eq:GradTAP1}\\
\frac{\partial \cF_{\sTAP}}{\partial \tbr_a}(\br,\tbr) & = -\sqrt{\beta} (\bX\br)_a+\tbm_a -\frac{\beta}{d}\sum_{i=1}^d\<\tbr_a,\br_i\>\, \br_i
+\frac{1}{d}\sum_{i=1}^d\sG(\bm_i,\bQ)\tbr_a\nonumber\\
& = -\sqrt{\beta} (\bX\br)_a+\tbm_a +\sqrt{\beta}\bOmega\tbr_a\, ,\label{eq:GradTAP2}
$$ where $\bm_i = \bm(\br_i,(\beta/d)\sum_{a\le n}\tbr_a^{\otimes 2})$, $\tbm_a = \tbm(\tbr_a,(\beta/d)\sum_{i\le d}\br_i^{\otimes 2})$, are defined as above, $\bQ=(\beta/d)\sum_{a\le n}\tbr_a^{\otimes 2}$, $\bQ=(\beta/d)\sum_{i\le d}\tbr_i^{\otimes 2}$, and $$\begin{aligned}
\bOmega &= \frac{1}{d\sqrt{\beta}}\sum_{i=1}^d\big\{\sG(\bm_i,\bQ)-\sF(\bm_i,\bQ)^{\otimes 2}\big\}\, ,\\
\tbOmega &= \frac{1}{d\sqrt{\beta}}\sum_{a=1}^n\big\{\tsG(\tbm_a,\tbQ)-\tsF(\bm_a,\tbQ)^{\otimes 2}\big\}\, .
$$
We can express $\br$, $\tbr$ in terms of $\bm$, $\tbm$ in Eqs. (\[eq:GradTAP1\]), (\[eq:GradTAP2\]) by using Eq. (\[eq:mrApp\]) $$\begin{aligned}
\bm &= \bX^{\sT}\,\tsF(\tbm;\tbQ)-\sF(\bm;\bQ) \tbOmega\, ,\;\;\;\;\;\;
\tbm = \bX\,\sF(\bm;\bQ)-\tsF(\tbm;\tbQ) \bOmega\, ,\label{eq:CriticalTAP_1}\\
\bQ &= \frac{1}{d}\sum_{a=1}^n \tsF(\tbm_a;\tbQ)^{\otimes 2}\, ,\;\;\;\; \;\;\;\; \;\;\;\tbQ = \frac{1}{d}\sum_{i=1}^d \sF(\bm_i;\bQ)^{\otimes 2}\, . \label{eq:CriticalTAP_2}
$$ These coincide with the fixed point of the AMP algorithm in Section \[sec:TAP-Topic\].
Uninformative critical point: Proof of Lemma \[lemma:Uninf\_TAP\] {#app:UninformativeTAP}
-----------------------------------------------------------------
Consider the stationarity conditions (\[eq:CriticalTAP\_1\]) and (\[eq:CriticalTAP\_2\]), together with the definitions of Eqs. (\[eq:OmegaTAP1\]), (\[eq:OmegaTAP2\]). Since these are invariant under permutations of the topics, they admit a solution of the form $\bm = \bv \bfone^{\sT}_k$, $\tbm = \tbv \bfone^{\sT}_k$, $\bQ = q_0\bJ_k+q_0'\id_k$, $\tbQ = \tq_0\bJ_k+\tq_0'\id_k$. Using Eq. (\[eq:CriticalTAP\_2\]) and Lemma \[lemma:UsefulFormulae\], Eqs. (\[eq:sfsimplifiedsymm\]), (\[eq:tsfsimplifiedsymm\]), we get $q_0' = \tq'_0=0$.
Substituting this in Eqs. (\[eq:OmegaTAP1\]), (\[eq:OmegaTAP2\]), and using again Lemma \[lemma:UsefulFormulae\], we get $$\begin{aligned}
\bOmega = \sqrt{\beta}\, \id_k\, ,\;\;\;\;\;\;\;\; \tbOmega = \frac{\sqrt{\beta}\delta}{k(k\nu+1)} \bPp\, ,
$$ where we recall that $\bPp= \id_k -\bfone_k\bfone_k/k$. Substituting these in Eq. (\[eq:CriticalTAP\_1\]), we obtained that this is satisfied provided $\bv, \tbv$ are given as in Eqs. (\[eq:UninfTAP\_1\]), (\[eq:UninfTAP\_2\]). Finally, $q_0$, $\tq_0$ are fixed by substituting in Eq. (\[eq:CriticalTAP\_2\]).
State evolution analysis {#state-evolution-analysis}
========================
State evolution equations
-------------------------
Note that there is an alternative way to express the state evolution recursion in Eqs. (\[eq:FirstSE\]), (\[eq:SecondSE\]). Given a probability measure $p$ on $\reals^k$ and a matrix $\bM\succeq 0$, $\bM\in\reals^{k\times k}$, we define the minimum mean square error $$\begin{aligned}
\mmse(\bM;p)\equiv \inf_{\hbx(\, \cdot\, )}\,\E\Big\{[\bx-\hbx(\by)][\bx-\hbx(\by)]^{\sT}\Big\}\, ,
$$ where the expectation is with respect to $\bx\sim p(\,\cdot\,)$ and $\by = \bM^{1/2}\bx+\bz$ for $\bz\sim\normal(0,\id_k)$. The infimum is understood in the positive semidefinite order, and it is achieved by $\hbx(\by ) = \E \{\bx|\by\}$. We then rewrite Eqs. (\[eq:FirstSE\]), (\[eq:SecondSE\]) as $$\begin{aligned}
\bM_{t+1} & =\beta\delta\, \Big\{\mmse(0;\tq_0)-\mmse(\tbM_t;\tq_0)\Big\}\, ,\label{eq:FirstSE_b}\\
\tbM_{t} & = \beta\, \Big\{\mmse(0;q_0)-\mmse(\bM_t;q_0)\Big\}\, .\label{eq:SecondSE_b}
$$
Uninformative fixed point {#app:SE_FP}
-------------------------
\[lemma:symmfixedpointSE\] The state evolution recursion in , admit uninformative fixed point of the form $$\begin{aligned}
\label{eq:fixedpoint}
\begin{split}
&\tbM^* = \rho_0\bJ_k,\quad\quad \rho_0 = \frac{\delta\beta^2}{k\delta\beta + k^2},\\
&\bM^* = \frac{\delta\beta}{k^2}\bJ_k.
\end{split}\end{aligned}$$
First note that for this value of $\tbM^*$, $\tbM^*\bw+\tbM^{*^{1/2}}\bz = y\bfone_k$ for some (random) $y$. Hence, using Eq. (\[eq:tsfsimplifiedsymm\]) $$\begin{aligned}
\delta\, \E\Big\{\tsF(\tbM^*\bw+\tbM^{*^{1/2}}\bz;\tbM^*)^{\otimes 2}\Big\} = \frac{\delta\beta}{k^2}\bJ_k = \bM^*.\end{aligned}$$ In addition, using the explicit form (\[eq:sFexplicit\]) $$\begin{aligned}
\E\Big\{\sF(\bM^*\bh+\bM^{*^{1/2}}\bz;\bM^*)^{\otimes 2}\Big\} = \beta(\id_k+\bM^*)^{-1}\bM^* = \frac{\beta^2\delta}{k^2}\left(\id_k+\frac{\delta\beta}{k^2}\bJ_k\right)^{-1}\bJ_k = \rho_0\bJ_k = \tbM^*.\end{aligned}$$ Hence, the pair $\bM^*,\tbM^*$ in is a fixed point for the iterations in , .
Stability of state evolution and proof of Theorem \[thm:StateEvolStable\] {#sec:StabilitySE}
-------------------------------------------------------------------------
The following theorem characterizes the region of parameters in which the uninformative fixed point of the state evolution iterations in Lemma \[lemma:symmfixedpointSE\] is stable.
Consider the state evolution equations in , . The uninformative symmetric fixed point of these equations is stable if and only if $$\begin{aligned}
\beta < \beta_{\sp} = \frac{k(k\nu+1)}{\sqrt{\delta}}.\end{aligned}$$
We linearize Eqs. , around the fixed point in by setting $\bM_t = \bM_*+\bDelta_t$, $\tbM_t = \tbM_*+\tbDelta_t$ and expanding Eqs. , to first order in $\bDelta, \tbDelta_t$. First note that Eq. takes the explicit form $$\begin{aligned}
\tbM_t = \beta(\id_k+\bM_t)^{-1}\bM_t\, .
$$ Hence, expanding to linear order we get $$\begin{aligned}
\tbDelta_t = \beta\left(\id_k + \frac{\delta\beta}{k^2}\bJ_k\right)^{-1}\bDelta_t \left(\id_k + \frac{\delta\beta}{k^2}\bJ_k\right)^{-1}+o(\bDelta_t)\, . \label{eq:LinearizationTilde}
$$ In the following, we shall decompose $\bDelta_t$ and $\tbDelta_t$ in the components along $\bfone_k$ and the ones orthogonal $$\begin{aligned}
\begin{split}
\bDelta_t & = \delta_t\, \bP+\bDelta_t^{(1)}+\bDelta_t^{(2)}\, ,\\
\bDelta_t^{(1)} & = \bP\bDelta_t\bPp+ \bPp\bDelta_t\bP\, ,\\
\bDelta_t^{(2)} & = \bPp\bDelta_t\bPp\, ,
\end{split}\label{eq:DecompositionDelta}
$$ and similarly for $\tbDelta_t$. Note that the linearization (\[eq:LinearizationTilde\]) preserves these subspaces $$\begin{aligned}
\tdelta_t &= \beta \left(1+\frac{\delta\beta}{k}\right)^{-2} \delta_t+o(\bDelta_t)\, ,\label{eq:Linearization1}\\
\tbDelta^{(1)}_t &= \beta \left(1+\frac{\delta\beta}{k}\right)^{-1} \bDelta^{(1)}_t+o(\bDelta_t)\, ,\\
\tbDelta^{(2)}_t &= \beta \, \bDelta^{(2)}_t+o(\bDelta_t)\, . \label{eq:Linearization3}
$$ Next we consider Eq. (\[eq:FirstSE\]). We compute the value of $$\begin{aligned}
f_{\bw,\bz} &= \tsF(\tbM_t\bw+\tbM_t^{1/2}\bz;\tbM_t)\\
&= \sqrt{\beta}\frac{\int\bw_1\exp\left\{\left\langle \tbM_t\bw+\tbM_t^{1/2}\bz,\bw_1\right\rangle -\frac{1}{2}\left\langle\bw_1,\tbM_t\bw_1\right\rangle\right\}\tq_0(\de \bw_1)}{\int\exp\left\{\left\langle \tbM_t\bw+\tbM_t^{1/2}\bz,\bw_1\right\rangle -\frac{1}{2}\left\langle\bw_1,\tbM_t\bw_1\right\rangle\right\}\tq_0(\de \bw_1)} = \sqrt{\beta}\frac{A_{\bw,\bz}}{B_{\bw,\bz}}.\end{aligned}$$ for $\bw\in \sP_1(k)$. We have $$\begin{aligned}
&\tbM_{t}\bw = \rho_0\bfone_k + \tbDelta^t\bw,\\
&\left\langle\bw_1, \tbM_{t}\bw_1\right\rangle = \rho_0 +\left\langle\bw_1, \tbDelta^t\bw_1\right\rangle.\end{aligned}$$ Hence, $$\begin{aligned}
A_{\bw,\bz} &=\int\bw_1\exp\left\{\left\langle \rho_0\bfone_k + \tbDelta^t\bw+\left(\rho_0 \bJ_k + \tbDelta^t\right)^{1/2}\bz, \bw_1\right\rangle - \frac{\rho_0}{2} - \frac{1}{2}\left\langle\bw_1, \tbDelta^t\bw_1\right\rangle\right\}\tq_0(\de \bw_1)\\
&= \int\bw_1\exp\left\{\frac{\rho_0}{2} + \left\langle\bw_1,\tbDelta^t\bw\right\rangle - \frac{1}{2}\left\langle \bw_1, \tbDelta^t\bw_1\right\rangle + \sqrt{\rho_0/ k} \left\langle \bJ_k \bz, \bw_1\right\rangle + \left\langle\bC_\bDelta^t\bz, \bw_1\right\rangle \right\}\tq_0(\de \bw_1)\end{aligned}$$ where $\bC_\bDelta^t \equiv \left(\rho_0\bJ_k + \tbDelta^t\right)^{1/2} - (\rho_0/k)^{1/2}\bJ_k$. Therefore, we have $$\begin{aligned}
A_{\bw,\bz} = a \int\bw_1\exp\left\{\left\langle\bw_1,\tbDelta^t\bw\right\rangle - \frac{1}{2}\left\langle \bw_1, \tbDelta^t\bw_1\right\rangle +\left\langle\bC_\bDelta^t\bz, \bw_1\right\rangle \right\}\tq_0(\de \bw_1)\end{aligned}$$ where $a = \exp\left\{ \rho_0/2 + \sqrt{\rho_0/k}\left\langle\bz, \bfone_k\right\rangle\right\}$. Expanding the exponential, we get $$\begin{aligned}
A_{\bw,\bz} = a\int\bw_1\left\{1+\left\langle\bw_1, \tbDelta^t\bw\right\rangle - \frac{1}{2}\left\langle\bw_1, \tbDelta^t\bw_1\right\rangle +\left\langle\bz, \bC_\bDelta^t\bw_1\right\rangle + \frac{1}{2}\left\langle\bz,\bC_\bDelta^t\bw_1\right\rangle^2 + o\left(\tbDelta^t\right)\right\}\tq_0(\de \bw_1).\end{aligned}$$ Thus, $$\begin{aligned}
A_{\bw,\bz} = a\left(\frac{1}{k}\bfone_k + \bS\tbDelta^t\bw -
\frac{1}{2}\begin{pmatrix} \left\langle \tbDelta^t, \bT_1\right\rangle \\ \left\langle \tbDelta^t, \bT_2\right\rangle \\ \vdots \\ \left\langle \tbDelta^t, \bT_k\right\rangle \end{pmatrix}
+ \bS\bC_\bDelta^t\bz + \frac{1}{2} \begin{pmatrix} \left\langle \bC_\bDelta^t\bz^{\otimes2}\bC_\bDelta^t, \bT_1\right\rangle \\ \left\langle \bC_\bDelta^t\bz^{\otimes2}\bC_\bDelta^t, \bOmega_2^\prime\right\rangle \\ \vdots \\ \left\langle \bC_\bDelta^t\bz^{\otimes2}\bC_\bDelta^t, \bT_k\right\rangle \end{pmatrix} + o\left(\tbDelta^t\right)\right) \end{aligned}$$ where $\bS, \bT \in \reals^{k\times k}$ are the moment tensors $$\begin{aligned}
&\bS = \int\bw_1^{\otimes 2}\tq_0(\de \bw_1) = \frac{\nu}{k\nu(k\nu+1)}\left(\id_k+\nu\bJ_k\right)= \frac{1}{k(k\nu+1)}\bPp+ \frac{1}{k}\bP\, ,\label{eq:S_Formula}\\
&\bT = \int\bw_1^{\otimes 3}\tq_0(\de \bw_1), \;\label{eq:bSdef}\\
&(T_i)_{jl} = \frac{1}{k\nu(k\nu+1)(k\nu+2)}.
\begin{cases}
\nu(\nu+1)(\nu+2)\quad \text{if}\; j = l = i,\\
\nu^2(\nu+1)\quad \text{if}\; j = i,\, l\neq i\; \text{or}\; l=i,\, j\neq i\; \text{or}\; l=j, j\neq i,\\
\nu^3\quad \text{otherwise}.
\end{cases}\label{eq:bTdef}\end{aligned}$$ Similarly, we have $$\begin{aligned}
B_{\bw,\bz} = a\int\left\{1 + \left\langle \bw_1, \tbDelta^t\bw\right\rangle - \frac{1}{2}\left\langle\bw_1, \tbDelta^t\bw_1\right\rangle + \left\langle\bz, \bC_\bDelta^t\bw_1\right\rangle + \frac{1}{2}\left\langle\bz, \bC_\bDelta^t\bw_1\right\rangle^2 + o\left(\tbDelta^t\right)\right\}\tq_0(\de \bw_1).\end{aligned}$$ Therefore, $$\begin{aligned}
B_{\bw,\bz} = a\left(1 + \frac{1}{k}\left\langle\bfone_k\otimes\bw, \tbDelta^t\right\rangle - \frac{1}{2}\left\langle\bS, \tbDelta^t\right\rangle + \frac{1}{k}\left\langle\bfone_k\otimes\bz,\bC_\bDelta^t\right\rangle + \frac{1}{2}\left\langle\bz, \bC_\bDelta^t\bS\bC_\bDelta^t\bz\right\rangle + o\left(\tbDelta^t\right)\right).\end{aligned}$$ Hence, we can write $$\begin{aligned}
f_{\bw,\bz} = \sqrt{\beta}\frac{A_{\bw,\bz}}{B_{\bw,\bz}}
&= \sqrt{\beta}\Bigg(\frac{1}{k}\bfone_k
+ \bS\tbDelta^t\bw
- \frac{1}{2}\begin{pmatrix} \left\langle \tbDelta^t, \bT_1\right\rangle \\ \left\langle \tbDelta^t, {\rule{0pt}{2.6ex}}_2\right\rangle \\ \vdots \\ \left\langle \tbDelta^t, \bT_k\right\rangle \end{pmatrix}
+ \bS\bC_\bDelta^t\bz + \frac{1}{2}\begin{pmatrix} \left\langle \bC_\bDelta^t\bz^{\otimes2}\bC_\bDelta^t, \bT_1\right\rangle \\ \left\langle \bC_\bDelta^t\bz^{\otimes2}\bC_\bDelta^t, \bT_2\right\rangle \\ \vdots \\ \left\langle
\bC_\bDelta^t\bz^{\otimes2}\bC_\bDelta^t, \bT_k\right\rangle\end{pmatrix} \\
&- \frac{1}{k^2}\left\langle\bfone_k\otimes\bw, \tbDelta^t\right\rangle\bfone_k
+ \frac{1}{2k}\left\langle\bS, \tbDelta^t\right\rangle\bfone_k \nonumber
- \frac{1}{k^2}\left\langle\bfone_k\otimes\bz,\bC_\bDelta^t\right\rangle\bfone_k
- \frac{1}{2k}\left\langle\bz, \bC_\bDelta^t\bS\bC_\bDelta^t\bz\right\rangle\bfone_k \\
&- \frac{1}{k} \left\langle\bfone_k\otimes \bz, \bC_\bDelta^t\right\rangle\bS\bC_\bDelta^t\bz
- \frac{1}{k^3}\left\langle\bfone_k\otimes \bz, \bC_\bDelta^t\right\rangle^2\bfone_k
+ o\left(\tbDelta^t\right)\Bigg).\end{aligned}$$ Therefore, linearizing Eq. (), we get (below, we denote by $[\bA]_s$ the symmetric part of matrix $\bA$, namely $[\bA]_s = (\bA+\bA^{\sT})/2$) $$\begin{aligned}
\bDelta_{t+1}&= \delta\E_{\bw,\bz}\left(f_{\bw,\bz}^{\otimes2}\right) -\frac{\delta\beta}{k^2}\bJ_k \\
&=
\delta\beta\Bigg(
\frac{2}{k^2}\big[\bS(\tbDelta^t-(\bC_{\bDelta}^t)^2)\bJ_k\big]_s
- \frac{1}{2k}\begin{pmatrix} \left\langle \tbDelta^t-(\bC_{\bDelta}^t)^2, \bT_1\right\rangle \\ \left\langle \tbDelta^t-(\bC_{\bDelta}^t)^2, \bT_2\right\rangle \\ \vdots \\ \left\langle \tbDelta^t-(\bC_{\bDelta}^t)^2, \bT_k\right\rangle \end{pmatrix}\otimes\bfone_k
- \frac{1}{2k}\bfone_k\otimes\begin{pmatrix} \left\langle \tbDelta^t-(\bC_{\bDelta}^t)^2, \bT_1\right\rangle \\ \left\langle \tbDelta^t-(\bC_{\bDelta}^t)^2, \bT_2\right\rangle \\ \vdots \\ \left\langle {\tbDelta^t}-(\bC_{\bDelta}^t)^2, \bT_k\right\rangle \end{pmatrix}\label{eq:DeltaIteration}\\
&- \frac{2}{k^4}\left\langle\bJ_k,\tbDelta^t\right\rangle\bJ_k
+ \frac{1}{k^2}\left\langle\bS,\tbDelta^t-(\bC_\bDelta^t)^2\right\rangle\bJ_k\nonumber
\\
&- \frac{2}{k^4}\left\langle\bJ_k, (\bC_{\bDelta}^t)^2\right\rangle\bJ_k
+ \bS(\bC_{\bDelta}^t)^2\bS
- \frac{2}{k^2}\big[\bS(\bC_{\bDelta}^t)^2\bJ_k \big]_s
+ \frac{1}{k^4}\left\langle\bJ_k, (\bC_{\bDelta}^t)^2\right\rangle\bJ_k
+ o(\tbDelta_t)\Bigg).\nonumber\end{aligned}$$
We next decompose $\tbDelta_t$ in the component along $\bJ_k$ and the one orthogonal, as per Eq. (\[eq:DecompositionDelta\]), and note that $$\begin{aligned}
\bC_{\bDelta}^t &=\Big((k\rho_0+\tdelta_t)\bP+\tbDelta_t^{(1)}+\tbDelta^{(2)}_t\Big)^{1/2}-(k\rho_0)^{1/2}\bP\\
&=\sqrt{k\rho_0+\tdelta_t}\, \bP +\big(\tbDelta_t^{(2)}\big)^{1/2}-\sqrt{k\rho_0}\, \bP+O(\tbDelta_t) -(k\rho_0)^{1/2}\bP = \big(\tbDelta_t^{(2)}\big)^{1/2}+O(\tbDelta_t) \, ,
$$ whence $$\begin{aligned}
(\bC_{\bDelta}^t)^2 = \tbDelta_t^{(2)}+o(\bDelta)\, .
$$ Using this identity together with Eqs. (\[eq:bSdef\]), (\[eq:bTdef\]) in Eq. (\[eq:DeltaIteration\]) we get $$\begin{aligned}
\delta_{t+1} & = o(\tbDelta_t)\, ,\\
\bDelta^{(1)}_{t+1} &= o(\tbDelta_t)\, ,\\
\bDelta^{(2)}_{t+1} &= \frac{\beta\delta}{k^2(k\nu+1)^2}\, \tbDelta^{(2)}_t+o(\tbDelta_t)\, .
$$
Together with Eqs. (\[eq:Linearization1\]) to (\[eq:Linearization3\]), these yield $$\begin{aligned}
\delta_{t+1} & = o(\bDelta_t)\, ,\\
\bDelta^{(1)}_{t+1} &= o(\bDelta_t)\, ,\\
\bDelta^{(2)}_{t+1} &= \frac{\beta^2\delta}{k^2(k\nu+1)^2}\, \bDelta^{(2)}_t +o(\tbDelta_t)\, .
$$ Hence the uninformative fixed point is stable if and only if $$\begin{aligned}
\beta \leq \frac{k(k\nu+1)}{\sqrt{\delta}}.\end{aligned}$$ Note that this is the same condition as the spectral threshold.
Stability of the uninformative point: Proof of Theorem \[thm:StabilityTAP\]
---------------------------------------------------------------------------
In this section we compute the Hessian of the TAP free energy around the uninformative stationary point. We will establish a second order approximation of $\tcF_{\sTAP}(\br,\tbr)$ near the stationary point. Namely, we denote by $\br^*_i = r^*_i\bfone_k$, $\tbr^*_a = \tr^*_a\bfone_k$ the uninformative stationary point, and by $\bm^*_i = m^*_i\bfone_k$, $\tbm^*_a = \tm^*_a\bfone_k$ the dual variables, where $$\begin{aligned}
m^*_i &= \frac{\sqrt{\beta}}{k} (\bX^{\sT}\bfone_n)_i\, ,\;\;\;\;\;\;\; \tm^*_a = \frac{\beta}{k(1+kq_0)} (\bX\bX^{\sT}\bfone_n)_a-\frac{\beta}{k+\delta\beta}\, ,\\
r^*_i &=\frac{\sqrt{\beta}}{k(1+kq_0^*)} (\bX^{\sT}\bfone_n)_i\, ,\;\;\;\;\;\;\; \tr^*_a =\frac{1}{k}\,.
$$ For any other assignment of the variables, $\br,\tbr$, $\bm,\tbm$, we introduce the decomposition $$\begin{aligned}
\br_i &= r_i^s\bfone_k+\bdelta_i \, ,\;\;\;\;\;\;\;\tbr_a= \tr_a^s\bfone_k+\tbdelta_a\, ,\label{eq:ExpansionP1}\\
r_i^s &= r_i^*+\delta^s_i\, ,\;\;\;\;\;\;\;\;\;\; \tr_a^s = \tr_a^*+\tdelta^s_a\, ,\\
\bm_i &= m_i^s\bfone_k+\bfeta_i \, ,\;\;\;\;\;\;\;\tbm_a= \tm_a^s\bfone_k+\tbfeta_a\, ,\\
m_i^s &= m_i^*+\eta^s_i\, ,\;\;\;\;\;\;\;\;\;\; \tm_a^s = \tm_a^*+\teta^s_a\, ,\label{eq:ExpansionP2}
$$ where $\<\bdelta_i,\bfone_k\>=\<\tbdelta_a,\bfone_k\>=\<\bfeta_i,\bfone_k\>=\<\tbfeta_a,\bfone_k\>=0$. Note that, by construction $\tr_a^s = 1/k$.
We will establish an expansion of the form $$\begin{aligned}
\cF_{\sTAP}(\br,\tbr) = \tcF_{\sTAP}(\br^*,\tbr^*) +
\cF^{(2)}_{\sTAP}(\bdelta,\tbdelta,\delta^s,\tdelta^s) +o(\delta^2)\, ,\label{eq:TAP_Expansion}
$$ where $\cF^{(2)}_{\sTAP}$ is a quadratic function, and when using the $O(\,\cdot\, )$ notation, we implicitly consider all $\delta,\eta$ parameters to be of the same order and use $\delta$ for denoting that order. Notice that the first-order term is missing from this expansion since $(\br^*,\tbr^*)$ is a stationary point.
The crucial step in obtaining the expansion (\[eq:TAP\_Expansion\]) is to derive a second order expansion for the logarithmic moment generating functions $\phi$, $\tphi$, and subsequently for the entropy functions $\psi$, $\tpsi$.
Setting variables as per Eq. (\[eq:ExpansionP1\]), we have $$\begin{aligned}
\phi\left(\bm_i,\frac{\beta}{d}\sum_{a=1}^n\tbr_a^{\otimes 2}\right) =& -\frac{1}{2}\log(1+ka_0) +\frac{k(m_i^s)^2}{2(1+ka_0)}+\frac{\beta^2(1+\beta\delta/k+k(m_i^*)^2)}{2d^2k(1+\beta\delta/k)^2}
\left\|\sum_{a=1}^n\tbdelta_a\right\|^2_2\label{eq:PhiFormula}\\
&-\frac{\beta m_i^*}{d(1+\beta\delta/k)}
\sum_{a=1}^n\<\bfeta_i,\tbdelta_a\>+\frac{1}{2}\|\bfeta_i\|_2^2 -\frac{\beta}{2d}\sum_{a=1}^n\|\tbdelta_a\|_2^2+o(\delta^2)\, ,\nonumber
$$ where $a_0 = (\beta/d)\sum_{a=1}^n (\tr^s_a)^2$.
Let $\bQ = (\beta/d)\sum_{a=1}^n\tbr_a^{\otimes 2}$, and define the orthogonal decomposition $\bQ = \bQ_0 +\bQ_1+\bQ_2$, where $\bQ_0= \bP\bQ\bP$, $\bQ_1= \bP\bQ\bPp+\bPp\bQ\bP$, $\bQ_2= \bPp\bQ\bPp$. Using the representation (\[eq:ExpansionP1\]), we get $$\begin{aligned}
\bQ_0 & =a_0\bfone_k\bfone_k^{\sT}\, ,\;\;\;\;\;\;\;\;\;\;\; a_0 = \frac{\beta}{d}\sum_{a=1}^n (\tr^s_a)^2\, ,\\
\bQ_1 & =\bfone_k\ba_1^{\sT}+\ba_1\bfone_k^{\sT}\, ,\;\;\;\;\;\;
\ba_1 = \frac{\beta}{d}\sum_{a=1}^n \tr^s_a \tbdelta_a\, ,\\
\bQ_2 & =\frac{\beta}{d}\sum_{a=1}^n \tbdelta_a\tbdelta_a^{\sT}\, .
$$ By Gaussian integration, we have $$\begin{aligned}
\phi(\bm_i,\bQ) = -\frac{1}{2}\Tr \log\big(\id+\bQ\big)+\frac{1}{2}\<\bm_i,(\id+\bQ)^{-1}\bm_i\>\, .\label{eq:PhiExact}
$$ Expanding the logarithm, we get $$\begin{aligned}
\Tr \log\big(\id+\bQ\big) =&\Tr \log\big(\id+\bQ_0\big) +\Tr\big\{(\id+\bQ_0)^{-1}(\bQ_1+\bQ_2)\big\}\nonumber\\
&-\frac{1}{2} \Tr\big\{(\id+\bQ_0)^{-1}\bQ_1 (\id+\bQ_0)^{-1}\bQ_1\big\} +o(\delta^2)\nonumber\\
=&\Tr \log\big(\id+\bQ_0\big)+\Tr(\bQ_2)- \, \<\ba_1,(\id+\bQ_0)^{-1}\ba_1\>\, \<\bfone,(\id+\bQ_0)^{-1}\bfone\> +o(\delta^2)\nonumber\\
= &\log(1+ka_0) + \frac{\beta}{d}\sum_{a=1}^n\|\tbdelta_a\|_2^2-\frac{k}{1+ka_0}\left\|\frac{\beta}{d}\sum_{a=1}^n \tr^s_a \tbdelta_a\right\|_2^2 +o(\delta^2)\nonumber\\
= &\log(1+ka_0) +\frac{\beta}{d}\sum_{a=1}^n\|\tbdelta_a\|_2^2-\frac{\beta^2}{kd^2(1+kq^*_0)}\left\|\sum_{a=1}^n \tbdelta_a\right\|_2^2 +o(\delta^2)
$$ Considering next the second term in Eq. (\[eq:PhiExact\]), we get $$\begin{aligned}
\<\bm_i,(\id+\bQ)^{-1}\bm_i\> = & (m_i^s)^2\<\bfone,(\id+\bQ_0+\bQ_1+\bQ_2)^{-1}\bfone\>+2 m_i^s\<\bfeta_i,(\id+\bQ_0+\bQ_1)^{-1}\bfone\>\nonumber\\
&+\<\bfeta_i,(\id+\bQ_0)^{-1}\bfeta_i\>+o(\delta^2)\nonumber \\
=& (m_i^s)^2\<\bfone,(\id+\bQ_0)^{-1}\bfone\>+(m_i^s)^2\<\bfone,(\id+\bQ_0)^{-1}\bQ_1 (\id+\bQ_0)^{-1}\bQ_1 (\id+\bQ_0)^{-1}\bfone\>\nonumber\\
&-2 m_i^s\<\bfeta_i,(\id+\bQ_0)^{-1}\bQ_1 (\id+\bQ_0)^{-1}\bfone\>+\|\bfeta_i\|_2^2+o(\delta^2)\nonumber\\
=& \frac{k(m_i^s)^2}{1+ka_0}+\frac{(km_i^s)^2}{(1+ka_0)^2}\|\ba_1\|^2_2-\frac{2km_i^s}{(1+ka_0)}\<\bfeta_i,\ba_1\>+\|\bfeta_i\|_2^2+o(\delta^2)\nonumber\\
=& \frac{k(m_i^s)^2}{1+ka_0}+\frac{(\beta m_i^s)^2}{d^2(1+kq^*_0)^2}
\left\|\sum_{a=1}^n\tbdelta_a\right\|^2_2-\frac{2\beta m_i^s}{d(1+kq^*_0)}
\sum_{a=1}^n\<\bfeta_i,\tbdelta_a\>+\|\bfeta_i\|_2^2+o(\delta^2)\nonumber
\, .
$$
Setting variables as per Eq. (\[eq:ExpansionP1\]), we have $$\begin{aligned}
\tphi\left(\tbm_a,\frac{\beta}{d}\sum_{i=1}^d\br_i^{\otimes 2}\right) & = \tm_a^s -\frac{1}{2}b_0 +\frac{1}{2k(k\nu+1)}\left\|\tbfeta_a-\frac{\beta}{d}\sum_{i=1}^d r^*_i \bdelta_i\right\|_2^2
-\frac{\beta}{2dk(k\nu+1)}\sum_{i=1}^d \|\bdelta_i\|_2^2 +o(\delta^2)\,,
$$ where $b_0 = (\beta/d)\sum_{i=1}^d (r^s_i)^2$.
Let $\tbQ = (\beta/d)\sum_{i=1}^d\br_i^{\otimes 2}$ and, as in the previous proof, define the orthogonal decomposition $\tbQ = \tbQ_0 +\tbQ_1+\tbQ_2$, where $\tbQ_0= \bP\tbQ\bP$, $\tbQ_1= \bP\tbQ\bPp+\bPp\tbQ\bP$, $\tbQ_2= \bPp\tbQ\bPp$. Using the representation (\[eq:ExpansionP1\]), we get $$\begin{aligned}
\tbQ_0 & =b_0\bfone_k\bfone_k^{\sT}\, ,\;\;\;\;\;\;\;\;\;\;\; b_0 = \frac{\beta}{d}\sum_{i=1}^d (r^s_i)^2\, ,\\
\tbQ_1 & =\bfone_k\bb_1^{\sT}+\bb_1\bfone_k^{\sT}\, ,\;\;\;\;\;\;
\bb_1 = \frac{\beta}{d}\sum_{i=1}^d r^s_i \bdelta_i\, ,\\
\tbQ_2 & =\frac{\beta}{d}\sum_{i=1}^d \bdelta_i\bdelta_i^{\sT}\, .
$$ For $\bw\in\supp(\tq_0)$, we have $\<\bfone,\bw\>=1$ and therefore $$\begin{aligned}
\tphi(\tbm_a,\tbQ) & = \log\left\{\int e^{\<\tbm,\bw\>-\frac{1}{2} \<\bw,\tbQ\bw\>}\tq_0(\de\bw)\right\}\\
& = \tm_a^s -\frac{1}{2}b_0+\log\left\{\int e^{\<\tbfeta_a-\bb_1,\bw\>-\frac{1}{2} \<\bw,\tbQ_2\bw\>}\tq_0(\de\bw)\right\}\\
& = \tm_a^s -\frac{1}{2}b_0+ \frac{1}{2}\<(\tbfeta_a-\bb_1)(\tbfeta_a-\bb_1)^{\sT}-\tbQ_1,\bS_{\perp}\> +o(\delta^2)\, ,
$$ where, cf. Eq. (\[eq:S\_Formula\]), $$\begin{aligned}
\bS_{\perp} = \int (\bPp\bw)^{\otimes 2}\tq_0(\de\bw) = \frac{1}{k(k\nu+1)} \bPp\, .
$$ Hence, we obtain immediately the claim.
We next transfer the above results on the moment generating functions $\phi$, $\tphi$, to analogous results on the entropy functions $\psi$, $\tpsi$.
\[lemma:PsiExp\] Setting variables as per Eq. (\[eq:ExpansionP1\]), we have $$\begin{aligned}
\psi\left(\br_i,\frac{\beta}{d}\sum_{a=1}^n\tbr_a^{\otimes 2}\right) =& \frac{1}{2}\log(1+ka_0) + \frac{1}{2}k(1+ka_0)(r_i^s)^2-\frac{\beta^2(1+\beta\delta/k+k(m_i^*)^2)}{2d^2k(1+\beta\delta/k)^2}
\left\|\sum_{a=1}^n\tbdelta_a\right\|^2_2\\
&+\frac{1}{2}\left\|\bdelta_i+\frac{\beta m_i^*}{d(1+\beta\delta/k)}\sum_{a=1}^n\tbdelta_a\right\|_2^2
+\frac{\beta}{2d}\sum_{a=1}^n\|\tbdelta_a\|_2^2+o(\delta^2)\, ,\nonumber
$$ where $a_0 = (\beta/d)\sum_{a=1}^n (\tr^s_a)^2$.
By definition $$\begin{aligned}
\psi(\br_i,\bQ) = \max_{m_i^s,\bfeta_i}\big\{km_i^sr_i^s+\<\bfeta_i,\bdelta_i\>- \phi(\bm_i,\bQ)\big\}\, .
$$ Since $\phi(\,\cdot\,,\bQ)$ is strongly convex, the maximum is realized when $\eta_i^s,\bfeta_i = O(\delta)$ and can be computed order-by-order in $\delta$. Hence, substituting (\[eq:PhiFormula\]) we obtain the claim.
\[lemma:TPsiExp\] Setting variables as per Eq. (\[eq:ExpansionP1\]), we have $$\begin{aligned}
\tpsi\left(\tbr_a,\frac{\beta}{d}\sum_{i=1}^d\br_i^{\otimes 2}\right) & = \frac{1}{2}b_0
+\frac{1}{2}k(k\nu+1)\|\tbdelta_a\|^2_2+\frac{\beta}{d}\sum_{i=1}^d r^*_i \<\bdelta_i,\tbdelta_a\>
+\frac{\beta}{2dk(k\nu+1)}\sum_{i=1}^d \|\bdelta_i\|_2^2 +o(\delta^2)\, ,
$$ where $b_0 = (\beta/d)\sum_{i=1}^d (r^s_i)^2$.
By definition $$\begin{aligned}
\tpsi(\tbr_i,\tbQ) = \max_{\tm_i^s,\tbfeta_i}\big\{k\tm_i^s\tr_i^s+\<\tbfeta_i,\tbdelta_i\>- \tphi(\tbm_i,\tbQ)\big\}\, .
$$ The proof is again obtained by maximizing order by order in $\delta$, and using $\tr_a^s = 1/k$.
Setting variables as per Eq. (\[eq:ExpansionP1\]), and introducing the vectors $\br^s = (r_i^s)_{i\le d}\in\reals^d$, $\tbr^s = (\tr_a^s)_{a\le n}\in\reals^n$, we obtain $$\begin{aligned}
\cF_{\sTAP}(\br,\tbr)= &\cF_{\sTAP}^{(s)}(\br^s,\tbr^s) + \cF_{\sTAP}^{(a)}(\bdelta,\tbdelta) +o(\delta^2)\, ,\label{eq:F_SecondOrder}\\
\cF_{\sTAP}^{(s)}(\br^s,\tbr^s)= & \frac{d}{2}\log\Big(1+\frac{\beta \delta}{k}\Big) + \frac{1}{2}k\Big(1+\frac{\beta \delta}{k}\Big)\|\br^s\|_2^2
-k\sqrt{\beta}\<\bfone, \bX\br^s\>\label{eq:SecondSymmetric}\, ,\\
\cF_{\sTAP}^{(a)} (\bdelta,\tbdelta) & = \frac{1}{2}\left(1+\frac{\beta\delta}{k(k\nu+1)}\right)\|\bdelta\|_F^2+\frac{1}{2}\big(\beta+k(k\nu+1)\big) \|\tbdelta\|_F^2-
\frac{\beta^2}{2dk(1+\beta\delta/k)}\left\|\sum_{a\le n}\tbdelta_a\right\|_2^2\nonumber\\
&-\sqrt{\beta}\Tr(\bX\bdelta\tbdelta^{\sT})+\frac{\beta}{d(1+\beta\delta/k)}\sum_{i\le d, a\le n} m_i^*\<\bdelta_i,\tbdelta_a\>\, .
$$
Using the decomposition (\[eq:ExpansionP1\]), we get $$\begin{aligned}
\Tr(\bX\br\tbr^{\sT}) &= k \Tr\big(\bX\br^{s}(\tbr^s)^{\sT}\big) + \Tr(\bX\bdelta\tbdelta^{\sT})\, , \\
\sum_{i\le d, a\le n} \<\br_i,\tbr_a\>^2 & = k^2\sum_{i\le d, a\le n} (r^s_i)^2(\tr_a^s)^2+ 2k \sum_{i\le d, a\le n} (r^s_i\tr_a^s)\<\bdelta_i,\tbdelta_a\> +o(\delta^2)\\
& = k^2\sum_{i\le d, a\le n} (r^s_i)^2(\tr_a^s)^2+ 2 \sum_{i\le d, a\le n} r^s_i\<\bdelta_i,\tbdelta_a\> +o(\delta^2)\, ,
$$ where we used the fact that $\tr^s_a=1/k$. Using these, together with Lemma \[lemma:PsiExp\], \[lemma:TPsiExp\] in Eq. (\[eq:FreeEnergy\_TAP\_TM\]), we get the decomposition (\[eq:F\_SecondOrder\]) where $$\begin{aligned}
\cF_{\sTAP}^{(s)}(\br^s,\tbr^s)= & \frac{d}{2}\log\Big(1+\frac{\beta k}{d}\,\|\tbr^s\|_2^2\Big) + \frac{1}{2}k^2\Big(1+\frac{\beta k}{d}\,\|\tbr^s\|_2^2\Big)\|\br^s\|_2^2
+\frac{1}{2}\beta\delta\|\br^s\|^2_2\nonumber\\
&-k\sqrt{\beta}\Tr(\bX\br^s(\tbr^s)^{\sT})-\frac{\beta k^2}{2d}\|\br^s\|_2^2\|\tbr^s\|_2^2\, ,
$$ Substituting $\tbr^s= \bfone_n/k$, we obtain Eq. (\[eq:SecondSymmetric\]).
Notice that $\cF_{\sTAP}^{(s)}(\br^s,\tbr^s)$ is a positive definite quadratic function in $\br^s$, minimized at $\br^s = \br^*$. Hence, in order to establish the stability of the uninformative stationary point, it is sufficient to check that the quadratic form $\cF_{\sTAP}^{(a)}(\bdelta,\tbdelta)$ is positive definite. The matrix representation of this quadratic form yields $$\begin{aligned}
\Hess = \left[\begin{matrix}
\Big(1+\frac{\delta\beta}{k(k\nu+1)}\Big)\id_d & -\sqrt{\beta}\bX^{\sT}\Big(\id_n-\frac{\beta}{d(k+\delta \beta)}\allone_n\Big)\\
-\sqrt{\beta}\Big(\id_n-\frac{\beta}{d(k+\delta \beta)}\allone_n\Big)\bX &
\big(\beta+k(k\nu+1)\big)\id_n-\frac{\beta^2}{d(k+\delta\beta)}\allone_n
\end{matrix}\right]\, . \label{eq:HessianFinalFormula}
$$ We are left with the task of proving that $\Hess\succ \bzero$ for $\beta< \beta_{\sp}(k,\delta,\nu)$. We will use the following random matrix theory lemma.
\[lemma:RMT\_lemma\] Let $\bu\in\reals^n$, $\bv\in\reals^d$ be vectors with $\|\bu\|_2=\|\bv\|_2=1$, $\gamma,\alpha_{\|}$, $\alpha_{\perp}, \lbar\in\reals$ be numbers, and let $\bPu = \bu\bu^{\sT}$ be the orthogonal projector onto $\bu$, and $\bPup = \id-\bu\bu^{\sT}$ be its orthogonal complement. Denote by $\bZ\in\reals^{n\times d}$ random matrices with $(Z_{ij})_{i\le n,j\le d}\sim\normal(0,1/d)$, with $n/d\to \delta\in(0,\infty)$ as $n\to\infty$, and define the matrix $$\begin{aligned}
\bM = \gamma\bu\bv^{\sT} +\alpha_{\|}\bPu\bZ+\alpha_{\perp} \bPup\bZ\, .\label{eq:M_lemma}
$$ Finally define $\gamma_*^2 \equiv (1+\sqrt{\delta})\alpha_{\perp}^2-\alpha_{\|}^2$, and $$\begin{aligned}
\lambda_*^2\equiv \begin{cases}
\frac{(\gamma^2+\alpha_{\|}^2)(\gamma^2+\alpha_{\|}^2-\alpha_{\perp}^2(1-\delta))}{\gamma^2+\alpha_{\|}^2-\alpha_{\perp}^2}
& \mbox{ if $\gamma^2>\gamma_*^2$,}\\
\alpha^2_{\perp}(1+\sqrt{\delta})^2 & \mbox{ otherwise.}\label{eq:LimitSingValue}
\end{cases}
$$ Then, denoting by $s_{\max}(\bM)$ the largest singular value of $\bM$, we have $\lim_{n\to\infty}s_{\max}(\bM) =\lambda_*$ in probability.
By rotational invariance of $\bZ$, we can and will assume $\bu=\be_1$, and will denote by $\tbZ\in\reals^{(n-1)\times d}$ the matrix containing the last $(n-1)$ rows of $\bZ$. We further let $\bw = \gamma\bv+\alpha_{\|}\bZ^{\sT}\bu$. With these definitions, $$\begin{aligned}
\bM\bM^{\sT}
= \left[\begin{matrix}
\|\bw\|_2^2 & \alpha_{\perp}(\tbZ\bw)^{\sT}\\
\alpha_{\perp}(\tbZ\bw)& \alpha_{\perp}^2\tbZ\tbZ^{\sT}
\end{matrix}\right]\, .
$$ Note that, almost surely, $\lim_{n\to\infty}\lambda_{\max}(\tbZ\tbZ^{\sT}) = (1+\sqrt{\delta})^2$ [@BaiSilverstein], and therefore $\lim\inf_{n\to\infty} s_{\max}(\bM)^2\ge \alpha_{\perp}^2(1+\sqrt{\delta})^2$ almost surely.
Recall that, as long as $s_n^2$ is not an eigenvalue of $\alpha_{\perp}^2\tbZ\tbZ^{\sT}$, we have $$\begin{aligned}
\det(s_n^2\id -\bM\bM^{\sT}) = \det(s_n^2\id -\alpha_{\perp}^2\tbZ\tbZ^{\sT})\, \Big\{s_n^2 - \|\bw\|_2^2-
\alpha_{\perp}^2\<\bw,\tbZ^{\sT}(s_n^2\id-\alpha_{\perp}^2\tbZ\tbZ^{\sT})^{-1}\tbZ\bw\>\Big\}\
$$ It is immediate to see that (unless $\alpha_{\perp}=0$ or $\bv=0$), $s_n^2> \lambda_{\max}(\alpha_{\perp}^2\tbZ\tbZ^{\sT})$ almost surely, and therefore $s_n$ is given by the largest solution of the equation $$\begin{aligned}
s_n^2 = \|\bw\|_2^2+\alpha_{\perp}^2\<\bw,\tbZ^{\sT}(s_n^2\id-\alpha_{\perp}^2\tbZ\tbZ^{\sT})^{-1}\tbZ\bw\>\, .\label{eq:FiniteN}
$$ Note that, almost surely, $\lim_{n\to\infty}\|\bw\|_2^2=\gamma^2+\alpha_{\|}^2 \equiv \tgamma^2$. Further, $\bw$ is independent of $\tbZ$. Hence, by a standard random matrix theory argument [@Guionnet; @BaiSilverstein], for any $s^2>\alpha_{\perp}^2(1+\sqrt{\delta})^2$, the following limits hold almost surely $$\begin{aligned}
\lim_{n\to\infty}\frac{\alpha_{\perp}^2}{\|\bw\|_2^2}\<\bw,\tbZ^{\sT}(s^2\id-\alpha_{\perp}^2\tbZ\tbZ^{\sT})^{-1}\tbZ\bw\> &=
\lim_{n\to\infty}\frac{1}{d}\Tr\Big[\tbZ^{\sT}\big((s^2/\alpha_{\perp}^2)\id-\tbZ\tbZ^{\sT}\big)^{-1}\tbZ\Big] \\
& = -\delta - \frac{s^2\delta}{\alpha_{\perp}^2}\lim_{n\to\infty}\frac{1}{n}\Tr\Big[\big(\tbZ\tbZ^{\sT}-(s^2/\alpha_{\perp}^2)\id\big)^{-1}\Big] \\
& = -\delta -\frac{s^2\delta}{\alpha_{\perp}^2} R\Big(\frac{s^2}{\alpha_{\perp}^2}\Big)\, ,
$$ where $R(t)$ is the Stieltjes transform of the limit eigenvalues distribution of a Wishart matrix, which is given by the Marcenko-Pastur law [@BaiSilverstein] $$\begin{aligned}
R(z) & = \frac{-z-\delta+1+\sqrt{(z+\delta-1)-4\delta z}}{2\delta z}\, .
$$ Recall that $z\mapsto R(z)$ is increasing on $[z_v,\infty)$, $z_c\equiv (1+\sqrt{\delta})^2$, with $R(z_c+u) = R(z_c)-c\sqrt{u}+O(u)$ (for a constant $c>0$) as $u\downarrow 0$, and $R(z) = -1/z+O(1/z^2)$ as $z\to \infty$. We therefore can consider the following asymptotic version of Eq. (\[eq:FiniteN\]): $$\begin{aligned}
&\frac{s^2}{\tgamma^2} =\hR\left(\frac{s^2}{\alpha_{\perp}^2}\right)\, ,\;\;\;\;\;\;\;\;\;\;
\hR(z) = 1-\delta -\delta z\, R(z)\, .\label{eq:FixedPointEigenvalue}
$$ Note that $\hR(z)$ is monotone decreasing on $[z_c,\infty)$ with $\hR(z_c) = (1+\sqrt{\delta})$, $\hR(z_c+u) =\hR(z_c)-c\sqrt{u}+O(u)$, and $\hR(z) = 1+O(1/z)$ as $z\to\infty$. For $\tgamma^2>(1+\sqrt{\delta})\alpha_{\perp}^2$, this equation has a unique solution $s_*^2$ with $s^2/\tgamma^2 <\hR(s^2\alpha_{\perp}^2)$ for $s^2\in [\alpha_{\perp}^2(1+\sqrt{\delta})^2,s_*^2)$ and $s^2/\tgamma^2 >\hR(s^2\alpha_{\perp}^2)$ for $s^2>s_*^2$. Hence, the largest solution $s_n^2$ of (\[eq:FiniteN\]) converges almost surely to $s_*^2$ as $n\to\infty$.
For $\tgamma^2>(1+\sqrt{\delta})\alpha_{\perp}^2$, we have $s^2/\tgamma^2 >\hR(s^2\alpha_{\perp}^2)$ for all $s^2>\alpha_{\perp}^2(1+\sqrt{\delta})^2$ and therefore $\lim\sup_{n\to\infty} s^2_n\le \alpha_{\perp}^2(1+\sqrt{\delta})^2$ almost surely. Since we have a matching lower bound, we conclude that $\lim_{n\to\infty} s^2_n\le \alpha_{\perp}^2(1+\sqrt{\delta})^2$ in this case.
Finally, the expression (\[eq:LimitSingValue\]) follows by solving rewriting Eq. (\[eq:FixedPointEigenvalue\]) as $\hR^{-1}(s^2/\tgamma^2)= s^2/\alpha_{\perp}^2$, whereby the inverse of $\hR$ in $(1,1+\sqrt{\delta})$ is given by $$\begin{aligned}
\hR^{-1}(x) = \frac{x(x+\delta-1)}{x-1}\, .
$$
We next state a general lemma that can be used to check whether a matrix of the form (\[eq:HessianFinalFormula\]) is positive semidefinite.
\[lemma:GeneralBlock\] Let $\bZ\in\reals^{n\times d}$ be random matrices with $(Z_{ij})_{i\le n, j\le d}\sim\normal(0,1/d)$, and $\bu\in\reals^n$, $\bv\in\reals^d$ be unit vectors, with $n/d\to\delta$ as $n\to\infty$. Define the projectors $\bPu = \bu\bu^{\sT}$ and $\bPup =\id-\bu\bu^{\sT}$. For $a,b r,s,\beta,\xi\in\reals$ with $\beta\ge 0$ and $r> s$, let $$\begin{aligned}
\obX & = \xi \,\bu\bv^{\sT} + \bZ\, ,\label{eq:RankOneLemma}\\
\oHess & =\left[\begin{matrix}
a\,\id_d & -\sqrt{\beta}\, \obX^{\sT}(\id_n-b\bPu)\\
-\sqrt{\beta}\, (\id_n-b\bPu) \obX & (r\id_n-s\bPu)
\end{matrix}\right]\, .\label{eq:HGeneral}
$$ Assume that one of the following two conditions holds:
1. $(1-b)^2(1+\xi^2)/(r-s)\ge (1+\sqrt{\delta})/r$ and $$\begin{aligned}
a (r-s) >\beta \frac{(1-b)^2(1+\xi^2)\big[(1-b)^2(1+\xi^2)r-(1-\delta)(r-s)\big]}{(1-b)^2(1+\xi^2)r-(r-s)}\, .
\label{eq:aCond_1}
$$
2. $(1-b)^2(1+\xi^2)/(r-s)< (1+\sqrt{\delta})/r$ and $$\begin{aligned}
a> \frac{\beta}{r}(1+\sqrt{\delta})^2\, . \label{eq:aCond_2}
$$
Then, there exists a constant $\eps>0$ such that, almost surely, $\Hess\succeq \eps\id$ for all $n$ large enough.
Let us first prove that, under the stated conditions, $\Hess\succeq\bzero$. Since $r\id_n-s\bPu\succ \bzero$, we have $\Hess\succ \bzero$ if and only if $$\begin{aligned}
a\id_d\succ \beta\obX^{\sT}(\id-b\bPu) (r-s\bPu)^{-1}(\id-b\bPu)\obX\, . \label{eq:ConditionSchur}
$$ Notice that $$\begin{aligned}
(\id-b\bPu) (r-s\bPu)^{-1}(\id-b\bPu) = \frac{1}{r}\, \bPup+\frac{(1-b)^2}{r-s}\, \bPu\, .
$$ Hence, condition (\[eq:ConditionSchur\]) is equivalent to $a>\lambda_{\max}(\bM^{\sT}\bM) = s_{\max}(\bM)^2$, where $$\begin{aligned}
\bM = \sqrt{\beta}\left[\frac{1-b}{\sqrt{r-s}}\bPu+\frac{1}{\sqrt{r}}\bPup\right]\obX\, .
$$ Note that $\bM$ is of the form of Lemma \[lemma:RMT\_lemma\], with $$\begin{aligned}
\gamma = \sqrt{\frac{\beta\xi^2(1-b)^2}{r-s}}\, ,\;\;\;\;\;\; \alpha_{\|} = \sqrt{\frac{\beta(1-b)^2}{r-s}}\, ,
\;\;\;\;\;\; \alpha_{\perp} = \sqrt{\frac{\beta}{r}}\, .
$$ The claim that $\Hess\succ\bzero$ then follows by using the asymptotic characterization of $s_{\max}(\bM)$ in Lemma \[lemma:RMT\_lemma\].
We next prove that in fact $\Hess\succeq \eps\id$. If the stated conditions hold, there exists $\eps$ small enough such that they hold also after replacing $a$ with $a'=a-\eps$ and $r$ with $r'=r-\eps$. Let us write $\Hess(a,r)$ for the matrix of Eq. (\[eq:HGeneral\]), where we emphasized the dependence on the parameters $a,r$. We have $\Hess(a,r)= \Hess(a',r')+\eps\id$, and hence the thesis follows since $\Hess(a',b')\succeq \bzero$.
In order to apply the last lemma, we will show that, for $\beta<\beta_{\sp}$, the LDA model of Eq. (\[eq:LDAModel\]) is equivalent for our purposes to a simpler model.
\[lemma:Contiguity\] Let $\bX\in\reals^{n\times d}$ be distributed according to the LDA model (\[eq:LDAModel\]) and let $\bR_1\in \reals^{n\times n}$, $\bR_2\in \reals^{d\times d}$ be uniformly random (Haar distributed) orthogonal matrices conditional to $\bR_1\bfone = \bfone$, with $\{\bX,\bR_1,\bR_2\}$ mutually independent. Denote by $\prob_{1,n}$ the law of $\bX_{R}\equiv\bR_1\bX\bR_2$.
Define $\obX = \xi \,\bu\bv^{\sT} + \bZ$ as per Eq. (\[eq:RankOneLemma\]), with $\bu =\bfone_n/\sqrt{n}$, $\bv$ be a vector with i.i.d. entries $v_i\sim\normal(0,1/d)$, independent of $\bZ$, and $\xi = \sqrt{\beta\delta/k}$, and denote by $\prob_{0,n}$ the law of $\obX$.
If $\beta<\beta_{\sp}(k,\nu,\delta)$, then $\prob_{1,n}$ is contiguous to $\prob_{0,n}$.
Recalling that $\bP = \bfone_k\bfone_k^{\sT}/k$, $\bPp=\id_k\bP$, and letting $\bv_0 = \bH\bfone_k/\sqrt{d k}$, we have $$\begin{aligned}
\bX =\xi\, \bu\bv_0^{\sT} + \frac{\sqrt{\beta}}{d}\bW_{\perp}\bH^{\sT}_\perp+\bZ \equiv \xi\, \bu\bv_0^{\sT} +\tbZ\, ,
$$ where $\bW_{\perp} =\bW\bPp$ and $\bH_{\perp} =\bH\bPp$. Since $\bv_0$ is distributes as $\bv$, and independent of $\tbZ$, it is sufficient to prove that the law of $\tbZ_R= \bR_1\tbZ\bR_2$ is contiguous to the law of $\bZ$.
Note that by the law of large numbers, almost surely (see Eq. (\[eq:S\_Formula\])) $$\begin{aligned}
\lim_{n\to\infty}\frac{1}{n}\|\bW_{\perp}\|_{\op}^2 & = \lim_{n\to\infty}\frac{1}{n}\|\bW_{\perp}^{\sT}\bW_{\perp}\|_{\op} =
\left\|\int (\bPp\bw)^{\otimes 2}\tq_0(\de\bw)\right\|_{\op} = \frac{1}{k(k\nu+1)}\, ,\\
\lim_{d\to\infty}\frac{1}{d}\|\bH_{\perp}\|_{\op}^2 & = \lim_{d\to\infty}\frac{1}{d}\|\bH_{\perp}^{\sT}\bH_{\perp}\|_{\op} =1\, .
$$ Hence $$\begin{aligned}
\lim\sup_{n\to\infty}\left\|\frac{\sqrt{\beta}}{d}\bW_{\perp}\bH^{\sT}_\perp\right\|_{\op}\le \sqrt{\frac{\beta\delta}{k(k\nu+1)}} \equiv \sqrt{\beta_{\perp}}\, .\end{aligned}$$ For $\beta<\beta_{\sp}$, we have $\beta_{\perp}<\sqrt{\delta}$, and therefore the rank-$k$ perturbation in $\tbZ$ does not produce an outlier eigenvalue [@benaych2012singular].
In order to prove that the law of $\tbZ_R= \bR_1\tbZ\bR_2$ is contiguous to the law of $\bZ$, note that $\tbZ_R \stackrel{{\rm d}}{=} (\sqrt{\beta}/d)\bR_1\bW_{\perp}\bH_{\perp}\bR_2+\bZ$. Let $\bbQ_{1,n}$ be the law of $\bW_1 = \bR_1\bW_{\perp}$ and $\bbQ_{2,n}$ the law of $\bW_2=\tilde{\bR_1}\bW_{\perp}$, where $\tilde{\bR_1}$ is a uniformly random orthogonal matrix (not Haar distributed). We claim that $\lim_{n\to\infty}\|\bbQ_{1,n}-\bbQ_2\|_{\sTV}=0$. Indeed both $\bbQ_1$ and $\bbQ_1$ are uniform conditional on $\bW^{\sT}\bW/\sqrt{n} = \bQ$ and $\bW^{\sT}\bfone/\sqrt{n} = \bb$. However, the joint laws of $(\bQ,\bb)$ converge in total variation to the same Gaussian limit by the local central limit theorem.
It is therefore sufficient to show that the law of $\tbZ_{RR}= \tilde{\bR_1}\tbZ\bR_2$ is contiguous to the law of $\bZ$. This follows by second moment method and follows exactly as in [@montanari2017limitation].
\[lemma:EigContig\] Let $\obX$ as per Eq. (\[eq:RankOneLemma\]), with $\bu =\bfone_n/\sqrt{n}$, $\bv$ be a vector with i.i.d. entries $v_i\sim\normal(0,1/d)$, independent of $\bZ$, and $\xi = \sqrt{\beta\delta/k}$, and define $$\begin{aligned}
\oHess = \left[\begin{matrix}
\Big(1+\frac{\delta\beta}{k(k\nu+1)}\Big)\id_d & -\sqrt{\beta}\obX^{\sT}\Big(\id_n-\frac{\beta}{d(k+\delta \beta)}\allone_n\Big)\\
-\sqrt{\beta}\Big(\id_n-\frac{\beta}{d(k+\delta \beta)}\allone_n\Big)\obX &
\big(\beta+k(k\nu+1)\big)\id_n-\frac{\beta^2}{d(k+\delta\beta)}\allone_n
\end{matrix}\right]\, .\label{eq:SimplifiedHessian}
$$ If $\beta<\beta_{\sp}(k,\nu,\delta)$, then the law of the eigenvalues of the Hessian $\Hess$ defined in Eq. (\[eq:HessianFinalFormula\]) is contiguous to the law of the eigenvalues of $\oHess$.
Consider the random orthogonal matrix $\bR\in\reals^{(n+d)\times (n+d)}$ $$\begin{aligned}
\bR = \left[
\begin{matrix}
\bR_2^{\sT} & \bzero\\
\bzero& \bR_1
\end{matrix}
\right]
$$ where $\bR_1\in \reals^{n\times n}$, $\bR_2\in \reals^{d\times d}$ be uniformly random (Haar distributed) orthogonal matrices conditional to $\bR_1\bfone = \bfone$. Notice that the eigenvalues of $\Hess$ are the same as the ones of $\bR\Hess\bR^{\sT}$. Further, we have $$\begin{aligned}
\bR\Hess\bR^{\sT} = \left[\begin{matrix}
\Big(1+\frac{\delta\beta}{k(k\nu+1)}\Big)\id_d & -\sqrt{\beta}\bX_R^{\sT}\Big(\id_n-\frac{\beta}{d(k+\delta \beta)}\allone_n\Big)\\
-\sqrt{\beta}\Big(\id_n-\frac{\beta}{d(k+\delta \beta)}\allone_n\Big)\bX_R &
\big(\beta+k(k\nu+1)\big)\id_n-\frac{\beta^2}{d(k+\delta\beta)}\allone_n
\end{matrix}\right] \, ,
$$ where $\bX_R = \bR_{1}\bX\bR_2$ is defined as in the statement of Lemma \[lemma:Contiguity\]. Applying that lemma, we obtain that the law of $\bR\Hess\bR^{\sT}$ is contiguous to the one of $\oHess$, and therefore we obtain the desired contiguity for the laws of eigenvalues.
The next lemma establishes that the simplified Hessian $\oHess$ is positive semidefinite.
Let $\oHess$ be defined as per Eq. (\[eq:SimplifiedHessian\]) where $\obX = \xi \,\bu\bv^{\sT} + \bZ$ with $\bu =\bfone_n/\sqrt{n}$, $\bv$ be a vector with i.i.d. entries $v_i\sim\normal(0,1/d)$, independent of $(Z_{ij})_{i\le n,j\le d}\sim_{i.i.d.}\normal(0,1/d)$, and $\xi = \sqrt{\beta\delta/k}$.
If $\beta<\beta_{\sp}(k,\delta,\nu)$, then there exists $\eps>0$ such that, almost surely, $\oHess\succeq \eps\,\id$ for all $n$ large enough.
The matrix $\obX$ fits the setting of Lemma \[lemma:GeneralBlock\] with $$\begin{aligned}
a &= 1+\frac{\delta \beta}{k(k\nu+1)}\, ,\;\;\;\;\;\;\;\;\;\; b = \frac{\beta\delta}{k+\delta\beta}\, ,\\
r &= \beta+k(k\nu+1)\, ,\;\;\;\;\;\;\;\;\;\; s = \frac{\beta^2\delta}{k+\delta\beta}\, .
$$ The claim follows by checking that condition 2 in Lemma \[lemma:GeneralBlock\] holds. Indeed we have $$\begin{aligned}
A \equiv \frac{(1-b)^2(1+\xi^2)}{r-s} = \frac{1}{\beta+(k\nu+1)(k+\beta\delta)}\, .
$$ Hence $A<(1+\sqrt{\delta}/r)$. Further, setting $q=k(k\nu+1)$, we have $$\begin{aligned}
a-\frac{\beta}{r}(1+\sqrt{\delta})^2 &= 1+\frac{\delta\beta}{q} - \frac{\beta(1+\sqrt{\delta})^2}{\beta+q}\\
&=\frac{1}{\beta+q}\Big(\frac{\delta\beta^2}{q}-2\sqrt{\delta} \beta+q\Big)\\
& = \frac{\delta}{q(\beta+q)}\Big(\beta-\frac{q}{\sqrt{\delta}}\Big)>0\, .
$$ (The last inequality follows since $\beta_{\sp} = q/\sqrt{\delta}$.) This completes the proof.
The proof of Theorem \[thm:StabilityTAP\] follows immediately from the above lemmas. Since the law of the eigenvalues of $\Hess$ is contiguous to the law of the eigenvalues of $\oHess$ (by Lemma \[lemma:EigContig\]), and $\oHess\succeq \eps\id$ with high probability, we have $$\begin{aligned}
\lim_{n\to\infty}\prob(\lambda_{\min}(\Hess)<\eps/2) = 0\, .
$$
TAP free energy: Numerical results
==================================
Damped AMP {#app:Damped}
----------
AMP turns out to converge poorly near the spectral threshold, i.e. for $\beta\approx \beta_{\sp}$. Note that this appears to be an algorithmic problem, rather than a problem related to the free energy approximation. To alleviate this issue, we used damped AMP for our numerical simulations. Damped AMP iterations are as follows $$\begin{aligned}
\bm^{t+1} &=& (1-\gamma) \bm^{t} +\gamma \bX^{\sT}\,\tsF(\tbm^t;\tbQ^t) -\gamma^ 2 \sF(\bm^t;\bQ^t) \bK^t_W\,, \\
\tbm^t &=& (1 - \gamma) \tbm^{t-1} +\gamma \bX\,\sF(\bm^t;\bQ^t) - \gamma^2 \tsF(\tbm^{t-1};\tbQ_{t-1}) \bK_H^t\,, \\
\bQ^{t+1} &=& \frac{1}{d}\sum_{a=1}^n \tsF(\tbm_a^t;\tbQ^t)^{\otimes 2}\, ,\\
\tbQ^t &=& \frac{1}{d} \sum_{i=1}^d \sF(\bm_i^t;\bQ^t)^{\otimes 2}\, .\end{aligned}$$ The matrices $\bK_H^t$ and $\bK_W^t$ are smoothed sum of Jacobian matrices and are computed as $$\begin{aligned}
\bK_H^{t+1}&=& \sum_{i = 1}^{t+1} (1 - \gamma)^{t - i + 1}\sB_t\,, \\
\bK_W^t &=& \sum_{i = 1}^t (1-\gamma)^{t - i}\sC_t\end{aligned}$$ where $$\begin{aligned}
(\sB_t)_{rs} &=& \frac{1}{d}\sum_{i=1}^d\frac{\partial\sF_s}{\partial (\bm^t_i)_{r}}(\bm_i^t;\bQ^t)\,, \\
(\sC_t)_{rs} &=& \frac{1}{d}\sum_{a=1}^n\frac{\partial\tsF_s}{\partial (\tbm^t_i)_{r}}(\tbm_a^t;\tbQ^t)\,. \end{aligned}$$ In these calculations, $\gamma$ is the smoothing parameter that throughout our simulations is fixed to $\gamma = 0.8$.
The specific choice of this damping scheme (and –in particular– the construction of matrices $\bK_H^{t+1}$, $\bK_W^{t+1}$) is dictated by the fact that this specific choice admits a state evolution analysis, analogous to the one holding on the undamped case.
Approximate Message Passing: Numerical results for $k=3$ {#app:AMPk3}
========================================================
![Normalized distances $\Norm(\hbH)$, $\Norm(\hbW)$ of the AMP estimates from the uninformative fixed point. Here $k=3$, $d = 1000$ and $n= d\delta$: each data point corresponds to an average over $400$ random realizations.[]{data-label="fig:AMP_norm_k_3"}](new_k_3_amp_norm-eps-converted-to.pdf){height="5.5in"}
![Empirical fraction of instances such that $\Norm(\hbW)\ge \eps_0=5\cdot 10^{-3}$ (left) or $\Norm(\hbH)\ge \eps_0$ (right), where $\hbW, \hbH$ are the AMP estimates. Here $k=3$, $d=1000$, and for each $(\delta,\beta)$ point on a grid we ran AMP on $400$ random realizations.[]{data-label="fig:AMP_norm_k_3_HM"}](new_k_3_amp_norm_heatmap_fractions-eps-converted-to.pdf){height="2.66in"}
![Binder cumulant for the correlation between AMP estimates $\hbW,\hbH$ and the true weights and topics $\bW, \bH$. Here $k=3$, $d=1000$, $n=d\delta$ and estimates are obtained by averaging over $400$ realizations.[]{data-label="fig:AMP_corrs_k_3"}](new_k_3_amp_binder-eps-converted-to.pdf){height="5.5in"}
![Binder cumulant for the correlation between AMP estimates $\hbW$, $\hbH$ and the true weights and topics $\bW, \bH$. Here $k=3$, $d=1000$ and estimates are obtained by averaging over $400$ realizations.[]{data-label="fig:AMP_corrs_k_3_HM"}](new_k_2_amp_Binder_heatmap-eps-converted-to.pdf){height="2.66in"}
In Figures \[fig:AMP\_norm\_k\_3\] to \[fig:AMP\_corrs\_k\_3\_HM\] we report our numerical results using damped AMP for the case of $k=3$ topics. These simulations are analogous to the one presented in the main text for $k=2$, cf. Section \[sec:TAP\_numerical\].
Figures \[fig:AMP\_norm\_k\_3\] and \[fig:AMP\_norm\_k\_3\_HM\] report results on the normalized distance from the uninformative subspace $\Norm(\hbH)$, $\Norm(\hbW)$. These are consistent with the claim that AMP converges to a fixed point that is significantly distant from this subspace only if $\beta >\beta_{\sBayes}(k,\nu,\delta)=\beta_{\sp}(k,\nu,\delta)$. In Figures \[fig:AMP\_corrs\_k\_3\] and \[fig:AMP\_corrs\_k\_3\_HM\] we present our results on the correlation between the AMP estimates $\hbH$, $\hbW$ and the true factors $\bH$, $\bW$. We measure this correlation through the same Binder parameter introduced in Section \[sec:NMF\_k3\].
Uniqueness of the solution to {#app:Uniqueness}
==============================
In this appendix, we prove that the solution to is unique under the following conjecture
\[conj:normsquared\]
Let $q>0$ and $\bw\in \reals^k$ be a random variable with density $p(\bw) \propto \exp\left\{-q\left\|\bw\right\|_2^2\right\}\tq_0(\bw)$. Then $$\begin{aligned}
\sigma(q)\gamma(q) \leq \frac{2}{q}\end{aligned}$$ where $\sigma(q)$ and $\gamma(q)$ are the standard deviation and skewness of $\left\|\bw\right\|_2^2$.
For a Gaussian random vector $\bz \sim \mathcal N(0, (2q)^{-1}\id_k)$ so that $p(\bz) \propto \exp\left\{-q\left\|\bz\right\|_2^2\right\}$, $$\begin{aligned}
\tilde \sigma(q) \tilde\gamma(q) = \frac{2}{q}\end{aligned}$$ where $\tilde \sigma(q), \tilde\gamma_1(q)$ are the standard deviation and the skewness of $\left\|\bz\right\|_2^2$.
Using the above conjecture, it can be shown that the solution to is unique.
Let $V(q)$ be the variance of $X = \|\bw\|_2^2$, when $\bw$ is distributed with density $p(\bw) \propto \exp\left\{-q\left\|\bw\right\|_2^2\right\}\tq_0(\bw)$. Define $$\begin{aligned}
f(q) = \frac{k\beta\delta}{k-1}\, \left\{\sE\left(\frac{\beta}{1+q};\nu\right) - \frac{1}{k^2}\right\}\, .\end{aligned}$$ Note that using the proof of Lemma , $f(q)$ is non-negative, continuous and monotone increasing for $q>0$. Further, $$\begin{aligned}
f^\prime(q) = \frac{\beta^2\delta}{(k-1)(1+q)^2}V\left(\frac{\beta}{1+q}\right).\end{aligned}$$ Since $f(0) > 0$, if we show that $f^\prime(q)$ is decreasing, then for $q>q^*$ where $q^*$ is the smallest solution to $f(q)=q$, $f^\prime(q) < 1$. This will imply that $f(q) < q$ for $q > q^*$ that proves the uniqueness. We have $$\begin{aligned}
f^{\prime\prime} (q) = \frac{\beta^2\delta}{(k-1)(1+q)^4} \left[-\frac{\beta}{(1+q)^2}V^\prime\left(\frac{\beta}{1+q}\right)(1+q)^2 - 2(1+q)V\left(\frac{\beta}{1+q}\right)\right]\end{aligned}$$ Hence, $f^\prime(q)$ is decreasing if and only if $$\begin{aligned}
- V^\prime\left(\frac{\beta}{1+q}\right) \leq 2\left(\frac{1+q}{\beta}\right)V\left(\frac{\beta}{1+q}\right).\end{aligned}$$ Therefore, it is sufficient to show that for $q>0$, $$\begin{aligned}
\frac{-V^\prime(q)}{V(q)} \leq \frac{2}{q}.\end{aligned}$$ Note that if we let $X=\|\bw\|_2^2$ where $\bw$ is as in Conjecture \[conj:normsquared\], we have $$\begin{aligned}
V(q) = \E (X^2) - (\E X)^2.\end{aligned}$$ Further, $$\begin{aligned}
V^\prime(q) &= -\E X^3 + (\E X)(\E X^2) - 2(\E X)\left[- \E X^2 + (\E X)^2\right]\\
&= -\E X^3 + 3(\E X^2)(\E X) - 2 (\E X)^3.\end{aligned}$$
Hence, $$\begin{aligned}
\frac{-V^\prime(q)}{V(q)} = \frac{ \E (X^3) - 3(\E X^2)(\E X) + 2 (\E X)^3}{- \E X^2 + (\E X)^2} = \sigma(q)\gamma(q) \leq \frac{2}{q}\end{aligned}$$ using Conjecture \[conj:normsquared\]. Therefore, $f(q)$ is concave and has a unique solution in $q \in (0, \infty)$.
[^1]: Department of Electrical Engineering, Stanford University
[^2]: Department of Electrical Engineering and Department of Statistics, Stanford University
|
---
abstract: 'Adoption of innovations, products or online services is commonly interpreted as a spreading process driven to large extent by social influence and conditioned by the needs and capacities of individuals. To model this process one usually introduces behavioural threshold mechanisms, which can give rise to the evolution of global cascades if the system satisfies a set of conditions. However, these models do not address temporal aspects of the emerging cascades, which in real systems may evolve through various pathways ranging from slow to rapid patterns. Here we fill this gap through the analysis and modelling of product adoption in the world’s largest voice over internet service, the social network of Skype. We provide empirical evidence about the heterogeneous distribution of fractional behavioural thresholds, which appears to be independent of the degree of adopting egos. We show that the structure of real-world adoption clusters is radically different from previous theoretical expectations, since vulnerable adoptions—induced by a single adopting neighbour—appear to be important only locally, while spontaneous adopters arriving at a constant rate and the involvement of unconcerned individuals govern the global emergence of social spreading.'
author:
- 'Márton Karsai [^1]'
- Gerardo Iñiguez
- Riivo Kikas
- Kimmo Kaski
- János Kertész
title: |
**Local cascades induced global contagion:\
How heterogeneous thresholds, exogenous effects, and unconcerned behaviour govern online adoption spreading**
---
**keywords:** cascading behaviour, social spreading phenomena, complex contagion, adoption thresholds
Introduction {#introduction .unnumbered}
============
Spreading of opinions, frauds, behavioural patterns, and product adoptions are all examples of social contagion phenomena where collective patterns emerge due to correlated decisions of a large number of individuals. Although these choices are personal, they are not independent but potentially driven by several processes such as social influence [@Centola2010Spread], homophily [@McPherson2001], and information arriving from external sources like news or mass media [@Toole2011Modeling]. Social contagion evolves over networks of interconnected individuals, where links associated with social ties transfer influence between peers [@RevModPhys.81.591]. Several earlier studies aimed to identify the dominant mechanisms at play in social contagion processes [@Rogers2003Diffusion; @Granovetter1978Threshold; @schelling1969models; @Axelrod1997Dissemination]. One key element, termed behavioural threshold by Granovetter [@Granovetter1978Threshold], is defined as *“the number or proportion of others who must make one decision before a given actor does so”*. Following this idea various network models have been introduced [@Watts2002Simple; @Handjani1997Survival; @valente-thresholds-1996; @Watts2007Influentials; @Melnik2013Multistage; @Gomez2010Modeling] to understand the threshold-driven spreading, commonly known as *complex contagion* [@Centola2007Complex]. Although these models are related to a larger set of collective dynamics, they are particularly different from *simple contagion* where the exposure of nodes is driven by independent contagion stimuli [@Barrat2008Dynamical; @Bass1969]. In addition, collective adoption patterns may appear as a consequence of homophilic structural correlations, where connected individuals adopt due to their similar interests and not due to direct social influence. Distinguishing between the effects of social influence and homophily at the individual level remains as a challenge [@Aral2009; @Shalizi2011]. Furthermore, in real social spreading phenomena all these mechanisms are arguably present. However, while in the case of homophily the adoption behaviour is only seemingly correlated, and for simple contagion only the number of exposures matters, in complex contagion the fraction of adopting neighbours relative to the total number of partners determines whether a node adopts or not, capturing the natural mechanisms involved in individuals’ decision makings [@Holt06; @bikhchandani-hirshleifer-welch-92; @Karsai2014Complex]. Due to this additional complexity, threshold models are able to emulate system-wide adoption patterns known as global cascades.
Behavioural cascades are rare but potentially stupendous social spreading phenomena, where collective patterns of exposure emerge as a consequence of small initial perturbations. Some examples are the rapid emergence of political and grass-root movements [@GonzalezBailon2011Dynamics; @BorgeHolthoefer2011Structural; @EllisInformation], fast spreading of information [@Dow2013Anatomy; @Gruhl2004Information; @Banos2013Role; @Watts2007Influentials; @Hale2013Regime; @Leskovec2005Patterns; @Leskovec2007Dynamics; @Goel2012Structure] or behavioural patterns [@Fowler2009Cooperative], etc. The characterisation [@Goel2012Structure; @BorgeHolthoefer2013Cascading; @Hackett2013Cascades; @Gleeson2008Cascades; @Brummitt2011; @GhoshCascadesArxiv2010] and modelling [@Watts2002Simple; @Hurd2013Watts; @Singh2013Thresholdlimited; @Gleeson2007Seed] of such processes have received plenty of attention and provide some basic understanding of the conditions and structure of empirical and synthetic cascades. However, these studies commonly fail in addressing the temporal dynamics of the emerging cascades, which may vary considerably between different cases of social contagion. Moreover, they have not answered why real-world cascades can evolve through various dynamic pathways ranging from slow to rapid patterning, especially in systems where the threshold mechanisms play a role and social phenomena spread globally. Besides the case of rapid cascading mentioned above, an example of the other extreme is the propagation of products in social networks [@Bass1969], where adoption evolves gradually even if it is driven by threshold mechanisms and may cover a large fraction of the total population [@Karsai2014Complex]. This behaviour characterises the adoption of online services such as Facebook, Twitter, LinkedIn and Skype (Fig.\[fig:1\]a), since their yearly maximum relative growth of cumulative adoption [@SocialMedia] (for definition see Material and Methods (MM)) is lower than in the case of rapid cascades as suggested e.g. by the Watts threshold (WT) model.
To fill this gap in the modelling of social diffusion, here we will analyse and model real-world examples of social contagion phenomena. Our aim is to identify the crucial mechanisms necessary to consider in models of complex contagion to match them better with reality, and define a model that incorporates these mechanisms and captures the possible dynamics leading to the emergence of real-world global cascades. We follow the adoption dynamics of the Skype paid service “buy credit” for $89$ months since 2004, which evolves over the social network of one of the largest voice over internet providers in the world. Data includes the time of first payment of each user, an individual and conscious action that tracks adoption behaviour. In addition we follow the “subscription" service over $42$ months since 2008 (for results see Supplementary Information \[SI\]). In contrast to other empirical studies where incomplete knowledge about the underlying social network leads to unavoidable bias [@Karsai2014Complex], we use here the largest connected component of the aggregated free Skype service as the underlying structure, where nodes are Skype users and links confirmed contacts between them. This is a good approximation since it maps all connections in the Skype social network without sampling, and the paid service is only available for individuals already enrolled in the Skype network. The underlying structure is an aggregate from September 2003 to November 2011 (i.e. over $99$ months) and contains roughly 4.4 billion links and 510 million registered users worldwide [@SkypeIPO]. The data is fully anonymised and considers only confirmed connections between users (for more data details see SI).
In what follows we first provide empirical evidence of the distribution of individual adoption thresholds and other structural and dynamical features of a worldwide adoption cluster. We incorporate the observed structural and threshold heterogeneities into a dynamical threshold model where multiple nodes adopting spontaneously (i.e. firstly among their neighbours) are allowed [@Ruan2015]. We find that if the fraction of users who reject to adopt the product is large, the system enters a quenched state where the evolution and structure of the global adoption cluster is very similar to our empirical observations. Model calculations and the analysis of the real social contagion process suggest that the evolving structure of an adoption cluster differs radically from what has been proposed earlier [@Watts2002Simple], since it is triggered by several spontaneous adoptions arriving at a constant rate, while stable adopters who are initially resisting exposure, are actually responsible for the emergence of global social adoption (Fig. \[fig:1\]b and c).
Results {#results .unnumbered}
=======
Social contagion phenomena can be modelled as binary-state processes evolving on networks and driven by threshold mechanisms. In these systems individuals are represented by nodes, each being either in a susceptible (0) or adopter (1) state and influencing each other by transferring information via social ties [@Granovetter1978Threshold]. Nodes are connected in a network with degree distribution $P(k)$ and average degree $z = \langle k \rangle$. In addition, each node has an individual threshold $\phi \in [0, 1]$ drawn from a distribution $P(\phi)$ with average $w = \langle \phi \rangle$. This threshold determines the minimum fraction of exposed neighbours that triggers adoption and captures the resistance of an individual against engaging in spreading behaviour. Once a node reaches its threshold, it switches state from $0$ to $1$ and keeps it until the end of the dynamics. In his seminal paper about threshold dynamics, Watts [@Watts2002Simple] classified nodes into three categories based on their threshold and degree. He identified [*innovator*]{} nodes that spontaneously change state to $1$, thus starting the process. Such nodes have a trivial threshold $\phi=0$. Then there are nodes with threshold $0 < \phi \leq 1/k$, called [*vulnerable*]{}, which need one adopting neighbour before their own adoption. Finally, there are more resilient nodes with threshold $\phi>1/k$, denoted as [*stable*]{}, referring to individuals in need of strong social influence to follow the actions of their acquaintances.
In the WT model [@Watts2002Simple], small perturbations (like the spontaneous adoption of a single seed node) can trigger global cascading patterns. However, their emergence is subject to the so-called [*cascade condition*]{}: the innovator seed has to be linked to a percolating vulnerable cluster, which adopts immediately afterwards and further triggers a global cascade (i.e. a set of adopters larger than a fixed fraction of the finite network). The cascade condition is satisfied if the network is inside a bounded regime in $(w, z)$-space [@Watts2002Simple]. This regime depends on degree and threshold heterogeneities [@Watts2002Simple] and may change its shape if several innovators start the process [@Singh2013Thresholdlimited].
Empirical observations {#empirical-observations .unnumbered}
----------------------
Degree and threshold heterogeneities are indeed present in the social network of Skype. The degree distribution $P(k)$ is well approximated by a lognormal function $P(k) \propto k^{-1} e^{-(\ln k - \mu_D)^2/(2\sigma_D^2)}$ ($k \geq \kmin$) with parameters $\mu_D=1.2$, $\sigma_D=1.39$ and $\kmin=1$ (Fig. \[fig:1\]d), giving an average degree $z = 8.56$ (for goodness of fit see SI). Moreover, at the time of adoption we can measure the threshold $\phi=\Phi_k/k$ of a user by counting the number $\Phi_k$ of its neighbours who have adopted the service earlier. We then group users by degree and calculate the distribution $P(\Phi_k)$ of the integer threshold $\Phi_k$ [@Gleeson2008Cascades] (Fig. \[fig:1\]e). By using the scaling relation $P(\Phi_k, k) = k P(\Phi_k/k)$ all distributions collapse to a master curve well approximated by a lognormal function $P(\phi) \propto \phi^{-1} e^{-(\ln\phi - \mu_T)^2/(2\sigma_T^2)}$, with parameters $\mu_T=-2$ and $\sigma_T=1$ as constrained by the average threshold $w = 0.19$ (see MM and SI). Note that we observe qualitatively the same scaling and lognormal shape of the threshold distribution for another service (see SI). These empirical observations, in addition to the broad degree distribution, provide quantitative evidence about the heterogeneous nature of adoption thresholds.
{width="90.00000%"}
Since we know the complete structure of the online social network, as well as the first time of service usage for all adopters, we can follow the temporal evolution of the adoption dynamics. By counting the number of adopting neighbours of an ego, we identify innovators ($\Phi_k=0$), and vulnerable ($\Phi_k=1$) or stable ($\Phi_k>1$) nodes. The adoption rates for these categories behave rather differently from previous suggestions [@Watts2002Simple] (Fig. \[fig:1\]f). First, there is not only one seed but an increasing fraction of innovators in the system who, after an initial period, adopt approximately at a constant rate. Second, vulnerable nodes adopt approximately with the same rate as innovators suggesting a strong correlation between these types of adoption. Third, the overall adoption process accelerates due to the increasing rate of stable adoptions induced by social influence. At the same time a giant adoption cluster grows and percolates through the whole network (Fig. \[fig:3\]a, main panel). Despite of this expansion dynamics and connected structure of the service adoption cluster, the service reaches less than $6\%$ of the total number of active Skype users over a period of $7$ years [@SkypeIPO]. Therefore we ask whether one can refer to these adoption clusters as cascades. They are not triggered by a small perturbation but induced by several innovators; their evolution is not instantaneous but ranges through several years; and although they involve millions of individuals, they reach only a reduced fraction of the whole network. To answer we incorporate the above mentioned features into a dynamical threshold model [@Ruan2015] with a growing group of innovators and investigate their effect on the evolution of global social adoption. Note that we also perform a null model study to demonstrate, on the system level, that social influence dominates the contagion process, but not homophily (see section S3 of the SI, together with another empirical spreading scenario in S7.1).
Model {#model .unnumbered}
-----
Our modelling framework is an extension to conventional threshold dynamics on networks studied by Watts, Gleeson, Singh, and others, where all nodes are initially susceptible and innovators are only introduced as an initial seed of arbitrary size [@Watts2002Simple; @Gleeson2007Seed; @Singh2013Thresholdlimited]. Apart from the threshold rule discussed above, our model considers two additional features: (i) a fraction $r$ of ‘immune’ nodes that never adopt, indicating lack of interest in the service; (ii) due to external influence, susceptible nodes adopt the innovation spontaneously (i.e. become innovators) throughout time with constant rate $p_n$, rather than only at the beginning of the dynamics. In this way, the dynamical evolution of the system is completely defined by the online social network, the distribution $P(\phi)$ and the parameters $r$, $p_n$. For the sake of simplicity we consider a configuration-model network, i.e., we ignore correlations in the social network and characterise it solely by its degree distribution $P(k)$. Furthermore, node degrees and thresholds are considered to be independent [@Gleeson2008Cascades; @gleeson2013binary; @gleeson2011high].
Our threshold model, which has also been introduced in [@Ruan2015], can be studied analytically by extending the framework of approximate master equations (AMEs) for monotone binary-state dynamics recently developed by Gleeson [@Gleeson2008Cascades; @gleeson2013binary; @gleeson2011high], where the transition rate between susceptible and adoption states only depends on the number $m$ of network neighbours that have already adopted. We describe a node by the property vector $\kvec = (k, c)$, where $k = k_0, k_1, \ldots k_{M-1}$ is its degree and $c = 0, 1, \ldots, M$ its type, i.e. $c = 0$ is the type of the fraction $r$ of immune nodes, while $c \neq 0$ is the type of all non-immune nodes that have threshold $\phi_c$. In this way $P(\phi)$ is substituted by the discrete distribution of types $P(c)$ (for $c > 0$). The integer $M$ is the maximum number of degrees (or non-zero types) considered in the AME framework, which can be increased to improve the accuracy of the analytical approximation at the expense of speed in its numerical computation (see S4.2). Under these conditions, the AME system describing the dynamics of the threshold model is reduced to the pair of ordinary differential equations (see SI),
\[eq:reducedAMEs\] $$\begin{aligned}
\dot{\rho} &= h(\nu, t) - \rho, \\
\dot{\nu} &= g(\nu, t) - \nu,\end{aligned}$$
where $\rho(t)$ is the fraction of adopters in the network, $\nu(t)$ is the probability that a randomly chosen neighbour of a susceptible node is an adopter, and the initial conditions are $\rho(0) = \nu(0) = 0$. Here, $$\label{eq:hTerm}
h = (1 - r) \Big[ \ft + (1 - \ft) \sum_{\kvec | c \neq 0} P(k) P(c) \sum_{m \geq k\phi_c} \Bkm(\nu) \Big],$$ and, $$\label{eq:gTerm}
g = (1 - r) \Big[ \ft + (1 - \ft) \sum_{\kvec | c \neq 0} \frac{k}{z} P(k) P(c) \sum_{m \geq k\phi_c} \Bkom(\nu) \Big],$$ where $\ft = 1 - (1 - \pr) e^{-\pr t}$, $\pr = p_n / (1 - r)$, and $\Bkm(\nu) = \binom{k}{m} \nu^m (1 - \nu)^{k - m}$ is the binomial distribution. The fraction of adopters $\rho$ is then obtained by solving Eq. (\[eq:reducedAMEs\]) numerically. Since susceptible nodes adopt spontaneously with rate $p_n$, the fraction of innovators $\rho_0(t)$ in the network is given by (see S4.3), $$\label{eq:innovFrac}
\rho_0 = \pr \int_0^t (1 - r - \rho) dt.$$
![[**Threshold model for the adoption of online services.**]{} [**(a-b)**]{} Surface plot of the normalised fraction of adopters $\rho / (1 - r)$ in $(w, z)$-space, for $r = 0.73$ and $t = 89$. Contour lines signal parameter values for which $20\%$ of non-immune nodes have adopted, for fixed $r$ and varying time (a), and for fixed time and varying $r$ (b). The continuous contour line and dot indicate parameter values in the last observation of Skype s3. A regime of maximal adoption ($\rho \approx 1 - r$) grows as time goes by, and shrinks for larger $r$. [**(c)**]{} Time series of the fraction of adopters $\rho$ for fixed $p_n = 0.00019$ and varying $r$ (main), and for fixed $r = 0$ and varying $p_n$ (inset). These curves are well approximated by the solution of Eq. (\[eq:reducedAMEs\]) for $k_0 = 3$, $k_{M-1} = 150$ and $M = 25$ (dashed lines). The dynamics is clearly faster for larger $p_n$. As $r$ increases, the system enters a regime where the dynamics is slowed down and adopters are mostly innovators. [**(d)**]{} Final fraction of innovators $\rho_0(\infty)$ and time $t_c$ when $50\%$ of non-immune nodes have adopted as a function of $r$, both simulated and theoretical. The crossover to a regime of slow adoption is characterised by a maximal fraction of innovators and time $t_c$. Unless otherwise stated, $p_n=0.00019$ and we use $N=10^4$, $\mu_D=1.09$, $\sigma_D=1.39$, $\kmin=1$, $\mu_T=-2$, and $\sigma_T=1$ to obtain $z = 8.56$ and $w = 0.19$ as in Skype s3. The difference in $\mu_D$ between data and model is due to finite-size effects (see Materials and Methods). Numerical results are averages over $10^2$ (a-b) and $10^3$ (c-d) realisations. \[fig:2\]](Fig2.pdf){width="60.00000%"}
We also implement the threshold model numerically via a Monte Carlo simulation in a network of size $N$, with a lognormal degree distribution and a lognormal threshold distribution as observed empirically. Thus, we can explore the behaviour of $\rho$ and $\rho_0$ as a function of $z$, $w$, $p_n$ and $r$, both in the numerical simulation and in the theoretical approximation given by Eqs. (\[eq:reducedAMEs\]) and (\[eq:innovFrac\]). For $p_n > 0$ some nodes adopt spontaneously as time passes by, leading to a frozen state characterised by a final fraction $\rho(\infty) = 1 - r$ of adopters. However, the time needed to reach such state depends heavily on the distribution of degrees and thresholds, as signalled by a region of large adoption ($\rho \approx 1 - r$) that grows in $(w, z)$-space with time (contour lines in Fig. \[fig:2\]a). If we fix a time in the dynamics and vary the fraction of immune nodes instead, this region shrinks as $r$ increases (contour lines in Fig. \[fig:2\]b). In other words, the set of networks (defined by their average degree and threshold) that allow the spread of adoption is larger at later times in the dynamics, or when the fraction of immune nodes is small. When both $t$ and $r$ are fixed, the normalised fraction of adopters $\rho / (1 - r)$ gradually decreases for less connected networks with larger thresholds (surface plot in Fig. \[fig:2\]a and b).
For $r \approx 0$ the critical fraction of innovators necessary to trigger a cascade of fast adoption throughout all susceptible nodes may be identified as the inflection point in the time series of $\rho$ (Fig. \[fig:2\]c, inset). The adoption cascade appears sooner for larger $p_n$, since this parameter regulates how quickly the critical fraction of innovators is reached. Yet as we increase $r$ above a threshold $r_c$, the system enters a regime where rapid cascades disappear and adoption is slowed down. The crossover between these regimes is gradual, as seen in the shape of $\rho$ for increasing $r$ (Fig. \[fig:2\]c, main panel). We may identify $r_c$ in various ways: by the maximum in both the final fraction of innovators $\rho_0(\infty)$ and the critical time $t_c$ when $\rho = (1-r)/2$ (Fig. \[fig:2\]d), or as the $r$ value where the inflection point in $\rho$ disappears. These measures indicate $r_c \approx 0.8$ for the chosen parameters. All global properties of the dynamics (like the functional dependence of $\rho$ and $\rho_0$) are very well approximated by the solution of Eqs. (\[eq:reducedAMEs\]) and (\[eq:innovFrac\]) (dashed lines in Fig. \[fig:2\]c and d). Indeed, the AME framework is able to capture the shape of the $\rho$ time series, the crossover between regimes of fast and slow adoption, as well as the maximum in $\rho_0(\infty)$ and $t_c$.
Validation {#validation .unnumbered}
----------
To better understand how innovation spreads throughout real social networks, we take a closer look at the internal structure of the service adoption process. By taking into account individual adoption times we construct an evolving adoption network with links between users who have adopted the service before time $t$ and are connected in the social structure. In order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only consider links between nodes who are neighbours in the underlying social network and whose adoption did not happen at the same time. This way links in the adoption graph indicate ties where social influence among individuals could have existed. The size distribution $P(s_a)$ of connected components in the adoption network shows the emergence of a giant percolating component over time (Fig. \[fig:3\]a), along with several other small clusters. Moreover, after decomposition we observe that the giant cluster does not consist of a single innovator seed and percolating vulnerable tree [@Watts2002Simple], but builds up from several innovator seeds that induce small vulnerable trees locally (Fig. \[fig:3\]b), each with small depth (Fig. \[fig:3\]d) [@Bakshy11; @Goel2012Structure]. At the same time the stable adoption network (considering connections between all stable adopters at the time) has a giant connected component, indicating the emergence of a percolating stable cluster with size comparable to the largest adoption cluster (Fig. \[fig:3\]a, inset). These observations suggest a scenario for the evolution of the global adoption component different from earlier threshold models [@Watts2002Simple]. It appears that here multiple innovators adopt at different times and trigger local vulnerable trees (Fig.\[fig:1\]b), which in turn induce a percolating component of connected stable nodes that holds the global adoption cluster together (Fig.\[fig:1\]c). Consequently, in the structure of the adoption network primary triggering effects are important only locally, while external and secondary triggering mechanisms seem to be responsible for the emergence of global-scale adoption.
![**Empirical cluster statistics and simulation results.** **(a)** Empirical connected-component size distribution at different times for the adoption \[$P(s_a)$, main panel\] and stable adoption \[$P(s_s)$, inset\] networks, with $s_a$ and $s_s$ relative to system size. **(b)** Empirical connected-component size distribution $P(s_v)$ for the relative size of innovator-induced vulnerable trees at different times. **(c)** Average size of the largest ($LC$) and 2nd largest ($LC^{2nd}$) components of the model network (‘Net’), adoption network (‘Casc’), stable network (‘Stab’), and induced vulnerable trees (‘Vuln’) as a function of $r$. Dashed lines show the observed relative size of the real $LC$ of the adopter network in $2011$ \[see main panel in (a)\] and the predicted $r$ value. **(d)** Distribution $P(d)$ of depths of induced vulnerable trees in both data and model for several $r$ values, showing a good fit with the data for $r=0.73$. The difference in the tail is due to finite-size effects. **(e)** Correlation $\langle s_v \rangle (k)$ between innovator degree and average size of vulnerable trees in both data and model with the same $r$ values as in (e). Model calculations for (d) and (e) correspond to networks of size $N=10^6$ and are averaged over $10^2$ realisations. \[fig:3\]](Fig3.pdf){width="60.00000%"}
To model the observed dynamics and explore the effect of immune nodes, we perform extensive numerical simulations of the threshold model with parameters determined directly from the data (see MM and SI). We use a network structure with empirical degree and threshold distributions and fix $p_n=0.00019$ as the constant rate of innovators, implying that the time scale of a Monte Carlo iteration in the model is 1 month. We measure the average size of the largest ($LC$) and second largest ($LC^{2nd}$) connected components of the background social network, and of the stable, vulnerable and global adoption networks, as a function of the fraction of immune nodes $r$. After $T=89$ iterations (matching the length of the real observation period) we identify three regimes of the dynamics (Fig. \[fig:3\]c): if $0<r<0.6$ (dark-shaded area) the spreading process is very rapid and evolves in a global cascade, which reaches most of the nodes of the shrinking susceptible network in a few iteration steps. About $10\%$ of adopters are connected in a percolating stable cluster, while vulnerable components remain very small in accordance with empirical observations. In the crossover regime $0.6<r<0.8$ (light-shaded area), the adoption process slows down considerably (Fig. \[fig:2\]d, lower panel), as stable adoptions become less likely due to the quenching effect of immune nodes. The adoption process becomes the slowest at $r_c=0.8$ (Fig. \[fig:2\]d, lower panel) when the percolating stable cluster falls apart, as demonstrated by a peak in the corresponding $LC^{2nd}$ curve (Fig. \[fig:3\]c, lower panel). Finally, around $r=0.9$ the adoption network becomes fragmented and no global diffusion takes place. We repeat the same calculations for another service and find qualitatively the same picture, but with the crossover regime shifted towards larger $r$ values due to the different parametrisation of the model process. Note that another possible reason for the slow adoption could be the time users wait between their threshold has been reached and actual adoption. We test for the effect of this potential scenario on the empirical curves but find no qualitative change in the dynamics (see SI).
We can use these calculations to estimate the only unknown parameter (the fraction $r$ of immune nodes in Skype) by matching the size of the largest component ($LC_{Net}$) between real and model adoption networks at time $T$. Empirically, this value is the relative size corresponding to the last point on the right-hand side of the distribution for $2011$ (Fig. \[fig:3\]a, main panel). The corresponding value in the model is $r = 0.73$ (dashed lines in Fig. \[fig:3\]c; also Fig. \[fig:2\]a and b), suggesting that the real adoption process lies in the crossover regime. The other analysed service turns out to lie right of the crossover regime, which explains its large innovator adoption rate and reduced size of stable and vulnerable adoption clusters (see SI).
To test the validity of the prediction of $r$ we perform three different calculations. First we measure the maximum relative growth rate of cumulative adoption and find a good match between model and data (Skype s3 and Model Skype s3 in Fig. \[fig:1\]a). In other words, the model correctly estimates the speed of the adoption process. Second, we measure the distribution $P(d)$ of depths of induced vulnerable trees (Fig. \[fig:3\]d). Finally, in order to verify earlier theoretical suggestions [@Singh2013Thresholdlimited], we look at the correlation $\langle s_v \rangle (k)$ between the degree of innovators and the average size of vulnerable trees induced by them (Fig. \[fig:3\]e). We perform the last two measurements on the real data and in the model process for $r=0.6$ and $0.9$, as well as for the predicted value $r=0.73$. In the case of $\langle s_v \rangle (k)$, we find a strong positive correlation in the data, explained partially by degree heterogeneities in the underlying social network, but surprisingly well emulated by the model. More importantly, although both quantities appear to scale with $r$, measures for the estimated $r$ value fit the empirical data remarkably well, confirming our earlier validation based on the matching of relative component sizes (for further discussion see SI).
Discussion {#discussion .unnumbered}
==========
Although some products and innovations diffusing in society may cover a large fraction of the population, their spreading tends to follow slow cascading patterns, the dynamics of which have been modelled before by simple diffusion models like that of Bass [@Bass1969]. However, this approach neglects threshold mechanisms that arguably drive the decision making of single individuals. On the other hand, threshold models study the conditions for cascades in global diffusion but do not address their temporal evolution, which is clearly a relevant factor in real-world adoption processes. These models are commonly used to predict rapid cascading patterns of adoption, which is a more realistic scenario for the spreading of information, opinions, or behavioural patterns but are not observed in the case of product or innovation diffusion where adoption requires additional efforts, e.g., free or paid registration. Here we provide a solution for this conundrum by analysing and modelling the worldwide spread of an online service in the techno-social communication network of Skype. Beyond the novel empirical evidence about heterogeneous adoption thresholds and non-linear dynamics of the adoption process, we identify two additional components necessary to introduce in the modelling of product adoption, namely: (a) a constant flow of innovators, which may induce rapid adoption cascades even if the system is initially out of the cascading regime; and (b) a fraction of immune nodes that forces the system into a quenched state where adoption slows down. These features are responsible for a critical structure of empirical adoption components that radically differs from previous theoretical expectations. We incorporate these mechanisms into a threshold model controlled by the rate of innovators and the fraction of immune nodes. The model is able to reproduce several pathways ranging from cascading behaviour to more realistic dynamics of innovation adoption. By constraining the model with empirically determined parameters, we provide an estimate for the real fraction of susceptible agents in the social network of Skype, and validate this prediction through correlated structural features matching empirical observations.
Our aim in this study was to provide empirical observations as well as methods and tools to model the dynamics of social contagion phenomena with the hope it will foster thoughts for future research. One possible direction would be the observation of the reported structure and evolution of the global adoption cluster in other systems similar to the ones studied in [@BorgeHolthoefer2011Structural; @Dow2013Anatomy; @Gruhl2004Information; @Goel2012Structure; @BorgeHolthoefer2013Cascading; @Bakshy11]. Other promising directions could be the consideration of homophilic or assortative structural correlations, the evolving nature of the underpinning social structure as studied in [@Karsai2014Complex], interpersonal influence, or the effects of leader-follower mechanisms on the social contagion process. Finally, we hope that the reported results may improve efficiency in the strategies of enhancing the diffusion of products and innovations, by shifting attention from the creation of short-lived perturbations to the sustenance of external input.
### Competing interest statement {#competing-interest-statement .unnumbered}
The authors have no competing interests.
### Authors’ contributions statement {#authors-contributions-statement .unnumbered}
M.K., G.I., R.K., K.K, and J.K designed the research and participated in writing the manuscript. R.K. and M.K. analysed the empirical data, G.I. made the analytical calculations, and M.K. and G.I. performed the numerical simulations.
### Acknowledgements {#acknowledgements .unnumbered}
The authors gratefully acknowledge the support of M. Dumas, A. Saabas, and A. Dumitras from STACC and Microsoft/Skype Labs as well as constructive comments by J. Saramäki and T. N[ä]{}si.
### Funding statement {#funding-statement .unnumbered}
G.I. acknowledges the Academy of Finland, and J.K. the CIMPLEX FET Open H2020 EU project for support. This research was partly funded by Microsoft/Skype Labs.
Material and Methods {#material-and-methods .unnumbered}
====================
Data description {#data-description .unnumbered}
================
We use a static representation of the Skype social network aggregated over 99 months between September 2003 and November 2011. We follow the adoption of the “buy credit” paid service for $89$ months starting from 2004, and the paid service “subscription” for $42$ months starting from 2008 (for further details about the network and service see SI). By considering the online social structure and adoption times, we identify users as innovator, vulnerable, or stable nodes based on the number $\Phi_k$ of adopting neighbours at the time of exposure. Thresholds are calculated as $\phi=\Phi_k/k$ for users with $k$ contacts. The adoption network is constructed by considering confirmed social links between users who adopted the service earlier than $t$. In order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only consider links between nodes who are neighbours in the underlying social network and whose adoption did not happen at the same time. Note that for the categorisation of nodes we use only the adoption time and the state of their peers, and thus real categories may differ slightly. For example, an innovator may appear as a vulnerable or stable node, even if its decision was not driven by social influence but some of its peers adopted earlier. To consider this bias we measure effective rates of adoption for the model process as well, just like for the empirical case (Fig.\[fig:1\]) and section S3.
Maximum relative growth rate {#maximum-relative-growth-rate .unnumbered}
============================
This measure is obtained by taking the maximum of the yearly adoption rate (yearly count of adoptions) normalised by the final observed adoption number of a given service. It characterises the maximum speed of adoption a service experienced during its history and takes values between 0 (no cascade) and 1 (instantaneous cascade). We repeat this measurement for the estimated number of registered users of Facebook, Twitter, and LinkedIn [@SocialMedia], as well as for the number of active users of Skype and three paid Skype services. Adoption rates for Facebook, Twitter, and LinkedIn correspond to the period between 2006 and 2012, and for Skype and its services to the interval from release date until 2011.
Empirical parameter estimation {#empirical-parameter-estimation .unnumbered}
==============================
We use the Skype data to directly determine all model parameters, apart from the fraction $r$ of immune nodes. To best approximate the degree distribution of the real network, after testing different candidate functions (see SI) we select a lognormal function $P(k) = e^{ -(\ln k-\mu_D)^2 / (2\sigma_D^2) } / (k\sigma_D\sqrt{2\pi})$ with parameters $\mu_D=1.2$ and $\sigma_D=1.39$ and minimum degree $\kmin = 1$, leading to the average degree $z = 8.56$. To account for finite-size effects in the model results for low $N$ (Fig. \[fig:2\]), we decrease $\mu_D$ slightly to obtain the same value of $z$ as in the real network.
The threshold distribution of each degree group collapses to a master curve after normalisation by using the scaling relation $P(\Phi_k,k)=k P (\Phi_k/k)$. This master curve can be well approximated by the lognormal distribution $P(\phi) = e^{ -(\ln \phi-\mu_T)^2 / (2\sigma_T^2) } / (\phi\sigma_T\sqrt{2\pi})$, with parameters $\mu_T=-2$ and $\sigma_T=1$ as determined by the empirical average threshold $w = 0.19$ and standard deviation $0.233$ (for further details see SI). We estimate a rate of innovators $p_n = 0.00019$ by fitting a constant function to $R_i(t)$ for $t > 2\tau$ (Fig. \[fig:1\]f). The fit to $\pn$ also matches the time scale of a Monte Carlo iteration in the model to 1 month. Model results (Fig. \[fig:3\]d and e) are calculated with $r = 0.73$ and $p_n = 0.00019$. Simulation results in Fig. \[fig:3\]c (d and e) are averaged over $100$ configuration-model networks of size $N=10^5$ ($10^6$) after $T=89$ iterations, matching the length of the observation period in Skype.
Model description {#model-description .unnumbered}
=================
We characterise the static social network by the extended distribution $\Pk$, where $\Pk = r P(k)$ for $c = 0$ and $\Pk = (1 - r) P(k) P(c)$ for $c > 0$. Non-immune, susceptible nodes with property vector $\kvec$ adopt spontaneously at a constant rate $\pn$, else they adopt only if a fraction $\phi_c$ of their $k$ neighbours have adopted before. These rules are condensed in the probability $\Fkm dt$ that a node will adopt in a small time interval $dt$, given that $m$ of its neighbours are already adopters, $$\label{eq:thresRule}
\Fkm =
\begin{cases}
\pr & \text{if} \quad m < k \phi_c \\
1 & \text{if} \quad m \geq k \phi_c
\end{cases}, \quad \forall m \; \text{and} \; k, c \neq 0,$$ with $F_{(k,0),m} = 0$ $\forall k, m$ and $F_{(0,c),0} = \pr$ $\forall c \neq 0$ (for immune and isolated nodes, respectively). The dynamics of adoption is well described by an AME for the fraction $\skm(t)$ of $\kvec$-nodes that are susceptible at time $t$ and have $m=0,\ldots,k$ adopting neighbours [@porter2014; @gleeson2013binary; @gleeson2011high], $$\label{eq:AMEsThres}
\dskm = -\Fkm \skm -\bs (k - m) \skm + \bs (k - m + 1) \skmo,$$ where $\bs(t) = \frac{\sumk \Pk \summ (k - m) \Fkm \skm(t)}{\sumk \Pk \summ (k - m) \skm(t)}$. To reduce the dimensionality of Eq. (\[eq:AMEsThres\]) we consider the ansatz $\skm = \Bkm (\nu) e^{-\pr t}$ for $m < k\phi_c$, leading to the condition $\dot{\nu} = \bs (1 - \nu)$. With $\rho = 1 - \sumk \Pk \summ \skm$ and some algebra, this condition is reduced to Eq. (\[eq:reducedAMEs\]) (see SI).
[10]{}
Centola, D. . , 1194–1197 (2010).
McPherson, M., Smith-Lovin, L. & Cook, JM. . , 415-444 (2001).
Toole. J. L., Cha. M. & Gonz[á]{}lez, M. C. . , e29528 (2012).
Castellano, C., Fortunato, S. & Loreto, V. . , 591–646 (2009).
Rogers, E. M. . (Simon & Schuster), 5th edition (2003).
Granovetter, M. . , 1420–1443 (1978).
Schelling, T. C. . , 488–493 (1969).
Axelrod, R. . , 203–226 (1997).
Watts, D. J. . , 5766–5771 (2002).
Handjani, S. . , 737–746 (1997).
Valente, T. W. . , 69–89 (1996).
Watts, D. J. & Dodds, P. S. . , 441–458 (2007).
Melnik, S., Ward, J. A., Gleeson, J. P. & Porter, M. A. . , 013124 (2013).
Gómez, V., Kappen, H. J., & Kaltenbrunner, A. . (HT’11, ACM, New York, NY, USA), pp. 181–190 (2010).
Centola, D. & Macy, M. . , 702–734 (2007).
Barrat A., Barthélemy, M. & Vespignani, V. . (Cambridge University Press, 2008).
Bass, F. M. . , 215–227 (1969).
Aral, S., Muchnika, L., Sundararajana, A. . , 21544–21549 (2009).
Shalizi, C. R., Thomas, A. C. . , 211–239 (2011).
Holt, C. A. . (Addison Wesley, 2006).
Bikhchandani, S., Hirshleifer D. & Welch, I. . , 992–1026 (1992).
Karsai, M., Iñiguez, G., Kaski, K. & Kertész, J. . , 20140694 (2014).
Ruan, Z., Iñiguez, G., Karsai, M. & Kertész, J. . , 218702 (2015).
González-Bailón, S., Borge-Holthoefer, J., Rivero, A. & Moreno, Y. . , 197 (2011).
Borge-Holthoefer, J., et. al. . , e23883 (2011).
Ellis, C. J. & Fender, J. . , 763–792 (2011).
Dow, P. A., Adamic, L. A. & Friggeri, A. . (ICWSM, AAAI, Boston, MA, USA), pp. 145–154 (2013).
Gruhl, D., Guha, R., Nowell, D. L. & Tomkins, A. . (WWW ’04, ACM, New York, NY, USA), pp. 491–501 (2004).
Baños, R. A., Borge-Holthoefer, J. & Moreno, Y. . , 6 (2013).
Hale, H. E. . , 331–353 (2013).
Leskovec, J., Singh, A. & Kleinberg, J. . (PAKDD ’06, Singapore), pp. 380–389 (2006).
Leskovec, J., Adamic, L. A. & Huberman, B. A. . (TWEB, ACM, New York, NY, USA), vol. 1, pp. 5 (2007).
Goel, S., Watts, D. J. & Goldstein, D. G. . (EC ’12, ACM, New York, NY, USA), pp. 623–638 (2012).
Fowler, J. H. & Christakis, N. A. . , 5334–5338 (2009).
Borge-Holthoefer, J., Baños, R. A., González-Bailón, S. & Moreno, Y. . , 1–22 (2013).
Hackett, A. & Gleeson, J. P. . , 062801 (2013).
Gleeson, J. P. . , 046117 (2008).
Brummitt, C. D., D’Souza, R. M. & E. A. Leicht, . , E680–E689 (2011).
Ghosh, R. &Lerman, K. , WSDM ’11. (WSDM ’11, ACM, New York, NY, USA), pp. 665–674 (2010).
Hurd, T. R. & Gleeson, J. P. . , 25-43 (2013).
Singh, P., Sreenivasan, S., Szymanski, B. K. & Korniss, Gy. . , 2330 (2013).
Gleeson, J. P. & Cahalane, D. J. . , 050101(R) (2007).
White, D. S. (2013). Date of access: 2015.01.29.
Morrissey, R. C., Goldman, N. D. & Kennedy, K. P. (2011). Date of access: 2014.10.14.
Porter, M. A. & Gleeson, J. P. .
Gleeson, J. P. . , 021004 (2013).
Gleeson, J. P. . , 068701 (2011).
Bakshy, E., Hofman, J. M., Mason, W. A. & Watts, D. J. . (WSDM ’11, ACM, New York, NY, USA), pp. 65–74 (2011).
{#section .unnumbered}
Detailed data description {#sec:ddescr}
=========================
This study has been conducted on a dataset of the social network of Skype. The centrepiece of the dataset is the *contact network*, where nodes represent users and edges between pairs of users exist if they are in each other’s contact lists. A user’s contact list is composed of *friends*. If user $u$ wants to add another user $v$ to his/her contact list, $u$ sends $v$ a contact request, and the edge is established at the moment $v$ approves the request (or not, if the contact request is rejected). Each edge is labelled with a time stamp indicating the moment the contact request was approved. As the underpinning social structure we consider the static representation of the Skype social network, aggregated for $99$ months between September 2003 and November 2011. The largest connected component of this structure includes roughly 510 million users and 4.4 billion edges.
As the chosen service evolving on the Skype network, we follow how users purchase “credits” for calling phones. For each user, the dataset includes the date when he/she first adopted the paid product “buy credit” (first credit purchase, for all purposes). We select this service since its lifetime of $89$ months is considerably long (it was introduced in 2004), and it can be adopted by registered Skype users only. This way the aggregated Skype network provides a complete description of the mediating social structure, which allows us to calculate the correct degree and adoption threshold for all individuals. To make additional observations and to further test our model, we perform calculations on a second paid service called “subscription”, which was introduced in April 2008, lasts for over $42$ months, and can also be adopted by registered Skype users only. Results regarding this service are presented in Section \[sec:addserv\].
By considering the online social structure and the adoption times we identify users as innovator, vulnerable, or stable nodes based on the number $\Phi_k$ of adopting neighbours at the time of exposure. Thresholds are calculated as $\phi=\Phi_k/k$ for users with $k$ contacts. The adoption network is constructed by considering confirmed social links between users who adopted the service earlier than the time of observation $t$. In order to avoid the effect of instantaneous group adoptions (evidently not driven by social influence), we only consider links between nodes who are neighbours in the underlying social network and whose adoption did not happen at the same time. Note that for the categorization of nodes we use only the adoption time and the state of their peers, and thus ‘real’ categories may differ slightly. For example, an innovator may appear as a vulnerable or stable node, even if its decision was not driven by social influence but some of its peers adopted earlier. To consider this bias we also measure ‘effective’ rates of adoption for the model process, just like for the empirical case (Fig.1, main text) and section S3.
The dataset does not include identity information. All usernames are anonymized and there is no way of inferring a user’s identity solely from the profile. The dataset does not contain any information about interpersonal interactions, apart from the contact list.
Empirical determination of model parameters {#sec:pars}
===========================================
Parameters in the model are the rate of innovators $p_{n}$, the degree distribution $P(k)$, the threshold distribution $P(\phi)$, and the fraction of immune nodes $r$. Other than $r$, all of them can be estimated from the data as follows.
Rate of innovators
------------------
As discussed in the main text, the rate of spontaneous adoption saturates approximately to a constant value after an initial transition period, which allows us to determine the rate of innovators by fitting a constant function on the curve after time $2\tau$. We estimate this rate to be $p_{n} = 0.00019$, as demonstrated in Fig. S\[fig:NullModelRate\]a where the dashed line assigns the fitted constant function.
Degree distribution {#sec:degDistrFit}
-------------------
Degrees in the aggregated Skype network are broadly distributed with a fat tail corresponding to strong degree heterogeneities. To characterize this distribution analytically we select two candidate distribution functions. The first is a shifted power-law distribution function of the form, $$P(k)=\frac{\gamma-1}{C+k_{min}}\left( \frac{C+k}{C+k_{min}} \right)^{-\gamma} \hspace{.2in} \mbox{for} \hspace{.2in} k_{min}\leq k,
\label{eq:shSF}$$ where $k$ denotes the degree, $\gamma$ is the power-law exponent scaling the tail of the distribution, and $k_{min}$ is the minimum degree (in our case 1). $C$ is a constant scaling the shift of the distribution, which can be determined as $C=z(\gamma-2)-k_{min}(\gamma-1)$ since we know the average degree $z = 8.56$ of the empirical network. This way our only free parameter during the fit is the degree exponent $\gamma$. After fitting this function by using the non-linear least-square method, we obtain a relatively good match with the empirical distribution (Fig. S\[fig:degdistr\]a) for exponent $\gamma=3.61$.
\
Our second candidate function is a lognormal distribution function of the form, $$P(k)=\frac{1}{k\sigma_D\sqrt{2\pi}}e^{-\frac{(\ln k-\mu_D)^2}{2\sigma_D^2}} \hspace{.2in} \mbox{for} \hspace{.2in} k_{min}\leq k,
\label{eq:logn}$$ where $\mu_D$ and $\sigma_D$ are the scaling parameters. After fitting this function by using the non-linear least square method with two free parameters ($\mu_D$ and $\sigma_D$), we obtain an excellent fit with the empirical distribution for parameters $\mu_D=1.2$ and $\sigma_D=1.39$.
To select the best candidate function, we calculate the corresponding Jensen-Shannon ($JS$) divergence values [@Lin1991] between the empirical and fitted distributions. As a result we find that for the shifted power-law function the best fit provides $JS=0.0257$, while for the lognormal distribution we get $JS=0.0051$. Thus we select the lognormal distribution as the best analytical function describing the degree distribution of the empirical network.
Threshold distribution {#sec:thrDistr}
----------------------
The adoption threshold $\phi$ of a node is defined as $\phi = \Phi_k / k$, i.e. the fraction of adopting neighbours that trigger the adoption of the central node. Therefore it can only take certain fractional values determined by the degree $k$. Although thresholds are defined as a fraction, by considering nodes of the same degree we can focus on the integer threshold $\Phi_k$, defined as the number of a node’s neighbours who have adopted the service earlier.
In our method we first group nodes of the same degree, record their integer thresholds, and then calculate the threshold distribution for each degree group, as shown in the main text (Fig. 1e, inset). These distributions collapse to a master curve after normalization by using the scaling relation $P(\Phi_k,k)=k P (\Phi_k/k)$ (Fig. 1e, main panel). Moreover, this master curve can be well approximated by a lognormal distribution of the form, $$P(\phi)=\frac{1}{\phi \sigma_T\sqrt{2\pi}}e^{-\frac{(\ln\phi-\mu_T)^2}{2\sigma_T^2}},
\label{eq:thrLogn}$$ where $\mu_T=-2$ and $\sigma_T=1$, as determined by the empirical average threshold $w = 0.19$ and standard deviation (STD) $0.233$.
These findings indicate that although individual thresholds are strongly determined by degree, their distribution is degree-invariant, suggesting that the fraction of adopting friends rather than its absolute number is relevant during the service adoption process. The estimated empirical values of parameters are summarized in Table \[table:pars\].
$p_{n}$ $\langle k \rangle$ $\mu_D$ $\sigma_D$ $w$ $STD(\phi)$ $\mu_T$ $\sigma_T$
----------- --------------------- --------- ------------ -------- ------------- --------- ------------
$0.00019$ $8.56$ $1.2$ $1.39$ $0.19$ $0.233$ $-2$ $1$
: Estimated empirical parameters for service “buy credit”.[]{data-label="table:pars"}
Social influence - null model study {#sec:sinf}
===================================
Studies of social contagion phenomena assume that social influence is responsible for the correlated adoption of connected people. However, an alternative explanation for the observed correlated adoption patterns is homophily: a link creation mechanism by which similar egos get connected in a social structure. In the latter case, the correlated adoption of a connected group of people would be explained by their similarity and not necessarily due to social influence. Homophily and influence are two processes that may simultaneously play a role during the adoption process. Nevertheless, distinguishing between them on the individual level is very difficult using our or any similar datasets [@Shalizi2011]. Fortunately, at the system level one may decide which process is dominant in the empirical data. To do that first we need to elaborate on the differences between these two processes.
Influence-driven adoption of an ego can happen once one or more of its neighbours have adopted, since then their actions may influence the decision of the central ego. Consequently, the time ordering of adoptions of the ego and its neighbours matters in this case. Homophily-driven adoption is, however, different. Homophily drives social tie formation such that similar people tend to be connected in the social structure. In this case connected people may adopt because they have similar interests, but the time ordering of their adoptions would not matter. Therefore, it is valid to assume that adoption could evolve in clusters due to homophily, but these adoptions would appear in a random order.
To test our hypothesis we define a null model where we take the adoption times of each adopter and shuffle them randomly among all adopting egos. This way a randomly selected time is assigned to each adopter, while the adoption rate and the final set of adopters remain the same. Moreover, this procedure only destroys correlations between adoption events induced by social influence, but keeps the social network structure and node degrees unchanged. In this way, during the null model process the same egos appear as adopters, but the time series of adoption may in principle change (or not), corresponding to social influence (or homophily) as a dominant factor during the adoption process. If adoption is mostly driven by homophily, the rates of adoption would not change considerably beyond statistical fluctuations. On the other hand, if social influence plays a role in the process, rates of adoption in the null model should be very different from the empirical curves, implying that the time ordering of events matters in the adoption process. In this case, the rate of innovators should be higher than in the empirical data, since nodes that are in the adoption cluster originally but not directly connected would have a greater chance to appear as innovators, due to a random adoption time that is not conditional to the time ordering of the adopting neighbours.
After calculating the adoption rates of different user groups in the shuffled null model sequence we observe the latter situation: the rate of innovators becomes dominant, while the rate of stable and vulnerable adoptions drops considerably as they appear only by chance. This suggests that the temporal ordering of adoption events matters a lot in the evolution of the observed adoption patterns, and thus social influence may play a strong role here. Of course one cannot decide whether influence is solely driving the process or homophily has some impact on it; in reality it probably does to some extent. However, we can use this null model measure to demonstrate the presence and importance of the mechanism of influence during the adoption process.
Threshold model {#sec:thresModel}
===============
Model description {#ssec:modDesc}
-----------------
This model emulates the rise and temporal evolution of system-wide adoption cascades in complex social networks [@watts2002; @singh2013; @Ruan2015]. Note that this model has been introduced in [@Ruan2015], where its general scaling behaviour has been explored. In a system of fixed size, a node has social interactions with $k$ other agents and is characterized by a continuous adoption threshold $\phi$. When faced with the prospect of adopting a given innovation, product, or fad, susceptible individuals adopt spontaneously with rate $\pn$. Otherwise, the node adopts if at least a fraction $\phi$ of its $k$ neighbours have adopted before (the so-called ‘threshold rule’). Moreover, a fraction $r$ of the system is ‘immune’ to the innovation, in the sense that these agents never adopt regardless of their values of $k$ and $\phi$. The distributions of degrees and thresholds, $P(k)$ and $P(\phi)$ (as well as the values of $\pn$ and $r$), thus determine the average topological state and dynamical evolution of the system.
The model may be implemented numerically via a Monte Carlo simulation of the rules described above in a system of size $N$. Here, the dynamical state of the system is determined by the adoption state (0 or 1) of all nodes, which change in asynchronous random order in a series of time steps. Once an agent adopts and its state changes from 0 to 1, it remains so for the rest of the dynamics, thus ensuring a frozen final state for the finite system where no more adoptions arise. Each time step consists of $N$ node updates: In each node update, a randomly selected node (non-immune and in state 0) adopts spontaneously with probability $\pr = \pn / (1 - r)$ [^2]; else it adopts only if the threshold rule is satisfied. The rescaled rate $\pr$ is necessary if we wish to obtain a rate $\pn$ of innovators in early times of the dynamics, regardless of the value of $r$. Finally, we assume that agents with $k=0$ receive no social influence (for any value of $\phi$), and therefore can only adopt spontaneously. We will now explore this dynamics with numerical simulations and a rate equation formalism.
Stochastic binary-state dynamics {#ssec:stocDynam}
--------------------------------
Here we extend an approximate master equation (AME) formalism for stochastic binary-state dynamics as developed recently by Gleeson [@porter2014; @gleeson2013; @gleeson2008; @gleeson2011]. In a stochastic binary-state dynamics, each node in the network can take one of two possible states (susceptible or adopter in the language of innovation adoption) at any point in time, and state-switching happens randomly with probabilities that only depend on the current state of the updating agent and on the states of its neighbours. This general definition includes the threshold model described above as a special case. Such formalism considers configuration-model networks, that is, an ensemble of networks specified by the degree distribution $P(k)$ but otherwise maximally random [@newman2010].
All relevant properties used to describe a node are included in the vector $\kvec = (k, c)$, where $k = k_0, k_1, \ldots k_{M-1}$ is the degree of the node and $c = 0, 1, \ldots, M$ a dummy variable that labels its ‘type’, i.e. any other property that characterizes the node apart from its degree. In the case of our threshold model, $c = 0$ is the type of the fraction $r$ of immune nodes, while $c \neq 0$ labels the type of all non-immune nodes with given threshold $\phi_c$. The various values of $c \neq 0$ correspond then to different adoption thresholds $\phi_c$. The integer $M$ is the maximum number of degrees/types considered in the AME framework, which can be increased to improve the accuracy of the analytical approximation at the expense of speed in its numerical computation[^3]. Any pair of nodes with identical values of $\kvec$ are considered equivalent in this level of description, forming a node class with the same average dynamics. Moreover, $P(k)$ and $P(\phi)$ can be generalized to the joint distribution $\Pk$ giving the probability that a randomly selected node has property vector $\kvec$ (i.e. degree $k$ and type $c$). Here it is useful to define $P(c)$ as the distribution of all non-zero types, $c = 1, \ldots, M$. If degrees and thresholds are chosen independently, like in our model, then $\Pk = r P(k)$ for $c = 0$ and $\Pk = (1 - r) P(k) P(c)$ for $c > 0$. The distribution $P(c)$ is, in other words, a discrete, rescaled version of the continuous threshold distribution $P(\phi)$.
In the language of innovation adoption, the dynamics of a node is determined by the number $m = 0, 1, \ldots k$ of its neighbours that have already adopted when the node is deciding whether to adopt or not. During a small time interval $dt$, a susceptible node (in state 0) adopts with probability $\Fkm dt$, while an adopter (in state 1) becomes susceptible with probability $R_{\kvec, m} dt$. The functions $\Fkm$ and $R_{\kvec, m}$, known as infection and recovery rates, respectively, determine the temporal evolution of the node class $\kvec$. In the particular case of threshold models, a so-called monotone dynamics, $R_{\kvec, m} = 0$ $\forall\, \kvec, m$ (since no adopters become susceptible again). As for $\Fkm$, the rules of spontaneous and threshold adoption imply, $$\label{eq:thresRule}
\Fkm =
\begin{cases}
\pr & \text{if} \quad m < k \phi_c \\
1 & \text{if} \quad m \geq k \phi_c
\end{cases}, \quad \forall m \; \text{and} \; k, c \neq 0,$$ that is, a node adopts the innovation either spontaneously with rate $\pr$, or with probability 1 if its number of adopting neighbours equals or exceeds the integer threshold $\Phi_k = \lceil k \phi_c \rceil$. Immune nodes ($c = 0$) have an infection rate of $F_{(k,0),m} = 0$ $\forall k, m$, while for isolated nodes ($k = 0$) $F_{(0,c),0} = \pr$ $\forall c \neq 0$. In other words, immune nodes never adopt, and isolated nodes can only adopt spontaneously. We note that $\Fkm$ is written in terms of $\pr = \pn / (1 - r)$, not $\pn$, in order to counter the trivial decrease in the rate of spontaneous adoption for non-zero $r$.
Let us now turn to the rate equations for our threshold model, called AMEs in the formalism by Gleeson. We denote by $\skm(t)$ the fraction of $\kvec$-class nodes that are susceptible at time $t$ and have $m$ adopting neighbours. Therefore, the fraction of agents with property vector $\kvec$ that are adopters at time $t$ is $\pk(t) = 1 - \sum_{m=0}^k \skm (t)$, and the fraction of adopters in the system is $\rho (t) = \sumk \Pk \pk(t)$. Here, the sum over classes means a sum over all degrees and types, i.e. $\sumk \bullet = \sum_k \sum_c \bullet$. Assuming a monotone dynamics ($R_{\kvec, m} = 0$), the AMEs for $\skm$ can be written as [@porter2014; @gleeson2013; @gleeson2011], $$\label{eq:AMEsThres}
\frac{d \skm}{dt} = -\Fkm \skm -\bs (k - m) \skm + \bs (k - m + 1) \skmo,$$ where $m = 0, \ldots, k$, $s_{\kvec, -1} \equiv 0$, $\Fkm$ follows Eq. (\[eq:thresRule\]), and $\bs(t)$ (the rate at which edges between pairs of susceptible nodes transform to edges between a susceptible agent and an adopter) is given by, $$\label{eq:rateBs}
\bs(t) = \frac{\sumk \Pk \summ (k - m) \Fkm \skm(t)}{\sumk \Pk \summ (k - m) \skm(t)}.$$ If at time $t = 0$ we randomly choose a fraction $\rho (0) = \sumk \Pk \pk(0)$ of nodes as seed for the adoption process, the initial conditions for Eq. (\[eq:AMEsThres\]) are $\skm (0) = [1 - \pk(0)] \Bkm [\rho(0)]$, with $\pk(0)$ the initial fraction of adopters in class $\kvec$ and $\Bkm$ a binomial factor, $$\label{eq:BinomFac}
\Bkm(\rho) = \binom{k}{m} \rho^m (1 - \rho)^{k - m}.$$
The solution $\skm(t)$ of the AME system in Eq. (\[eq:AMEsThres\]) provides a very accurate description of the dynamics of our model, yet its dimension (i.e. number of equations to solve) grows quadratically with the number of degrees and linearly with the number of threshold values considered. Fortunately, the AMEs for our model can be mapped to a reduced-dimension system with a derivation similar to the one used by Gleeson in the case of the Watts threshold model [@watts2002; @singh2013].
Reduced-dimension AMEs {#ssec:redAMEs}
----------------------
To reduce the dimension of Eq. (\[eq:AMEsThres\]), we need to consider system-wide quantities that are more aggregated than $\skm$. One of them is the probability that a randomly chosen node is an adopter, $\rho(t) = 1 - \sumk \Pk \summ \skm (t)$, i.e. the fraction of adopters in the network. The other one is the probability that a randomly chosen neighbour of a susceptible node is an adopter, $\nu(t) = \sumk \Pk \summ m \skm(t) / \summ k \skm(t)$.
We start by proposing an exact solution for the AME system in terms of the following ansatz, $$\label{eq:AMEansatz}
\skm(t) = [1 - \pk(0)] \Bkm [\nu(t)] e^{-\pr t}
\quad \text{for} \; m < k\phi_c \; \text{and} \; c \neq 0,$$ and $s_{(k,0),m} = \Bkm(\nu)$ for $c = 0$, where $\Bkm$ follows Eq. (\[eq:BinomFac\]). The meaning of the ansatz in Eq. (\[eq:AMEansatz\]) is quite intuitive and considers two processes. First, a susceptible agent with degree $k$ and $m$ adopting neighbours is not selected as part of the initial adoption seed with probability $1 - \pk(0)$ and is connected to $m$ adopters with the binomially distributed probability $\Bkm(\nu)$. Second, for $m < k\phi_c$ a susceptible node does not fulfill the threshold rule and can only adopt spontaneously with probability $e^{-\pr t}$, since the system is progressively been filled by adopters. Considering these two processes as independent we end up with the product in Eq. (\[eq:AMEansatz\]). Finally, since immune nodes do not adopt and are distributed randomly over the network, $s_{(k,0),m}$ is determined only by a binomial factor.
The next step is to insert the ansatz (\[eq:AMEansatz\]) into the AME system (\[eq:AMEsThres\]) and derive a set of differential equations for the aggregated quantities $\rho$ and $\nu$. Taking the time derivative $\dskm$ of Eq. (\[eq:AMEansatz\]) (i.e. the left-hand side of Eq. (\[eq:AMEsThres\])) we get, $$\label{eq:ansatzINames1}
\dskm = \left( \left[ \frac{m}{\nu} - \frac{k-m}{1-\nu} \right] \dot{\nu} - \pr \right) \skm.$$ Then, we use the threshold rule (\[eq:thresRule\]) for $m < k\phi_c$, the ansatz (\[eq:AMEansatz\]) and the binomial identity, $$\label{eq:binomIdent}
\Bkmo(\nu) = \frac{1-\nu}{\nu} \frac{m}{k-m+1} \Bkm(\nu),$$ in the right-hand side of Eq. (\[eq:AMEsThres\]) to obtain, $$\label{eq:ansatzINames2}
-\Fkm \skm -\bs (k - m) \skm + \bs (k - m + 1) \skmo = \left[ -\pr + \bs \left( m - k + \frac{1-\nu}{\nu}m \right) \right] \skm.$$ Equating Eqs. (\[eq:ansatzINames1\]) and (\[eq:ansatzINames2\]) as in the AME system (\[eq:AMEsThres\]) leads to, $$\label{eq:condNu}
\dot{\nu} = \bs (1 - \nu),$$ a condition on $\nu$ so that the ansatz (\[eq:AMEansatz\]) is a solution of Eq. (\[eq:AMEsThres\]). This differential equation has the initial condition $\nu(0) = \rho(0)$, obtained by evaluating Eq. (\[eq:AMEansatz\]) at $t = 0$ and comparing with the expression $[1 - \pk(0)] \Bkm [\rho(0)]$, which corresponds to a random distribution of initial adopters among $\kvec$ classes. Furthermore, by assuming a (yet to be determined) function $g(\nu, t)$ such that $\dot{\nu} = g(\nu, t) - \nu$, Eq. (\[eq:condNu\]) reduces to, $$\label{eq:condBeta}
\bs = \frac{g(\nu, t) - \nu}{1 - \nu}.$$
Now, we consider the following general result derived by Gleeson in [@gleeson2013] (Eqs. (F6)–(F10) therein), $$\label{eq:GleesonEq}
\sumk \Pk \summ (k - m) \skm = z (1 - \nu)^2,$$ with $z = \sum_k k P(k)$ the average degree in the network. Eq. (\[eq:GleesonEq\]) is valid for functions $\skm$ and $\nu$ that satisfy Eqs. (\[eq:AMEsThres\]) and (\[eq:condNu\]) respectively, for any $\Fkm$ and random initial conditions on $\skm$ and $\nu$, and is thus applicable in our case. Our goal here is to use Eq. (\[eq:GleesonEq\]) to find an expression for $g(\nu)$ and therefore write the differential equation (\[eq:condNu\]) explicitly. Noting that the left-hand side of Eq. (\[eq:GleesonEq\]) is the denominator in the definition (\[eq:rateBs\]) of $\bs$ and that $F_{(k,0),m} = 0$ (i.e. immune nodes do not adopt), Eq. (\[eq:rateBs\]) gives, $$\begin{aligned}
\label{eq:betaExpl1}
\bs &= \frac{1 - r}{z (1 - \nu)^2} \left[ \pr \sumkc P(k) P(c) \summLess (k - m) \skm + \sumkc P(k) P(c) \summMore (k - m) \skm \right] \nonumber\\
&= \frac{1}{z (1 - \nu)^2} \left[ \sumk \Pk \summ (k - m) \skm - r \sum_k P(k) \summ (k - m) s_{(k,0),m} \right. \nonumber\\
&\quad \left. - (1 - r) (1 - \pr) \sumkc P(k) P(c) \summLess (k - m) \skm \right],\end{aligned}$$ where we have written $\Pk$ explicitly as $\Pk = r P(k)$ for $c = 0$ and $\Pk = (1 - r) P(k) P(c)$ for $c > 0$, in order to notice the dependence on $r$. Then, we insert the ansatz (\[eq:AMEansatz\]) (with its special case $s_{(k,0),m} = \Bkm(\nu)$ for immune nodes), as well as the identities $(k - m) \Bkm(\nu) = k (1 - \nu) \Bkom(\nu)$ and $\summLess \Bkom(\nu) = 1 - \summMore \Bkom(\nu)$ to obtain, $$\begin{aligned}
\label{eq:betaExpl2}
\bs &= \frac{1}{1 - \nu} \Bigg( (1 - r) \Bigg[ 1 - (1 - \pr) e^{-\pr t} \Bigg. \Bigg. \nonumber\\
&\quad \left. \left. + (1 - \pr) e^{-\pr t} \sumkc \frac{k}{z} P(k) P(c) \left( \pk(0) + [1 - \pk(0)] \summMore \Bkom(\nu) \right) \right] - \nu \right).\end{aligned}$$
A comparison of Eqs. (\[eq:condBeta\]) and (\[eq:betaExpl2\]) gives us the following expression for $g(\nu, t)$, $$\label{eq:gFactor}
g(\nu, t) = (1 - r) \left( \ft + (1 - \ft) \sumkc \frac{k}{z} P(k) P(c) \left[ \pk(0) + [1 - \pk(0)] \sum_{m \geq k\phi_c} \Bkom(\nu) \right] \right),$$ where we define $\ft$ as $\ft = 1 - (1 - \pr) e^{-\pr t}$. Thus, the AME system (\[eq:AMEsThres\]) gets reduced to the differential equation $\dot{\nu} = g(\nu, t) - \nu$, with $g(\nu, t)$ given explicitly by Eq. (\[eq:gFactor\]).
Even though the equation $\dot{\nu} = g(\nu, t) - \nu$ is closed and in this sense equivalent to Eq. (\[eq:AMEsThres\]), we can also derive the corresponding equation for $\rho$, since we are mainly interested in the temporal evolution of the fraction of adopters in the network. From the definition of $\rho$ and Eq. (\[eq:AMEsThres\]) we have, $$\begin{aligned}
\label{eq:rhoDeriv1}
\dot{\rho} = - \sumk \Pk \summ \dskm &= \sumk \Pk \summ \Fkm \skm \nonumber\\
&\quad + \bs \sumk \Pk \summ \big[ (k - m) \skm - (k - m + 1) \skmo \big],\end{aligned}$$ where the second term in the right-hand side telescopes to zero. Then, we use an algebraic manipulation similar to that of Eqs. (\[eq:betaExpl1\]) and (\[eq:betaExpl2\]) to obtain, $$\begin{aligned}
\label{eq:rhoDeriv2}
& \sumk \Pk \summ \Fkm \skm = (1 - r) \left( \pr \sumkc P(k) P(c) \summLess \skm + \sumkc P(k) P(c) \summMore \skm \right) \nonumber\\
&= (1 - r) \left( 1 - (1 - r) (1 - \pr) \sumkc P(k) P(c) \summLess \skm \right) - \rho \nonumber\\
&= (1 - r) \left( \ft + (1 - \ft) \sumkc P(k) P(c) \left[ \pk(0) + [1 - \pk(0)] \sum_{m \geq k\phi_c} \Bkm(\nu) \right] \right) - \rho.\end{aligned}$$ In this way, Eqs. (\[eq:rhoDeriv1\]) and (\[eq:rhoDeriv2\]) can be rewritten as $\dot{\rho} = h(\nu, t) - \rho$, where, $$\label{eq:hFactor}
h(\nu, t) = (1 - r) \left( \ft + (1 - \ft) \sumkc P(k) P(c) \left[ \pk(0) + [1 - \pk(0)] \sum_{m \geq k\phi_c} \Bkm(\nu) \right] \right).$$
Joining all of these results, the AME system (\[eq:AMEsThres\]) gets reduced to the system of two ordinary differential equations,
\[eq:reducedAMEs\] $$\begin{aligned}
\dot{\rho} &= h(\nu, t) - \rho, \\
\dot{\nu} &= g(\nu, t) - \nu,\end{aligned}$$
with the quantities $g(\nu)$ and $h(\nu)$ given explicitly by Eqs. (\[eq:gFactor\]) and (\[eq:hFactor\]).
The system (\[eq:reducedAMEs\]) can be solved numerically to obtain $\rho(t)$ and thus characterize the temporal evolution of the adoption process. Let us further separate the fraction of adopters as $\rho(t) = \rho_0(t) + \rho_1(t)$, where $\rho_0$ and $\rho_1$ are the fractions of innovators and induced adopters (i.e. vulnerable and stable nodes), respectively. Now consider the identity, $$\label{eq:suscepIdent}
1 - \rho = \sumk \Pk \summ \skm = r + (1 - r) \sumkc P(k) P(c) \summ \skm = r + \rho_s,$$ where $\rho_s(t)$ is the fraction of non-immune, susceptible nodes that can eventually adopt, either spontaneously or not. Since such susceptible nodes adopt spontaneously at a rate $\pr$, the rate equation for innovators is $\dot{\rho}_0 = \pr \rho_s$. Then, with Eq. (\[eq:suscepIdent\]) we obtain, $$\label{eq:innovRateEq}
\rho_0(t) = \pr \int_0^t [1 - r - \rho(t)] dt,$$ which can be calculated explicitly with the numerical solution of Eq. (\[eq:reducedAMEs\]).
Waiting time of adoption {#sec:tw}
========================
\
Another reason behind the non-rapid evolution of the adoption process could be the time users wait after their personal adoption threshold is reached and before adopting the service. This lag in adoption can be due to individual characteristics, or come from the fact that social influence does not spread instantaneously (as commonly assumed in threshold models, including ours). However, the waiting time $\tau_w$ can be estimated by measuring the time difference between the last adoption in a user’s egocentric network and the time of adoption. We define $\tau_w=0$ for innovators, but $\tau_w$ can take any positive value for vulnerable and stable adopters up to the length of the observation period.
Waiting times are broadly distributed for adopters (Fig. S\[fig:WaitingT\]a), meaning that many users adopt the service shortly after their personal threshold is reached, but a considerable fraction waits long before adopting the service. The heterogeneous nature of waiting times may be a key element behind the observed adoption dynamics. One way to figure out the effect of waiting times on the speed of cascade evolution is by removing them. We can extract waiting times from adoption times and thus calculate rescaled adoption times. The rescaled adoption time of a user is the last time when his/her fraction of adopting neighbours changed and the adoption threshold was (hypothetically) reached. After this procedure we can calculate a new adoption rate function by using rescaled adoption times and compare it to the original. From Fig. S\[fig:WaitingT\]b we can conclude that although adoption becomes faster, the rescaled adoption dynamics is still not rapid. On the contrary, it suggests that the rescaled adoption dynamics is still very slow and quite similar to the original. Consequently, waiting times cannot explain the observed dynamics of adoption.
Note that long waiting times can have a further effect on the measured dynamics. After the ‘real’ threshold of a user is reached and he/she waits to adopt, some neighbours may adopt the product. Hence all observed measures are in this sense ‘effective’: observed thresholds are larger or equal than real thresholds; the innovator rate is smaller or equal; the vulnerable and stable rates will be larger or equal; and waiting times will be shorter or equal than the real values. Consequently the process may be actually faster than that we observe in Fig. S\[fig:WaitingT\]b after removing effective waiting times. However, this bias becomes important only after the majority of individuals in the social network has adopted the service and the spontaneous emergence of adopting neighbours becomes more frequent. As the fraction of adopters in our dataset is always less than $6\%$ [@SkypeIPO], we expect minor effects of this observational bias on measurements.
Empirical and model cluster statistics {#sec:clustStats}
======================================
\
As described in the main text, we perform extensive model calculations using empirically determined parameters to estimate the only unknown parameter, the fraction of immune nodes $r$. We match the relative size of the largest connected component of the real adoption network with its corresponding measure in the model at the end of the observation period, and estimate the fraction of immune nodes in the real system as $r=0.73$. To support our estimation we also measure the distribution $P(d)$ of the depth of induced vulnerable trees and the correlation $\langle s_v \rangle (k)$ between the degree of innovator nodes and the average size of induced vulnerable trees in the model, and match them with the equivalent empirical measures. To provide further support for the estimated $r$ value we show the dependence of these quantities of different $r$ values.
We measure $P(d)$ and $\langle s_v \rangle (k)$ for $r=0.6$ and $0.9$, as well as for the predicted value $r=0.73$ (Fig. S\[fig:ClusStat\]). It is clear that both quantities scale with $r$. For smaller $r$ more nodes are susceptible for adoption, allowing deeper and larger vulnerable trees, while for larger $r$ no large induced cluster can emerge as the system is forced into a quenched state. Moreover, measures for the estimated $r$ value fit the empirical data considerably well. This collapse is remarkable, since we neglect any higher-order structural and temporal correlations in the model (like assortative mixing, community structure, bursty adoption patterns, periodic activity fluctuations, etc.), which are present in the empirical system. Differences in the tails of the measures are due to finite-size effects since the modelled network is two orders of magnitude smaller than the empirical social structure. Note that although we can look for an $r$ fraction that produces a better fit between model and data in terms of $P(d)$ and $\langle s_v \rangle (k)$, the collapse in Fig. S\[fig:ClusStat\] demonstrates the quality of an independent procedure of estimating $r$ (i.e. by matching the relative size of components). Therefore, these results are intended for validation only and not as a method to estimate the correct value of $r$.
Calculations for additional service {#sec:addserv}
===================================
\
Empirical observations
----------------------
In order to support our empirical observations and modelling of the social spreading of Skype, we examine the adoption dynamics of an additional paid service called “subscription”, introduced in April 2008 and with adoption data for over $42$ months until the end of the observation period. This service is only available for registered Skype users, and we can therefore use the accumulated static Skype network as background social structure. In order to investigate the adoption of this service we repeat all calculations described previously. First we measure the decoupled rate of innovator, vulnerable, and stable adopters (Fig. S\[fig:srv4RateTh\]a). We see that after a short initial period innovators adopt approximately with a constant rate, setting the model parameter to $p_n=0.00012$. Moreover, here innovators dominate social spreading since the rate of vulnerable and stable adoptions is relatively low.
We also measure the integer threshold distribution for different degree groups (Fig. S\[fig:srv4RateTh\]b, inset) just as described in Section \[sec:thrDistr\]. These distributions scale together after normalization with the scaling relation $P(\Phi_k,k)=k P (\Phi_k/k)$ (Fig. S\[fig:srv4RateTh\]b, main panel) and are well approximated by a lognormal distribution \[Eq. (\[eq:thrLogn\])\] with parameters $\mu_T=-3.73$ and $\sigma_T=1.39$, as determined by the average threshold $w=0.063$ and STD $0.153$. Note that since the adoption dynamics of this service is dominated by innovators, the average threshold $w$ is smaller than in the case of the “buy credit” service. All parameters are summarized in Table \[table:parssrv4\]. Since the background network is the same for both services, network parameters are those of Table.\[table:pars\].
\
Although the adoption process is dominated by innovators, a giant connected component evolves in the adoption network (Fig. S\[fig:srv4RateTh\]a, main panel). On the other hand, its relative size is considerable smaller than for the “buy credit” service. The stable adoption network is also dominated by a giant component, but its relative size is even smaller when compared to the adoption network (Fig. S\[fig:srv4RateTh\]a, inset). Moreover, the largest vulnerable trees are only two orders of magnitude smaller than the stable giant cluster (Fig. S\[fig:srv4RateTh\]b). For comparison, this difference is five order of magnitude for the “buy credit” service.
$p_{n}$ $w$ $STD(\phi)$ $\mu_T$ $\sigma_T$
----------- --------- ------------- --------- ------------
$0.00012$ $0.063$ $0.153$ $-3.73$ $1.39$
: Estimated empirical parameters for service “subscription”.[]{data-label="table:parssrv4"}
Model and validation
--------------------
![ Average size of the largest ($LC$) and 2nd largest ($LC^{2nd}$) components of the model network (‘Net’), model adoption network (‘Casc’), model stable network (‘Stab’), and induced vulnerable trees (‘Vuln’) as a function of $r$. Dashed lines show the observed relative size of the real $LC$ of the adopter network in 2011 (Fig. S\[fig:srv4adClust\], main panel) and the predicted $r$ value. The lower panel depicts the time $t_{50\%}$ when the adoption process has reached $50\%$ of the susceptible network as a function of $r$. We use $100$ realizations of configuration-model networks with size $N=10^5$ and lognormal degree distribution parametrized as described in Section \[sec:degDistrFit\]. Model calculations correspond to the parameters of Table \[table:parssrv4\] for $42$ iteration steps (matching the length of the observation period).[]{data-label="fig:srv4model"}](LC12_R_types_T100_fig_lgn_srv4.pdf){width="110mm"}
We repeat all model calculations with the parameters of the “subscription” service to see whether we can recover its adoption dynamics by using the dynamical threshold model introduced in the main text and in Section \[sec:thresModel\]. We check the dependence on $r$ of the average size of the largest connected component of the network ($LC$) of susceptible nodes available for the adoption process, the adoption network, the stable adoption network, and of vulnerable trees (Fig. S\[fig:srv4model\], upper panel). In addition we record the average size $LC^{2nd}$ of the second largest connected component (Fig. S\[fig:srv4model\], middle panel). Finally we show the time when the adoption process has reached the $50\%$ of available susceptible nodes in the adoption network (Fig. S\[fig:srv4model\], lower panel).
The $r$ dependence of the adoption process appears to be qualitatively similar to our earlier calculations on the “buy credit” service, but there are remarkable differences. Firstly, the crossover regime (depicted by the light grey area in Fig. S\[fig:srv4model\]) is shifted towards larger $r$ values due to the different threshold distribution and innovator adoption rate. Secondly, after matching the relative size of the largest connected component of the empirical adoption network (last point on the right-hand side of Fig. S\[fig:srv4adClust\], main panel), the predicted $r=0.928$ is out of the crossover regime. At this point the background social network is still not fragmented (as evidenced by the black line in Fig. S\[fig:srv4model\], which has not reached its maximum yet) and it allows for the emergence of large connected adoption clusters. It is very sparse, however, which explains: (a) the dominating innovator adoption rate observed empirically; (b) the reduced size of the giant component of the adoption and stable adoption networks; and (c) the relatively large innovator trees as compared to the stable adoption network components. We observe that the largest vulnerable trees are smaller than the largest stable clusters in the empirical data, while the opposite is true for the model. A possible explanation of this difference is the assumption in the model that the network is degree-uncorrelated. This is a necessary approximation in order to treat the model analytically, but it might not hold for the empirical network. All in all, this picture suggests that the “subscription” service is out of the rapid and even the crossover cascading regimes, and that its dynamics is mostly driven by independent innovators rather than social influence, on a network of which a large majority is not susceptible to innovation.
[9]{}
White D. S., [*Social Media Growth 2006 to 2012*]{} (2013). Date of access: 2015.01.29.
Lin J., Divergence measures based on the Shannon entropy. [*Trans. Inf. Theory*]{} [**37**]{}, 145 (2009).
Shalizi C. R. and Thomas A. C., Homophily and Contagion Are Generically Confounded in Observational Social Network Studies. [*Sociol. Methods Res.*]{} [**40**]{}(2), 211–239 (2011).
Watts D. J., A simple model of global cascades on random networks. [*Proc. Natl. Acad. Sci. USA*]{} [**99**]{}, 5766–5771 (2002).
Singh P., Sreenivasan S., Szymanski B. K., Korniss Gy., Threshold-limited spreading in social networks with multiple initiators. [*Sci. Rep.*]{} [**3**]{}, 2330 (2013).
Ruan Z., Iñiguez G., Karsai M., Kertész J., Kinetics of social contagion. [*Phys. Rev. Lett.*]{} [**115**]{}, 218702 (2015).
Porter M. A., Gleeson J. P., Dynamical systems on networks: A tutorial. Eprint arXiv 1403.7663 (2014).
Gleeson J. P., Binary-state dynamics on complex networks: Pair approximation and beyond. [*Phys. Rev. X*]{} [**3**]{}, 021004 (2013).
Gleeson J. P., Cascades on correlated and modular random networks. [*Phys. Rev. E*]{} [**77**]{}, 046117 (2008).
Gleeson J. P., High-accuracy approximation of binary-state dynamics on networks. [*Phys. Rev. Lett.*]{} [**107**]{}, 068701 (2011).
Newman M. E. J., [*Networks: An Introduction*]{}. (Oxford University Press) (2010).
Morrissey R. C., Goldman N. D., Kennedy K. P,. (2011). Date of access: 2014.10.14.
[^1]: Corresponding author: marton.karsai@ens-lyon.fr
[^2]: We define $\pr = 1$ for $\pn > 1 - r$.
[^3]: Explicitly, rather than using $k_0 = k_{\text{min}}$, $k_{M-1} = N - 1$ and $M = N - k_{\text{min}}$ (i.e, considering all possible degrees in the empirical/simulated network), we take a small $k_0 > k_{\text{min}}$ and large $k_{M-1} < N - 1$, $M < N - k_{\text{min}}$, with the other $M - 2$ degree values equidistantly distributed between $k_0$ and $k_{M-1}$, thus disregarding some degrees and gaining speed in the computation of the AMEs. Similarly, the $M$ threshold values corresponding to nonzero types are uniformly distributed in the open interval $(0, 1)$.
|
---
abstract: 'This paper addresses the noisy label issue in audio event detection (AED) by refining strong labels as sequential labels with inaccurate timestamps removed. In AED, strong labels contain the occurrence of a specific event and its timestamps corresponding to the start and end of the event in an audio clip. The timestamps depend on subjectivity of each annotator, and their label noise is inevitable. Contrary to the strong labels, weak labels indicate only the occurrence of a specific event. They do not have the label noise caused by the timestamps, but the time information is excluded. To fully exploit information from available strong and weak labels, we propose an AED scheme to train with sequential labels in addition to the given strong and weak labels after converting the strong labels into the sequential labels. Using sequential labels consistently improved the performance particularly with the segment-based F-score by focusing on occurrences of events. In the mean-teacher-based approach for semi-supervised learning, including an early step with sequential prediction in addition to supervised learning with sequential labels mitigated label noise and inaccurate prediction of the teacher model and improved the segment-based F-score significantly while maintaining the event-based F-score.'
address: |
$^{1}$Department of Electronic Engineering, Sogang University, South Korea\
$^{2}$Naver Corporation, South Korea
bibliography:
- 'mybib.bib'
title: Overcoming label noise in audio event detection using sequential labeling
---
**Index Terms**: audio event detection, weak label, noisy label, semi-supervised learning, sequential label
Introduction {#intro}
============
Audio event detection (AED) refers to the task of recognizing when and which audio events occur in an audio recording [@turpault2019sound]. In order to train an AED system, one may need a dataset with strong labels that annotate timestamps corresponding to the start and end of event occurrences in addition to their presence or absence. Unfortunately, annotating the strong labels is too laborious to develop a large-sized strongly labeled dataset. Furthermore, many kinds of audio events *e.g.* footsteps, wind blowing, or burning fire, etc, have a vague start or end in time, and some labeled datasets have no verification between inter-annotators, as mentioned in the previous DCASE challenge [@mesaros2016tut]. Therefore, the timestamps depend on the subjectivity of each annotator, which are frequently inaccurate or noisy. Therefore, a number of recent audio event detection (AED) researches have used weakly labeled data, such as AudioSet [@gemmeke2017audio] and FSD [@fonseca2017freesound], where do not have the timestamps. In particular, training on weakly labeled data have been widely researched using multiple instance learning recently [@turpault2019sound; @wang2019comparison; @serizel2018large].
Despite various training approaches, the weak label has an apparent limitation that there is no time information. To improve the performance of AED systems based on the weak labels only, additional use of sequential labels describing temporal sequential relationship of events was proposed by [@wang2019connectionist], where the start and end points of audio events were mapped as individual label symbols. As shown in Figure \[fig:labeltype\], the sequential label provides only the sequence of event boundaries instead of indicating their timestamps. The author focused on the possibility of connectionist-temporal-classification(CTC)-based framework being applied to AED. Given strong, weak, and sequential labels, the author trained three models individually using each label set and training with sequential labels showed the mid-level performance between strong and weak labels. Although strong labels were available for all data in [@piczak2015esc; @temko2006clear; @nakamura1999data], very limited data are annotated by strong labels in many practical datasets such as the DCASE challenges [@turpault2019sound]. In this situation, strong labels are very important to match a specific sound to its event class in an AED model from the scratch even though their timestamps are noisy. In addition, it is noteworthy that sequential labels are noise-robust information which can be obtained from strong labels. To fully exploit information from available strong and weak labels, we propose an AED scheme to train with sequential labels in addition to the given strong and weak labels after converting the strong labels into the sequential labels. Since noise-robust information of the strong labels are refined in the sequential labels, the sequential labels may provide consistent cues to train an AED model. In addition, unlike weak labels, the sequential labels contain temporal sequential relationship of events that is useful to guide the model, which may result in performance improvement when using the sequential, strong, and weak labels simultaneously.
![Comparison of strong, sequential, and weak labeling.[]{data-label="fig:labeltype"}](labeltype.pdf){width="0.89\columnwidth"}
Furthermore, the proposed approach can be applied for semi-supervised learning, such as teacher-student learning. Like the annotator’s error, noise (or inaccuracy) also exists in pseudo-labels generated from the teacher model during training phase, and the label noise is more severe on strong labels than weak labels. Therefore, we also propose an approach to use teacher-student learning based on sequential labels which are more noise-robust than the strong labels and more informative than the weak labels.
Related Works
=============
The main topic of this paper is how to use sequential label for training better model. To train the sequential label without temporal alignment, preliminary researches have been conducted on CTC in AED [@wang2019connectionist; @wang2017first]. Based on its effective application, we also used CTC and Connectionist Temporal Localization (CTL) frameworks [@wang2019connectionist] for the sequential label training.
Connectionist temporal classification
-------------------------------------
CTC was first proposed in speech recognition [@graves2006connectionist]. Rather than frame-wise supervision, the CTC framework only requires phone sequences of the training utterances without an alignment between phonemes and frames. CTC considers all possible phone sequence cases through a phone combination with blank labels. To merge them into a single output, many-to-one mapping is used for removing blank or repeated labels. Using many-to-one mapping, the total output probability can be derived as the sum of the all possible probabilities.
As for the output label of AED, the author of [@wang2017first] suggested to use boundaries of sound events. For example, if the audio clip contains speech, car and cat sound as shown in Figure 1, sequence-label is “speech-start, speech-end, car-start, cat-start, car-end, speech-start, cat-end, car-start, speech-end, car-end" as ground truth. Therefore, $n$-class system has $2n + 1$ output nodes, two for each sound event and one for blank token. Through various experiments, the author showed the feasibility of novel labeling structure for AED.
Connectionist temporal localization
-----------------------------------
After the preliminary AED research using CTC, the author of [@wang2019connectionist] proposed the CTL system to address a issue called “peak clustering". As shown in Figure 2 in [@wang2019connectionist], the CTC system tends to predict on/offset repeatedly next to each other for a long single event input. This consecutive event boundaries made misdetected event clusters, and was named “peak clustering". The author analyzed that this issue is mainly due to the multiple purpose of using a blank label. The blank label in CTC serves two purpose: (1) emitting “no event" at a frame and (2) separation of same event repetition. These inconsistent objectives interfere with training blank label properly, therefore occurs frequent blank-output which can lead to consecutive event peaks. To address this issue, the author proposed the CTL framework that eliminates both blank and on/offset label in the output layer of model.
In the CTL framework, the model estimates the probability of event classes for each frame, and derives the event boundary probability by estimating on/offset using rectified delta operator. Let $y_t(E)$ is the probability that event $E$ being active at frame $t$, and $z_t(\acute{E})$ and $z_t(\grave{E})$ are the probability of on/offset of the event *E* at frame *t*. $z_t(\acute{E})$ and $z_t(\grave{E})$ are calculated as follows : $$z_t(\acute{E}) = \max[0, y_t(E) - y_{t-1}(E)].$$ $$z_t(\grave{E}) = \max[0, y_{t-1}(E) - y_t(E)].$$
CTL assumes the probabilities of different event boundaries at the same frame as mutually independent instead of mutually exclusive. In this way, the probability of no event boundaries occurring at frame is calculated by:
$$\label{eq:epsilon}
\epsilon_t = \textstyle\prod_{l}[1-z_t(l)],$$
where $l$ goes over all event boundaries. The probability of emitting a single event boundary $l$ at frame $t$ is then:
$$\label{eq:p_sl}
p_t(l) =\textstyle z_t(l) \cdot \prod_{l'\neq l}[1-z_t(l')].$$
If we define
$$\label{eq:delta}
\delta_t(l) =\frac{z_t(l)}{1-z_t(l)},$$
then, we can get $$\label{eq:p_rd}
p_t(l) = \textstyle\epsilon_t \cdot \delta_t(l).$$ From Eq. (\[eq:epsilon\]), the blank label can be eliminated, and based on the modification of many-to-one mapping function and forward algorithm of CTC, the blank label is no longer used for separating repetition. More details on CTL can be found in [@wang2019connectionist].
The CTL can allow the multiple event occurrence at the same frame. However, this is rare in practice and the result of [@wang2019connectionist] showed AED performance deterioration. Therefore, we assume that there is no event co-occurrence in this work.
Proposed Approach
=================
We propose to use sequential labeling for two-types of label noise : strong label noise by human annotator and inaccuracy of strong prediction by a teacher model.
Sequential labeling using noisy label
-------------------------------------
Sequential labeling is quite intuitive. As shown in Figure 1, on/offset of each event segment are extracted from strong label, and sorted in chronological order. We used the CTC/CTL framework for training the sequential label. The AED system can be trained only with sequential label, but like recent challenges [@turpault2019sound; @serizel2018large; @fonseca2019audio], other types of label also can be used jointly for training. If we use the CTL framework for sequential label, computing frame-wise probabilities is same way as other labels, therefore the strong, weak, and sequential labeling system can be combined with no additional effort. The weight for individual losses was heuristically selected through intensive experiments, *e.g.* strong : weak : sequential label = 4 : 2 : 1. We verified the performance comparison in detail when the sequential or strong label is used alone and with other types of labels.
Expansion to semi-supervised learning
-------------------------------------
To exploit the unlabeled data effectively, sequential labeling is applied to the teacher-student learning. In this work, we used a mean-teacher-based approach [@tarvainen2017mean] widely used in AED [@turpault2019sound; @jiakai2018mean]. The main concept of this approach is averaging model weights over training steps to produce a better model. The teacher model updated by a moving average of student model weights, and the consistency cost, in addition to the classification cost, is used for comparing the prediction between the student and teacher models. After the training, the teacher model is used for evaluation.
The consistency cost for both strong and weak-prediction has already been used for AED [@jiakai2018mean]. This strong-prediction on unlabeled data by teacher model is similar to labeling process by human annotator. However, using three-types of label (strong, weak and sequential) from beginning of training showed unstable learning curve, since prediction inaccuracy in model training is much worse than label noise caused by annotator’s subjectivity. Therefore, we developed mean-teacher learning scheme for using sequential label. The algorithm is described by pseudo-code in Figure 2. We used Mean Squared Error as strong and weak consistency cost function and CTL as sequential consistency cost function. Also, we used sigmoid ramp-up function for weighting consistency cost as proposed in [@tarvainen2017mean]. The maximum value of ramp-up function was set to 1 through intensive experiments.
From the beginning to half-point of training schedule, sequential label is used for loss instead of strong label, and vice versa for rest of the schedule. We found that training performance also showed unstable when strong label used first. We consider this result is due to minimizing sequential-consistency loss is relatively easier than strong-consistency. In other word, the strong-prediction difference between student and teacher model is huge and inaccurate in the beginning of training. Therefore, using sequential-prediction can mitigate instability of model training in early stage.
![Python-style pseudo-code for the semi-supervised learning with sequential label.[]{data-label="fig:pseudocode"}](pseudocode.pdf){width="0.92\columnwidth"}
Experiments and Results
=======================
In this section, we describe our experiment setup and results. Our proposed method was described in Figure \[fig:Exschematic\], evaluated and compared with the DCASE 2019 baseline.
Experimental setup
------------------
We have used the datasets of DCASE 2019 Task 4 and DCASE 2016 Task 3. The DCASE 2019 Task 4 dataset to classify 10 sound classes in the domestic environment consists of 2,045 strongly-labeled synthetic data, 1,578 weakly-labeled data, and 14,412 unlabeled data. The weakly-labeled and unlabeled data are real-recorded data. The DCASE 2016 Task 3 dataset to classify 17 sound event classes consists of real-recorded data with strong annotations. The strong annotations were conducted by two research assistants trained through several example recordings, and more information is described in [@mesaros2016tut].
We used the DCASE 2019 Task 4 Training set and the DCASE 2016 Task 3 Development set for training, and the DCASE 2019 Task 4 Public Evaluation set and the DCASE 2016 Task 3 Evaluation set for evaluation. To evaluate our proposed method, the DCASE 2019 Task 4 Baseline [@turpault2019sound] was used as our baseline model, and the evaluation metrics were event-based F-score (macro average) and segment-based F-score (macro average) computed using the sed\_eval library [@mesaros2016metrics].
Experimental results
--------------------
### CTC vs CTL {#sec:CTCCTL}
We conducted an experiment to compare the performance of models trained by CTC and CTL when using strong or sequential labels. In case of using the CTC, we trained our model by multi-conditional learning because the model should predict the event boundary probabilities. Table \[tab:CTCvsCTL\] summarizes the F-scores of models trained by the CTC and CTL. For comparison, a model trained by strong labels only is also evaluated. The performance of using the CTC only was significantly lower than the others, which might be due to the “peak clustering" problem. Using the CTL showed comparable performance with the loss based on the strong labels only. When using both the sequential and strong labels, training based on the CTC provided a higher segment-based F-score than the case using either one while CTL-based training improved both the event- and segment-based F-scores. It demonstrates the effectiveness of the proposed method to supplement the strong labels with the sequential labels without causing the “peak clustering" problem even though the sequential labels were obtained from the strong labels. Therefore, we conducted subsequent experiments by using the CTL as the sequential loss instead of the CTC.
[|c|c|c|]{}
---------------
Training Loss
---------------
: F-scores (%) of models trained with losses based on CTC/CTL or strong labels for the DCASE 2019 Task 4 Public Evaluation set.[]{data-label="tab:CTCvsCTL"}
&
-------------
Event-based
F-score
-------------
: F-scores (%) of models trained with losses based on CTC/CTL or strong labels for the DCASE 2019 Task 4 Public Evaluation set.[]{data-label="tab:CTCvsCTL"}
&
------------
Seg.-based
F-score
------------
: F-scores (%) of models trained with losses based on CTC/CTL or strong labels for the DCASE 2019 Task 4 Public Evaluation set.[]{data-label="tab:CTCvsCTL"}
\
CTC only & 1.34 & 9.15\
CTL only & 17.27 & 38.24\
strong-label loss only & 15.24 & 41.62\
strong-label and CTC losses & 14.21 & 42.55\
strong-label and CTL losses & **17.71** & **44.19**\
### Sequential labeling for noisy strong labels {#sec:method1}
We evaluated our proposed approach based on sequential labels refined from strong labels, using the DCASE 2016 Task 3 Development set or strongly labeled data in the DCASE 2019 Task 4 Training set for training.
Table \[tab:2016result\] shows the F-scores on the DCASE 2016 Task 3 dataset. Considering that the model trained with both strong and sequential labels achieved better performance than that trained with the strong labels only, sequential information was helpful for improving the performance. In particular, performance improvement in the segment-based F-score was greater than in the event-based F-score since the sequential information focused on occurrences of events to provide consistent and noise-robust cues useful for training.
[|c|c|c|]{}
---------------------
Labels for training
---------------------
: F-scores (%) of models trained with strong labels with and without sequential labels on the DCASE 2016 Task 3 dataset.[]{data-label="tab:2016result"}
&
-------------
Event-based
F-score
-------------
: F-scores (%) of models trained with strong labels with and without sequential labels on the DCASE 2016 Task 3 dataset.[]{data-label="tab:2016result"}
&
------------
Seg.-based
F-score
------------
: F-scores (%) of models trained with strong labels with and without sequential labels on the DCASE 2016 Task 3 dataset.[]{data-label="tab:2016result"}
\
strong labels only & 6.17 & 17.87\
strong and sequential labels & **7.86** & **23.38**\
\[tab:M1\_result\_woMT\]
Table \[tab:M1\_result\_woMT\] summarizes the F-scores of models trained with various combination of strong, weak, and sequential labels when using strongly labeled data in the DCASE 2019 Task 4 Training set for training. To conduct experiments with weak or sequential labels, given strong labels were converted into weak and sequential labels. Training with the weak labels only containing the least information provided the worst performance. Consistent with the results in Table \[tab:CTCvsCTL\], training with the sequential labels had a lower segment-based F-score and a higher event-based F-score than that with the strong labels because of label noise in the strong labels. Adding the sequential labels to the strong or weak labels consistently improved both the event- and segment-based F-scores. As mentioned in Section \[intro\], that is because the sequential labels provided consistent cues to train the model by refining noise-robust information of the strong labels and retained temporal sequential relationship of events that the weak labels did not have. In contrast, the model trained with the strong and weak labels showed the mid-level performance between training with the strong and weak labels only because the weak and strong labels were so different in label characteristics that they could not supplement each other and had independently influenced the model. Moreover, the model trained with all the strong, weak, and sequential labels achieved the best segment-based F-score and an event-based F-score comparable with the model trained with both the strong and sequential labels by fully exploiting available information.
### Sequential labeling for semi-supervised learning {#sec:method2}
Table \[tab:Method2\] shows the F-scores on the mean-teacher-based approach for semi-supervised learning with the DCASE 2019 Task 4 Training set for training. Including the sequential mean-teacher step in the conventional approach improved the segment-based F-score with a slightly degraded event-based F-score because the early step with sequential predictions mitigated inaccurate strong predictions and focused on occurrences of events rather than their timestamps. Furthermore, adding supervised learning with sequential labels to the previous approach improved both the event- and segment-based F-scores with significant improvement on the segment-based F-score, which was consistent with the results in Table \[tab:M1\_result\_woMT\]. In particular, both the sequential labels and predictions were helpful for improving the segment-based F-score by focusing on occurrences of events consistently.
Conclusion
==========
In this paper, we have proposed to use sequential information for mitigating label noise and inaccurate prediction in an early step of semi-supervised learning. Through the experiments on the recent public datasets, we demonstrated that using sequential information could improve AED performance.
|
---
abstract: |
Using a time of flight technique, the maximal values of kinetic energy as a function of primary mass of fragments from low energy fission of $^{234}U$ and $^{236}U$ were measured by Signarbieux From calculations of scission configurations, one can conclude that, for those two fissioning systems, the maximal value of total kinetic energy corresponding to fragmentations ($_{42}$Mo$_{62}$, $_{50}$Sn$_{80}$) and ($_{42}$Mo$_{64}$, $_{50}$Sn$_{80}$), respectively, are equal to the available energies, and their scission configurations are composed by a spherical heavy fragment and a prolate light fragment both in their ground state.
*Keywords*: Low energy fission; $^{234}U$; $^{236}U$; fragment kinetic energy; cold fission\
PACS: 21.10.Gv; 25.85.Ec; 24.10.Lx\
\
author:
- 'M. Montoya'
date: 'September 18th, 2008 '
title: 'The most compact scission configuration of fragments from low energy fission of $^{234}$U and $^{236}$U '
---
Introduction {#intro}
============
One of the most studied quantities to understand the fission process is the fragment mass and distribution, which is very closely related to the topological features in the multi-dimensional potential energy surface [@moller]. Structures on the distribution of mass and may be interpreted by shell effects on potential energy of the fissioning system, determined by the Strutinsky prescription and discussed by Dickmann [@dick] and Wilkins [@wilkins].
In order to investigate the fragments with very low excitation energy, using the time of flight method, Signarbieux [@Signarbieux] measured the fragment mas ($A$) distribution for high values of fragment kinetic energy. Because in that kinetic energy region there is no neutron emission, the time of flight technique permits separate neighboring fragment masses. In this work one calculates the deformations of those fragments which must correspond to the most compact scission configurations, i.e. to the highest values of Coulomb interaction energy between the two fragments.
The most compact scission configurations {#sec:compact}
========================================
In the process of thermal neutron induced fission of $^{233}U$, a composed nucleus $^{234}U^*$ with excitation energy equal to neutron separation energy ($\epsilon_n$) is formed first. Then, this nucleus splits in two complementary light and heavy fragments having $A_L$ and $A_H$ as mass numbers, and $E_L$ and $E_H$ as kinetic energies, respectively.
The Q-value of the this reaction is given by the relation $$Q (Z_L,A_L, Z_H, A_H) = M(92,234)- M(Z_L, A_L) - M(Z_H, A_H),
\label{eq:Q}$$
where M(Z,A) is the mass of nucleus with $Z$ and $A$ as proton number and mass number, respectively.
This available energy at scission configuration is spend in prescission total kinetic energy ($TKE_0$), fragments interaction Coulomb energy, $CE$, fragments deformation energy,
$$TDE = DE_L + DE_H,
\label{eq:TDE}$$
where ($DE_L$) and ($DE_H$) are the light and heavy fragment deformation energy, respectively, and in fragments intrinsic excitation energy,
$$TXE = XE_L + XE_H,
\label{eq:TXE}$$
where ($XE_L$) and ($XE_H$) are the light and heavy fragment intrinsic excitation energy, respectively.
Then the balance energy at scission configuration results
$$Q + \epsilon_n = TKE_0 + CE + TDE + DXE.
\label{eq:QETDX}$$
If there is no neutron emission, the light and heavy fragments reach the detectors with their primary kinetic energies equal to $KE_L$ and $KE_H$, respectively. The total primary fragments kinetic energy will be
$$TKE = KE_L + KE_H = TKE_0 + CE = Q + \epsilon_n - TDE - TXE.
\label{eq:TKEQDX}$$
The maximal value of total kinetic energy is reached when the sum of TDE and TDX is minimal, i.e.
$$TKE_{max} = (TKE_0 + CE)_{max} = Q + \epsilon_n - (TDE - TXE)_{min}.
\label{eq:TKEQDXmax}$$
The most compact scission configuration occurs when maximal value of coulomb energy is equal to the available energy, i.e.
$$CE_{max} = Q + \epsilon_n.
\label{eq:CEmaxQ}$$
In this case, from eq. \[eq:TKEQDX\] one obtains the relations
$$TKE_{max} = CE_{max} = Q + \epsilon_n,
\label{eq:TKEmaxCQ}$$
and
$$DE_{min} = 0,
DX_{min} = 0,
TKE_0 = 0.
\label{eq:DEDXmin}$$
Not always this situation is possible to occur. Nevertheless we can assume that for each mass fragmentation the maximal value of total kinetic energy is obtained for similar condition, i.e. $TKE_0 = 0$, $TXE = 0$ and $TDE=TDE_{min}$.
Deformation energy
==================
![Deformation energy for nuclei $^{106-108}$Mo calculated by a drop liquid model with pairing and shell correction [@Myers]. See text.[]{data-label="fig:106108Mo"}](1.eps){height="50.00000%"}
![Deformation energy for nuclei $^{106-108}$Tc calculated by a drop liquid model with pairing and shell correction [@Myers]. See text.[]{data-label="fig:106108Tc"}](2.eps){height="50.00000%"}
![Deformation energy for nuclei $^{130-132}$Sn calculated by a drop liquid model with pairing and shell correction [@Myers]. See text.[]{data-label="fig:130132Sn"}](3.eps){height="45.00000%"}
The total energy of a nucleus is calculated at first approximation by a liquid drop model type ($\widetilde W$), using the mass formula of Myers and Swiatecki [@Myers]. The shell correction ($\delta U$) is calculated by the Strutinsky’s method [@Strutinsky], using Nilsson Hamiltonian [@Quentin]:
$$V_{corr}= -\kappa [\vec{l} \cdot \hat{s} + \mu ({\vec{l}}^2 -<{\vec l}^2>_N)],
\label{eq:NilssonV}$$
where $\kappa$ and $\mu$ are the Nilsson’s constants.
The pairing correction is calculated using the BCS method [@BCS].
Then, the relation for the total energy of the nucleus (Z,N) results:
$$DE(Z,N,\epsilon)= \widetilde W (Z,N, \epsilon) - \widetilde W_S(Z,N) + \delta U_N + \delta U_Z+\delta P_N + \delta P_Z,
\label{eq:DEW}$$
where $\widetilde W(Z,N,D)$ is the energy of a nucleus $(Z,N)$ having deformation $D$, and $\widetilde W_S(Z,N)$ the energy in its spherical shape.
The constant of the harmonic oscillator was the suggested by Nilsson [@Nilsson]:
$$\hbar w_0 = 41A^{-1/3}.
\label{eq:Harconst}$$
As one said, the total fragments kinetic energy is close to the available energy for light and heavy complementary fragments with masses around $A=104$ and $A=132$, respectively. Let us relate this result to the deformation for nuclei in this mass neighborhood.
The energies of nuclei $^{106-108}$Mo and $^{106-108}$Tc as a function of their corresponding deformations ($\epsilon$)are presented on Figs. \[fig:106108Mo\] and \[fig:106108Tc\], respectively. The assumed Nilsson’s constants [@Nilsson] for these nuclei are $\kappa_N = 0.678$, $\kappa_P = 0.07$, $\mu_N = 0.33 $ and $\mu_P = 0.35$.
As we can see, those nuclei have a prolate shape with to $\epsilon = 0.3$ in their ground state. If the fragment deformation changes from $\epsilon = 0$ to $\epsilon = 0.3$ the deformation energy will decreases by around 2 MeV, while a change from $\epsilon = 0.3$ to $\epsilon = 0.4$ increases of deformation energy by 4 MeV. This result suggests that these nuclei are prolate and soft between $\epsilon = 0$ to $\epsilon = 0.3$ and became stiff for higher prolate deformations.
The energy as a function of deformation for nuclei $^{130-132}$Sn are presented on Fig. \[fig:130132Sn\]. The assumed Nilsson’s constants for these nuclei are $\kappa_N = 0.635$, $\kappa_P = 0.067$, $\mu_N = 0.43$ and $\mu_P = 0.54$. One can see that $^{130}$Sn is softer than $^{132}$Sn. For a deformation from $\epsilon = 0.0$ to $\epsilon = 0.2$ the nucleus $^{130}$Sn spends around 5 MeV while the nucleus $^{132}$Sn, for the same deformation, spends 10 MeV. The neutron number number $N=82$ and proton numbers around $Z=50$ correspond to spherical hard nuclei.
The above characteristics of light fragments, corresponding to masses from $A = 100$ to $A = 106$, and their complementary fragments, corresponding to mass from $A = 130$ to $A = 132$, makes possible that their maximal values of the total kinetic energy of complementary fragments ($TKE$) are close to the available energy.
![Equipotential curves for scission configuration of fragments $_{42}$Mo$_{62}$, $_{50}$Sn$_{80}$ as a function of their deformation. $\epsilon_L$ and $\epsilon_H$ are the light and heavy fragment deformation. []{data-label="fig:EDECScission"}](4.eps){height="50.00000%"}
For the the case of $^{233}$U(n$_{th}$, f), the total kinetic energy of the couple $_{42}$Mo$_{62}$, $_{50}$Sn$_{80}$ is almost equal to the available energy. This results means that the corresponding scission configuration is composed by fragments in their ground state, i.e. $DE = 0.$ for each one. On the Fig. \[fig:EDECScission\] we can see the several equipotential energy of the scission configuration composed by those fragments given by the relation
$$V = CE + DE_{H}+ DE_{L},
\label{eq:VDHE}$$
where $\epsilon_H$ and $\epsilon_L$ are the heavy and light fragment deformation energy, respectively, calculated using the Nilsson model[@Nilsson] and $CE$ is the interaction Coulomb energy between the two fragments separated by 2 fm. On this curve one obtains that for$\epsilon_H = 0$ and $\epsilon_L=0.3$ the Coulomb energy is equal to the available energy to 204 MeV.
The results are similar to complementary fragments corresponding to the deformed transitional nuclei with $A_L$ between 100 and 106 ($N$ between 60 y 64) and to the spherical nuclei with $A_H$ around 132 ($Z = 50$ y $N = 82$).
For the complementary fragments $_{42}$Mo$_{62}$ and $_{50}$Sn$_{80}$, the maximal value of CE corresponds to ground state nuclei or close to that. This case is unique. Other configurations will need deformation energy, which will be higher for the harder nuclei. On the Fig. \[fig:130132Sn\] is presented the deformation energy for the spherical nuclei $^{130}$Se, $^{131}$Se and $^{132}$Se, respectively. We can see that the double magic nucleus $^{132}$Se need 2 MeV more than $^{130}$Se for going from the spherical state $\epsilon = 0$ to the slightly deformed $\epsilon = 0.05$. The fact that $^{130}$Se is no so hard as $^{132}$Se explain which the highest values of Coulomb interaction energy correspond to values close to the available energy for $^{233}$U(n$_th$,f) as well as for $^{235}$U(n$_th$,f).
Conclusion
==========
From calculations of scission configurations from thermal neutron induced fission of $^{233}U$ and $^{235}U$, one can conclude that the highest value of Coulomb interaction energy between complementary fragments correspond to fragmentations ($_{42}$Mo$_{62}$, $_{50}$Sn$_{80}$) and ($_{42}$Mo$_{64}$, $_{50}$Sn$_{80}$, respectively. For both cases the calculated maximal value of Coulomb interaction energy are equal to the available energy of the reaction for spherical ($\epsilon_H=0$) heavy fragments and prolate ($\epsilon_L=0.3$) complementary light fragments, which correspond to their ground states. Moreover the light fragments are soft between $\epsilon_L=0.0$ and $\epsilon_L=0.3$ and hard if they go to more prolate shapes; while the heavy fragment $_{50}$Sn$_{80}$ is no so hard as $_{50}$Sn$_{82}$. The calculated maximal value of Coulomb interaction energy is equal to the measured maximal value of total kinetic energy of fragments. The prescission kinetic energy and intrinsic excitation energy of fragments are assumed to be null. These results suggest that fission process take time to explore all energetically permitted scission configurations.
[99]{} P. M$\ddot{o}$ller, D. G. Madland, A. J. Sierk, A. Iwamoto, [*Nature*]{} [**409**]{} (2001) 785. F. Dickmann and K. Dietrich, [*Nucl. Phys. A*]{} [**129**]{} (1969) 241. B. D.Wilkins, E.P. Steinberg and R.R. Chasman, [*Phys. Rev. C*]{} [**14**]{} (1976) 1832.
C. Signarbieux, M. Montoya, M. Ribrag, C. Mazur, C. Guet, P. Perrin and M. Maurel [*J. Phys. Lett. (Paris) /*]{} [**42**]{}(1976)L-437 - L-440.
C. G. Nilsson, [*Mat. Fys. Medd. Dan. Vid. SelsK /*]{} [**29**]{}(1955) n.16.
W. D. Myers and W.S. Swiatecky, [*Nucl. Phys./*]{} [**81**]{}(1966)1.
V. M. Strutinsky, [*Nucl. Phys./*]{} [**A95**]{}(1967)420.
P. Quentin and R. Babinet [*Nucl. Phys./*]{} [**A159**]{}(1970)365-384 and references therein.
S.T. Belyaev,[*Mat. Fys. Medd. Vid. Selsk/*]{}, [**31**]{}(1959)n. 1.
|
---
abstract: 'Controlling waves in complex media has become a major topic of interest, notably through the concepts of time reversal and wavefront shaping. Recently, it was shown that spatial light modulators can counter-intuitively focus waves both in space and time through multiple scattering media when illuminated with optical pulses. In this letter we transpose the concept to a microwave cavity using flat arrays of electronically tunable resonators. We prove that maximizing the Green’s function between two antennas at a chosen time yields diffraction limited spatio-temporal focusing. Then, changing the photons’ dwell time inside the cavity, we modify the relative distribution of the spatial and temporal degrees of freedom (DoF), and we demonstrate that it has no impact on the field enhancement: wavefront shaping makes use of all available DoF, irrespective of their spatial or temporal nature. Our results prove that wavefront shaping using simple electronically reconfigurable arrays of reflectors is a viable approach to the spatio-temporal control of microwaves, with potential applications in medical imaging, therapy, telecommunications, radar or sensing. They also offer new fundamental insights regarding the coupling of spatial and temporal DoF in complex media.'
author:
- 'Philipp del Hougne, Fabrice Lemoult, Mathias Fink, Geoffroy Lerosey'
title: 'Spatio-temporal wavefront shaping in a microwave cavity'
---
Wave propagation in complex media is known to cause a complete scrambling of the input wavefronts, due to multiple scattering or reverberation of waves. As a result, the wave fields in these media resemble optical speckles both in space and in time [@goodman_speckle; @NatPhotReview]. Yet information transfer through complex media is crucial for many applications in telecommunications, imaging or medical therapies. Examples of a complex medium at optical frequencies include highly scattering opaque ones as well as biological tissues or multimode fibers [@choi2015WSforBioMed; @park2013WSforOCT; @choi_biomed; @pap_biomed; @bianchi_biomed; @cizmar_biomed]; in the microwave domain, forests or cities can be considered multiple scattering media, while reverberating media are also very common, ranging from reverberation chambers for electromagnetic compatibility tests, via open disordered cavities for computational imaging to indoor environments trapping wireless communication signals [@hill_electromagnetic_2009; @DavidSmith_CompImag_APL; @PhaselessCompImag_DavidSmith; @SMM_PoC]. Numerous techniques, notably time reversal and wave front shaping [@TR_fink; @EMTR_prl; @mosk_SLM; @Mosk_WS_OptTrans; @EigChan_Choi_NPhot; @BretagnePRE; @Anlage_syntheticTR; @Anlage_ExpCompTR; @Anlage_NLlossyTR; @kuhl_1DWS; @publikation1; @Gigan_NonInvImagPR; @Gigan_NonInvTM_PA; @LD_binDMD; @TM_RevMed], have been proposed to take advantage of the multiple scattering and reveberation occuring during propagation. A common ground of these approaches is that they make use of the secondary sources offered by scatterers and reflectors, which provide additional degrees of freedom (DoF), to the point that they can even outperform focusing in homogeneous media [@EMTR_science; @park2013subwavelength; @mosk_disorder4perfectFOC; @choi2011subwavelengthFOC; @Mosk_vis_subwavelength_foc_byWS].
Time reversal, which is naturally a broadband approach, results in spatio-temporal focusing of waves. As a consequence, it provides a maximum of acoustic or electromagnetic intensity at a given time and a given location that may be employed for various applications such as medical therapy, electronic warfare and wireless or underwater communications [@TR4Lithotripsy; @acoustic_bazooka; @EM_bazooka; @kuperman_oceanTR; @Kuperman_TR4underwaterComm; @Geof_TR4comm; @naqvi_TRcomm]. On the other hand, wavefront shaping, because it acts in the spatial domain, is originally a monochromatic concept that results in maxima of deposited energy at desired foci. Nevertheless, the mixing of spatial and temporal DoF in complex media was studied and exploited in acoustics, to use temporal DoF for spatial focusing [@fab_ST]. Similarly, exploiting the fact that a spatial control over a transient wave field can also act on the time dependence of the transmission through a complex medium, demonstrations of spatio-temporal focusing by wavefront shaping were recently reported in optics, initially based on closed-loop iterative optimization schemes and more recently also based on open-loop transmission matrix approaches [@aulbach_STF; @katz_STF; @gigan_STF; @dariaSciRep; @mikael_STF].
{width="\textwidth"}
In this letter, we transpose this concept back to the microwave domain, using spatial microwave modulators (SMMs), that is, arrays of resonators that are electronically reconfigurable with simple logical controls [@SMM_design]. To do so, we work in a reverberant cavity, whose surface has been partly covered by a SMM. We first prove that shaping the wave field inside the cavity in the time domain, by maximizing the envelope of the transient Green’s function between two antennas at a chosen time, results in spatio-temporally focused microwaves. We then present a parametric investigation of the enhancement in spatio-temporal focusing under well-controlled conditions. The choice of a microwave cavity as a complex medium ensures a fixed volume and different dwell times are easily explored by changing the cavity’s quality factor $Q$. As the latter governs the distribution of the cavity’s DoF between space and time, this system is a good candidate to accurately study the interplay between spatial and temporal effects. We examine the characteristics in time and space of spatio-temporal focusing for different cavity dwell times and subsequently quantify both the instantaneous signal enhancement as well as the total energy deposited at the target position by spatio-temporal wavefront shaping.
The typical experiment, illustrated schematically in Fig. 1, consists in measuring the transmission between two monopole antennas placed in a metallic reverberant cavity that is disordered (volume $1.1 \ \mathrm{m^3}$, surface area $6.6 \ \mathrm{m^2}$). One emits a pulse centered on $f_0=2.47 \ \mathrm{GHz}$ with a bandwidth of $\Delta f_{in} = 66 \ \mathrm{MHz}$, chosen to be smaller than that of the SMM. Using the other one, we record the corresponding transient Green’s function with a sampling rate oscilloscope. It contains the ballistic coherent signal, originating from the direct path, and a long exponentially decaying coda originating from multiple reverberations, whose typical duration $\tau$ equals the dwell time of the photons inside the cavity.
To achieve our objective of maximizing the envelope of the received signal at a desired time, exemplary indicated by the vertical green line in Fig. 1(a,b), we cover $7\%$ of the cavity walls with our SMM, consisting of $102$ elements whose dimensions are on the order of $\lambda_0 /2$. Each element is electronically controllable to be in either of the two states explained in the inset of Fig. 1 [@SMM_design]. At and around $f_0$, there is a difference of $\pi$ in the phase of the respective reflection coefficients. By modifying the SMM configuration, we change the distribution of times of pulse arrivals and hence the shape of the Green’s function. Iteratively we identify step by step the best configuration to satisfy our objective, using the feedback obtained from the oscilloscope [@moskWSalgo].
As exemplary shown in Fig. 1(a,b), with the above setup and procedure we can focus the transient electromagnetic energy at a desired time. The iterative procedure eventually identifies a SMM configuration that matches the phases of different paths such that they interfere constructively at this time. When working with complex media, a single realization is yet usually not at all representative of the system’s average behavior; one might for instance start from a minimum, resulting in an extraordinary enhancement. To obtain statistically significant results, many realizations over disorder are required. We conveniently achieve this by rotating the mode-stirrer shown in Fig. 1 in steps of $12^{\circ}$, which results in a completely uncorrelated “new” cavity; and we repeat the experiment for two independent positions of the emitter. This allows us to average the results of our experiments over up to $90$ realizations of the disordered cavity.
![Green’s function envelopes averaged over 60 realizations of the disordered cavities, before (thin black line) and after (thick color lines) temporal focusing of the waves using the SMM and the iterative algorithm on a set of chosen times. The series of experiments has been realized for 3 different quality factors $Q$ of the cavity. The duration of the focused peaks is always the same: that of the emitted pulse.[]{data-label="fig2"}](Fig2){width="8.6cm"}
To comparatively investigate different cavity dwell times $\tau$, electromagnetic absorbers are glued to the cavity walls (in small pieces, distributed approximately isotropically) to control the independent variable $Q$. For three different quality factors, we display in Fig. 2 averaged Green’s function envelopes before and after optimization, for a selection of optimization times. A clear temporal focusing effect is visible, at any time of the Green’s functions between the two antennas. We observe that the duration of the optimized pulse is $15 \ \mathrm{ns}$, depending neither on the chosen optimization time nor on the cavity dwell time. Similarly to time reversal experiments, we hence verify here that the temporal length of the focused wave field is of the order of that of the pulse emitted by the source antenna.
The question that arises now is: what happens spatially? To explore this, we probe the field spatially by replacing the receiving antenna with a line of antennas, separated by about $\lambda_0/10$, and of reduced length to minimize near-field coupling. Displacing a single antenna in the cavity is not feasible as it would invasively impact on the cavity’s boundary conditions. Fig. 3 illustrates averaged focusing examples (similar curves as in Fig. 2 but simultaneously probed off the objective antenna) for all three values of $Q$. Those maps exhibit again the time focusing, but surprisingly we note that the energy enhancement is also concentrated spatially around the target antenna. The temporal focusing has thus simultaneously carried out spatial focusing, with a focal width which reaches the diffraction limit of $\lambda_0/2$.
This effect, even if it appears counter-intuitive, is actually a direct consequence of the speckle-like nature of the wave field within a disordered cavity. Each frequency content excited within the cavity when the antenna emits a pulse is also spatially distributed onto speckle grains. And, in a cavity, because the boundaries isotropically surround the receiving antennas, their spatial extent is half a wavelength. The SMM allows to match the phases of the uncorrelated speckle grains at the desired position but acts randomly at every other uncorrelated position, overall resulting in the spatial focusing. Note that we have also verified that similar spatio-temporal maps are obtained whatever the optimization time, thus guaranteeing that all of the temporal results presented in Fig. 2 gave rise as well to a spatial focus onto $\lambda_0/2$ wide spots. This has a profound interest in terms of applications since one only needs to record the wave field at a given position and a single time to ensure that the waves are also spatially focused thanks to the complex nature of the propagation medium.
![Spatio-temporal maps of the Green’s functions envelopes around a given time and position chosen for wavefront shaping, for three different quality factors, and averaged over 30 realizations of disorder. Each temporal wavefront shaping experiment results in a spatio-temporally focused wave field, onto spots of dimensions $\lambda_0 /2$ in space and $1/\Delta f_{in}$ in time.[]{data-label="fig3"}](Fig3){width="8.6cm"}
Having characterized the effect of spatio-temporal focusing in space and time, we now quantify the attainable enhancements for different dwell times $\tau$ of the photons in the cavities. Firstly, we consider what we define as instantaneous spatio-temporal enhancement $\eta_A$ at the chosen optimization time $t_{\mathrm{opt}}$: $$\label{eqnEta}
\eta_A = \frac{\langle h_{\mathrm{fin}}(t_{\mathrm{opt}}) \rangle}{\langle h_{\mathrm{init}}(t_{\mathrm{opt}}) \rangle},$$ where $h$ is the measured Green’s function envelope. In Fig. 4(a) we show the average instantaneous enhancements $\eta_A$ achieved at different optimization times, for the three different values of $Q$. Note that the curves are displayed in time units normalized by $\tau$, for ease of comparison. All three curves superpose and a plateau, for which $\eta_A=6$, is obtained in all three cases after the ballistic signal but before attenuation becomes too important.
![Quantification of the impact of different cavity dwell times $\tau$ on the enhancement obtained by spatio-temporal wavefront shaping at different optimization times $t_{\mathrm{opt}}^{\mathrm{norm}}$ (for clarity in units of the cavity dwell time $\tau$ corresponding to each quality factor). (a) Instantaneous spatio-temporal enhancement $\eta_A$. (b) Deposited energy enhancement $\xi_E$. All results are based on 60 independent realizations of the disordered cavity.[]{data-label="fig4"}](Fig4){width="8.6cm"}
At this point, it is instructive to evaluate the number of spatial $N_S$ and temporal $N_T$ DoF that participate in the spatio-temporal focusing achieved for various $Q$ of the cavity. Analogous to time reversal experiments, the spatio-temporal focus is indeed the coherent sum of $N_T$ uncorrelated frequency components, each of which results from the coherent sum of $N_S$ independent overlapping eigenmodes of the cavity within a correlation frequency $f_{corr}$ [@fab_ST; @derodePRE1; @derodePRE2; @publikation1].
The number of spatial DoF within a correlation frequency is given by [@publikation1] [^1]: $$N_S = 4 \pi V \frac{f_0^2}{c^3} \times f_{corr} = 4 \pi V \frac{f_0^3}{c^3} \times \frac{1}{Q}.$$ The amount of temporal DoF is simply the number of statistically independent frequencies within the bandwidth $\Delta f_{in}$: $$N_T = \frac{\tau}{\Delta t_{in}} = \frac{\Delta f_{in}}{f_{corr}} = \frac{\Delta f_{in}}{f_0} \times Q.$$ Hence, the total number of spatio-temporal DoF can be evaluated as: $$N = N_S \times N_T = 4 \pi V \frac{f_0^2}{c^3} \times \Delta f_{in}.$$
It is important to note that, for each experiment, the total number of DoF is the same, being independent of the cavity’s quality factor $Q$. Yet we clearly see in these formulae that $Q$ controls the repartition of the system’s DoF between space and time. As a consequence, since the average behavior of the instantaneous spatio-temporal enhancement $\eta_A$ is the same for the three cases with very different distributions of DoF, we conclude from our results that spatio-temporal wavefront shaping tends to use all DoF available in the system, whether they are of temporal or of spatial nature. Stated differently, in a problem where there are few uncorrelated frequencies available, the operation will tend to focus waves spatially similarly to a monochromatic experiment, while with a large number of independent frequencies in the bandwidth, wavefront shaping will mainly result in the synchronization of these frequencies at a chosen time.
This can be underlined by quantifying how the total amount of energy received at the target position is affected by the spatio-temporal focusing. Indeed, by considering the ratio between the deposited energy after and before optimization: $$\label{eqnXi}
\xi_E = \frac{\langle \int \! h^{2}_{\mathrm{fin}}(t) \, \mathrm{d}t \rangle}{\langle \int \! h^{2}_{\mathrm{init}}(t) \, \mathrm{d}t \rangle}.$$ This removes any temporal coherent effect. We are left only with the focusing enhancement in space, hence revealing to what extent the spatio-temporal focusing profited from focusing in time.
We show in Fig. 4(b) the average energy enhancements $\xi_E$ achieved at different optimization times, for the three different values of $Q$. Now, the quality factor matters: the cavity with the lowest dwell time achieves the highest values of $\xi_E$. The enhancement of the total energy deposited at the target position by wavefront shaping thus decreases as a higher proportion of DoF is of temporal nature. This last comment allows us to point out a clear difference between time reversal and spatio-temporal wavefront shaping. Indeed, when using time reversal to focus waves in space and time, a long transient Green’s function is recorded and sent back into the medium. This generates very energetic foci, because a very long time-varying wave-field is compressed in time at the collapse time. Hence, longer dwell times of the photons inside the cavity result in higher deposited energies. This is not the case using wavefront shaping since, in any case before and after shaping the wave field, the same pulse and consequently the same amount of energy entering the medium was emitted. We finally note that total energy and spontaneous enhancements have very different temporal shapes; this is easily understandable since, when considered the total energy deposited, optimizing at early times within the Green’s function has a much bigger effect than at late times.
To conclude, in this work we have introduced the concept of spatio-temporal focusing by wavefront shaping in the microwave domain, in a cavity. We have parametrically investigated the influence of the distribution of the degrees of freedom between space and time. While the same instantaneous signal enhancement was observed for different cavity dwell times, the use of spatial DoF was more important the lower the cavity’s quality factor was, leading to higher enhancements of the total energy delivered to the target. We believe that our set of experiments carried out in the microwave domain lends new insights into related works with complex media, notably in optics where a precise control of the independent variables of the problem might be difficult or unfeasible. Moreover, transposing the concept of spatio-temporal wavefront shaping from optics to the microwave range, using very simple electronically controllable SMMs, may offer a wealth of applications. These could be found in the domains of radars, antennas, high power sources, imaging devices, or wireless communications for instance [@acoustic_bazooka; @EM_bazooka; @DavidSmith_CompImag_APL; @PhaselessCompImag_DavidSmith; @Geof_TR4comm; @naqvi_TRcomm].
P.d.H. acknowledges funding from the French “Ministère de la Défense, Direction Générale de l’Armement”. This work is supported by LABEX WIFI (Laboratory of Excellence within the French Program “Investments for the Future”) under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL\* and by Agence Nationale de la Recherche under reference ANR-13-JS09-0001-01.
[49]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ** (, ).
, , , , ****, ().
, , , , , ****, ().
, , , , , , , , , , ****, ().
, , , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, **, vol. (, ).
, , , , , , , ****, ().
, , , , , ().
, , , , ****, ().
, ****, ().
, , , , , , ****, ().
, ****, ().
, ****, ().
, , , , , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, ().
, , , , , ****, ().
, , , , ****, ().
, , , , , , ****, ().
, , , , , , , , ****, ().
, , , , ().
, , , , ****, ().
, , , , , , , , , , ****, ().
, , , ****, ().
, , , , , , , , ****, ().
, , , , , , ****, ().
, , , ****, ().
, , , , , ****, ().
, , , , ****, (), ISSN .
, , , , , , ****, ().
, , , , , , ****, ().
, , , , , , **** ().
, , , , , , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, , , , ****, ().
, , , , , , , ****, ().
, , , , , , ****, ().
, , , , , , , ****, ().
, , , , ****, ().
, ****, ().
, , , ****, ().
, , , ****, ().
[^1]: Note that we consider cases where $\Delta f_{in} \ll f_0$; hence our estimate of $N_S$ made for the central frequency $f_0$ is appropriate for any frequency within the considered bandwidth.
|
---
abstract: |
We carry out an investigation of the existence of infinitely many solutions to a fractional $p$-Kirchhoff type problem with a singularity and a superlinear nonlinearity with a homogeneous Dirichlet boundary condition. Further the solution(s) will be proved to be bounded and a weak comparison principle has also been proved. A [*‘$C^1$ versus $W_0^{s,p}$’*]{} analysis has also been discussed.
[**Keywords**]{}: singularity, non-Ambrosetti-Rabinowitz condition, Cerami condition, multiplicity, symmetric Mountain-Pass theorem.\
[**AMS Classification**]{}: 35J35, 35J60.
author:
- |
Debajyoti Choudhuri$^\ddagger$\
\
title: 'Existence and Hölder regularity of infinitely many solutions to a $p$-Kirchhoff type problem involving a singular and a superlinear nonlinearity without the Ambrosetti-Rabinowitz (AR) condition'
---
Introduction
============
Off late, the problems involving a nonlocal and fractional operators have become hugely popular area of investigation owing to its manifold applications, viz. stratified materials, population dynamics, continuum mechanics, water waves, minimal surface problems etc. Interested readers may refer to [@fisval; @ian; @barr; @moli1; @moli2; @tang; @serva1] and the references therein.\
The problem addressed in this article is as follows. $$\begin{aligned}
\label{main}
\left(a+b\int_{\mathbb{R}^N}|u(x)-u(y)|^{p}K(x-y)dxdy\right)\mathfrak{L}_p^su-\lambda g(x) u^{p-1}&=&\mu h(x)u^{-\gamma}+f(x,u),~\text{in}~\Omega\nonumber\\
u&>&0,~\text{in}~\Omega\nonumber\\
u&=&0,~\text{in}~\mathbb{R}^N\setminus\Omega\end{aligned}$$ where $\lambda,\mu>0$, $f,g\geq 0$ are functions defined and bounded over $\Omega$, $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipshitz boundary $\partial\Omega$, $a,b>0$, $0<\gamma<1$, $1<p<\infty$, $ps<N$, $s\in(0,1)$. The function $f$ is a carathéodory function and the operator $\mathfrak{L}_p^s$ is defined as $$\mathfrak{L}_p^su=2\underset{\epsilon\rightarrow 0^+}{\lim}\int_{\mathbb{R}^N\setminus B_{\epsilon}(x)}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x-y)dy$$ for all $x\in\mathbb{R}^N$ where $B_{\epsilon}=\{y:|y-x|<\epsilon\}$. The function $K:\mathbb{R}^N\setminus\{0\}\rightarrow(0,\infty)$ is measurable with the following properties $(P)$: $$\begin{aligned}
\label{kernel}
&(i)&~\mathfrak{\rho}K\in L^1(\mathbb{R}^N)~\text{where}~\mathfrak{\rho}(x)=\min\{|x|^p,1\} \nonumber\\
&(ii)&~\text{There exists}~ \delta>0~\text{such that}~K(x)\geq \delta|x|^{-N-ps}~\text{for all}~x\in\mathbb{R}^N\nonumber\\
&(iii)&~K(x)=K(-x)~\text{for all}~x\in\mathbb{R}^N\setminus\{0\}.\nonumber\end{aligned}$$ One can retrieve the fractional $p$-Laplacian operator if the [*‘kernel’*]{} $K$ is chosen to be $K(x)=|x-y|^{-N-ps}$. The discussion in Section 4.1 uses following condition. $$(P'):~\delta_1|x|^{-N-ps}\geq K(x)\geq \delta_2|x|^{-N-ps}$$ for all $x\in\mathbb{R}^N$ where $\delta_1,\delta_2>0$. In general, it is a practice to denote the Kirchhoff function as $\mathfrak{M}$. In the current case $\mathfrak{M}(t)=a+bt$. Therefore when $\mathfrak{M}(t)\equiv 1$, $p=2$, $\lambda=0$, $g(x)=1$ a.e. in $\Omega$, we reduce to the problem in to $$\begin{aligned}
\label{submain}
\begin{split}
(-\Delta)^su&=\mu u^{-\gamma}+f(x,u),~\text{in}~\Omega\\
u&>0,~\text{in}~\Omega \\
u&=0,~\text{in}~\mathbb{R}^N\setminus\Omega.
\end{split}\end{aligned}$$ For further details on the problem in one may refer to [@serva1]. The authors have used a variational technique to guarantee the existence of multiple solutions. Further results on existence of multiple solutions can be found in [@ghanmi2016nehari; @mukherjee2016dirichlet]. In most of these studies, the authors obtained two distinct weak solutions. For the existence of infinite number of solutions but with a sublinear growth function can be found in [@ghosh1].\
Meanwhile, we direct the readers to a variety of forms for the function $\mathfrak{M}$ [@bin; @alv; @xiang1; @moli3; @zuo1; @zuo2; @xiang2; @ming1; @ming2]. With the advent of these refrences and with the help of fountain theorem, the authors in [@zuo3] have proved the existence of infinitely many solutions for a fractional $p$-Kirchhoff problem. In [@nya], the authors showed the existence and multiplicity of solutions to a degenrate fractional $p$-Kirchhoff problem. Recently Ghosh [@ghosh1] has proved the existence of infinitely many solutions to a system of fractional Laplacian Kirchhoff type problem with a sublinear growth. Motivated from the work due to Ren et al [@ren1] we will show the existence of the existence of infinitely many solutions to a $p$-Kirchhoff type problem with a superlinear growth without the AR condition. It will also be proved that the solution(if exists) is in $L^{\infty}(\Omega)$. A weak comparison principle has also been proved. A little bit of history about the AR condition - this condition was first introduced by Ambrosetti and Rabinowitz (refer [@amb1]) in 1973. Thereafter this condition formed a formidable tool in the analysis of elliptic PDEs, especially to prove the boundedness of the Palais-Smale (PS) sequences for the associated energy functional to the problem. To our knowledge there is no evidence in the literature that considered a $p$-Kirchhoff type problem with a singular nonlinearity. Therefore the problem considered and the results obtained here are new.
A simple physical motivation
============================
This section is devoted to a physical motivation to the problem considered in this article. The explanation is physically heuristic but nevertheless gives a strong mathematical motivation motivation to take-up this problem. We will confine ourselves to the one dimensional case of the moel of an elastic string of finite length fixed at both the ends. The vertical displacement of the string will be represented by $u:[-1,1]\times[0,\infty)\rightarrow\mathbb{R}$. Then, mathematically, the end point constraints can beexpressed as $$u(0,t)=u(2,t)=0$$ for all $t\geq 0$. In order to identify this finite string with an infinite string one can consider $$u(x,t)=0$$ for all $x\in\mathbb{R}\setminus[0,2]$, $t\geq 0$. Thus,the acceleration $\frac{\partial^2u}{\partial t^2}u$ of the vertical displacement $u$ of the vibrating string must be balanced (thanks to Newton’s laws) by the elastic force of the string and by the external force field $f$. Therefore we have $$\frac{\partial^2u(x)}{\partial t^2}=m\frac{\partial^2u(x)}{\partial x^2}+f(x),~\text{for all}~x~\text{in}~[0,2],~ t\geq 0.$$ When the steady case is considered, we have $$m\frac{\partial^2u(x)}{\partial x^2}=f(x),~\text{in}~[0,2].$$ We quote here G.F. Carrier [@carrier] which says that - [*it is well known that the classical linearized analysis of the vibrating string can lead to results which are reasonably accurate only when the minimum (rest position) tension and the displacements are of such magnitude that the relative change in tension during the motion is small*]{}. Taking this into account one can suppose that the tension due to small deformation is linear in form, then we have the following expression. $$\mathfrak{M}(l)=m_0+2Cl$$ where $l$ is the increment in the length of the string with respect to its mean position, i.e. $$l=\int_{0}^{2}\sqrt{1+\left(\frac{\partial u}{\partial x}\right)^2}dx-2,$$ $C>0$ is a constant of proportionality. Thus for small deformations we have $$\sqrt{1+\left(\frac{\partial u}{\partial x}\right)^2}=1+\frac{1}{2}\left(\frac{\partial u}{\partial x}\right)^2.$$ Hence, $$l=\frac{1}{2}\left(\frac{\partial u}{\partial x}\right)^2.$$ Therefore, the problem now boils down to a Kirchhoff type problem $$\begin{aligned}
\left(m_0+C\frac{1}{2}\left(\frac{\partial u}{\partial x}\right)^2\right)\frac{\partial^2u(x)}{\partial x^2}&=&f(x),~\text{for all}~x~\text{in}~[0,2],~ t\geq 0\nonumber\\
u(x)&=&0,~\text{in}~\mathbb{R}\setminus[0,2].\end{aligned}$$ In other words it models the vibration of a string.
Technical preliminaries and functional analytic set up
======================================================
We begin by giving the conditions of the function $f$ which is is assumed to have a superlinear growth. Note that there are functions which are superlinear yet not satisfying the AR condition. Before that we quickly will recall the AR condition. $$\begin{aligned}
\label{ARcond}
(AR):~&\text{there exists constants}~r>0, \theta>\eta>1~\text{such that}\nonumber\\
&0<\theta F(x,t)\leq tf(x,t)~\text{for any}~x\in\Omega, t\in\mathbb{R}~\text{and}~|t|\geq r.\end{aligned}$$ Here $F(x,t)=\int_{0}^{t}f(x,t)dt$. For example, $f(x,t)=t^{r-1}\sin(t)$. From we have that $F(x,t)\geq c_1|t|^{\theta}-c_2$. for any $(x,t)\in\Omega\times\mathbb{R}$, where $\theta>\eta$ for $c_1, c_2>0$ constants. Another form is given by $$\begin{aligned}
\label{ARanotherform}
\lim_{|t|\rightarrow\infty}\frac{F(x,t)}{|t|^{\theta}}=\infty~\text{uniformly for}~x\in\Omega.\end{aligned}$$ Under the condition the function $f$ is superlinear at infinity. Observe that the example cited above satisfies but not $F(x,t)\geq c_1|t|^{\theta}-c_2$. Some important works that has proved the existence of infinitely many solutions but without the AR condition can be found in [@thin1]. we now give the asusmptions made on the function $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$. $$\begin{aligned}
\label{f_cond}
&(f_1)& \exists C>0~\text{and}~q\in(p,p_s^*)~\text{such that}~|f(x,t)|\leq C(1+|t|^{q-1})\nonumber\\
&(f_2)& f(x,-t)=f(x,t) \forall (x,t)\in\Omega\times\mathbb{R}\nonumber\\
&(f_3)& \lim_{|t|\rightarrow\infty}\frac{F(x,t)}{|t|^{2p}}=\infty~\text{uniformly for all}~x\in\Omega\nonumber\\
&(f_4)& \lim_{|t|\rightarrow 0}\frac{f(x,t)}{|t|^{p-1}}=0~\text{uniformly for all}~x\in\Omega\nonumber\\
&(f_5)& \exists \overline{t}>0~\text{such that}~t\mapsto \frac{f(x,t)}{t^{2p-1}}~\text{is decreasing if}~t\leq-\overline{t}<0\nonumber\\
& &~\text{and increasing if}~t\geq\overline{t}>0\forall x\in\Omega\nonumber\\
&(f_6)& \exists \sigma\geq 1~\text{and}~T\in L^1(\Omega)~\text{satifying}~T(x)\geq 0~\text{such that}~\mathfrak{G}(x,t)\leq\sigma\mathfrak{G}(x,t)+T(x)\nonumber\\
& &\forall x\in\Omega~\text{and}~0\leq|s|\leq|t|,~\text{where}~\mathfrak{G}(x,t)=\frac{1}{2p}tf(x,t)-F(x,t).\nonumber\end{aligned}$$
\[condprop\] The condition $(f_6)$ was assumed by Jeanjean [@jean]. When $\sigma=1$ one can see that the conditions $(f_5)$ and $(f_6)$ are equivalent. In general, there are functions (for example $f(x,t)=2p|t|^{2p-2}t\ln(1+t^{2p})+p\sin t$) that satisfy $(f_6)$ but not $(f_5)$.
We assign $Q=\mathbb{R}^{2N}\setminus C(\Omega)\times C(\Omega)\subset\mathbb{R}^{2N}$ where $C(\Omega)=\mathbb{R}^N\setminus\Omega$.The space $X$ will denote the space of Lebesgue measurable functions from $\mathbb{R}^N$ to $\mathbb{R}$ such that its restriction to $\Omega$ of any function $u$ in $X$ belongs to $L^p(\Omega)$ and $$\int_{Q}|u(x)-u(y)|^{p}K(x-y)dxdy<\infty.$$ The space $X$ is equipped with the norm $$\begin{aligned}
\|u\|_X&=&\|u\|_{L^p(\Omega)}+\left(\int_{Q}|u(x)-u(y)|^{p}K(x-y)dxdy\right)^{\frac{1}{p}}.\nonumber\end{aligned}$$ We define the subspace $X_0$ of $X$ as $$X_0=\{u\in X:u=o~\text{a.e. in}~\mathbb{R}^N\setminus\Omega\}$$ equipped with the norm $$\|u\|=\left(\int_{Q}|u(x)-u(y)|^{p}K(x-y)dxdy\right)^{\frac{1}{p}}.$$ The space $X_0$ is a Banach and a reflexive space ( refer Lemma 2.4 of [@xiang1] and Theorem 1.2 of [@adam]). We will denote the usual fractional Sobolev space by $W_0^{s,p}(\Omega)$ equipped with norm $$\|u\|_{W^{s,p}(\Omega)}=\|u\|_{L^p(\Omega)}+\left(\int_{\Omega\times\Omega}|u(x)-u(y)|^{p}K(x-y)dxdy\right)^{\frac{1}{p}}.$$ Note that the norms $\|\cdot\|_{X}$ and $\|\cdot\|_{W^{s,p}}$ are not equivalent when $K(x)=|x|^{-N-ps}$ as $\Omega\times\Omega$ is strictly contained in $Q$. This makes the space $X_0$ different from the usual fractional Sobolev space. Thus the fractional Sobolev space is insufficient for dealing with our problem from the variational method point of view.\
We now define the definition of a solution to in a weaker sense.
\[weaksoln\] A function $u\in X_0$ is a weak solution to if $hu^{-\gamma}\phi\in L^1(\Omega)$ and $$\begin{aligned}
(a+b\|u\|^p)\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))K(x-y)dxdy\nonumber\\
-\lambda\int_{\Omega}g(x)|u|^{p-2}u\phi dx-\mu\int_{\Omega}h(x)u^{-\gamma}\phi dx-\int_{\Omega}F(x,u)dx=0\nonumber\end{aligned}$$ for all $\phi\in X_0$.
Henceforth, we will mean a weak solution whenever we use the term solution.
Throughout the article any constant will be denoted by alphabets $C$ or $K$.
With these developements, we are now in a position to state our main result(s).
\[mainthm1\] Let $K:\mathbb{R}^N\setminus\{0\}\rightarrow (0,\infty)$ be a function as above and let conditions $(f_1)-(f_5)$ hold. Then, for any $\lambda\in\mathbb{R}$ and for small enough $\mu\in\mathbb{R}$ the problem in has infinitely many solutions in $X_0$ with unbounded energy.
\[mainthm2\] Let $K:\mathbb{R}^N\setminus\{0\}\rightarrow (0,\infty)$ be a function as above and let conditions $(f_1)-(f_4)$ hold. If condition $(f_6)$ is considered instead of $(f_5)$ then also the conclusion of the Theorem \[mainthm1\] holds.
\[regularity holder\] Let $u_0\in C^1(\overline{\Omega})$ which satisfies $$\label{bord}
u_0\geq K\mbox{d}(x,\partial\Omega)^s\mbox{ for some }\eta>0$$ be a local minimizer of $I$ (defined later in this section) in $C^1(\overline{\Omega})$ topology; that is, there exists $\epsilon>0$ such that, $u\in C^1(\overline{\Omega}),\;\|u-u_0\|_{C^1(\overline{\Omega})}<\epsilon$ implies $I(u_0)\leq I(u)$. Then, $u_0$ is a local minimum of $I$ in $W^{s,p}_0(\Omega)$ as well.
We quickly recall that $$\begin{aligned}
\label{eigenprob}
\mathfrak{L}_p^su&=&\lambda|u|^{p-2}u~\text{in}~\Omega\nonumber\\
u&=&0~\text{in}~\mathbb{R}^N\setminus\Omega.\end{aligned}$$ This has a divergent sequence of positive eigen values $$0\leq \lambda_1\leq \lambda_2\leq...\leq\lambda_n\leq...$$ which has eigen values say $(e_n)_{n\in\mathbb{N}}$. Refer to Proposition $9$ of [@serva3] where it has been shown that this sequence provides an orthonormal basis in $L^p(\Omega)$ and an orthogonal basis in $X_0$.\
We first define $$\begin{aligned}
\label{energy_functional}
I(u)&=&A(u)-B(u)-C(u)\end{aligned}$$ where $$\begin{aligned}
\begin{split}
A(u)&=\frac{a}{p}\|u\|^p+\frac{b}{2p}\|u\|_{X_0}^{2p}\\
B(u)&=\frac{\lambda}{p}\int_{\Omega}g(x)|u|^pdx\\
C(u)&=\frac{\mu}{1-\gamma}\int_{\Omega}h(x)u^{1-\gamma}dx+\int_{\Omega}F(x,u)dx.
\end{split}\end{aligned}$$ As said earlier $F(x,t)=\int_{0}^tf(x,t)dx$. From here owards we will denote $\|\cdot\|_{L^p(\Omega)}$ as $\|\cdot\|_p$. To our dissatisfaction, the functional $I$ is not a $C^1$ functional. However we redefine this functional $I$ as follows. Define $$\overline{f}(x,t) =
\begin{cases}
\mu h(x)t^{-\gamma}+f(x,t), &~\text{if}~t>\underline{u}_{\mu}\\
\mu h(x)\underline{u}_{\mu}^{-\gamma}+f(x,\underline{u}_{\mu}),&~\text{if}~t\leq \underline{u}_{\mu}
\end{cases}$$ where $f(x,t)=\frac{\mu g(x)}{t^{\gamma}}+f(x,t)$ and $\underline{u}_{\mu}$ is a solution to $$\begin{aligned}
\label{auxprob}
\left(a+b\int_{\mathbb{R}^N}|u(x)-u(y)|^{p}K(x-y)dxdy\right)\mathfrak{L}_p^su-\lambda g(x)u^{p-1}&=&f(x,u),~\text{in}~\Omega\nonumber\\
u&>&0,~\text{in}~\Omega\nonumber\\
u&=&0,~\text{in}~\mathbb{R}^N\setminus\Omega\nonumber\\\end{aligned}$$ whose existence can be guaranteed from [@ren1]. Let $\overline{F}(x,s)=\int_{0}^{s}\overline{f}(x,t)ds$. Now the functional $$\begin{aligned}
\label{modi_func}
\overline{I}(u)&=&A(u)-B(u)-\overline{C}(u)
\end{aligned}$$ where $\overline{C}(u)=\int_{\Omega}\overline{F}(x,u)dx$. The way the functional has been defined, it is easy to see that the critical points of the functional in are also the critical points of the functional . Most importantly, the functional $\overline{I}$ which is defined in is $C^1$ which allows us to use the variational methods. Further $$\begin{aligned}
\label{der_modi_func}
\langle \overline{I}'(u),\phi \rangle&=&(a+b\|u\|^p)\int_{Q}|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))K(x-y)dxdy\nonumber\\
& &-\lambda\int_{\Omega}g(x)|u|^{p-2}u\phi dx-\int_{\Omega}h(x)\overline{f}(x,u)\phi dx
\end{aligned}$$ for all $\phi\in X_0$.\
The following lemma will be used in this work.
\[embres\] (Refer Lemma 2.4 [@ghosh_jmp]) Let the kernel $K$ be as above. We then have the following.
1. For any $r\in[1,p_s^*)$, the embedding $X_0\hookrightarrow L^r(\Omega)$ is compact when $\Omega$ is bounded with smooth enough boundary.
2. For all $r\in[1,p_s^*]$, the embedding $X_0\hookrightarrow L^r(\Omega)$ is continuous.
Following is the definition of Cerami condition (Definition 2.1 [@soni1]).
\[cerami\] \[Cerami condition\] Let $I$ be a $C^1(X_0,\mathbb{R})$ functional. $I$ is said to satisfy the $(Ce)_c$ at level $c\in\mathbb{R}$, if any sequence $(u_n)\subset X_0$ for which $I(u_n)\rightarrow c$ in $X_0$, $I'(u_n)\rightarrow 0$ in $X_0'$, the dual space of $X_0$, as $n\rightarrow\infty$, then there exists a strongly convergent subsequence of $(u_{n_k})$ of $(u_n)$ in $X_0$.
Henceforth, a subsequence of any sequence, say $(v_n)$, will also be denoted by $(v_n)$.
We now give the symmetric mounain pass theorem [@col].
\[symmMPT\] (Symmetric mountain pass theorem) Let $X$ be an infinite dimensional Banach space. $\overline{Y}$ is a finite dimensional Banach space and $X=Y \bigoplus Z$. For any $c>0$ if $I\in C^1(X,\mathbb{R})$ satisfies $(Ce)_c$ and
1. $I$ is even and $I(0)=0$ for all $u\in X$
2. There exists $r>0$ such that $I(u)\geq R$ for all $u\in B_r(0)=\{u\in X:\|u\|_X\leq r\}$
3. For any finite dimensional subspace $\overline{X}\subset X$, there exists $r_0=r(\overline{X})>0$ such that $I(u)\leq 0$ on $\overline{X}\setminus B_{r_0}(0_{\overline{X}})$, where $0_{\overline{X}}$ is the null vector in $\overline{X}$
then there exists an unbounded sequence of critical values of $I$ characterized by a minimax argument.
Auxilliary and main result
==========================
We begin this section by proving a few auxilliary lemmas.
\[Ce\_lemma\] Let $(f_1)$ hold. Then any bounded sequence $(u_n)$ in $X_0$ which satisfies $(1+\|u_n\|)I'(u_n)\rightarrow 0$ as $n\rightarrow\infty$ possesses a strongly convergent subsequence in $X_0$.
Let $(u_n)$ be a bounded sequence in $X_0$. Since $X_0$ is reflexive we have $$\begin{aligned}
\label{conv1}
\begin{split}
& u_n\rightharpoonup u~\text{in}~X_0,\\
& u_n\rightarrow u~\text{in}~L^r(\Omega), 1\leq r< p_s^*,\\
& u_n\rightarrow u~\text{a.e. in}~\Omega.
\end{split}\end{aligned}$$ All we need to prove is that $u_n\rightarrow u$ strongly in $X_0$. By the Hölder’s inequality we have $$\begin{aligned}
\label{holdineq1}
\begin{split}
0\leq \int_{\Omega}|f(x,u_n)|(u_n-u)dx&\leq\int_{\Omega}C(1+|u_n|^{q-1})(u_n-u)dx\\
&\leq C (|\Omega|^{\frac{q-1}{q}}+\|u_n\|_q^{q-1})\|u_n-u\|_{q}.
\end{split}\end{aligned}$$ By we obtain $$\underset{n\rightarrow\infty}{\lim}\int_{\Omega}|f(x,u_n)|(u_n-u)dx=0.$$ Further we have from the Hölder’s inequality that $$\int_{\Omega}|u_n|^{p-2}u_n(u_n-u)dx\leq\|u_n\|_p^{p-1}\|u_n-u\|_p.$$ Therefore again from we have $$\underset{n\rightarrow\infty}{\lim}\int_{\Omega}|u_n|^{p-2}u_n(u_n-u)dx=0.$$ Note that $$\underset{n\rightarrow\infty}{\lim}|u_n^{1-\gamma}-uu_n^{-\gamma}| =
\begin{cases}
\underset{n\rightarrow\infty}{\lim}|u_n^{1-\gamma}-uu_n^{-\gamma}| , &~\text{if}~u_n>\underline{u}_{\mu}\\
\underset{n\rightarrow\infty}{\lim}|\underline{u}_{\mu}^{-\gamma}u_n-u\underline{u}_{\mu}^{-\gamma}| ,&~\text{if}~u_n\leq \underline{u}_{\mu}
\end{cases}$$ This $$\underset{n\rightarrow\infty}{\lim}\int_{\Omega}u_n^{-\gamma}(u_n-u)dx=0$$ in either cases. We now define a linear functional $$J_v(u)=\int_{\mathbb{R}^N}|v(x)-v(y)|^{p-2}(v(x)-v(y))(u(x)-u(y))K(x-y)dxdy.$$ The Hölder inequality yields $$|J_{v}(u)|\leq\|v\|_{X_0}^{p-1}\|u\|_{X_0}$$ thereby implying that $J$ is a continuous linear functional on $X_0$. Therefore $$\underset{n\rightarrow\infty}{\lim}J_v(u_n-u)=0.$$ From the discussion so far in the proof of this theorem and since $u_n\rightharpoonup u$ in $X_0$, we conclude that $\langle\overline{I}'(u_n),u_n-u\rangle\rightarrow 0$ as $n\rightarrow\infty$. Also $(1+\|u_n\|_{X_0})\overline{I}'(u_n)\rightarrow 0$ in $X_0'$. Therefore, we have $$\begin{aligned}
\begin{split}
o(1)=&\langle\overline{I}'(u_n),u_n-u\rangle-\lambda\int_{\Omega}g(x)|u_n|^{p-2}u_n(u_n-u)dx\\
&-\mu\int_{\Omega}h(x)u_n^{-\gamma}(u_n-u)dx-\int_{\Omega}f(x,u_n)(u_n-u)dx\\
=&(a+b\|u_n\|^p)J_{u_n}(u_n-u)+o(1)~\text{as}~n\rightarrow\infty.
\end{split}\end{aligned}$$ Hence, by the boundedness of $(u_n)$ in $X_0$ and $\underset{n\rightarrow\infty}{\lim}J_v(u_n-u)=0$ we have $$\begin{aligned}
\underset{n\rightarrow\infty}{\lim}(J_{u_n}(u_n-u)-J_{u}(u_n-u))&=&0.\end{aligned}$$ Recall the Simon inequalities which is as follows. $$\begin{aligned}
\label{simon}
\begin{split}
|U-V|^{P}&\leq C_p(|U|^{p-2}U-|V|^{p-2}V).(U-V),~p\geq 2\\
|U-V|^{P}&\leq C_p'(|U|^{p-2}U-|V|^{p-2}V)^{\frac{p}{2}}.(|U|^p+|V|^p)^{\frac{2-p}{2}},~1<p<2
\end{split}\end{aligned}$$ for all $U,V\in\mathbb{R}^N$, $C_p, C_p'>0$ are constants.\
When $p\geq 2$, we have $$\begin{aligned}
\begin{split}
\|u_n-u\|^p\leq& C_p\int_{\mathbb{R}^{2N}}[|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))-|u(x)-u(y)|^{p-2}(u(x)-u(y))]\\
&\times [(u_n(x)-u_n(y))-(u(x)-u(y))]K(x-y)dxdy\\
=&C_p[J_{u_n}(u_n-u)-J_u(u_n-u)]\rightarrow 0~\text{as}~n\rightarrow\infty.
\end{split}\end{aligned}$$ When $1<p<2$ $$\begin{aligned}
\begin{split}
\|u_n-u\|^p\leq& C_p'[J_{u_n}(u_n-u)-J_{u}(u_n-u)]^{\frac{p}{2}}(\|u_n\|^p+\|u\|^p)^{\frac{2-p}{2}}\\
\leq&C_p'[J_{u_n}(u_n-u)-J_{u}(u_n-u)]^{\frac{p}{2}}(\|u_n\|^{\frac{2-p}{2}}+\|u\|^{\frac{2-p}{2}})\rightarrow 0~\text{as}~n\rightarrow\infty.
\end{split}\end{aligned}$$ Thus $u_n\rightarrow u$ strongly in $X_0$ as $n\rightarrow\infty$.
Let $(f_1), (f_3), (f_5)$ hold. Then the functional $\overline{I}$ satisfies the $(Ce)_c$ condition.
Let $(f_5)$ hold. From the Remark \[condprop\] there exists $C_1>0$ such that $$\begin{aligned}
\mathfrak{G}(x,s)&\leq&\mathfrak{G}(x,t)+C,\end{aligned}$$ for all $x\in\Omega$ and $0\leq |s|\leq |t|$. Refer for the definition of $\mathfrak{G}(.,.)$. Let $(u_n)$ be a Cerami sequence in $X_0$. Thus $$\begin{aligned}
\label{Ce_cond1}
\overline{I}(u_n)\rightarrow c~\text{in}~X_0\nonumber\\
(1+\|u_n\|)\overline{I}'(u_n)\rightarrow 0~\text{in}~X_0'\nonumber\end{aligned}$$ as $n\rightarrow\infty$. All we need to show is that the sequence $(u_n)$ is bounded in $X_0$. The conclusion will follow from the Lemma \[Ce\_lemma\].\
Suppose not, i.e. then upto a subsequence atleast we have $\|u_n\|_{X_0}\rightarrow\infty$. By the second condition of we have $$\overline{I}'(u_n)\rightarrow 0$$ as $n\rightarrow\infty$. Hence $$\|u_n\|\left\langle \overline{I}'(u_n),\frac{u_n}{\|u_n\|} \right\rangle\rightarrow 0$$ as $n\rightarrow\infty$. We define $\xi_k=\frac{u_n}{\|u_n\|}$ so that $\|\xi_n\|=1$. Thus $(\xi_n)$ is a bounded sequence and hence $$\xi_n\rightarrow \xi~\text{in}~L^{p}(\Omega)$$ and upto a subsequence $$\xi_n\rightarrow \xi~\text{a.e. in}~ \Omega.$$ Further, from Lemma A.1 of [@xiang5] there exists a function $\alpha(x)$ such that $$|\xi_n(x)|\leq\alpha(x)~\text{in}~\mathbb{R}^N.$$ This lead to the consideration of two cases, [*viz.*]{} $\xi=0$ and $\xi\neq 0$. Since $\overline{I}$ is a $C^1$ functional, therefore $\overline{I}(\alpha_n u_n)=\underset{\alpha l\in[0,1]}{\max}\overline{I}(\alpha u_n)$.\
Let $\xi=0$ and define $h_T=\left(\frac{4p T}{b}\right)^{\frac{1}{2p}}$ such that $\frac{h_{T}}{\|u_n\|}\in(0,1)$ for $T\in\mathbb{N}$ and sufficinetly large $n$. Since $\xi=0$ and $\xi_n\rightarrow \xi~\text{a.e. in}~ \Omega$ we have $$\begin{aligned}
\label{conv2}
\int_{\Omega}|h(T)\xi_n|^pdx\rightarrow 0.\end{aligned}$$ By the continuity of $F$ we obtain $$\begin{aligned}
\label{conv3}
F(x,h_T\xi_n(x))\rightarrow F(x,h_T\xi(x))=F(x,0)~\text{in}~\Omega~\text{as}~n\rightarrow\infty.\end{aligned}$$ From $(f_1)$ and $|\xi_n(x)|\leq\alpha(x)~\text{in}~\mathbb{R}^N$ in combination with the Hölder inequality we get $$\begin{aligned}
\label{con4}
|F(x,h_T\xi_n)|&\leq&C|h_T\alpha(x)|+\frac{C}{q}|h_T\alpha(x)|^q~\in L^1(\Omega)\end{aligned}$$ for any $n,T\in\mathbb{N}$. Therefore from the Lebesgue dominated convergence theorem we get $$\begin{aligned}
\label{conv5}
F(.,h_T\xi_n(.))\rightarrow F(.,h_T\xi(.))~\text{in}~L^1(\Omega)~\text{as}~n\rightarrow\infty.\end{aligned}$$ for any $T\in\mathbb{N}$. Thus $$\int_{\Omega}F(x,h_T\xi_n(x))dx\rightarrow 0~\text{as}~n\rightarrow\infty.$$ We further have $$\begin{aligned}
\label{conv6}
\overline{I}(\alpha_nu_n)&\geq&\overline{I}\left(\frac{h_T}{\|u_n\|}u_n\right)\nonumber\\
&=&\overline{I}(h_T\xi_n)\nonumber\\
&\geq&\frac{a}{p}\|h_T\xi_n\|^p+\frac{b}{2p}\|h_T\xi_n\|^{2p}-\frac{\lambda\|f\|_{\infty}}{p}\int_{\Omega}|h_T\xi_n|^pdx\nonumber\\
& &-\frac{\mu\|g\|_{\infty}}{1-\gamma}\int_{\Omega}|h_T\xi_n|^{1-\gamma}dx-\int_{\Omega}F(x,h_T\xi_n)dx\nonumber\\
&\geq&\frac{b}{2p}\|h_T\xi_n\|^{2p}=2T.\end{aligned}$$ Therefore $$\begin{aligned}
\label{conv6'}\overline{I}(\alpha_nu_n)\rightarrow\infty\end{aligned}$$ as $n\rightarrow\infty$. We now show that $$\underset{n\rightarrow\infty}{\lim}\sup \overline{I}(\alpha_nu_n)\leq\beta$$ for some $\beta>0$. Observe that $\int_{\Omega}\frac{|u_n|^p}{\|u_n\|^p}dx\leq C$. Since the boundary $\partial\Omega$ is Lipshitz continuous and $\frac{1}{\|u_n\|}\rightarrow 0$ as $n\rightarrow\infty$ we have that $$\begin{aligned}
\label{conv7}
\begin{split}
&\left|\frac{\lambda}{2p}\int_{\Omega}g(x)|u_n|^pdx\right|\leq C(\lambda,p)\|g\|_{\infty}\\
&\left|\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_nu_n)^{1-\gamma}dx\right|\leq C(\mu,\gamma,p)\|h\|_{\infty}.
\end{split}\end{aligned}$$ Since $\frac{d}{d\alpha}|_{\alpha=\alpha_n}\overline{I}(\alpha u_n)=0$ for all $n$ we have $$\begin{aligned}
\label{conv8}
\langle \overline{I}'(\alpha_nu_n),\alpha_nu_n \rangle&=&\alpha_n\frac{d}{d\alpha}|_{\alpha=\alpha_n}\overline{I}(\alpha u_n)=0.\end{aligned}$$ We further have $$\begin{aligned}
\label{conv9}
\overline{I}(\alpha_nu_n)&=&\overline{I}(\alpha_nu_n)-\frac{1}{2p}\langle \overline{I}'(\alpha_nu_n),\alpha_nu_n \rangle\nonumber\\
&=&\frac{a}{2p}\|\alpha_nu_n\|^p-\frac{\lambda}{2p}\int_{\Omega}g(x)|\alpha_nu_n|^pdx-\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_nu_n)^{1-\gamma}dx\nonumber\\
& &-\int_{\Omega}F(x,\alpha_nu_n)dx+\frac{1}{2p}\int_{\Omega}f(x,\alpha_nu_n)\alpha_nu_ndx\nonumber\\
&\leq&\frac{a}{2p}\|\alpha_nu_n\|^p+C(\lambda,p)+C(\mu,\gamma,p)+\int_{\Omega}\mathfrak{G}(x,\alpha_nu_n)dx\nonumber\\
&\leq&\frac{a}{2p}\|\alpha_nu_n\|^p+C(\lambda,p)+C(\mu,\gamma,p)+\int_{\Omega}\mathfrak{G}(x, u_n)dx+C|\Omega|\nonumber\\
&=&\frac{a}{2p}\|\alpha_nu_n\|^p+C(\lambda,p)+C(\mu,\gamma,p)+\int_{\Omega}\mathfrak{G}(x, u_n)dx+C|\Omega|\nonumber\\
& &-\frac{\lambda}{2p}\int_{\Omega}f(x)|\alpha_nu_n|^pdx+\frac{\lambda}{2p}\int_{\Omega}f(x)|\alpha_nu_n|^pdx\nonumber\\
& &-\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}g(x)(\alpha_nu_n)^{1-\gamma}dx
+\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}g(x)(\alpha_nu_n)^{1-\gamma}dx\nonumber\nonumber\\
& &+C|\Omega|\nonumber\\
&\leq&\overline{I}(\alpha_nu_n)-\frac{1}{2p}\langle \overline{I}'(\alpha_nu_n),\alpha_nu_n+2C(\lambda,p)+2C(\mu,\gamma,p)+C|\Omega|\nonumber\\
&=&c+o(1)+2C(\lambda,p)+2C(\mu,\gamma,p)+C|\Omega|<\infty~\text{as}~n\rightarrow\infty.\end{aligned}$$ where $|\Omega|$ is the lebesgue measure of $\Omega$. This is a contradiction to . Thus $(u_n)$ is bounded $X_0$.\
Let $\xi_n\neq 0$. Define $$A=\{x\in\Omega:\xi(x)\neq 0\}.$$ Therefore e have $$\begin{aligned}
\label{conv10}
|u_n(x)|&=&|\xi_n(x)|\|u_n\|\rightarrow\infty~\text{in}~A~\text{as}~n\rightarrow\infty.\end{aligned}$$ Further from $(f_3)$ $$\begin{aligned}
\label{conv11}
\frac{G(x,u_n(x))}{\|u_n\|^{2p}}&=&\frac{G(x,u_n(x))}{|u_n(x)|^{2p}}\frac{|u_n(x)|^{2p}}{\|u_n\|^{2p}}\nonumber\\
&=&\frac{G(x,u_n(x))}{|u_n(x)|^{2p}}|\xi_n|^{2p}\rightarrow\infty~\text{in}~A~\text{as}~n\rightarrow\infty.\end{aligned}$$ By the Fatou’s lemma $$\begin{aligned}
\label{conv12}
\int_{\Omega}\frac{G(x,u_n(x))}{\|u_n\|^{2p}}dx\rightarrow\infty.\end{aligned}$$ Let us now analyse over $\Omega\setminus A$. Thus from $(f_3)$ again we have $$\underset{|t|\rightarrow\infty}{\lim}F(x,t)=\infty$$ for $x\in\Omega$. Therefore for arbitrary $M>$ there exists $t'$ such that $$\begin{aligned}
\label{conv13}
F(x,t)\geq M~\text{whenever}~|t|\geq t', x\in\Omega. \end{aligned}$$ Hence $$\begin{aligned}
\label{conv14}
F(x,t)&\geq\min\left\{M, \underset{(x,t)\in\Omega\times[-t',t']}{\min}\{F(x,t)\}\right\}=M'~\text{say}.\end{aligned}$$ So we get $$\begin{aligned}
\label{conv15}
\underset{|t|\rightarrow\infty}{\lim}\int_{\Omega\setminus A}\frac{F(x,u_n(x))}{\|u_n\|^{2p}}dx\geq 0.\end{aligned}$$ Also $$\begin{aligned}
\label{conv15'}
0\leq\frac{\mu}{1-\gamma}\int_{\Omega}h(x)\frac{u_n^{1-\gamma}}{\|u_n\|^{2p}}dx&\leq&\frac{\mu\|h\|_{\infty}}{1-\gamma}\|u_n\|^{1-\gamma-2p}=o(1).\end{aligned}$$ By the variational characterization of $\lambda_j$, the $j$-th eigen value of $\mathfrak{L}_p^s$, i.e. $$\lambda_j=\underset{u\in X_0\setminus\{0\}}{\min}\left\{\frac{\int_{\mathbb{R}^{2N}}|u(x)-u(y)|^{p}K(x-y)dxdy}{\int_{\Omega}f(x)|u(x)|^pdx}\right\}$$ where $g>0$ and bounded in $\Omega$. Now since $$\begin{aligned}
\label{conv16}
\begin{split}
o(1)&=\frac{\overline{I}(u_n)}{\|u_n\|^{2p}}\\
=&\frac{a}{p\|u_n\|^{p}}+\frac{b}{2p}-\frac{\lambda}{p}\int_{\Omega}f(x)\frac{|u_n(x)|^p}{\|u_n\|^{2p}}dx\\
&-\frac{\mu}{1-\gamma}\int_{\Omega}g(x)\frac{u_n^{1-\gamma}}{\|u_n\|^{2p}}dx-\int_{\Omega}\frac{F(x,u_n)}{\|u_n\|^{2p}}dx-\int_{\Omega\setminus A}\frac{F(x,u_n)}{\|u_n\|^{2p}}dx\\
\leq&o(1)+\frac{b}{2p}+\frac{\|g\|_{\infty}\lambda}{p\lambda_j}\frac{1}{\|u_n\|^{p}}-\frac{\mu}{1-\gamma}\int_{\Omega}h(x)\frac{u_n^{1-\gamma}}{\|u_n\|^{2p}}dx-\int_{\Omega}\frac{F(x,u_n)}{\|u_n\|^{2p}}dx-\int_{\Omega\setminus A}\frac{F(x,u_n)}{\|u_n\|^{2p}}dx\\
\leq&-\infty
\end{split}\end{aligned}$$ where the last step is due to , , . This is absolutely a contradiction!. Thus the sequence $(u_n)$ is bounded in $X_0$ and hence by the Lemma \[Ce\_lemma\] we conclude that $(u_n)$ possesses a strongly convergent subsequence in $X_0$.
We now prove the results stated in the Theorems \[mainthm1\] and \[mainthm2\]. For this let us develope some prerequisites. It is well known that the space $X_0$ is a Banach space and we have that $$X_0=\underset{i\geq 1}{\bigoplus}X_i$$ where $X_{i}=\text{span}\{e_j\}_{j\geq i}$. Define $$Y_m=\underset{1\leq j \leq m}{\bigoplus}X_j$$ $$Z_m=\underset{j\geq m}{\bigoplus}X_j.$$ Clearly $Y_m$ is a finite dimensional sub space of $X_0$.
\[conv17\] Let $\kappa\in[1,p_s^*)$. We have $$\zeta(\kappa)=\sup\{\|u\|_{\kappa}:u\in Z_m,\|u\|=1\}\rightarrow 0$$ as $m\rightarrow\infty$.
From the definition of $(Z_m)$ we have that $Z_m\subset Z_{m+1}$ and thus $0\leq\zeta_m\leq\zeta_{m+1}$. This implies that $\zeta_m\rightarrow\zeta\geq 0$ as $m\rightarrow\infty$. Further by the definition of [*supremum*]{} for every $m$ there exists $u_m\in Z_m$ such that $\|u_m\|$ and $\|u_m\|_{\kappa}>\frac{\zeta_m}{2}$. By the reflexivity of $X_0$ the sequence $(u_m)$ is weakly convergent to, say, $u$. Obviously $u=0$ as $\langle u_m,v \rangle\rightarrow 0$ for every $v\in Y_m$. By the embedding results from Lemma \[embres\] we have that $u_m\rightarrow 0$ in $L^{\kappa}(\Omega)$ for any $\kappa\in[1,p_s^*)$. Thus we conclude that $\zeta=0$.
[*Proof of Theorem \[mainthm1\]*]{}: Since $Y_m$s are all finite dimensional spaces, hence the norms $\|\cdot\|$ and $\|\cdot\|_{\kappa}$ are equivalent for $\kappa\in[1,p_s^*)$.\
Observe that $\overline{I}(u)=0$. If $\frac{\lambda}{a}>\lambda_1$, then from the definition of $\lambda_j$ given in the previous theorem, there exists $\lambda_j$ such that $\frac{\lambda}{a}\in[\lambda_{j-1},\lambda_j)$, $\frac{\mu}{a}\in[\lambda_{k-1},\lambda_k)$. Further, from $(f_1)$ and $(f_4)$, for every $\epsilon>0$ there exists $C_{\epsilon}>0$ such that $$\begin{aligned}
\label{conv18}
F(x,t)&\leq&\frac{\epsilon}{p}|t|^{p}+\frac{C_{\epsilon}}{q}|t|^q,\end{aligned}$$ for any $(x,t)\in\Omega\times\mathbb{R}$. By the definiton of $\zeta_m$ in Lemma \[conv17\], fix $\epsilon>0$ and choose $m'\geq 1$ such that $$\begin{aligned}
\label{conv19}
\begin{split}
\|u\|_p^p&\leq\frac{a-\frac{\lambda}{\lambda_j}}{2\|f\|_{\infty}\epsilon}\|u\|^p\\
\|u\|_q^q&\leq\frac{q(a-\frac{\lambda}{\lambda_j})}{2p C_{\epsilon}}\|u\|^q~\text{for every}~u\in Z_{m'}.
\end{split}\end{aligned}$$ Choose $r<1$, in Lemma \[symmMPT\]. Since $q>p$, we have $$\begin{aligned}
\label{conv20}
\begin{split}
\overline{I}(u)&=\frac{a}{p}\|u\|^p+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p}\int_{\Omega}f(x)|u(x)|^pdx-\frac{\mu}{1-\gamma}\int_{\Omega}g(x)u^{1-\gamma}dx-\int_{\Omega}F(x,u)dx\\
&\geq \frac{a}{p}\|u\|^p-\frac{\lambda}{p\lambda_j}\|u\|^p-\frac{\mu\|g\|_{\infty}}{1-\gamma}\|u\|^{1-\gamma}-\frac{\epsilon}{p}\|u\|_p^{p}-\frac{C_{\epsilon}}{q}\|u\|_q^q\\
&\geq \left(\frac{a}{p}-\frac{\lambda}{p\lambda_j}\right)\|u\|^p-\frac{\mu\|g\|_{\infty}}{1-\gamma}\|u\|^{1-\gamma}-\frac{a-\frac{\lambda}{\lambda_j}}{2p}\|u\|^{p}-\frac{a-\frac{\lambda}{\lambda_j}}{2p}\|u\|^q\\
&\geq \frac{a-\frac{\lambda}{\lambda_j}}{2p}\|u\|^p-\frac{a-\frac{\lambda}{\lambda_j}}{2p}\|u\|^q-\frac{C\mu\|g\|_{\infty}}{1-\gamma}\|u\|^{1-\gamma}\\
&=\frac{a-\frac{\lambda}{\lambda_j}}{2p}(r^p-r^q)-\frac{\mu\|g\|_{\infty}}{1-\gamma}r^{1-\gamma}=R>0
\end{split}\end{aligned}$$ for sufficiently small $\mu>0$. Finally, from $(f_3)$, there exists $A>\frac{b}{2pC^{2p}}$ (a possible choice of $C$, as we shall see later, is a Sobolev constant), $B>0$ such that $$\begin{aligned}
\label{conv21}
F(x,t)&\geq&A|t|^{2p}\end{aligned}$$ for any $x\in\Omega$ and $|t|>B$. From $(f_1)$ we have $$\begin{aligned}
\label{conv22}
|F(x,t)|&\leq&C(1+B^{q-1})|t|~\text{for every}~x\in\Omega~{and}~|t|\leq B.\end{aligned}$$ Let $C'=C(1+B^{q-1})>0$. Then we get $$\begin{aligned}
\label{conv23}
F(x,t)&\geq&A|t|^{2p}-C'|t|,~\text{for}~(x,t)\in\Omega\times\mathbb{R}.\end{aligned}$$ By the equivalence of norm in $Y_m$ and , we have $$\begin{aligned}
\label{conv24}
\begin{split}
\overline{I}(u)=&\frac{a}{p}\|u\|^p+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p}\int_{\Omega}f(x)|u|^pdx -\frac{\mu}{1-\gamma}\int_{\Omega}g(x)u^{1-\gamma}dx\\
&-\int_{\Omega}F(x,u)dx\\
\leq&\frac{a}{p}\|u\|^p+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p}\int_{\Omega}f(x)|u|^pdx -\int_{\Omega}F(x,u)dx\\
\leq&\frac{a}{p}\|u\|^p+\frac{b}{2p}\|u\|^{2p}-\frac{\lambda}{p\lambda_j}\|u\|^p-A\|u\|_{2p}^{2p}+C'\|u\|_1\\
\leq&\frac{a}{p}\|u\|^p+\left(\frac{b}{2p}-AC^{2p}\right)\|u\|^{2p}+C_2C'\|u\|.
\end{split}\end{aligned}$$ Thus for a sufficiently large $r_0=r(\overline{X})$, we have $\overline{I}(u)\leq 0$ whenever $\|u\|\geq r_0$.\
\
[*Proof of Theorem \[mainthm2\]*]{}: Suppose now $(f_6)$ holds. The proof follows [*verbatim*]{} of the Theorem \[mainthm1\], except that we need to prove the inequality in . Thus we have $$\begin{aligned}
\label{conv25}
\begin{split}
\frac{1}{\sigma}\overline{I}(\alpha_nu_n)=&\frac{1}{\sigma}\left(\overline{I}(\alpha_nu_n)-\frac{1}{2p}\langle \overline{I}'(\alpha_nu_n),\alpha_nu_n \rangle\right)\\
=& \frac{1}{\sigma}\left[\left(\frac{a}{2p} \right)\|\alpha_nu_n\|^p+\frac{\lambda}{2p}\int_{\Omega}g(x)|\alpha_nu_n|^pdx-\mu\left(\frac{1}{1-\gamma}-\frac{1}{2p}\right)\int_{\Omega}h(x)(\alpha_nu_n)^{1-\gamma}dx\right. \\
&\left.-\int_{\Omega}F(x,\alpha_nu_n)dx+\frac{1}{2p}\int_{\Omega}f(x,\alpha_nu_n)\alpha_nu_ndx\right]\\
\leq&\frac{1}{\sigma}\left[\left(\frac{a}{2p} \right)\|\alpha_nu_n\|^p+\int_{\Omega}\mathfrak{G}(x,\alpha_nu_n)dx\right]+C(\lambda,p)\|g\|_{\infty}\\
\leq&\left(\frac{a}{2p} \right)\|\alpha_nu_n\|^p+\int_{\Omega}\mathfrak{G}(x,\alpha_nu_n)dx+\frac{1}{\sigma}\int_{\Omega}T(x)dx+C(\lambda,p)\|g\|_{\infty}\\
\leq&\left(\frac{a}{2p} \right)\|\alpha_nu_n\|^p+\int_{\Omega}\mathfrak{G}(x,\alpha_nu_n)dx+\frac{1}{\sigma}\int_{\Omega}T(x)dx+C(\lambda,p)\|g\|_{\infty}\\
&-\frac{\lambda}{2p}\int_{\Omega}|u_n|^pdx+\frac{\lambda}{2p}\int_{\Omega}|u_n|^pdx\\
\leq&\overline{I}_{u_n}-\frac{1}{2p}\langle \overline{I}'(\alpha_nu_n),\alpha_nu_n \rangle +2C(\lambda,p)\|g\|_{\infty}+\frac{1}{\sigma}\int_{\Omega}T(x)dx\\
\leq& c+o(1)+2C(\lambda,p)\|g\|_{\infty}+\frac{1}{\sigma}\int_{\Omega}T(x)dx<\infty.
\end{split}\end{aligned}$$ Hence the proof.\
\
We will now show that the solution $u$ to is in $L^{\infty}(\Omega)$, i.e. bounded. Firstly, let us recall the following elementary inequality needed for the proof of the $L^\infty$ estimate.
\[ineq1\] (Lemma 5.1 in [@ghosh_jmp]) For all $c$, $d\in\mathbb{R}$, $\rho\geq p$, $p\geq 1$, $k>0$ we have $$\begin{aligned}
\begin{split}
\frac{p^p(\rho+1-p)}{\rho^p}&(c|c|_k^{\frac{\rho}{p}-(p-1)}-d|d|_k^{\frac{\rho}{p}-(p-1)})^p\\&\leq (c|c|_k^{\rho-p(p-1)}-d|d|_k^{\rho-p(p-1)})(c-d)^{p-1}
\end{split}
\end{aligned}$$ with the assumption that $c \geq d$.
Let us define $$m(t)=\begin{cases}
\text{sgn}(t)|t|^{\frac{\rho}{p}-1}, & |t|<k \\
\frac{p}{\rho}\text{sgn}(t)k^{\frac{\rho}{p}-1}, & |t|\geq k.
\end{cases}$$ Note that $$\begin{aligned}
\begin{split}
\int_{d}^{c}m(t)dt&=\frac{p}{\rho}(c|c|_k^{\frac{\rho}{p}-(p-1)}d|d|_k^{\frac{\rho}{p}-(p-1)}).
\end{split}
\end{aligned}$$ Similarly, $$\begin{aligned}
\begin{split}
\int_{d}^{c}m(t)^pdt&\leq\frac{1}{\rho+1-p}(c|c|_k^{\rho-p(p-1)}-d|d|_k^{\rho-p(p-1)}).
\end{split}
\end{aligned}$$ Using the Cauchy-Schwartz inequality we obtain $$\begin{aligned}
\begin{split}
\left(\int_{d}^{c}m(t)dt\right)^p&\leq(c-d)^{p-1}\int_{d}^{c}h(t)^pdt.
\end{split}
\end{aligned}$$ Thus $$\begin{aligned}
\begin{split}
&\frac{p^p}{\rho^p}(c|c|_k^{\frac{\rho}{p}-(p-1)}-d|d|_k^{\frac{\rho}{p}-(p-1)})^p\\
&=\left(\int_{d}^{c}m(t)dt\right)^p\\
&\leq(c-d)^{p-1}\int_{d}^{c}m(t)^pdt\\
&\leq\frac{(c-d)^{p-1}}{\rho+1-p}(c|c|_k^{\rho-p(p-1)}-d|d|_k^{\rho-p(p-1)}).
\end{split}
\end{aligned}$$
\[bounded\] Let $f:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be as defined in $(f_1)$, then for any weak solution $u\in X_0$ we have $u\in L^{\infty}(\Omega)$.
Let $1\leq q<p_s^*$ and let $u$ be a weak solution to the given problem and $\alpha=\left(\frac{p_s^*}{p}\right)^{\frac{1}{p}}$. For every $\rho\geq p(p-1)$, $k>0$, the mapping $t\mapsto t|t|_k^{r-p}$ is Lipschitz in $\mathbb{R}$. Therefore, $u|u|_k^{\rho-p}\in X_0$. In general for any $t$ in $\mathbb{R}$ and $k>0$, we have defined $t_k=\text{sgn}(t)\min\{|t|,k\}$. We apply the embedding results due to Theorem \[embres\], the previous lemma \[ineq1\], test with the test function $u|u|_k^{\rho-p}$ and on using the growth condition of $f$ which is given in $(f_1)$ to get $$\begin{aligned}
\label{crit_subcrit}
\begin{split}
\|u|u|_k^{\frac{\rho}{p}-1}\|_{p_s^*}^p\leq& C \|u|u|_k^{\frac{\rho}{p}-1}\|^p\\
\leq & C\frac{\rho^p}{\rho+1-p}\langle u,u|u|_k^{\rho-p} \rangle_{}\\
\leq & \rho^pC'\frac{1}{\left(a+b\|u|u|_k^{\frac{\rho}{p}-1}\right)} \left(\lambda\int_{\Omega}g(x)|u|^{p-1}u|u|_k^{\rho-p}dx+\mu\int_{\Omega}h(x)u^{-\gamma}|u||u|_k^{\rho-p}\right.\\
&\left.+\int_{\Omega}|f(x,u)||u||u|_k^{\rho-p}dx\right)\\
\leq & C'\rho^{p}\int_{\Omega}\left(g(x)|u|^{p-1}u|u|_k^{\rho-p}+h(x)|u|^{1-\gamma}|u|_k^{\rho-p}+|u||u|_k^{\rho-p}+|u|^q|u|_k^{\rho-p}\right)dx
\end{split}
\end{aligned}$$ for some $C>0$ independent of $\rho\geq p$ and $k>0$. On applying the Fatou’s lemma as $k\rightarrow\infty$ gives $$\begin{aligned}
\label{eq0}
\begin{split}
\|u\|_{\alpha^p \rho}&\leq C'\rho^{\frac{p}{\rho}}\left\{\int_{\Omega}(|u|^{\rho-(p-1)}+|u|^{\rho+q-p}\right.\\
&+\left.|u|^{\rho-p-\gamma+1})dx\right\}^{1/\rho}.
\end{split}
\end{aligned}$$ The idea is to try and develop an argument to guarantee that $u\in L^{p_1}(\Omega)$ for all $p_1\geq 1$. Therefore define a recursive sequence $(\rho_n)$ by choosing $\mu>\mu_0$ and setting $\rho_0=\mu$, $\rho_{n+1}=\alpha^p \rho_n+p-q$. The proof follows [*verbatim*]{} of the proof of the Theorem 5.2 in [@ghosh_jmp] hereafter which guarantees that the solution is in $L^{\infty}(\Omega)$.
\[weak comparison\] Let $u, v\in X_0$. Suppose, $(a+b\|v\|^p)\mathfrak{L_p}^sv-h(x)\frac{\mu}{v^{\gamma}}\geq(a+b\|u\|^p)\mathfrak{L_p}^su-h(x)\frac{\mu}{u^{\gamma}}$ weakly with $v=u=0$ in $\mathbb{R}^N\setminus\Omega$. Then $v\geq u$ in $\mathbb{R}^N.$
Since, $(a+b\|v\|^p)\mathfrak{L_p}^sv-h(x)\frac{\mu}{v^{\gamma}}\geq(a+b\|u\|^p)\mathfrak{L_p}^su-h(x)\frac{\mu}{u^{\gamma}}$ weakly with $u=v=0$ in $\mathbb{R}^N\setminus\Omega$, we have [$$\begin{aligned}
\label{compprinci}
\langle(a+b\|v\|^p)\mathfrak{L_p}^sv,\phi\rangle-\int_{\Omega}h(x)\frac{\mu\phi}{v^{\gamma}}dx&\geq\langle(a+b\|u\|^p)\mathfrak{L_p}^su,\phi\rangle-\int_{\Omega}h(x)\frac{\mu\phi}{u^{\gamma}}dx
\end{aligned}$$]{} $\forall{\phi\geq 0\in X_0}$.\
In particular choose $\phi=(u-v)^{+}$. To this choice, the inequality in looks as follows. [$$\begin{aligned}
\label{compprinci1}
&\langle(a+b\|v\|^p)\mathfrak{L_p}^sv-(a+b\|u\|^p)\mathfrak{L_p}^su,(u-v)^{+}\rangle \nonumber\\
&~~~~~~-\int_{\Omega}\mu h(x)(u-v)^{+}\left(\frac{1}{v^{\gamma}}-\frac{1}{u^{\gamma}}\right)dx\geq 0.
\end{aligned}$$]{} Define $m(t)=(a+bt^p)\geq a>0$ for $t\geq 0$ and $$M(t)=\int_{0}^{t}m(t)dt.$$ By the Cauchy-Schwartz inequality we have $$\begin{aligned}
\label{csineq}|(u(x)-u(y))(v(x)-v(y))|&\leq& |u(x)-u(y)||v(x)-v(y)|\nonumber\\
& \leq &\frac{|u(x)-u(y)|^2+|v(x)-v(y)|^2}{2}.\end{aligned}$$ Consider $I_1=\langle M'(u),u\rangle-\langle M'(u),v\rangle-\langle M'(v),u\rangle+\langle M'(v),v\rangle$ and let $|u(x)-u(y)| \geq|v(x)-v(y)|$. Therefore using we get $$\begin{aligned}
\label{first_ineq}
I_1&=&p m(\|u\|^p)\left(\int_{Q}|u(x)-u(y)|^{p-2}\{|u(x)-u(y)|^2\right.\nonumber\\
& &\left.-(u(x)-u(y))(v(x)-v(y))\}dxdy\right)\nonumber\\
& &+p m(\|v\|^p)\left(\int_{q}|v(x)-v(y)|^{p-2}\{|v(x)-v(y)|^2\right.\nonumber\\
& &\left.-(u(x)-u(y))(v(x)-v(y))\}dxdy\right)\nonumber\\
&\geq&\frac{p}{2} m(\|u\|^p)\left(\int_{Q}|u(x)-u(y)|^{p-2}\{|u(x)-u(y)|^2\right.\nonumber\\
& &\left.-|v(x)-v(y)|^2\}dxdy\right)\nonumber\\
& &+\frac{p}{2} m(\|v\|^p)\left(\int_{Q}|v(x)-v(y)|^{p-2}\{|v(x)-v(y)|^2\right.\nonumber\\
& &\left.-|u(x)-u(y)|^2\}dxdy\right)\nonumber\\
&\geq&pm(\|u\|^p)\left(\int_{Q}(|u(x)-u(y)|^{p-2}-|v(x)-v(y)|^{p-2})(|u(x)-u(y)|^2-|v(x)-v(y)|^2)dx\right).\nonumber\\
&\geq&pa\left(\int_{Q}(|u(x)-u(y)|^{p-2}-|v(x)-v(y)|^{p-2})(|u(x)-u(y)|^2-|v(x)-v(y)|^2)dx\right).\nonumber\\\end{aligned}$$ When $|u(x)-u(y)| \leq|v(x)-v(y)|$, we interchange the roles of $u$, $v$ to get $$\begin{aligned}
\label{second_ineq}
I_1&\geq&pa\left(\int_{Q}(|u(x)-u(y)|^{p-2}-|v(x)-v(y)|^{p-2})(|u(x)-u(y)|^2-|v(x)-v(y)|^2)dx\right).\nonumber\\\end{aligned}$$ Thus $$\begin{aligned}
\langle M'(u)-M'(v),u-v\rangle&=&I_1\geq 0.\end{aligned}$$ Thus $M'$ is a monotone operator. This monotonicity sufficient for our work.\
Coming back to , we have $$\begin{aligned}
0&\geq&-\langle(a+b\|u\|^p)\mathfrak{L_p}^su-(a+b\|v\|^p)\mathfrak{L_p}^sv,(u-v)^{+}\rangle\nonumber\\
&=&\langle(a+b\|v\|^p)\mathfrak{L_p}^sv-(a+b\|u\|^p)\mathfrak{L_p}^su,(u-v)^{+}\rangle\geq 0.\nonumber\\\end{aligned}$$ Therefore, $|\{x:u(x)>v(x)\}=0|$. Hence $u\geq v$ a.e. in $\Omega$.
$C^1$ versus $W^{s,p}$ local minimizers of the energy
-----------------------------------------------------
This section is devoted towards discussing [*‘$C^1$ versus $W^{s,p}$’*]{} analysis for a particular class of Kernel satisfying $(P')$. Let us begin with some well-known results and prove a few lemmas towards which a geometrical property of a general bounded domain $\Omega$ with $C^{1, 1}$ boundary is stated and is as follows.
\[geo\] Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with a $C^{1, 1}$ boundary $\partial\Omega$. Then, there exists $\rho>0$ such that for all $x_0\in\partial\Omega$ there exists $x_1, x_2\in\mathbb{R}^N$ on the normal line to $\partial\Omega$ at $x_0$, with the following properties
1. $B_{\rho}(x_1)\subset\Omega$, $B_{\rho}(x_2)\subset\Omega^c$;
2. $\bar{B}_{\rho}(x_1)\cap\bar{B}_{\rho}(x_2)=\{x_0\}$;
3. $d(x)=|x-x_0|$ for all $x\in[x_0, x_1]$.
Using the Lemma \[geo\], we generalize two of the results from Iannizzotto [@iannizzotto2014global]. Before that, we set $\forall~R>0$, $x_0\in\mathbb{R}^N$ $$\begin{aligned}
\begin{split}
Q(u; x_0, R)&=\|u\|_{L^{\infty}(B_R(x_0))}+ \text{Tail}(u; x_0, R)\\Q(u, R)&=Q(u; 0, R)\\d(x,\partial\Omega)&=\inf_{y\in\Omega}\{d(x,y)\}.
\end{split}\end{aligned}$$
\[ianbdd\] For any $r>0$, there exists $K>0$ such that $|\mathfrak{L}_p^su|\leq K$ in $B_r(x)$, where $u$ is a weak solution to the problem .
By definition, [$$\begin{aligned}
\begin{split}
|\mathfrak{L}_p^su(x)|&\leq C\int_{B_r(x)} |u(x)-u(y)|^{p-1}K(x-y)dy\\
&\leq C\int_{B_r(x)}\frac{|u(x)-u(y)|^{p-1}}{|x-y|^{N+(p-1)s}}\frac{dy}{|x-y|^{ps-(p-1)s}}dy.
\end{split}
\end{aligned}$$]{} Converting this into polar coordinates and applying the Hölder’s inequality we obtain $|\mathfrak{L}_p^su(x)|\leq K$ in $B_r(x)$. It follows from this that $|(a+b\|u\|^p)\mathfrak{L}_p^su(x)|\leq K'$ in $B_r(x)$.
\[calpha1\] There exists $0<\alpha\leq s$ such that any weak solution $u$ to the problem we have $[u/d^s]_{C^{\alpha}(\overline{\Omega})}\leq K$.
Note that, $$\begin{aligned}
\label{hold1}
&\text{Tail}(u/d^s;x,R_0)^{p-1}\nonumber\\&=R_0^{ps}\int_{B_{R_0}(x)^c}\frac{|u(y)|^{p-1}}{d(y)^{s(p-1)}|x-y|^{N+sp}}dy\nonumber
\end{aligned}$$ $$\begin{aligned}
&\leq CR_0^{ps}\left(\int_{B_{2R_0}(x_0)\setminus B_{R_0(x_0)}}\frac{\|u/d^s\|_{L^{\infty}(B_{2R_0}(x_0))}^{p-1}}{|x-y|^{N+ps}}dy\right.\nonumber\\
&\left.+\int_{B_{2R_0(x_0)^c}}\frac{|u(y)|^{p-1}}{d(y)^{s(p-1)}|x-y|^{N+ps}}dy\right)\nonumber\\
&\leq C\left(\|u/d^s\|_{L^{\infty}(B_{2R_0}(x_0))}^{p-1}\right.\nonumber\\&+\left.R_0^{ps}\int_{B_{2R_0(x_0)^c}}\frac{|u(y)|^{p-1}}{d(y)^{s(p-1)}|x-y|^{N+ps}}dy\right)\nonumber\\
&=CQ(u/d^s;x_0,2R_0).
\end{aligned}$$ The reader may refer [@iannizzotto2014global] for a definition of [*Tail*]{}. Thus we obtained $Q(u/d^s;x_0,R_0)\leq CQ(u/d^s;x_0,2R_0)$, which implies the following Hölder seminorm estimate. $$\begin{aligned}
\label{estimate holder1}
\begin{split}
&[u/d^s]_{C^{\alpha}(B_{R_0}(x_0))}\\&\leq C[(KR_0^{ps})^{1/(p-1)}+Q(u/d^s;x_0,2R_0)]R_0^{-\alpha}.
\end{split}
\end{aligned}$$ Here, $K$ is the bound of $\mathfrak{L}_p^su$ in $B_{2R_0}(x_0)$. Let $\alpha\in(0,s]$ and $\Omega'\Subset\Omega$. Then we have through the compactness of $\Omega'$ and the estimate (\[estimate holder1\]) that $\|u/d^s\|_{C^{\alpha}(\overline{\Omega'})}\leq C$.\
Let $\Pi:V\rightarrow \partial\Omega$ be a metric projection map defined as $\Pi(x)=\text{Argmin}_{y\in\partial\Omega^c}\{|x-y|\}$ where $V=\{x\in\overline{\Omega}:d(x,\partial\Omega)\leq \rho\}$. By (\[estimate holder1\]) we have $$\begin{aligned}
\label{estimate holder 2}
\begin{split}
[u/d^s]_{C^{\alpha}(B_{r/2}(x))}&\leq C[(Kr^{ps})^{1/(p-1)}+\|u/d^s\|_{L^{\infty}(B_r(x))}\\&+\text{Tail}(u/d^s;x,r)]r^{-\alpha}.
\end{split}
\end{aligned}$$ The first term in (\[estimate holder 2\]) is trivially controlled since $\alpha\leq s\leq \frac{sp}{p-1}$. The other terms are controlled uniformly due to the compactness of the set $V$.
From [@finebdry] we can say that $u\in C^1(\overline{\Omega})$. Futher, it can be proved that for any $r>0$, there exists $K>0$ such that $|\mathfrak{L}_p^s Du|\leq K$ in $B_r(x)$, where $u$ is a weak solution to the problem . The proof follows the Lemma \[ianbdd\].
\[calpha2\] There exists $0<\alpha\leq s$ such that for any weak solution $u$ of the problem we have $[Du]_{C^{\alpha}(\overline{\Omega})}\leq C$.
Observe that $$\begin{aligned}
\label{hold2}
\begin{split}
&\text{Tail}(Du;x,R_0)^{p-1}\\
&=R_0^{ps}\int_{B_{R_0}(x)^c}\frac{|Du(y)|^{p-1}}{|x-y|^{N+sp}}dy\\
&\leq CR_0^{ps}\left(\int_{B_{2R_0}(x_0)\setminus B_{R_0(x_0)}}\frac{\|Du\|_{L^{\infty}(B_{2R_0}(x_0))}^{p-1}}{|x-y|^{N+ps}}dy\right.\\
&\left.+\int_{B_{2R_0(x_0)^c}}\frac{|Du(y)|^{p-1}}{|x-y|^{N+ps}}dy\right)\\
&\leq C\left(\|Du\|_{L^{\infty}(B_{2R_0}(x_0))}^{p-1}+R_0^{ps}\int_{B_{2R_0(x_0)^c}}\frac{|Du(y)|^{p-1}}{|x-y|^{N+ps}}dy\right)\\
&=CQ(Du;x_0,2R_0).
\end{split}
\end{aligned}$$ Therefore we obtained $Q(Du;x_0,R_0)\leq CQ(Du;x_0,2R_0)$ which implies the following Hölder seminorm estimate. [$$\begin{aligned}
\label{estimate holder2}
\begin{split}
[Du]_{C^{\alpha}(B_{R_0}(x_0))}&\leq C[(KR_0^{ps})^{1/(p-1)}+Q(Du;x_0,2R_0)]R_0^{-\alpha}.
\end{split}
\end{aligned}$$]{} Here $K$ is the bound of $|\mathfrak{L}_p^sDu|$ in $B_{2R_0}(x_0)$. Let $\alpha\in(0,s]$ and $\Omega'\Subset\Omega$. Then we have through the compactness of $\Omega'$ and the the estimate (\[estimate holder2\]) that $\|Du\|_{C^{\alpha}(\overline{\Omega'})}\leq C$.\
Let $\Pi:V\rightarrow \partial\Omega$ be a metric projection map defined as $\Pi(x)=\text{Argmin}_{y\in\partial\Omega^c}\{|x-y|\}$ where $V=\{x\in\overline{\Omega}:d(x,\partial\Omega)\leq \rho\}$. By (\[estimate holder2\]) we have $$\begin{aligned}
\label{estimate holder_2}
\begin{split}
[Du]_{C^{\alpha}(B_{r/2}(x))}&\leq C[(Kr^{ps})^{1/(p-1)}+\|Du\|_{L^{\infty}(B_r(x))}\\
&+\text{Tail}(Du;x,r)]r^{-\alpha}.
\end{split}
\end{aligned}$$ We now try to control the growth of the terms on the right hand side of (\[estimate holder\_2\]). The first term is trivially controlled since $\alpha\leq s\leq \frac{sp}{p-1}$. The other terms are controlled uniformly due to the compactness of the set $V$.
We will now prove the Theorem \[regularity holder\]. The main tool to prove this result requires an application of the lagrange multiplier rule which is given in the form of a theorem from [@lagrange_1].
\[lagrangeSuff\] Let $I$ and $J$ be real $C^{1}$ functionals on a real Banach space say $X$. If $z_0\in X$ satisfies the following problem: $$\text{minimizing}~I(z)~\text{under the constratint}~K(z)=0.$$ Then there exists $\mu\in\mathbb{R}$ such that $I'(z_0)=\mu J'(z_0)$.
For a more generalized version of the result, one may refer to [@lagrange_2] and the references therein.\
[**Proof of Theorem \[regularity holder\]**]{}: Let $\Omega'\Subset\Omega$. We will only consider the subcritical case i.e. when $q<p_s^*-1$.We prove by contradiction, i.e. suppose $u_0$ is not a local minimizer. Let $r\in (q,p_s^*-1)$ and define $$\begin{aligned}
\label{aux1}
\begin{split}
J(w)&=\frac{1}{r+1}\int_{\Omega'}|w-u_0|^{r+1}dx, (w\in W^{s,p}(\Omega')).
\end{split}\end{aligned}$$ [**Case i**]{}: [Let $J(v_{\epsilon})<\epsilon$]{}.\
Define $S_{\epsilon}=\{v\in W_0^{s,p}(\Omega):0\leq J(v)\leq \epsilon\}$. Consider the problem $I_{\epsilon}=\inf_{v\in S_{\epsilon}}\{\bar{I}(v)\}$. The infimum exists since the set $S_{\epsilon}$ is bounded and the functional $\bar{I}$ is $C^1$. Furthermore, $\bar{I}$ is also weakly lower semicontinuous and $S_{\epsilon}$ is closed, convex. Thus $I_{\epsilon}$ is actually attained, at say $v_{\epsilon}\in S_{\epsilon}$, and $I_{\epsilon}=\bar{I}(v_{\epsilon})<\bar{I}(u_0)$.\
[*Claim*]{}: We now show that $\exists \eta>0$ such that $v_{\epsilon}\geq \eta \phi_1$.\
[*Proof*]{}: Define $v_{\eta}=(\eta\phi_1-v_{\epsilon})^{+}$. We prove the claim by contradiction, i.e. $\forall\eta>0$ let $|\Omega_{\eta}|=|supp\{(\eta\phi_1-v_{\epsilon})^{+}\}|>0$. For $0<t<1$, define $\xi(t)=\bar{I}(v_{\epsilon}+tv_{\eta})$. Thus [$$\begin{aligned}
\begin{split}
\xi'(t)&=\langle\bar{I}_{\lambda}'(v_{\epsilon}+tv_{\eta}),v_{\eta}\rangle\\
&=\langle(a+b\|u\|^p)\mathfrak{L}_p^s(v_{\epsilon}+tv_{\eta})-\lambda g(x) (v_{\epsilon}+tv_{\eta})^{p-1}-\mu h(x)(v_{\epsilon}+tv_{\eta})^{-\gamma}-f(x,v_{\epsilon}+tv_{\eta}),v_{\eta}\rangle.
\end{split}
\end{aligned}$$]{} Similarly, $$\begin{aligned}
\begin{split}
\xi'(1)=&\langle\bar{I}'(v_{\epsilon}+v_{\eta}),v_{\eta}\rangle\\
=&\langle\bar{I}'(\eta\phi_1),v_{\eta}\rangle\\
=&\langle(a+b\|\eta\phi_1\|^p)\mathfrak{L}_p^s(\eta\phi_1)-\lambda g(x) (\eta\phi_1)^{p-1}-\mu h(x)(\eta\phi_1)^{-\gamma}\\
&-f(x,\eta\phi_1),v_{\eta}\rangle<0
\end{split}\end{aligned}$$ for sufficiently small $\eta>0$. Moreover, $$\begin{aligned}
\begin{split}
-\xi'(1)+\xi'(t)=&\langle(a+b\|u\|^p)\mathfrak{L}_p^s(v_{\epsilon}+tv_{\eta})-(a+b\|u\|^p)\mathfrak{L}_p^s(v_{\epsilon}+v_{\eta})\\
&+\lambda g(x)((v_{\epsilon}+v_{\eta})^{p-1}-(v_{\epsilon}+tv_{\eta})^{p-1})\\
&+\mu h(x)((v_{\epsilon}+v_{\eta})^{-\gamma}-(v_{\epsilon}+tv_{\eta})^{-\gamma})\\
&+(f(x,v_{\epsilon}+v_{\eta})-f(v_{\epsilon}+tv_{\eta}),v_{\eta}\rangle.
\end{split}\end{aligned}$$ since $\lambda g(x)t^{p-1}+\mu h(x)t^{-\gamma}+f(.,t)$ is a uniformly nonincreasing function with respect to $x\in\Omega$ for sufficiently small $t>0$. From the monotonicity of $(a+b\|u\|^p)\mathfrak{L}_p^s$ (refer Theorem \[weak comparison\]) we have that, for sufficiently small $\eta>0$, $0\leq \xi'(1)-\xi'(t)$. From the Taylor series expansion and the fact that $J(v_{\epsilon})<\epsilon$, $\exists~ 0<\theta<1$ such that $$\begin{aligned}
\begin{split}
0&\leq \bar{I}(v_{\epsilon}+v_{\eta})-\bar{I}(v_{\epsilon})\\
&=\langle\bar{I}'(v_{\epsilon}+\theta v_{\eta}),v_{\eta}\rangle\\
&=\xi'(\theta).
\end{split}\end{aligned}$$ Thus for $t=\theta$ we get $\xi'(\theta)\geq 0$ which is a contradiction to $\xi'(\theta)\leq\xi'(1)<0$ as obtained above. Thus $v_{\epsilon}\geq \eta\phi_1$ for some $\eta>0$.\
In fact, from the Lemmas \[calpha1\] and \[calpha2\] we have $ \sup_{\epsilon\in(0,1]}\{\|v_{\epsilon}\|_{C^{1,\alpha}(\overline{\Omega})}\}\leq C$. By the compact embedding $C^{1,\alpha}(\overline{\Omega})\hookrightarrow C^{1,\kappa}\overline{\Omega})$, for any $\kappa<\alpha$, we have $v_{\epsilon}\rightarrow u_0$ which contradicts the assumption made.\
[**Case ii**]{}: [$K(v_{\epsilon})=\epsilon$.]{}\
Let $v_{\eta}=(\eta\phi_1-v_{\epsilon})^+$ and $\xi(t)=\overline{I}(v_{\epsilon}+tv_{\eta})$. Then by arguments as in Case i, we have that $\xi$ is decreasing. This implies that $\overline{I}(v_{\epsilon})>\overline{I}(v_{\epsilon}+tv_{\eta})$. Since the functionals $\overline{I}$, $J$ are $C^1$, hence in this case from the Lagrange multiplier rule (refer Theorem \[lagrangeSuff\]) there exists $\mu_{\epsilon}\in\mathbb{R}$ such that $\bar{I}'(v_{\epsilon})=\mu_{\epsilon}J'(v_{\epsilon})$. We will first show that $\mu_{\epsilon}\leq 0$. Suppose $\mu_{\epsilon}>0$, then $\exists~ \phi\in X_0$ such that $$\begin{aligned}
\begin{split}
\langle\bar{I}'(v_{\epsilon}),\phi \rangle<0~ \text{and} ~\langle J'(v_{\epsilon}),\phi \rangle<0.
\end{split}\end{aligned}$$ Then for small $t>0$ we have $$\begin{aligned}
\begin{split}
\bar{I}(v_{\epsilon}+t \phi)&<\bar{I}_{\lambda}(v_{\epsilon})\\
J(v_{\epsilon}+t \phi)&<J(v_{\epsilon})=\epsilon
\end{split}\end{aligned}$$ which is a contradiction to $v_{\epsilon}$ being a minimizer of $\bar{I}_{\lambda}$ in $S_{\epsilon}$.\
We now consider the following two cases.\
[**Case a**]{}: ($\mu_{\epsilon}\in(-l,0)$ where $l>-\infty$).\
Now consider the sequence of problems $$\begin{aligned}
\begin{split}
(P_{\epsilon}):~(a+b\|u\|^p)\mathfrak{L}_p^su&=\lambda g(x)u^{p-1}+\mu h(x) u^{-\gamma}+f(x,u)+\mu_{\epsilon}|u-u_0|^{r-1}(u-u_0)
\end{split}\end{aligned}$$ Observe that $u_0$ is a weak solution to ($P_{\epsilon}$). From the weak comparison principle (Theorem \[weak comparison\]) we have $v_{\epsilon}\geq \eta\phi_1$ for some $\eta>0$ small enough, independent of $\epsilon$ since $\eta\phi_1$ is a strict subsolution to $(P_{\epsilon})$. Further, since $-l\leq \mu_{\epsilon}\leq 0$, there exists $M$, $c$ such that $$\begin{aligned}
\begin{split}
(a+b\|u\|^p)\mathfrak{L}_p^s(v_{\epsilon}-1)^{+}&\leq M+c((v_{\epsilon}-1)^{+})^r.
\end{split}\end{aligned}$$ Using the Moser iteration technique as in Theorem \[bounded\] we obtain $\|v_{\epsilon}\|_{\infty}\leq C'$. Therefore $\exists L>0$ such that $\eta\phi_1\leq v_{\epsilon}\leq L\phi_1$. By using the arguments previously used, we end up getting $|v_{\epsilon}|_{C^{\alpha}(\overline{\Omega})}\leq C'$. The conclusion follows as in the previous case of $J(v_{\epsilon})<\epsilon$.\
[**Case b**]{}: $\inf_{\mu_{\epsilon}}=-\infty$\
Let us assume $\mu_{\epsilon}\leq -1$. As above, we can similarly obtain $v_{\epsilon}\geq \eta\phi_1$ for $\eta>0$ small enough and independent of $\epsilon$. Further, there exists a constant $M>0$ such that $\lambda g(x)t^{p-1}+\mu h(x)t^{-\gamma}+f(x,t)+\tau|t-u_0(x)|^{r-1}(t-u_0(x))<0$, $\forall (\tau,x,t)\in(-\infty,-1]\times\Omega\times(M,\infty)$.\
From the weak comparison principle on $(a+b\|u\|^p)\mathfrak{L}_p^s$, we get $v_{\epsilon}\leq M$ for $\epsilon>0$ sufficiently small. Since $u_0$ is a local $C^1$ - minimizer, $u_0$ is a weak solution to and hence $$\begin{aligned}
\begin{split}
\langle (a+b\|u_0\|^p)\mathfrak{L}_p^su_0,\phi \rangle&=\lambda\int_{\Omega}g(x)u_0^{p-1}\phi dx+\mu\int_{\Omega}h(x)u_0^{-\gamma}\phi dx+\int_{\Omega}f(x,u_0)\phi dx
\end{split}\end{aligned}$$ $\forall\phi\in C_c^{\infty}(\Omega)$. Also, $u_0$ satisfies $$\begin{aligned}
\label{s1}
\begin{split}
\langle (a+b\|u_0\|^p)\mathfrak{L}_p^su_0,w \rangle&=\lambda\int_{\Omega}g(x)u_0^{p-1}w+\mu\int_{\Omega}h(x)u_0^{-\gamma}w dx+\int_{\Omega}f(x,u_0)w dx.
\end{split}\end{aligned}$$ Similarly, $$\begin{aligned}
\label{s2}
\begin{split}
\langle (a+b\|v_{\epsilon}\|^p)\mathfrak{L}_p^sv_{\epsilon},w \rangle&=\lambda\int_{\Omega}g(x)v_{\epsilon}^{p-1}w+\mu\int_{\Omega}h(x)v_{\epsilon}^{-\gamma}w dx+\int_{\Omega}f(x,v_{\epsilon})w dx.
\end{split}\end{aligned}$$ On subtracting from and testing with $w=|v_{\epsilon}-u_0|^{\beta-1}(v_{\epsilon}-u_0)$, where $\beta\geq 1$, we obtain $$\begin{aligned}
\begin{split}
0\leq& \beta\langle (a+b\|u_{\epsilon}\|^p)\mathfrak{L}_p^sv_{\epsilon}-(a+b\|u_0\|^p)\mathfrak{L}_p^su_0,|v_{\epsilon}-u_0|^{\beta-1}(v_{\epsilon}-u_0) \rangle\\
&-\lambda\int_{\Omega}g(x)(v_{\epsilon}^{p-1}-u_0^{p-1})|v_{\epsilon}-u_0|^{\beta-1}(v_{\epsilon}-u_0)dx\\
&-\mu\int_{\Omega}h(x)(v_{\epsilon}^{-\gamma}-u_0^{-\gamma})|v_{\epsilon}-u_0|^{\beta-1}(v_{\epsilon}-u_0)dx\\
=&\int_{\Omega}(f(x,v_{\epsilon})-f(x,u_0))|v_{\epsilon}-u_0|^{\beta-1}(v_{\epsilon}-u_0)dx\\
&+\mu_{\epsilon}\int_{\Omega}|v_{\epsilon}-u_0|^{\beta+r}dx.
\end{split}\end{aligned}$$ By the Hölder’s inequality and the bounds of $v_{\epsilon}$, $u_0$ we obtain $$\begin{aligned}
\begin{split}
-\mu_{\epsilon}\|v_{\epsilon}-u_0\|_{\beta+r}^{r}&\leq C|\Omega|^{\frac{r}{\beta+r}}.
\end{split}\end{aligned}$$ Here $C$ is independent of $\epsilon$ and $\beta$. On passing the limit $\beta\rightarrow\infty$ we get $-\mu_{\epsilon}\|v_{\epsilon}-u_0\|_{\infty}\leq C$. Working on similar lines we end up getting $v_{\epsilon}$ is bounded in $C^{\alpha}(\overline{\Omega})$ independent of $\epsilon$ and the conclusion follows.
Conclusions {#conclusions .unnumbered}
===========
Existence of infinitely many solutions to the problem in has been shown. In addition, a weak comparison result has been proved. It has also been shown that the solutions are in $L^{\infty}(\Omega)$. Further it has also been proved that the $C^1$ minimizers are the $W_0^{s,p}$ minimizers as well. Some future scope of work on this line would be to prove that the $C^1$ minimizers are the $W_0^{s,p}$ minimizers as well for a general Kernel.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author thanks S. Ghosh for the numerous discussion sessions and the constructive criticisms on the article. The author also dedicates this article to thousands of labourers and workers of India who, during this COVID19 pandemic, have lost their lives travelling on foot for thousands of kilometers to reach their respective homes.
Data availability statement {#data-availability-statement .unnumbered}
===========================
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
[10]{} A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical poiunt theory and applications, , 14, 349-381, 1973.
A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, , 94, 156-170 , 2014.
A. Fiscella, P. Pucci, $p$-fractional Kirchhoff equations involving critical nonlinearities, , 35, 350-378 , 2017.
A. Ghanmi and K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, , 6(2), 201-217, 2016.
A. Iannizzotto, M. Squassina, $\frac{1}{2}$-Laplacian problems with exponential nonlinearity, , 414(1), 372-385, 2014.
A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, , 32(4), 1353-1392, 2016.
A. Iannizzotto., S. Mosconi and M. Squassina, Fine boundary regularity for the degenerate fractional $p$-Laplacian, , 2018.
A. Soni, D. Choudhuri, Existence of multiple solutions to an elliptic problem with measure data, , 4, 369-388, 2020.
B. Barrios, E, Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, , 252(2), 6133-6161, 2012.
C.O. Alves, F.J. Corr$\hat{e}a$ and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, , 49(1), 85-93, 2005.
F. Colasuonno, P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, , 74(17), 5962-5974, 2011.
G.F. Carrier On the non-linear vibration problem of the elastic string, , 3, 157-165, 1945.
K. Saoudi, S. Ghosh, D. Choudhuri, Multiplicity and Hölder regularity of solutions for a nonlocal elliptic PDE involving singularity, , 60, 101509 1-28, 2019.
S. Ghosh, Fractional Kirchhoff-Schrödinger-Poisson system involving singularity,
G. Autori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, , 225(8), 2340-2362, 2013.
G. Molica Bisci and B.A. Pansera, Three weak solutions for nonlocal fractional equaltions, , 14(3), 619-629, 2014.
G. Molica Bisci, Sequences of weak solutions for fractional equations, , 21(2), 241-253, 2014.
G. Molica Bisci , Fractional equations with bounded primitive, , 27, 53-58, 2014.
J. Zuo, T. An, W. Liu, A variational inequlaity of Kirchhoff type in $\mathbb{R}^N$, , 21(2), 241-253, 2014.
J. Zuo, T. An and M. Li Superlinear Kirchhoff-type problems of the fractional $p$-Laplacian without the $(AR)$ condition, , 2018, 1-13, 2018.
J. Zuo, T. An, L. Yang, X. Ren, The Nehari manifold for a fractional $p$-Kirchhoff system involving sign-changing weight function and concave-convex nonlinearities, , Art. ID 7624373, 9 Pages, 2019.
Khaled, M., Rhoudaf, M. & Sabiki, H., Lagrange Multiplier Rule to a Nonlinear Eigenvalue Problem in Musielak-Orlicz Spaces, , 2019, DOI: 10.1080/01630563.2019.1615945.
K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operator, , 14(1), 867-898, 2013.
L. Jeanjean, On the existence of bounded Palais-SAmale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, , 129(4), 787-809, 1999.
L. Wang, K. Xie, B. Zhang, Existence and multiplicity of solutions for critical Kirchhoff-type $p$-Laplacian problems, , 458, 361-378, 2018.
M. Caponi, P. Pucci, Existence theorem for entire soluitions of stationary Kirchhoff fractional $p$-Laplacian equations, ,195(6), 2099-2129, 2016.
M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the nonlocal fractional $p$-Laplacian, , 424(2), 1021-1041, 2015.
M. Xiang, B. Zhang, Degenerate Kirchhoff problems involving the fractionl $p$-Laplacian without the $(AR)$ condition, ,60(9), 1277-1287, 2015.
M. Xiang, B. Zhang, X. Guo, Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem , , 120, 299-313, 2015.
M. Xiang, B. Zhang, V.D. Radulescu, Existence of solutions for a bi-nonlocal fractional $p$-Kirchhoff type problem, , 71(1), 255-266, 2016.
M. Xiang, B. Zhang, H. Qiu, Existence of solutions for a critical fractional Kirchhoff type problem in $\mathbb{R}^N$, , 60(9), 1647-1660, 2017.
N. Pan, B. Zhang, J, Cao, Degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian, , 37, 56-70, 2017.
N. Nyamoradi, L.I. Zaidan, Existence of solutions for degenerate Kirchhofftype problems with fractional $p$-Laplacian, , 115, 1-13, 2017.
N. V. Thin, Nontrivial solutions of some fractional problems, , 38, 146-170, 2017.
P.K. Mishra, K. Sreenadh, Fractional $p$-Kirchhoff system with sign-changing nonlinearities, ,III(1), 281-296, 2017.
P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, , 32(1), 1-22, 2016.
R.A. Adams, J.J. Fournier, Sobolev spaces, , second edition, 2003.
R. Servadei and E. Valdinoci, Mountain pass solutions for nonlocal elliptic operators, , 389(2), 887-898, 2012.
R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by nonlocal operators, , 29(3), 1019-1126, 2013.
R. Servadei, E. Valdinoci, Variational methods for nonlocal operaors of elliptic type, , 33(5), 2105-2137, 2013.
T. Mukherjee and K. Sreenadh, On Dirichlet problem for fractional $p$-Laplacian with singular non-linearity, , 2016.
X. Mingqi, G. M. Bisci, G. Tian, B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, , 29(2), 357-374, 2016.
X. Mingqi, V.D. Radulescu, B. Zhang, Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities, , 24(3), 1249-1273, 2018.
X. Mingqi, V.D. Radulescu, B. Zhang, Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions, , 31(7), 3228-3250, 2018.
M. Ren, J. Zuo, Z. Qiao, L. Zhu, Infinitely many solutions for a superlinear fractional $p$-Kirchhoff-type problem without the $(AR)$ condition, , Art ID: 1353961, 2019.
Zowe, J. & Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces. , 5, 47-62, 1979.
Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, , 28(7), 2247-2264, 2015.
|
---
abstract: |
Semiclassical approximation to the Wheeler-DeWitt equation which corresponds to gravity with a minimally coupled scalar field has been performed. To the leading order, vacuum Einstein’s equation along with the functional Schrödinger equation for the matter field, propagating in the background of classical curved space are obtained. The Schrödinger equation is solved for a quartic potential. It is observed that the wave-functional admits the wormhole boundary condition even for large negative values of the coupling constant $\epsilon$. For conformal coupling $\epsilon = {1\over 6}$, the Hawking-Page wormhole solution is recovered.\
author:
- Abhik Kumar Sanyal
title: 'Semiclassical gravity with a nonminimally coupled scalar field.'
---
Dept. of Physics, Jangipur College, Murshidabad, West Bengal, India-742213\
and\
Relativity and Cosmology Research Centre, Dept. of Physics, Jadavpur University, Calcutta, India-700032\
Keywords: scalar-tensor theory of gravity; graceful exit; wormhole.
Introduction
============
It has been well established [@1; @2; @3; @4] that much below Planck’s scale, the Wheeler-DeWitt equation for gravity in the presence of a minimally coupled scalar field can be approximated to obtain equations corresponding to quantum field theory in curved space-time, along with the vacuum Einstein’s equation with possible back reaction. For this purpose, one has to expand the WKB phase of the wave-functional $\psi$ in the power series of Planck’s mass $m_{pl}$, rather than the Planck’s constant $\hbar$. This method of approximation has also been successfully applied by the present author [@5] to a recent theory - “A Non-singular Universe" [@6; @7] - which is essentially a constrained system of the non-standard cosmological model that includes a higher order curvature invariant term. So far there has been little effort to apply the said method of approximation to gravity with non-minimally coupled scalar field.\
In recent years it has been observed that the non-minimal coupling might play an important role in improving the inflationary models [@8; @9; @10; @11]. On one hand the extended inflation obtained by La and Steinhardt [@12] based on the solutions obtained by Mathiazhagan and Johri [@13] in the Brans-Dicke theory of gravity and that obtained by Accetta and Trester [@14] in induced theory of gravity, raised the hope that non-minimally coupled scalar field might help to solve the “graceful exit" problem suffered by most of the inflationary models. On the other hand, recent works of Fakir and Unruh [@15] and also of Fakir, Habib and Unruh [@16] suggest that non-minimal coupling might help solving the long standing problem of density perturbation.\
One of the greatest successes of the inflationary models is that these can give rise to initial perturbations which can provide explanation for structure (galaxy, clusters, super-clusters etc.) formation. However, the trouble with all the inflationary models is that, they give rise to density perturbations which are five to six order of magnitudes in excess of that required. In order to fit the observed density perturbation, the self consistent parameter $\lambda$ of the scalar field is usually tuned to an extremely low value ($\lambda \le 10^{-12}$) which is about ten order of magnitude smaller than the values calculated from standard field theory.\
In a couple of papers [@15; @16] Fakir, Unruh and Habib have shown that instead of tuning the self coupling parameter $\lambda$, it is indeed possible to solve this long standing problem in principle, simply by constraining the non-minimal coupling constant $\epsilon$ to a large negative value ($\sim -10^{-3}$). The outcome of their analysis has proved that the large negative $\epsilon$ may lead to well-behaved, self-consistent classical solutions which admit inflation, despite previous belief to the contrary. Further, they have demonstrated rigorously that within the frame work of chaotic inflation [@11], one can produce density perturbations of amplitudes consistent with the large scale behaviour, keeping $\lambda$ within the order ($\sim 10^{-2}$) of the ordinary GUT range.\
These encouraging aspects of non-minimally coupled scalar fields led Fakir [@17] to study it in the context of quantum cosmology. He obtained the first order WKB solutions to the modified Wheeler-DeWitt equation and showed that the generic features of Vilenkin [@18] Hartle-Hawking [@19] wave-functions remain preserved in non-minimally coupled case. He also tried to present a probabilistic interpretation of the wave-function by arguing that instead of being the wave-function of the universe, the minisuperspace wave-function could describe the quantum creation of inflationary sub-universes in Linde’s chaotic inflationary model [@11].\
So far we have discussed the success of the non-minimal coupling in the context of the inflationary scenario. However, interest in the field has increased following a recent paper of Coule [@20], who has shown that a non-minimally coupled scalar field admits wormhole solutions both for real and imaginary fields.\
Wormholes [@21; @22; @23; @24] are usually treated as solutions to the classical Euclidean field equations. For such wormholes to exists, the Ricci tensor ($R_{\mu\nu}$) should have negative eigenvalues, and as such classical wormholes do not exist for minimally coupled real massless scalar fields. The situation changes in the case of non-minimal coupling ($\epsilon \ne 0$), since in that case $R_{\mu\nu} \simeq (M_{pl} - 6\epsilon \phi^2) \partial_{\mu}\phi \partial_{\nu}\phi$, where, $M_{pl}$ is the Planck’s mass. Therefore, classical wormholes can exist only for positive values of $\epsilon$, provided the field takes some appropriate large value to make $R_{\mu\nu} < 0$. Hawking and Page [@25] have shown that wormholes can be represented in a more general manner as solutions of Wheeler-DeWitt equations with appropriate boundary conditions. Such boundary conditions can be formulated sketchily in the following manner. The wave-function should be damped for large three-volume, while it should tend to a constant value for small three-volume and of-course should be regular everywhere. Hawking and Page [@25] have also produced such quantum wormhole solutions for massless minimal and conformal scalar fields. Coule [@20] on the other hand, studied wormhole solutions for non-minimally coupled scalar fields from the view point of the sign of the potential term appearing in the Wheeler-DeWitt equation, instead of solving the equations explicitly. The outcome of his work is that, like classical wormholes, quantum wormholes also exist, but for positive value of $\epsilon$ in the case of real massless scalar field. However, for an imaginary scalar fields, wormholes exist for all positive and negative values of $\epsilon$, and of course for $\epsilon = 0$, i.e. for minimally coupled fields.\
Since macroscopic wormholes are supposed to provide a mechanism for the evaporation and complete disappearance of black-holes, while microscopic wormholes might play an important role for the vanishing of the cosmological constant, it is potentially an important scenario rather than a model, that might have played a dominant role in the evolution of the very early universe. Nevertheless, the fact that only positive value of $\epsilon$ seems to provide wormhole solutions for a real massless scalar field, while large negative $\epsilon$ can solve the density perturbation problem, is surely worrisome. However, the issue of wormholes has not been settled as yet. Coule [@20] has suggested that while relating quantum wormholes to the classical ones, it might predict a minimum size of the proper volume of the universe, below which the wave-function would vanish.\
In view of the above discussions, we are motivated to study the cosmological aspect of the theory of gravity when it is non-minimally coupled to a real scalar field, yet from a new direction. As already mentioned, here we apply semi-classical technique to the corresponding Wheeler-DeWitt equation by expanding the phase of the wave-function in the power series of the Planck’s mass. In the process we have derived the Tomonaga-Schwinger equation, which is essentially the functional Schrödinger equation for the scalar field propagating in the background of classical curved space. We then solve it under the choice of quartic potential $V(\phi) = \lambda \phi^4$, and ultimately presented the wave-functional with higher order correction term. Next, we are motivated to check whether our solution satisfies the wormhole boundary condition for large negative value of $\epsilon \sim (-10^{-3})$. We have in the process confirmed that at least one such wormhole solution exist for a real free scalar field, and one for a self-coupled field in the closed ($k = + 1$) universe model.
Semiclassical Wave-functional and Wormhole Configuration
========================================================
We start with the following non-minimally coupled gravitational action,
\[2.1\] S = d\^4 x - [18G]{}\_d\^3 x K(1 - [4G]{}\^2),where, $\epsilon$ is the coupling constant, and while $\epsilon = 0$ corresponds to minimal coupling, $\epsilon = {1\over 6}$ corresponds to conformal coupling. The surface term contains the determinant of the three-metric $h$ and the trace $K$ of the extrinsic curvature tensor $K_{ij}$. In the Robertson-Walker line element
\[2.2\] ds\^2 = -dt\^2 + a(t)\^2the action reduces to
\[2.3\] S = dt,where, $M = {3\pi\over 2G} = {3\over 2}\pi m_{pl}^2$, $m_{pl}$ being the Planck’s mass. Thus the $\left(^0_0\right)$ component of Einstein’s field equations, viz. the classical Hamiltonian constraint equation in configuration space variables reads
\[2.4\] -[12]{}([a\^2a\^2]{} + [ka\^2]{}) + [1M - 6\^2]{}=0.In terms of phase-space variables, the corresponding Hamiltonian constraint equation reads
\[2.5\] [1M - 36\^2\^2- 6\^2]{}(-[p\_a\^2 2a]{} + [M - 6\^22 a\^3]{}p\_\^2 + [6a\^2]{}p\_ap\_) - [M - 6\^22]{}ka + a\^3 V() = 0.To follow the standard canonical quantization scheme, we can now express the Wheeler-DeWitt equation after carefully removing the operator ordering ambiguities between $\hat a$ and $\hat{p_a}$, $\hat\phi$ and $\hat{p_{\phi}}$. For this purpose, we replace $\hat p_a$ by $a^{-n}\hat p_a a^n$, $\hat p_a^2$ by $a^{-n}\hat p_a a^n \hat p_a$; $\hat p_{\phi}$ by $\phi^{-l}\hat p_{\phi}\phi^l$, and $\hat p_{\phi}^2$ by $\phi^{-l}\hat p_{\phi}\phi^l\hat p_{\phi}$, where $n$ and $l$ are operator ordering indices. Next, momentum operators are replaced by corresponding gradient operators and thus the Wheeler-DeWitt equation reads
\[2.6\] &|> = 0
We can now carry out semi-classical approximation by writing $\psi$ as $\psi = \exp\left({iS\over \hbar}\right)$ and expanding the phase in the power series of $M$ as $S = MS_0 + S_1 + M^{-1} S_2 + ....$ etc. Finally, substituting all these in the above Wheeler-DeWitt equation , and comparing expressions having the same order in $M$, we obtain to the highest order $M^3$, the following equation,
\[2.7\] [S\_0]{} = 0,which clearly states that $S_0$ is purely a function of the gravitational field variable. To the next order $M^2$, we obtain
\[2.8\] [12a]{}([S\_0]{})\^2 + [ka2]{} = 0,which essentially is the Hamilton-Jacobi equation for source-free gravity. This equation reduces to the vacuum Einstein’s equation $\dot a^2 + k = 0$, if we identify $\rho_a$ with $M{\partial S_0\over \partial a}$, or equivalently under the choice of the time parameter [@5]
\[2.9\] [t]{}=-[1a]{}[S\_0a]{}[a]{}.At this juncture we recall that the validity of WKB approximation under consideration requires $|{d\lambda_a\over da}|\ll 1$, where $\lambda_a$ is the de Broglie wavelength for pure gravity. Identifying the derivative of the phase $S_0$ with the canonical momentum, we observe that the above statement reduces to $|{d\over da}\big({\hbar \over MS_0}\big)|\ll 1$. In view of equation this implies that $a\gg\sqrt{55} \times 10^{-29}$ cm. This gives a lower limit to $a$, beyond which semiclassical approximation fails. However, it is only a few order of magnitude larger than the Planck’s length and as such both microscopic and macroscopic wormholes would possibly exist. Hence this method of approximation is well posed to address both the problems of vanishing of the cosmological constant and the final stage of evaporation and complete disappearance of cosmological black holes. Although in the present work we have not studied the evolution of this finite resolution limit of the scale factor, we think that there might exist appropriate conditions under which a microscopic wormhole would evolve to a macroscopic one. Next, to the following order $M$, we obtain
\[2.10\]& [i2a]{}[\^2S\_0a\^2]{}-[1a]{}[S\_0a]{}[S\_1a]{}+ [i2a\^2]{}(n - 12l)[S\_0a]{}+[6a\^2]{}[S\_0a]{}[S\_1]{}-[i2a\^3]{}[\^2 S\_1\^2]{}\
& +[12a\^3]{}([S\_1]{})\^2-[il2a\^3 ]{}[S\_1]{}+6\^2 (1-3)k a + a\^3 V()= 0.
Now if we define a functional $f(a,\phi)$ as $f(a,\phi) = D(a)\exp[{iS_1(a,\phi)\over \hbar}]$, where $D(a)$ is chosen to satisfy the equation
\[2.11\] D(a) = 0,then equation may be recast in the following form
\[2.12\]-[ia]{}([S\_0a]{})[f(a,)a]{}=f(a,).
Now under the choice of the time parameter, the left hand side of equation reduces to ${\partial f\over \partial t}$. Further, in view of the Hamiltonian it is apparent that for minimal coupling there is no operator ordering ambiguity between $\hat \phi$ and $\hat p_{\phi}$, and so one can set $l = 0$. Hence the right hand side of reduces to the Hamiltonian operator for the minimally coupled field operating on $f(a,\phi)$, in the background of classical curved space. Thus for $\epsilon = 0$, reduces to the functional Schrödinger equation propagating in the background of classical curved space. It is now left to show that for arbitrary coupling $\epsilon \ne 0$, the right hand side of is essentially the Hamiltonian operator for non-minimally coupled field $\phi$ operating on the function $f(a,\phi)$, in the background of classical curved space. For this pupose, we first recall that, to find the Hamiltonian for minimally coupled matter field in the background of curved space, it is required to start with the action $\int d^4x\sqrt{-g} L_m = \int\big[{1\over 2}\dot\phi^2 - V(\phi)\big]a^3 dt$, and to carry out the variation with respect to field $\phi$ keeping $a$ fixed, since $a$ has already been determined by the vacuum Einstein’s equation,- $\dot a^2 + k = 0$. Likewise, here we start with the action
\[2.13\] A = d\^4x +[12\^2]{}\_d\^3 x K\^2.Remember that pure gravity has already been separated following equation , so here we have considered only the part of the action which does not include pure gravity. In the process, action reduces to the action for induced theory of gravity. In the Robertson-Walker line element under consideration, and using equation , equation takes the form
\[2.14\] A .Hence the Hamiltonian $H = p_{\phi}\dot \phi - L$, reads
\[2.15\] H = [p\_\^22a\^3]{} +[6a\^2]{}([S\_0a]{})p\_ + 6k(1-3)a \^2 + a\^3 V().To take up canonical quantization scheme, we must remove some of the operator ordering ambiguities between $\hat \phi$ and $\hat p_{\phi}$, by replacing $\hat p_{\phi}^2$ by $\phi^{-l}\hat p_{\phi} \phi^l \hat p_{\phi}$. The functional Schrödinger equation for the non-minimally coupled matter field propagating in the background of curved space then takes the form
\[2.16\] i=f,which is essentially equation . Therefore we have achieved our goal. It is now possible to integrate equation to find the solution of $D(a)$ as
\[2.17\] D(a) = m a\^[[12(1+n-12l)]{}]{},$m$ being the constant of integration. Hence up-to this order of approximation $M$, the semiclassical wave-function $\psi = \exp{i\over \hbar}[MS_0+S_1]$ turns out to be
\[2.18\] = [m a\^[[12(1+n-12l)]{}]{}]{}f(a,).We have considered negative sign in the exponent in order to ensure that $\psi$ should exponentially damp for large three-geometry. However, the fate of the wave-function at small three-geometry remains obscure, until the solution for $f(a,\phi)$ is found in view of equation or as well. For this purpose, we use equation in equation . Thus for the closed model ($k = +1$), under the choice $f(a,\phi) = f(\alpha)$, where, $\alpha = a\phi$, equation reduces, for a quartic potential $V(\phi) = \lambda\phi^4$ to
\[2.19\] f\_[,aa]{} + f\_[,a]{} - [2\^2]{}\[6(1-3)+a\^2\]\^2 f,where $f_{,\alpha}$ stands for the derivative of $f$ with respect to $\alpha$. For the sake of convenience, only the positive sign has been considered. It should be noted that $\alpha = 0$ only gives a non-essential regular singularity. Next, we make a transformation,
\[2.20\] f = \^[-[12]{}]{} g() ,so that the first order derivative term in equation does not appear. Thus we obtain
\[2.21\] g\_[,]{} +g() = 0.Equation admits a series solution $$g(\alpha) = \sum_{n=0}^\infty g_n \alpha^{n+s},$$ corresponding to which we obtain a recurrence relation involving four terms as
\[2.22\] g\_[j+6]{}-[(1-6)(1+l)]{}g\_[j+4]{} -[1\^2]{}g\_[j+2]{} -[2\^2]{}g\_j = 0.Hence two sets of coefficients may be obtained in terms of $g_0$, which remains arbitrary: one set is for $s = {l\over 2}$, and the other for $s = 1-{l\over 2}$. Note that for $l = 1$, only one independent solution of equation would emerge. The coefficients are,
$$g_2 = {1-6\epsilon\over 2\hbar}g_0;\hspace{1.15 in}g_4 = {(1+l)(1-6\epsilon)^2 +2 \over 8\hbar^2(l+3)}g_0;$$ \[2.23a\] g\_6 = [16(l+5)]{}g\_0,while, $g_1 = g_3 = g_5 = ......= 0$, for $s={l\over 2}$, where $l\ne -3, -5, -7 ...$ etc. and
$$g_2 = {1-6\epsilon \over 2\hbar}\left({1+l\over 3-l}\right) g_0;\hspace{0.4 in}g_4 = \left[{(1+l)^2(1-6\epsilon)^2+ 2(3-l)\over 8\hbar^2(5-l)(3-l)}\right]g_0;$$ \[2.23b\] g\_6 = [16(7-l)]{}g\_0,etc. while $g_1 = g_3 = g_5 = ......= 0$, for $s=1-{l\over 2}$, where $l\ne 3, 5, 7 ...$ etc. Now for the set of coefficients , the wave-function $\psi$ of takes the form in view of the transformation
\[2.24\]& = [g\_0m]{}
The above wave-function will be damped at $a\rightarrow \infty$, if $\epsilon$ is not too large a positive quantity. If $\epsilon\gg {1\over 6}$, then the argument of the second exponent becomes positive. The series in that case might play the dominant role, for which the wave-function might diverge for large value of $a$. As $a\rightarrow 0$, the numerator becomes a constant, and so the denominator decides the behaviour of $\psi$. For $l = 0$, a wormhole solution is admissible for $n = -1$, irrespective of the sign of $\epsilon$. Hence, we find that even for large negative value of $\epsilon (\sim 10^{-3})$, as required for obtaining the observed order of magnitude of density perturbation ${\partial\rho\over \rho}\sim 10^{-4}$, keeping $\lambda$ within the GUT range $(\sim 10^{-2})$, a wormhole configuration exists. This is definitely a new wormhole solution. In the conformally coupled case, $\epsilon = {1\over 6}$, the Hawking-Page [@25] wormhole solution is recovered for $l=1$ and $n=1$. For large negative $\epsilon$, the wave-function $\psi$ diverges as $a\rightarrow 0$, for $l >0$. One can in that case choose $l < 0$, to regulate $\psi$. However, if one considers the validity range of semiclassical approximation ($a \gg \sqrt{55}\times 10^{-29}$ cm.), then at the limit, one might obtain other wormhole configurations, as $\psi$ would remain constant in the limit, below which $\psi \rightarrow \infty$. This was speculated by Coule [@20]. It is to be noted that the situation remains unaltered even for $\lambda = 0$, i.e. for free field.\
In a similar manner it is possible to construct the wave-function for the set of coefficients . In that case,
\[2.25\]& = [g\_0m]{}\
&
As before, if $\psi$ is not too large a positive quantity, then $\psi \rightarrow 0$ as $a \rightarrow \infty$ and $\psi \rightarrow$ constant as $a \rightarrow 0$ for $l = 0$ and $n = 1$. This is yet another wormhole configuration that exists even for arbitrary large negative value of $\epsilon$. For conformal coupling $\epsilon = {1\over 6}$, the Hawking-Page wormhole configuration [@25] can be recovered for $n = 1$, even without restricting $l$. Wormhole configuration may be obtained as before for large negative value of $\epsilon$ and $l < 0$, if one restricts the scale factor to $a \gg \sqrt{55}\times 10^{-29}$ cm., which is the limit for the validity of semi-classical approximation, as already mentioned. The situation here too remains unaltered for free scalar field $\lambda = 0$.
Conclusion
==========
The WKB phase for the Wheeler-DeWitt wave-functional for gravity with non-minimally coupled scalar field has been expanded in the inverse power series of the Planck’s mass to obtain the semi-classical wave-functional. In the process, the Tomonaga-Schwinger equation has been obtained and solved for quartic potential. The wave-functional has been found to produce the Hawking-Page wormhole solution for the conformally coupled scalar field $\epsilon = {1\over 6}$. Special attention has been paid in studying the case of arbitrarily large negative $\epsilon$. This is because, the recent results of Fakir, Unruh and Habib have explored the fact that, large negative $\epsilon$ can give rise to well-behaved, self-consistent classical solutions, which admit inflationary behaviour, and within which in a certain class of cases, the density perturbation problem may be solved. Our analysis has revealed that at least one wormhole solution exists even for arbitrarily large negative value of $\epsilon$. The possibility of obtaining a class of wormhole solutions for large negative $\epsilon$ has also been explored. Further, it has been noted that for every large positive $\epsilon$, the wave-functional might possibly diverge, as the scale factor goes over to infinity. This definitely gives a further physical motivation for considering large negative values of the non-minimal coupling constant. It is noteworthy that the forms of the wave-functional and remain unaltered for $\lambda = 0$. Hence the wormhole configurations which have been found to exist in the case of quartic potential will also exist even for a free field.\
**Acknowledgement**\
Thanks are due to the referee for his useful comments.
[20]{} T. Padmanavan, Int. J. Mod. Phys. **4**, 4735 (1989). T. Padmanavan and T. P. Singh, Class. Quantum Grav. **7**, 411 (1990). T. P. Singh, Class. Quantum Grav. **7**, L149 (1990). C. Keifer, Class. Quantum Grav. **9**, 147 (1992). A. K. Sanyal, unpublished. [^1] V. Mukhanov and R. Brandenberger, Phys. Rev. Lett. **68**, 1969 (1992). R. Brandenberger, V. Mukhanov and A. Sornborger, preprint, Brown-Het 891 (Feb. 1993). A. H. Guth, Phys. Rev. D **23**, 347 (1981). A. D. Linde, Phys. Lett. B **108**, 389 (1982). A. Albrechet and P. J. Steinhardt, Phys. Rev. Lett. **48**, 1220 (1982). A. D. Linde, Phys. Lett. B **129**, 177 (1983); B **175**, 395 (1986). D. La and P. J. Steinhardt, Phys. Rev. Lett. **62**, 376 (1989). C. Mathiazhagan and V. B. Johri, Class. Quantum Grav. **1**, L29 (1984). F. S. Accetta and J. J. Trester Phys. Rev. D **39**, 2854 (1989). R. Fakir and W. G. Unruh, Phys. Rev. D **41**, 1783 (1990). R. Fakir, S. Habib and W. G. Unruh, Ap. J. **394**, 396 (1992). R. Fakir, Phys. Rev. D **41**, 3012 (1990). A. Vilenkin, Phys. Rev. D **27**, 2848 (1983). J. B. Hartle and S. W. Hawking, Phys. Rev. D **28**, 2960 (1983). D. H. Coule, Class. Quantum Grav. **9**, 2353 (1992). S. B. Giddings and A. Strominger, Nucl. Phys. B **306**, 890 (1988). S. W. Hawking, Phys. Rev. D **37**, 904 (1988). S. Coleman, Nucl. Phys. B **310**, 643 (1988). J. Preskill, Nucl. Phys. B **232**, 141 (1989). S. W. Hawking and D. N. Page, Phys. Rev. D **42**, 2655 (1990).
[^1]: Published later in PMC Physics A, 2009, 3-5 (2009), may also be found in arXiv:0910.2302 \[gr-qc\].
|
---
address:
- 'Robinson College, Cambridge CB3 9AN, United Kingdom'
- 'Gonville & Caius College, Cambridge CB2 1TA, United Kingdom'
author:
- 'Tim$^\introdagger$ and Vladimir Dokchitser'
date: 'March 6, 2008'
title: 'A note on Larsen’s conjecture and ranks of elliptic curves'
---
[^1] [^2]
Introduction
============
In [@Lar] M. Larsen proposed the following:
\[larsen\] Let $K$ be a finitely generated infinite field, $K_s$ its separable closure, $G$ a finitely generated subgroup of $\Gal(K_s/K)$, and $K_{\!s}{}\!^G$ its fixed field. If $E/K$ is an elliptic curve, then the group $E(K_{\!s}{}\!^G)\tensor\Q$ is infinitely generated.
When $G$ is generated by one element, this conjecture is known for $K=\Q$ [@Im] and in a number of other cases [@BI; @Im2; @Im3]; see also [@Loz] for some results when $G$ has two generators. We give evidence for the full conjecture for $K=\Q$ by showing that it is implied both by the Shafarevich-Tate and the (first part of) the Birch–Swinnerton-Dyer conjecture for elliptic curves over number fields:
\[main1\] Let $E/\Q$ be an elliptic curve and $G\subset \Gal(\bar\Q/\Q)$ a finitely generated subgroup. The analytic rank and the $p^\infty$-Selmer rank for every odd $p$ are unbounded in number fields contained in $\bar{\Q}^G$.
Recall that for an elliptic curve $E$ over a number field $F$, the [*analytic rank*]{} $\rk_{an} E/F$ is defined as $\ord_{s=1}L(E/F,s)$. Even for $E$ defined over $\Q$ the analytic continuation of $L(E/F,s)$ to $s=1$ is not known for general $F$, but in the proof of the theorem we will choose number fields where it is. The [*$p^\infty$-Selmer rank*]{} $\rk_p E/F$ is the usual Mordell-Weil rank $\rk E/F=\dim E(F)\tensor\Q$ plus the $\Z_p$-corank of $\sha(E/F)$, the number of copies of $\Q_p/\Z_p$ in $\sha(E/F)$. We remind the reader of the usual conjectures concerning these ranks:
Let $K$ be a number field, and $E/K$ an elliptic curve.
- $L(E/K,s)$ is entire, and $\rk E/K\!=\!\rk_{an}\!E/K$ (Birch–Swinnerton-Dyer).
- $\sha(E/K)$ is finite; thus $\rk E/K\!=\!\rksel EKp$ for all $p$ (Shafarevich-Tate).
- $(-1)^{\rk E/K}=w(E/K)$ (Parity conjecture).
The global root number $w(E/K)$ is defined as a product of local root numbers $w(E/K_v)=\pm 1$ over all places of $K$, and is the conjectural sign in the functional equation for $L(E/K,s)$. Thus it determines $\rk_{an}(E/K)\mod 2$, so the parity conjecture is a weaker version of the Birch–Swinnerton-Dyer conjecture.
In fact, the proof of Theorem \[main1\] shows that the parity conjecture also implies Conjecture \[larsen\] for elliptic curves over $K=\Q$. When $K$ is a general number field, the analytic continuation of $L(E/K,s)$ to $s=1$ is not known so we cannot say anything about $\rk_{an}E/K$. However, we still have the following
\[main2\] Let $K$ be a number field, $E/K$ an elliptic curve and $G$ a finitely generated subgroup of $\Gal(\bar K/K)$. Then both the parity and the Shafarevich-Tate conjectures imply Conjecture \[larsen\], provided
- $K$ has a real place, or
- $E$ has non-integral $j$-invariant.
Slightly more generally, the theorem applies if there is a place $v$ of $K$ and some quadratic extension $L_w/K_v$ where $w(E/L_w)=-1$. (See [@Evilquad] for a classification of such $E/K_v$.)
The proofs of both theorems rely on an elementary root number argument in $\F_p^r\rtimes C_2$-extensions and some version of the parity conjecture. For the second theorem we will need the following result, which is a slight variation of [@Squarity] Thm. 1.3.
\[gensq\] Let $K$ be a number field, $E/K$ an elliptic curve and $M/K$ a quadratic extension. Suppose for every prime $w|6$ of $M$ which is not split in $M/K$, $E$ has semistable reduction at $w$, and the reduction is not good supersingular if $w|2$. If $\sha(E/M(E[2]))[6^\infty]$ is finite, then $$(-1)^{\rk E/M}=w(E/M).$$
This has a consequence, which is of interest in itself:
\[useful\] Let $M/K$ be a quadratic extension of number fields, $E/K$ an elliptic curve and assume that $\sha(E/M(E[2]))$ is finite. If all primes of bad reduction of $E$ split in $M/K$, then the parity conjecture holds for $E/M$, $$(-1)^{\rk E/M} = w(E/M) = (-1)^{\text{{\rm\#Archimedean places of $M$}}}.$$
[**Notation.**]{} Throughout the note $K$ denotes a number field and $E$ an elliptic curve defined over $K$. We write $w(E/K_v)=\pm 1$ for the local root number of $E$ at $v$, and $w(E/K)=\prod_{v} w(E/K_v)$ for the global root number. For an Artin representation $\tau$ of $\Gal(\bar K/K)$ we write $w(E/K_v,\tau)$ and $w(E/K,\tau)=\prod_{v} w(E/K_v,\tau)$ for the local and global root numbers of the twist of $E$ by $\tau$. See e.g. [@RohG] for properties of root numbers of elliptic curves and their twists.
[**Acknowledgments.**]{} We would like to thank A. Jensen for helpful discussions.
Proof of Theorem \[main1\]
==========================
\[lem\] Let $M/K$ be a quadratic extension of number fields, $E/K$ an elliptic curve and $G\subset \Gal(\bar K/K)$ a finitely generated subgroup. For every odd prime $p$ and $r\ge 1$ there exists a Galois extension $F/K$ containing $M$ such that
1. $\Gal(F/K)\iso\F_p^r\rtimes C_2$, with $C_2$ acting by $-1$.
2. The image of $G$ in $\Gal(F/K)$ has order at most 2.
3. The primes of $M$ above primes of bad reduction for $E/K$ split completely in $F$.
For such an extension, $w(E/K,\rho)=w(E/M)$ for every irreducible 2-dimensional representation $\rho$ of $\Gal(F/K)$.
Pick primes $\p_1,\ldots,\p_n$ of $K$ that split completely in $M(\zeta_p)$, and consider $\m=\prod_i\p_i$ as a modulus of $M$ ($\zeta_p$ denotes a primitive $p$th root of unity). Write $I^\m$ for the group of fractional ideals of $M$ that are coprime to $\m$, and $P^\m$ for the subgroup of principal ideals that can be generated by an element congruent to 1 mod $\m$.
Let $Q$ be the largest $\F_p$-vector space quotient of the group $I^\m/P_\m$. Then $2n-\delta \le \dim Q \le 2n$ with $\delta=\dim\O_M^*/\O_M^{*p}+\dim{\rm Cl}_M[p]$. Comparing this with the corresponding group of $K$, we see that the $\Gal(M/K)$-antiinvariant part of $Q$ has dimension $d\ge n-\delta$. By class field theory it yields a Galois extension $F_n/K$ with $\Gal(F_n/K)\iso\F_p^d\rtimes C_2$, $C_2$ acting by $-1$.
Consider $H=\Gal(F_n/M)=\F_p^d$. The group $G\cap\Gal(\bar M/M)$ is of index at most 2 in $G$, and the $\F_p$-dimension of its image in $H$ is bounded by the number of generators of $G$. Also, the decomposition subgroup in $H$ of any prime of $M$ has size bounded by a constant independent of $n$. Now let $H_0\<H$ be generated by the image of $G\cap\Gal(\bar M/M)$ and the decomposition subgroups of primes of $M$ that lie above primes of bad reduction for $E/K$. Then $F_n^{H_0}$ satisfies (2) and (3), and letting $n\to \infty$ gives the desired extensions $F/K$.
For the last claim, let $\epsilon$ be the non-trivial character of $\Gal(M/K)$. The complex irreducible representations of $\Gal(F/K)$ are $\triv$, $\epsilon$ and 2-dimensional representations each of which factors through a $D_{2p}$-quotient.
Recall that $w(E/M)=w(E/K,\triv\oplus\epsilon)$ by inductivity of global root numbers, so it suffices to check that $w(E/K_v,\triv\oplus\epsilon)=w(E/K_v,\rho)$ for all places $v$ of $K$. This holds for places $v$ whose decomposition group $D$ in $F/K$ has order at most 2, because $\rho$ and $\triv\oplus\epsilon$ are isomorphic as $D$-representations. In particular, this includes Archimedean places and places of bad reduction for $E$. On the other hand, if $E$ has good reduction at $v$, then for every 2-dimensional self-dual Artin representation $\tau$ of $\Gal(\bar K/K)$, $$w(E/K_v,\tau) = w(\tau/K_v)^2 = \det(\tau(-1)),$$ by the unramified twist formula [@TatN] 3.4.6 and the determinant formula [@TatN] 3.4.7. Here we implicitly use the local reciprocity map at $v$ to evaluate $\tau$ at $-1$. As $\det\rho=\epsilon=\det(\triv\oplus\epsilon)$, the claim follows.
We now prove Theorem \[main1\]. Take an imaginary quadratic field $M\!=\!\Q(\sqrt{\!-d})$ where all bad primes for $E$ split. Note that $w(E/M)=-1$, as the contribution from bad primes is $(\pm 1)^2$, and the local root number is $+1$ for primes of good reduction and $-1$ for infinite places.
Let $p$ be an odd prime. By the $p$-parity conjecture for $E/\Q$ and its quadratic twist by $-d$ ([@Squarity] Thm. 1.4), we have $(-1)^{\rk_p(E/M)}=w(E/M)$, so $\rk_p(E/M)$ is odd. Let $F$ be as in Lemma \[lem\] for some $r\ge 1$, and let $V\subset\F_p^r$ be any index $p$ subgroup, so $V\triangleleft\Gal(F/\Q)$ and $\Gal(F^V/\Q)\iso D_{2p}$. The image of $G$ in this Galois group has order at most 2, so there is a degree $p$ extension $L/\Q$ inside $F^V$ fixed by $G$. As all bad primes for $E$ split in $M$, by [@Squarity] Prop. 4.17, $$\rksel E{M}p+\tfrac{2}{p-1}(\rksel E{L}p-\rksel E{\Q}p)
\equiv
0
\pmod2.$$ In particular, $\rksel E{L}p>\rksel E{\Q}p$.
Write $X$ for the dual $p^\infty$-Selmer group $\Hom(\Sel_{p^\infty}(E/F),\Q_p/\Z_p)$ and $\X=X\tensor_{\Z_p}\Q_p$. Thus $\X$ is a $\Q_p$-valued representation of $\Gal(F/\Q)$, and $\rksel Ekp=\dim\X^{\Gal(F/k)}$ for every $k\subset F$, see e.g. [@Squarity] Lemma 4.14.
The irreducible $\Q_p$-representations of $\Gal(F^V/\Q)\iso D_{2p}$ are trivial $\triv$, sign $\epsilon$ and $(p-1)$-dimensional irreducible $\rho_V$. Their invariants under an element of order 2 of $D_{2p}$ are $1$-, $0$- and $\tfrac{p-1}2$-dimensional, respectively. It follows that $\X$ contains a copy of $\rho_V$ for every $V$, and $\dim \X^G$ is therefore at least $\tfrac{p-1}2$ times the number of hyperplanes $V\subset\F_p^r$ (which is $\tfrac{p^r-1}{p-1}$). Letting $r\to\infty$ we deduce that the $p^\infty$-Selmer rank of $E$ is unbounded for number fields inside $\bar\Q^G$.
To prove that the analytic rank is unbounded, let $p=3$ for simplicity and take the same $F$ as above. The irreducible complex representations of $\Gal(F/\Q)$ are $\triv, \epsilon$, and $\rho_V$ for varying $V$. The curve $E/\Q$ is modular, so the $L$-functions $L(E/\Q,s)$, $L(E,\epsilon,s)$ and $L(E,\rho_V,s)$ are analytic and satisfy the expected functional equation. (The twists by $\rho_V$ are Rankin-Selberg products.) The same applies to $L(E/k,s)$ for every $k\subset F$; indeed, by Artin formalism, $$L(E/k,s) \>=\> L(E,\Ind_k^\Q\triv,s) \>=\>
\!\!\!\!\prod_{\scriptscriptstyle {\tau\in\{\triv,\epsilon\}\cup\{\rho_V\}_V}}\!\!\!\!
L(E,\tau,s)^{\langle \tau, \Ind_k^\Q\triv \rangle},$$ with $\Ind_k^\Q$ a shorthand for $\Ind_{\Gal(F/k)}^{\Gal(F/\Q)}$. Finally, $w(E,\rho_V)=-1$ by Lemma \[lem\], so each $L(E,\rho_V,s)$ vanishes at $s=1$. Hence $$\rkan Ek\ge\frac{[k:\Q]-2}2$$ as $\Ind_k^\Q\triv$ contains at most one copy of $\triv$ and $\epsilon$. Now take $k=F^G$ and let $r\to\infty$ as before.
Proof of Theorem \[main2\]
==========================
We claim there is a quadratic extension $M/K$ where $w(E/M)=-1$ and $\rk E/M$ is odd. This will suffice, as we can then use exactly the same construction as in the proof of Theorem \[main1\] with $K$ in place of $\Q$ and any odd $p$. Fix a place $v$ of $K$ which is either real or with $\ord_v j(E)<0$. We take $M/K$ to be any quadratic extension where
- $v$ becomes complex if $K_v=\R$.
- There is a unique prime $v'$ above $v$, and $E$ has split multiplicative reduction at $v'$ if $\ord_v j(E)<0$.
- If $w\ne v$ is either Archimedean, a place of bad reduction of $E$ or divides 2 then $w$ splits in $M/K$.
Such a quadratic $M$ exists by the weak approximation theorem, as we prescribe only its local behaviour at finitely many places. The existence of an appropriate local extension in the second case follows from the theory of the Tate curve, see e.g. [@Sil2] Thm 5.3.
Write $v'$ for the unique prime of $M$ above $v$. Then $w(E/M_{v'})=-1$ (see e.g. [@RohG] Thm. 2) and all other Archimedean places and bad primes for $E$ contribute $(\pm 1)^2=1$, so $w(E/M)=-1$. Finally $\rk E/M$ is odd, by either the assumed parity conjecture or the Shafarevich-Tate conjecture together with Proposition \[gensq\].
Proof of Proposition \[gensq\]
==============================
\[lemma2\] Let $K$ be a number field and $E/K$ an elliptic curve with a $K$-rational 2-torsion point. Let $\Sigma$ be the set of primes $v|2$ of $K$ where $E$ has additive or good supersingular reduction. Suppose $M/K$ is a quadratic extension where every prime $v\in\Sigma$ either splits or becomes a place of multiplicative or good ordinary reduction for $E$. Then $(-1)^{\rksel EM2}=w(E/M)$.
Choose a model for $E/K$ of the form $$E: \>\> y^2 = x^3 + ax^2 + bx \>,$$ and let $\phi:E\to E'$ be the $2$-isogeny with $(0,0)$ in the kernel.
Let $\sigma_w(E/M)=(-1)^{\ord_2\coker\phi_w-\ord_2\ker\phi_w}$, where $\phi_w: E(M_w)\to E'(M_w)$ is the induced map on local points. By Cassels’ formula (see [@Isogroot] §1), $$(-1)^{\rksel EM2}=\prod_w \sigma_w(E/M).$$ Write $(\cdot,\cdot)_w=\pm 1$ for the Hilbert symbol at $w$. Then $$\prod_{w|v} w(E/M_w) = \prod_{w|v} \sigma_w(E/M) (a,-b)_w(-2a,a^2-4b)_w$$ for every place $v$ of $K$. Indeed, this trivially holds for all $v$ that split in $M/K$, and it holds at all other primes by [@Isogroot], proof of Thm. 4 in §7. The result follows by the product formula for Hilbert symbols.
\[lemma3\] Let $K$ be a number field and $E/K$ an elliptic curve. Let $\Sigma$ be the set of primes $v|6$ of $K$ where $E$ has additive reduction. Suppose $M/K$ is a quadratic extension where every prime $v\in\Sigma$ either splits or becomes a place of semistable reduction for $E$. Let $F/K$ be any $S_3$-extension not containing $M$, and write $N=F^{C_2}M$, $N'=F^{C_3}M$. Then $$w(E/M)w(E/N)w(E/N') = (-1)^{\rksel EM3+\rksel EN3+\rksel E{N'}3}.$$
Fix a global differential $\omega\ne 0$ for $E/K$. For an extension $k/K$ and a prime $w$ of $k$ write $C_w(E/k) = c_w(E/k) |{\omega}/{\neron{w}}|_w$, where $c_w$ is the local Tamagawa number, $\neron{w}$ a Néron differential at $w$, and $|\cdot|_w$ the normalised $w$-adic absolute value. By [@Squarity] Thm. 4.11 with $p=3$, $$\rksel EM3+\rksel EN3+\rksel E{N'}3 \equiv
\ord_3
\frac{\prod_z C_z(E/MF)}{\prod_u C_u(E/N')}
\mod 2,$$ the products taken over the primes of $MF$ and $N'$ respectively. It suffices to show that for every place $v$ of $K$, $$\prod\limits_{s|v} w(E/M_s) \prod\limits_{t|v} w(E/N_t) \prod\limits_{u|v} w(E/N'_u)
=
\frac{\prod_{z|v}(-1)^{\ord_3 C_z(E/MF)}}{\prod_{u|v}(-1)^{\ord_3 C_u(E/N')}}\>,$$ where we interpret the right-hand side as 1 for Archimedean places. If $v$ splits in $M/K$ the formula trivially holds as each term occurs an even number of times. For all other $v$, this is proved in [@Squarity] Prop 3.3 (the $G$-set argument in Case 1 also covers Archimedean places).
We now prove Proposition \[gensq\].
The assumption on $\sha$ also forces $\sha(E/k)[6^\infty]$ to be finite for all intermediate fields $K\subset k\subset M(E[2])$, see e.g. [@Squarity] Rmk. 2.10. Now we proceed as in the proof of [@Squarity] Thm. 3.6.
Write $F=K(E[2])$. If $E(K)[2]\ne 0$, apply Lemma \[lemma2\]. Otherwise $\Gal(F/K)$ is either $C_3$ or $S_3$. In the former case, $\Gal(FM/M)\iso C_3$ as well, thus $\rk E/M$ and $\rk E/FM$ have the same parity. It is also well-known that global root numbers are unchanged in odd degree cyclic extensions (this follows from [@TatN] 3.4.7, 4.2.4), so $w(E/FM)=w(E/M)$ and the result again follows from Lemma \[lemma2\] applied to $FM/F$.
In the last case $\Gal(F/K)\iso S_3$, write $N=F^{C_2}M$, $N'=F^{C_3}M$. If $M\not\subset F$, the above argument shows that $w(E/N)=(-1)^{\rk E/N}$ and similarly for $N'$; now apply Lemma \[lemma3\]. If $M\subset F$, then $F/M$ is a Galois cubic extension, so we may again show that $w(E/F)=(-1)^{\rk E/F}$. But this holds by Lemma \[lemma2\] applied to the quadratic extension $F/N$.
Finally, Corollary \[useful\] is immediate from the fact that $w(E/K_v)=-1$ for Archimedean $v$.
[9]{}
F. Breuer, B.-H. Im, Heegner points and the rank of elliptic curves over large extensions of global fields, arxiv:math/0604107.
T. Dokchitser, V. Dokchitser, Parity of ranks for elliptic curves with a cyclic isogeny, J. Number Theory 128 (2008), 662–679.
T. Dokchitser, V. Dokchitser, On the Birch–Swinnerton-Dyer quotients modulo squares, 2006, arxiv: math.NT/0610290. T. Dokchitser, V. Dokchitser, Elliptic curves with all quadratic twists of positive rank, 2008, arxiv: 0802.4027.
B.-H. Im, Heegner points and Mordell-Weil groups of elliptic curves over large fields, to appear in Trans. Amer. Math. Soc.
B.-H. Im, Mordell-Weil groups and the rank of elliptic curves over large fields, arxiv:math/0411533.
B.-H. Im, Heegner points and Mordell-Weil groups of elliptic curves over large fields, arxiv:math/0411534.
M. Larsen, Rank of elliptic curves over almost algebraically closed fields, Bull. London Math. Soc. 35 (2003), 817-820.
Á. Lozano-Robledo, On the product of twists of rank two and a conjecture of Larsen, preprint.
D. Rohrlich, Galois Theory, elliptic curves, and root numbers, Compos.Math. 100 (1996), 311–349.
J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, Springer-Verlag 1994.
J. Tate, Number theoretic background, in: Automorphic forms, representations and L-functions, Part 2 (ed. A. Borel and W. Casselman), Proc. Symp. in Pure Math. 33 (AMS, Providence, RI, 1979) 3-26.
[^1]: [*MSC 2000:*]{} Primary 11G05; Secondary 11G40, 14G25
[^2]: $^\dagger$Supported by a Royal Society University Research Fellowship
|
---
abstract: 'We study the correlations of classical and quantum systems from the information theoretical points of view. We analyze a simple measure of correlations based on entropy (such measure was already investigated as [*the degree of entanglement*]{} by Belavkin, Matsuoka and Ohya). Contrary to naive expectation, it is shown that separable state might possesses stronger correlation than an entangled state.'
author:
- |
Yuji HIROTA\
Department of Mathematics, Tokyo University of Science, Tokyo, 162-8601, Japan
- |
Dariusz CHRUŚCIŃSKI\
Institute of Physics, Nicolaus Copernicus University, Toruń, 87-100, Poland
- |
Takashi MATSUOKA\
Department of Business Administration and Information,\
Tokyo University of Science, Suwa, Nagano, 391-0292, Japan
- |
and\
Masanori OHYA\
Department of Information Science, Tokyo University of Science, Chiba, 278-8510, Japan
title: |
**On correlations and mutual entropy\
in quantum composite systems**
---
Introduction
============
Correlations play a key role both in classical and quantum physics. In particular the study of correlations is crucial in many-body physics and classical and quantum statistical physics. Recently, it turned out that correlations play prominent role in quantum information theory and many modern applications of quantum technologies and there are dozens of papers dealing with this problem (for the recent review see e.g. [@HHHH]).
The aim of this paper is to analyze classical and quantum correlations encoded in the bi-partite quantum states. Beside quantum entanglement we analyze a new measure – so called $D$-correlations – and the quantum discord. We propose to compare correlations of different bi-partite states with the same reduces states, i.e. locally they contain the same information. It is shown that surprisingly a separable state may be more correlated that an entangled one. Analyzing simple examples of Bell diagonal states we illustrate the behavior of various measures of correlations. We also provide an introduction to bi-partite states and entanglement mappings introduced by Belavkin and Ohya and recall basic notions from classical and quantum information theory. An entanglement mapping encodes the entire information about a bi-partite quantum state and hence it provides an interesting way to deal with entanglement theory. Interestingly, it may be applied in infinite-dimensional case and in the abstract $\mathbb{C}^*$-algebraic settings. Therefore, in a sense, it provides a universal tool in entanglement theory.
The paper is organized as follows: in the next section we recall basic facts from the theory of composite quantum systems and introduce the notion of entanglement mappings. Moreover, we recall an interesting construction of quantum conditional probability operators. Section \[E\] recall classical and quantum entropic quantities and collects basic facts from classical and quantum information theory. In particular it contains the new measure of correlation called $D$-correlation. Section \[DISCORD\] recalls the notion of *quantum discord* which was intensively analyzed recently in the literature. In section \[CIRC\] we recall the notion of a circulant state and provide several examples of states for which one is able to compute various measures of correlations. Final conclusions are collected in the last section.
Throughout the paper, we use standard notation: $\mathcal{H},\,\mathcal{K} $ for complex separable Hilbert spaces and denote the set of the bounded operators and the set of all states on $\mathcal{H}$ by $\mathbf{B}(\mathcal{H})$ and $\mathbf{S}(\mathcal{H})$, respectively. In the $d$-dimensional Hilbert space, the standard basis is denoted by $\{e_0,e_1,\cdots,e_{d-1}\}$ and the inner product is denoted by $\langle \cdot,\,\cdot\rangle$. We write $e_{ij}$ for $|e_i\rangle\langle e_j|$. Given any state $\theta$ on the tensor product Hilbert space $\mathcal{H}\otimes\mathcal{K}$, we denote by $\mathrm{Tr}_{\mathcal{K}}\theta$ the partial trace of $\theta$ with respect to $\mathcal{K}$.
Quantum states and entanglement maps
====================================
Consider a quantum system living in the Hilbert space $\mathcal{H}$. In this paper we consider only finite dimensional case. However, as we shall see several results may be nicely generalized to the infinite-dimensional setting. Denote by ${\mathcal{T}(\mathcal{H})}$ a set of trace class operators in $\mathcal{H}$, meaning that $\rho \in {\mathcal{T}(\mathcal{H})}
$ if $\rho \geq 0$ and $\mathrm{Tr}\, \rho < \infty$, which is always true in finite-dimensional case. Finally, let $${\mathbf{S}(\mathcal{H})} = \{\, \rho \in {\mathcal{T}(\mathcal{H})}\ |\
\mathrm{Tr}\, \rho =1\, \}\ ,$$ Consider now a composite system living in ${\mathcal{H} \otimes \mathcal{K}}$ and denote by $\mathbf{S}_{\mathrm{SEP}} \subset \mathbf{S}({\mathcal{H}
\otimes \mathcal{K}})$ a convex subset of separable states in ${\mathcal{H}
\otimes \mathcal{K}}$. Recall that $\rho \in \mathbf{S}({\mathcal{H} \otimes
\mathcal{K}})$ is separable if $\rho = \sum_\alpha \, p_\alpha\, \eta_\alpha {\,\otimes\,} \sigma_\alpha$, where $\eta_\alpha \in {\mathbf{S}(\mathcal{H})}$ and $\sigma_\alpha \in {\mathbf{S}(\mathcal{K})}$, and $p_\alpha$ denotes probability distribution: $p_\alpha \geq 0$ an $\sum_\alpha p_\alpha =1$. A state $\rho \in \mathbf{S}({\mathcal{H} \otimes \mathcal{K}})$ is called positive partial transpose (PPT) if its partial transpose satisfies $(\mathrm{id}_\mathcal{H} {\,\otimes\,} \tau)\rho \geq 0$, where $\mathrm{id}_\mathcal{H}$ denotes an identity map in ${\mathbf{B}(\mathcal{H})}$. It means that $\rho$ is PPT if $(\mathrm{id}_\mathcal{H} {\,\otimes\,} \tau)\rho \in \mathbf{S}({\mathcal{H} \otimes \mathcal{K}})$. Denote by $\mathbf{S}_{\mathrm{PPT}}$ a convex subset of PPT states. It is well known [@Pe] that $\mathbf{S}\left( \mathcal{H}\otimes\mathcal{K}\right)\supset \mathbf{S}_{\mathrm{PPT}}\supset \mathbf{S}_{\mathrm{SEP}}$. In general, the PPT condition is not sufficient for separability.
Interestingly, due to the well known duality between states living in ${\mathcal{H} \otimes \mathcal{K}}$ and linear maps ${\mathbf{B}(\mathcal{K})}
{\, \rightarrow\, } {\mathbf{B}(\mathcal{H})}$, one may translate the above setting in terms of linear maps. Let us recall basic facts concerning completely positive maps [@Paulsen]. A linear map $\chi:\mathbf{B} (\mathcal{K}) \to \mathbf{B}(\mathcal{H})$ is said to be completely positive (CP) if, for any $n\in\mathbb{N}$, the map $$\chi _n: M_n(\mathbb{C})\otimes \mathbf{B}(\mathcal{K}) \longrightarrow
M_{n}(\mathbb{C})\otimes \mathbf{B}(\mathcal{H}), \quad (a_{i,j})_{i,j}
\longmapsto \bigl(\chi(a_{i,j})\bigr)_{i,j}$$ is positive, where $\mathbf{B}(\mathcal{H})$ denotes bounded operators in $\mathcal{H}$ and $M_{n}(\mathbb{C})$ stands for $n\times n$ matrices with entries in $\mathbb{C}$. A linear map $\chi:\mathbf{B}(\mathcal{K}) \to
\mathbf{B}(\mathcal{H})$ is said to be completely copositive (CCP) if composed with transposition $\tau$, i.e. $\tau\circ\chi$, is CP.
Consider now a state $\theta \in \mathbf{S}({\mathcal{H} \otimes \mathcal{K}})$ and let $\phi: \mathbf{B}(\mathcal{K})\to \mathbf{B}(\mathcal{H} )$ be a linear map defined by $$\phi(b):= \mathrm{Tr}_{\mathcal{K}}\, [(1_\mathcal{H}\otimes b)\theta]\ ,$$ for any $b \in {\mathbf{B}(\mathcal{K})}$. The dual map $\phi^*$ reads $$\phi^*(a) = \mathrm{Tr}_{\mathcal{H}}\,[(a\otimes 1_\mathcal{K}) \theta] \ ,$$ for any $b \in {\mathbf{B}(\mathcal{H})}$. It should be stressed that the above construction is perfectly well defined also in the infinite-dimensional case if wew assume that $\theta$ is a normal state, that is, it is represented by the density operator. Note, that a state $\theta$ and the linear map $\phi$ give rise a linear functional $\omega :
\mathbf{B}({\mathcal{H} \otimes \mathcal{K}}) {\, \rightarrow\, } \mathbb{C}$ $$\omega (a\otimes b) : = \mathrm{Tr}(a\otimes b) \theta,$$ for any $a\in \mathbf{B}(\mathcal{H}),\,b\in \mathbf{B}(\mathcal{K})$. This formula may be equivalently rewritten as follows $$\omega (a\otimes b) =\mathrm{Tr}_{\mathcal{H}}~a\phi(b) = \mathrm{Tr}_{\mathcal{K}} ~ \phi^\ast(a) b. \label{trace}$$ It is clear that the marginal states read $$\mathrm{Tr}_{\mathcal{K}}\theta = \phi(1_\mathcal{K})\in \mathbf{B}(\mathcal{\ H}),\quad \mathrm{Tr}_{\mathcal{H}}\theta =\phi^\ast(1_\mathcal{H})\in
\mathbf{B}(\mathcal{K}).$$ Belavkin and Ohya observed [@BO1; @BO2] that if $\theta \in \mathbf{S}({\mathcal{H} \otimes \mathcal{K}})$, then both $\phi$ and its dual $\phi^*$ are CCP. We denote by $\mathbf{B}\mathcal{(H)}$ the dual space to the algebra $\mathbf{B}\mathcal{(H)}$.
A CCP map $\phi:\mathbf{B}\mathcal{(K)}\rightarrow\mathbf{B}\mathcal{(H)} $ normalized as $\mathrm{Tr}_{\mathcal{H}}\phi (1_{\mathcal{K}}) =1$ is called the entanglement map from $\rho:=\phi ^{\ast}(1_{\mathcal{H}})\in \mathbf{B}\mathcal{(K)}$ to $\sigma:=\phi (1_{\mathcal{K}})\in \mathbf{B}(\mathcal{H}
) $.
A density operator $\theta _{\phi }$ corresponding to the entanglement map $\phi$ with its marginals $\phi ^{\ast}(1_{\mathcal{H}})$ and $\phi (1_{
\mathcal{K}})$ can be represented as follows: let $\psi^+_\mathcal{K}$ denotes a maximally entangled state in $\mathcal{K}{\,\otimes\,} \mathcal{K}$. Then $$\theta_\phi := (\phi {\,\otimes\,} \tau) P^+_\mathcal{K}\ ,$$ with $P^+_\mathcal{K} = d_\mathcal{K}\,|\psi^+_\mathcal{K}\>\<\psi^+_\mathcal{K}|$, where $d_\mathcal{K} = \mathrm{dim}\,\mathcal{K}$. If $\{ e_k
\}$ stands for an orthonormal basis in $\mathcal{K}$, then $$P^+_\mathcal{K} = \sum_{i,j=1}^{d_\mathcal{K}} e_{ij} {\,\otimes\,} e_{ij}\ ,$$ with $e_{ij}:= |e_i\>\<e_j|$, and hence $$\theta_{\phi}= \sum_{i,j=1}^{d_\mathcal{K}}\, \phi(e_{ji}) \otimes e_{ij} \ .
\label{def1}$$ The map assigning $\theta_\phi$ to $\phi$ is usually called a Choi-Jamiołkowski isomorphism. It should be stressed that $\theta_\phi$ does not depend upon the choice of $\{e_k\}$.
\[w-criterion\] A linear map $\phi : {\mathbf{B}(\mathcal{K})}\rightarrow {\mathbf{B}(\mathcal{H})}$ is CCP if and only if $\theta _{\phi
}\geq 0$. Clearly, $\phi $ is CP if and only if $\phi \circ \tau $ is CCP.
Due to Lemma \[w-criterion\], we have the following criterion.
[@JMO; @MMO] A state $\theta _{\phi }$ is a PPT state if and only if its entanglement map $\phi $ is CP.
Recently, Kossakowski et al.[@AKMS] proposed the following construction: for $\theta \in \mathbf{S}(\mathcal{H} \otimes\mathcal{K})$ one defines the bounded operator $$\pi_{\theta}:= \bigl(\rho^{-\frac{1}{2}}\otimes 1_\mathcal{K}\bigr)\,
\theta\,\bigl(\rho^{-\frac{1}{2}}\otimes 1_\mathcal{K}\bigr),$$ where $\rho :=\mathrm{Tr}_{\mathcal{K}}\theta$. It is verified that $\pi_\theta$ satisfies $$\begin{aligned}
\pi_{\theta }&\geq 0, \label{con1} \\
\mathrm{Tr}_{\mathcal{K}}\pi_\theta &= 1_\mathcal{H} \in \mathbf{B}(\mathcal{\ H}). \label{con2}\end{aligned}$$ In what follows we assume that $\rho$ is a faithful state, i.e. $\rho >0$. It follows from ( \[con1\]) and (\[con2\]) that the operator $\pi_\theta$ is the quantum analogue of a classical conditional probability. Indeed, if $\mathbf{B}( \mathcal{H}\otimes\mathcal{K})$ is replaced by commutative algebra, then $\pi_\theta$ coincides with a classical conditional probability.
An operator $\pi\in \mathbf{B}(\mathcal{H}\otimes\mathcal{K})$ is called the quantum conditional probability operator (QCPO, for short) if $\pi$ satisfies condition (\[con1\]) and (\[con2\]).
It is easy to verify[@AKMS] that for any CP unital map $\varphi : {\mathbf{B}(\mathcal{K})} {\, \rightarrow\, }{\mathbf{B}(\mathcal{H})}$ and an orthonormal basis in $\mathcal{K}$ the following operator $$\pi_\varphi=\sum_{k,l=1}^{d_\mathcal{K}}\varphi(e_{kl}) \otimes e_{kl} \ ,
\label{def2}$$ defines QCPO. From Lemma \[w-criterion\] and unitality of $\varphi$, it follows that $\pi _{\varphi }$ satisfies conditions (\[con1\]) and ([con2]{}). For a given $\pi_{\varphi }$ and any faithful marginal state $\rho
\in \mathbf{S}(\mathcal{H })$, one can construct a state $\theta$ of the composite system $$\theta_\varphi = \sum_{k,l=1}^{d_\mathcal{K}} \rho^{\frac{1}{2}}\,\varphi(e_{kl}) \rho^{\frac{1}{2}}\otimes e_{kl}\ .$$ It is clear that $\theta_\varphi$ is a PPT state if and only if the map $\varphi$ is a CCP. There exists a simple relation between the density operator $\theta_\phi$ in (\[def1\]) and the QCPO $\pi_\varphi$ in ([def2]{}) due to the following decomposition of the entanglement map $\phi $.
[@BD] Every entanglement map $\phi $ with $\phi(1_\mathcal{K}) =\rho$ has a decomposition $$\phi \left( \cdot \right) =\rho ^{\frac{1}{2}}\varphi \circ \tau \left(
\cdot \right) \rho ^{\frac{1}{2}},$$ where $\varphi $ is a CP unital map to be found as a unique solution to $$\varphi(\cdot ) =\rho ^{-\frac{1}{2}}\phi \circ \tau(\cdot) \rho ^{-\frac{1}{
2}}.$$
[@CKMO] If a composite state $\theta _{\phi }$ given by (\[def1\]) has a faithful marginal state $\rho =\phi(1_\mathcal{K})$, then $\theta_\phi$ is represented by $$\theta_\phi=\bigl(\rho^{\frac{1}{2}}\otimes 1_\mathcal{K}\bigr)\,
\pi_{\phi}\,\bigl(\rho^{\frac{1}{2}}\otimes 1_\mathcal{K}\bigr),
\label{FundaQ}$$ where $\pi_\phi=\sum_{k,l}\rho^{-\frac{1}{2}}\,\phi(e_{kl})\,\rho^{-\frac{1}{2}}\otimes e_{kl}$.
Classical and quantum information {#E}
=================================
In classical description of a physical composite system its correlation can be represented by a joint probability measure or a conditional probability measure. In classical information theory we have proper criteria to estimate such correlation, which are so-called the mutual entropy and the conditional entropy given by Shannon [@Sh]. Here we review Shannon’s entropies briefly.
Let $X=\{x_{i}\}_{i=1}^{n}$ and $Y=\{y_{j}\}_{j=1}^{m}$ be random variables with probability distributions $p_{i}$ and $q_{j}$, respectively, and let $p_{i|j}$ denotes conditional probability $P(X=x_{i}|Y=y_{j})$. The joint probability $r_{ij}=P(X=x_{i},Y=y_{j})$ is given by $$r_{ij}=p_{i|j}\,q_{j}\ . \label{FundaC}$$Let us recall definitions of mutual entropy $I(X:Y)$ and conditional entropies $S(X\mid Y),\,S(Y\mid X)$: $$I(X:Y)=\sum_{i,j}\,r_{ij}\log \frac{r_{ij}}{p_{i}q_{j}}\ ,$$and $$S(X\mid Y)=-\sum_{j}q_{j}\sum_{i}p_{i|j}\log p_{i|j}\ ,\ \ \ \ \ S(Y\mid
X)=-\sum_{i}p_{i}\sum_{j}p_{j|i}\log p_{j|i}\ .$$ Using (\[FundaC\]), we can easily check that the following relations $$I(X:Y)=S(X)+S(Y)-S(XY)\ , \label{relation1}$$ and $$\begin{aligned}
S(X\mid Y)& =S(XY)-S(Y)=S(X)-I(X:Y)\ , \label{relation2} \\
S(Y\mid X)& =S(XY)-S(X)=S(Y)-I(X:Y)\ , \label{relation3}\end{aligned}$$ where $S(X)=-\sum_{i}p_{i}\log p_{i}\,$, and $S(XY)=-\sum_{ij}r_{ij}\log r_{ij}$. Note, that $p_{i|j}$ gives rise to a stochastic matrix $T_{ij}:=p_{i|j}$ and hence it defines a classical channel $$p_{i}=\sum_{j}T_{ij}q_{j}\ .$$ Note, that data provided by $r_{ij}$ are the same as those provided by $T_{ij}$ and $p_{j}$. Hence one may instead of $I(X:Y)$ use the following notation $I(P,T)$, where $P$ represent an *input* state and $T$ the classical channel. One interprets $I(P,T)$ as a information transmitted *via* a channel $T$. The fundamental Shannon inequality $$0\leq I(P;T)\leq \min \bigl\{S(X),\,S(Y)\bigr\}\ , \label{FundaI}$$ gives the obvious bounds upon the transmitted information.
Now, we extend the classical mutual entropy to the quantum system using the Umegaki relative entropy.[@Ume] Let $\theta \in \mathbf{S}({\mathcal{H}
\otimes \mathcal{K}})$ with marginal states $\rho \in \mathbf{S}(\mathcal{H}) $ and $\sigma\in \mathbf{S}(\mathcal{K})$. One defines quantum mutual entropy as a relative entropy between $\theta$ and the product of marginals $\rho {\,\otimes\,} \sigma$: $$I(\theta) = S(\theta\,||\,\rho {\,\otimes\,} \sigma) = \mathrm{Tr}\, \{
\theta\bigl(\log \theta - \log [\rho \otimes\sigma] \bigr) \}\ .$$ As in the classical case one shows that $$I(\theta) = S(\rho) + S(\sigma) - S(\theta)\ .$$ Introducing quantum conditional entropy $$S_\theta(\rho\,|\,\sigma) := S(\theta) - S(\sigma)\ ,$$ one finds $$I(\theta) = S(\rho) - S_\theta(\rho\,|\,\sigma)\ ,$$ or, equivalently $$I(\theta) = S(\sigma) - S_\theta(\sigma\,|\,\rho)\ .$$
[@BO1; @BO2; @Cerf; @Gro] For any entanglement map $\phi:\mathbf{B}(\mathcal{K})\to\mathbf{B}(\mathcal{H})$ with $\rho = \phi(1_\mathcal{K})$ and $\sigma
=\phi^\ast(1_\mathcal{H})$, the quantum mutual entropy $I_\phi(\rho :\sigma)$ is defined by $$I_\phi(\rho :\sigma) := S(\theta_\phi\, ||\, \rho \otimes \sigma) = \mathrm{
Tr}\, \{\theta_\phi\bigl(\log \theta _\phi - \log [\rho \otimes\sigma]
\bigr) \}\ ,$$ where $S(\cdot\, ||\, \cdot) $ is the Umegaki relative entropy.
One easily finds $$\begin{aligned}
I_\phi(\rho : \sigma) = S(\rho) + S(\sigma) -S(\theta_\phi)\ .
\label{relation4}\end{aligned}$$ The above relation (\[relation4\]) is a quantum analog of (\[relation1\]). One defines the quantum conditional entropies as generalizations of (\[relation2\]), (\[relation3\]) [@BO1; @BO2; @Cerf; @HH]: $$\begin{aligned}
S_\phi(\sigma\,|\,\rho) := S(\sigma) -I_\phi(\rho :\sigma) =S(\theta_\phi) -
S(\rho)\ .\end{aligned}$$ It is usually assumed that $I_\phi(\rho:\sigma)$ measures all correlations encoded into the bipartite state $\theta_\phi$ with marginals $\rho$ and $\sigma$.
For the entanglement map $$\phi(b) := \rho\, \mathrm{Tr}_\mathcal{K}(\sigma b)\ ,$$ one finds $\theta _{\phi }=\rho \otimes \sigma$, and hence $$\begin{aligned}
I_\phi(\rho : \sigma) = 0 \ , \ \ S_{\theta }\bigl(\sigma\,|\,\rho \bigr) =
S(\sigma)\ , \ \ S_{\theta }\bigl(\rho\,|\,\sigma \bigr) = S(\rho) \ ,\end{aligned}$$ which recover well known relations for a product state $\rho {\,\otimes\,}
\sigma$.
\[example:pure entangled\] Let $\{\lambda _{i}\}$ be the sequence of complex numbers satisfying $\sum_{i}|\lambda _{i}|^{2}=1$. For entanglement mappings $$\phi (b)=\sum_{i,j=1}^{r}\, \lambda _{i}\overline{\lambda }_{j}\, e_{ij}\,
\langle f_{j},\,bf_{i}\rangle \ ,$$ where $\{ e_k\}$ and $\{f_l\}$ are orthonormal basis in $\mathcal{H}$ and $\mathcal{K}$, respectively, the state $\theta _{\phi }$ can be written in the following form $$\theta _{\phi }=\sum_{i,j=1}^{r}\lambda _{i}\,\overline{\lambda }_{j}\,
e_{ij} \otimes f_{ij} =\bigl\vert\Psi \rangle \langle \Psi \bigr\vert\ ,$$ where $$\bigl\vert\Psi \bigr\rangle=\sum_{i=1}^r\lambda _{i}\,e_{i}\otimes f_{i} \in
{\mathcal{H} \otimes \mathcal{K}}\ .$$ Note, that $$r \leq \min\{ d_\mathcal{H},d_\mathcal{K}\} \ ,$$ equals to the Schmidt rank of $\Psi \in {\mathcal{H} \otimes \mathcal{K}}$. One finds for the reduced states $$\rho =\phi (1_{\mathcal{K}})=\sum_{i=1}^r|\lambda _{i}|^{2} e_{ii}\ , \ \ \
\ \sigma =\phi ^{\ast }(1_{ \mathcal{H}})=\sum_{i=1}^r|\lambda _{i}|^{2}
f_{ii}\ ,$$ and hence $$\begin{aligned}
I_\phi(\rho : \sigma) =S(\rho )+S(\sigma )-S(\theta) =2S(\rho )>\min \bigl\{S(\rho ),S(\sigma )\bigr\}\ ,\end{aligned}$$ together with $$\begin{aligned}
S_{\theta}(\sigma | \rho ) =S_{\theta }(\rho | \sigma )=-S(\rho )<0,\end{aligned}$$ where $S(\rho )=S(\sigma )=-\sum_{i=1}^r|\lambda _{i}|^{2}\log |\lambda
_{i}|^{2} $.
As is mentioned in Section 2, the classical mutual entropy always satisfies the Shannon’s fundamental inequality, i.e. it is always smaller than its marginal entropies, and the conditional entropy is always positive. Note that separable state has the same property. It is no longer true for pure entangled states.
Now we introduce another measure for correlation of composite states.[BO1,BO2,CKMO,MO]{}
For the entanglement map $\phi: \mathbf{B}(\mathcal{K})\to\mathbf{B}(\mathcal{H})$, we define the $D$-correlation $D(\theta)$ of $\theta$ as $$\begin{aligned}
D(\theta) &:= -\frac{1}{2}\left\{ S_{\theta}(\sigma|\rho) + S_\theta(\rho|
\sigma)\right\} = \frac 12 ( S(\rho) + S(\sigma)) - S(\theta) \ .
\label{DEN}\end{aligned}$$
Note that the $D$-correlation with the opposite convention $-D(\theta)$ is called the degree of entanglement.[@BO1; @BO2; @CKMO; @MO] One proves the following:
\[section4:purecase\] [@AMO; @MO] If $\theta_\phi$ is a pure state, then the following statements hold:
1. 2.
It is well-known that if $\theta$ is a PPT state, then $$S(\theta) -S(\rho) \geq 0,\quad S(\theta)-S(\sigma) \geq 0,$$ where $\rho $ and $\sigma $ are the marginal states of $\theta$.[@VW]
\[section4:mixedcase\] If $\theta$ is a PPT state, then $$D(\theta)\leq 0.$$
Suppose now that we have two entanglement mappings $\phi_k:\mathbf{B}(\mathcal{K})\to\mathbf{B}(\mathcal{H}),\,(k=1,2)$ such that $\phi_1(1_\mathcal{K})=\phi_2(1_\mathcal{K})$ and $\phi_1^\ast(1_\mathcal{H})=\phi_2^\ast(1_\mathcal{H})$. Let $\theta_1, \theta_2 \in \mathbf{S}({\mathcal{H} \otimes \mathcal{K}})$ be the corresponding states. We propose the following:
$\theta_1$ is said to have stronger $D$-correlations than $\theta_2$ if $$D(\theta_1) > D(\theta_2)\ . \label{order}$$
Several measures of correlation based on entropic quantities were already discussed by Cerf and Adami[@Cerf], Horodecki[@HH], Henderson and Vedral[@Vedral], Groisman et al.[@Gro].
Quantum discord {#DISCORD}
===============
Let us briefly recall the definition of quantum discord [@Zurek; @Vedral]. Recall, that mutual information may be rewritten as follows $$\mathcal{I}(\theta) = S(\sigma) - S_\theta(\sigma|\rho) \ .$$ An alternative way to compute the conditional entropy $S_\theta(\sigma|\rho)$ goes as follows: one introduces a measurement on $\mathcal{H}$-party defined by the collection of one-dimensional projectors $\{\Pi_k\}$ in $\mathcal{H}$ satisfying $\Pi_1 + \Pi_2 + \ldots = 1_\mathcal{H}$. The label ‘$k$’ distinguishes different outcomes of this measurement. The state after the measurement when the outcome corresponding to $\Pi_k$ has been detected is given by $$\theta_{\mathcal{K}|k} = \frac{1}{p_k} (\Pi_k {\,\otimes\,} 1_\mathcal{K})\theta (\Pi_k {\,\otimes\,} 1_\mathcal{K})\ ,$$ where $p_k$ is a probability that $\mathcal{H}$-party observes $k$th result, i.e. $p_k =
\mathrm{Tr}(\Pi_k \rho)$, and $\theta_{\mathcal{K}|k}$ is the (collapsed) state in ${\mathcal{H} \otimes \mathcal{K}}$, after $\mathcal{H}$-party has observed $k$th result in her measurement. The entropies $S(\theta_{\mathcal{K}|k})$ weighted by probabilities $p_k$ yield the conditional entropy of part $\mathcal{K}$ given the complete measurement $\{\Pi_k\}$ on the part $\mathcal{H}$ $$S(\theta|\{\Pi_k\}) = \sum_k p_k S(\theta_{\mathcal{K}|k})\ .$$ Finally, let $$\mathcal{I}(\theta|\{\Pi_k\}) = S(\sigma) - S(\theta|\{\Pi_k\}) \ ,$$ be the corresponding measurement induced mutual information. The quantity $$\label{C-sup}
\mathcal{C}_{\mathcal{H}}(\theta) = \sup_{\{\Pi_k\}} \mathcal{I}(\theta|\{\Pi_k\})\ ,$$ is interpreted [@Zurek; @Vedral] as a measure of classical correlations. Now, these two quantities – $\mathcal{I}(\theta)$ and $\mathcal{C}_\mathcal{H}(\theta)$ – may differ and the difference $$\mathcal{D}_{\mathcal{H}}(\theta) = \mathcal{I}(\theta) - \mathcal{C}_\mathcal{H}(\theta)$$ is called a quantum discord.
Evidently, the above definition is not symmetric with respect to parties $\mathcal{H}$ and $\mathcal{K}$. However, one can easily swap the role of $\mathcal{H}$ and $\mathcal{K}$ to get $$\mathcal{D}_{\mathcal{K}}(\theta) = \mathcal{I}(\theta) - \mathcal{C}_\mathcal{K}(\theta) \ ,$$ where $$\label{C-sup}
\mathcal{C}_{\mathcal{K}}(\theta) = \sup_{\{\widetilde{\Pi}_\alpha\}}
\mathcal{I}(\theta|\{\widetilde{\Pi}_\alpha\})\ ,$$ and $\widetilde{\Pi}_\alpha$ is a collection of one-dimensional projectors in $\mathcal{K}$ satisfying $\widetilde{\Pi}_1 + \widetilde{\Pi}_2 + \ldots
= 1_\mathcal{K}$. For a general mixed state $\mathcal{D}_\mathcal{H}(\theta)
\neq \mathcal{D}_\mathcal{K}(\theta)$. However, it turns out that $\mathcal{D}_\mathcal{H}(\theta),\, \mathcal{D}_\mathcal{K}(\theta) \geq 0$. Moreover, on pure states, quantum discord coincides with the von Neumann entropy of entanglement $S(\rho) = S(\sigma)$. States with zero quantum discord – so called classical-quantum states – represent essentially a classical probability distribution $p_k$ embedded in a quantum system. One shows that $\mathcal{D}_\mathcal{H}(\theta)=0$ if and only if there exists an orthonormal basis $|k\>$ in $\mathcal{H}$ such that $$\label{Q=0}
\theta = \sum_k p_k\, |k\>\<k| {\,\otimes\,} \sigma_k \ ,$$ where $\sigma_k$ are density matrices in $\mathcal{K}$. Similarly, $\mathcal{D}_\mathcal{K}(\theta)=0$ if and only if there exists an orthonormal basis $|\alpha\>$ in $\mathcal{K}$ such that $$\label{Q=0-B}
\theta = \sum_\alpha q_\alpha\, \rho_\alpha {\,\otimes\,} |\alpha\>\<\alpha|
\ ,$$ where $\rho_\alpha$ are density matrices in $\mathcal{H}$. It is clear that if $\mathcal{D}_\mathcal{H}(\theta)=\mathcal{D}_\mathcal{K}(\theta)=0$, then $\theta$ is diagonal in the product basis $|k\> {\,\otimes\,} |\alpha\>$ and hence $$\label{Q=0-B}
\theta = \sum_{k,\alpha} \lambda_{k\alpha}\, |k\>\<k| {\,\otimes\,}
|\alpha\>\<\alpha| \ ,$$ is fully encoded by the classical joint probability distribution $\lambda_{k\alpha}$.
Finally, let us introduce a symmetrized quantum discord $$\label{D-symm}
\mathcal{D}_{\mathcal{H}:\mathcal{K}}(\theta) := \frac{1}{2} \Big[ \mathcal{D}_\mathcal{H}(\theta) + \mathcal{D}_\mathcal{K}(\theta) \Big] \ .$$ Let us observe that there is an intriguing relation between (\[D-symm\]) and (\[DEN\]). One has $$D(\theta) = I(\theta) - \frac 12 [ S(\rho) + S(\sigma) ] \ ,$$ whereas $$\mathcal{D}_{\mathcal{H}:\mathcal{K}}(\theta) = I(\theta) - \mathcal{C}_{\mathcal{H}:\mathcal{K}}(\theta)\ .$$ Note, that $\mathcal{D}_{\mathcal{H}:\mathcal{K}}(\theta) \geq 0$ but $D(\theta)$ can be negative (for PPT states). It is assumed that $\mathcal{D}_{\mathcal{H}:\mathcal{K}}(\theta)$ measures perfectly quantum correlations encoded into $\theta$.
\[example:separable\] For the entanglement map given by $$\phi(b)= \sum_{i} \lambda_i\,\rho_i\mathrm{Tr}\sigma_i b,\quad \phi^\ast(a)
= \sum_{i} \lambda_i\,\sigma_i\mathrm{Tr}\rho_i a, \quad \biggl( \,\sum_i\lambda_i = 1,\,\lambda_i\geq 0\,\forall i\,\biggr),$$ the corresponding state $\theta$ can be written in the form $$\theta = \sum_{i} \lambda_i\,\rho_i\otimes \sigma_i,$$ with $\rho =\phi(1_\mathcal{K}) =\sum_i \lambda_i\rho_i$ and $\sigma=\phi^\ast(1_\mathcal{H}) = \sum_i \lambda_i\sigma_i$. Then, we have the following inequalities.[@AMO2; @BO1; @BO2] $$\begin{aligned}
&0\leq I(\theta) \leq \min \bigl\{S(\rho), S(\sigma)\bigr\}, \\
&S_\theta(\sigma | \rho) \geq 0,\quad S_\theta(\rho|\sigma) \geq 0.\end{aligned}$$
Let $\{e_i\}_i$ and $\{f_j\}_j$ be the complete orthonormal systems in $\mathcal{H}$ and $\mathcal{K}$, respectively. For the entanglement map given by $$\phi(b) = \sum_i \lambda_{i}\vert e_i\rangle\langle e_i\vert\langle
f_i,bf_i\rangle,\quad \phi ^\ast(a) = \sum \lambda_i\vert f_i\rangle\langle
f_i\vert\langle e_i,ae_i\rangle,$$ the corresponding state $\theta$ can be written in the form $$\theta = \sum \lambda_{i}\vert e_i\rangle\langle e_i\vert \otimes \vert
f_i\rangle\langle f_i\vert\ ,$$ with $\rho =\phi(1_\mathcal{K}) =\sum \lambda_{i}\vert e_{i}\rangle \langle
e_i\vert,\, \sigma=\phi^\ast(1_\mathcal{H}) =\sum_i\lambda_i\,\vert
f_i\rangle\langle f_i\vert$. It is clear that $\mathcal{D}_{\mathcal{H}:\mathcal{K}}(\theta)=0$. Moreover, one obtains $$\begin{aligned}
I(\theta) &=S(\rho) +S(\sigma) -S(\theta_\phi) =S(\rho), \\
S_\theta(\sigma |\rho) &= S_\theta(\rho |\sigma)=0,\end{aligned}$$ where $S(\rho)=S(\sigma)=S(\theta_\phi) =-\sum \lambda_i\log \lambda_i$. This correlation corresponds to a perfect correlation in the classical scheme.
Quantum correlations for circulant states {#CIRC}
=========================================
In this section, we analyze correlations encoded into the special family of so called *circulant states*.
A circulant state
-----------------
We start this section by recalling the definition of a circulant state introduced in [@CKcir07] (see also [@Art]). Consider the finite dimensional Hilbert space $\mathbb{C}^d$ with the standard basis $\{e_0,\,e_1,\,\cdots,\,e_{d-1}\}$. Let $\Sigma_0$ be the subspace of $\mathbb{C}^d\otimes\mathbb{C}^d$ generated by $e_i\otimes
e_i~(i=0,\,1,\,\cdots,\,d-1):$
$$\Sigma_0 = \mathrm{span}\{e_0\otimes e_0,\,e_1\otimes e_1,\,\cdots,
e_{d-1}\otimes e_{d-1}\}.$$
Define a shift operator $S^\alpha : \mathbb{C}^d {\, \rightarrow\, } \mathbb{C}^d$ by $$S^\alpha e_k = e_{k+\alpha} \ , \ \ \ \mathrm{mod}\ d$$ and let $$\Sigma_\alpha := (1_d {\,\otimes\,} S^\alpha)\Sigma_0 \ .$$ It turns out that $\Sigma_\alpha$ and $\Sigma_\beta~(\alpha\ne\beta)$ are mutually orthogonal and one has the following direct sum decomposition $$\mathbb{C}^d\otimes \mathbb{C}^d \cong
\Sigma_0\oplus\Sigma_1\oplus\cdots\oplus\Sigma_{d-1}.$$ This decomposition is called a circulant decomposition.[@CKcir07] Let $a^{(0)},\,a^{(1)},\,\cdots,\,a^{(d-1)}$ be positive $d\times d$ matrices with entries in $\mathbb{C}$ such that $\rho_\alpha$ is supported on $\Sigma_\alpha$. Moreover, let $$\mathrm{tr}(a^{(0)}+\cdots +a^{(d-1)})=1\ .$$ Now, for each $a^{(\alpha)} \in M_d(\mathbb{C})$ one defines a positive operator in $\mathbb{C}^d {\,\otimes\,} \mathbb{C}^d$ be the following formula $$\vartheta_\alpha = \sum_{i,j=0}^{d-1} a^{(\alpha)}_{ij}\, e_{ij}\otimes
S^\alpha e_{ij} S^{\alpha\dagger }.$$ Finally, let us introduce $$\vartheta := \vartheta_0 \oplus \cdots \oplus \vartheta_{d-1} \ .$$ One proves[@CKcir07] that $\rho$ defines a legitimate density operators in $\mathbb{C}^d {\,\otimes\,} \mathbb{C}^d$. One calls it a *circulant state*. For further details of circulant states we refer to Refs. [CKcir07,Art]{}.
Now, let consider a partial transposition of the circulant state. It turns out that $\rho^\tau = ({\mathchoice{\rm 1\mskip-4mu l}{\rm 1\mskip-4mu
l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5mu l}} {\,\otimes\,} \tau)\rho$ is again circulant but it corresponds to another cyclic decomposition of the original Hilbert space $\mathbb{C}^d {\,\otimes\,} \mathbb{C}^d$. Let us introduce the following permutation $\pi$ from the symmetric group $S_d$: it permutes elements $\{0,1,\ldots,d-1\}$ as follows $$\pi(0) = 0 \ , \ \ \ \ \pi(i) = d-i \ , \ \ i=1,2,\ldots,d-1\ .$$ We use $\pi$ to introduce $$\widetilde{{\Sigma}}_0 = \mbox{span}\left\{ e_0 {\,\otimes\,} e_{\pi(0)}\, ,
e_1 {\,\otimes\,} e_{\pi(1)}\, , \ldots\, , e_{d-1} {\,\otimes\,}
e_{\pi(d-1)} \right\} \ ,$$ and $$\widetilde{{\Sigma}}_\alpha = ({\mathchoice{\rm 1\mskip-4mu l}{\rm
1\mskip-4mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5mu l}} {\,\otimes\,}
S^\alpha) \widetilde{{\Sigma}}_0\ .$$ It is clear that $\widetilde{\Sigma}_\alpha$ and $\widetilde{\Sigma}_\beta$ are mutually orthogonal (for $\alpha\neq \beta$). Moreover, $$\label{D-new}
\widetilde{\Sigma}_0 \oplus \ldots \oplus \widetilde{\Sigma}_{d-1} = \mathbb{C}^d {\,\otimes\,} \mathbb{C}^d \ ,$$ and hence it defines another circulant decomposition. Now, the partially transformed state $\vartheta^\tau$ has again a circulant structure but with respect to the new decomposition (\[D-new\]): $$\label{ro-C-new}
\vartheta^\tau = \widetilde{\vartheta}^{(0)} + \cdots + \widetilde{\vartheta}^{(d-1)} \ ,$$ where $$\widetilde{\vartheta}^{(\alpha)} = \sum_{i,j=0}^{d-1} \widetilde{a}^{(\alpha)}_{ij} \ e_{ij} {\,\otimes\,} S^\alpha e_{\pi(i)\pi(j)} S^{\dag
\alpha} \ ,\ \ \ \ \ \alpha=0,\ldots,d-1 \ ,$$ and the new $d \times d$ matrices $[\widetilde{a}^{(\alpha)}_{ij}]$ are given by the following formulae: $$\label{a-tilde}
\widetilde{a}^{(\alpha)} \, =\, \sum_{\beta=0}^{d-1}\, a^{(\alpha+\beta)}
\circ ({\Pi} {S}^\beta)\ , \ \ \ \ \ \ \ \ \mbox{mod $d$}\ ,$$ where “$\circ$" denotes the Hadamard product,[^1] and $\Pi$ being a $d \times d$ permutation matrix corresponding to $\pi$, i.e. $\Pi_{ij} := \delta_{i,\pi(j)}$. It is therefore clear that our original circulant state is PPT iff all $d$ matrices $\widetilde{a}^{(\alpha)}$ satisfy $$\widetilde{a}^{(\alpha)} \geq 0 \ , \ \ \ \ \alpha=0,\ldots,d-1\ .$$
Generalized Bell diagonal states
--------------------------------
The most important example of circulant states is provided by Bell diagonal states [@BH1; @BH2; @BH3] defined by $$\label{Bell}
\rho = \sum_{m,n=0}^{d-1} p_{mn} P_{mn}\ ,$$ where $p_{mn}\geq 0$, $\ \sum_{m,n}p_{mn}=1$ and $$P_{mn} = (\mathbb{I} {\,\otimes\,} U_{mn}) \,P^+_d\, (\mathbb{I} {\,\otimes\,} U_{mn}^\dagger)\ ,$$ with $U_{mn}$ being the collection of $d^2$ unitary matrices defined as follows $$\label{U_mn}
U_{mn} e_k = \lambda^{mk} S^n e_k = \lambda^{mk} e_{k+n}\ ,$$ with $$\lambda= e^{2\pi i/d} \ .$$ The matrices $U_{mn}$ define an orthonormal basis in the space $M_d(\mathbb{C})$ of complex $d \times d$ matrices. One easily shows $$\mathrm{Tr}(U_{mn} U_{rs}^\dagger) = d\, \delta_{mr} \delta_{ns} \ .$$ Some authors call $U_{mn}$ generalized spin matrices since for $d=2$ they reproduce standard Pauli matrices: $$\label{U-sigma}
U_{00} = \mathbb{I}\ , \ U_{01} = \sigma_1\ , \ U_{10} = i \sigma _2\ , \
U_{11} = \sigma_3\ .$$ Let us observe that Bell diagonal states (\[Bell\]) are circulant states in $\mathbb{C}^d {\,\otimes\,} \mathbb{C}^d$. Indeed, maximally entangled projectors $P_{mn}$ are supported on $\Sigma_n$, that is, $$\label{Pi_n}
\Pi_n = P_{0n} + \ldots + P_{d-1,n} \ ,$$ defines a projector onto $\Sigma_n$, i.e. $$\Sigma_n = \Pi_n ( \mathbb{C}^d {\,\otimes\,} \mathbb{C}^d) \ .$$ One easily shows that the corresponding matrices $a^{(n)}$ are given by $$a^{(n)}= H D^{(n)} H^* \ ,$$ where $H$ is a unitary $d\times d$ matrix defined by $$H_{kl} := \frac{1}{\sqrt{d}}\, \lambda^{kl} \ ,$$ and $D^{(n)}$ is a collection of diagonal matrices defined by $$D^{(n)}_{kl} := p_{kn} \delta_{kl}\ .$$ One has $$a^{(n)}_{kl} = \frac 1d \sum_{m=0}^{d-1} p_{mn} \lambda^{m(k-l)}\ ,$$ and hence it defines a circulant matrix $$a^{(n)}_{kl} = f^{(n)}_{k-l}\ ,$$ where the vector $f^{(n)}_m$ is the inverse of the discrete Fourier transform of $p_{mn}$ ($n$ is fixed).
A family of Horodecki states
----------------------------
Let $\mathcal{H}=\mathcal{K}= \mathbb{C}^{3}$. For any $\alpha \in \lbrack
0,5]$, one defines[@HHHmix01] the following state $$\begin{aligned}
\theta _{1}(\alpha ) & = \frac{2}{7}\, P^+_3 + \frac\alpha 7\, \Pi_1 + \frac{5-\alpha }{7} \, \Pi_2\ .\end{aligned}$$ The eigenvalues of $\theta _{1}(\alpha )$ are calculated as $0,\frac{2}{7},\,3 \times \frac{\alpha }{21}$ and $3 \times \frac{5-\alpha }{21}$ and hence one obtains for the $D$-correlations $$D\bigl(\theta _{1}(\alpha )\bigr)= \log 3 + \frac{2}{7}\log \frac{2}{7} +
\frac{\alpha }{7}\log \frac{\alpha }{21} + \frac{5-\alpha }{7}\log \frac{5-\alpha }{21}\ .$$
\[section5:thm1\] [@HHHmix01] The family $\theta_1(\alpha)$ satisfies:
1. $\theta_1(\alpha)$ is PPT if and only $\alpha \in [1,4]$
2. $\theta_1(\alpha)$ is separable if and only if $\alpha \in [2,3]$
3. $\theta_1(\alpha)$ is both entangled and PPT if and only if $\alpha
\in [1,2) \cup (3,4]$
4. $\theta_1(\alpha)$ is NPT if and only if $\alpha \in [0,1) \cup (4,5]$.
Due to this Theorem, one can find that the $D(\theta _{1}(\alpha ))$ does admit a natural order. That is, the $D$-correlation for any entangled state is always stronger than $D$-correlation for an arbitrary separable state. Similarly, one observes that $D$-correlation for any NPT state is always stronger than $D$-correlation for an arbitrary PPT state. The graph of $D\bigl(\theta _{1}(\alpha )\bigr)$ is shown in Fig. \[section5:fig1\]. Actually, one finds that the minimal value of $D$-correlations corresponds to $\alpha = 2.5$, that is, it lies in the middle of the separable region.
On the other hand, we can also compute the symmetrized discord $\mathcal{D}_{\mathbb{C}^{3};\mathbb{C}^{3}}\left( \theta _{1}\left(\alpha \right) \right)$ and have obtained Fig. \[section5:fig1\]. It is easy to find that the graph is symmetric with respect to $\alpha =2.5$. As in Fig. 2, the value of the symmetrized discord satisfies the following inequality; $$0 < \mathcal{D}_{\mathbb{C}^{3};\mathbb{C}^{3}}\left(\theta
_{1}(\alpha)\right) \,\leq\, \mathcal{D}_{\mathbb{C}^{3};\mathbb{C}^{3}}\left(\theta _{1}(\beta)\right) \,\leq\, \mathcal{D}_{\mathbb{C}^{3};\mathbb{C}^{3}}\left(\theta _{1}(\gamma)\right),$$ where $\alpha \in \left[ 2,3\right],\,\beta \in \left[1,2\right] \cup \left[3,4\right]$ and $\gamma \in \left[0,1\right] \cup \left[4,5\right]$.
The family of $\theta _{1}\left( \alpha \right) $ has the quantum correlation even in separable states corresponding to $\alpha \in \left[ 2,3\right] $ in the sense of discord. We know that the above two types of criteria give the similar order of correlation.
Notice that $D\left( \theta _{1}\left( \alpha \right) \right) $ is always negative even in NPT sates and the positivity of $D$-correlation represents a true quantum property (see Example 3.3 and Proposition 3.5). In this sense the quantum correlation of $\theta _{1}\left( \alpha \right) $ is not so strong.
![Left — the graph of $D(\protect\theta _{1}(x))\,$ with $x \in
[0,5]$. The minimal value of $D$ corresponds to $x=2.5$. Right — the graph of $\mathcal{D}_{\mathbb{C}^{3};\mathbb{C}^{3}}\left( \protect\theta _{1}\left( \protect\alpha \right) \right)$. []{data-label="section5:fig1"}](D-correlation.eps "fig:"){width="6.0cm"} ![Left — the graph of $D(\protect\theta _{1}(x))\,$ with $x \in
[0,5]$. The minimal value of $D$ corresponds to $x=2.5$. Right — the graph of $\mathcal{D}_{\mathbb{C}^{3};\mathbb{C}^{3}}\left( \protect\theta _{1}\left( \protect\alpha \right) \right)$. []{data-label="section5:fig1"}](discord_for_Horodecki_2.eps "fig:"){width="6.0cm"}
This family may be generalized to $\mathbb{C}^d {\,\otimes\,} \mathbb{C}^d$ as follows: consider the following family of circulat 2-qudit states $$\label{rho}
\theta(\alpha) = \sum_{i=1}^{d-1} \lambda_i \Pi_i + \lambda_d P^+_d\ ,$$ with $\lambda_n\geq 0$, and $\lambda_1 + \ldots +\lambda_{d-1} + \lambda_d=1$. Let us take the following special case corresponding to $$\begin{aligned}
\label{lll}
\lambda_1 = \frac{ \alpha}{\ell} \ , \ \ \lambda_{d-1} = \frac{(d-1)^2+1
-\alpha}{\ell} \ , \ \ \lambda_d = \frac{d-1}{\ell} \ .\end{aligned}$$ and $\lambda_2 = \ldots = \lambda_{d-2} = \lambda_d$, with $$\ell = (d-1)(2d-3) +1\ .$$ One may prove the following[@Adam]
The family $\theta(\alpha)$ satisfies:
1. $\theta(\alpha)$ is PPT if and only $\alpha \in [1,(d-1)^2]$
2. $\theta(\alpha)$ is separable if and only if $\alpha \in
[d-1,(d-1)(d-2)+1]$
3. $\theta_1(\alpha)$ is both entangled and PPT if and only if $\alpha
\in [1,d-1) \cup ((d-1)(d-2)+1,(d-1)^2]$
4. $\theta_1(\alpha)$ is NPT if and only if $\alpha \in [0,1) \cup
((d-1)^2,(d-1)^2+1]$.
For example if $d=4$ one obtains the following picture of $D(\theta(\alpha))$ (see Fig. 4)
![The graph of $D(\protect\theta(x))\,$ with $x \in [0,10]$. The minimal value of $D$ corresponds to $x=5$.[]{data-label="section5:fig1"}](D-correlation-d=4.eps){width="6.0cm"}
Again, one finds that the $D(\theta(\alpha ))$ does admit a natural order. That is, the $D$-correlation for any entangled state is always stronger than $D$-correlation for an arbitrary separable state. Similarly, one observes that $D$-correlation for any NPT state is always stronger than $D$-correlation for an arbitrary PPT state.
Example: a family of Bell diagonal states
-----------------------------------------
Consider the following class of Bell-diagonal states in $\mathbb{C}^3 {\,\otimes\,} \mathbb{C}^3$: $$\theta_2(\varepsilon) = \frac{1}{\Lambda } ( 3P^+_3 + \varepsilon \Pi_1 +
\varepsilon^{-1} \Pi_2 ) \ ,$$ with $\Lambda =1+\varepsilon +\varepsilon ^{-1}$. One easily finds for its $D $-correlations $$\begin{aligned}
D\bigl(\theta _{2}(\varepsilon )\bigr) = \frac 1\Lambda \left( \log \frac
1\Lambda + \varepsilon^{-1} \log \frac{\varepsilon^{-1}}{\Lambda} +
\varepsilon \log \frac{\varepsilon}{\Lambda} + \log 3 \right) \ .\end{aligned}$$
The following theorem gives us a useful characterization of $\theta_{2}(\varepsilon )$ [@JCRacl09].
\[section3:thm2\] The states of $\theta_1(\varepsilon)$ are classified by $\varepsilon$ as follows:
1. $\theta_2(\varepsilon)$ is separable if $\varepsilon =1$
2. $\theta_2(\varepsilon)$ is both PPT and entangled for $\varepsilon
\neq 1$.
The graph of $D\bigl(\theta _{2}(\varepsilon )\bigr)$ is shown in Fig. \[section3:fig2\]. $D\bigl(\theta _{2}(\varepsilon )\bigr)$ is rapidly decreasing with $\varepsilon$ approaching $1$ from $0$ and increases when $\varepsilon$ is over $1$. That is, $D\bigl(\theta _{2}(\varepsilon )\bigr)$ takes the minimal value at $\varepsilon =1$ and it is approximated about $D\bigl(\theta _{2}(1)\bigr)=-\frac{2}{3}\log 3\approx -0.7324$. As is the case of $\theta_1(\alpha)$, the $D$-correlation $D\bigl(\theta
_{2}(\varepsilon )\bigr)$ for an entangled state is always stronger than the one for a separable state. As $\varepsilon\rightarrow 0$ or $\infty $, $\theta _{2}\left( \varepsilon \right) $ converges to a separable perfectly correlated state which can be recognized as a classical state $$\lim_{\varepsilon \rightarrow 0 }\theta _{2}(\varepsilon )=\frac{1}{3}\Bigl(e_{00}\otimes e_{22}+e_{11}\otimes e_{00}+e_{22}\otimes e_{11}\Bigr) =
\Pi_2\ , \label{section3:eqn2}$$
$$\lim_{\varepsilon \rightarrow \infty}\theta _{2}(\varepsilon )= \frac{1}{3}\Bigl(e_{00}\otimes e_{11}+e_{11}\otimes e_{22}+e_{22}\otimes e_{00}\Bigr) =
\Pi_1\ ,$$
and for every $\varepsilon >0$, $$D\bigl(\theta _{2}(\varepsilon )\bigr)<0=\lim_{\varepsilon \rightarrow 0}D\bigl(\theta _{2}(\varepsilon )\bigr)=\lim_{\varepsilon \rightarrow \infty }D\bigl(\theta _{2}(\varepsilon )\bigr).$$It shows that a correlation of a PPT entangled state $\theta _{2}\left(
\varepsilon \neq 1\right) $ is weaker than that of the (classical) separable perfectly correlated states in the sense of (\[order\]).
Now, since $\theta _{1}(\alpha )$ and $\theta _{2}(\varepsilon )$ have common marginal states, we can compare the order of quantum correlations for them. One has, for example, $$D\bigl(\theta _{2}(1)\bigr)\approx -0.7324>-0.7587\approx D\bigl(\theta
_{1}(3.1)\bigr). \label{section3:ineq1}$$Accordingly Theorem \[section5:thm1\] and \[section3:thm2\], however, $\theta _{2}(1)$ is separable while $\theta _{1}(3.1)$ is entangled state. Incidentally, this means that the correlation for the separable state $\theta _{2}(1)$ is stronger than the entangled state $\theta _{1}(3.1)$ in the sense of (\[order\]).
![Left — the graph of $D(\protect\theta _{2}(x))$. Note that $D$ is minimal for $x=1$ which correspond to the separable state. Right — the graph of $\mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon)$ for $\varepsilon\in (0,1]$. Note that $\,\mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon))= \mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon^{-1}))$. []{data-label="section3:fig2"}](D-correlation-epsilon.eps "fig:"){width="6.0cm"} ![Left — the graph of $D(\protect\theta _{2}(x))$. Note that $D$ is minimal for $x=1$ which correspond to the separable state. Right — the graph of $\mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon)$ for $\varepsilon\in (0,1]$. Note that $\,\mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon))= \mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon^{-1}))$. []{data-label="section3:fig2"}](discord-e-states.eps "fig:"){width="6.0cm"}
On the other hand one finds the following plot of the quantum discord Fig. 3.
It is clear that $$\lim_{\varepsilon \rightarrow 0} \mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon)) = \lim_{\varepsilon \rightarrow \infty}\mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon)) = 0 \ ,$$ since both $\Pi_1$ and $\Pi_2$ are perfectly classical states. Note, that $\mathcal{D}_{\mathbb{C}^3:\mathbb{C}^3}(\theta_2(\varepsilon=1))> 0$ which shows that separable state $\theta_2(\varepsilon=1)$ does contain quantum correlations.
Conclusions
===========
We provided several examples of bi-partite quantum states and computed two types of correlations for them. It turned out that the correlation for a separable state can be stronger than the one for an entangled state in the sense of (\[order\]). This observation is inconsistent with the conventional understanding of quantum entanglement. However, we also showed that the discord of such separable states might strictly positive. This means that these states have a non-classical correlation. From this point of view, it is no longer unusual that the correlation for a separable state is stronger than the one for an entangled state.
Acknowledgments {#acknowledgments .unnumbered}
===============
T.M. is grateful to V.P.Belavkin for fruitful discussions on mutual entropy and entanglement maps. D.C. was partially supported by the National Science Centre project DEC-2011/03/B/ST2/00136. We would like to acknowledge the supports of QBIC (Quantum Bio-Informatics Center) grant of Tokyo University of Science.
[99]{} L.Accardi, D. Chruściński, A. Kossakowski, T. Matsuoka and M. Ohya, “On classical and quantum liftings”, Open. Syst. Info. Dyn. **17**, 361–386 (2010).
L. Accardi, T. Matsuoka, M. Ohya, “Entangled Markov chains are indeed entangled”, Infin. Dim. Anal. Quantum Probab. Top. **9**, 379–390 (2006).
L.Accardi, T. Matsuoka, M. Ohya,“Entangled Markov chain satisfying entanglement condition”, RIMS **1658**, 84–94 (2009).
L.Accardi, M. Ohya,“Composite channels, transition expectation and liftings”, J. Appl. Math. Optim., **39**, 33–59 (1999).
M. Asorey, A. Kossakowski, G. Marmo, E.C.G. Sudarshan, “Relation between quantum maps and quantum state”, Open. Syst. Info. Dyn. **12**, 319–329 (2006).
B. Baumgartner, B. Hiesmayer and H. Narnhofer, “State space for two qutrits has a phase space structure in its cone”, Phys. Rev. A **74**, 032327 (2006).
B. Baumgartner, B. Hiesmayer and H. Narnhofer, “A special simplex in the state space for entangled qutrits”, J. Phys. A: Math. Theor., **40**, 7919 (2007).
B. Baumgartner, B. Hiesmayer and H. Narnhofer, “The geometry of biparticle qutrits including bound entanglement”, Phys. Lett. A, ** 372**, 2190 (2008).
V. P. Belavkin, “Optimal filtering of Markov signals with quantum white noise”, Radio Eng. Electron. Phys., **25**, 1445-1453 (1980).
V. P. Belavkin, “Nondemolition principle of quantum measurement theory”, Found. Phys., **24**, 685–714 (1994).
V. P. Belavkin, M. Ohya, “Quantum entropy and information in discrete entangled state”, Infin. Dim. Anal. Quantum Probab. Top. **4**, 33-59 (2001).
V. P. Belavkin, M. Ohya, “Entanglement, quantum entropy and mutual information”, Proc. R. Soc. London A **458**, 209–231 (2002).
V. P. Belavkin, X. Dai, “An operational algebraic approach to quantum channel capacity”, Int. J. Quantum Inf. **6**, 981 (2008).
N. J. Cerf and C. Adami, “Negative entropy and information in quantum mechanics”, Phys. Rev. Lett. **79**, 5194–5197 (1997).
M. D. Choi, “Completely positive maps on complex matrix”, Lin. Alg. Appl., **10**, 285 (1975).
D. Chruściński, Y. Hirota, T. Matsuoka and M. Ohya, “Remarks on the degree of entanglement” QP- PQ Quantum Probab. & White Noise Anal. **28**, 145–156 (2011).
D. Chru$\mathrm{\acute{s}}$ci$\mathrm{\acute{n}}$ski and A. Kossakowski: “Circulant states with positive partial transpose”, Physical Review A 76 (2007), 14 pp.
D. Chruściński and A. Pittenger, *Generalized Circulant Densities and a Sufficient Condition for Separability*, J. Phys. A: Math. Theor. **41** (2008) 385301.
D. Chruściński, A. Kossakowski, T. Matsuoka, K. Młodawski, “A class of Bell diagonal states and entanglement witnesses”, Open. Syst. Info. Dyn. **17**, 213–231 (2010).
D. Chruściński, A. Kossakowski, T. Matsuoka and M. Ohya, “Entanglement mapping vs. quantum conditional probability operator”, QP- PQ Quantum Probab. & White Noise Anal. **28**, 223–236 (2011).
D. Chruściński and A. Rutkowski, *Entanglement witnesses for $d \otimes d$ systems and new classes of entangled qudit states*, Eur. Phys. J. D **62**, 273 (2011).
B. Groisman, S. Popescu and A. Winter, “Quantum, classical, and total amount of correlations in a quantum state” Phys. Rev A **72** , 0323187 (2005).
L. Henderson and V. Vedral, “Classical, quantum and total correlation”, J. Phys. A **34** 6913 (2001)
M. Horodecki and R. Horodecki, “Information-theoretical aspect of quantum inseparability of mixed states”, Phys. Rev. A **54**, 1838-1843 ( 996).
M. Horodecki, P. Horodecki and R. Horodecki, “Separability of mixed states: necessary and sufficient condition”, *Phy. Lett.* , **A** **223**, 1 (1996).
M. Horodecki, P. Horodecki and R. Horodecki: “Mixed state entanglement and quantum condition”, In Quantum Information, Springer Tracts in Modern Physics 173 (2001), pp 151–195.
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement”, Rev. Mod. Phys. **81**, 865 (2009).
A. Jamiołkowski, “Linear transformation which preserve trace and positive semidefiniteness of operators”, Rep. Math. Phys. **3**, 275 (1072).
A. Jamiołkowski, T. Matsuoka and M. Ohya, “Entangling operator and PPT condition”, TUS preprint (2007).
J. Jurkowski, D. Chru$\mathrm{\acute{s}}$ci$\mathrm{\acute{n}}$ski and A. Rutkowski: “A class of bound entangled states of two qutrits”, Open Syst. Inf. Dyn., 16 (2009), no 2-3, pp 235–242.
G. Kimura, A. Kossakowski, “A note on positive maps and classification on states”, Open Sys. & Information Dyn. **12**, 1(2005).
W. A. Majewski, T. Matsuoka and M. Ohya, “Characterization of partial positive transposition states and measures of entanglement”, J. Math. Phys. **50**, 113509 (2009).
T. Matsuoka, “On generalized entanglement”, QP- PQ Quantum Probab. & White Noise Anal. **21**, 170–180 (2007).
T. Matsuoka, M. Ohya, “Quantum entangled state and its characterization”, Foud. Probab. Phys. 3 **750**, 298–306 (2005).
K. Modi, T. Paterek, W. Son, V. Vedral and M. Williamson, “Unified view of quantum and classical correlations”, Phys. Rev. Lett., **104**, 080501 (2010).
M. Ohya, “On composite state and mutual information in quantum information theory”, IEEE Info. Theory, **29**, 77–774 (1983).
M. Ohya, “Note on quantum probability”, Nuovo Cimento, **38**, 402-406 (1983).
M. Ohya, I. V. Volovich, *Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-systems*, Springer, New York, 2011.
H. Ollivier, W. Z. Zurek, “Quantum discord: A measure of the quantumness of correlations”, Phys. Rev. Lett., **88**, 017901 (2002).
V. Paulsen, *Completely Bounded Maps and Operator Algebras*, Cambridge University Press, 2003.
A. Peres, “Separability criterion for density matrix”, *Phys.Rev.Lett.*, **77,** 1413 (1996).
C. E. Shannon, “A mathematical theory of communication”, Urbana, IL: Univ. Illinois (1949).
H. Umegaki, “Conditional expectation in an operator algebras IV”, Kodai Math. Sem. Rep., **14**, 59–85 (1962).
K.G.H. Vollbrecht, M.M. Wolf, “Conditional entropies and their relation to entanglement criteria”, J. Math. Phys. **43**, 4299 (2002).
S. L. Woronowicz, “Positive maps of low dimensional matrix algebra”, Rep. Math. Phys., **10**, 165 (1976)
[^1]: A Hadamard (or Schur) product of two $n \times n$ matrices $A=[A_{ij}]$ and $B=[B_{ij}]$ is defined by $$(A \circ B)_{ij} = A_{ij} B_{ij}\ .$$
|
---
abstract: 'We demonstrate that the attractive interaction measured between like-charged colloidal spheres near a wall can be accounted for by a nonequilibrium hydrodynamic effect. We present both analytical results and Brownian dynamics simulations which quantitatively capture the one-wall experiments of Larsen and Grier (Nature [**385**]{}, 230, 1997).'
address: |
$^1$Department of Physics, Harvard University, Cambridge, MA 02138\
$^2$Department of Mathematics, MIT, Cambridge, MA 02139
author:
- 'Todd M. Squires$^1$ and Michael P. Brenner$^2$'
title: 'Like-Charge Attraction through Hydrodynamic Interaction'
---
-0.5in
Colloidal spheres provide a simple model system for understanding the interactions of charged objects in a salt solution. Hence, it came as a great surprise when it was observed that two like-charged spheres can attract each other when the spheres are confined by walls [@kep94; @car96; @croc96; @lar97]. Since both the charge densities and sizes of the spheres in question are in the range of large proteins, it would be expected that a change in sign of this interaction would have important implications for biological systems [@honig]. Theorems by Sader and Chan[@sad99] and Neu[@neu99] demonstrate that under very general conditions the Poisson-Boltzmann equation for the potential between like-charged spheres in a salt solution will not admit attractive interactions. Explanations for the observed attraction have thus exclusively focused on deviations from the classical Derjaguin, Landau, Verwey and Overbeek (DLVO) theory.
Herein, we propose that an attractive interaction of two like-charged colloidal spheres measured in the presence of a single wall can arise from a non-equilibrium hydrodynamic effect. The idea is that the relative motion between two spheres depends on [*both*]{} the forces acting between them and in addition, their hydrodynamic coupling. In a bulk solution, far from solid boundaries, an external force acting on two identical spheres cannot change their relative positions. This is a consequence of the kinematic reversibility of Stokes flow and of the symmetries inherent in the problem.
However, these symmetries are broken in confined geometries, where the hydrodynamic effect of boundaries is important. In this situation, relative motion between the particles could stem from [*either*]{} an interparticle force, [*or*]{} from a hydrodynamic coupling caused by forces acting on each of the particles individually. In a typical experiment with charged colloidal spheres, the charge density on the walls of the cell is of order the charge density on the spheres[@grier]. We demonstrate that the hydrodynamic coupling between two spheres caused by their repulsion from a wall leads to motion which, if interpreted as an equilibrium property, is consistent with an effective potential between the spheres with an attractive well. Our calculations quantitatively reproduce the experimental measurements of these potentials.
The response of a particle to an external force is significantly changed near a wall because the flow field must vanish identically on the wall. For point forces, Lorentz determined this wall-corrected flow field [@lor], which Blake later expressed using the method of image forces [@blake], analogous to image charges used in electrostatics. Images of the appropriate strength on the opposite side of the wall exactly cancel out the fluid flow on the wall. When two particles are pushed away from a wall, the flow field from one particle’s image tends to pull the other particle towards it, and vice versa (Fig. 1). This decreases the distance between the particles.
=3.5in
The attractive interaction between two charged spheres in the presence of a wall can now be understood with a simple picture. When the spheres are sufficiently close to the wall, they are electrostatically repelled from it. The net force on each sphere thus includes both their mutual electrostatic repulsion and their repulsion from the wall. How the spheres respond depends on their hydrodynamic mobility: when the spheres are close together (Fig. 2a), their mutual repulsion overwhelms any hydrodynamic coupling, and the spheres will separate as expected for like-charged bodies. However, when they are beyond some critical separation (Fig. 2b), the hydrodynamic coupling due to the wall force overcomes the electrostatic repulsion, so that the particles move together as they move away from the wall.
=2.0in =2.0in
Although this decrease in mutual separation is a non-equilibrium kinetic effect, it could be interpreted as the result of an attractive equilibrium pair-potential. This is most clearly understood without Brownian motion. Two particles initially located a distance $r$ apart move because of both interparticle forces and the repulsive force from the wall. The response of these two particles to forces ${\bf F}_{1}$ and ${\bf F}_{2}$ is expressed by the hydrodynamic mobility tensor $ {\bf b}\left( {\bf X}_{1},{\bf X}_{2}\right) $, defined by $${\bf v}={\bf b}\left( {\bf X}_{1},{\bf X}_{2}\right) \cdot {\bf F},$$ where ${\bf v}=({\bf \dot{X}}_{1},{\bf \dot{X}}_{2})$ are the particle velocities and ${\bf F}=({\bf F}_{1},{\bf F}_{2})$ are the forces on the particles. Thus, the distance between the spheres (measured in the plane parallel to the walls) will change by an amount $\Delta r = \Delta x_2 - \Delta x_1$ in a small time $\Delta t$, where we denote the $x$-direction to be along the line connecting the spheres, and the $z$-direction to be perpendicular to the wall. Utilizing symmetries of the mobility tensor, it is straightforward to show that $\Delta r$ will be $$\Delta r=\left\{ 2 (b_{X_2 X_2}-b_{X_2 X_1}) |F_p| + 2 b_{X_2 Z_1} F_w
\right\} \Delta t,$$ where $F_p$ and $F_w$ are respectively the repulsive electrostatic sphere-sphere and sphere-wall forces. The tensor component $b_{X_2Z_1}$ refers to the $x$-motion of particle 2 due to a force in the $z$-direction on particle 1, and so on.
If this system were assumed to be in equilibrium, then the relative motion would be interpreted as the result of an effective potential, so that an effective force $F_{eff}=-\partial_rU_{eff}$ $$\Delta r=\left\{ 2 (b_{X_2X_2}-b_{X_2 X_1}) |F_{eff}| \right\} \Delta t,$$ so that one would determine this effective potential to be given by $$U_{eff}(r,h)=U_{p}(r)-
F_w\int_{\infty }^{r}\frac{b_{X_2 Z_1}(r,h)}{b_{X_2 X_2}(h)-b_{X_2 X_1}(r,h)}dr, \label{analytic}$$ where $U_p(r)$ is the interparticle thermodynamic pair potential, $r$ is the separation between particles, and $h$ is their distance from the wall.
In order to compare our results with experiments, we determine the hydrodynamic mobilities in the point-force limit, using Blake’s solution [@blake]. We use the DLVO potential [@saville; @der; @ver48] for the electrostatic interaction of two spheres in the form presented by Larsen and Grier [@lar97], $$\frac{U_{DLVO}}{k_{B}T}=Z^{2}\lambda _{B}\left( \frac{e^{\kappa a}}{1+\kappa
a}\right) ^{2}\frac{e^{-\kappa r}}{r},
\label{dlvogrier}$$ where $a$ and $Z$ are respectively the radius and effective charge of each sphere, the Bjerrum length $\lambda _{B}=e^{2}/\varepsilon k_{B}T$, and the Debye-Hückel screening length $\kappa ^{-1}=(4\pi n\lambda _{B})^{-1/2},$ with a concentration $n$ of simple ions in the solution. This formula is obtained using effective point charges in a linear superposition approximation. To determine the repulsive electrostatic force between each sphere and the wall, we used the same effective point-charge approach to obtain $$\frac{U_{wall}}{k_{B}T}=Z\sigma _g\lambda _{B}\frac{e^{\kappa a}}{\kappa
(1+\kappa a)}e ^{-\kappa h},
\label{wallforce}$$ where $\sigma_g$ is the effective charge density on the glass wall. We note that while the functional form of this equation is correct, it is not clear that the effective charges in equations (\[dlvogrier\]) and (\[wallforce\]) will be exactly the same, as geometric factors buried in each effective charge will vary from situation to situation. A more reliable description of sphere-sphere and wall-sphere interactions will be necessary for quantitative comparisons with independently measured charge densities.
Using all of Larsen and Grier’s experimental parameters as inputs to the theory, we numerically integrate (\[analytic\]) to obtain this apparent effective potential. The only necessary parameter not given is the surface charge density of the glass walls $\sigma_g$, which we take to be $\sigma_g = 5 \sigma_p$, consistent with Kepler and Fraden’s measurements [@kep94]. Fig. 3 shows this effective potential for various sphere-wall separations. The hydrodynamic coupling of collective motion away from the wall with relative motion in the plane of the wall leads to an attractive component. It is important to emphasize that this hydrodynamic coupling is a [*kinematic*]{} effect, and has no thermodynamic significance–all forces acting on the spheres are purely repulsive.
=3.5in
We note as well that a simple approximate expression exists for the hydrodynamic term in the effective potential (\[analytic\]), since $b_{X_2 X_2}(h)/b_{X_2 X_1}(r,h)\sim O(h/a) >>1.$ Approximating the denominator in the integrand as simply $b_{X_2 X_2}$, we explicitly evaluate the integral to give $$U_{eff}(r,h)=U_{p}(r)-\frac{F_w}{1-\frac{9 a}{16 h}}\frac{3 h^3 a}{(4h^2 +
r^2)^{3/2}}.$$
As a complement to this analytic approach, we simulate the dynamics of this system, using (\[dlvogrier\]) and (\[wallforce\]) for the sphere-sphere and wall-sphere forces, respectively. We account for Brownian motion of the particles in the standard Stokes-Einstein fashion, whereby the diffusion tensor is proportional to the mobility tensor, ${\bf D}=k_{B}T{\bf b}$ [@bat76; @erm78; @hinch]. Using all experimental parameters and $\sigma_g = 5 \sigma_p$ as explained above, we performed a computer version of Larsen and Grier’s experiment, and analyzed the resulting data using their methods[@croc96b]. Our results suggest that this approach includes all of the essential ingredients necessary for quantitatively understanding their observations.
In Fig. 4, we present simulations for the two cases presented by Larsen and Grier: the first with the spheres $2.5$ microns from the wall, so that they interact significantly with the charge double layer of the wall, and the second starting $9.5$ microns from the wall, well outside of the wall’s charge double layer.
=2.5in =2.5in
Our theoretical picture agrees quantitatively with measured data. Moreover, there are many consequences of the theory that can be tested experimentally: (1) Effective kinetic potentials can be predicted for different sets of conditions and quantitatively compared with experiments; (2) The hydrodynamic mechanism requires a net drift of the particles away from the wall, which could be independently measured. (3) Finally, the theory provides a simple explanation for the observation that the attraction disappears when the salt concentration is increased. While this at first seems counterintuitive–the particles are mutually attractive only when they are mutually repulsive–the significance of the wall-driven hydrodynamic coupling makes this clear.
Several pieces of experimental evidence have been collected which seemed to suggest the existence of an attractive minimum in the thermodynamic pair potential of like-charged colloidal particles in confined geometries. Besides the one wall experiment under discussion, attractive pair potentials have been observed for two spheres trapped between two walls[@croc96], and for a suspension of spheres trapped between two walls[@kep94; @car96]. In addition, it has been shown that metastable colloidal crystals take orders of magnitude longer to melt than would be expected without a thermodynamic attraction[@largrier96]. Similarly, voids in colloidal crystals take much longer to close than expected[@ise]. It is not clear how the theory presented here will bear upon these experiments.
The theory presented in this paper offers a non-equilibrium hydrodynamic explanation for the attractive potential in the single-wall experiments without invoking a novel thermodynamic attraction. We have found quantitative agreement with experimental results when the effective wall charge density is chosen to be $\sigma_g = 5\sigma_p$, which is in the ballpark of measured estimates. Without a quantitative measurement of this parameter, this work does not strictly rule out the possibility that a novel attraction exists. This situation can be definitively resolved by more quantitative comparisons with experiments.
[**Acknowledgments:**]{} We are indebted to D. Grier and E. Dufresne for introducing us to their experiments, and for a stimulating collaboration. Useful discussions with J. Crocker, H. Stone, and D. Weitz are gratefully acknowledged. This research was supported by the Mathematical Sciences Division of the National Science Foundation, the A.P. Sloan Foundation, and the NDSEG Fellowship Program (TS).
[10]{}
G.M. Kepler and S. Fraden. Attractive potential between confined colloids at low ionic strength. , 73:356–359, 1994.
M.D. Carbajal-Tinoco, F. Castro-Román, and J.L. Arauz-Lara. Static properties of confined colloidal suspensions. , 53:3745–3749, 1996.
J.C. Crocker and D.G. Grier. When like charges attract: the effects of geometrical confinement on long-range colloidal interactions. , 77:1897–1900, 1996.
A.E Larsen and D.G. Grier. Like-charge attractions in metastable colloidal crystallites. , 385:230–233, 1997.
B. Honig and A. Nicholls. Classical electrostatics in biology and chemistry. , 268:1144–1149, 1995.
J.E Sader and D.Y.C. Chan. Long-range electrostatic attractions between identically charged particles in confined geometries: An unresolved problem. , 213:268–269, 1999.
J.C. Neu. Wall-mediated forces between like-charged bodies[ in an electrolyte]{}. , 82:1072–1074, 1999.
D.G. Grier, Personal Communication, 1999.
H.A. Lorenz. Ein allgemeiner satz, die bewegung einer reibenden flüssigkeit betreffend, nebst einigen anwendungen desselben. , 1:23–42, 1906.
J.R. Blake. A note on the image system for a stokeslet in a no-slip boundary. , 70:303–310, 1971.
W.B. Russel, D.A. Saville, and W.R. Schowalter. . Cambridge University Press, Cambridge, 1989.
B.V. Derjaguin and L. Landau. Theory of the stability of strongly charged lyophobic sols and the adhesion of strongly charged particles in solutions of electrolytes. , 14:633–662, 1941.
E.J. Verwey and J. Th. G. Overbeek. . Elsevier, Amsterdam, 1948.
G.K. Batchelor. Brownian diffusion of particles with hydrodynamic interaction. , 74:1–29, 1976.
D.L. Ermak and J.A. McCammon. Brownian dynamics with hydrodynamic interactions. , 69:1352–1360, 1978.
P.S. Grassia, E.J. Hinch, and L.C. Nitsche. Computer simulations of brownian motion of complex systems. , 282:373–403, 1995.
J.C. Crocker and D.G. Grier. Methods of digital video microscopy for colloidal studies. , 179:298–310, 1996.
A. E. Larsen and D.G. Grier. Melting of metastable crystallites in charge-stabilized colloidal suspensions. , 76:3862–3865, 1996.
K. Ito, H. Yoshida, and N. Ise. Void structures in colloidal dispersions. , 263:66–68, 1994.
|
---
abstract: 'In this work we study a system of interacting fermions with large spin and SP(N) symmetry. We contrast their behaviour with the case of SU(N) symmetry by analysing the conserved quantities and the dynamics in each case. We also develop the Fermi liquid theory for fermions with SP(N) symmetry. We find that the effective mass and inverse compressibility are always enhanced in the presence of interactions, and that the N-dependence of the enhancement is qualitatively different in distinct parameter regimes. The Wilson ratio can be enhanced, indicating that the system can be made closer to a magnetic instability, in contrast to the SU(N) scenario. We conclude discussing what are the experimental routes to SP(N) symmetry within cold atoms and the exciting possibility to realize physics in higher dimensions in these systems.'
author:
- Aline Ramires
title: The Symplectic Fermi Liquid and its realization in cold atomic systems
---
Introduction
============
Symmetries have always played an important role in physics. Some can occur naturally, being intrinsic to the system of interest, others require fine-tuning of parameters in order to be present. The experimental control we have over cold atoms allow us to construct systems with symmetries which are larger than the ones naturally present in matter. The study of systems with enlarged symmetries is in principle very attractive: these have larger degeneracies, which can be used to the experimentalists’ advantage in adiabatic cooling in order to achieve low temperatures more effectively [@Geo; @Bla; @Hof]; also, the enhancement of quantum fluctuations and the presence of more degrees of freedom provide theorists with the possibility to study new phases of matter [@Wu10].
The realization of SU(N) symmetry has already been explored both theoretically and experimentally. This symmetry has been identified in alkaline-earth and Yb cold atomic systems [@Caz; @Gor]. These atoms have a completely full outer electronic shell of s-character, so total electronic spin and angular momentum equal to zero. As a consequence, the nuclear spin is effectively decoupled from the electronic degrees of freedom and the s-wave scattering lengths for all nuclear spin configurations are equal. The effective Hamiltonian describing the interacting alkaline-earth atoms is SU(N) symmetric, where $N=2f+1$ and $f$ is the hyperfine spin (which for these atoms is equal to the nuclear spin). Experimental realizations of systems with SU(N) symmetry were already reported, with ultracold Yb isotopes trapped in one dimension [@Pag] or loaded in a 3D lattice [@Tai]. Theoretically several aspects of the SU(N) symmetry were already explored, including the characterization of the Fermi Liquid behaviour [@Yip; @Caz09], magnetism [@Che; @Kat; @Her; @Xu10; @Gua], superconductivity [@Ho; @Cap], multipolar orders [@Tu], staggered flux order [@Hon] and topology [@Szi; @Yan; @Aro].
The presence of SU(N) symmetry is restricted to alkaline-earth atoms with a non-zero nuclear spin, what in principle gives us a small number of options amongst all isotopes available in nature. The question that follows is: are there different enlarged symmetries which can be realized with other atomic isotopes? The answer is yes, and SP(N) symmetry is a good candidate since it is a subgroup of SU(N), therefore less restrictive. An early work by Wu et al. [@Wu03] on cold atoms with hyperfine spin $f=3/2$ explore a *hidden SP(4) symmetry*. More recently[@Wu05] the presence of symplectic symmetry in higher-spin Hubbard models within cold fermions was discussed, and it is pointed out that the symplectic symmetry does not require any fine tuning for $f=3/2$, while higher hyperfine spin systems require certain tuning of the interactions for the symmetry to be present. Another subgroup of SU(N) is SO(N), which can also be realized in cold atomic systems with bosonic isotopes[@Jia].
The main aspect that makes cold atom systems with SP(N) symmetry distinct from the ones with SU(N) symmetry is related to the dynamics of each spin component. In systems with SU(N) symmetry one can understand each spin component as a different colour or flavour, and the interactions allow only for colour-preserving scattering, as depicted in Fig. \[SUNSPN\] a) and c). On the other hand, in case of SP(N) symmetry, it is more intuitive to label the spin components with a colour and an arrow, which can be either up or down. The colour can be understood as the different magnitudes of the spin component, and the up and down arrows as their sign. The form of the interactions in this case allows for a very special kind of scattering, which takes a pair of atoms with the same colour (up and down) and transmute it to a pair of a different colour, as shown in Fig. \[SUNSPN\] b) and e). These points are reviewed and discussed in detail in Section \[SPN\]. More generally, one would have spin-flip scattering processes which do not preserve each nuclear spin component but only the total angular momentum of the colliding atoms. This has been observed and controlled experimentally with the long-lived alkaline radioisotope $^{40} K$ in an optical lattice [@Kra].
SP(N) symmetry can only be realized with fermions ($N=2f+1$ can only acquire even values for the symplectic group, requiring half-integer hyperfine spins). At low enough temperatures, fermions can reach quantum degeneracy and behave as a Fermi liquid (FL). Fermi liquid behaviour is ubiquitous in condensed matter systems and a very robust state of matter. The original FL theory was developed for spin-1/2 fermions [@Lif] and recently there was a generalization for fermions with larger spin and SU(N) symmetry [@Yip]. In Section \[FL\] we develop the Fermi liquid theory for SP(N) cold fermions, analyzing the effective mass, compressibility and susceptibility and contrast these results with the SU(N) Fermi liquid [@Yip]. The most interesting aspect of the analysis concerns magnetism. We distinguish between two kinds of susceptibilities: a generalized and a physical susceptibility. Both are renormalized in the same fashion in the presence of interactions and can be either enhanced or suppressed, depending on the parameter regime. In Section \[Con\] we discuss the possible routes to realize SP(N) symmetry within cold atomic systems, and highlight exciting directions for future work which allows us to explore experimentally issues only thought to be in the theoretical realm, as physics in higher dimensions.
SU(N) and SP(N) Symmetries in Cold Atoms {#SUSPN}
========================================
We start with a general model for cold atoms with hyperfine-spin $f$. We assume a dilute gas with contact interactions so at low energies only the s-wave scattering channel is relevant [@Lee; @Caz]. We can write the effective Hamiltonian as: $$\begin{aligned}
H= H_0 + H_{I},\end{aligned}$$ where $H_0$ is the kinetic part: $$\begin{aligned}
H_0 = \int_\br \sum_{\alpha=-f}^f \Psi^\dagger_\alpha(\br) \left( -\frac{1}{2m}\nabla^2 + V (\br) \right) \Psi_\alpha (\br),\end{aligned}$$ which describes moving atoms under a trapping potential $V(\br)$. Here $\Psi_\alpha^\dagger (\br)$ and $\Psi_\alpha (\br)$ are creation and annihilation operators, respectively, for atoms with hyperfine spin component $\alpha$ located at $\br$ which at this stage can be either bosons or fermions. The interacting part reads: $$\begin{aligned}
H_{I} = \frac{1}{2} \int_\br \sum_{\alpha,\beta,\mu,\nu=-f}^f \!\!\! \Psi_\beta^\dagger(\br) \Psi_\alpha^\dagger (\br) \Gamma_{\alpha\beta;\mu\nu} \Psi_\mu(\br) \Psi_\nu (\br),\end{aligned}$$ where the interaction vertex can be decomposed in different total angular momentum channels as[@Ho98]: $$\begin{aligned}
\label{IntVer}
\Gamma_{\alpha\beta;\mu\nu} = \sum_{F=0}^{2f } g_F \sum_{M=-F}^F \langle f \alpha, f \beta | F M\rangle \langle F M | f \mu, f \nu\rangle.\end{aligned}$$ Here $F$ is the total angular momentum of the two interacting atoms, $M$ its component and $ \langle f \alpha, f \beta | F M\rangle$ are Clebsch-Gordan coefficients (CGC). $g_F$ is the strength of the interaction in the channel with total angular momentum $F$. The model could similarly be written for atoms in an optical lattice, and the discussion below, concerning symmetries, should follow in an analogous fashion.
It can be shown by analyzing $H_I$, that only even-$F$ channels contribute to scattering. One can take $\alpha \leftrightarrow \beta$, use the properties of the CGC shown in Appendix \[AppCGC\] and the fact that the (fermionic) bosonic operators (anti-) commute to rewrite the interaction term explicitly as: $$\begin{aligned}
H_{I} &=& \frac{1}{4} \int_\br \sum_{\alpha,\beta,\mu,\nu=-f}^f \sum_F g_F (1+\eta (-1)^{2f-F}) \\ \nonumber &\times& \sum_M \Psi_\beta^\dagger \Psi_\alpha^\dagger \langle f \alpha, f \beta | F M\rangle \langle F M | f \mu, f \nu\rangle \Psi_\mu \Psi_\nu \end{aligned}$$ where $\eta=+1$ for bosons and $\eta=-1$ for fermions. Note that for either bosons or fermions the factor $(1+\eta (-1)^{2f-F})$ simplifies to $(1+ (-1)^{F})$, which is zero for odd-F and equal to 2 for even-F. This is a consequence of the compensation of the factors $\eta$ and $(-1)^{2f}$, which product is always equal to one since $f$ is an integer for bosons and a half-integer for fermions. Out of the total $2f + 1$ scattering channels, only $f+1$ for bosons or $f+1/2$ for fermions actually contribute to scattering.
SU(N) symmetry {#SUN}
--------------
We start the discussion towards SP(N) symmetry showing first that SU(N) can be realized in this system under the special condition of $g_F=g$, meaning that the interactions in all scattering channels are the same. In order to prove the presence of the symmetry, we can evaluate the commutator of the generators of SU(N) group with the Hamiltonian. SU(N) has $N^2-1$ generators, which can be written as: $$\begin{aligned}
\label{SUGen}
O_{\alpha\beta} = \int_\br \Psi_\alpha^\dagger (\br) \Psi_\beta (\br),\end{aligned}$$ where each index $\alpha$ and $\beta$ can run over $N=2f+1$ values. Note that not all the generators are linearly independent since the Casimir operator $C=\sum_\alpha O_{\alpha\alpha}$ is a constant. These generators follow the SU(N) commutation relation: $$\begin{aligned}
[O_{\alpha\beta}, O_{\mu\nu}] = O_{\alpha\nu} \delta_{\mu\beta} - O_{\mu\beta} \delta_{\alpha\nu}.\end{aligned}$$
The commutator with the non-interacting part of the Hamiltonian is rather trivial and equal to zero. Evaluating now the commutator with the interacting part, we find after some manipulation: $$\begin{aligned}
\label{HIO}
\left[H_I, O_{\alpha'\beta' } \right] &=&\int_\br \Big( \Gamma_{\alpha\beta;\mu \alpha' } \Psi_\beta^\dagger (\br) \Psi_\alpha^\dagger (\br) \Psi_\mu (\br) \Psi_{\beta ' }(\br) \\ \nonumber && - \Gamma_{\alpha \beta ' ;\beta \mu} \Psi_\beta^\dagger (\br) \Psi_{\alpha' }^\dagger (\br) \Psi_\alpha (\br) \Psi_\mu (\br) \Big),\end{aligned}$$ which is generally not equal to zero. In the equation above the sum over repeated indexes is implied. Under the consideration that the interactions in all scattering channels are equal, $g_F=g$, we can use the orthogonality condition of the CGC (Eq. \[CGCOrt1\]) to simplify the interaction vertex to: $$\begin{aligned}
\Gamma_{\alpha\beta;\mu\nu}^{SU(N)} = g \delta_{\alpha\mu}\delta_{\beta\nu},\end{aligned}$$ so we can write the explicit form of the interaction part of the Hamiltonian in case of SU(N) symmetry: $$\begin{aligned}
\label{SUInt}
H_{I}^{SU(N)} = \frac{g}{2} \int_\br \sum_{\alpha,\beta=-f}^f \!\!\! \Psi_\alpha^\dagger(\br) \Psi_\beta^\dagger (\br) \Psi_\beta(\br) \Psi_\alpha (\br).\end{aligned}$$
It is a simple task to show that $$\begin{aligned}
\label{HIO}
\left[H_{I}^{SU(N)}, O_{\alpha'\beta' } \right] &=&0,\end{aligned}$$ what proves the SU(N) symmetry. Note that this result is independent of the bosonic or fermionic character of the atoms.
The diagonal generators: $$\begin{aligned}
O_{\alpha\alpha} = \int_\br \Psi_\alpha^\dagger (\br) \Psi_\alpha (\br) =n_\alpha\end{aligned}$$ commute with the Hamiltonian, so these are conserved quantities. The total number of particles with a given spin-component, or flavour, is preserved if the interaction, is the same for all channels and SU(N) symmetry is realised.
SP(N) symmetry {#SPN}
--------------
Given the discussion above, now we move to the study of the SP(N) case. One subtlety about the SP(N) generalization is that the group is only defined for even-N and its realization is possible only with fermionic atoms. We can check the presence of SP(N) symmetry by evaluating the commutator of the generators of SP(N) with the Hamiltonian. SP(N) is a subgroup of SU(N) and has $N(N+1)/2$ generators, which can be written as specific linear combinations of the SU(N) generators defined above in Eq. \[SUGen\]: $$\begin{aligned}
\label{SPGen}
S_{\alpha\beta} &=& O_{\alpha\beta} +(-1)^{\alpha+\beta} O_{-\beta-\alpha},\end{aligned}$$ where again $\alpha$ and $\beta$ can run over $N$ values. Note that these generators are not all linearly independent given the relation $S_{\alpha\beta} = (-1)^{\alpha+\beta}S_{-\beta-\alpha}$.
The commutator with the non-interacting part of the Hamiltonian is again trivial. Concerning the interacting part, given that SP(N) is a subgroup of SU(N), if a Hamiltonian has SU(N) symmetry (if $g_F=g$), it will also commute with the generators of SP(N). Note, though, that this is not what we are looking for since the actual symmetry of the system is still SU(N) in this case. We need to look for a way to break the full SU(N) symmetry down to SP(N). From the strongly correlated systems perspective, it is known that SP(N) was introduced in order to deal with valence bonds in frustrated magnetism [@Rea3; @Fli09] and singlet pairing [@Fli08; @Fli12]. It is suggestive then, that the zero total angular momentum channel $g_{F=0}$ is the important one to distinguish SU(N) from SP(N) symmetry.
We can use the results obtained for SU(N), before the assumption that all channels have the same interaction strength, given by Eq. \[HIO\], and look at the less restrictive condition of having $g_0\neq g_{F>0}=g$. In this case we can combine all terms with the same magnitude of the interaction and part of $g_0$ to use the orthogonality condition of the CGC, leading to zero contribution to the commutator, as found in the SU(N) discussion above. We are left with a term proportional to $\Delta g=g_0-g$ in the $F=0$ channel to be evaluated. Now the interaction vertex simplifies to $$\begin{aligned}
\Gamma_{\alpha\beta;\mu\nu}^{SP(N)} = \Gamma_{\alpha\beta;\mu\nu}^{SU(N)} - \frac{\Delta g}{N} (-1)^{\alpha+\mu} \delta_{\alpha,-\beta} \delta_{\mu,-\nu}, \end{aligned}$$ after using Eq. \[CGC0\], identifying $2f+1=N$, and remembering that we are dealing only with fermions, so $2f$ is always an odd number. Under these considerations the interacting part of the Hamiltonian for the case of SP(N) symmetry can be written explicitly as: $$\begin{aligned}
\label{SPInt}
&&H_{I}^{SP(N)} = \frac{g}{2} \int_\br \sum_{\alpha,\beta=-f}^f \!\!\! \Psi_\alpha^\dagger(\br) \Psi_\beta^\dagger (\br) \Psi_\beta(\br) \Psi_\alpha (\br)\\ \nonumber
&+&\frac{\Delta g}{2N} \int_\br \sum_{\alpha,\beta=-f}^f \!\!\! (-1)^{\alpha+\beta}\Psi_\alpha^\dagger(\br) \Psi_{-\alpha}^\dagger (\br) \Psi_\beta(\br) \Psi_{-\beta} (\br).\end{aligned}$$ Note that the first term is the same as the one present in the case of SU(N) symmetry. The second term, proportional to the detuning of the $F=0$ channel, is the part of the interaction which breaks SU(N) down to SP(N).
We can now evaluate the commutator of the interacting part of the Hamiltonian under the condition $g_0\neq g_{F>0}=g$ with the SU(N) generators to find: $$\begin{aligned}
\left[H_{I}^{SP(N)}, O_{\alpha'\beta'} \right]&=&- \int_\br \sum_{\alpha=-f}^f \frac{\Delta g}{2N} (-1)^{\alpha}\\
\nonumber &&\times\left[ (-1)^{\alpha'} \Psi_\alpha^\dagger (\br) \Psi_{-\alpha}^\dagger (\br) \Psi_{-\alpha'} (\br) \Psi_{\beta' } (\br) \right. \\ \nonumber &&- \left. (-1)^{\beta' }\Psi_{-\beta' }^\dagger (\br) \Psi_{\alpha' }^\dagger (\br) \Psi_\alpha (\br) \Psi_{-\alpha} (\br) \right],\end{aligned}$$ what is generally not equal to zero. Note though that: $$\begin{aligned}
\left[H_{I}^{SP(N)}, (-1)^{\alpha'+\beta'} O_{-\beta'-\alpha'} \right]
&=& - \left[H_{I}^{SP(N)}, O_{\alpha'\beta'} \right]\nonumber \\ \end{aligned}$$ so for the SP(N) generators: $$\begin{aligned}
\left[H_{I}^{SP(N)}, S_{\alpha'\beta'}\right] = 0,\end{aligned}$$ what indicates that the model with the interactions satisfying $g_0\neq g_{F>0}=g$ has SP(N) symmetry and not the larger SU(N) symmetry.
The diagonal generators are now: $$\begin{aligned}
S_{\alpha\alpha} &=& \int_\br \left( \Psi_\alpha^\dagger(\br) \Psi_\alpha(\br) -\Psi_{-\alpha}^\dagger(\br) \Psi_{-\alpha}(\br) \right)\\ \nonumber&=& n_\alpha-n_{-\alpha}=m_\alpha\end{aligned}$$ and commute with the Hamiltonian, so these are conserved quantities. The magnetization for a given magnitude of the spin-component, or the colour-magnetization, is preserved if the interaction, is the same for all but the $F=0$ channel and SP(N) symmetry is realised.
Comparison of SU(N) and SP(N) symmetric systems
-----------------------------------------------
At this point a discussion on the physical implications of SU(N) and SP(N) symmetries is interesting. From the explicit form of the Hamiltonians in Eqs. \[SUInt\] and \[SPInt\], it becomes clear that in the first case two particles with components $\alpha$ and $\beta$ can only scatter into states with the same spin components, so the number of particles with each spin component is a constant. This is illustrated pictorially in Fig. \[SUNSPN\] c). On the other hand, the SP(N) interaction has a second contribution which allows the spin components, or colours, to change. Now two particles with the same colour and up and down arrows ($\beta$ and $-\beta$, for example) can scatter into a pair of states with opposite arrows and a different colour ($\alpha$ and $-\alpha$, for example). Fig. \[SUNSPN\] e) illustrates this point.
![Schematic comparison of SU(4) and SP(4) systems. An SU(4) system has 4 different flavours, represented by the 4 different colours in (a). In contrast, the SP(4) system represented in (b) has an extra label as “up" or “down" for two colours, in a total of four different flavours. In (c) we depict a SU(4) representative scattering process, which is colour-preserving. In (d) we represent the analogous scattering for SP(4). In the symplectic case there is also the possibility for processes as depicted in (e), where “up" and “down" pairs of fermions of a given colour can transmute into a pair of a different colour.[]{data-label="SUNSPN"}](FigSUNSPN.pdf){width="\linewidth"}
Another difference between SU(N) and SP(N) symmetries concerns the dynamics of the system. Given an initial state with a specific occupancy of the different spin components, the SU(N) and SP(N) cases will evolve differently. In particular, if we have alkaline-earth atoms with hyperfine spin $f$ and all the levels are occupied, the system has SU($N=2f+1$) symmetry. On the other hand, if only a certain subset $n<2f+1$ of the flavours is occupied, the symmetry which is realized is SU(n$<$N). This is a consequence of the fact that the number of particles in each flavour is conserved. In contrast, in the symplectic scenario, the interaction allows for spin component transmutation. Even if we load the system with a single colour (with up and down arrows), the system will equilibrate to the lowest energy state with same occupation number to each flavour, and the symmetry is actually the maximal SP($N=2f+1$). Interestingly enough, if one initially traps only positive components, so they cannot pair with their complements in order to transmute into other flavours, then the symmetry is lowered to SU($n\leq N/2$), where $n$ is the number of unpaired components trapped.
In the introduction it was already mentioned that SP(4) symmetry has been pointed out for systems with $f=3/2$ [@Wu03; @Wu05]. What is special about $f=3/2$ is the fact that it naturally satisfies the condition for SP(4), without any fine-tuning. We know that, by symmetry, only even channels contribute to scattering, so only $F=0,2$ are the allowed scattering channels for $f=3/2$. Even if the interactions in all channels are different, we fall under the condition with the $F=0$ channel different from all other channels (here only one, $F=2$). For larger spins we would have extra channels present, with different interactions. For instance, for $f=5/2$ there are three channels $F=0,2,4$. To realise SP(N) symmetry in this case one needs to tune the interactions for the $F=2$ and $F=4$ channels to be the same. A compilation of results for the particular case of SP(4) can be found in Wu [@Wu], with more recent results on magnetism[@Kole; @Xu09] and superconductivity [@Cap]. As we will discuss in more detail in Sec. \[ExpRea\], unfortunately nature does not provide us with atoms with hyperfine spin-3/2 which have no dipole-dipole interactions and are not alkaline-earths (these realize the larger SU(4) and not SP(4) symmetry), so a smart experimental setup is necessary in order to realize even the first non-trivial case of SP(N) with $N=4$.
From the discussion above it is suggestive that the presence of new scattering processes that allow for spin-flip scattering (or colour transmutation) brings new aspects that should be investigated and the possibility of more interesting physics, mainly concerning magnetism, to be found. As a first exploration of these consequences, in this work we focus on the effects on the Fermi liquid behaviour.
The Symplectic Fermi Liquid {#FL}
===========================
Given the motivation above for the realization of SP(N) symmetry within cold fermions, now we analyse the Fermi liquid behaviour of a system of fermions with symplectic symmetry. This is the state that would be accessible in experiments if the temperature is below quantum degeneracy, but not low enough so that order is able to develop. From this section on we will focus on the following Hamiltonian, already Fourier transformed to momentum space:
$$\begin{aligned}
\label{SPFL}
H_{FL}^{SP(N)} &=& \sum_\bk \sum_{\alpha} \Psi_{\bk\alpha}^\dagger\left(\frac{\bk^2}{2m}-\mu\right)\Psi_{\bk\alpha}
+ \frac{g}{2} \sideset{}{' }\sum_{\{\bk\}} \sum_{\substack{\alpha,\beta\\ \alpha \neq \beta} } \Psi_{\bk_1,\alpha}^\dagger \Psi_{\bk_2\beta}^\dagger \Psi_{\bk_3\beta} \Psi_{\bk_4\alpha} \\ \nonumber
&+& \frac{\Delta g}{2N}\sideset{}{' } \sum_{\{\bk\}}\sum_{ \alpha, \beta } (-1)^{\alpha+\beta}\Psi_{\bk_1\alpha}^\dagger \Psi_{\bk_2-\alpha}^\dagger \Psi_{\bk_3\beta} \Psi_{\bk_4-\beta},\end{aligned}$$
where $\Psi_{\bk\alpha}^\dagger$ ($\Psi_{\bk\alpha}$) creates (annihilates) a fermion with momentum $\bk$ and spin component $\alpha$, which can assume half-integer values between $-f$ and $f$. The first term describes free fermions with mass $m$ and chemical potential $\mu$. Here we ignore the trapping potential, assuming the fermions explore a region in space with an almost constant potential. The second term introduces part of the interactions which is also present in the SU(N) case (here we make explicit that the sum does not allow $\alpha=\beta$ since we are dealing with fermions). The third term introduces a new interaction vertex, which is particular to the SP(N) case, where $\Delta g=g_0-g$. The primed sum over $\{\bk\}=\bk_1,\bk_2,\bk_3,\bk_4$ indicate the sum over all momenta, subject to momentum conservation $\bk_1+\bk_2=\bk_3+\bk_4$.
We can construct a FL theory, following the lines of Lifshitz and Pitaevskii[@Lif], treating the quasiparticle distribution function and the quasiparticle energy as $N \times N$ matrixes, in which each index corresponds to a spin component running from $-f$ to $f$, where $f$ is a half-integer number. In Yip et al.[@Yip], the authors generalize the FL theory to SU(N) symmetry and compute the effective mass, magnetic susceptibility and compressibility in terms of the new Landau parameters. Here we will comment on the generalization to the symplectic case, and how physical quantities depend on the parameter $N$ in this new scenario.
The change in the quasiparticle energy $\delta \epsilon_{\alpha\beta} (\bk)$ due to an infinitesimal change in the quasiparticle distribution function $\delta n_{\alpha\beta} (\bk)$ can be written as: $$\begin{aligned}
\delta \epsilon_{\alpha\beta} (\bk)=\sum_{\bk'}\sum_{\mu,\nu} f_{\alpha\mu,\beta\nu}(\bk,\bk') \delta n_{\nu\mu} (\bk'),\end{aligned}$$ where $f_{\alpha\mu,\beta\nu}(\bk,\bk')$ is the interaction function[@Lif]. The specific form of the interaction function depends on the actual interactions between the particles and it will be worked out in Sec. \[IntFun\] below. Given SP(N) symmetry, we can parametrize the interaction function as follows: $$\begin{aligned}
\label{FLPar}
f_{\alpha\mu,\beta\nu}(\bk,\bk') &=& f_s(\bk,\bk') \delta_{\alpha\beta}\delta_{\mu\nu} + f_\epsilon(\bk,\bk') \epsilon_{\alpha\mu}\epsilon_{\beta\nu}\\ \nonumber&+& f_a (\bk,\bk') \sum_A \Gamma^A_{\alpha\beta}\Gamma^A_{\mu\nu}.\end{aligned}$$ Note that, differently from the SU(N) case now we have three different parameters: $f_s(\bk,\bk')$, $f_a(\bk,\bk')$ and $f_\epsilon(\bk,\bk')$. This reflects the fact that there are three independent 4-indexed invariants under SP(N) transformations. This point is discussed in more detail in Appendix \[AppSym\]. Here $\epsilon$ is an antisymmetric matrix, $\Gamma^A$ are the generators of the specific symmetry group. The label $A$ runs from $1$ to the total number of generators, which is equal to $N(N+1)/2$ for SP(N). The generators are traceless and we choose them to be normalized as $Tr[\Gamma^A \Gamma^B] = \delta_{AB}$. Note that the generators introduced in Section \[SPN\] do not satisfy this condition, but in Appendix \[AppGen\] we show these can be redefined such that this normalization holds, making the following calculations more straightforward.
Effective Mass and Compressibility {#Sus}
----------------------------------
The effective mass and compressibility for the SP(N) FL can be computed in the same fashion as for the SU(N) or SU(2) FL, so we simply state and comment on the results in this section. The effective mass reads: $$\begin{aligned}
\frac{m^*}{m} = 1+ N \overline{F_s(\theta) \cos\theta},\end{aligned}$$ where $$\begin{aligned}
F_s(\theta) = \rho^*(E_f) \left( f_s(\theta)+\frac{1}{N} f_\epsilon(\theta) \right),\end{aligned}$$ with $\theta$ the angle between $\bk$ and $\bk'$, which are at the Fermi surface. The overline denotes average over the solid angle. Here we introduce the density of states per spin component at the Fermi energy $\rho^*(E_f) = m^* k_f/(2\pi^2)$, with $k_f$ the Fermi momentum, defined from the total particle density $\rho_T=N_T/V = N\frac{k_f^3}{6 \pi^2}$.
Analogously, the inverse compressibility $u^2 = \frac{N_T}{m}\frac{d\mu}{dN_T}$ is modified in the presence of interactions as: $$\begin{aligned}
\frac{u^{*2}}{u^2} = \frac{1+ N \overline{F_s(\theta)}}{1+ N \overline{F_s(\theta) \cos\theta}}.\end{aligned}$$ These results are similar in form to the SU(N) results, in which case $f_\epsilon(\theta)=0$.
The Generalized and the Physical Magnetic Susceptibilities {#Sus}
----------------------------------------------------------
Now we would like to focus the discussion on the magnetic susceptibility. Here we will make a distinction between two kinds of susceptibility: a generalized susceptibility $\chi_G$, and a physical susceptibility $\chi_P$.
For the generalized susceptibility we define a generalized magnetization with components $m^A$, associated with a generalized magnetic field with components $h^A$ which couple to the respective generator as $- \mu_B h^A \Gamma^A$. This is the natural generalization of the SU(2) case, in which there are 3 generators (the three Pauli matrices), each one coupling to one component of the magnetic field in 3-dimensional space as $- \mu_B h^i \cdot \sigma^i$. This is a formal definition, and it is what was evaluated for the SU(N) FL as a generalized susceptibility[@Yip].
We can perform a similar calculation, following Lifshitz and Pitaevskii[@Lif], and evaluate the change in energy due to the presence of an external generalized magnetic field as follows: $$\begin{aligned}
\delta \epsilon_{\alpha\beta}(\bk) &=& -\mu_B \sum_A h^A \Gamma^A_{\alpha\beta} \\ \nonumber&+& \sum_{\mu,\nu}\int d\tau' f_{\alpha\mu,\beta\nu} (\bk,\bk') \delta n_{\nu\mu}(\bk'),\end{aligned}$$ where the first term accounts for the change in energy due to the presence of the field, while the second takes into account feedback effects due to interactions. We transformed the sum over $\bk'$ into an integral introducing $d\tau' = \frac{d\bk' }{(2\pi)^3}$. We use the ansatz $$\begin{aligned}
\delta \epsilon_{\alpha\beta}(\bk) &=& -\mu_B \frac{\gamma}{2}\sum_A h^A \Gamma^A_{\alpha\beta},\end{aligned}$$ where $\gamma$ is a parameter to be determined self-consistently. Using the fact that the generators are traceless, normalized and follow the symplectic condition $\epsilon \Gamma^A \epsilon = (\Gamma^A)^T$, we find: $$\begin{aligned}
\gamma = \frac{2}{1+ \overline{F_a(\theta)}},\end{aligned}$$ where $$\begin{aligned}
F_a(\theta) = \rho^*(E_f) \left( f_a(\theta)-f_\epsilon(\theta) \right).\end{aligned}$$ Finally, the generalized susceptibility is defined as: $$\begin{aligned}
\chi_G h^A = m^A = \mu_B \sum_{\alpha\beta} \int d\tau \Gamma^A_{\alpha\beta} \delta n_{\beta\alpha} (\bk),\end{aligned}$$ and has the form: $$\begin{aligned}
\chi_G = \frac{2 \mu_B^2 \rho^*(E_f)}{1+ \overline{F_a(\theta)} }.\end{aligned}$$ Note that the non-interacting susceptibility computed in this fashion is independent of N. This is in fact the result for what was defined as the generalized susceptibility $\chi_G$, but it goes against the physical intuition that if we have a Fermi gas with many spin components, all susceptible to a magnetic field, the susceptibility should depend on the number of components. Also, this computation assumes the existence of a generalized magnetic field with as many components as generators, so $N(N+1)/2$ components. This suggests that one would need to be in higher spacial dimensions in order to realize it. We are going to comment further on this aspect in Sec \[highd\].
Based on this discrepancy we evaluate now what we call the physical susceptibility $\chi_P$. The physical point of view asks the following question: what happens when we apply actual magnetic field (assuming a 3-dimensional space) to a system with an enlarged symmetry? The standard estimation for the magnetization of a pair of spin components $\alpha$ and $-\alpha$ is $m_\alpha=2\alpha \mu_B (n_\alpha-n_{-\alpha})$, and one can approximate $n_\alpha-n_{-\alpha}\approx \rho(E_f) 2 \alpha g \mu_B h^z$, where $h^z$ is now a physical magnetic field chosen to be in the z-direction. The total magnetization can then be written as: $$\begin{aligned}
m^z= \sum_\alpha m_\alpha &=& 4 \rho(E_f) g \mu_B^2 \sum_\alpha \alpha^2 h^z\end{aligned}$$ so from $m^z=\chi_P h^z$ we can identify the physical susceptibility as: $$\begin{aligned}
\chi_P&=&\frac{2 \mu_B^2 \rho^*(E_f)}{1+ \overline{F_a(\theta)} } \frac{N (N^2-1)}{6},\end{aligned}$$ which now depends on $N$. Note that in the case $N=2$ we recover the known SU(2) result, with the last fraction equal to one and $f_\epsilon(\theta)=0$. This result should be valid for both SU(N) and SP(N) Fermi liquids, which will differ on the specific form of the renormalization factors, which we treat explicitly below.
Note that in both cases, for the generalized or physical susceptibilities, the effects of interactions lead to the same renormalization: $$\begin{aligned}
\left(\frac{\chi^*_{G,P}}{\chi_{G,P}}\right)^{-1}&=& \frac{1+ \overline{F_a(\theta)}}{1+ N \overline{F_s(\theta) \cos\theta}}.\end{aligned}$$
Explicit form of the interaction function {#IntFun}
-----------------------------------------
The interaction function is defined as the second variation of the total energy with respect to occupation numbers: $$\begin{aligned}
f_{\alpha\mu,\beta\nu}(\bk,\bk') = \frac{\delta^2 E}{\delta n_{\beta\alpha}(\bk)\delta n_{\nu\mu}(\bk')}.\end{aligned}$$
If we approach the interacting problem perturbatively, starting from the Fermi gas as the non-interacting problem, we have that the ground state over which averages are going to be taken has only diagonal non-zero occupation numbers $\delta n_{\alpha\beta}(\bk) = \delta n_{\alpha\alpha}(\bk) \delta_{\alpha,\beta}\equiv \delta n _\alpha (\bk) \delta_{\alpha\beta}$, so the only non-zero interaction functions have the form: $$\begin{aligned}
f_{\alpha\beta}(\bk,\bk') = \frac{\delta^2 E}{\delta n_{\alpha}(\bk)\delta n_{\beta}(\bk')},\end{aligned}$$ where we defined $f_{\alpha\beta}(\bk,\bk') \equiv f_{\alpha\beta,\alpha\beta}(\bk,\bk')$.
In first order, the contribution of the interactions to the total energy can be written as: $$\begin{aligned}
E^{(1)} &=& \sum_{\bk_1,\bk_2} \sum_{\alpha,\beta} n_{\alpha}(\bk_1) n_{\beta}(\bk_2) \\ \nonumber &&\times \Bigg[ \frac{g}{2}(1-\delta_{\alpha\beta}) +\frac{\Delta g}{2N}\delta_{\alpha,-\beta}\Bigg].\end{aligned}$$
Note that the last term in the first order correction is not present in the SU(N) scenario since $\Delta g=g_0-g$ is zero in that case. This term appears with a factor of $1/N$, but for small values of $N$ and significant $\Delta g$ it can play an important role. The interaction function in first order in the interactions is then: $$\begin{aligned}
f_{\alpha\beta}^{(1)}(\bk,\bk') = \frac{g}{2}(1-\delta_{\alpha\beta}) +\frac{\Delta g}{2N}\delta_{\alpha,-\beta}.\end{aligned}$$
![Quasiparticle interaction vertices in first order in $g/2$ (circle) and in $\Delta g/2N$ (square), respectively, depicted as Feynman diagrams. Note that $\alpha\neq \beta$ in the first diagram.[]{data-label="FigVer"}](DiagramsVertices.pdf){width="0.75\linewidth"}
In second order, the contribution to the total energy is more involved. In favour of a concise notation we write: $$\begin{aligned}
H_I^{SP(N)} =&&\sideset{}{' }\sum_{\{\bk\}} \sum_{\alpha,\beta,\mu,\nu} G_{\alpha\beta\mu\nu} \Psi^\dagger_{\bk_1\alpha} \Psi^\dagger_{\bk_2\beta} \Psi_{\bk_3\mu} \Psi_{\bk_4\nu},\nonumber\\\end{aligned}$$ where $$\begin{aligned}
G_{\alpha\beta\mu\nu} &=& \frac{g}{2} (1-\delta_{\alpha\beta})\delta_{\mu\beta}\delta_{\alpha\nu} \\ \nonumber&&+ \frac{\Delta g}{2N} (-1)^{\alpha+\mu} \delta_{\beta,-\alpha}\delta_{\mu,-\nu}.\end{aligned}$$
With this definition we can write the second order contribution to the total energy as: $$\begin{aligned}
E^{(2)} &=& \sideset{}{' }\sum_{\{\bk\}} \sum_{\alpha\beta\mu\nu} (G_{\alpha\beta\mu\nu})^2 \frac{n_{\bk_4\nu} n_{\bk_3\mu}(1-n_{\bk_2\beta})(1-n_{\bk_1\alpha}) }{(\bk_4^2 + \bk_3^2-\bk_2^2-\bk_1^2)/2m}.\nonumber\\\end{aligned}$$ As we are interested in the interaction function, not in the total energy, we evaluate the sums after we vary the energy with respect to the occupation numbers. The result is the following:
$$\begin{aligned}
f^{(2)}_{\alpha\beta} (\bk,\bk') = &-& \left(\frac{g}{2}\right)^2 \Big[ (1-\delta_{\alpha\beta}) \left[I_1(\bk,\bk')+I_2(\bk,\bk')\right]+ \delta_{\alpha\beta} (N-1) I_1(\bk,\bk') \Big] \\ \nonumber
&-& 2\left(\frac{g}{2}\right)\left(\frac{\Delta g}{2N}\right) \Big[\delta_{\alpha,-\beta} \left[I_1(\bk,\bk')+I_2(\bk,\bk')\right]+ \delta_{\alpha\beta} I_1(\bk,\bk') \Big] \\ \nonumber
&-& \left(\frac{\Delta g}{2N}\right)^2 \left[ I_1(\bk,\bk')+ \delta_{\alpha,-\beta} \frac{N}{2} I_2(\bk,\bk') \right] ,\end{aligned}$$
where $I_{1,2}(\bk,\bk')$ are the sums: $$\begin{aligned}
I_1(\bk,\bk') = \sum_{\bk_1,\bk_2} \frac{n_{\bk_1}-n_{\bk_2}}{(\bk_1^2-\bk_2^2)/2m} \delta_{\bk_1+\bk,\bk_2+\bk'},\end{aligned}$$
$$\begin{aligned}
I_2(\bk,\bk') = \sum_{\bk_1,\bk_2} \frac{n_{\bk_1}+n_{\bk_2}}{(2 k_f^2-\bk_1^2-\bk_2^2)/2m} \delta_{\bk_1+\bk_2,\bk+\bk'},\end{aligned}$$
which can be evaluated as integrals in order to obtain the familiar closed forms: $$\begin{aligned}
I_1(\bk,\bk') = -\frac{4 k_f m}{(2\pi)^2} \left[ 1 + \frac{1-s^2}{2s} \ln \left(\frac{1+s}{1-s}\right) \right],\end{aligned}$$
$$\begin{aligned}
I_2(\bk,\bk') = -\frac{8 k_f m}{(2\pi)^2} \left[ 1 - \frac{s}{2} \ln \left(\frac{1+s}{1-s}\right) \right],\end{aligned}$$
where $s=\sin(\theta/2)$ and $\theta$ is the angle between $\bk$ and $\bk'$, both assumed to be at the Fermi surface.
We can look now at specific cases of the interaction function in second order in the interactions. First, for particles with the same spin: $$\begin{aligned}
f_{\alpha\alpha} (\bk,\bk') &=& -\left[ \left(\frac{g}{2}\right)^2 (N-1) \right. \\ \nonumber &+& \left. 2 \left(\frac{g}{2}\right)\left(\frac{\Delta g}{2N}\right)+\left(\frac{\Delta g}{2N}\right)^2\right] I_1(\bk,\bk'),\end{aligned}$$ which we can identify with the diagrams in Fig. \[DiagAA\]. Note that the diagram with two circular vertexes, proportional to $\left(\frac{g}{2}\right)^2$, is the only one with internal lines which need to be summed over all the possible spin indexes but $\alpha$, which gives the factor of $N-1$ above. Note also that there is no first order correction in the case of particles with same spin component.
![Feynmann diagrams related to the interaction function in second order for particles with equal spin.[]{data-label="DiagAA"}](DiagramsAA.pdf){width="\linewidth"}
For particles with opposite spin components: $$\begin{aligned}
f_{\alpha,-\alpha} (\bk,\bk') &=& \frac{g}{2}+\frac{\Delta g}{2N}\\ \nonumber&-& \left(\frac{g}{2}\right)^2\left[ I_1(\bk,\bk') + I_2(\bk,\bk')\right] \\\nonumber
&-&2 \left(\frac{g}{2}\right) \left(\frac{\Delta g}{2N}\right)\left[ I_1(\bk,\bk') + I_2(\bk,\bk')\right] \\\nonumber
&-& \left(\frac{\Delta g}{2N}\right)^2\left[I_1(\bk,\bk')+\frac{N}{2} I_2(\bk,\bk') \right],\end{aligned}$$ which second order terms can be identified with the diagrams in Fig. \[DiagA-A\]. Note that this time only one of the diagrams with two square vertexes, proportional to $(\frac{\Delta g}{2N})^2$, have internal lines that need to be summed over, giving rise to the factor of $N/2$ in the last line.
![Diagrams for the interaction function up to second order for particles with opposite spin components.[]{data-label="DiagA-A"}](DiagramsA-A.pdf){width="\linewidth"}
Finally, for the case of $\alpha\neq \pm\beta$ we find: $$\begin{aligned}
f_{\alpha,\beta \neq \pm\alpha} (\bk,\bk') &=& \frac{g}{2} - \left(\frac{g}{2}\right)^2\left[ I_1(\bk,\bk') + I_2(\bk,\bk')\right]
\\\nonumber &-& \left(\frac{\Delta g}{2N}\right)^2 I_1(\bk,\bk'),\end{aligned}$$ related to the diagrams pictured in Fig. \[DiagAB\].
![Diagrams depicting the interaction function up to second order for particles with spins satisfying $\alpha\neq \pm \beta$.[]{data-label="DiagAB"}](DiagramsAB.pdf){width="\linewidth"}
Note that when $\Delta g=g_0-g=0$ we recover the SU(N) results[@Yip]. Under this condition $f_{\alpha,-\alpha}=f_{\alpha,\beta\neq \alpha}$ since there are only two FL parameters in the SU(N) case, in contrast to three in the SP(N) case. Due to the different parametrization of the interaction function in terms of $f_{s,a,\epsilon}(\bk,\bk')$ given the different group properties, in particular the completeness relation, the corrections to the physical observables are going to be different, as will be shown explicitly in the following subsection.
Explicit correction to physical quantities {#Corr}
------------------------------------------
Given the results above we can now explicitly write the Fermi liquid parameters $f_{s,a,\epsilon}(\bk,\bk')$. Using the completeness relation for SP(N): $$\begin{aligned}
\sum_A \Gamma^A_{\alpha\beta}\Gamma^A_{\mu\nu} = \delta_{\alpha\nu} \delta_{\beta\mu} - \epsilon_{\alpha\mu}\epsilon_{\beta\nu},\end{aligned}$$ and the fact that the only non-zero interaction functions have the form: $f_{\alpha\beta}(\bk,\bk') \equiv f_{\alpha\beta,\alpha\beta}(\bk,\bk')$, Eq. \[FLPar\] can be rewritten as: $$\begin{aligned}
f_{\alpha,\beta}(\bk,\bk') &=& f_s(\bk,\bk')+ (f_a-f_m)(\bk,\bk')\delta_{\alpha,-\beta} \\\nonumber&+& f_m(\bk,\bk')\delta_{\alpha\beta},\end{aligned}$$ so we are able to identify: $$\begin{aligned}
f_s(\bk,\bk')&=& f_{\alpha,\beta\neq\pm\alpha} (\bk,\bk'), \\ \nonumber
f_m(\bk,\bk')&=& (f_{\alpha,\alpha}- f_{\alpha,\beta\neq\pm\alpha})(\bk,\bk'), \\ \nonumber
f_a(\bk,\bk')&=& (f_{\alpha,\alpha}+ f_{\alpha,-\alpha} - 2f_{\alpha,\beta\neq\pm\alpha})(\bk,\bk').\end{aligned}$$
The effect of interactions on physical quantities appear in terms of $f_{s,m,a}(\bk,\bk')$, averaged over the Fermi surface. The calculation involves then averages of combinations of $I_1(\bk,\bk')$ and $I_2(\bk,\bk')$ over the Fermi surface, what leads to well known integrals. The effective mass reads: $$\begin{aligned}
\frac{m^*}{m} = 1+ \frac{16}{15}\Bigg[ && \!\!\!\!\!\left(\frac{\tilde{g}}{2}\right)^2 \left((5+N) \ln2-5+2N\right) \nonumber\\
&+& 2 \left(\frac{\tilde{g}}{2}\right) \left(\frac{\Delta \tilde{g}}{2N}\right) (7\ln2-1)\nonumber\\
&+& \left(\frac{\Delta \tilde{g}}{2N}\right)^2 N (7\ln2-1)\Bigg]\end{aligned}$$ and the compressibility has a similar form: $$\begin{aligned}
\left(\frac{u^{*}}{u}\right)^2 = 1&+& \frac{N}{2}\left[(N-1)\left(\frac{\tilde{g}}{2}\right)^2 \frac{8}{3}(2\ln2+1) \right.\nonumber\\ &+&
2 \left(\frac{\tilde{g}}{2}\right) \left(\frac{\Delta \tilde{g}}{2N}\right) \frac{16}{3} (\ln2 +2)\\ \nonumber&+&\left. \left(\frac{\Delta \tilde{g}}{2N}\right)^2 \frac{16}{3} \big((2-N)\ln2 + N +1\big)\right].\end{aligned}$$ Here we defined the dimensionless quantities $\tilde{g} = \rho(E_f) g$ and $\Delta\tilde{g} = \rho(E_f) \Delta g$. By inspection one can see that for $N>2$ the corrections to the effective mass and compressibility are always positive, even in the case $\Delta \tilde{g}<0$, leading to an enhancement of both quantities due to interactions. Interestingly enough, the behaviour of the enhancement of the effective mass and compressibility as a function of N varies for different parameter regions, as shown in Fig. \[Mass\]. We focus on the small $\tilde{g}$ and $\Delta \tilde{g}$ parameter region since the calculation is perturbative in these parameters. For $ \tilde{g}\ll \Delta \tilde{g}$ the enhancement decreases as a function of N, while for $\tilde{g} \gtrsim \Delta \tilde{g}$ the enhancement increases as a function of N. For intermediate regimes a non-monotonic dependence on N can be observed. These qualitative features are valid for both $\Delta \tilde{g}>0$ and $\Delta \tilde{g} <0$.
![Renormalization of the effective mass as a function of N. Note the different qualitative behaviour for different parameter regimes. The curves above have different ranges and parameters. The blue circles were plotted for $\tilde{g}=0.005$ and $\Delta \tilde{g} = 0.5$ with the curve ranging up to 1.08; the orange squares used $\tilde{g}=0.1$ and $\Delta \tilde{g} = 0.5$ and range up to 1.125; the green diamonds were plotted for $\tilde{g}=0.1$ and $\Delta \tilde{g} =0.1$, with the curve ranging up to 1.10. The lines are guides to the eye.[]{data-label="Mass"}](MassN.pdf){width="\linewidth"}
Concerning the correction to the magnetic susceptibility, we can look at the Wilson ratio, as a measure of the enhancement of the susceptibility due to exchange interactions: $$\begin{aligned}
W= \frac{\chi_{G,P}^*/m^*}{\chi_{G,P}/m}=\frac{1}{1+ \overline{F_a(\theta)}}.\end{aligned}$$ which takes the explicit form: $$\begin{aligned}
W= 1 &-& \frac{\Delta \tilde{g}}{2N} - 8 \left(\frac{\tilde{g}}{2}\right) \left(\frac{\Delta \tilde{g}}{2N}\right) -\left(\frac{\Delta \tilde{g}}{2N}\right)^2 \frac{8N}{3}(1-\ln 2).\nonumber\\\end{aligned}$$ Considering the Wilson ratio in first order in the interactions, there can be an instability for $\Delta \tilde{g}>0$ at $\Delta \tilde{g}/2N=1$. Note that there is an N-dependence for the instability in first oder, different from the SU(N) case[@Yip]. The larger the N, the harder is for the system to reach the instability, or the larger the interaction needed for the magnetic order to set in.
The second order contribution to the Wilson ratio puts the system closer to an instability, assuming that $\tilde{g}>0$ as we have originally repulsive interactions. This is in contrast with the findings for SU(N), since for $N>2$ the second order corrections always take the system away from a magnetic instability[@Yip].
![Wilson ratio as a function of N. For this plot the parameters used are $\tilde{g}=0.5$ and $\Delta \tilde{g}=\pm 0.5$. []{data-label="Susceptibility"}](SusceptibilityN.pdf){width="\linewidth"}
Here we would like to notice that one cannot benchmark these results with the SU(N) case by directly taking the limit $\Delta \tilde{g}=0$ since we are using a different parametrization (given the different completeness relations for the generators of the different symmetry groups). The benchmarking needs to go one step back, eliminating the parameters $f_a(\theta)$ from the evaluation of the corrections above and taking $\Delta \tilde{g} =0$, in which case the SU(N) results are recovered[@Yip]. We also note that the limit with $N=2$ cannot be directly taken since in this case $\tilde{g}_0$ is the only one scattering channel. Again one needs to go one step back and make $\tilde{g}=0$ in order to recover the SU(2) results.
Discussion and Conclusion {#Con}
=========================
Experimental realization and verification {#ExpRea}
-----------------------------------------
We now discuss what are the candidates for the realization of symplectic symmetry within ultra-cold fermionic systems. We start restricting ourselves to atoms whose scattering properties can be well described by contact interaction in the limit of ultra-cold temperatures, so we neglect atoms with sizeable dipole-dipole interactions[@MaiT]. We also focus on stable or long lived isotopes which can be actually handled in an experimental setup. These restrictions eliminate some of the transition metals, lanthanides and actinides which have large dipole-dipole interactions and elements heavier than Pb. We now go over the different families in the periodic table, exploring what are the possibilities to realize enlarged symmetries. The discussion below is summarized in Table \[TabEl\].
*Alkali Metals*: These have the outer electronic shell with configuration $ns^1$ (here $n$ is the principal quantum number). The total electronic angular momentum is $J=1/2$. As we are interested in atoms which are fermions, we need to look for isotopes with integer nuclear spin. This leaves us with $^2$H, $^6$Li and $^{40}$K, with nuclear spin $I$ equal to 1, 1 and 4, respectively. Due to the hyperfine interaction these combine into a total hyperfine spin $f$ equal to 1/2, 1/2 and 9/2. The first two cases are trivial in the sense that SP(2) is isomorphic to SU(2). In conclusion, $^{40}$K is the promising candidate among the alkali metals.
*Alkaline-Earth Metals*: These have a full outer electronic shell with configuration $ns^2$. In this case $J=0$ and the electronic degrees of freedom are therefore decoupled from the (possibly non-zero) nuclear spin. In order to obtain a fermionic isotope we need a nucleus with half-integer spin. Now we have the following options: $^9$Be, $^{25}$Mg, $^{41,43}$Ca, $^{87}$Sr and $^{135,137}$Ba, with nuclear spins ranging from 3/2 to 9/2, as summarized in Table \[TabEl\]. These are known to realize SU(N) symmetry, so in principle one could realize enlarged symmetries ranging from SU(4) up to SU(10) with alkaline-earth atoms.
*Transition Metals*: One would expect that there would be also some candidates amongst the transition metals. For the family IB, with $ns^1$ electronic configuration, one would need a nucleus with an even number of nucleons, but nature does not provide us with stable isotopes of this kind. The atoms in family IIB have a full electronic shell of s-character, so we look for isotopes with half-integer nuclear spin. There are several: $^{67}$Zn, $^{111,113}$Cd and $^{199,201}$Hg, with nuclear spins ranging from 1/2 to 5/2. Since these have a full electronic shell, they also realize SU(N) symmetry. Other transition metals have large dipole-dipole interactions or do not have stable fermionic isotopes.
*Lanthanides*: Amongst the Lanthanides Yb is one of the few elements with no dipole-dipole interaction due to its complete electronic shell. There are two isotopes which are fermions: $^{171,173}$Yb, with nuclear spin equal to 1/2 and 5/2, respectively. These would realize SU(2) and SU(6) symmetries, respectively.
*Families IIIA-VIIA*: Elements in the families VA-VIIA have a substantial multipolar character, so we are not going to consider them. Elements in the families IIIA have an odd number of electrons, therefore we should look for integer nuclear spin isotopes. The only stable isotope is $^{10}$B with nuclear spin equal to 3. Elements of the families IVA have an even number of electrons, therefore we are interested in isotopes with half-integer spin. There are several isotopes available in nature: $^{13}$C, $^{29}$Si, $^{73}$Ge, $^{115,117}$Sn and $^{207}$Pb with nuclear spins equal to 1/2, with the exception of $^{73}$Ge which has nuclear spin equal to 9/2.
*Noble Gases*: These have a complete electronic shell, so $J=0$ and we should look for isotopes with half-integer nuclear spin. The stable isotopes are $^{3}$He, $^{21}$Ne, $^{83}$Kr, $^{129,131}$Xe, with hyperfine spins ranging between 1/2 and 9/2. These also realize SU(N) symmetry.
From the analysis above we can conclude that nature is quite unfair towards the realization of symplectic symmetry. The most interesting case would be to have an isotope with hyperfine spin $f=3/2$ and electronic angular momentum $J\neq 0$, which would not require fine-tuning of the scattering channels. In this case there are only two interaction channels satisfying $g_0\neq g_2$. Unfortunately there is no such isotope (at least not on its ground state and without substantial dipole-dipole interaction). The only $f=3/2$ cases are within the elements with $J=0$, so they actually realize the larger SU(4) symmetry and not SP(4) symmetry. Note that the discussion in terms of the interaction strengths in different channels, $g_F$, is analogous to the discussion in terms of scattering lengths, $a_F$, since these are related by the identity $\rho(E_f) g_F = 2 k_f a_F/\pi \hbar$, where as before $k_f$ is the Fermi momentum and $\rho(E_f)$ the density of states at the Fermi level.
[ | c | c | c | c | c | c |]{}
---------------
Electronic
Configuration
---------------
& Family & Isotope & Nuclear Spin & Hyperfine Spin & Symmetry\
& & $^9$Be&3/2&3/2&SU(4)\
&& $^{25}$Mg&5/2&5/2&SU(6)\
&& $^{41,43}$Ca&7/2&7/2&SU(8)\
&& $^{87}$Sr&9/2&9/2&SU(10)\
&& $^{135,137}$Ba&3/2&3/2&SU(4)\
& & $^{173}$Yb&5/2&5/2&SU(6)\
& & $^{67}$Zn&5/2&5/2&SU(6)\
&& $^{201}$Hg&5/2&3/2&SU(4)\
& & $^{21}$Ne&3/2&3/2&SU(4)\
&& $^{83}$Kr&9/2&9/2&SU(10)\
&& $^{131}$Xe&3/2&3/2&SU(4)\
& & $^{40}$K&4&9/2&SP(10)$^*$\
&Family IIIA&$^{10}$B &3&5/2&SP(6)$^*$\
& & $^{73}$Ge&9/2&7/2&SP(8)$^*$\
There are elements with larger hyperfine spins which have $J\neq 0$. These are: $^{40}$K, $^{10}$B and $^{73}$Ge, and can realize SP(N) symmetry in case the scattering lengths are fine tuned such that $a_0\neq a_{F>0}$. We focus on $^{40}$K, the only one amogst these that was already taken to ultra-low temperatures. Interestingly, it has a very large hyperfine spin ($f=9/2$) so the associated symmetry is SP(10). Note that the symmetry will be present only if the scattering lengths $a_{F=2,4,6,8}$ are all made equal. To fine tune four parameters in the system looks like a challenge, but in fact $^{40}$K is already very close to naturally satisfy this condition. From Krauser et al.[@Kra], one can see that the scattering lengths for $F>0$ are the same within about $12\%$ ($a_0\sim120$, $a_2=147.83$, $a_4= 161.11$, $a_6=166.00$, $a_8= 168.53$ in units of the Bohr radius). Given the fact that we are naturally close to the fine-tuned point with all $g_{F>0}$ equal, it might be interesting to explore how one could tune the system towards better satisfying this condition, perhaps by the use of Feshbach resonances.
As already discussed in Sec. \[SPN\], for SP(N) systems we cannot simply initially load it with a few of the states in order to realize a smaller SP(n$<$N) symmetry, as can be done in the SU(N) case. If we load the system with one flavour and its complement, it can scatter to another flavour and its component, $\alpha,-\alpha\rightarrow \beta,-\beta$, as sketched in Fig. \[SUNSPN\] e). If that was possible, one could take $^{40}$K and load the system only with the states $\{\pm1/2,\pm3/2\}$, realizing SP(4) symmetry. In this direction, one could think on engineering a way to block the scattering to other states.
Another possibility to realize SP(N) symmetry would be to think on the other way around: it might be possible to detune the $F=0$ channel away from the remaining channels in an isotope which has SU(N) symmetry so we are able to break it down to SP(N) symmetry. In principle this can be achieved by connecting the low lying states with excited states by external fields.
One could also explore the atoms we neglected above, with strong dipole-dipole interactions, by tuning them with Feshbach resonances such that the contact interactions are much stronger than the dipole-dipole interactions. One of the atoms with the strongest dipole-dipole interaction is Dy and there are two fermionic isotopes: $^{161,163}$Dy, both with nuclear spin equal to 5/2. Another atom with significant dipolar character is Er, which has only one stable fermionic isotope, $^{167}$Er. Recently Er and Dy have been shown to display a very dense Feshbach spectrum with signatures of chaotic behaviour [@Mai]. This suggests that one could scan the system as a function of magnetic field in order to find a value which gives the suitable scattering lengths following $a_0\neq a_{F\neq 0}$, in similar fashion to what was done by Lahaye at al.[@Lah]. $^{53}$Cr is another dipolar isotope which was already brought to a degenerate state [@Nay].
In order to verify the presence of SP(N) symmetry one could do more than simply measuring the scattering lengths in different channels. One can track the evolution of the occupation number of each spin component. For the SU(N) case each component is preserved independently, so there should be no changes in $n_\alpha$ over time. For SP(N) the color-magnetizations $m_\alpha=n_\alpha - n_{-\alpha}$ are the conserved quantities, so even though $n_{\alpha}$ can change over time, this specific difference does not. This can be verified by Stern-Gerlach experiments, similar to the one performed by Krauser et al.[@Kra].
Realization of Physics in Higher Dimensions {#highd}
-------------------------------------------
Symmetries dictate the kinds of quantum states, or particle types, that can exist. If we are in isotropic space in d-dimensions, the physics should be invariant under SO(d) transformations, for massive particles with positive definite energy[@Wig; @Wei]. The different types of particles that exist correspond to the different irreducible projective representations of SO(d), or to the distinct irreducible representations of its covering group, the Spin(d) group[@Moo; @Law]. The representations of Spin(d) which are not representations of SO(d) are spinor representations, associated with fermionic particles. Interestingly enough, for low dimensions there is a series of interesting isomorphisms, in particular, Spin(3) $\cong$ SU(2) $\cong$ SP(2) and Spin(5) $\cong$ SP(4)[@Moo; @Law]. These isomorphisms suggest the following correspondence: if we start with a 3-dimensional world, in which case the physics is invariant under SO(3) rotations, we can look for the irreducible representations of SO(3) and we find that they correspond to all integer angular momentum states. If we now look at representations of SU(2), its double-cover, we find extra representations which are related to half-integer spin particles, or fermions. Interestingly enough SP(2) has 3 generators, corresponding to the three different spin components, which couple, correspondingly, to the three different magnetic field components in three dimensional space. In the same fashion, if we are in $d=5$, the double-cover of SO(5) is Spin(5) $\cong$ SP(4). If we are able to realize a system with effective SP(4) symmetry, in particular in the fundamental representation (the smallest faithful representation), we are in fact realizing the analogous of spin-1/2 particles in 3-dimensions, but now in 5- dimensions. SP(4) has $10$ generators, corresponding to 10 different “spin components". Note that in five dimensions the number of magnetic field components is also 10. Defining a generalized magnetic field as the possible antisymmetric pair-wise combinations of the electromagnetic tensor components $F_{ij}$, with $i,j=1,...,d$, we have in total $d(d-1)/2$ magnetic field components.
One could think of these particles confined to three dimensions in the same way as we talk about SU(2) spin-1/2 particles in one- or two-dimensional systems. Even though it is not clear how to manipulate the fictitious magnetic field in higher dimensions one could still measure fluctuations of the system and determine its response functions by the use of the fluctuation-dissipation theorem. This is an interesting direction for future work.
Final Remarks
-------------
In conclusion we have reviewed the problem of interacting fermions with SP(N) symmetry within cold atoms and contrasted its behaviour with the SU(N) scenario. We characterized the main properties of the Fermi Liquid state: effective mass, compressibility and magnetic susceptibility. We find that both the effective mass and inverse compressibility are enhanced in the presence of interactions following SP(N) symmetry. The magnetic susceptibility can be either enhanced or suppressed, depending on the sign of the detuning parameter $\Delta \tilde{g}=\rho(E_f) (g_0-g)$. We conclude discussing what are the possible routes to realize SP(N) symmetry within cold atoms, which apparently always requires some degree of fine-tunning, setting up an interesting challenge for experimentalists. The correspondence of SP(4) systems to physics in 5-dimensions is a fascinating direction for future work.
I thank Ana Maria Rey, Tilman Esslinger, Gordon Baym, Carlos Sa de Melo and Gregory W. Moore for illuminating discussions. I also thank Sungkip Yip for bringing to my attention his work on the SU(N) version and Rémi Desbuquois for very insightful discussions on the possible experimental paths to realize SP(N) symmetry. I also thank Manfred Sigrist and Yehua Liu for carefully reading and commenting on a preliminary version of this manuscript. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293. This work was also supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation.
Generators of the symplectic group {#AppGen}
==================================
The definition of the generators for the symplectic-N generalization in reference [@Fli08] is different from the one we use in this manuscript. Here we define: $$\begin{aligned}
\label{DefGen}
S_{\alpha\beta} = \Psi_\alpha^\dagger \Psi_\beta+ (-1)^{\alpha+\beta} \Psi_{-\beta}^\dagger \Psi_{-\alpha},\end{aligned}$$ with $\alpha$ and $\beta$ ranging from $-f$ to $f$ for $N=2f+1$.
We can show that the above operator form in fact gives a set of generators of SP(N) by looking at the properties of their matrix form in a specific basis. There are $N(N+1)/2$ linearly independent $S_{\alpha\beta}$ which are related to $N \times N$ matrices $M_i$, $i=1,...,N(N+1)/2$, which follow the symplectic condition: $$\begin{aligned}
M_i^T \Omega + \Omega M_i =0,\end{aligned}$$ where $\Omega$ is an antisymmetric matrix.
It can be shown that these are the generators of the symplectic group by using the explicit matrix forms, in the basis $(\Psi_{3/2}, \Psi_{1/2},\Psi_{-1/2},\Psi_{-3/2})$: $$\begin{aligned}
[S_{\alpha\beta}]_{mn} &=& \delta_{m, s-\alpha+1}\delta_{n,s-\beta+1} \\ \nonumber&+& (-1)^{\alpha+\beta} \delta_{m,s+\beta+1}\delta_{n,s+\alpha+1},\end{aligned}$$ and the antisymmetric form $\Omega=AntiDiag[1,-1,1,-1,...]$: $$\begin{aligned}
[\Omega]_{mn} = (-1)^m \delta_{m, 2s-n+2}.\end{aligned}$$ Verifying the symplectic condition: $$\begin{aligned}
&&[S_{\alpha\beta}^T]_{mn} [\Omega]_{np} + [\Omega]_{mn} [S_{\alpha\beta}]_{np} = 0,\end{aligned}$$ using the fact that $[S_{\alpha\beta}^T]_{mn}=[S_{\alpha\beta}]_{nm}$ and the explicit matrix forms given above we find: $$\begin{aligned}
&& (-1)^{n+1} \delta_{n,s-\alpha+1} \delta_{m, s-\beta+1} \delta_{n,2s-p+2}\\ \nonumber
&+&
(-1)^{\alpha+\beta+n+1} \delta_{n, s+\beta+1}\delta_{m,s+\alpha+1} \delta_{n,2s-p+2} \\ \nonumber
&+& (-1)^{m+1} \delta_{m, 2s-n+2} \delta_{n, s-\alpha+1}\delta_{p, s-\beta+1}\\ \nonumber
&+& (-1)^{\alpha+\beta + m +1} \delta_{m, 2s-n+2} \delta_{n,s+\beta+1}\delta_{p,s+\alpha+1}=0,\end{aligned}$$ which is zero if we are working with fermions and $s$, $\alpha$ and $\beta$ are half-integers.
Here we note that we would get the same matrix form for the generators as in Flint et al. [@Fli08] if we use a different basis: $(\Psi_{3/2}, \Psi_{-1/2},\Psi_{1/2},\Psi_{-3/2})$, in which case the antisymmetric matrix has a different form: $\tilde{\Omega}=AntiDiag[1,1,1,...,-1,-1,-1,...]$. In Flint et al. [@Fli08] the basis used is $(\Psi_{3/2}, \Psi_{1/2},\Psi_{-1/2},\Psi_{-3/2})$ with the same measure.
Note that the generators as presented above are traceless but not properly orthonormalized. For the development of the FL theory it is convenient to work with generators which are orthonormal. This can be achieved by rescaling these generators as $\tilde{S}_{\alpha\beta}=S_{\alpha\beta}/\sqrt{2}$ and combining them as follows:
A\) Generators of the form $\tilde{S}_{\alpha\alpha}$, with both indexes equal. Given the relation $\tilde{S}_{\alpha\alpha}=-\tilde{S}_{-\alpha-\alpha}$, we need to consider only the generators with positive indexes. There are $N/2$ of those and these are already properly orthonormalized such that $Tr[\tilde{S}_{\alpha\alpha} \tilde{S}_{\beta\beta}]=\delta_{\alpha\beta}$.
B\) There are also $N$ linearly independent generators with opposite indexes as $\tilde{S}_{\alpha-\alpha}$. To guarantee orthonormality these should be reorganized as $(\tilde{S}_{\alpha-\alpha} + \tilde{S}_{-\alpha\alpha})/2$ and $(\tilde{S}_{\alpha-\alpha} - \tilde{S}_{-\alpha\alpha})/2i$.
C\) The missing generators have the form $\tilde{S}_{\alpha\beta}$ with $\alpha\neq \pm \beta$. Given again the relation $\tilde{S}_{\alpha\beta}=(-1)^{\alpha+\beta}\tilde{S}_{-\beta-\alpha}$, if both indexes are positive there are $\frac{N}{2}\left(\frac{N}{2}-1\right)$, and we do not need to consider the generators with both negative indexes. If one index is positive and the other negative, it is linearly dependent of another generator of the same form, so again there are $\frac{N}{2}\left(\frac{N}{2}-1\right)$. This totals $N\left(\frac{N}{2}-1\right)$ generators of the form $\tilde{S}_{\alpha\beta}$ with $\alpha\neq \pm \beta$. These should be combined in such a way that they are all summed but one, which is subtracted, what leads to $N\left(\frac{N}{2}-1\right)$ independent combinations.
Note that the total number of generators is still $N(N+1)/2$. In the main text we refer to this set of properly orthonormalized generators as $\Gamma^A$, such that $Tr[\Gamma^A\Gamma^B]=\delta_{AB}$, without writing them explicitly.
Parametrization of the interaction function for the SP(N) FL {#AppSym}
============================================================
In order to understand the parametrization of the interaction function, we can look at the total change in energy due to a change in the occupation number $\delta n_{\alpha\beta}(\bk)$: $$\begin{aligned}
\delta E &=&\sum_{\bk, \alpha\beta} \delta\epsilon_{\alpha\beta}(\bk) \delta n_{\beta\alpha}(\bk)\\ \nonumber
&=& \sum_{\bk, \bk',\alpha\beta\mu\nu} f_{\alpha\mu,\beta\nu}(\bk,\bk' )\delta n_{\nu\mu}(\bk') \delta n_{\beta\alpha}(\bk).\end{aligned}$$ We can now go back to the operator form in order to analyze the symmetries more explicitly, identifying $n_{\alpha\beta}(\bk) = \langle\Psi_{\bk\beta}^\dagger\Psi_{\bk\alpha}\rangle$, we have that the change in the total energy has the form $$\begin{aligned}
\sim \Psi_{\bk\alpha}^\dagger \Psi_{\bk' \mu}^\dagger f_{\alpha\mu,\beta\nu}(\bk,\bk' ) \Psi_{\bk\beta} \Psi_{\bk' \nu},\end{aligned}$$ before taking the averages, with the sum over repeated indexes implied in the equation above and in the following. If there is a unitary symmetry group whose transformations are denoted by $U$, we can rotate the operators in spin space such that this form is invariant. We can write $\Psi_{\alpha} = \sum_a U_{\alpha a} \Psi_a$ and rewrite the equation above as: $$\begin{aligned}
\sim \Psi_{\bk a}^\dagger U_{a\alpha}^\dagger \Psi_{\bk' c}^\dagger U_{c\gamma}^\dagger f_{\alpha\gamma,\beta\delta}(\bk,\bk' ) U_{\beta b}\Psi_{\bk b} U_{\delta d}\Psi_{\bk' d},\end{aligned}$$ so in order for this term to be invariant: $$\begin{aligned}
U_{a\alpha}^\dagger U_{c\gamma}^\dagger f_{\alpha\gamma,\beta\delta}(\bk,\bk' ) U_{\beta b} U_{\delta d} = f_{ac,bd}(\bk,\bk' ).\end{aligned}$$ If the transformation is unitary $U U^\dagger= I$, the identity above can be satisfied if $f_{\alpha\gamma,\beta\delta} = \delta_{\alpha\beta}\delta_{\gamma\delta}$: $$\begin{aligned}
U_{a\alpha}^\dagger U_{c\gamma}^\dagger( \delta_{\alpha\beta}\delta_{\gamma\delta}) U_{\beta b} U_{\delta d} &=& U_{a\alpha}^\dagger U_{c\gamma}^\dagger U_{\alpha b} U_{\gamma d}\\ \nonumber
&=& (U^\dagger U )_{ab} (U^\dagger U )_{cd} \\ \nonumber &=& \delta_{ab}\delta_{cd},\end{aligned}$$ and also for $f_{\alpha\gamma,\beta\delta} = \delta_{\alpha\delta}\delta_{\gamma\beta}$: $$\begin{aligned}
U_{a\alpha}^\dagger U_{c\gamma}^\dagger( \delta_{\alpha\delta}\delta_{\gamma\beta}) U_{\beta b} U_{\delta d} &=& U_{a\alpha}^\dagger U_{c\gamma}^\dagger U_{\gamma b} U_{\alpha d}\\ \nonumber
&=& (U^\dagger U )_{ad} (U^\dagger U )_{cb} \\ \nonumber &=& \delta_{ad}\delta_{cb}.\end{aligned}$$ For the SU(N) case these are the only possible constructions based on the unitarity of the transformations.
In the SP(N) case, given the symplectic condition for the transformations $\epsilon U^{T} = U^\dagger\epsilon$, there is one more possibility $f_{\alpha\gamma,\beta\delta} = \epsilon_{\alpha\gamma}\epsilon_{\beta\delta}$: $$\begin{aligned}
U_{a\alpha}^\dagger U_{c\gamma}^\dagger(\epsilon_{\alpha\gamma}\epsilon_{\beta\delta}) U_{\beta b} U_{\delta d} &=& U_{a\alpha}^\dagger\epsilon_{\alpha\gamma} U_{\gamma c}^* U_{b\beta }^T \epsilon_{\beta\delta} U_{\delta d} \\ \nonumber
&=& \epsilon_{a\alpha}U_{\alpha\gamma}^T U_{\gamma c}^* \epsilon_{b\beta}U_{ \beta\delta}^\dagger U_{\delta d}\\ \nonumber
&=& \epsilon_{a\alpha}\delta_{\alpha c} \epsilon_{b\beta}\delta_{\beta d}\\ \nonumber
&=& \epsilon_{ac}\epsilon_{bd}.\end{aligned}$$
Some properties of the Clebsch-Gordan coefficients {#AppCGC}
==================================================
Below are some useful properties of the CGC $ \langle j_1 m_1, j_2 m_2 | J m \rangle = \langle J m | j_1 m_1, j_2 m_2 \rangle $ used in the main text: $$\begin{aligned}
\label{CGC1}
\langle j_1 m_1, j_2 m_2 | J m \rangle = (-1 )^{j_1+j_2-J } \langle j_1 -m_1, j_2 -m_2 | J -m \rangle,\end{aligned}$$ $$\begin{aligned}
\label{CGC2}
\langle j_1 m_1, j_2 m_2 | J m \rangle = (-1 )^{j_1+j_2-J } \langle j_2 m_2, j_1 m_1 | J m \rangle,\end{aligned}$$ $$\begin{aligned}
\label{CGC0}
\langle j_1 m_1, j_2 m_2 | 0 0 \rangle = \delta_{j_1,j_2} \delta_{m_1-m_2}\frac{(-1 )^{j_1-m_1 }}{\sqrt{2 j_1+1}},\end{aligned}$$ and the orthogonality relations: $$\begin{aligned}
\label{CGCOrt1}
\sum_{F,M} \langle f \alpha, f \beta | F M \rangle \langle F M | f \mu, f \nu\rangle = \langle f \alpha, f \beta | f \mu, f \nu\rangle =\delta_{\alpha \mu}\delta_{\beta \nu},\end{aligned}$$ $$\begin{aligned}
\label{CGCOrt2}
\sum_{\alpha,\beta} \langle F M | f \alpha, f \beta \rangle \langle f \alpha, f \beta | F'M'\rangle = \langle F M | F'M'\rangle =\delta_{F, F'}\delta_{M, M}.\end{aligned}$$
[99]{}
A. Georges, *Lectures given at the Enrico Fermi Summer School on Ultracold Fermi Gases*, organized by M. Inguscio, W. Ketterle and C. Salomon, Varenna, Italy (2006).
P. B. Blakie and A. Bezett, Phys. Rev. A 71, 033616 (2005).
W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 89, 220407 (2002).
C. Wu, Physics 3, 92 (2010).
M. A. Cazalila and A. M. Rey, Rep. Prog. Phys. 77, 124401 (2014).
A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J.Ye, P. Zoller, E. Demler, M. D. Lukin and A. M. Rey, Nature Physics 6, 289 (2010).
G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J. Catani, C. Sias, M. Inguscio and Leonardo Fallani, Nature Physics 10, 198 (2014).
S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki, T. Tsujimoto, R. Murakami and Y. Takahashi, Phys. Rev. Lett. 105, 1 (2010).
S. K. Yip, B. L. Huang and J. S. Kao, Phys. Rev. A 89, 043610 (2014).
M. A. Cazalilla, A. F. Ho and M. Ueda, New J. of Phys. 11, 103033 (2009).
S. Chen, C. Wu, S.-C. Zhang and Y. Wang, Phys. Rev. B 72, 214428 (2005).
K. Kataoka, S. Hattori and I. Ichinose, Phys/ Rev. B 83, 174449 (2011).
M. Hermele, V. Gurarie, and A. M. Rey, Phys. Rev. Lett. 103, 135301 (2009).
C. Xu, Phys. Rev. B 81, 144431 (2010).
X.W. Guan, M. T. Batchelor, C. Lee and H.-Q. Zhou, Phys. Rev. Lett. 100, 200401 (2008).
T. L. Ho and S. K. Yip, Phys. Rev. Lett. 82, 247 (1999).
S. Capponi, G. Roux, P. Azaria, E. Boulat, and P. Lecheminant, Phys. Rev. B 75, 100503(R) (2007).
H. H. Tu, G. M. Zhang and L. Yu, Phys. Rev. B 74, 144404 (2006); Phys. Rev. B 76, 014438 (2007).
C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. 92, 170403 (2004).
E. Szirmai and M. Lewenstein, Europhys. Lett. 93 66005 (2011).
K. Yang, D. Das Sarma and A. H. MacDonald, Phys. Rev. B 74, 075423 (2006).
D. P. Arovas, A. Karlhede and D. Lilliehook, Phys. Rev. B 59, 13147 (1998).
C. Wu, J.-P. Hu and S.-C. Zhang, Phys. Rev. Lett. 91, 186402 (2003).
C. Wu and S.-C. Zhang, Phys. Rev. B 71, 155115 (2005).
Y. Jiang, J. Cao and Y. Wang, J. Phys. A: Math. Theor. 44, 345001 (2011).
J. S. Krauser et al. Nature Physics 8, 813 (2012).
E. M. Lifshitz and L. P. Pitaevskii, *Statistical Physics - Part 2*, Pergamon Press, Oxford (1980).
T. D. Lee, K. Huang and C. N. Yang, Phys. Rev. 106, 1135 (1957).
T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998).
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991).
R. Flint and P. Coleman, Phys. Rev. B 79, 014424 (2009).
R. Flint, M. Dzero and P. Coleman, Nature Physics 4, 643 (2008).
R. Flint and P. Coleman, Physical Review B 86, 184508 (2012).
C. Wu, Mod. Phys. Lett. B 20, 1707 (2006).
A. K. Kolezhuk and T. Vekua, Phys. Rev. B 83, 014418 (2011).
C. Xu, Phys. Rev. B 80, 184407 (2009).
T. Maier, PhD Thesis, Stuttgart University (2015).
T. Maier, H. Kadau, M. Schmitt, M. Wenzel, I. Ferrier-Barbut, T. Pfau, A. Frisch, S. Baier, K. Aikawa, L. Chomaz, M. J. Mark, F. Ferlaino, C. Makrides, E. Tiesinga, A. Petrov, S. Kotochigova, Phys. Rev. X 5, 041029 (2015).
T. Lahaye, T. Koch, B. Frohlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi and Tilman Pfau, Nature 448, 672 (2007).
B. Naylor, A. Reigue, E. Mare?chal, O. Gorceix, B. Laburthe-Tolra, and L. Vernac, Phys. Rev. A 91, 011603 (2015).
E. P. Wigner, *Group Theory and its Application to the Quantum Mechanics of Atomic Spectra*, Academic Press, New York (1959).
S. Weinberg, *The Quantum Theory of Fields*, Chapter 2, Cambridge University Press, Cambridge (1995).
G. Moore, *Lecture Notes on Group Theory, Chapter 11: The Spin Group* (2009); *Lecture Notes on Quantum Symmetries and Compatible Hamiltonians, Chapter 17: Pin and Spin* (2013).
H. B. Lawson Jr. and M-L Michelsohn, *Spin Geometry*, Princeton University Press, Princeton (1989).
|
---
abstract: 'In this paper we recall a simple formulation of the stationary electrovacuum theory in terms of the famous complex Ernst potentials, a pair of functions which allows one to generate new exact solutions from known ones by means of the so–called nonlinear hidden symmetries of Lie–Bäcklund type. This formalism turned out to be very useful to perform a complete classification of all 4D solutions which present two spacetime symmetries or possess two Killing vectors. Curiously enough, the Ernst formalism can be extended and applied to stationary General Relativity as well as the effective heterotic string theory reduced down to three spatial dimensions by means of a (real) matrix generalization of the Ernst potentials. Thus, in this theory one can also make use of nonlinear matrix hidden symmetries in order to generate new exact solutions from seed ones. Due to the explicit independence of the matrix Ernst potential formalism of the original theory (prior to dimensional reduction) on the dimension $D$, in the case when the theory initially has $D\ge 5$, one can generate new solutions like [*charged*]{} black holes, black rings and black Saturns, among others, starting from uncharged field configurations.'
---
-10mm [H]{} §[[S]{}]{} Ł[[L]{}]{}
[Nonlinear hidden symmetries in General Relativity and String\
0.25cm Theory: a matrix generalization of the Ernst potentials]{}
[****]{}
[****]{}
0.1cm $^\ddagger$Facultad de Ingeniería Eléctrica, Universidad Michoacana de San Nicolás de Hidalgo.\
Edificio B, Ciudad Universitaria, C.P. 58040, Morelia, Michoacán, México.\
$^\natural$Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo.\
Edificio C–3, Ciudad Universitaria, C.P. 58040, Morelia, Michoacán, México.\
$^*$School of Physics, Nuclear and Elementary Particle Physics Department,\
Aristotle University of Thessaloniki (AUTH), 54124 Thessaloniki, Greece.
Introduction
============
It is well known that both, the stationary action and the coupled field equations of the Einstein–Maxwell theory can be formulated in terms of a pair of very simple complex functions that were called [*Ernst potentials*]{} after their inventor [@e; @iw]. In the language of these potentials, the black holes of Schwarzschild and Kerr, Reissner–Nordströn and Kerr–Newmann adopt a very simple form, as well as some cosmological models, among other exact solutions [@e; @kramer]. Indeed, this formalism facilitates the general study of the symmetries of the theory and, hence, the construction of new exact solutions by means of very well-known solution–generating techniques (see, for instance, [@kinnersleyetal]).
It turns out that the Ernst formalism can be generalized to low–energy effective string theories and General Relativity with extra dimensions in terms of matrix potentials instead of complex functions (see [@kramer],[@ggk]–[@belver], for instance). This matrix formalism also enables one to study the complete symmetry group of the underlying theory and to apply generalized solution–generating techniques with matrix charges involved [@CSEMDA]–[@hk5]. In particular, this matrix formalism can be applied to the classification and construction of charged black holes, black rings and black Saturns in 5D and multiple black rings in $D\ge 6$ in the framework of such theories [@emparanetal]–[@elvangfigueras].
In this paper we first recall the derivation of the Ernst potentials for the stationary Einstein–Maxwell theory and write both field equations and the effective action in their language. We further refer to the stationary formulation of the low–energy heterotic string theory, and the corresponding field equations, in terms of a pair of matrix Ernst potentials that closely resembles the formulation of the stationary theory of electrovacuum in the language of the complex Ernst potentials. A fact that, in principle, allows one to generalize all the so far obtained results in the stationary Einstein–Maxwell theory to the realm of the stationary heterotic string theory.
As an extra bonus, within the framework of higher dimensional General Relativity and the low energy limit of heterotic string theory, the matrix Ernst potentials can be used to classify and construct exact solutions that corresponds to higher dimensional objects like black holes, black rings, black Saturns and multiple black rings. A sketch of how this program can be performed is given at the end of this paper.
Ernst potentials in the stationary Einstein–Maxwell theory
==========================================================
In this section we briefly review the derivation of the Ernst potentials within the framework of the stationary Einstein–Maxwell theory basically following the work given by [@iw].
Let us consider the 4D action of the electrovacuum theory §\_[EM]{}= d\^4xG\^(\^4R-F\^2\_[mn]{}), where $G$ is the determinant of the metric $G_{mn}$, $F_{mn}=\pa_mA_{n} -
\pa_nA_{m}$, $A_{m}$ is the gauge field, $^4\!R$ is the scalar curvature in 4D and $m,n,=0,1,2,3;$ $\mu, \nu=1,2,3.$
Consider now the stationary ansatz for the metric ds\^2=G\_[mn]{}dx\^mdx\^n=-f(dt+\_dx\^)\^2+ f\^[-1]{}\_dx\^dx\^, where $f$, $\gamma_{\mu\nu}$ and $\omega_{\mu}$ are quantities independent on $t$.
Indices of spatial coordinates are raised and lowered with the aid of the metric tensor $\gamma_{\mu\nu}$ and its inverse $\gamma^{\mu\nu}$, unless otherwise indicated through a left superindex $^{(0)}$.
Thus, if $F_{mn}$ is a covariant tensor, then $$F^{\a\beta}=\gamma^{\a\mu}\gamma^{\beta\nu}F_{\mu\nu}\quad \mbox{\rm and} \quad
^{(0)}\!F^{\a\beta}= g^{\a m}g^{\beta n}F_{mn}.$$
The three–dimensional vector $\omega_{\mu}$ can always be dualized through an invariant torsion vector in the following form f\^[-2]{}\^=-\^[-1/2]{}\^ \_\_ \[dual\] or, equivalently, f\^[-2]{}=-, by making use of the three–dimensional vectorial calculus which employs $\gamma_{\mu\nu}dx^{\mu}dx^{\nu}$ as background metric.
Let us now consider a stationary electromagnetic field $F_{mn}=\pa_m
A_n-\pa_n A_m$ with the given metric.
The stationarity condition $\pa_0 A_m=0$ for the electric field implies F\_[0]{}=-\_A\_0, \[elec\] while the sourceless Maxwell equations \_=0 \[max\] in the case when $m=\mu$ provide us with the magnetic components \^[(0)]{}F\^=f\^[-1/2]{}\^\_, \[mag\] in terms of the scalar magnetic potential $\psi$.
It turns out that all the remaining components can be expressed as functions of these six magnitudes; for instance, \^[(0)]{}F\^[0]{}=\_\^[(0)]{}F\^+\^F\_[0]{}, \[id\] is an identity that is directly inferred from the stationary metric.
By substituting the relations (\[id\]), (\[mag\]), (\[elec\]) and (\[dual\]) in the Maxwell equations (\[max\]) with $m=0$ one gets (f\^[-1]{}A\_0)=-f\^[-2]{}. \[m=0\]
By rewriting $F_{\mu\nu}\equiv\pa_{\mu}A_{\nu}-\pa_{\nu}A_{\mu}$ with the aid of the relations (\[elec\]) and (\[mag\]), and making use of the expression for the cyclic identity $\epsilon^{\mu\nu\rho}\pa_{\rho}F_{\mu\nu}=0,$ one obtains (f\^[-1]{})=f\^[-2]{}A\_0. \[idcic\]
Now one is able to introduce the scalar complex potential =A\_0+i, \[peme\] which is precisely the electromagnetic Ernst potential.
By combining (\[m=0\]) and (\[idcic\]) one obtains a single complex equation (f\^[-1]{})=if\^[-2]{}. \[ecmax\]
Thus, in this way we have reduced the stationary Maxwell equations to a single equation in terms of the complex electromagnetic Ernst potential.
On the other side, within the framework of the Einstein equations for the gravitational field, it turns out convenient to express the Ricci tensor R\_[mn]{}= \_m\^a\_[na]{}-\_a\^a\_[mn]{}+\^a\_[bm]{}\^b\_[an]{}- \^a\_[ba]{}\^b\_[mn]{} in terms of a complex three–dimensional vector $\vec G$ defined by 2fG=f+i\[2fg\] for the general case of the stationary metric.
In this way we can obtain the following relations -f\^[-2]{}R\_[00]{}=G+(G\^\*-G)G, \[r00\] -2if\^[-2]{}\^[(0)]{}R\_0\^=\^[-1/2]{}\^ (\_G\_+G\_G\^\*\_), \[r0mu\] f\^[-2]{}(\_\_\^[(0)]{}R\^- \_R\_[00]{})=R\_() +G\_G\^\*\_+G\^\*\_G\_, \[rmn\] where $R_{\rho\sigma}(\gamma)$ stands for the Ricci tensor calculated through the three–dimensional metric $\gamma_{\mu\nu}dx^{\mu}dx^{\nu}$.
Thus, from the above obtained formulas, for the energy–momentum tensor of the electromagnetic field -4T\_[mn]{}=g\^[ab]{}F\_[ma]{}F\_[nb]{}-g\_[mn]{}F\_[ab]{}F\^[ab]{} one gets the following relations F\_[mn]{}F\^[mn]{}=()\^2- (A\_0)\^2, 8f\^[-1]{}T\_[00]{}=()\^2+(A\_0)\^2, \[t00\] 4f\^[-1]{}\^[(0)]{}T\^\_0=\^[-1/2]{}\^ (\_)(\_A\_0), \[t0m\] -4f\^[-1]{}\^[(0)]{}T\^= (\^)(\^)+ (\^A\_0)(\^A\_0) -\^ , \[tmn\] where $\pa^{\mu}=\gamma^{\mu\nu}\pa_{\nu}$.
By making use of the Einstein equations R\_[mn]{}=-8T\_[mn]{}, from the relations (\[r0mu\]) and (\[t0m\]) one obtains =-4A\_0= i(\^\*-\^\*).
In this way, the following equation +i(\^\*-\^\*)=\[xi\] defines the scalar potential $\chi$ up to an additive constant.
Now let us define the complex scalar potential E=f-\^\*+i, \[gep\] called gravitational Ernst potential.
This potential allows one to obtain, from the relations (\[2fg\]) and (\[xi\]), the following equality fG=E+\^\*. \[fG\]
By substituting (\[fG\]) in the gravitational field equations (\[r00\]) and (\[t00\]), and making use of the Maxwell equations (\[ecmax\]), we obtain a single equation f\^2E=(E+2\^\*)E; \[gp\] on the other hand, the relation (\[ecmax\]) can be expressed in the following way: f\^2=(E+2\^\*). \[emp\]
It is evident that from the definition (\[gep\]), one can obtain the following expression for the function $f:$ f=(E+E\^\*)+\^\*. \[f\]
Thus, relations (\[gp\]) and (\[emp\]) are the well–known differential Ernst equations for the stationary electrovacuum.
Finally, the gravitational field equations (\[rmn\]) and (\[tmn\]) reduce to the following expression -f\^2R\_=E,\_[(]{}E\^\*\_[,)]{}+E,\_[(]{}\^\*\_[,)]{}+\^\*E\^\*,\_[(]{}\_[,)]{}- (E+E\^\*),\_[(]{}\^\*\_[,)]{}, \[einstein\] where the symmetrization of indices are defined in the following form 2E,\_[(]{}E\^\*\_[,)]{}(\_E)(\_E\^\*)+ (\_E)(\_E\^\*).
In this way, the field equations for the Ernst potentials (\[gp\]) and (\[emp\]), together with the Einstein equations (\[einstein\]), determine the dynamics of the field system of the stationary Einstein–Maxwell theory.
This system of self–consistent second order differential equations, despite their apparent simplicity, has no general solution at the moment. Only particular solutions are known in the literature and it is of great relevance to obtain new solutions possessing a coherent and consistent physical interpretation. It is worth noticing that precisely at this point is where the solution–generating techniques (which make use of nonlinear hidden symmetries to construct new solutions starting from seed ones) can be of great help towards this aim.
Effective action of the stationary EM theory and Ernst potentials
-----------------------------------------------------------------
Now let us express the effective action of the stationary Einstein–Maxwell theory from which one can derive both the Einstein equations (\[einstein\]), and the Ernst equations (\[gp\]) and (\[emp\]) by the variational method.
By redefining the electromagnetic Ernst potential as follows F, the effective stationary action of the Einstein–Maxwell theory adopts the following form \^4§\_[EM]{}= d\^3xg\^(-\^3R+\^3Ł\_[EM]{}), where the matter Lagrangian $^3\!\L_{EM}$ is given by \^3Ł\_[EM]{}= |E+F\^\*F|\^2- |F|, where now $f=\frac{1}{2}(E+E^*+FF^*)$. It is a straightforward exercise to vary this action and obtain the above quoted Einstein and Ernst equations.
Low energy effective action of heterotic string and matrix Ernst potentials
===========================================================================
The effective action of the low–energy limit of the heterotic string at tree level takes into account just the massless modes of the theory and possesses the form [@ms; @s] §=dxG\^e\^[-]{}(R+ \_[;M]{}\^[(D);M]{}.- .H\_[MNP]{}H\^[(D)MNP]{}- F\^[(D)I]{}\_[MN]{}F\^[(D)IMN]{}), \[accion\] where F\^[(D)I]{}\_[MN]{}=\_MA\^[(D)I]{}\_N-\_NA\^[(D)I]{}\_M, I=1,2,...,n; H\_[MNP]{}=\_MB\_[NP]{}-A\^[(D)I]{}\_MF\^[(D)I]{}\_[NP]{}+ Here $G\D_{MN}$ is the metric, $B\D_{MN}$ is the anti–symmetric Kalb–Ramond tensor field, $\p\D$ is the dilaton and $A^{(D)I}_M$ is a set of $U(1)$ vector fields ($I=1,\,2,\,...,n$). $D$ is the dimensionality of the spacetime and $M,N,P=1,2,3,...,10$. In the consistent critical case (where the quantum theory is free of anomalies) $D=10$ and $n=16$, but we shall leave these parameters arbitrary in our analysis for the sake of generality.
By following Maharana and Schwarz [@ms], and Sen [@s], we further perform the dimensional reduction of this model on a $D-3=d$–torus. Thus, the resulting three–dimensional, stationary theory possesses the $SO(d+1,d+1+n)$ symmetry group and describes gravity in terms of the metric tensor g\_=e\^[-2]{}(G\_- G\_[p+3,]{}G\_[q+3,]{}G\^[pq]{}), \[g3D\] where the subscripts $p,q=1,2,...,d$; coupled to the following set of three–dimensional fields:
a\) scalar fields G=(G\_[pq]{}= G\_[p+3,q+3]{}),B=( B\_[pq]{}=B\_[p+3,q+3]{}), A=( A\^I\_p=A\^[(D)I]{}\_[p+3]{}) ,=-G|. \[scalars\]
b\) antisymmetric tensor field of second rank B\_=B\_-4B\_[pq]{}A\^p\_A\^q\_- 2(A\^p\_A\^[p+d]{}\_-A\^p\_A\^[p+d]{}\_), (hereafter we shall set $B_{\mu\nu}=0$ in order to remove the effective three–dimensional cosmological constant from our consideration).
c\) vector fields $A^{(a)}_{\mu}=
\left((A_1)^p_{\mu},(A_2)^{p+d}_{\mu},(A_3)^{2d+I}_{\mu}\right)$ ($a=1,...,2d+n$) (A\_1)\^p\_=G\^[pq]{}G\_[q+3,]{},(A\_3)\^[I+2d]{}\_=-A\^[(D)I]{}\_+A\^I\_qA\^q\_,(A\_2)\^[p+d]{}\_=B\_[p+3,]{}-B\_[pq]{}A\^q\_+ A\^I\_[p]{}A\^[I+2d]{}\_.
In three dimensions all vector fields $A^{(a)}_{\mu},$ can be dualized on–shell with the aid of the pseudoscalar potentials $u,$ $v$ and $s$ in the following form: $$\begin{aligned}
\nabla\times\overrightarrow{A_1}&=&\frac{1}{2}e^{2\p}G^{-1}
\left(\nabla u+(B+\frac{1}{2}AA^T)\nabla v+A\nabla s\right),
\nonumber \\
\nabla\times\overrightarrow{A_3}&=&\frac{1}{2}e^{2\p} (\nabla
s+A^T\nabla v)+A^T\nabla\times\overrightarrow{A_1}, \nonumber
\\
\nabla\times\overrightarrow{A_2}&=&\frac{1}{2}e^{2\p}G\nabla v-
(B+\frac{1}{2}AA^T)\nabla\times\overrightarrow{A_1}+
A\nabla\times\overrightarrow{A_3}.\end{aligned}$$ Thus, the resulting effective three–dimensional theory describes the scalars $G$, $B$, $A$ and $\p$ and the pseudoscalars $u$, $v$ and $s$ coupled to the metric $g_{\mu\nu}$.
We further define the so–called [*matrix*]{} Ernst potentials (MEP) from all these scalar and pseudoscalar potentials in order to express the low–energy effective action of the heterotic string in a similar form to the formulation of the stationary Einstein–Maxwell theory in terms of the complex Ernst potentials [@hk3]: = ( -e\^[-2]{}+v\^TXv+v\^TAs+s\^Ts&v\^TX-u\^T Xv+u+As&X ) =( s\^T+v\^TA A ), \[XA\]where $X=G+B+\frac{1}{2}AA^T$. These potentials are of dimensions $(d+1) \times (d+1)$ and $(d+1) \times
n,$ respectively.
The physical meaning of their components are as follows: The relevant information about the gravitational field is encoded in the potential $X$, while its rotational nature is parameterized by the pseudoscalar $u$; $\p$ is the dilatonic field; $v$ is related to the multi–dimensional components of the antisymmetric tensor field of Kalb–Ramond. Finally, $A$ and $s$ represent electric and magnetic potentials.
Stationary effective action of heterotic string and field equations in the language of MEP
------------------------------------------------------------------------------------------
In terms of MEP the effective three–dimensional theory adopts the form [@hk3]: \^3§=d\^3x g\^{-\^3R+\^3Ł\_[\_[HS]{}]{}}, \[ES3D\]where the matter Lagrangian is given by Ł\_[\_[HS]{}]{}= [Tr]{}, \[matter3D\]$^3\!R$ is the three–dimensional curvature scalar and the matrix potential $\X$ is defined by $\X=\G+\B+\frac{1}{2}\A\A^T$.
The symmetric part of the potential is given by the matrix $\G=\frac{1}{2}\left(\X+\X^T-\A\A^T\right)$ and the antisymmetric one by $\B=\frac{1}{2}\left(\X-\X^T\right);$ these matrices are parameterized as follows: = ( -e\^[-2]{}+v\^TGv&v\^TG Gv&G ) =( 0&v\^TB-u\^T Bv+u&B ).
By making use of the conventional method of variations, from the effective action (\[ES3D\]) one obtains both the [*Einstein equations*]{} \^3R\_= [Tr]{}, as well as the [*Ernst equations*]{} for the potentials $\X$ and $\A$ which represent the matter sector of the theory: \^2-2(-\^T)(+\^T-\^T)\^[-1]{}=0, \^2-2(-\^T)(+\^T-\^T)\^[-1]{}=0, as a matrix version of the equations of the stationary Einstein–Maxwell theory.
As we have pointed out above, these differential equations are not so simple to solve in a closed form. However, one can make use of the similarity which exists with respect to the equations of the stationary Einstein–Maxwell theory in order to guess and write down the solutions in a direct way or to perform nonlinear symmetries to generate new exact solutions from known ones (for some examples see [@HSsolns]).
Heterotic string [*vs.*]{} Einstein–Maxwell
===========================================
Thus, it has been shown that there exists a close relation between the stationary effective actions of the heterotic string and the Einstein–Maxwell theory: -E, F, \[mapa\] .
One can realize that the relation (\[mapa\]) allows us to generalize in a straightforward way the results obtained within the framework of the Einstein–Maxwell theory to the realm of the heterotic string (where a suitable physical interpretation will be needed since more fields are involved) by making use of the MEP formalism. Actually, the four–dimensional Einstein–Maxwell theory, being reduced to three dimensions, can be written as a special case of the MEP formalism with some peculiarities in terms of the complex Ernst potentials $E$ and $F$ [@hk5].
Let us rewrite them in a less conventional form -\_[\_[EM]{}]{}=[Re]{}E+\_2[Im]{}E, \_[\_[EM]{}]{}=[Re]{}F+\_2[Im]{}F, \_2= ( 0&-1 1&0 ). \[xa\]We can treat these matrices as the matrix Ernst potentials (\[XA\]) of the $D=4$ theory (\[accion\]) with $\phi^{(4)}=B^{(4)}_{MN}=0$. Then we conclude that we need two Abelian gauge fields $n=2$ and that they should satisfy the following constraint s\^1=A\^2=[Re]{}F, -s\^2=A\^1=[Im]{}F. Note, that $s^I$ $(I=1,2)$ describe the magnetic potentials, whereas $A^I$ are the electric ones. Thus, both Maxwell fields arising in the framework of the representation (\[XA\])–(\[matter3D\]) and (\[xa\]) turn out to be mutually conjugated (i.e. $F_{MN}^{(4)2}=\tilde F_{MN}^{(4)1}$ in four dimensions). Next, for the single extra metric component one has: G=-(E+E\^\*+FF\^\*)f, u=[Im]{}E. \[f=G\]By taking into account that $\G=G$, and by substituting equations (\[xa\]) and (\[f=G\]) into the matter Lagrangian (\[matter3D\]), we obtain Ł\_[\_[EM]{}]{}=|E+F\^\*F|\^2 - f\^[-1]{}|F|\^2. As we already have seen, this is precisely the matter Lagrangian of the stationary Einstein–Maxwell theory. Thus, our MEP formulation of the heterotic string theory includes the Einstein–Maxwell theory as a special case.
It is worth noticing as well that the higher dimensional General Relativity theory can also be written in terms of a matrix Ernst potential when reduced to three dimensions. This fact corresponds to a special case in which the matter degrees of freedom of the low–energy heterotic string theory (\[accion\]) vanish: the anti–symmetric Kalb–Ramond tensor field $B\D_{MN}=0,$ the dilaton $\p\D=0$ and the Abelian gauge fields $A^{(D)I}_M=0,$ so that the matrix Ernst potential is symmetric $\X=\G$ and $\B=\A=0.$ It should also be mentioned that the three-dimensional dilaton field must remain nontrivial since it is identified with the determinant of the extra dimensional metric according to the definitions (\[g3D\]) and (\[scalars\]).
Thus, this parametrization of the above mentioned higher dimensional theories in terms of the MEP can be very useful when performing a complete classification of the higher dimensional ($D\ge 5$) black objects (holes, rings, Saturns, etc.) obtained in the literature during last years (see [@emparanetal] for a review).
Nonlinear hidden symmetries and their possible applications in $D\ge 5$
=======================================================================
One of the advantages of the (matrix) Ernst potential formalism is that the study of symmetries (conservation laws) of the stationary effective action can be performed in a very straightforward way. It turns out that the complete symmetry group, apart from rescalings and shifts of the Ernst potentials, involves nonlinear symmetries that were initially called [*hidden*]{} in the framework of General Relativity; moreover, an infinite–dimensional double hidden symmetry structure was revealed for string effective actions [@Gao]. In particular, these symmetries act nontrivially in the charge space of a seed solution and can be used to generate new charged solutions from uncharged ones. There also other effects when applying this symmetries (see, for instance, [@kinnersleyetal; @ggk; @hk5; @hamf; @haptv].
Here we shall quote just the symmetries which preserve the asymptotic properties of the (matrix) Ernst potentials for physically meaningful field configurations of both the stationary Einstein–Maxwell and low–energy heterotic string theories. These symmetries possess the same form for both theories and allow one to generate similar solutions in both realms [@hk5].
For the stationary Einstein–Maxwell theory we have: EE, Fe\^[i]{}F; ([EMT]{}) E, FF; ([NET]{}) E, F, ([NHT]{}) where EMT stands for Electric–Magnetic Transformation, NET for Normalized Ehlers Transformation and NHT for Normalized Harrison Transformation, the parameter $\lambda _{\H}$ is complex while the parameters $\alpha$ and $\epsilon$ are real. It is easy to check that when the parameters $\lambda _{\H},$ $\alpha$ and $\epsilon$ vanish, one recovers the original (seed) potentials.
On the other hand, for the stationary low–energy effective action of the heterotic string we have the following matrix symmetries: +\_, \_\^T=-\_ +\_, + \_\^T+\_\_\^T , , \^T=1 §\^T§, §\^T, §(§\^T)\^[-1]{}. &&(1+\_) (1+\_)\^[-1]{},\
&&(1+\_) (1+\_)\^[-1]{}(1-\_)+ \_. (1+\_\_\^T) (1-\_\^T+\_\_\^T)\^[-1]{}(A-\_)+\_, (1+\_\_\^T) (1-\_\^T+\_\_\^T)\^[-1]{}+\_\_\^T. where $\lambda_{\E}^T=-\lambda_{\E}$ and $\lambda_{\H}$ is a real rectangular matrix of dimension $(d+1)\times n.$
The last pair of nonlinear symmetries can be applied to construct new exact solutions starting from known (sometimes quite simple) field configurations in both theories. As an example one can cite the construction of the of the Reissner–Nordström solution starting from the Schwarzschild black hole one in the 4D Einstein–Maxwell theory.
We finally quote a procedure to construct new charged field configurations from known neutral solutions within the framework of theories like General Relativity and the effective low–energy action of the heterotic string with more than four dimensions (in the spirit of [@hamf; @haptv]). Thus, this procedure can be applied to the construction of charged black holes, black rings and black Saturns if $D=5$, and charged multiple black rings in $D=6$:
1. Write the exact solution of the uncharged field configuration (black ring or black Saturn, for instance) in the form of a generalized Weyl metric [@emparanreallWeyl; @harmark] by making use of a suitable coordinate system.
2. Identify the symmetric and antisymmetric parts of the matrix Ernst potential $\X$.
3. Perform the nonlinear hidden symmetry NHT on the matrix Ernst potentials $\X$ and $\A$.
4. Write the new higher–dimensional charged exact solution with the aid of $\X$ and $\A$.
5. Physically interpret the new solution with the aid of the behaviour of the fields and their properties.
This procedure can be performed also in a wider class of higher–dimensional field configurations that have the form of a stationary axisymmetric seed solution (the so–called Weyl–Papapetrou class) [@othersolutions] and it is interesting to see what kind of physical configurations arise after applying the MEP symmetry method.
Acknowledgements
================
One of the authors (AHA) is really grateful to the organizers of the Miniworkshop [*Symmetries 2010*]{}, held at UAZ, for providing a friendly and warm environment for developing fruitful and illuminating discussions. KK would like to thank the whole staff of IFM, UMSNH. He acknowledges support from CONACYT 60060–J. This research was supported by grants CIC–UMSNH–4.16 and CONACYT 60060–J. AHA is also grateful to SNI.
[100]{}
F.J. Ernst, Phys. Rev. [**168**]{} 1415 (1968); [**167**]{} 1175 (1968); J. Math. Phys. [**12**]{} 2395 (1971).
W. Israel and G.A. Wilson, J. Math. Phys. [**13**]{} 865 (1972).
D. Kramer, H. Stephani, M. McCallum and E. Herlt, [*Exact Solutions of the Einstein’s Field Equations*]{}, (Deutcher Verlag der Wissenschaften, Berlin, 1980).
W. Kinnersley, J. Math. Phys. [**18**]{} 1529 (1977); W. Kinnersley and D.M. Chitre, J. Math. Phys. [**18**]{} 1538 (1977); [**19**]{} 1926 (1978); Phys. Rev. Lett. [**40**]{} 1608 (1978).
D.V. Gal’tsov and O. Kechkin, Phys. Rev. [**D50**]{} 7394 (1994); D. Gal’tsov, A.A. García, and O. Kechkin, J. Math. Phys. [**36**]{} 5023 (1995).
A. Herrera–Aguilar and O. Kechkin, Int. J. Mod. Phys. [**A12**]{} 1573 (1997); [**A13**]{} 393 (1998); [**A14**]{} 1345 (1999); N. Barbosa–Cendejas and A. Herrera–Aguilar, Gen. Rel. Grav. [**35**]{} 449 (2003).
V. Belinski and E. Verdaguer, [*Gravitational Solitons*]{}, (Cambridge University Press, Cambridge, 2001).
A. Herrera–Aguilar and O. Kechkin, Mod. Phys. Lett. [**A13**]{} 1907 (1998).
A. Herrera–Aguilar and O. Kechkin, Phys. Rev. [**D59**]{} 124006 (1999).
R. Emparan and H.S. Reall, Phys. Rev. Lett. [**88**]{} 4877 (2002); Class. Quant. Grav. [**23**]{} R169 (2006); Living Rev. Rel. [**11**]{} 6 (2008); R. Emparan and P. Figueras, JHEP [**1011**]{} 022 (2010).
H. Elvang and P. Figueras, JHEP [**0705**]{} 050 (2007).
N. Marcus and J.H. Schwarz, Nucl. Phys. [**B228**]{} 145 (1983); J. Maharana and J.H. Schwarz, Nucl. Phys. [**B390**]{} 3 (1993).
A. Sen, Nucl. Phys. [**B434**]{} 179 (1995).
A. Herrera–Aguilar and O. Kechkin, Mod. Phys. Lett. [**A13**]{} 1629 (1998); 1979 (1998); Class. Quant. Grav. [**16**]{} 1745 (1999); Mod. Phys. Lett. [**A16**]{} 29 (2001); Int. J. Mod. Phys. [**A17**]{} 2485 (2002); J. Math. Phys. [**45**]{} 216 (2004); R. Becerril and A. Herrera–Aguilar, J. Math. Phys. [**46**]{} 052503 (2005); A. Herrera–Aguilar and M. Nowakowski, Class. Quant. Grav. [**21**]{} 1015 (2004).
Y.–J. Gao, Gen. Rel. Grav. [**35**]{} 1573 (2003); Phys. Rev. [**D76**]{} 044025 (2007); Rept. Math. Phys. [**62**]{} 1 (2008); Mod. Phys. Lett. [**A24**]{} 311 (2009).
A. Herrera–Aguilar and H.R. Márquez–Falcon, in [*ICHEP 2006: Proceedings, 2 Vols.*]{} Eds. A. Sissakian, G. Kozlov and E. Kolganova, (Singapore, World Scientific, 2007).
A. Herrera–Aguilar, Rev. Mex. Fis. [**49S2**]{} 141 (2003); Mod. Phys. Lett. [**A19**]{} 2299 (2004); A. Herrera–Aguilar, J.O. Téllez–Vázquez and J.E. Paschalis, Regular Chaot. Dyn. [**14**]{} 526 (2009).
R. Emparan and H.S. Reall, Phys. Rev. [**D65**]{} 084025 (2002).
T. Harmark, Phys. Rev. [**D70**]{} 124002 (2004).
G.T. Horowitz and A. Sen, Phys. Rev. [**D54**]{} 808 (1996); M. Cvetic and D. Youm, Nucl. Phys. [**B476**]{} 118 (1996); A.A. Tseytlin, Mod. Phys. Lett. A [**11**]{} 689 (1996); J.C. Breckenridge, D.A. Lowe, R.C. Myers, A.W. Peet, A. Strominger and C. Vafa, Phys. Lett. [**B381**]{} 423 (1996); R. Kallosh, A. Rajaraman and W.K. Wong, Phys. Rev. [**D55**]{} 3246 (1997); J.C. Breckenridge, R.C. Myers, A.W. Peet and C. Vafa, Phys. Lett. [**B391**]{} 93 (1997); C.A.R. Herdeiro, Nucl. Phys. [**B582**]{} 363 (2000); T. Matos, D. Núñez, G. Estévez and M. Ríos, Gen. Rel. Grav. [**32**]{} 1499 (2000); H. Elvang, Phys. Rev. [**D68**]{} 124016 (2003); A. Herrera–Aguilar and R.R. Mora–Luna, Phys. Rev. [**D69**]{} 105002 (2004); S.S. Yazadjiev, Phys. Rev. [**D72**]{} 104014 (2005); Class. Quant. Grav. [**22**]{} 3875 (2005); Phys. Rev. [**D73**]{} 064008 (2006); Phys. Rev. [**D73**]{} 124032 (2006); Phys. Rev. [**D77**]{} 127501 (2008).
|
[The $C_{\ell}$-free process]{}
[Lutz Warnke]{}
[ Mathematical Institute, University of Oxford\
24–29 St. Giles’, Oxford OX1 3LB, UK\
[warnke@maths.ox.ac.uk]{}]{}
<span style="font-variant:small-caps;">Abstract.</span> The $C_{\ell}$-free process starts with the empty graph on $n$ vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of $C_{\ell}$ is created. For every $\ell \geq 4$ we show that, with high probability as $n \to \infty$, the maximum degree is $O( (n \log n)^{1/(\ell-1)})$, which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the $C_{\ell}$-free process typically terminates with $\Theta(n^{\ell/(\ell-1)}(\log n)^{1/(\ell-1)})$ edges, which answers a question of Erd[ő]{}s, Suen and Winkler. This is the first result that determines the final number of edges of the more general $H$-free process for a non-trivial *class* of graphs $H$. We also verify a conjecture of Osthus and Taraz concerning the average degree, and obtain a new lower bound on the independence number. Our proof combines the differential equation method with a tool that might be of independent interest: we establish a rigorous way to ‘transfer’ certain decreasing properties from the binomial random graph to the $H$-free process.
Introduction
============
The *random graph process* was introduced by Erd[ő]{}s and R[é]{}nyi [@ErdosRenyi1959] in 1959. It starts with the empty graph on $n$ vertices and adds new edges one by one, where each edge is chosen uniformly at random among all edges not yet present. Since then it has been studied extensively, and many tools and methods for investigating its typical properties have been developed, see e.g. [@Bollobas2001RandomGraphs; @Durrett2007; @JLR2000RandomGraphs]. In this work we consider a natural variant of the above process which has very recently received a considerable amount of attention [@Bohman2009K3; @BohmanKeevash2010H; @GerkeMakai2010K3; @Picollelli2010K4Minus; @Picollelli2010C4; @Warnke2010H; @Warnke2010K4; @Wolfovitz2009H; @Wolfovitz2010K4; @Wolfovitz2009K3Subgraph].
The *$H$-free process* was suggested by Bollob[á]{}s and Erd[ő]{}s [@Bollobas2010PC] in 1990, as a way to generate an interesting probability distribution on the set of maximal $H$-free graphs with potential applications to Ramsey Theory. Given some fixed , it is a modification of the classical random graph process, where each new edge is chosen uniformly at random subject to the condition that no copy of $H$ is formed. It was first described in print in 1995 by Erd[ő]{}s, Suen and Winkler [@ErdoesSuenWinkler1995], who asked how many edges the final graph typically has (this also appears as a problem in [@ChungGraham98]). The main difficulty when analysing this process is that there is a complicated dependence among the edges; the order in which they are inserted is also relevant.
The first results addressed certain special graphs, determining the typical final number of edges up to logarithmic factors. The case $H=C_3$ was studied in 1995 by Erd[ő]{}s, Suen and Winkler [@ErdoesSuenWinkler1995], and in 2000 Bollob[á]{}s and Riordan [@BollobasRiordan2000] considered $H \in \{K_4,C_4\}$. In fact, a result of Ruci[ń]{}ski and Wormald [@RucinskiWormald1992] predates those mentioned above: in 1992 they considered the (much simpler) maximum degree $d$-process, which corresponds to the case $H=K_{1,d+1}$, and showed that whp[^1] it ends with $\lfloor nd/2\rfloor$ edges. The general $H$-free process was first analysed independently by Bollob[á]{}s and Riordan [@BollobasRiordan2000] and Osthus and Taraz [@OsthusTaraz2001] in 2000. In fact, they assumed that $H$ satisfies a certain density condition (strictly $2$-balanced), which holds for many interesting graphs, including cycles and complete graphs. Osthus and Taraz determined the typical final number of edges up to logarithmic factors and conjectured that whp the average degree in the final graph of the $C_{\ell}$-free process is $\Theta((n \log n)^{1/(\ell-1)})$.
The next improvements came about ten years later. In a breakthrough in 2009, Bohman [@Bohman2009K3] obtained the first matching bounds: he proved that the $C_3$-free process ends whp with $\Theta(n^{3/2} \sqrt{\log n})$ edges, confirming a conjecture of Spencer [@Spencer1995]. Next, Wolfovitz [@Wolfovitz2009H] slightly improved the lower bound on the expected final number of edges for a range of graphs $H$. Very recently, for the class of strictly $2$-balanced , Bohman and Keevash [@BohmanKeevash2010H] obtained new lower bounds that hold whp, which they conjectured to be tight up to the constants. In fact, their conjecture is for the maximum degree: for the $C_{\ell}$-free process they conjectured that the maximum degree is whp at most $D (n \log n)^{1/(\ell-1)}$ for some $D>0$.
As one can see, the typical final number of edges in the $H$-free process has attracted a lot of attention, and for a large class of interesting bounds are known. However, not much progress has been made in obtaining good upper bounds. After Bohman’s result for $C_3$, the next case to be resolved was $H=K_4$, for which matching bounds have been obtained by the author [@Warnke2010K4], and, independently, by Wolfovitz [@Wolfovitz2010K4]. During the preparation of this paper Picollelli [@Picollelli2010K4Minus; @Picollelli2010C4] also resolved the cases $H \in \{C_4,K_{4}^{-}\}$. But despite this progress, since the upper bound for the maximum degree $d$ process in [@RucinskiWormald1992] is immediate, one can argue that non-trivial matching upper bounds have not been determined for any *class* of graphs.
The $H$-free process is nowadays considered a model of independent interest as well. For strictly $2$-balanced $H$, the early evolution of various graph parameters, including the degree and the number of small subgraphs, has been investigated in [@BohmanKeevash2010H; @Wolfovitz2009K3Subgraph]. These results suggest that, perhaps surprisingly, during this initial phase the graph produced by the $H$-free process is very similar to the uniform random graph with the same number of edges, although it contains no copy of $H$. Studying the typical structural properties, e.g. the degree, in the later evolution of the $H$-free process is an intriguing problem, and so far only some preliminary results are known, cf. [@GerkeMakai2010K3; @Warnke2010H].
Motivation for studying the $H$-free process also comes from extremal combinatorics, where its analysis has produced several new results. For example, improved lower bounds on the Tur[á]{}n numbers of certain bipartite graphs and Ramsey numbers $R(s,t)$ with $s \geq 4$ have been established in [@Bohman2009K3; @BohmanKeevash2010H; @Wolfovitz2009H], and Bohman [@Bohman2009K3] reproved the famous lower bound for $R(3,t)$ obtained by Kim [@Kim95]. One of the key ingredients for these results is an upper bound on the independence number of the $H$-free process, cf. [@Bohman2009K3; @BohmanKeevash2010H]. So far only for the special cases $H \in \{C_3,C_4\}$ are these estimates known to be best possible, and it would be interesting to obtain good lower bounds for other graphs.
Main result
-----------
In this paper we prove a new upper bound on the final number of edges of the $C_{\ell}$-free process. In fact, we give a new upper bound for the maximum degree, which confirms a conjecture of Bohman and Keevash [@BohmanKeevash2010H] and improves previous upper bounds by Osthus and Taraz [@OsthusTaraz2001].
\[thm:main\_result\] For every $\ell \geq 4$ there exists $D>0$ such that whp the maximum degree in the final graph of the $C_{\ell}$-free process is at most $D (n \log n)^{1/(\ell-1)}$.
Up to the constant our upper bound is best possible, since the results of Bohman and Keevash [@BohmanKeevash2010H] imply that for some $c > 0$, whp the minimum degree is at least $c (n \log n)^{1/(\ell-1)}$. The special case $\ell=4$ was proved independently by Picollelli [@Picollelli2010C4]; since this manuscript was submitted Picollelli [@Picollelli2011Cl] has independently also proved the case $\ell \geq 4$. So, combining our findings with [@BohmanKeevash2010H], we not only verify the mentioned conjecture of Osthus and Taraz [@OsthusTaraz2001], but establish the following stronger result.
\[cor:main\_result:degree\_edge\] For every $\ell \geq 4$ there exist $c,D > 0$ such that in the final graph of the $C_{\ell}$-free process whp the number of edges is between $cn^{\ell/(\ell-1)} (\log n)^{1/(\ell-1)}$ and $Dn^{\ell/(\ell-1)} (\log n)^{1/(\ell-1)}$, and whp the degree of every vertex is between $c (n \log n)^{1/(\ell-1)}$ and $D (n \log n)^{1/(\ell-1)}$.
This is a natural extension of the main result of Bohman [@Bohman2009K3] for the $C_{3}$-free process, and answers a question of Erd[ő]{}s, Suen and Winkler for the $C_{\ell}$-free process (see [@ChungGraham98; @ErdoesSuenWinkler1995]): whp the final graph has $\Theta(n^{\ell/(\ell-1)} (\log n)^{1/(\ell-1)})$ edges. Since this question was asked for the $H$-free process in 1995, this is the first result that determines (up to constants) the final number of edges for a *class* of graphs.
We also obtain a new lower bound on the independence number of the $C_{\ell}$-free process. Indeed, as pointed out to us by Picollelli, using Corollary $2.4$ of Alon, Krivelevich and Sudakov [@AlonKrivelevichSudakov1999], Corollary \[cor:main\_result:degree\_edge\] implies the following bound conjectured in an earlier version of this paper (together with a proof of a weaker bound).
\[cor:main\_result:independence:number\] For every $\ell \geq 4$ there exists $c > 0$ such that whp the independence number in the final graph of the $C_{\ell}$-free process is at least $c (n \log n)^{(\ell-2)/(\ell-1)}$.
Up to the constant this matches the upper bound established by Bohman and Keevash [@BohmanKeevash2010H]. We infer that whp the independence number in the final graph of the $C_{\ell}$-free process is $\Theta(n \log n)^{(\ell-2)/(\ell-1)})$.
Comparison with previous work
-----------------------------
The basic idea of the proof is similar to [@OsthusTaraz2001]: we show that, after a certain number of steps, every pair $(\tilde{v},U)$ with $\tilde{v} \notin U$ and $|U|=D (n \log n)^{1/(\ell-1)}$ has some property that prevents $U \subseteq \Gamma(\tilde{v})$ in the final graph of the $C_{\ell}$-free process. Osthus and Taraz [@OsthusTaraz2001] establish their $O(n^{1/(\ell-1)} \log n)$ bound for the maximum degree using a ‘static’ point of view: they couple the $C_{\ell}$-free process (or more generally the $H$-free process) with the classical random graph process and then show that even after deleting all edges contained in a copy of $C_{\ell}$, every $(\tilde{v},U)$ has the desired property. By contrast, we obtain the better $O((n \log n)^{1/(\ell-1)})$ bound by tracking the step-by-step effects of each edge added in the $C_{\ell}$-free process, and our main tool is the differential equation method used in [@Warnke2010K4].
Our argument relates to the proof of Bohman for the $C_{3}$-free process as follows. In [@Bohman2009K3] it is shown that every large set of vertices contains at least one edge, which implies a bound on the maximum degree, since the neighbourhood of each vertex is an independent set. In other words, the upper bound follows from a bound on the independence number. For the $C_{\ell}$-free process, $\ell \geq 4$, the maximum degree is a separate question. In particular, we need to consider a more involved event, and thus must study the combinatorial structure of large sets more precisely.
To this end we track several random variables for every $(\tilde{v},U)$. But, when applying the differential equation method, there are significant technical difficulties, and a simple refinement of the approach used in [@Warnke2010K4] for the $K_4$-free process does not suffice to overcome them. Here one crucial ingredient is a new connection between the $H$-free process and the Erd[ő]{}s–R[é]{}nyi random graph, which might be of independent interest. More precisely, we develop a ‘transfer theorem’, which enables us to prove certain results for the $H$-free process using the *much* simpler binomial random graph model. This is a key tool for establishing properties of the $C_{\ell}$-free process which otherwise seem difficult to derive. We believe that it will also aid in proving new upper bounds for the $H$-free process.
Organization of the paper
-------------------------
We start by collecting the relevant properties of the $C_{\ell}$-free process in Section \[sec:CL-free\]. In Section \[sec:tools\] we then introduce several probabilistic tools and the differential equation method. Section \[sec:bounding\_max\_degree\] is devoted to the proof of Theorem \[thm:main\_result\]. Our argument relies on two key statements, whose proofs are deferred to Sections \[sec:trajectory\_verification\] and \[sec:good\_configurations\_exist\]. We apply the differential equation method in Section \[sec:trajectory\_verification\], and introduce the ‘transfer theorem’ in Section \[sec:transfer\]. Next, in Section \[sec:binomial\_results\] we collect properties of the binomial random graph, which are then used to complete the proof in Section \[sec:good\_configurations\_exist\].
The $C_{\ell}$-free process: preliminaries and notation {#sec:CL-free}
=======================================================
In this section we introduce some notation and briefly review properties of the $C_{\ell}$-free process needed in our argument. We closely follow [@BohmanKeevash2010H] and the reader familiar with the results of Bohman and Keevash may wish to skip this section.
Terminology and notation {#sec:notation}
------------------------
Let $G(i)$ denote the graph with vertex set $[n]=\{1,\ldots, n\}$ after $i$ steps of the $C_{\ell}$-free process. Its edge set $E(i)$ contains $i$ edges; we partition the remaining non-edges $\binom{[n]}{2} \setminus E(i)$ into two sets, $O(i)$ and $C(i)$, which we call *open* and *closed* pairs, respectively. We say that a pair $uv$ of vertices is *open* in $G(i)$ if $G(i) \cup \{uv\}$ contains no copy of $C_{\ell}$. So, the $C_{\ell}$-free process always chooses the next edge $e_{i+1}$ uniformly at random from $O(i)$. In addition, for $uv \in O(i) \cup C(i)$ we write $C_{uv}(i)$ for the set of pairs $xy \in O(i)$ such that adding $uv$ and $xy$ to $G(i)$ creates a copy of $C_{\ell}$ containing both $uv$ and $xy$. Note that $uv \in O(i)$ would become closed, i.e., belong to $C(i+1)$, if $e_{i+1} \in C_{uv}(i)$.
With a given graph in mind, we denote the *neighbourhood* of a vertex $v$ by $\Gamma(v)$, where, as usual, $\Gamma(v)$ does not include $v$. For $S \subseteq [n]$ we define $\Gamma(S) = \bigcup_{v \in S} \Gamma(v)$. Furthermore, for $A,B \subseteq [n]$, let $e(A,B)$ denote the number of edges that have one endpoint in $A$ and the other in $B$, where an edge with both ends in $A \cap B$ is counted once. If the graph under consideration is $G(i)$ we simply write $\Gamma_i(\cdot)$, but usually we omit the subscript if the corresponding $i$ is clear from the context. Given a set $S$ and an integer $k \geq 0$, we write $\binom{S}{k}$ for the set of all $k$-element subsets of $S$.
We use the symbol $\pm$ in two different ways, following [@Bohman2009K3; @BohmanKeevash2010H]. First, we denote by $a \pm b$ the interval $\{ a + x b : -1 \leq x \leq 1\}$. Multiple occurrences are treated independently; for example, $\sum_{i \in [j]}(a_i \pm b_i)$ and $\prod_{i \in [j]}(a_i \pm b_i)$ mean $\{ \sum_{i \in [j]}(a_i + x_i b_i) : -1 \leq x_1, \ldots, x_{j} \leq 1\}$ and $\{ \prod_{i \in [j]}(a_i + x_i b_i) : -1 \leq x_1, \ldots, x_{j} \leq 1\}$, respectively. For brevity we also use the convention that $x=a \pm b$ means $x \in a \pm b$. Second, when considering pairs of random variables and functions, e.g. $Y^{+}$, $Y^{-}$ and $y^{+}$, $y^{-}$, we use the superscript $\pm$ to denote two different statements: one with $\pm$ replaced by $+$, and the other with $\pm$ replaced by $-$. For example, $Y^{\pm}(i)=y^{\pm}(t)$ means $Y^{+}(i)=y^{+}(t)$ and $Y^{-}(i)=y^{-}(t)$. Finally, combinations of both ways are treated independently; for example, $Y^{\pm}(i)=y^{\pm}(t) \pm b$ means $Y^{+}(i)=y^{+}(t) \pm b$ and $Y^{-}(i)=y^{-}(t)\pm b$.
Parameters, functions and constants
-----------------------------------
In the remainder of this paper we fix $\ell \geq 4$. Following [@BohmanKeevash2010H], we introduce constants ${\varepsilon}$, $\mu$ and $W$. We choose $W$ sufficiently large and afterwards ${\varepsilon}$ and $\mu$ small enough such that, in addition to the constraints implicit in [@BohmanKeevash2010H] for $H=C_{\ell}$, we have $$\label{eq:Cl-constants:Wepsmu}
W \geq \ell^{2} 2^{\ell+1} \geq 50
, \qquad
{\varepsilon}\leq 1/\big(2^{15}\ell^3\big)
\qquad \text{ and } \qquad
2W\mu^{\ell-1} \leq {\varepsilon}.$$ Since the additional constraints in [@BohmanKeevash2010H] only depend on $H=C_{\ell}$, we deduce that $\mu$ is an absolute constant (depending only on $\ell$). Next, similar as in [@BohmanKeevash2010H] we set $$\label{eq:Cl-parameters}
p = n^{-1+1/(\ell-1)} , \quad t_{\max} = \mu (\log n)^{1/(\ell-1)} \quad \text{and} \quad m = n^2 p t_{\max} = \mu n^{\ell/(\ell-1)} (\log n)^{1/(\ell-1)} .$$ Formally, $m$ (a number of steps) should be defined as $\lfloor n^2 p t_{\max} \rfloor$, say, but, as usual, we will henceforth ignore the irrelevant rounding to integers. For every step $i$ we define $t = t(i) = i/(n^2p)$, where, for the sake of brevity, we simply write $t$ if the corresponding $i$ is clear from the context. Next we introduce the functions $$\label{eq:Cl-functions}
q(t) = e^{-(2t)^{\ell-1}} \qquad \text{ and } \qquad f(t) = e^{(t^{\ell-1} + t)W} .$$ Now, using , for every $0 \leq t \leq t_{\max}$, for $n$ large enough we readily obtain $$\label{eq:Cl-functions-estimates}
1 \geq q(t) \geq n^{-{\varepsilon}/4} \qquad \text{ and } \qquad 1 \leq f(t) q(t)^{\ell} \leq f(t) \leq n^{{\varepsilon}} .$$
Previous results for the $C_{\ell}$-free process
------------------------------------------------
The results of Bohman and Keevash [@BohmanKeevash2010H] imply that a wide range of random variables are dynamically concentrated throughout the of the $C_{\ell}$-free process. For our argument the key properties are estimates on the number of open pairs as well as bounds for the degree and certain closed pairs. So, for the reader’s convenience we state their results here in a simplified form.
\[thm:BohmanKeevash2010H\][[@BohmanKeevash2010H]]{} Set $s_e = n^{1/(2\ell)-{\varepsilon}}$. Let ${{\mathcal T}}_j$ denote the event that for every $0 \leq i \leq j$, we have $|O(i)| > 0$ as well as $$\begin{aligned}
\label{eq:open-estimate}
|O(i)| &= \left(1 \pm 3f(t)/s_e\right) q(t)n^2/2 && \text{and}\\
\label{eq:degree-estimate}
|\Gamma_i(v)| & \leq 3 np t_{\max} && \text{for all vertices $v \in [n]$.} \end{aligned}$$ Let ${{\mathcal J}}_j$ denote the event that for every $0 \leq i \leq j$ we have $$\begin{aligned}
\label{eq:closed-estimate}
&|C_{uv}(i)| = \left((\ell-1) (2t)^{\ell-2} q(t) \pm 7\ell f(t)/s_e\right) p^{-1} && \text{for all $uv \in O(i) \cup C(i)$ and } \\
\label{eq:closed-intersection-estimate}
& |C_{u'v'}(i) \cap C_{u''v''}(i)| \leq n^{-1/\ell} p^{-1} && \text{for all distinct $u'v',u''v'' \in O(i)$.} \end{aligned}$$ Then ${{\mathcal J}}_m \cap {{\mathcal T}}_m$ holds whp in the $C_{\ell}$-free process.
After some simple estimates, both and follow directly from Theorem $1.4$ in [@BohmanKeevash2010H]. Now, using $\mathrm{aut}(C_{\ell}) = 2\ell$ and $(2t)^{\ell-2} q(t) \leq 1$, which follow from elementary considerations, Corollary $6.2$ and Lemma $8.4$ in [@BohmanKeevash2010H] imply and . (Because the ‘high probability events’ of [@BohmanKeevash2010H] in fact hold with probability at least $1-n^{-\omega(1)}$, we may take the union bound over all steps and pairs.) We remark that there is a factor of $2$ difference in since we use unordered instead of ordered pairs.
In our argument we use two additional properties of the $C_{\ell}$-free process. The next lemma follows from Lemmas $4.2$ and $4.3$ in [@Warnke2010K4], which in turn are based on Lemmas $4.1$–$4.3$ in [@BohmanKeevash2010H].
\[lem:edges\_bounded\_and\_large\_degree\_bounded\] Let ${{\mathcal K}}_i$ denote the event that for all $a,b \geq 1$ and every $A, B \subseteq [n]$ with $|A|=a$ and $|B|=b$, in $G(i)$ we have $e(A,B) < \max\{4{\varepsilon}^{-1}(a+b),pabn^{2{\varepsilon}}\}$. Let ${{\mathcal L}}_i$ denote the event that for all $a \geq 1$ and $d \geq \max\{16{\varepsilon}^{-1}, 2apn^{2{\varepsilon}}\}$, for every $A \subseteq [n]$ with $|A| = a$ we have $|D_{A,d}(i)| < 16{\varepsilon}^{-1}d^{-1}a$, where $D_{A,d}(i) \subseteq [n]$ contains all vertices $v \in [n]$ with $|\Gamma(v) \cap A| \geq d$ in $G(i)$. Then the probability that ${{\mathcal T}}_m$ holds and ${{\mathcal K}}_m \cap {{\mathcal L}}_m$ fails is $o(1)$.
Probabilistic tools {#sec:tools}
===================
In this section we introduce several probabilistic tools that we will use in our argument.
Concentration inequalities
--------------------------
The following Chernoff bounds, see e.g. Section $2.1$ of [@JLR2000RandomGraphs], provide estimates for the probability that a sum of independent indicator variables deviates substantially from its expected value.
\[lem:chernoff\]Let $X = \sum_{i\in[n]} X_i$, where the $X_i$’s are independent Bernoulli-distributed random variables. Set $\mu = {{\mathbb E}}[X]$. Then for all $t \geq 0$ we have $$\label{eq:chernoff:lower}
{{\mathbb P}}[X \leq \mu - t] \leq e^{-t^2/(2\mu)} .$$ Furthermore, for all $t \geq 7 \mu$ we have $$\label{eq:chernoff:upper:simple}
{{\mathbb P}}[X \geq t] \leq e^{-t} .$$
In our argument we need to estimate the probability that in $G_{n,p}$ some subset contains ‘too many’ copies of a certain graph. R[ö]{}dl and Ruciński [@RoedlRucinski1995] showed that exponential upper-tail bounds can be obtained if we allow for deleting a few edges; this is usually referred to as the Deletion Lemma [@JansonRucinski2004Deletion].
\[lem:deletion\_lemma\]Suppose $0 < p < 1$ and that ${{\mathcal S}}$ is a family of subsets from $\binom{[n]}{2}$. We say that a graph $G$ *contains* $\alpha \in {{\mathcal S}}$ if all the edges of $\alpha$ are present in $G$. Let $\mu$ denote the expected number of elements in ${{\mathcal S}}$ that are contained in $G_{n,p}$. Let ${{\mathcal D}}{{\mathcal L}}(b,k,{{\mathcal S}})$ denote the event that there exists ${{\mathcal I}}_0 \subseteq {{\mathcal S}}$ with $|{{\mathcal I}}_0| \leq b$ such that, setting $E_0 = \bigcup_{\alpha \in {{\mathcal I}}_0} \alpha$, $G(n,p) \setminus E_0$ contains at most $\mu+k$ elements from ${{\mathcal S}}$. Then for every $b,k>0$ the probability that ${{\mathcal D}}{{\mathcal L}}(b,k,{{\mathcal S}})$ fails is at most $$\label{eq:lem:deletion_lemma}
\left(1+\frac{k}{\mu}\right)^{-b} \leq \exp\left\{-\frac{b k}{\mu+k}\right\} .$$
In [@Warnke2010K4] a slightly weaker variant of the above lemma was proven for the $H$-free process, where $H$ is strictly $2$-balanced. The results of Section \[sec:transfer\] will shed some light on this intriguing phenomenon.
Differential equation method {#sec:dem}
----------------------------
A crucial ingredient of our analysis is the differential equation method, which was developed by Wormald [@Wormald1995DEM; @Wormald1999DEM] to show that in certain discrete stochastic processes a collection ${{\mathcal V}}$ of random variables is whp approximated by the solution of a suitably defined system of differential equations. Developing ideas of Bohman and Keevash [@BohmanKeevash2010H], the following variant was introduced in [@Warnke2010K4]. It will be an important tool for showing that certain random variables are dynamically concentrated throughout the evolution of the $C_{\ell}$-free process.
\[lem:dem\]Suppose that $m=m(n)$ and $s=s(n)$ are positive parameters. Let ${{\mathcal C}}={{\mathcal C}}(n)$ and ${{\mathcal V}}={{\mathcal V}}(n)$ be sets. For every $0 \leq i \leq m$ set $t=t(i)=i/s$. Suppose we have a filtration ${{\mathcal F}}_0 \subseteq {{\mathcal F}}_1 \subseteq \cdots$ and random variables $X_{\sigma}(i)$ and $Y^{\pm}_{\sigma}(i)$ which satisfy the following conditions. Assume that for all $\sigma \in {{\mathcal C}}\times {{\mathcal V}}$ the random variables $X_{\sigma}(i)$ are non-negative and ${{\mathcal F}}_i$-measurable for all $0 \leq i \leq m$, and that for all $0 \leq i < m$ the random variables $Y^{\pm}_{\sigma}(i)$ are non-negative, ${{\mathcal F}}_{i+1}$-measurable and satisfy $$\label{eq:lem:dem:rv_relation}
X_{\sigma}(i+1) - X_{\sigma}(i) = Y^{+}_{\sigma}(i) - Y^{-}_{\sigma}(i) .$$ Furthermore, suppose that for all $0 \leq i \leq m$ and $\Sigma \in {{\mathcal C}}$ we have an event ${{{\mathcal B}}_{{i}}(\Sigma)}\in {{\mathcal F}}_i$. Then, for all $0 \leq i \leq m$ we define ${{{\mathcal B}}_{\leq {i}}(\Sigma)}= \bigcup_{0 \leq j \leq i} {{{\mathcal B}}_{{j}}(\Sigma)}$. In addition, suppose that for each $\sigma \in {{\mathcal C}}\times {{\mathcal V}}$ we have positive parameters $u_{\sigma}=u_{\sigma}(n)$, $\lambda_{\sigma}=\lambda_{\sigma}(n)$, $\beta_{\sigma}=\beta_{\sigma}(n)$, $\tau_{\sigma}=\tau_{\sigma}(n)$, $s_{\sigma} = s_{\sigma}(n)$ and $S_{\sigma}=S_{\sigma}(n)$, as well as functions $x_{\sigma}(t)$ and $f_{\sigma}(t)$ that are smooth and non-negative for $t \geq 0$. For all $0 \leq i^* \leq m$ and $\Sigma \in {{\mathcal C}}$, let ${{\mathcal G}}_{i^*}(\Sigma)$ denote the event that for every $0 \leq i \leq i^*$ and $\sigma=(\Sigma,j)$ with $j \in {{\mathcal V}}$ we have $$\label{eq:lem:dem:parameter_trajectory}
X_{\sigma}(i) = \left(x_{\sigma}(t) \pm \frac{f_{\sigma}(t)}{s_{\sigma}} \right) S_{\sigma} .$$ Next, for all $0 \leq i^* \leq m$ let ${{\mathcal E}}_{i^*}$ denote the event that for every $0 \leq i \leq i^*$ and $\Sigma \in {{\mathcal C}}$ the event ${{{\mathcal B}}_{\leq {i-1}}(\Sigma)} \cup {{\mathcal G}}_{i}(\Sigma)$ holds. Moreover, assume that we have an event ${{\mathcal H}}_i \in {{\mathcal F}}_i$ for all $0 \leq i \leq m$ with ${{\mathcal H}}_{i+1} \subseteq {{\mathcal H}}_{i}$ for all $0 \leq i < m$. Finally, suppose that the following conditions hold:
1. (Trend hypothesis) For all $0 \leq i < m$ and $\sigma = (\Sigma,j) \in {{\mathcal C}}\times {{\mathcal V}}$, whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{\leq {i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds we have $$\label{eq:lem:dem:parameter_martingale_property}
{{\mathbb E}}\big[Y^{\pm}_{\sigma}(i) \mid {{\mathcal F}}_i \big] = \left( y^{\pm}_{\sigma}(t) \pm \frac{h_{\sigma}(t)}{s_{\sigma}} \right) \frac{S_{\sigma}}{s} ,$$ where $y_{\sigma}^{\pm}(t)$ and $h_{\sigma}(t)$ are smooth non-negative functions such that $$\label{eq:lem:dem:derivative}
x'_{\sigma}(t) = y^+_{\sigma}(t) - y^-_{\sigma}(t) \qquad \text{ and } \qquad f_{\sigma}(t) \geq 2\int_{0}^{t} h_{\sigma}(\tau) \ d\tau + \beta_{\sigma} .$$
2. (Boundedness hypothesis) For all $0 \leq i < m$ and $\sigma = (\Sigma,j) \in {{\mathcal C}}\times {{\mathcal V}}$, whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{\leq {i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds we have $$\label{eq:lem:dem:parameter_max_change}
Y^{\pm}_{\sigma}(i) \leq \frac{\beta_{\sigma}^2}{s_{\sigma}^2 \lambda_{\sigma} \tau_{\sigma}} \cdot \frac{S_{\sigma}}{u_{\sigma}} .$$
3. (Initial conditions) For all $\sigma \in {{\mathcal C}}\times {{\mathcal V}}$ we have $$\label{eq:lem:dem:initial_condition}
X_{\sigma}(0) = \left(x_{\sigma}(0) \pm \frac{\beta_{\sigma}}{3s_{\sigma}} \right) S_{\sigma} .$$
4. (Bounded number of configurations and variables) We have $$\label{eq:lem:dem:bounded_parameters}
\max\left\{|{{\mathcal C}}|,|{{\mathcal V}}|\right\} \leq \min_{\sigma \in {{\mathcal C}}\times {{\mathcal V}}} e^{u_{\sigma}} .$$
5. (Additional technical assumptions) For all $\sigma \in {{\mathcal C}}\times {{\mathcal V}}$ we have $$\begin{gathered}
\label{eq:lem:dem:technical_assumptions:sm}
s \geq \max\{15 u_{\sigma} \tau_{\sigma} (s_{\sigma} \lambda_{\sigma}/\beta_{\sigma})^2, 9s_{\sigma} \lambda_{\sigma}/\beta_{\sigma}\} , \qquad s/(18 s_{\sigma} \lambda_{\sigma}/\beta_{\sigma}) < m \leq s \cdot \tau_{\sigma}/1944 ,\\
\label{eq:lem:dem:technical_assumptions:xy}
\sup_{0 \leq t \leq m/s} y^{\pm }_{\sigma}(t) \leq \lambda_{\sigma} , \qquad
\int_0^{m/s} |x''_{\sigma}(t)| \ dt \leq \lambda_{\sigma} , \\
\label{eq:lem:dem:technical_assumptions:h}
h_{\sigma}(0) \leq s_{\sigma} \lambda_{\sigma} \qquad \text{ and } \qquad \int_0^{m/s} |h'_{\sigma}(t)| \ dt \leq s_{\sigma} \lambda_{\sigma} .\end{gathered}$$
Then we have $$\label{eq:lem:dem}
\mathbb{P}[\neg{{\mathcal E}}_m \cap {{\mathcal H}}_m] \leq 4 \max_{\sigma \in {{\mathcal C}}\times {{\mathcal V}}}e^{-u_{\sigma}} .$$
An important feature of Lemma \[lem:dem\] is that the variables in ${{\mathcal V}}$ are tracked for every *configuration $\Sigma \in {{\mathcal C}}$*. However, it only gives approximation guarantees for the variables that ‘belong’ to $\Sigma$ as long as the ‘local’ *bad event* ${{{\mathcal B}}_{\leq {i}}(\Sigma)}$ fails. For more details we refer to Section $5.3$ and Appendix A.$1$ in [@Warnke2010K4]. Here we just remark that if the above conditions $1$–$5$ are satisfied for $n$ large enough, ${{\mathcal H}}_m$ holds whp and $u_{\sigma} = \omega(1)$ for all $\sigma \in {{\mathcal C}}\times {{\mathcal V}}$, then Lemma \[lem:dem\] implies that ${{\mathcal E}}_m$ holds whp.
Bounding the maximum degree {#sec:bounding_max_degree}
===========================
In this section we prove our main result, namely that whp the maximum degree in the final graph of the $C_{\ell}$-free process is $O((n \log n)^{1/(\ell-1)})$. In Sections \[sec:motivation\] and \[sec:formal\_setup\] we first discuss the main proof ideas and introduce the formal setup used. Section \[sec:finishing\_the\_proof\] is then devoted to the proof of Theorem \[thm:main\_result\], which in turn relies on two involved statements that are proved in subsequent sections.
Sketch of the proof {#sec:motivation}
-------------------
The following definition plays a crucial role in our proof. Given $(\tilde{v},U)$, where $\tilde{v} \in [n]$ and $U \subseteq [n] \setminus \{\tilde{v}\}$, a *$C_{\ell}$-extension for $(\tilde{v},U)$* is a path on $\ell-1$ vertices whose end vertices are in $U$ and whose remaining vertices are disjoint from $U \cup \{\tilde{v}\}$. Clearly, for every vertex $\tilde{v} \in [n]$, in the final graph of the $C_{\ell}$-free process $(\tilde{v},\Gamma(\tilde{v}))$ must not have a $C_{\ell}$-extension. Set $$\label{eq:Cl-parameters2}
\delta = \frac{1}{60^2 \ell! \ell^{\ell}} , \qquad \gamma = \max\left\{\frac{3^{\ell+1}}{\delta \mu^{\ell-1}},180\right\} \qquad \text{and} \qquad u = \gamma n p t_{\max} = \gamma \mu (n\log n)^{1/(\ell-1)} ,$$ again ignoring the irrelevant rounding to integers in the definition of $u$. In order to bound the maximum degree by $u = D (n\log n)^{1/(\ell-1)}$, where $D=\gamma \mu$, it is enough to prove that whp every $(\tilde{v},U) \in [n] \times \binom{[n]}{u}$ with $\tilde{v} \notin U$ has at least one $C_{\ell}$-extension after the first $m$ steps. The same basic idea was used in [@OsthusTaraz2001], but our proof takes a different route, inspired by [@Warnke2010K4]. After $i$ steps, we denote by $O_{\tilde{v},U}(i)$ the set of open pairs which would complete a $C_{\ell}$-extension for $(\tilde{v},U)$ if chosen as the next edge. It seems plausible that it in order prove Theorem \[thm:main\_result\], it suffices to show that, after some initial number of steps, $|O_{\tilde{v},U}(i)|$ is always not too small. Indeed, this implies a reasonable probability of completing such an extension in each step, which in turn suggests that the probability of avoiding a $C_{\ell}$-extension in all of the first $m$ steps is very small.
We now illustrate our approach for establishing a good lower bound on $|O_{\tilde{v},U}(i)|$ for the case when $\ell=5$. For ease of exposition, we ignore $n^{{\varepsilon}}$ factors whenever these are not crucial and also assume that the number of steps $i$ is large. So, in our rough calculations we will e.g. ignore whether an edge is open or not, since $|O(i)| = \omega(n^{2-{\varepsilon}})$ by and . Note that in this case we have $p=n^{-3/4}$, $m \approx n^{5/4}$, $|C_{xy}(i)| \approx p^{-1}$ and $|U| \approx np = n^{1/4}$ by , and .
### The random variables used
We define $O'_{\tilde{v},U}(i)$ as the set of pairs $xy \in O_{\tilde{v},U}(i)$ with $x \in U$ and $y \notin U \cup \{\tilde{v}\}$. Observe that for every $xy \in O'_{\tilde{v},U}(i)$ there exists a path $v_0v_1v_2=y$ with $v_{0} \in U \setminus \{x\}$ and $v_1 \notin U \cup \{\tilde{v},x,y\}$, cf. Figure \[fig:open:sketch\]. The ‘last’ edge completing a $C_{5}$-extension for $(\tilde{v},U)$ could be any one of the edges of the path, so we expect that $O'_{\tilde{v},U}(i)$ contains constant proportion of $O_{\tilde{v},U}(i)$.
(83.20, 54.69)(0,0) (0,0)[![\[fig:open:sketch\]A pair $xy \in O'_{\tilde{v},U}(i)$. Solid lines represent edges and dotted lines open pairs.](rv_open "fig:")]{} (28.72,42.12) (15.43,24.94) (28.72,9.52) (54.65,42.12) (65.74,9.52)
Let $Z_{\tilde{v},U}(i)$ contain all quadruples $(v_0,v_1,v_2,v_3) \in U \times [n]^2 \times U$ with $\{v_0v_1,v_1v_2\} \subseteq E(i)$, $v_2v_3 \in O(i)$ and $\{v_1,v_2\} \cap (U \cup \{\tilde{v}\}) = \emptyset$. Using random graphs as a guide, we expect that $G(i)$ shares many properties with the binomial random graph $G_{n,p}$, since its edge density is roughly $2tp \approx n^{-3/4} = p$. So, given $y$, the expected number of $v_0 \in U$ for which there exists a path $v_0v_1v_{2}=y$ should be roughly $n|U|p^2 = o(1)$. Hence on average $xy \in O'_{\tilde{v},U}(i)$ is contained in only one such path ending in $U$, which suggests that up to constants $|Z_{\tilde{v},U}(i)| \approx |O'_{\tilde{v},U}(i)|$. To sum up, our discussion indicates that a reasonable lower bound for $|Z_{\tilde{v},U}(i)|$ suffices to prove that $|O_{\tilde{v},U}(i)|$ is large. For this we intend to use the differential equation method and so we introduce additional variables in order to control the one-step changes of $|Z_{\tilde{v},U}(i)|$. To this end let $Y_{\tilde{v},U}(i)$ be the set of all $(v_0,v_1,v_2,v_3) \in U \times [n]^2 \times U$ with $\{v_1,v_2\} \cap (U \cup \{\tilde{v}\}) = \emptyset$ that satisfy $v_0v_1 \in E(i)$, $\{v_1v_2,v_2v_3\} \subseteq O(i)$, and, similarly, let $X_{\tilde{v},U}(i)$ contain all such quadruples with $\{v_0v_1,v_1v_2,v_2v_3\} \subseteq O(i)$.
### Technical difficulties {#sec:difficulties}
One of the main problems with the approach described above is the bound on the one-step changes. It can happen that in one step up to $p^{-1}$ quadruples are removed from $Z_{\tilde{v},U}(i)$, which turns out to be too large for applying the differential equation method directly. Indeed, pick $\tilde{v},U$ such that $\{v_0\} \cup \Gamma_i(w) \subseteq U$, $|\Gamma_i(w)| \approx |U|$ and $\tilde{v} \notin \{w\} \cup U \cup \Gamma_i(U)$; taking the random graph $G_{n,p}$ as a guide, for $e_{i+1}=wv_0$ it is easy to see that about $(np)^2|U| \approx p^{-1}$ quadruples $(v_0,v_1,v_2,v_3)$ with $v_3 \in \Gamma_i(w)$ are removed from $Z_{\tilde{v},U}(i)$. For the $C_{4}$-free process this can be resolved using ad-hoc arguments (e.g. exploiting that every $v \neq \tilde{v}$ satisfies $|\Gamma_i(v) \cap U| \leq 1$ if no $C_{4}$-extension for $(\tilde{v},U)$ exists), but for larger cycles the situation is more delicate. To overcome this issue, we consider a different random variable $T_{\tilde{v},U}(i)$, which is an approximation of $Z_{\tilde{v},U}(i)$ and is defined in such a way that the one-step changes are automatically not too large. Roughly speaking, this can be achieved by ‘ignoring’ the steps where the one-step changes would be too large; similar ideas have been used e.g. in [@Bohman2009K3; @BohmanKeevash2010H; @KimVu2004; @Warnke2010K4]. Clearly, this introduces a new difficulty: we need to ensure that we do not ignore ‘too much’, so that on the one hand the expected one-step changes are still ‘correct’, and on the other hand $|Z_{\tilde{v},U}(i)| \approx |T_{\tilde{v},U}(i)|$ holds. Consequently, we refine the tracked variables and use more sophisticated rules for ignoring tuples.
There is another significant obstacle when applying the differential equation method: adding $e_{i+1}=v_1v_2$ to $(v_0,v_1,v_2,v_3) \in Y_{\tilde{v},U}(i)$ does *not* always result in an element of $Z_{\tilde{v},U}(i+1)$, since $e_{i+1}=v_1v_2$ closes $v_2v_3$ whenever $v_2v_3 \in C_{v_1v_2}(i)$ holds. This is an important difference to the $C_{\ell}$-free process with $\ell \leq 4$, where this does not cause any problems when bounding the maximum degree. For example, whenever this happens for $\ell = 4$, it is not difficult to deduce that at least one $C_{4}$-extension for $(\tilde{v},U)$ already exists. Returning to the case $\ell=5$, using our random graph intuition we expect that $|Y_{\tilde{v},U}(i)| \approx |U|^2n^2p \approx n^{7/4}$. Similar calculations suggest that the expected number of quadruples in $Y_{\tilde{v},U}(i)$ with $v_2v_3 \in C_{v_1v_2}(i)$ should be negligible compared to $|Y_{\tilde{v},U}(i)|$. However, if we pick $U$ such that $\Gamma_i(w) \subseteq U$ and $|\Gamma_i(w)| \approx |U|$, for $\tilde{v} \notin \{w\} \cup U \cup \Gamma_i(U)$, it certainly can happen that there are $|U|^2 \cdot np \cdot n \approx |Y_{\tilde{v},U}(i)|$ quadruples in $Y_{\tilde{v},U}(i)$ with $v_2v_3 \in C_{v_1v_2}(i)$. In other words, it is simply *not* true that for all $(\tilde{v},U)$ the effect of these ‘bad’ quadruples is negligible. This is a new difficulty in comparison to the variables tracked in the analysis of the $H$-free process [@BohmanKeevash2010H]. To deal with this issue, we substantially refine the tracked random variables, developing ideas used in [@Warnke2010K4]. Intuitively, we show that for every $(\tilde{v},U)$ there exists a slightly altered set of random variables where the above extreme example (and other difficulties) can be avoided. Here the new ‘transfer theorem’ (Theorem \[thm:transfer:binomial\]) is an important ingredient, which allows us to use the *much* more tractable binomial random graph model for certain calculations (see Section \[sec:binomial\_results\]).
Formal setup {#sec:formal_setup}
------------
We now introduce the formal setup used in our argument. In the following it is useful to keep in mind that we intend to apply the differential equation method (Lemma \[lem:dem\]).
### Preliminaries: neighbourhoods and partitions {#sec:formal_setup:preliminaries}
Recall that by we have $u = \gamma n p t_{\max} = \gamma \mu (n\log n)^{1/(\ell-1)}$. We set $$\label{eq:Cl-parameters3}
k=u/60 = \gamma/60 \cdot n p t_{\max} = \gamma \mu /60 \cdot (n\log n)^{1/(\ell-1)} \qquad \text{ and } \qquad r=\lfloor n/(\ell-3)\rfloor .$$ Given $X \subseteq [n]$, we partition $\{1, \ldots, (\ell-3)r \} \setminus X$ as follows: for every $1 \leq j \leq \ell-3$ we set $$\label{eq:def:Vj}
V_{j}=V_{j}(X) = \{ v \in [n] \setminus X \;:\; (j-1)r < v \leq j r\} .$$ With a given graph in mind, which will later be $G(i)$ or the binomial random graph, for every $S \subseteq [n]$ we define its *neighbourhoods wrt. $X$* as $$\label{eq:def:Nj}
N^{(0)}(S,X) = S \qquad \text{ and } \qquad N^{(j+1)}(S,X) = \Gamma\big(N^{(j)}(S,X)\big) \cap V_{j+1}(X) ,$$ see also Figure \[fig:neighbourhoods:Lambda\].
(236.02, 100.79)(0,0) (0,0)[![\[fig:neighbourhoods:Lambda\]The neighbourhoods $N^{(j)}(S) = N^{(j)}(S,X)$ for $j \in [3]$, where $S$ may also intersect with $X$ and the vertex classes, i.e., with $X \cup V_1 \cup V_2 \cup V_3$. Furthermore, $S \cap N^{(j)}(S) \neq \emptyset$ is also possible.](neighbourhoods_S_X "fig:")]{} (44.12,84.34) (16.24,48.97) (70.26,47.90) (61.74,9.57) (108.01,84.34) (134.16,47.90) (171.90,84.34) (198.05,47.90)
Observe that all $N^{(j)}(S,X)$ are disjoint if $S \subseteq X$. Furthermore, $X \subseteq Y$ implies $$\label{eq:neighbourhood:monotone}
V_{j}(Y) \subseteq V_{j}(X) \qquad \text{ and } \qquad N^{(j)}(S,Y) \subseteq N^{(j)}(S,X) .$$ Finally, for the sake of brevity we define $N^{(\leq j)}(S,X) = \bigcup_{0 \leq j' \leq j} N^{(j')}(S,X)$.
### Configurations {#sec:main-proof:configs}
We define the set ${{\mathcal C}}$ of configurations to be the set of all $\Sigma=(\tilde{v},U,A,B,R)$ with $\tilde{v} \in [n]$, $U \in \binom{[n] \setminus \{\tilde{v}\}}{u}$, disjoint $A,B \in \binom{U}{k}$, and $R \subseteq [n]$ with $\{\tilde{v}\} \cup U \subseteq R$ and $|R| \leq kn^{10\ell{\varepsilon}}$. Given $\Sigma \in {{\mathcal C}}$, we then set $T_{\Sigma}=A \times V_1 \times \cdots \times V_{\ell-3} \times B$, where each $V_{j}=V_{j}(R)$ is given by .
Given $\Sigma \in {{\mathcal C}}$, distinct $x,y \in [n]$ and $j \in [\ell-1]$, let $C_{x,y,\Sigma}(i,j)$ contain all pairs $bw \in B \times N^{(\ell-3)}(A,R)$ for which there exist disjoint paths $b=w_1 \cdots w_{j}=x$ and $y=w_{j+1} \cdots w_{\ell}=w$ in $G(i)$. Note that adding $xy$ and $bw$ completes a copy of $C_{\ell}$ containing both $xy$ and $bw$. Furthermore, observe that $C_{x,y,\Sigma}(i,j)$ and $C_{y,x,\Sigma}(i,j)$ may differ. So, for all $xy \in O(i) \cup C(i)$ we see that the intersection of $C_{xy}(i)$ with $B \times N^{(\ell-3)}(A,R)$ is contained in $\bigcup_{j \in [\ell-1]} \big[ C_{x,y,\Sigma}(i,j) \cup C_{y,x,\Sigma}(i,j) \big]$. Finally, note that by monotonicity we have $C_{x,y,\Sigma}(i,j) \subseteq C_{x,y,\Sigma}(i+1,j)$.
### Random variables {#sec:main-proof:variables}
For every $\Sigma \in {{\mathcal C}}$ we track the sizes of several sets throughout the evolution of the $C_{\ell}$-free process. For brevity, given $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma}$, we set $f_{j} = v_{j-1}v_{j}$ for all $1 \leq j \leq \ell-2$. For every $0 \leq j \leq \ell-3$ we introduce sets $T_{\Sigma,j}(i)$, which for $0 \leq j < \ell-3$ will satisfy $$\label{eq:X:j:inclusion}
T_{\Sigma,j}(i) \subseteq \big\{ (v_0, \ldots, v_{\ell-2}) \in T_{\Sigma} \;:\; \{f_1, \ldots, f_j\} \subseteq E(i) \;\wedge\; \{f_{j+1}, \ldots, f_{\ell-2}\} \subseteq O(i) \big\} ,$$ and for the special case $j = \ell-3$ we will have $$\label{eq:X:l3:inclusion}
T_{\Sigma,\ell-3}(i) \subseteq \big\{ (v_0, \ldots, v_{\ell-2}) \in T_{\Sigma} \;:\; \{f_1, \ldots, f_{\ell-3}\} \subseteq E(i) \;\wedge\; f_{\ell-2} \in O(i) \cup C(i) \big\} ,$$ see also Figure \[fig:tuples:T\]. Note that $f_{\ell-2}$ can be in $O(i)$ or $C(i)$ for $T_{\Sigma,\ell-3}(i)$, but we will see later that the number of tuples with pairs in $C(i)$ is negligible. In the following we define the $T_{\Sigma,j}(i)$ inductively, starting with $T_{\Sigma,j}(0) = \emptyset$ for $j > 0$ and $T_{\Sigma,0}(0) = T_{\Sigma}$. Now suppose the process chooses $e_{i+1}=xy \in O(i)$ as the next edge in step $i+1$. For $j > 0$ a tuple $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j-1}(i)$ is *added* to $T_{\Sigma,j}(i+1)$, i.e., is in $T_{\Sigma,j}(i+1)$, if $f_j = e_{i+1}$, $\{f_{j+1}, \ldots, f_{\ell-2}\} \cap C_{f_{j}}(i) = \emptyset$, and in $G(i)$ there is no path $w_0 \cdots w_j=v_j$ with $w_0 \in A$. Furthermore, for $j < \ell-3$ a tuple $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j}(i)$ is *removed*, i.e., not in $T_{\Sigma,j}(i+1)$, if $e_{i+1} \in \{f_{j+1}, \ldots, f_{\ell-2}\}$ or $e_{i+1} \in C_{f_{j+1}}(i) \cup \cdots \cup C_{f_{\ell-2}}(i)$.
(114.66, 92.51)(0,0) (0,0)[![\[fig:tuples:T\]Tuples $(v_0, v_1,v_2, v_{3})$ in $T_{\Sigma,0}(i)$, $T_{\Sigma,1}(i)$ and $T_{\Sigma,2}(i)$ for $\ell=5$, where $\Sigma=(\tilde{v},U,A,B,R)$. Solid lines represent edges, dotted lines open pairs and dashed lines pairs that are open or closed. For the other pairs there is no restriction, i.e., they may be open, closed or an edge.](rv_l5_T0 "fig:")]{} (42.35,71.87) (93.56,50.52) (20.14,74.04) (69.74,74.04) (31.44,42.90) (62.05,15.41) (91.39,71.87) (93.56,22.74)
(114.66, 92.51)(0,0) (0,0)[![\[fig:tuples:T\]Tuples $(v_0, v_1,v_2, v_{3})$ in $T_{\Sigma,0}(i)$, $T_{\Sigma,1}(i)$ and $T_{\Sigma,2}(i)$ for $\ell=5$, where $\Sigma=(\tilde{v},U,A,B,R)$. Solid lines represent edges, dotted lines open pairs and dashed lines pairs that are open or closed. For the other pairs there is no restriction, i.e., they may be open, closed or an edge.](rv_l5_T1 "fig:")]{} (42.35,71.87) (93.56,50.52) (20.14,74.04) (69.74,74.04) (31.44,42.90) (62.05,15.41) (91.39,71.87) (93.56,22.74)
(114.66, 92.51)(0,0) (0,0)[![\[fig:tuples:T\]Tuples $(v_0, v_1,v_2, v_{3})$ in $T_{\Sigma,0}(i)$, $T_{\Sigma,1}(i)$ and $T_{\Sigma,2}(i)$ for $\ell=5$, where $\Sigma=(\tilde{v},U,A,B,R)$. Solid lines represent edges, dotted lines open pairs and dashed lines pairs that are open or closed. For the other pairs there is no restriction, i.e., they may be open, closed or an edge.](rv_l5_T2 "fig:")]{} (42.35,71.87) (93.56,50.52) (20.14,74.04) (69.74,74.04) (31.44,42.90) (62.05,15.41) (91.39,71.87) (93.56,22.74)
For the special case $j=\ell-3$, a tuple $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)$ is *removed*, i.e., not in $T_{\Sigma,\ell-3}(i+1)$, or *ignored*, i.e., remains in $T_{\Sigma,\ell-3}(i+1)$, according to the following rules:
Case 1.
: If $f_{\ell-2} = e_{i+1}$, then the tuple $(v_0, \ldots, v_{\ell-2})$ is removed,
Case 2.
: If $e_{i+1} \in C_{f_{\ell-2}}(i)$, then the tuple $(v_0, \ldots, v_{\ell-2})$ is
1. removed if there exists $j \in [\ell-1]$ and $x,y \in [n]$ such that $e_{i+1}=xy$, $f_{\ell-2} \in C_{x,y,\Sigma}(i,j)$ and $|C_{x,y,\Sigma}(i,j)| \leq p^{-1} n^{-30\ell{\varepsilon}}$, and
2. ignored otherwise.
The above definition clearly satisfies and . Intuitively, the rules for removing tuples from $T_{\Sigma,\ell-3}(i)$ ensure that the one-step changes are ‘by definition’ not too large. Furthermore, the way in which the tuples are added yields the following *extension property ${{\mathcal U}}_{T}$*.
\[lemma:extension:property\] Given $i \geq 0$, let ${{\mathcal U}}_T(i)$ denote the property that for all $\Sigma \in {{\mathcal C}}$ and $1 \leq j \leq \ell-3$, for every $(v_{j}, \ldots, v_{\ell-2}) \in V_j \times \cdots V_{\ell-3} \times B$ there exists at most one $(v_0, \ldots, v_{j-1}) \in A \times V_1 \times \cdots \times V_{j-1}$ such that $(v_0, \ldots, v_{\ell-2}) \in \bigcup_{i' \leq i} T_{\Sigma,j}(i')$. Then ${{\mathcal U}}_T = {{\mathcal U}}_T(i)$ holds for every $i \geq 0$.
The proof proceeds by induction on $i$ and $j$; we leave the straightforward details to the reader (it is helpful to observe that after $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j-1}(i)$ is added to $T_{\Sigma,j}(i+1)$, no further tuples containing $v_{j}$ can be added due to the $v_0 \cdots v_{j}$ path). Note that by ${{\mathcal U}}_T$ every $(v_{j}, \ldots, v_{\ell-2}) \in V_j \times \cdots V_{\ell-3} \times B$ is contained in at most one tuple in $\bigcup_{i' \leq i} T_{\Sigma,j}(i')$. This is an important ingredient of our argument, and we remark that a simpler variant of this property has previously been used in [@Warnke2010K4].
Recall that our goal is to show that there are many open pairs whose addition would complete a $C_{\ell}$-extension for $(\tilde{v},U)$. Given $\Sigma=(\tilde{v},U,A,B,R)$, note that for every $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)$, if $f_{\ell-2} \in O(i)$, then adding $f_{\ell-2}$ to $G(i)$ would complete such a $C_{\ell}$-extension. Now, since ${{\mathcal U}}_T$ implies that every pair $f_{\ell-2}=xy$ with $x \in V_{\ell-3}$ and $y \in B$ is contained in at most one such tuple in $T_{\Sigma,\ell-3}(i)$, our aim is to obtain a lower bound on the size of $$\label{eq:def:partial_open}
Z_{\Sigma,\ell-3}(i) = \big\{ (v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i) \;:\; f_{\ell-2} \in O(i) \big\} .$$
### Bad events {#sec:main-proof:bad_high_probability_events}
The following bad event ${{\mathcal B}}_{i}(\Sigma)$ is crucial for our argument: it addresses the two main technical difficulties outlined in Section \[sec:difficulties\]. For all $0 \leq i \leq m$ and $\Sigma \in {{\mathcal C}}$ we define ${{\mathcal B}}_{i}(\Sigma) = {{\mathcal B}}_{1,i}(\Sigma) \cup {{\mathcal B}}_{2,i}(\Sigma)$, where
1. in $G(i)$ there are more than $k^2(np)^{\ell-4}n^{-9{\varepsilon}}$ pairs $(b,w) \in B \times N^{(\ell-4)}(A,R)$ for which there exists a path $b=w_0 \cdots w_{\ell-2}=w$, and
2. in $G(i)$ we have $|L_{\Sigma}(i)| \geq p^{-1}n^{-1/(2\ell)}$, where $L_{\Sigma}(i)$ contains all $xy \in \binom{[n]}{2}$ with $\max_{j \in [\ell-1]}\{|C_{x,y,\Sigma}(i,j)|,|C_{y,x,\Sigma}(i,j)|\} \geq p^{-1}n^{-30\ell{\varepsilon}}$.
Clearly, ${{\mathcal B}}_{i}(\Sigma)$ depends only on the first $i$ steps and is increasing, i.e., ${{\mathcal B}}_{i}(\Sigma) \subseteq {{\mathcal B}}_{i+1}(\Sigma)$ holds.
We now briefly give some intuition for ${{\mathcal B}}_{1,i}(\Sigma)$ and ${{\mathcal B}}_{2,i}(\Sigma)$, which are important ingredients for estimating the number of tuples added to $T_{\Sigma,\ell-3}(i+1)$ and removed from $T_{\Sigma,\ell-3}(i)$. First, recall that $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-4}(i)$ can not be added to $T_{\Sigma,\ell-3}(i+1)$ if $f_{\ell-2} \in C_{f_{\ell-3}}(i)$. For such ‘useless’ tuples there exists a path $v_{\ell-2}=w_0 \cdots w_{\ell-2}=v_{\ell-4}$ with $(v_{\ell-2},v_{\ell-4}) \in B \times N^{(\ell-4)}(A,R)$ in $G(i)$, and whenever $\neg{{\mathcal B}}_{1,i}(\Sigma)$ holds there can not be ‘too many’ such pairs. As we shall see, from this we can deduce (using the extension property ${{\mathcal U}}_{T}$) that the number of ‘useless’ tuples is small compared to $|T_{\Sigma,\ell-4}(i)|$. Second, recall that not all tuples $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)$ are removed if $e_{i+1} \in C_{f_{\ell-2}}(i)$: some are are ignored. Here the key point is that $e_{i+1} \in C_{f_{\ell-2}}(i) \setminus L_{\Sigma}(i)$ is a sufficient condition for being removed, and, with in mind, that $\neg{{\mathcal B}}_{2,i}(\Sigma)$ essentially implies that $|L_{\Sigma}(i)|$ is small compared to $|C_{f_{\ell-2}}(i)|$. Intuitively, this will allow us to show that the ignored tuples have negligible impact, i.e., that $|Z_{\Sigma,\ell-3}(i)| \approx |T_{\Sigma,\ell-3}(i)|$.
Proof of Theorem \[thm:main\_result\] {#sec:finishing_the_proof}
-------------------------------------
In this section we prove Theorem \[thm:main\_result\] assuming the following two statements. Intuitively, the first lemma ensures that for ‘good’ configurations $\Sigma$ the variables $|T_{\Sigma,j}(i)|$ are dynamically concentrated, and the second lemma essentially guarantees that for every $(\tilde{v},U)$ there exists a good $\Sigma^*=(\tilde{v},U,A,B,R)$ for which $|T_{\Sigma^*,\ell-3}(i)| \approx |Z_{\Sigma^*,\ell-3}(i)|$. Now we give some intuition for the trajectories our variables follow. Using , we see that the proportion of pairs which are open or an edge in $G(i)$ roughly equals $q(t)$ or $2tp$, respectively, where $t=i/(n^2p)$. So, using random graphs as a guide, it seems plausible to expect $|T_{\Sigma,j}(i)| \approx c_j (2tp)^{j}{q(t)}^{\ell-2-j} k^2 r^{\ell-3}$, where the factor $c_j=1/j!$ takes into account that we only count tuples created in a certain order. In the following results the functions $q(t)$, $f(t)$ and parameters $k$, $m$, $p$, $r$, $u$ are defined by , , and .
\[lem:dem:trajectories\] For all $0 \leq i^* \leq m$ and $\Sigma \in {{\mathcal C}}$, let ${{\mathcal G}}_{i^*}(\Sigma)$ denote the event that for every $0 \leq i \leq i^*$ and all $0 \leq j \leq \ell-3$ we have $$\label{eq:lem:dem:trajectories:T}
|T_{\Sigma,j}(i)| = \left( (2t)^{j}{q(t)}^{\ell-2-j}/j! \; \pm \; f(t) {q(t)}^{\ell-3-j}/n^{2{\varepsilon}} \right) k^2 r^{\ell-3}p^{j} ,$$ and let ${{\mathcal E}}_{j}$ denote the event that for all $0 \leq i \leq j$ and $\Sigma \in {{\mathcal C}}$ the event ${{\mathcal B}}_{i-1}(\Sigma) \cup {{\mathcal G}}_{i}(\Sigma)$ holds. Then ${{\mathcal E}}_m$ holds whp in the $C_{\ell}$-free process.
\[lem:dem:config\] Let ${{\mathcal R}}_{j}$ denote the event that for all $0 \leq i \leq j$, for every $(\tilde{v},U) \in [n] \times \binom{[n]}{u}$ with $\tilde{v} \notin U$ there exists $\Sigma^*=(\tilde{v},U,A,B,R) \in {{\mathcal C}}$ such that $\neg{{\mathcal B}}_{i-1}(\Sigma^*)$ holds and $$\label{eq:lem:dem:config:ignored}
|T_{\Sigma^*,\ell-3}(i) \setminus Z_{\Sigma^*,\ell-3}(i)| \leq k^2 (rp)^{\ell-3}n^{-9{\varepsilon}} .$$ Then ${{\mathcal R}}_m$ holds whp in the $C_{\ell}$-free process.
The proofs of these lemmas are rather involved and therefore deferred to Sections \[sec:trajectory\_verification\] and \[sec:good\_configurations\_exist\]. With these results in hand, we are now ready to establish our main result.
For the sake of concreteness, we prove the theorem with $D=\gamma \mu$. Given $\tilde{v} \in [n]$, $U \subseteq [n] \setminus \{\tilde{v}\}$ and $i \leq m$, let ${{\mathcal X}}_{\tilde{v},U,i}$ denote the event that up to , there is no $C_{\ell}$-extension for $(\tilde{v},U)$ in the $C_{\ell}$-free process. By ${{\mathcal X}}_m$ we denote the event that there exists $(\tilde{v},U) \in [n] \times \binom{[n]}{u}$ with $\tilde{v} \notin U$ for which ${{\mathcal X}}_{\tilde{v},U,m}$ holds. Furthermore, for every $i \leq m$ we set ${{\mathcal A}}_i = {{\mathcal E}}_i \cap {{\mathcal R}}_i \cap {{\mathcal T}}_i$, where ${{\mathcal T}}_i$ is defined as in Theorem \[thm:BohmanKeevash2010H\] and ${{\mathcal E}}_i$, ${{\mathcal R}}_i$ as in Lemmas \[lem:dem:trajectories\] and \[lem:dem:config\]. If ${{\mathcal X}}_m$ fails, then, as discussed in Section \[sec:motivation\], the $C_{\ell}$-free process has maximum degree at most $u = D (n\log n)^{1/(\ell-1)}$. So, since ${{\mathcal A}}_m$ holds whp by Theorem \[thm:BohmanKeevash2010H\] and Lemmas \[lem:dem:trajectories\] and \[lem:dem:config\], to complete the proof it suffices to show $$\label{eq:thm:main_result:prob_no_extension_small}
{{\mathbb P}}[{{\mathcal X}}_m \cap {{\mathcal A}}_m] = o(1) .$$ Suppose that for $m/2 \leq i \leq m$ the event ${{\mathcal A}}_i = {{\mathcal E}}_i \cap {{\mathcal R}}_i \cap {{\mathcal T}}_i$ holds. Observe that ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i-1}}(\Sigma^*)}$ implies ${{\mathcal G}}_{i}(\Sigma)$, which is defined as in Lemma \[lem:dem:trajectories\]. Using we see that $m/2 \leq i \leq m$ implies $t=i/(n^2p)=\omega(1)$, so for $j=\ell-3$ the main term in the brackets of is $(2t)^{\ell-3}{q(t)}/(\ell-3)!$ since $f(t)/[n^{2{\varepsilon}}q(t)]=o(1)$ by . Thus, whenever ${{\mathcal E}}_i \cap {{\mathcal R}}_i$ holds, using , and $q(t) \geq n^{-{\varepsilon}/4}$, it follows that for every $(\tilde{v},U)$ with $U \in \binom{[n]\setminus\{\tilde{v}\}}{u}$ there exists $\Sigma^*=(\tilde{v},U,A,B,R) \in {{\mathcal C}}$ satisfying $$|T_{\Sigma^*,\ell-3}(i)| \geq k^2(2tpr)^{\ell-3}q(t)/(\ell-1)!
\quad \text{ and } \quad |T_{\Sigma^*,\ell-3}(i) \setminus Z_{\Sigma^*,\ell-3}(i)| \leq k^2 (2tpr)^{\ell-3}q(t)n^{-7{\varepsilon}} .$$ Note that ${{\mathcal T}}_i$ gives $q(t) \geq |O(i)|/n^2$ by and . So, combining our findings with $Z_{\Sigma^*,\ell-3}(i) \subseteq T_{\Sigma^*,\ell-3}(i)$, using $k = u/60$, $r \geq n/\ell$, and $t=i/(n^2p)$ we see that for such $\Sigma^*$ we crudely have $$\label{eq:thm:main_result:many_open:tuples}
\begin{split}
|Z_{\Sigma^*,\ell-3}(i)| &= |T_{\Sigma^*,\ell-3}(i)| - |T_{\Sigma^*,\ell-3}(i) \setminus Z_{\Sigma^*,\ell-3}(i)| \geq k^2(2tpr)^{\ell-3}q(t) / \ell! \\
&\geq \delta u^2(tpn)^{\ell-3} q(t) = \delta \frac{u^{2} i^{\ell-3}}{n^{\ell-3}} q(t) \geq \delta \frac{u^{2} i^{\ell-3}}{n^{\ell-1}} |O(i)| .
\end{split}$$ Recall that $O_{\tilde{v},U}(i) \subseteq O(i)$ denotes the set of open pairs which would complete a $C_{\ell}$-extension for $(\tilde{v},U)$ if chosen as the next edge $e_{i+1}$. Let $O_{\Sigma^*}(i)$ be the set of all $xy \in O(i)$ for which there exists $(v_{0}, \ldots, v_{\ell-2}) \in Z_{\Sigma^*,\ell-3}(i)$ with $f_{\ell-2}=xy$. As already discussed in Section \[sec:main-proof:variables\], by construction we have $O_{\Sigma^*}(i) \subseteq O_{\tilde{v},U}(i)$, and ${{\mathcal U}}_T$ implies $|O_{\Sigma^*}(i)| = |Z_{\Sigma^*,\ell-3}(i)|$. Together with this establishes $$\label{eq:thm:main_result:many_open}
|O_{\tilde{v},U}(i)| \geq \delta \frac{u^{2} i^{\ell-3}}{n^{\ell-1}} |O(i)| .$$ Using this estimate, we now prove . To this end fix $(\tilde{v},U) \in [n] \times \binom{[n]}{u}$ with $\tilde{v} \notin U$. We see that $$\label{eq:thm:main_result:prob_no_extension_prod}
\begin{split}
{{\mathbb P}}[{{\mathcal X}}_{\tilde{v},U,m} \cap {{\mathcal A}}_m] &= {{\mathbb P}}[{{\mathcal X}}_{\tilde{v},U,m/2} \cap {{\mathcal A}}_{m/2}] \prod_{m/2 \leq i \leq m-1} {{\mathbb P}}[{{\mathcal X}}_{\tilde{v},U,i+1} \cap {{\mathcal A}}_{i+1} \mid {{\mathcal X}}_{\tilde{v},U,i} \cap {{\mathcal A}}_i]\\
&\leq \prod_{m/2 \leq i \leq m-1} {{\mathbb P}}[e_{i+1} \notin O_{\tilde{v},U}(i) \mid {{\mathcal X}}_{\tilde{v},U,i} \cap {{\mathcal A}}_i] .
\end{split}$$ Note that ${{\mathcal X}}_{\tilde{v},U,i} \cap {{\mathcal A}}_i$ depends only on the first $i$ steps of the process, so given this, the process fails to choose $e_{i+1}$ from $O_{\tilde{v},U}(i)$ with probability $1-|O_{\tilde{v},U}(i)|/|O(i)|$. Now from and as well as the inequality $1-x \leq e^{-x}$ we deduce, with room to spare, $$\label{eq:prob_no_extension_closed}
{{\mathbb P}}[{{\mathcal X}}_{\tilde{v},U,m} \cap {{\mathcal A}}_m] \leq \exp\left\{- \delta \frac{u^{2} }{n^{\ell-1}} \sum_{m/2 \leq i \leq m-1} i^{\ell-3}\right\} \leq \exp\left\{- \frac{\delta}{2^{\ell}} \frac{u^{2} m^{\ell-2}}{n^{\ell-1}}\right\} .$$ Substituting the definitions of $m$, $u$, $p$ and $t_{\max}$ into we obtain $${{\mathbb P}}[{{\mathcal X}}_{\tilde{v},U,m} \cap {{\mathcal A}}_m] \leq \exp\left\{- \frac{\delta \gamma }{2^{\ell}} n^{\ell-2} p^{\ell-1} t_{\max}^{\ell-1} u \right\} = \exp\left\{- \gamma \frac{\delta \mu^{\ell-1}}{2^{\ell}} u \log n\right\} \leq n^{- 2 u} ,$$ where the last inequality follows from , i.e., the definition of $\gamma$. Finally, taking the union bound over all choices of $(\tilde{v},U)$ implies , which, as explained, completes the proof.
Trajectory verification {#sec:trajectory_verification}
=======================
This section is devoted to the proof of Lemma \[lem:dem:trajectories\]. Henceforth we work with the ‘natural’ filtration given by the $C_{\ell}$-free process, where ${{\mathcal F}}_i$ corresponds to the first $i$ steps, and tacitly assume that $n$ is sufficiently large whenever necessary. For every $0 \leq i \leq m$ we set ${{\mathcal H}}_i = {{\mathcal J}}_i \cap {{\mathcal T}}_i$, where ${{\mathcal J}}_i$, ${{\mathcal T}}_i$ are defined as in Theorem \[thm:BohmanKeevash2010H\]. Clearly, ${{\mathcal H}}_m$ holds whp. Furthermore ${{\mathcal H}}_{i+1} \subseteq {{\mathcal H}}_i$ and ${{\mathcal H}}_i \in {{\mathcal F}}_i$, since ${{\mathcal H}}_i$ is monotone decreasing and depends only on the first $i$ steps. We set $s=n^2p$ and apply the differential equation method (Lemma \[lem:dem\]) with ${{\mathcal V}}=\{0, \ldots, \ell-3\}$. Recalling that ${{{\mathcal B}}_{{i}}(\Sigma)}$ is monotone increasing, we see that ${{{\mathcal B}}_{{i}}(\Sigma)}= {{{\mathcal B}}_{\leq {i}}(\Sigma)}$. For all $\sigma \in {{\mathcal C}}\times {{\mathcal V}}$ we define $$\label{def:dem:parameters}
u_{\sigma} = k n^{15\ell{\varepsilon}} = \omega(1), \qquad \lambda_{\sigma} = \tau_{\sigma} = n^{{\varepsilon}}, \qquad \beta_{\sigma}=1, \qquad \text{ and } \qquad s_{\sigma} = s_{o} = n^{2{\varepsilon}} .$$ Formally, for all $\sigma = (\Sigma,j) \in {{\mathcal C}}\times {{\mathcal V}}$ we set $X_{\sigma}(i) = |T_{\Sigma,j}(i)|$ and $Y^{\pm}_{\sigma}(i) = |T^{\pm}_{\Sigma,j}(i)|$, where $T^+_{\Sigma,j}(i) = T_{\Sigma,j}(i+1) \setminus T_{\Sigma,j}(i)$ and $T^-_{\Sigma,j}(i) = T_{\Sigma,j}(i) \setminus T_{\Sigma,j}(i+1)$. But, for the sake of clarity, we will henceforth just use $|T_{\Sigma,j}(i)|$ and $|T^{\pm}_{\Sigma,j}(i)|$. Now, for every $\sigma = (\Sigma,j) \in {{\mathcal C}}\times {{\mathcal V}}$ we set $x_{\sigma}(t) = x_j(t)$, $y_{\sigma}^{\pm}(t) = x^{\pm}_j(t)$, $S_{\sigma} = S_j$, $f_{\sigma}(t) = f_j(t)$ and $h_{\sigma}(t) = h_j(t)$, where $$\begin{aligned}
\label{def:xj_Sj}
x_j(t) &= 1/j! \cdot (2t)^{j}{q(t)}^{\ell-2-j}, &
S_j &= k^2 r^{\ell-3}p^j , \\
\label{def:xpj_fj}
x^{+}_j(t) &= 2j/j! \cdot (2t)^{j-1}{q(t)}^{\ell-2-j}, &
f_j(t) &= f(t) {q(t)}^{\ell-3-j} , \\
\label{def:xmj_hj}
x^{-}_j(t) &= 2 (\ell-2-j) (\ell-1) (2t)^{\ell-2}x_j(t),
& h_j(t) &= f'_j(t)/2 . \end{aligned}$$ The definition of $ x^{+}_j(t)$ might seem overly complicated, but it conveniently ensures $x^{+}_0(t)=0$ and $x^{+}_{j}(t) = 2 x_{j-1}(t)/q(t)$ for $j > 0$. With the above parametrization we can restate as $$\label{eq:lem:dem:trajectories:T:simplified}
|T_{\Sigma,j}(i)| = \left( x_j(t) \pm f_j(t)/s_{o} \right) k^2 r^{\ell-3}p^{j} .$$ The remainder of this section is organized as follows. First, in Section \[sec:proof:verification:trend\] we verify the trend hypothesis of Lemma \[lem:dem\], and, next, the boundedness hypothesis in Section \[sec:proof:verification:bound\]. Finally, in Section \[sec:proof:verification:end\] we check the remaining conditions of the differential equation method.
Trend hypothesis {#sec:proof:verification:trend}
----------------
In order to establish , whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds, for every $j \in {{\mathcal V}}$ we have to prove $$\label{eq:Tj:th_goal}
{{\mathbb E}}[|T^{\pm}_{\Sigma,j}(i)| \mid {{\mathcal F}}_i ] = \left(x^{\pm}_j(t) \pm \frac{h_j(t)}{s_{o}} \right) \frac{k^2 r^{\ell-3}p^{j}}{n^2p} .$$
### Basic estimates {#sec:pm_inequalities}
The following inequalities were given in [@Warnke2010K4], and can easily be verified using elementary calculus. Recall that $a \pm b$ denotes the interval $\{a + xb : -1 \leq x \leq 1 \}$, see Section \[sec:notation\].
\[lem:pm\_inequalities\] Suppose $0 \leq x \leq 1/2$. Then $$\label{eq:pm_inverse}
(1 \pm x)^{-1} \subseteq 1 \pm 2 x .$$
\[lem:pm\_product\_inequality\_extended\] Suppose $x,y, f_x, f_y, g, h \geq 0$ and $g \leq 1$. Then $f_x + x g\leq h/2$ implies $$\label{eq:pm_product_ext1}
(1 \pm g) ( x \pm f_{x} ) \subseteq x \pm h .$$ Furthermore, $x f_y + y f_x + f_x f_y + x y g \leq h/2$ implies $$\label{eq:pm_product_ext2}
(1 \pm g) ( x \pm f_{x} ) ( y \pm f_{y} ) \subseteq xy \pm h .$$
### Triples added in one step. {#sec:proof:verification:trend:added}
In this section we verify for $T^{+}_{\Sigma,j}(i)$.
**The case $j=0$.** Clearly, adding an edge to $G(i)$ can not create new open tuples in $T_{\Sigma,0}(i)$. Thus we always have $|T^+_{\Sigma,0}(i)| = 0 = x^+_0(t)$, which settles this case.
**The case $j > 0$.** Recall that $e_{i+1} \in O(i)$ is added to $G(i)$. Let $P_{\Sigma,j-1}(i)$ contain all $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j-1}(i)$ for which there exists a path $w_0 \ldots w_j=v_j$ with $w_0 \in A$ in $G(i)$. Similarly, $D_{\Sigma,j-1}(i) \subseteq T_{\Sigma,j-1}(i)$ contains all tuples with $\{f_{j+1}, \ldots, f_{\ell-2}\} \cap C_{f_{j}}(i) \neq \emptyset$, where $f_{j'} = v_{j'-1}v_{j'}$. With these definitions in hand, note that $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j-1}(i)$ is added to $T_{\Sigma,j}(i+1)$, i.e., is in $T_{\Sigma,j}(i+1)$, if and only if $f_j = e_{i+1}$ and $(v_0, \ldots, v_{\ell-2}) \notin P_{\Sigma,j-1}(i) \cup D_{\Sigma,j-1}(i)$, see Section \[sec:main-proof:variables\]. Since the $C_{\ell}$-free process chooses $e_{i+1}$ uniformly at random from $O(i)$, whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds we have $$\label{eq:Tj:th+_0}
{{\mathbb E}}[|T^+_{\Sigma,j}(i)| \mid {{\mathcal F}}_i ] = \sum_{(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j-1}(i) \setminus [P_{\Sigma,j-1}(i) \cup D_{\Sigma,j-1}(i)]} \frac{1}{|O(i)|} .$$ We now bound the size of $P_{\Sigma,j-1}(i)$. Since ${{\mathcal H}}_i$ implies , the degree of every vertex is bounded by, say, $npn^{{\varepsilon}}$. So, using $|A| = k \leq npn^{{\varepsilon}}$, $j \leq \ell-3$, $(np)^{\ell-2}=n^{1- 1/(\ell-1)}$ and $r \geq n/\ell$, in $G(i)$ the number of $w_j$ for which there exists a path $w_0 \ldots w_j$ with $w_0 \in A$ is at most $$\label{eq:Tj:th:ineq}
|A| \cdot (npn^{{\varepsilon}})^{j} \leq (npn^{{\varepsilon}})^{\ell-2} \leq n^{1+\ell{\varepsilon}- 1/(\ell-1)} \leq r n^{-1/(2\ell)} .$$ Given $w_j$, we now bound the number of $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j-1}(i)$ with $w_j=v_j$. Observe that there are at most $k (npn^{{\varepsilon}})^{j-1}$ choices for such $v_1, \ldots, v_{j-1}$, and at most $r^{\ell-j-3}k$ choices for $v_{j+1}, \ldots, v_{\ell-2}$. Putting things together, we deduce that $$\label{eq:Tj:th:P_Sigma_size}
|P_{\Sigma,j-1}(i)| \leq r n^{-1/(2\ell)} \cdot k (npn^{{\varepsilon}})^{j-1} \cdot r^{\ell-j-3} k \leq k^2 r^{\ell-3} p^{j-1} n^{-1/(3\ell)} .$$ Turning to $D_{\Sigma,j-1}(i)$, we first consider the case where $0 < j < \ell-3$. Suppose that $f_h \in C_{f_{j}}(i)$. Depending on whether $h=j+1$ or $h > j+1$, there exists either a path $v_{j-1}=w_1 \cdots w_{\ell-1}=v_h$ with $j < h < \ell-2$, or a path $w_1 \cdots w_{\kappa}=v_{h-1}$ with $w_1 \in \{v_j,v_{j-1}\}$, $1 < \kappa \leq \ell-2$ and $j < h-1<\ell-2$, cf. Figure \[fig:tuples:selfclosing\].
(60.00, 50.00)(0,0) (0,-2.75)[![\[fig:tuples:selfclosing\]The solid lines represent paths such that adding both $f_{j}=v_{j-1}v_{j}$ and $f_{h}=v_{h-1}v_{h}$ completes a copy of $C_{\ell}$ consisting of those paths. In other words, adding $f_{j}$ closes $f_{h}$, i.e., $f_{h} \in C_{f_{j}}(i)$.](cycle_c1 "fig:")]{} (22.67,8.12)[$v_{h}$]{} (41.19,36.46)[$v_{j-1}$]{} (41.19,8.12)[$v_j\!=\!v_{h-1}$]{}
(82.00, 50.00)(0,0) (0,0)[![\[fig:tuples:selfclosing\]The solid lines represent paths such that adding both $f_{j}=v_{j-1}v_{j}$ and $f_{h}=v_{h-1}v_{h}$ completes a copy of $C_{\ell}$ consisting of those paths. In other words, adding $f_{j}$ closes $f_{h}$, i.e., $f_{h} \in C_{f_{j}}(i)$.](cycle_c2_a "fig:")]{} (6.67,36.46)[$v_h$]{} (6.67,8.12)[$v_{h-1}$]{} (49.19,36.46)[$v_{j-1}$]{} (49.19,8.12)[$v_{j}$]{}
(82.00, 50.00)(0,0) (0,0)[![\[fig:tuples:selfclosing\]The solid lines represent paths such that adding both $f_{j}=v_{j-1}v_{j}$ and $f_{h}=v_{h-1}v_{h}$ completes a copy of $C_{\ell}$ consisting of those paths. In other words, adding $f_{j}$ closes $f_{h}$, i.e., $f_{h} \in C_{f_{j}}(i)$.](cycle_c2_b "fig:")]{} (6.67,36.46)[$v_h$]{} (6.67,8.12)[$v_{h-1}$]{} (49.19,36.46)[$v_{j-1}$]{} (49.19,8.12)[$v_{j}$]{}
So, in both cases, there exists a path $w_1 \cdots w_{\kappa}=v_{x}$ with $w_1 \in \{v_j,v_{j-1}\}$, $1 < \kappa \leq \ell-1$ and $j < x < \ell-2$. With this observations in hand, we are now ready to estimate the number of tuples $(v_{0}, \ldots, v_{\ell-2}) \in D_{\Sigma,j-1}(i)$. Recall that by ${{\mathcal H}}_i$ the degree of every vertex is at most $npn^{{\varepsilon}}$. It follows that there are at most $k (npn^{{\varepsilon}})^{j-1}r$ choices for $v_0, \ldots, v_{j}$, and at most $\ell^2$ choices for $h$ and $x$. Given $v_0, \ldots, v_{j}$ as well as $h$ and $x$, there are at most $2\ell(npn^{{\varepsilon}})^{\ell-2} \leq r n^{-1/(3\ell)}$ choices for $v_{x}$ by . Since we already picked $v_x$ with $j < x < \ell-2$, for the remaining vertices among $v_{j+1}, \ldots, v_{\ell-2}$ we have at most $r^{\ell-j-4}k$ choices. Putting things together, we see that for $0 < j < \ell-3$ we have $$\label{eq:Tj:th:D_Sigma_size}
|D_{\Sigma,j-1}(i)| \leq k (npn^{{\varepsilon}})^{j-1} \cdot r \cdot \ell^2 \cdot r n^{-1/(3\ell)} \cdot r^{\ell-j-4}k \leq k^2 r^{\ell-3} p^{j-1} n^{-1/(4\ell)} .$$ Now we bound $|D_{\Sigma,j-1}(i)|$ for the remaining case $j = \ell-3$. Recall that $f_{\ell-3}= v_{\ell-4}v_{\ell-3}$. If $f_{\ell-2}=v_{\ell-3}v_{\ell-2} \in C_{f_{\ell-3}}(i)$, then, with a similar reasoning as in the previous case, there exists a path $v_{\ell-4}=w_{0} \cdots w_{\ell-2}=v_{\ell-2}$, where $v_{\ell-4} \in N^{(\ell-4)}(A,R)$ and $v_{\ell-2} \in B$. Since $\neg{{{\mathcal B}}_{{i}}(\Sigma)}$ holds, by $\neg{{{\mathcal B}}_{{1,i}}(\Sigma)}$ there are at most $k^2(np)^{\ell-4}n^{-9{\varepsilon}}$ such pairs $(v_{\ell-2}, v_{\ell-4}) \in B \times N^{(\ell-4)}(A,R)$ in $G(i)$. Recall that by the extension property ${{\mathcal U}}_{T}$ (cf. Lemma \[lemma:extension:property\]) every triple $(v_{\ell-4},v_{\ell-3}, v_{\ell-2})$ is contained in at most one tuple in $T_{\Sigma,\ell-4}(i)$. So, since there are at most $k^2(np)^{\ell-4}n^{-9{\varepsilon}}$ choices for $v_{\ell-4},v_{\ell-2}$, and at most $r$ choices for $v_{\ell-3} \in V_{\ell-3}$, using ${{\mathcal U}}_{T}$ we deduce that for $j=\ell-3$ we have $$\label{eq:Tj:th:D_Sigma_size_l3}
|D_{\Sigma,j-1}(i)| \leq k^2(np)^{\ell-4}n^{-9{\varepsilon}} \cdot r \leq k^2 r^{\ell-3} p^{\ell-4} n^{-8{\varepsilon}} = k^2 r^{\ell-3} p^{j-1} n^{-8{\varepsilon}} .$$ After these preparations, we now estimate whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds. Observe that ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}$ implies ${{\mathcal G}}_{i}(\Sigma)$, and so $|T_{\Sigma,j-1}(i)|$ satisfies . Furthermore, since ${{\mathcal H}}_i$ holds, this implies that $|O(i)|$ satisfies . In addition, note that $s_{e} = n^{1/(2\ell)-{\varepsilon}}$ and imply $f(t)/s_{e} = o(1)$ and $f_{j-1}(t) \geq 1$. Substituting the former estimates and – into , using $n^{1/(3\ell)} \geq n^{8{\varepsilon}} = \omega(s_{o})$, , $x^{+}_{j}(t) = 2 x_{j-1}(t)/q(t)$ and $f_{j}(t) = f_{j-1}(t) / q(t)$, we deduce that $$\label{eq:Tj:th+_1}
\begin{split}
{{\mathbb E}}[|T^+_{\Sigma,j}(i)| \mid {{\mathcal F}}_i ] &= \frac{( x_{j-1}(t) \pm f_{j-1}(t)/s_{o}) k^2 r^{\ell-3}p^{j-1} \pm 2k^2 r^{\ell-3} p^{j-1} n^{-8{\varepsilon}}}{(1 \pm 3f(t)/s_e)q(t)n^2/2}\\
& \subseteq \frac{( x_{j-1}(t) \pm 2f_{j-1}(t)/s_{o}) k^2 r^{\ell-3}p^{j-1}}{(1 \pm 3f(t)/s_e)q(t)n^2/2}\\
& \subseteq (1 \pm 6f(t)/s_e) \cdot (x_{j}^{+}(t) \pm 4f_{j}(t)/s_{o}) \cdot k^2 r^{\ell-3}p^{j}/(n^2p) .
\end{split}$$ Therefore the desired bound, i.e., for $T^{+}_{\Sigma,j}(i)$, follows if $$\label{eq:Tj:th+_sufficient_goal}
(1 \pm 6f(t)/s_e) \cdot (x_{j}^{+}(t) \pm 4f_{j}(t)/s_{o}) \subseteq x_{j}^{+}(t) \pm h_j(t)/s_{o} .$$ Now, using $f(t)=o(s_e)$ and Lemma \[lem:pm\_product\_inequality\_extended\], by writing down the assumptions of and multiplying both sides with $2 s_{o}$, observe that follows from $$\label{eq:Tj:th+__sufficient}
8 f_{j}(t) + 12 x_{j}^{+}(t)f(t) s_{o}/s_{e} \leq h_j(t) .$$ Using and we see that the second term on the left hand side is $o(1)$. So, it suffices if $$8 f_j(t) + 1 \leq h_j(t) ,$$ which is easily seen to be true, since $h_j(t) \geq W/4 \cdot (f_j(t) + 1)$ and $W \geq 50$ by , and .
### Triples removed in one step {#sec:proof:verification:trend:removed}
Next, we prove for $T^{-}_{\Sigma,j}(i)$. Since the rules for removing tuples from $T_{\Sigma,j}(i)$ are different for $j < \ell-3$ and $j = \ell-3$, we use a case distinction.
**The case $j < \ell-3$.** Recall that a tuple $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j}(i)$ is removed, i.e., not in $T_{\Sigma,j}(i+1)$, if $e_{i+1} \in \{f_{j+1}, \ldots, f_{\ell-2}\}$ or $e_{i+1} \in C_{f_{j+1}}(i) \cup \cdots \cup C_{f_{\ell-2}}(i)$. Since the edge $e_{i+1}$ is chosen uniformly at random from $O(i)$, whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds, using $|\{f_{j+1}, \ldots, f_{\ell-2}\}| \leq \ell$ we have $$\label{eq:Tj:th-_0}
{{\mathbb E}}[|T^-_{\Sigma,j}(i)| \mid {{\mathcal F}}_i ] = \sum_{(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j}(i)} \frac{|C_{f_{j+1}}(i) \cup \cdots \cup C_{f_{\ell-2}}(i)| \pm \ell}{|O(i)|} .$$ Note that ${{\mathcal H}}_i$ implies that the inequalities , and hold. In particular, using $n^{1/\ell} = \omega(s_e)$, $n^{-1/\ell} p^{-1} = \omega(1)$ and $f(t) \geq 1$, this yields $$\label{eq:Tj:th-closed}
\begin{split}
|C_{f_{j+1}}(i) \cup \cdots \cup C_{f_{\ell-2}}(i)| \pm \ell &\subseteq (\ell-j-2)[(\ell-1)(2t)^{\ell-2} q(t) \pm 7\ell f(t)/s_e]p^{-1} \pm \ell^2 n^{-1/\ell} p^{-1} \pm \ell\\
& \subseteq (\ell-j-2)[(\ell-1)(2t)^{\ell-2}q(t) \pm 9\ell f(t)/s_e]p^{-1} .
\end{split}$$ Since ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}$ implies ${{\mathcal G}}_{i}(\Sigma)$, it follows that $|T_{\Sigma,j}(i)|$ satisfies . In addition, as in Section \[sec:proof:verification:trend:added\], $f(t)/s_e = o(1)$ holds and $|O(i)|$ satisfies by ${{\mathcal H}}_i$. Substituting the former estimates into , and using as well as $x^{-}_{j}(t)/x_{j}(t) = 2 (\ell-j-2) (\ell-1) (2t)^{\ell-2}$, we obtain $$\label{eq:Tj:th-_1}
\begin{split}
& {{\mathbb E}}[|T^-_{\Sigma,j}(i)| \mid {{\mathcal F}}_i ] = \frac{( x_{j}(t) \pm f_j(t)/s_{o}) k^2 r^{\ell-3}p^{j} \cdot (\ell-j-2)[ (\ell-1) (2t)^{\ell-2} q(t) \pm 9\ell f(t)/s_e]p^{-1}}{(1 \pm 3f(t)/s_e)q(t)n^2/2}\\
& \qquad \subseteq (1 \pm 6f(t)/s_e ) \cdot ( x_{j}(t) \pm f_j(t)/s_{o} ) \cdot [x^{-}_{j}(t)/x_{j}(t) \pm 20 \ell^2 f(t)/({q(t)}s_e) ] \cdot k^2 r^{\ell-3}p^{j}/(n^2p) .
\end{split}$$ Therefore the desired bound, i.e., for $T^{-}_{\Sigma,j}(i)$, follows if $$\label{eq:open_triples:th_sufficient_goal}
(1 \pm 6f(t)/s_e ) \cdot ( x_{j}(t) \pm f_j(t)/s_{o} ) \cdot [ x^{-}_{j}(t)/x_{j}(t) \pm 20 \ell^2 f(t)/({q(t)}s_e) ] \subseteq x_j^-(t) \pm h_j(t)/s_{o} .$$ We now show using Lemma \[lem:pm\_product\_inequality\_extended\]. Similar as for the added tuples, by writing down the assumptions of , multiplying with $2 s_{o}$ and then noticing that all terms containing $s_{e}$ contribute $o(1)$, we see that it suffices if $$(\ell-2-j)(\ell-1)2^{\ell} t^{\ell-2} f_j(t) + 1 \leq h_j(t) ,$$ which is easily seen to be true, since $h_j(t) \geq W/2 \cdot ( t^{\ell-2} f_j(t) + 1)$ and $W/2 \geq \ell^2 2^{\ell}$ by and .
**The case $j = \ell-3$.** Recall that a tuple $(v_{0}, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)$ is removed, i.e., not in $T_{\Sigma,\ell-3}(i+1)$, if $e_{i+1}=f_{\ell-2}$, or in addition to $e_{i+1} \in C_{f_{\ell-2}}(i)$ it is not ignored. A moment’s thought reveals that for every $(v_{0}, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)$ with $e_{i+1} \in C_{f_{\ell-2}}(i)$, if $e_{i+1} \notin L_{\Sigma}(i)$ then (R2) holds, where $L_{\Sigma}(i)$ is as in the definition of ${{{\mathcal B}}_{{2,i}}(\Sigma)}$. In other words, for every $(v_{0}, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)$ we see that $e_{i+1} \in C_{f_{\ell-2}}(i) \setminus L_{\Sigma}(i)$ is a sufficient condition for being removed. Clearly, a necessary condition for being removed is $e_{i+1} \in \{f_{\ell-2}\} \cup C_{f_{\ell-2}}(i)$. Combining our previous findings and using that $e_{i+1}$ is chosen uniformly at random from $O(i)$, whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds we deduce that $$\label{eq:Tl3:th-_0}
{{\mathbb E}}[|T^-_{\Sigma,\ell-3}(i)| \mid {{\mathcal F}}_i ] = \sum_{(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)} \frac{|C_{f_{\ell-2}}(i)| \pm |L_{\Sigma}(i)| \pm 1}{|O(i)|} .$$ Recall that ${{\mathcal H}}_i$ implies the inequalities and . Furthermore, since $\neg{{{\mathcal B}}_{{2,i}}(\Sigma)}$ holds, we have $|L_{\Sigma}(i)| \leq p^{-1}n^{-1/(2\ell)}$. So, similar as in the previous case, using $n^{1/(2\ell)} = \omega(s_e)$, $n^{-1/(2\ell)} p^{-1} = \omega(1)$ and $f(t) \geq 1$, we obtain $$\label{eq:Tl3:th-closed}
\begin{split}
|C_{f_{\ell-2}}(i)| \pm |L_{\Sigma}(i)| \pm 1 &\subseteq [(\ell-1) (2t)^{\ell-2}q(t) \pm 7\ell f(t)/s_e]p^{-1} \pm p^{-1}n^{-1/(2\ell)} \pm 1\\
& \subseteq [(\ell-1) (2t)^{\ell-2}q(t) \pm 9\ell f(t)/s_e]p^{-1} ,
\end{split}$$ where the final estimate equals that of for $j=\ell-3$. It is not difficult to see that the remaining calculations of the case $j< \ell-3$ carry over word by word, which yields for $T^{-}_{\Sigma,\ell-3}(i)$. To summarize, we have verified the trend hypothesis .
Boundedness hypothesis {#sec:proof:verification:bound}
----------------------
Observe that in order to verify the boundedness hypothesis , using it suffices to show that whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds, for every $j \in {{\mathcal V}}$ we have $$\label{eq:dem:Tj:max_bound}
|T^{\pm}_{\Sigma,j}(i)| \leq k r^{\ell-3} p^{j} n^{-20\ell{\varepsilon}} .$$
### Triples added in one step. {#sec:proof:verification:bound:added}
In this section we verify for $T^{+}_{\Sigma,j}(i)$. Recall that $e_{i+1} \in O(i)$ is added to $G(i)$. By construction we always have $|T^{+}_{\Sigma,0}(i)| = 0$, and thus we henceforth consider the case $j > 0$. Note that a necessary condition for $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j-1}(i)$ being added to $T_{\Sigma,j}(i+1)$ is $f_j = e_{i+1}$. Observe that there are at most $k r^{\ell-3-j}$ choices for $(v_{j+1}, \ldots, v_{\ell-2}) \in V_{j+1} \times \cdots \times V_{\ell-3} \times B$. So, using the extension property ${{\mathcal U}}_T$ (cf. Lemma \[lemma:extension:property\]), we deduce that for each $e_{i+1}$ there are at most $k r^{\ell-3-j}$ tuples in $T_{\Sigma,j-1}(i)$ with $f_j = e_{i+1}$. Together with , , and $j \geq 1$ this implies $$\label{eq:dem:Tj:add_bound}
|T^{+}_{\Sigma,j}(i)| \leq k r^{\ell-3-j} = k r^{\ell-3}p^{j} \cdot (rp)^{-j} = o(k r^{\ell-3} p^{j} n^{-20\ell{\varepsilon}}) ,$$ as desired.
### Triples removed in one step {#sec:proof:verification:bound:removed}
Next we use case distinction to establish for $T^{-}_{\Sigma,j}(i)$.
**The case $j < \ell-3$.** We claim that whenever ${{\mathcal E}}_i \cap \neg{{{\mathcal B}}_{{i}}(\Sigma)}\cap {{\mathcal H}}_i$ holds, for all $(v_{0}, \ldots, v_{\ell-2}) \in T_{\Sigma,j}(i)$ and every $xy \in \{f_{j+1}, \ldots, f_{\ell-2}\}$, the number of tuples in $T_{\Sigma,j}(i)$ containing $xy$ is bounded by $$\label{eq:dem:Tj:max_contained}
kr^{\ell-4}p^{j}n^{\ell{\varepsilon}} .$$
First suppose that $xy = f_{j+1}$. For $(v_{j+2}, \ldots, v_{\ell-2}) \in V_{j+2} \times \cdots \times V_{\ell-3} \times B$ there are at most $k r^{\ell-4-j} \leq k r^{\ell-4}p^{j}$ choices, and so follows using the extension property ${{\mathcal U}}_T$ (cf. Lemma \[lemma:extension:property\]).
Next we consider the case $xy = f_{\ell-2}$. As usual, whenever ${{\mathcal H}}_i$ holds, the degree of every vertex is bounded by, say, $npn^{{\varepsilon}}$. Since for every $(v_{0}, \ldots, v_{\ell-2}) \in T_{\Sigma,j}(i)$ the vertices $v_{0}, \ldots, v_{j}$ form a path starting in $A$, we deduce that there are at most $k (npn^{{\varepsilon}})^{j}$ choices for such $v_{0}, \ldots, v_{j}$. Furthermore, there are most $r^{\ell-4-j}$ choices for $(v_{j+1}, \ldots, v_{\ell-4}) \in V_{j+1} \times \cdots \times V_{\ell-4}$. Therefore the number of tuples in $T_{\Sigma,j}(i)$ with $xy = f_{\ell-2}$ is bounded by $k (npn^{{\varepsilon}})^{j} \cdot r^{\ell-4-j} \leq k r^{\ell-4} p^{j} n^{\ell{\varepsilon}}$, as claimed by .
Finally we consider the case where $xy = f_{h}$ with $j+1 < h < \ell-2$. With a similar reasoning as in the previous case, there are at most $k (npn^{{\varepsilon}})^{j}$ choices for $v_0, \ldots, v_{j}$, at most $r^{h-j-2}$ choices for $v_{j+1}, \ldots, v_{h-2}$ and at most $k r^{\ell-h-3}$ choices for $v_{h+1}, \ldots, v_{\ell-2}$. To summarize, there are at most $$k (npn^{{\varepsilon}})^{j} \cdot r^{h-j-2} \cdot k r^{\ell-h-3} \leq k^2 r^{\ell-5} p^{j} n^{\ell{\varepsilon}} \leq k r^{\ell-4} p^{j}$$ tuples in $T_{\Sigma,j}(i)$ with $xy = f_{h}$, which establishes , with room to spare.
With the above estimate in hand, we are now ready to bound $|T^{-}_{\Sigma,j}(i)|$. Recall that $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma,j}(i)$ is removed, i.e., not in $T_{\Sigma,j}(i+1)$, if $e_{i+1} \in \{f_{j+1}, \ldots, f_{\ell-2}\}$ or $e_{i+1} \in C_{f_{j+1}}(i) \cup \cdots \cup C_{f_{\ell-2}}(i)$, which is equivalent to $\{f_{j+1}, \ldots, f_{\ell-2}\} \cap C_{e_{i+1}}(i) \neq \emptyset$. In other words, such a tuple is removed if for some $j+1 \leq h \leq \ell-2$ we have $f_{h}=e_{i+1}$ or $f_{h} \in C_{e_{i+1}}(i)$. Recall that whenever ${{\mathcal H}}_i$ holds, by we have, say, $|C_{e_{i+1}}(i)| \leq p^{-1}n^{{\varepsilon}}$. So, using that gives an upper bound for the number of tuples in $T_{\Sigma,j}(i)$ which contain $f_{h}$, we deduce that $$|T^{-}_{\Sigma,j}(i)| \leq (\ell+|C_{e_{i+1}}(i)|) \cdot kr^{\ell-4}p^{j}n^{\ell{\varepsilon}} \leq kr^{\ell-4}p^{j-1} n^{2\ell{\varepsilon}} \leq kr^{\ell-3}p^{j} \cdot n^{2\ell{\varepsilon}}/(rp) ,$$ which, with a similar reasoning as in , establishes for $T^{-}_{\Sigma,j}(i)$ with $j < \ell-3$.
**The case $j = \ell-3$.** Recall that a tuple $(v_{0}, \ldots, v_{\ell-2}) \in T_{\Sigma,\ell-3}(i)$ is removed, i.e., not in $T_{\Sigma,\ell-3}(i+1)$, according to different rules. In the following we bound the total number of tuples removed in one step by each rule, which were called cases $1$ and $2$ in Section \[sec:main-proof:variables\]. In case $1$ we have $f_{\ell-2}=e_{i+1}$ and so, given $e_{i+1}$, using ${{\mathcal U}}_T$ we deduce that at most one tuple is removed under case $1$.
Turning to case $2$, given $e_{i+1}=xy$, note that a necessary condition for being removed by (R2) is that for some $j \in [\ell-1]$ we have $f_{\ell-2} \in C_{x,y,\Sigma}(i,j)$ or $f_{\ell-2} \in C_{y,x,\Sigma}(i,j)$. Recall that by ${{\mathcal U}}_T$ every such pair $f_{\ell-2}$ is contained in at most one tuple in $T_{\Sigma,\ell-3}(i)$. So, since a tuple is only removed if the corresponding $C_{x,y,\Sigma}(i,j)$ or $C_{y,x,\Sigma}(i,j)$ has size at most $p^{-1}n^{-30\ell{\varepsilon}}$, we deduce that at most $2\ell \cdot p^{-1}n^{-30\ell{\varepsilon}}$ tuples are removed in one step by (R2).
Putting it all together, using $p^{-1} = (np)^{\ell-2}$ and $np \leq k$, for $j=\ell-3$ we obtain $$|T^{-}_{\Sigma,\ell-3}(i)| \leq 1 + 2\ell p^{-1}n^{-30\ell{\varepsilon}} \leq (np)^{\ell-2} n^{-25\ell{\varepsilon}} \leq k (np)^{\ell-3} n^{-25\ell{\varepsilon}} ,$$ which readily establishes the boundedness hypothesis .
Finishing the trajectory verification {#sec:proof:verification:end}
-------------------------------------
In this section we verify the remaining conditions of the differential equation method (Lemma \[lem:dem\]).
**Initial conditions.** Using , for $j > 0$ we clearly have $|T_{\Sigma,j}(0)|=0=x_{j}(0)$, which settles these cases. For the remaining case $j=0$ we crudely have $$|T_{\Sigma,0}(0)|=|T_{\Sigma}| = k^2(r \pm kn^{10\ell{\varepsilon}})^{\ell-3} = (1 \pm kn^{10\ell{\varepsilon}}/r)^{\ell-3} k^2r^{\ell-3} \subseteq (1 \pm o(1)/s_{o}) k^2r^{\ell-3} ,$$ which together with $x_0(0)=1$, $S_0 = k^2r^{\ell-3}$ and $\beta_{\sigma}=1$ establishes .
**Bounded number of configurations and variables.** Using $k=u/60$ and we obtain $$\label{eq:bound:nr_configs}
|{{\mathcal C}}| \leq n \cdot \binom{n}{u} \cdot 3^u \cdot \sum_{r \leq kn^{10\ell{\varepsilon}}} \binom{n}{r} \leq n^{2u+kn^{10\ell{\varepsilon}}} < e^{kn^{15\ell{\varepsilon}}} = e^{u_{\sigma}} ,$$ which together with $|{{\mathcal V}}| \leq \ell$ clearly establishes .
**Additional technical assumptions and the function $f_{\sigma}(t)$.** Using $s=n^2p$ as well as , and , straightforward calculations show that holds, with room to spare; we leave the details to the reader. Recall that by we have $t_{\max} = m/s = \Theta((\log n)^{1/(\ell-1)})$. Furthermore, using –, elementary calculus yields $x_{j}^{\pm }(t) = O(t_{\max}^{\ell+j-2})$ and $|x_{j}''(t)| = O(t_{\max}^{2\ell+j-4})$ for $t \leq t_{\max}$. Thus, since for all $\sigma =(\Sigma,j)\in {{\mathcal C}}\times {{\mathcal V}}$ we have $x_{\sigma}(t) = x_{j}(t)$ and $y_{\sigma}^{\pm }(t) = x_{j}^{\pm }(t)$, it follows that $$\sup_{0 \leq t \leq m/s} y_{\sigma}^{\pm }(t) = O(\log^2 n) \leq n^{\varepsilon}= \lambda_{\sigma} \qquad \text{ and } \qquad \int_0^{m/s} |x_{\sigma}''(t)| \ dt = O(\log n \cdot \log^3 n) \leq \lambda_{\sigma} .$$ Recall that for all $\sigma \in {{\mathcal C}}\times {{\mathcal V}}$ we have $h_{\sigma}(t) = f'_{\sigma}(t)/2$ and $f_{\sigma}(t) = f(t) {q(t)}^{\iota}$, where $\iota \in \{0,\ldots, \ell-3\}$. Hence, using $f_{\sigma}(0)=1=\beta_{\sigma}$, we see that $$f_{\sigma}(t) = 2 \int_{0}^{t} h_{\sigma}(\tau) \ d\tau + f_{\sigma}(0) = 2 \int_{0}^{t} h_{\sigma}(\tau) \ d\tau + \beta_{\sigma} .$$ Note that $h_{\sigma}(0)=O(1) \leq n^{3{\varepsilon}} = s_{\sigma} \lambda_{\sigma}$ and $h'_{\sigma}(t) \geq 0$. Pick $t^{*}=t^{*}(\ell) \geq 1$ large enough such that for all $t \geq t^{*}$ we have $t^{2\ell} \leq f(t)$. Observe that $h'_{\sigma}(t)$ is bounded by some constant for $t \leq t^{*}$, and note that for we have, say, $h'_{\sigma}(t) \leq W^3{f(t)}^2$. Putting things together, using and , i.e., $m/s = O(\log n)$ and $f(t) \leq n^{{\varepsilon}}$, we readily obtain $$\int_{0}^{m/s} |h'_{\sigma}(t)| \ dt \leq \int_{0}^{t^{*}} h'_{\sigma}(t) \ dt + \int_{t^{*}}^{m/s} W^3{f(t)}^2 \ dt \leq O(1) + O(\log n \cdot n^{2{\varepsilon}}) \leq n^{3{\varepsilon}} = s_{\sigma} \lambda_{\sigma} .$$ To summarize, we showed that as well as the additional technical assumptions – hold, and this completes the proof of Lemma \[lem:dem:trajectories\].
A ‘transfer theorem’ for the $H$-free process {#sec:transfer}
=============================================
In the $H$-free process there is a complicated dependency among the edges, and thus standard concentration inequalities are not directly applicable. In this section we show how to overcome this problem for decreasing properties by establishing a ‘transfer theorem’. Roughly speaking, this allows us to ‘transfer’ results for decreasing properties from the binomial random graph model to the $H$-free process, at the cost of only slightly increasing the ‘expected’ edge density. In our argument this will be a crucial tool for establishing Lemma \[lem:dem:config\].
Relating the $H$-free process with the uniform random graph
-----------------------------------------------------------
We start by relating the $H$-free process with the more familiar uniform random graph. In the $H$-free process the set of open pairs $O(i)$ is defined in the obvious way: it contains all pairs $xy \in \binom{[n]}{2} \setminus E(i)$ for which $G(i) \cup \{xy\}$ remains $H$-free. The following estimate is not best possible, but it suffices for our purposes and keeps the formulas simple.
\[lem:transfer:uniform\] Suppose ${{\mathcal Q}}$ is a decreasing graph property and that $\lambda=\lambda(n) \geq 2$ is a parameter. Then for every $1 \leq i \leq \binom{n}{2}/\lambda$, setting $M= i \lambda$, we have $$\label{eq:lem:transfer:uniform}
{{\mathbb P}}[G(i) \notin {{\mathcal Q}}\text{ and } |O(i)| \geq n^2/\lambda] \leq {{\mathbb P}}[G_{n,M} \notin {{\mathcal Q}}] + e^{-i/4} ,$$ where $G(i)$ denotes the graph produced by the $H$-free process after the first $i$ steps.
We sequentially generate the edges $e_{1},e_{2},\ldots$, where each edge $e_{j+1}$ is chosen uniformly at random from $E(K_n) \setminus \{e_{1},e_{2},\ldots, e_{j}\}$. On the one hand, the edge-set $\{e_{1},e_{2},\ldots, e_{M}\}$ clearly gives $G_{n,M}$. On the other hand, we obtain the graph produced by the $H$-free process by sequentially traversing the $e_j$ and only adding those edges which do not complete a copy of $H$. First, for every $1 \leq j \leq M$ we define the indicator variable $X_j$ for the event that $e_{j}$ is added to the graph of the $H$-free process, and, furthermore, define the random variable $$X^{j} = \sum_{1 \leq j' \leq j} X_{j'} ,$$ which counts the number of edges in the graph produced by the $H$-free process after traversing $e_1,\ldots,e_{j}$. Next, for every $1 \leq j \leq M$ we define $$Y_{j} = \begin{cases}
1, & \text{if $|O(X^{j-1})| < n^2/\lambda$,}\\
X_{j}, & \text{otherwise,}
\end{cases}
\qquad \text{ and } \qquad Y^j = \sum_{1 \leq j' \leq j} Y_{j'} .$$ If $|O(X^{j-1})| \geq n^{2}/\lambda$ holds, we have $Y_j = X_j$ by construction. In this case the next edge is added to the graph of the $H$-free process with probability at least $|O(X^{j-1})|/\binom{n}{2} \geq 2/\lambda$. Otherwise $Y_{j} = 1$ holds, and so we conclude that ${{\mathbb P}}[Y_{j}=1 \mid Y_1, \ldots, Y_{j-1}] \geq 2/\lambda$, which implies that $Y^{M}$ stochastically dominates a binomial random variable with $M$ trials and success probability $2/\lambda$. With this in mind, standard Chernoff bounds, see e.g. of Lemma \[lem:chernoff\], give $$\label{eq:chernoff}
{{\mathbb P}}[Y^{M} \leq 2i-t] \leq e^{-t^2/(4i)} .$$
In the remainder we prove . To this end first observe that $$\label{eq:lem:transfer:uniform:1}
{{\mathbb P}}[G(i) \notin {{\mathcal Q}}\text{ and } |O(i)| \geq n^{2}/\lambda] \leq {{\mathbb P}}[G(i) \notin {{\mathcal Q}}\text{ and } X^{M} \geq i] + {{\mathbb P}}[|O(i)| \geq n^{2}/\lambda \text{ and } X^{M} < i] .$$ Note that by construction $X^{M} \geq i$ implies $G(i) \subseteq G_{n,M}$, and, since ${{\mathcal Q}}$ is a decreasing graph property, in this case $G(i) \notin {{\mathcal Q}}$ implies $G_{n,M} \notin {{\mathcal Q}}$. It follows that $$\label{eq:prob:subset}
{{\mathbb P}}[G(i) \notin {{\mathcal Q}}\text{ and } X^{M} \geq i] \leq {{\mathbb P}}[G_{n,M} \notin {{\mathcal Q}}] .$$ Furthermore, since $O(i)$ is decreasing, if both $|O(i)| \geq n^{2}/\lambda$ and $X^{M} < i$ hold, then this implies $Y^{M} = X^{M} < i$. So, by we have $$\label{eq:prob:badevent}
{{\mathbb P}}[|O(i)| \geq n^{2}/\lambda \text{ and } X^{M} < i] \leq {{\mathbb P}}[Y^{M} < i] \leq e^{-i/4} .$$ Substituting these bounds into gives , completing the proof.
If we relax the additive error in Lemma \[lem:transfer:uniform\] to $o(1)$, then for $|O(i)| \geq \binom{n}{2}/\lambda$ a slight modification of the above proof works with $M=i\lambda +\omega(1) \lambda \sqrt{i}$; we leave these details to the interested reader.
A ‘transfer theorem’ for decreasing properties
----------------------------------------------
Using Theorem \[thm:BohmanKeevash2010H\] and , we see that $|O(m)| \geq n^{2-{\varepsilon}/2}$ holds whp in the $C_{\ell}$-free process. So, setting $\lambda = \lambda(n) = n^{{\varepsilon}/2}$ and using the ‘asymptotic equivalence’ of the uniform and the binomial random graph for monotone graph properties (see e.g. Section $1.4$ of [@JLR2000RandomGraphs]), Lemma \[lem:transfer:uniform\] readily gives the next theorem. A similar idea is used in [@Wolfovitz2010K4] for $H=K_4$. Observe that the edge-density of $G(m)$ is roughly $2pt_{\mathrm{max}} = \Theta(p (\log n)^{1/(\ell-1)})$ in the $C_{\ell}$-free process. Intuitively, the following theorem thus states that for decreasing properties, $G(m)$ is ‘comparable’ with the binomial random graph with only slightly larger edge density $pn^{{\varepsilon}}$.
\[thm:transfer:binomial\] Define $m=m(n)$ and $p=p(n)$ as in . Suppose that ${\varepsilon}$ is chosen as in and that ${{\mathcal Q}}$ is a decreasing graph property. Then for the $C_{\ell}$-free process we have $$\label{eq:thm:transfer:binomial}
{{\mathbb P}}[G(m) \notin {{\mathcal Q}}] \leq {{\mathbb P}}[G_{n,pn^{{\varepsilon}}} \notin {{\mathcal Q}}] + o(1) .
\vspace{-2.0em}$$
In fact, this result also holds for the $H$-free process, where $H$ is strictly $2$-balanced, if $m$, $p$ and ${\varepsilon}$ are chosen as in Sections $1.2$ and $1.3$ of [@BohmanKeevash2010H], since then $|O(m)| \geq n^{2-{\varepsilon}/2}$, with room to spare. We believe that the above ‘transfer theorem’ will significantly aid in the future analysis of the $H$-free process, since for decreasing properties it often allows us to work with the *much* easier binomial random graph model, which has been extensively studied and for which e.g. sophisticated concentration inequalities are available.
Properties of random graphs {#sec:binomial_results}
===========================
In this section we introduce several decreasing graph properties, which are key ingredients in our proof of Lemma \[lem:dem:config\]. Using the ‘transfer theorem’ of Section \[sec:transfer\], it suffices to prove that they hold whp for the binomial random graph $G_{n,p'}$ with $p' = p n^{{\varepsilon}}$, where $p$ is defined as in and ${\varepsilon}$ is chosen as in . We remark that essentially all results in this section are not best possible, but suffice for our purposes. For example, in an attempt to keep the formulas simple, we have not optimized the multiplicative $n^{{\varepsilon}}$ factors involved (their contribution in our later arguments will be negligible).
Basic properties {#sec:binomial_results:basic_properties}
----------------
\[lem:bounded\_codegree\]Let ${{\mathcal N}}$ denote the event that for all pairs of distinct vertices $x,y \in [n]$ we have $|\Gamma(x) \cap \Gamma(y)| \leq 9$. Then ${{\mathcal N}}$ holds whp in $G_{n,p'}$.
Using $\ell \geq 4$, and , i.e., $p=n^{-1+1/(\ell-1)} \leq n^{-2/3}$ and ${\varepsilon}\leq 1/20$, we deduce that $${{\mathbb P}}[\neg{{\mathcal N}}] \leq \binom{n}{2} \binom{n-2}{10} (pn^{{\varepsilon}})^{20} \leq n^2 (n p^2 n^{2{\varepsilon}})^{10} \leq n^2 (n^{-1/3+2{\varepsilon}})^{10} = o(1) ,$$ as claimed.
The following result states that every set of size at most $u$ contains a large independent subset. A similar argument was used by Bollobás and Riordan in [@BollobasRiordan2000].
\[lem:set:independent:subset\] Let ${{\mathcal I}}$ denote the event that for every $U \subseteq [n]$ with $|U| \leq u$ there exists an independent set $S \subseteq U$ with $|S| \geq |U|/6$. Then ${{\mathcal I}}$ holds whp in $G_{n,p'}$.
Let ${{\mathcal E}}$ denote the event that every $U \subseteq [n]$ with $|U| \leq u$ spans less than $3|U|$ edges. We have $${{\mathbb P}}[\neg{{\mathcal E}}] \leq \sum_{1 \leq x \leq u} \binom{n}{x} \binom{\binom{x}{2}}{3x} (pn^{{\varepsilon}})^{3x} \leq \sum_{1 \leq x \leq u} \left(\frac{ne}{x}\right)^x \left(\frac{xe}{6}\right)^{3x} (pn^{{\varepsilon}})^{3x} \leq \sum_{x \geq 1} \left(n u^2 p^3 n^{3{\varepsilon}}\right)^x .$$ Using $\ell \geq 4$, , and , i.e., $u \leq np n^{\varepsilon}$, $p=n^{-1+1/(\ell-1)} \leq n^{-2/3}$ and ${\varepsilon}\leq 1/60$, we see that $$n u^2 p^3 n^{3{\varepsilon}} \leq n^3 p^5 n^{5{\varepsilon}} \leq n^{-1/3+5{\varepsilon}} \leq n^{-1/4} ,$$ which implies ${{\mathbb P}}[\neg{{\mathcal E}}] = o(1)$. Suppose that ${{\mathcal E}}$ holds. Then every set of at most $u$ vertices induces a graph with minimum degree less than six. Given $U \subseteq [n]$ with $|U| \leq u$, we set $W = U$. Now, by iteratively selecting a vertex $v \in W$ with at most five neighbours in $G[W]$ and removing $\{v\} \cup \Gamma(v)$ from $W$, we obtain an independent set with at least $|U|/6$ vertices, and the proof is complete.
Bounding the numbers of certain paths {#sec:binomial_results:paths}
-------------------------------------
The results in this section give estimates for the numbers of certain paths. Their statements will contain certain exceptions, and, as we shall see, many of these complications are in fact necessary.
### Preliminaries: the size of certain neighbourhoods {#sec:binomial_results:paths:preliminaries}
The following crude upper bound on the degree of every vertex readily follows from standard Chernoff bounds (Lemma \[lem:chernoff\]) – we omit the straightforward details.
\[lem:set:neighbourhood:size\] Let ${{\mathcal D}}$ denote the event that for every $v \in [n]$ we have $|\Gamma(v)| \leq npn^{2{\varepsilon}}$. Then ${{\mathcal D}}$ holds whp in $G_{n,p'}$.
With similar reasoning it is also not difficult to see that whp for all large sets $S$, in $G_{n,p'}$ we have, say, $|\Gamma(S)| \geq |S| np$, which is much larger than $|S|$. Intuitively, the next lemma thus implies that for most reasonable sized $A \subseteq [n]$, only a small proportion of $\Gamma(S)$ is contained in $N^{(\leq \ell-3)}(A,S \cup A)$.
\[lem:edges:bounded\] Let ${{\mathcal M}}$ denote the event that for all disjoint $A,S \subseteq [n]$ with $|A|,|S| \leq kn^{5{\varepsilon}}$ we have $$\label{eq:lem:edges:bounded}
e\bigl(S,\; N^{(\leq \ell-3)}(A,S \cup A)\bigr) \leq k n^{4\ell{\varepsilon}} .$$ Then ${{\mathcal M}}$ holds whp in $G_{n,p'}$.
Let $\Psi$ contain all pairs $(A,S)$ with disjoint $A,S \subseteq [n]$ satisfying $|A|,|S| \leq kn^{5{\varepsilon}}$. Given $\psi = (A,S) \in \Psi$, let ${{\mathcal M}}_{\psi}$ denote the event that holds, and let ${{\mathcal Y}}_{\psi}$ contain all $Y \subseteq A \cup \bigcup_{1 \leq d \leq \ell-3} V_{d}(S \cup A)$ with $|Y| \leq (np n^{2{\varepsilon}})^{\ell-2}n^{5{\varepsilon}}$. Given $\psi = (A,S) \in \Psi$ and $Y \in {{\mathcal Y}}_{\psi}$, let ${{\mathcal N}}_{\psi,Y}$ denote the event that $N^{(\leq \ell-3)}(A,S \cup A) = Y$. Using $k \leq npn^{{\varepsilon}}$, it is not difficult to see that whenever ${{\mathcal D}}$ holds, then for every $\psi \in \Psi$ some ${{\mathcal N}}_{\psi,Y}$ with $Y \in {{\mathcal Y}}_{\psi}$ holds. Furthermore, $\neg{{\mathcal M}}$ clearly implies that some ${{\mathcal M}}_{\psi}$ with $\psi \in \Psi$ fails. So, we obtain $${{\mathbb P}}[\neg {{\mathcal M}}] \leq {{\mathbb P}}[\neg {{\mathcal D}}] + \sum_{\psi = (A,S) \in \Psi} \ \sum_{Y \in {{\mathcal Y}}_{\psi}} {{\mathbb P}}[\neg {{\mathcal M}}_{\psi} \cap {{\mathcal N}}_{\psi,Y}] .$$ Note that for every $\psi = (A,S) \in \Psi$ the events ${{\mathcal N}}_{\psi,Y}$ are mutually exclusive. So, using $|\Psi| \leq n^{2kn^{5{\varepsilon}}}$ and that ${{\mathcal D}}$ holds whp by Lemma \[lem:set:neighbourhood:size\], to finish the proof it is enough to show that for every $\psi = (A,S) \in \Psi$ and $Y \in {{\mathcal Y}}_{\psi}$ we have $$\label{eq:lem:edges:bounded:prob}
{{\mathbb P}}[\neg {{\mathcal M}}_{\psi} \mid {{\mathcal N}}_{\psi,Y}] \leq n^{-\omega(kn^{5{\varepsilon}})} .$$ Observe that we can find $Y=N^{(\leq \ell-3)}(A,S \cup A)$ by starting with $N^{(0)}(A,S \cup A) = A$, and then iteratively testing vertices in $V_{d}(S \cup A)$ to see whether they are adjacent to $N^{(d-1)}(A,S \cup A)$, up to $d=\ell-3$. Since $S$ is disjoint from $A$ and all $V_{d}(S \cup A)$ with $1 \leq d \leq \ell-3$, this exploration has not revealed any pairs between $S$ and $Y$. We deduce that, conditioned on ${{\mathcal N}}_{\psi,Y}$, all edges between $S$ and $Y=N^{(\leq \ell-3)}(A,S \cup A)$ are included independently with probability $p'=pn^{{\varepsilon}}$. Now, using $(np)^{\ell-2}=p^{-1}$ and $\ell \geq 4$, the expected number of these edges is bounded by $$|S| \cdot |Y| \cdot p' \leq kn^{5{\varepsilon}} \cdot (np n^{2{\varepsilon}})^{\ell-2}n^{5{\varepsilon}} \cdot p n^{{\varepsilon}} = k n^{(2\ell+7){\varepsilon}} \leq k n^{(4\ell-1){\varepsilon}} .$$ Thus standard Chernoff bounds, see e.g. of Lemma \[lem:chernoff\], imply , completing the proof.
### Paths ending in the neighbourhood of another set {#sec:binomial_results:paths:neighbourhood}
We start with a technical lemma, which will be used in the subsequent proofs of Lemmas \[lem:path:endpoints\] and \[lem:path:endpoints:pairs\].
\[lem:path:endpoints:shortest\] Let ${{\mathcal Q}}_1$ denote the event that for all $v \in [n]$ and $A,X \subseteq [n]$ with $A \subseteq X$ and $|A|,|X| \leq k n^{5\ell{\varepsilon}}$, for every $2 \leq j \leq \ell-1$ and $0 \leq d \leq \ell-3$ there are at most at most $(np)^{j-1} n^{9 \ell{\varepsilon}}$ vertices $w \in N^{(\leq d)}(A,X)$ for which there exists a path $$\label{eq:lem:path:endpoints:shortest}
v=w_0 \cdots w_{j}=w \qquad \text{ with } \qquad \{w_{0},\ldots,w_{j-1}\} \cap N^{(\leq d)}(A,X) = \emptyset .$$ Then ${{\mathcal Q}}_1$ holds whp in $G_{n,p'}$.
Let $\Psi$ contain all tuples $(v,A,X,j,d)$ with $v \in [n]$, $A,X \subseteq [n]$, $2 \leq j \leq \ell-1$ and $0 \leq d \leq \ell-3$ satisfying $A \subseteq X$ and $|A|,|X| \leq k n^{5\ell{\varepsilon}}$. Given $\psi = (v,A,X,j,d) \in \Psi$, by ${{\mathcal Q}}_{\psi}$ we denote the event that there are at most $(np)^{j-1} n^{9\ell{\varepsilon}}$ vertices $w \in N^{(\leq d)}(A,X)$ for which there exists a path satisfying . Clearly, $\neg{{\mathcal Q}}_1$ implies that some ${{\mathcal Q}}_{\psi}$ with $\psi \in \Psi$ fails.
Next, given $\psi = (v,A,X,j,d) \in \Psi$, we denote by ${{\mathcal Y}}_\psi$ the set of pairs $(Y,Z)$ with $Y \subseteq A \cup \bigcup_{1 \leq d' \leq d} V_{d'}(X)$ and $Z \subseteq [n] \setminus Y$ satisfying $|Y| \leq (np n^{2{\varepsilon}})^{d+1}n^{5\ell{\varepsilon}}$ and $|Z| \leq (npn^{2{\varepsilon}})^{j-1}$. Furthermore, for every $Y \subseteq [n]$ and $v \in [n]$ we inductively define $$\label{eq:def:gamma:vY}
\Gamma^{(0)}(v,Y) = \{v\} \setminus Y \qquad \text{ and } \qquad \Gamma^{(i+1)}(v,Y) = \Gamma(\Gamma^{(i)}(v,Y))\setminus Y .$$ Given $\psi = (v,A,X,j,d) \in \Psi$ and $\phi = (Y,Z) \in {{\mathcal Y}}_\psi$, let ${{\mathcal N}}_{\psi,\phi}$ be the event that $N^{(\leq d)}(A,X) = Y$ and $\Gamma^{(j-1)}(v,Y) = Z$. Whenever ${{\mathcal D}}$ holds, using $k \leq npn^{{\varepsilon}}$ it is easy to see that for every $\psi \in \Psi$ some ${{\mathcal N}}_{\psi,\phi}$ with $\phi \in {{\mathcal Y}}_\psi$ holds. Putting things together, we obtain $${{\mathbb P}}[\neg{{\mathcal Q}}_1] \leq {{\mathbb P}}[\neg {{\mathcal D}}] + \sum_{\psi = (v,A,X,j,d) \in \Psi} \ \sum_{\phi=(Y,Z) \in {{\mathcal Y}}_\psi} {{\mathbb P}}[\neg {{\mathcal Q}}_{\psi} \cap {{\mathcal N}}_{\psi,\phi}] .$$ Since ${{\mathcal D}}$ holds whp by Lemma \[lem:set:neighbourhood:size\], using $|\Psi| \leq n^{3kn^{5\ell{\varepsilon}}}$ and that for every $\psi \in \Psi$ the events ${{\mathcal N}}_{\psi,\phi}$ are mutually exclusive, to complete the proof it suffices to show that for every $\psi = (v,A,X,j,d) \in \Psi$ and $\phi= (Y,Z) \in {{\mathcal Y}}_\psi$ we have $$\label{eq:lem:path:endpoints:shortest:prob}
{{\mathbb P}}[\neg {{\mathcal Q}}_{\psi} \mid {{\mathcal N}}_{\psi,\phi}] \leq n^{-\omega(kn^{5\ell{\varepsilon}})} .$$ Recall that on ${{\mathcal N}}_{\psi,\phi}$ we have $Y=N^{(\leq d)}(A,X)$ and $Z=\Gamma^{(j-1)}(v,Y)$. Every $w \in Y$ for which there exists a path satisfying is contained in $\Gamma(Z)$, and so whenever ${{\mathcal Q}}_{\psi}$ fails we deduce $|\Gamma(Z) \cap Y| \geq (np)^{j-1} n^{9\ell{\varepsilon}}$, which in turn implies $$\label{eq:lem:path:endpoints:shortest:prob:badevent}
e(Y,Z) \geq (np)^{j-1} n^{9\ell{\varepsilon}} .$$ Next we analyse the distribution of the edges between $Y$ and $Z$ conditional on ${{\mathcal N}}_{\psi,\phi}$. We can iteratively determine $Y=N^{(\leq d)}(A,X)$ as in the proof of Lemma \[lem:edges:bounded\]. Then, given $Y$, we can similarly find $Z = \Gamma^{(j-1)}(v,Y)$; by this can clearly be done without testing any pairs between $Y$ and $Z$. It certainly can happen that during the first exploration, i.e., when determining $Y$, we have already revealed some pairs between $Y$ and $Z$, consider e.g. the case where $Z \cap V_1(X) \neq \emptyset$. However, by construction all such pairs are *non-edges*. Therefore the number of edges between $Y$ and $Z$ is stochastically dominated by a binomial distribution with $|Y| \cdot |Z|$ trials and success probability $p'=pn^{{\varepsilon}}$. Using $d \leq \ell-3$ and $j \leq \ell-1$ as well as $(np)^{d+1} \leq (np)^{\ell-2} = p^{-1}$, the expected value of the corresponding binomial random variable is at most $$|Y| \cdot |Z| \cdot p' \leq (np n^{2{\varepsilon}})^{d+1}n^{5\ell{\varepsilon}} \cdot (np n^{2{\varepsilon}})^{j-1} \cdot p n^{{\varepsilon}}
\leq (np)^{j-1} n^{(5\ell+2d+2j+1){\varepsilon}} \leq (np)^{j-1} n^{(9\ell-1){\varepsilon}} .$$ So, since $j \geq 2$ and $k \leq npn^{{\varepsilon}}$, standard Chernoff bounds show that holds with probability at most $e^{-kn^{8\ell{\varepsilon}}}$, see e.g. of Lemma \[lem:chernoff\]. This establishes and thus completes the proof.
Given a vertex $v \in [n]$, we expect that roughly $(np')^{\ell-2}$ vertices $w \in [n]$ are endpoints of a path $v=w_0 \cdots w_{\ell-2}=w$. Loosely speaking, the next lemma states that there are significantly fewer such vertices $w$ if we only count endpoints in a certain restricted set and forbid some exceptional paths. For the argument of Section \[sec:good\_configurations\_exist\] it is important to observe that ${{\mathcal P}}_1$ is monotone decreasing.
\[lem:path:endpoints\] Let ${{\mathcal P}}_1$ denote the event that for all disjoint $A,S \subseteq [n]$ with $|A|,|S|\leq k$ there exists $X \subseteq [n]$ with $|X| \leq k n^{5\ell{\varepsilon}}$, such that for every $v \in S$ there are at most $(np)^{\ell-3} n^{15\ell{\varepsilon}}$ vertices $w \in N^{(\ell-3)}(A,X)$ for which there exists a path $$\label{eq:lem:path:endpoints}
v=w_0 \cdots w_{\ell-2}=w \qquad \text{ with } \qquad w_1 \notin A .$$ Then ${{\mathcal P}}_1$ holds whp in $G_{n,p'}$.
By Lemmas \[lem:set:neighbourhood:size\], \[lem:edges:bounded\] and \[lem:path:endpoints:shortest\] the event ${{\mathcal D}}\cap {{\mathcal M}}\cap {{\mathcal Q}}_1$ holds whp. In the following we are going to argue that for every graph $G$ satisfying those properties, ${{\mathcal P}}_1$ holds as well. As this claim is purely deterministic, it suffices to prove it for fixed disjoint $A,S \subseteq [n]$ with $|A|,|S|\leq k$. By ${{\mathcal M}}$ there are at most $kn^{4\ell{\varepsilon}}$ edges between $S$ and $N^{(\leq \ell-3)}(A,S \cup A)$. Let $V_{S,A}$ contain the endpoints of those edges and define $$\label{eq:lem:path:endpoints:def:X}
X= A \cup S \cup V_{S,A} .$$ Note that $|X| \leq kn^{5\ell{\varepsilon}}$. Given $v \in [n]$, by $W_{v}$ we denote the set of $w \in N^{(\ell-3)}(A,X)$ for which there exists a path satisfying . To finish the proof, it suffices to show that for every $v \in S$ we have $$\label{eq:lem:path:endpoints:goal}
|W_{v}| \leq (np)^{\ell-3} n^{10\ell{\varepsilon}} .$$ Fix $v \in S \subseteq X$. Since $S \cap A = \emptyset$, for every path $v=w_0 \cdots w_{\ell-2}=w$ with $w \in N^{(\ell-3)}(A,X)$ there exists $1 \leq j \leq \ell-2$ such that $$\label{eq:lem:path:endpoints:condition}
\{w_{0},\ldots,w_{j-1}\} \cap N^{(\leq \ell-3)}(A,X) = \emptyset \qquad \text{ and } \qquad w_j \in N^{(\leq \ell-3)}(A,X) .$$ Recall that by assumption $w_1 \notin A$. So, by and we may restrict our attention to the case $j \geq 2$, since $S$ has no neighbours in $N^{(\leq \ell-3)}(A,X) \setminus A$. Now, as ${{\mathcal Q}}_1$ holds, considering $d \leftarrow \ell-3$, for every $2 \leq j \leq \ell-2$ we deduce that there are at most $(np)^{j-1}n^{9\ell{\varepsilon}}$ vertices $w_j \in N^{(\leq \ell-3)}(A,X)$ for which there exists a path $v=w_0 \cdots w_j$ satisfying . Recall that the degree of every vertex is at most $npn^{2{\varepsilon}}$ by ${{\mathcal D}}$. So, given $ w_j$, there are at most $(npn^{2{\varepsilon}})^{\ell-j-2}$ vertices $w \in N^{(\ell-3)}(A,X)$ for which there exists a path $w_j \cdots w_{\ell-2}=w$. Putting things together, we deduce that $$|W_{v}| \leq \sum_{2 \leq j \leq \ell-2} (np)^{j-1} n^{9\ell{\varepsilon}} \cdot (npn^{2{\varepsilon}})^{\ell-j-2} \leq (np)^{\ell-3} n^{15 \ell{\varepsilon}} .$$ As explained, this implies ${{\mathcal P}}_1$, and the proof is complete.
Note that in Lemma \[lem:path:endpoints\] a condition of the form $w_1 \notin A$ is necessary. Indeed, standard Chernoff bounds imply that whp every vertex has degree $\Omega(np')$. Furthermore, e.g. with a similar argument as in the proof of Lemma $10.6$ in [@Bollobas2001RandomGraphs], one can show that whp for all choices of $A,S,X$, for all $Z \subseteq A$ with $|Z|\geq np$ we have, say, $|N^{(\ell-3)}(Z,X)| \geq |Z| (np)^{\ell-3} \geq (np)^{\ell-2}$. So, by picking $A \in \binom{[n]}{k}$ such that it contains at least $np = o(k)$ neighbours of some vertex $v^*$, we have at least $(np)^{\ell-2}$ vertices $w \in N^{(\ell-3)}(A,X)$ which are endpoints of paths $v^*=w_0 \cdots w_{\ell-2}=w$ with $w_1 \in A$, violating the claimed bound.
### Paths connecting two sets {#sec:binomial_results:paths:sets}
Given $A,B,X \subseteq [n]$, for every $j \geq 1$ and $0 \leq d \leq \ell-3$, we say that $w_0 \cdots w_j = v_d \cdots v_0$ is a *$(j,d)$-path* wrt. $(A,B,X)$ if $v_0 \in A$, $w_0 \in B$ and $v_{d'} \in V_{d'}(X)$ for all $1 \leq d' \leq d$, cf. Figure \[fig:jdpaths\].
(154.02, 88.51)(0,0) (0,0)[![\[fig:jdpaths\]Examples of $(2,2)$-paths for $\ell=5$. As usual, solid lines represent edges; for the other pairs there are no restrictions. Note that $w_1$ may be in $A \cup B$ or the vertex classes $V_1 \cup V_2$.](jd_22_path "fig:")]{} (42.35,71.87) (93.56,50.52) (20.14,74.04) (68.60,74.04) (25.20,46.02) (59.22,12.29) (91.39,71.87) (93.56,22.74) (130.00,48.86)
(140.02, 88.51)(0,0) (0,0)[![\[fig:jdpaths\]Examples of $(2,2)$-paths for $\ell=5$. As usual, solid lines represent edges; for the other pairs there are no restrictions. Note that $w_1$ may be in $A \cup B$ or the vertex classes $V_1 \cup V_2$.](jd_22_path_equals_13 "fig:")]{} (48.33,71.87) (99.54,50.52) (26.12,75.17) (81.67,75.17) (19.84,43.75) (62.50,12.29) (97.37,71.87) (99.54,22.74) (53.86,43.75)[$w_1$]{}
Intuitively, the next technical result states that the number of $(j,d)$-paths is not ‘too large’ if we allow for deleting a few edges.
\[lem:path:endpoints:pairs:deletion\] Let ${{\mathcal Q}}_2$ denote the event that for all $A,B \subseteq [n]$ with $|A|,|B|\leq k$ there exists $F \subseteq \binom{[n]}{2}$ with $|F| \leq k n^{2{\varepsilon}}$, such that for every $1 \leq j \leq \ell-1$ and $0 \leq d \leq \ell-4$ the number of $(j,d)$-paths wrt. $(A,B,A \cup B)$ that are edge disjoint from $F$ is bounded by $k^2(np)^{j-3} n^{4 \ell{\varepsilon}}$. Then ${{\mathcal Q}}_2$ holds whp in $G_{n,p'}$.
Fix $A,B \subseteq [n]$ with $|A|,|B|\leq k$. Given $j$ and $d$, we denote by ${{\mathcal S}}_{j,d}={{\mathcal S}}_{j,d}(A,B)$ the family of edge-sets of all possible $(j,d)$-paths wrt. $(A,B,A \cup B)$. Clearly, $|V_{d'}(A \cup B)| \leq n$ for all $1 \leq d' \leq d$. So, using $p = (np)^{-(\ell-2)}$, $j \leq \ell-1$ and $d \leq \ell-4$, the expected number $\mu_{j,d}$ of such $(j,d)$-paths satisfies $$\mu_{j,d} \leq k^2 n^{j+d-1}(pn^{{\varepsilon}})^{d+j} \leq k^2 (np)^{j+d-1} p n^{2\ell{\varepsilon}} = k^2 (np)^{j+d+1-\ell} n^{2\ell{\varepsilon}} \leq k^2 (np)^{j-3} n^{2\ell{\varepsilon}} .$$ Set $\kappa_j = k^2 (np)^{j-3} n^{3\ell{\varepsilon}}$ and $b=kn^{{\varepsilon}}$. Using the Deletion Lemma (cf. Lemma \[lem:deletion\_lemma\]) the probability that ${{\mathcal D}}{{\mathcal L}}(b,\kappa_j,{{\mathcal S}}_{j,d})$ fails for some $1 \leq j \leq \ell-1$ and $0 \leq d \leq \ell-4$ is bounded by $$\sum_{1 \leq j \leq \ell} \ \sum_{0 \leq d \leq \ell-4} (1+\kappa_j/\mu_{j,d})^{-b} \leq \ell^2 \cdot n^{-\ell {\varepsilon}b} = n^{-\omega(k)} ,$$ with room to spare. Whenever ${{\mathcal D}}{{\mathcal L}}(b,\kappa_j,{{\mathcal S}}_{j,d})$ holds, we denote by $F_{j,d}$ the corresponding ‘ignored’ edge set $E_0$ as in Lemma \[lem:deletion\_lemma\]. If all ${{\mathcal D}}{{\mathcal L}}(b,\kappa_j,{{\mathcal S}}_{j,d})$ with $1 \leq j \leq \ell-1$ and $0 \leq d \leq \ell-4$ hold simultaneously, then defining $F$ as the union of all edge sets $F_{j,d}$ has the required properties. Finally, taking the union bound over all choices of $A$ and $B$ completes the proof.
For most large sets $B$ and $W$, we expect that the number of $(b,w) \in B \times W$ for which there exists a path $b=w_0 \cdots w_{\ell-2}=w$ should be roughly $|B||W|n^{\ell-3}p'^{\ell-2} = |B||W|n^{(\ell-2){\varepsilon}}/(np)$. Loosely speaking, the next lemma suggests that for most reasonable sized $A,B \subseteq [n]$, this upper bound holds for $W=N^{(\ell-4)}(A,X)$ if we forbid certain exceptional paths, as in this case $|W| \approx |A| (np')^{\ell-4}$.
\[lem:path:endpoints:pairs\] Let ${{\mathcal P}}_2$ denote the event that for all disjoint $A,B \subseteq [n]$ with $|A|,|B|\leq k$ there exists $X \subseteq [n]$ and $F \subseteq \binom{[n]}{2}$ with $|X| \leq k n^{5\ell{\varepsilon}}$ and $|F| \leq k n^{2{\varepsilon}}$, such that the number of pairs $(b,w) \in B \times N^{(\ell-4)}(A, X)$ for which there exists a path $b=w_0 \cdots w_{\ell-2}=w$ with $$\label{eq:lem:path:endpoints:pairs:condition}
w_1 \notin A \qquad \text{ and } \qquad \bigl(w_2 \not\in A \ \text{ or } \ \{w_0w_1,w_1w_2\} \cap F = \emptyset\bigr)$$ is at most $k^2(np)^{\ell-5} n^{15 \ell{\varepsilon}}$. Then ${{\mathcal P}}_2$ holds whp in $G_{n,p'}$.
Before turning to the proof, note that ${{\mathcal P}}_2$ is monotone decreasing.
By Lemmas \[lem:set:neighbourhood:size\], \[lem:edges:bounded\], \[lem:path:endpoints:shortest\] and \[lem:path:endpoints:pairs:deletion\] it is enough to show that ${{\mathcal P}}_2$ holds for every graph $G$ satisfying ${{\mathcal D}}\cap {{\mathcal M}}\cap {{\mathcal Q}}_1 \cap {{\mathcal Q}}_2$. As this claim is purely deterministic, it suffices to prove it for fixed disjoint $A,B \subseteq [n]$ with $|A|,|B|\leq k$. Given $X \subseteq [n]$ and $F \subseteq \binom{[n]}{2}$, we denote by $P_{j,d}(X,F)$ the set of $(j,d)$-paths wrt. $(A,B,X)$ that are edge disjoint from $F$. By ${{\mathcal Q}}_2$ there exists $F \subseteq \binom{[n]}{2}$ with $|F| \leq k n^{2{\varepsilon}}$ such that for all $1 \leq j \leq \ell-2$ and $0 \leq d \leq \ell-4$ we have $$\label{eq:lem:path:endpoints:pairs:bound:jr:paths}
|P_{j,d}(A \cup B,F)| \leq k^2(np)^{j-3} n^{4 \ell{\varepsilon}} .$$ Let $V_F$ contain all vertices outside $A$ that are endpoints of edges in $F$. Note that $|V_F| \leq 2 k n^{2{\varepsilon}}$. Considering $S \leftarrow B \cup V_F$, by ${{\mathcal M}}$ there are at most $kn^{4\ell{\varepsilon}}$ edges between $B \cup V_F$ and $N^{(\leq \ell-3)}(A,B \cup V_F \cup A)$. Let $V_{B,F}$ contain the endpoints of all those edges and set $$\label{eq:lem:path:endpoints:pairs:def:X}
X= A \cup B \cup V_F \cup V_{B,F} .$$ Observe that, say, $|X| \leq kn^{5\ell{\varepsilon}}$. Furthermore, using we see that $$\label{eq:lem:path:endpoints:pairs:properties:removal}
V_F \cap \bigcup_{1 \leq \kappa \leq \ell-4} V_{\kappa}(X) = \emptyset \qquad \text{ and } \qquad \Gamma\big(V_F \cup B\big) \cap \bigl(N^{(\leq \ell-4)}(A,X) \setminus A\bigr) = \emptyset .$$ For every $1 \leq j \leq \ell-2$ we define $W_j$ as the set of all pairs $(b,y) \in B \times N^{(\leq \ell-4)}(A,X)$ for which there exists a path $b=w_0 \cdots w_{j}=y$ satisfying and $$\label{eq:lem:path:endpoints:pairs:condition:j}
\{w_{0},\ldots, w_{j-1}\} \cap N^{(\leq \ell-4)}(A,X) = \emptyset \qquad \text{ and } \qquad w_j \in N^{(\leq \ell-4)}(A,X) .$$ We claim that in order to complete the proof, it suffices to show that for all $1 \leq j \leq \ell-2$ we have $$\label{eq:lem:path:endpoints:pairs:size:Wj}
|W_j| \leq k^2(np)^{j-3} n^{10\ell{\varepsilon}} .$$ Indeed, let $W$ contain all pairs $(b,w) \in B \times N^{(\ell-4)}(A,X)$ for which there exists a path $b=w_0 \cdots w_{\ell-2}=w$ satisfying . Note that for every such $b=w_0 \cdots w_{\ell-2}=w$ there exists $1 \leq j \leq \ell-2$ such that $b=w_0 \cdots w_{j}$ satisfies . Recall that by ${{\mathcal D}}$ the degree is bounded by $npn^{2{\varepsilon}}$. So, given $w_j$, there are at most $(npn^{2{\varepsilon}})^{\ell-j-2}$ vertices $w \in N^{(\ell-4)}(A,X)$ for which there exists a path $w_j \cdots w_{\ell-2}=w$. Putting things together, assuming we obtain $$\label{eq:lem:path:endpoints:pairs:size:W}
|W| \leq \sum_{1 \leq j \leq \ell-2} |W_j| \cdot (npn^{2{\varepsilon}})^{\ell-j-2}
\leq k^2 (np)^{\ell-5} n^{15\ell{\varepsilon}} ,$$ and so ${{\mathcal P}}_2$ holds, as claimed.
We shall now prove . Observe that for $j=1$ we need to consider paths $w_0w_1$ with $w_0 \in B$ and $w_1 \in N^{(\leq \ell-4)}(A,X) \setminus A$. Now, using the second part of we see that $w_1 \in \Gamma(w_0) \cap (N^{(\leq \ell-4)}(A,X) \setminus A)$ is impossible. This implies $|W_1|=0$, which clearly establishes for $j=1$.
For $j \geq 2$ we first consider $W_{j,F} \subseteq W_j$, which contains all pairs $(b,y) \in W_j$ for which there exists a path $b=w_0 \cdots w_j=y$ satisfying and $$\label{eq:lem:path:endpoints:pairs:condition:F:disjoint}
\{w_0w_1,\ldots, w_{j-1}w_{j}\} \cap F = \emptyset .$$ Clearly, for every $(b,y) \in W_{j,F}$ there exists $0 \leq d \leq \ell-4$ such that at least one $(j,d)$-path wrt. $(A,B,X)$ with $b=w_0$ and $w_{j}=y$ satisfies . We claim that the corresponding $(j,d)$-path $w_0 \cdots w_j = v_d \cdots v_0$ is edge-disjoint from $F$, i.e., contained in $P_{j,d}(X,F)$. To see this, observe that every $f \in \{v_{d}v_{d-1}, \cdots, v_1v_0\} \cap F$ has at least one vertex outside of $A$, say $v_{\kappa} \in V_{\kappa}(X)$ with $1 \leq \kappa \leq d$, which contradicts , since by construction $v_{\kappa} \in V_F$. In addition, by and we see that $P_{j,d}(X,F) \subseteq P_{j,d}(A \cup B,F)$. Putting things together, using our discussion yields $$\label{eq:lem:path:endpoints:pairs:bound:WjF:disjoint}
|W_{j,F}| \leq \sum_{0 \leq d \leq \ell-4} |P_{j,d}(X,F)| \leq \sum_{0 \leq d \leq \ell-4} |P_{j,d}(A \cup B,F)| \leq k^2(np)^{j-3} n^{5 \ell{\varepsilon}} .$$
It remains to estimate the number of pairs in $W^{*}_{j,F} = W_{j} \setminus W_{j,F}$, where the corresponding paths intersect with $F$. We start with the special case $j=2$, i.e., paths $b=w_0w_1w_2=y$ with $(b,y) \in W^{*}_{2,F}$ satisfying . Observe that every $f \in \{w_0w_1,w_1w_2\} \cap F$ contains $w_1 \in V_F$, since $w_1 \notin A$ by . Note that $w_2 \in A$ contradicts the second part of , and that $w_2 \in \Gamma(w_1) \cap (N^{(\leq \ell-4)}(A,X) \setminus A)$ is impossible by . To sum up, $|W^{*}_{2,F}|=0$, which together with implies for $j=2$.
Turning to $j \geq 3$, for every $1 \leq \varsigma \leq j$ we denote by $W^{*}_{j,F,\varsigma} \subseteq W^{*}_{j,F}$ the set of pairs $(b,y) \in W^{*}_{j,F}$ with $y \notin A$ where the corresponding path $b=w_0 \cdots w_j =y$ satisfies $w_{\varsigma-1}w_{\varsigma} \in F$ and . We claim that it is enough to show that for every $1 \leq \varsigma \leq j$ we have $$\label{eq:lem:path:endpoints:pairs:bound:WjF:intersection:notinA}
|W^{*}_{j,F,\varsigma}| \leq k^2(np)^{j-3}n^{8\ell{\varepsilon}} .$$ Indeed, since there are at most $|B| \cdot |A| \leq k^2 \leq k^2(np)^{j-3}$ pairs $(b,y) \in W^{*}_{j,F}$ with $y \in A$, we obtain $$|W^{*}_{j,F}| \leq k^2(np)^{j-3} + \sum_{1 \leq \varsigma \leq j} |W^{*}_{j,F,\varsigma}| \leq k^2(np)^{j-3}n^{9\ell{\varepsilon}} ,$$ which together with establishes , as claimed.
In the following we verify . First we show that $|W^{*}_{j,F,\varsigma}| = 0$ for $\varsigma \in \{j-1,j\}$. If $w_{j-1}w_{j} \in F$, then $w_j \notin A$ implies $w_j \in V_F$, but the remaining possibility $w_j \in N^{(\leq \ell-4)}(A,X) \setminus A$ contradicts . If $w_{j-2}w_{j-1} \in F$, then by we have $w_{j-1} \notin N^{(\leq \ell-4)}(A,X)$ and so $w_{j-1} \in V_F$. Since by assumption $w_j \notin A$ we must have $w_j \in \Gamma(w_{j-1}) \cap (N^{(\leq \ell-4)}(A,X) \setminus A)$, which is impossible by .
Now, suppose that $w_{\varsigma-1}w_{\varsigma} \in F$ with $1 \leq \varsigma \leq j-2$. Considering $v \leftarrow w_{\varsigma}$ and $d \leftarrow \ell-4$, by ${{\mathcal Q}}_1$ there are at most $(np)^{j-\varsigma-1}n^{6\ell{\varepsilon}}$ vertices $w_j \in N^{(\leq \ell-4)}(A,X)$ for which there exists a path $w_{\varsigma}=w'_0 \cdots w'_{j-\varsigma}=w_j$ with $\{w_{\varsigma}, \ldots, w_{j-1}\} \cap N^{(\leq \ell-4)}(A,X) = \emptyset$. So, using $|F| \leq kn^{2{\varepsilon}}$, since there are at most $|B|=k$ choices for $b \in B$, for $\varsigma \geq 2$ we deduce that $$|W^{*}_{j,F,\varsigma}| \leq |B| \cdot 2|F| \cdot (np)^{j-\varsigma-1}n^{6\ell{\varepsilon}} \leq k^2(np)^{j-\varsigma-1}n^{(6\ell+3){\varepsilon}} \leq k^2(np)^{j-3}n^{8\ell{\varepsilon}} ,$$ as claimed. Note that for the remaining case $\varsigma=1$ each (ordered) edge $w_0w_1 \in F$ also determines the vertex $b=w_0 \in B$. So, compared to the estimate above we win a factor of $|B|$, and a virtually identical calculation yields that also holds in this case, which completes the proof.
With very similar reasoning as for Lemma \[lem:path:endpoints\], one can argue that an extra condition for the case $w_2 \in A$ is needed in Lemma \[lem:path:endpoints:pairs\]: this time we can otherwise violate the claimed bound whp by fixing some vertex $v^*$ and then choosing disjoint $A,B \subseteq [n]$ such that each contains at least $np$ vertices from $\Gamma(v^*)$; we leave the details to the interested reader.
Very good configurations exist {#sec:good_configurations_exist}
==============================
In this section we prove Lemma \[lem:dem:config\]. Given a graph property ${{\mathcal Y}}$, let ${{\mathcal Y}}_i$ denote the event $G(i) \in {{\mathcal Y}}$, i.e., that $G(i)$ satisfies ${{\mathcal Y}}$. Now, for every $0 \leq i \leq m$ we set $${{\mathcal W}}_i = {{\mathcal I}}_i \cap {{\mathcal K}}_i \cap {{\mathcal L}}_i \cap {{\mathcal N}}_i \cap {{\mathcal P}}_{1,i} \cap {{\mathcal P}}_{2,i} \cap {{\mathcal T}}_i,$$ where ${{\mathcal K}}_i$, ${{\mathcal L}}_i$, ${{\mathcal T}}_i$ are defined as in Theorem \[thm:BohmanKeevash2010H\] and Lemma \[lem:edges\_bounded\_and\_large\_degree\_bounded\], and ${{\mathcal I}}$, ${{\mathcal N}}$, ${{\mathcal P}}_{1}$, ${{\mathcal P}}_{2}$ are defined as in Lemmas \[lem:bounded\_codegree\], \[lem:set:independent:subset\], \[lem:path:endpoints\] and \[lem:path:endpoints:pairs\]. It is not difficult to see that ${{\mathcal W}}_i$ is monotone decreasing and, using the ‘transfer theorem’ (Theorem \[thm:transfer:binomial\]), that ${{\mathcal W}}_m$ holds whp. Observe that by monotonicity ${{\mathcal W}}_m$ implies ${{\mathcal W}}_i$ for every $i \leq m$, and that $\neg{{{\mathcal B}}_{{i}}(\Sigma)}$ implies $\neg{{{\mathcal B}}_{{i-1}}(\Sigma)}$. So, to complete the proof it suffices to consider fixed $G(i)$ satisfying ${{\mathcal W}}_i$ and show that for every $(\tilde{v},U)$ with $U \in \binom{[n] \setminus \{\tilde{v}\}}{u}$ there exists $\Sigma^*=(\tilde{v},U,A,B,R) \in {{\mathcal C}}$ satisfying $\neg{{{\mathcal B}}_{{i}}(\Sigma^*)}$ and . In fact, since the above claim is purely deterministic, it is enough to also consider fixed $(\tilde{v},U)$. Our proof proceeds in several steps and we tacitly assume that $n$ is sufficiently large whenever necessary. First, in Section \[sec:good\_configurations\_exist:find\_config\] we choose a ‘special’ configuration $\Sigma^*=(\tilde{v},U,A,B,R)$ and collect some of its basic properties. In the remaining sections we verify that $\Sigma^*$ has the properties claimed by Lemma \[lem:dem:config\]. More precisely, in Section \[sec:good\_configuration\] we show that $\neg{{{\mathcal B}}_{{i}}(\Sigma^*)}$ holds, and in Section \[sec:good\_configuration:few\_ignored\] we establish .
Finding $\Sigma^*=(\tilde{v},U,A,B,R)$ {#sec:good_configurations_exist:find_config}
--------------------------------------
In the following we show how we pick $\Sigma^*=(\tilde{v},U,A,B,R)$. Along the way, we furthermore collect some immediate properties of the resulting $\Sigma^*$. We set $$\label{eq:Cl-parameters4}
\tau = 40\ell \qquad \text{ and } \qquad \vartheta = 20 \ell \tau = 800 \ell^2 .$$ For the main steps of our argument it is useful to keep in mind that $\vartheta \gg \tau \gg \ell$ and $\vartheta{\varepsilon}\ll 1/\ell$. First, we choose $S \subseteq U$ such that $$\label{eq:size:S}
\text{$S$ is an independent set} \qquad \text{ and } \qquad |S| \geq u/6 ,$$ which is possible since ${{\mathcal I}}_i$ holds. Henceforth we assume that $v_1, \ldots, v_n \in [n]$ are ordered so that $$\label{eq:vertex_ordering}
|\Gamma(v_1) \cap S| \geq |\Gamma(v_2) \cap S| \geq \cdots \geq |\Gamma(v_j) \cap S| \geq \cdots \geq |\Gamma(v_n) \cap S| .$$ We greedily choose first $\ell_A$, and afterwards $\ell_B$, such that they are the smallest indices for which $$N_A = \bigcup_{1 \leq j \leq \ell_A} \big(\Gamma(v_j) \cap S\big) \qquad \text{ and } \qquad N_B = \bigcup_{\ell_A < j \leq \ell_B} \big(\Gamma(v_j) \cap S\big) \setminus N_A$$ each have cardinality at least $2k$, where we set the corresponding index to $\infty$ if this is not possible. Recall that $k=\gamma/60 \cdot np t_{\max}$ by and $\gamma \geq 180$ by . So, since ${{\mathcal T}}_i$ holds, by the maximum degree is at most $3 np t_{\max} \leq k$. Using $k=u/60$, we deduce that $$\label{eq:size_neighbourhoods}
|N_A \cup N_B| \leq 6k \leq u/10 .$$
### Picking $A,B$
If $\ell_B=\infty$ or $\ell_B > n^{2\vartheta{\varepsilon}}$, we choose arbitrary disjoint sets, each of size $k=u/60$, satisfying $$A,B \subseteq S \setminus (N_A \cup N_B) ,$$ which is possible by and . For later usage, we furthermore set $I_A = \emptyset$ and $I_B = \emptyset$.
If $\ell_B \leq n^{2\vartheta{\varepsilon}} = o(k)$, we set $I_A = \{v_1, \ldots, v_{\ell_A}\}$ and $I_B = \{v_{\ell_A+1}, \ldots, v_{\ell_B}\}$. Since $G(i)$ satisfies ${{\mathcal N}}_i$, the codegrees are all bounded by nine, and thus $$\label{eq:size_common_neighbourhoods}
|\Gamma(I_B) \cap N_A| \leq |\Gamma(I_B) \cap \Gamma(I_A)| \leq 9 \cdot \ell_B \cdot \ell_A \leq 9 n^{4\vartheta{\varepsilon}} = o(k) .$$ Now we choose arbitrary sets, each of size $k$, satisfying $$A \subseteq N_A \setminus \bigl(I_B \cup \Gamma(I_B)\bigr) \qquad \text{ and } \qquad B \subseteq N_B ,$$ which is possible by . Clearly, $A$ and $B \cup I_B$ are disjoint.
Next we estimate the size of certain neighbourhoods. A similar argument can be found in [@Warnke2010K4].
\[lem:size:neighbourhoods\] We have $\Gamma(I_A) \cap B = \emptyset$ and $\Gamma(I_B) \cap A = \emptyset$. Given $Y \in \{A,B\}$, every $v \notin I_Y$ satisfies $$\label{eq:vnotinIAB:neighbourhood:bound}
|\Gamma(v) \cap Y| \leq np n^{-\vartheta{\varepsilon}} .$$
If $\ell_B=\infty$, then all vertices $v \in [n]$ satisfy the stronger bound $|\Gamma(v) \cap (A \cup B)| = 0$.
Next, we consider the case $n^{2\vartheta {\varepsilon}} < \ell_B < \infty$, where $I_A = I_B = \emptyset$. Since all vertices $v \in \{v_{1}, \ldots, v_{\ell_B}\}$ satisfy $|\Gamma(v) \cap (A \cup B)| = 0$, using it is not difficult to see that in order to prove , it suffices to show $|\Gamma(v_{x}) \cap S| \leq np n^{-\vartheta{\varepsilon}}$ for $x = n^{2\vartheta {\varepsilon}}$. Set $H = \{v_{1}, \ldots, v_{x}\}$. On the one hand, using we have $2 e(H,S) \geq x |\Gamma(v_{x}) \cap S|$. On the other hand, since $G(i)$ satisfies ${{\mathcal K}}_i$, using $|H| = n^{2\vartheta {\varepsilon}}$ and $|S|\leq npn^{{\varepsilon}}$, we have, say, $e(H,S) \leq npn^{2{\varepsilon}}$. So, we deduce $|\Gamma(v_{x}) \cap S| \leq np n^{-\vartheta{\varepsilon}}$, as claimed.
Finally, suppose that $\ell_B \leq n^{2\vartheta {\varepsilon}}$. Observe that $\Gamma(I_A) \cap B = \emptyset$ and $\Gamma(I_B) \cap A = \emptyset$ hold by construction. Fix $Y \in \{A,B\}$. Since by ${{\mathcal N}}_i$ all codegrees are at most nine, for every $v \notin I_Y$ we have $|\Gamma(v) \cap Y| \leq |\Gamma(v) \cap \Gamma(I_Y)| \leq 9 \ell_B$, which readily establishes , and thus completes the proof.
### Choosing $R$
Observe that $|I_B| \leq n^{2\vartheta{\varepsilon}}$. Considering $A$ and $S \leftarrow I_B$, we denote by $X_1$ the set $X$ whose existence is guaranteed by ${{\mathcal P}}_{1,i}$. Similarly, let $X_2$ and $F$ denote the sets $X$ and $F$ whose existence is guaranteed by ${{\mathcal P}}_{2,i}$ when considering $A$ and $B$. We have $|X_1|, |X_2| \leq k n^{5\ell{\varepsilon}}$ and $|F| \leq k n^{2{\varepsilon}}$. Now we set $$\label{eq:def:R}
R = \{\tilde{v}\} \cup U \cup X_1 \cup X_2 .$$ Clearly, $|R| \leq k n^{10\ell{\varepsilon}}$ holds, with room to spare. Next we collect several structural properties. By and and have $N^{(j)}(A,R) \subseteq N^{(j)}(A,X_1) \cap N^{(j)}(A,X_2)$. So, using $(\Gamma(I_B) \cup I_B) \cap A = \emptyset$, we immediately obtain the following statement:
\[lem:bound\_paths\_IB\] We have $|I_B| \leq n^{2\vartheta{\varepsilon}}$, and for every $v \in I_B$ there are at most $(np)^{\ell-3} n^{15 \ell{\varepsilon}}$ vertices $w \in N^{(\ell-3)}(A,R)$ for which there exists a path $v=w_0 \cdots w_{\ell-2}=w$.
In addition, using that $A \cup B$ is an independent set, we readily deduce the following result:
\[lem:bound\_pairs:BW\] We have $|F| \leq k n^{2{\varepsilon}}$, and there are at most $k^2(np)^{\ell-5} n^{15 \ell{\varepsilon}}$ pairs $(b,w) \in B \times N^{(\ell-4)}(A,R)$ for which there exists a path $b=w_0 \cdots w_{\ell-2}=w$ satisfying $w_2 \not\in A$ or $\{w_0w_1,w_1w_2\} \cap F = \emptyset$.
In the subsequent sections, the construction of $A$ and $B$ is irrelevant; all that we use is that $A$, $B$ are disjoint subsets of $U$ with size $k$, and there are sets $F$, $I_A$, $I_B$, $R$ such that the conclusions of Lemmas \[lem:size:neighbourhoods\]–\[lem:bound\_pairs:BW\] hold in $G(i)$.
The configuration $\Sigma^*$ is good {#sec:good_configuration}
------------------------------------
In this section we show that $\neg{{{\mathcal B}}_{{i}}(\Sigma^*)}= \neg{{{\mathcal B}}_{{1,i}}(\Sigma^*)} \cap \neg{{{\mathcal B}}_{{2,i}}(\Sigma^*)}$ holds.
### The bad event ${{{\mathcal B}}_{{1,i}}(\Sigma^*)}$ {#sec:good_configuration:bad:2}
In order to prove that ${{{\mathcal B}}_{{1,i}}(\Sigma^*)}$ fails, using Lemma \[lem:bound\_pairs:BW\] it suffices to show that there are at most $k^2(np)^{\ell-4}n^{-10{\varepsilon}}$ paths $w_0 \cdots w_{\ell-2}$ with $(w_0,w_2) \in B \times A$ satisfying $w_0w_1 \in F$ or $w_1w_2 \in F$. Let $P_{\Sigma^*}$ denote all such paths. For every $w_0w_1 \in F \cap E(i)$ with $w_0 \in B$, using Lemma \[lem:size:neighbourhoods\] we see that $w_1 \notin I_A$, which by implies that there are at most $np n^{-\vartheta{\varepsilon}}$ choices for $w_2 \in \Gamma(w_1) \cap A$. With a similar argument, for every $w_1w_2 \in F \cap E(i)$ with $w_2 \in A$ we have at most $np n^{-\vartheta{\varepsilon}}$ choices for $w_0 \in \Gamma(w_1) \cap B$. Furthermore, since the degree is bounded by $npn^{{\varepsilon}}$, given $w_2 \in A$ there are at most $(npn^{{\varepsilon}})^{\ell-4}$ paths $w_2 \cdots w_{\ell-2}$. So, using $np \leq k$, $|F| \leq k n^{2{\varepsilon}}$ and , i.e., $\vartheta \geq 20 \ell$, we deduce that $$|P_{\Sigma^*}| \leq np n^{-\vartheta{\varepsilon}} \cdot 2|F| \cdot (npn^{{\varepsilon}})^{\ell-4} \leq k^2 (np)^{\ell-4} n^{(\ell-\vartheta){\varepsilon}} < k^2 (np)^{\ell-4} n^{-10{\varepsilon}} ,$$ which, as explained, establishes $\neg{{{\mathcal B}}_{{1,i}}(\Sigma^*)}$.
### The bad event ${{{\mathcal B}}_{{2,i}}(\Sigma^*)}$ {#sec:good_configuration:bad:1}
In anticipation of the estimates in Section \[sec:good\_configuration:few\_ignored\], here we analyse the combinatorial structure of $L_{\Sigma^*}(i)$ much more precisely than needed. To this end we introduce the sets $L_{\Sigma^*}(i,j)$, where for every $j \in [\ell-1]$ we denote by $L_{\Sigma^*}(i,j)$ the set of all *ordered* pairs $xy$ with distinct $x,y \in [n]$ such that $|C_{x,y,\Sigma^*}(i,j)| \geq p^{-1}n^{-30\ell{\varepsilon}}$. We start by showing that we may restrict our attention to the case $j \in \{1,2\}$. Recall that $C_{x,y,\Sigma^*}(i,j)$ contains all pairs $bw \in B \times N^{(\ell-3)}(A,R)$ for which there exist disjoint paths $b=w_1 \cdots w_{j}=x$ and $y=w_{j+1} \cdots w_{\ell}=w$ in $G(i)$. Fix $x \neq y$. Since the degree is at most $npn^{{\varepsilon}}$ by , for $j \geq 3$ the number of choices for $w$ is at most $(npn^{{\varepsilon}})^{\ell-j-1} \leq (npn^{{\varepsilon}})^{\ell-4}$. Now, as there are at most $|B| \leq k \leq npn^{{\varepsilon}}$ ways to pick $b \in B$, using $(np)^{\ell-2}=p^{-1}$ we crudely have $$\label{eq:closed:xySigma:small:jl}
|C_{x,y,\Sigma^*}(i,j)| \leq npn^{{\varepsilon}} \cdot (npn^{{\varepsilon}})^{\ell-4} \leq p^{-1} n^{\ell{\varepsilon}}/(np) < p^{-1} n^{-30\ell{\varepsilon}} ,$$ which implies $xy \notin L_{\Sigma^*}(i,j)$. Therefore $L_{\Sigma^*}(i,j) = \emptyset$ for $j \geq 3$, so $$\label{eq:closed:xySigma:inequality}
|L_{\Sigma^*}(i)| \leq |L_{\Sigma^*}(i,1)| + |L_{\Sigma^*}(i,2)| .$$ With foresight, for all $j \geq 1$ we define $M^{(j)}(A)$ as the set of $v \in [n]$ with $|W^{(j)}(v,A)| \geq (np)^{j} n^{-\tau{\varepsilon}}$, where $W^{(j)}(v,A)$ contains all vertices $w \in N^{(\ell-3)}(A,R)$ for which there exists a path $v=w_0 \cdots w_{j}=w$ in $G(i)$. Now we claim that $$\label{eq:closed:xySigma:i2:inclusion}
L_{\Sigma^*}(i,2) \subseteq \left\{xy \;:\; x \in I_B \; \wedge \; y \in M^{(\ell-3)}(A) \right\} .$$ Note that $C_{x,y,\Sigma^*}(i,2)$ contains only pairs $bw \in B \times N^{(\ell-3)}(A,R)$ for which there exists paths $b=w_1w_{2}=x$ and $y=w_{3} \cdots w_{\ell}=w$ in $G(i)$. First suppose that $x \notin I_B$. Using Lemma \[lem:size:neighbourhoods\], by we have at most $npn^{-\vartheta{\varepsilon}}$ choices for $b \in \Gamma(x) \cap B$. Since the degree is at most $npn^{{\varepsilon}}$, we have at most $(npn^{{\varepsilon}})^{\ell-3}$ choices for $w$. So, using $(np)^{\ell-2}=p^{-1}$ and , i.e., $\vartheta \geq 40 \ell$, we deduce that $$|C_{x,y,\Sigma^*}(i,2)| \leq npn^{-\vartheta{\varepsilon}} \cdot (npn^{{\varepsilon}})^{\ell-3} \leq p^{-1} n^{(\ell-\vartheta){\varepsilon}} < p^{-1} n^{-30\ell{\varepsilon}} ,$$ which implies $xy \notin L_{\Sigma^*}(i,2)$. Next, we consider the case where $y \notin M^{(\ell-3)}(A)$. With a very similar reasoning as above, this time using $|W^{(\ell-3)}(y,A)| \leq (np)^{\ell-3} n^{-\tau{\varepsilon}}$ and , i.e., $\tau = 40 \ell$, we obtain $$|C_{x,y,\Sigma^*}(i,2)| \leq npn^{{\varepsilon}} \cdot (np)^{\ell-3} n^{-\tau{\varepsilon}} \leq p^{-1} n^{(1-\tau){\varepsilon}} < p^{-1} n^{-30\ell{\varepsilon}} ,$$ which implies $xy \notin L_{\Sigma^*}(i,2)$. This completes the proof of .
By a similar but simpler argument we furthermore see that $$\label{eq:closed:xySigma:i1:inclusion}
L_{\Sigma^*}(i,1) \subseteq \left\{xy \;:\; x \in B \; \wedge \; y \in M^{(\ell-2)}(A) \right\} .$$
Next we estimate the cardinality of $M^{(j)}(A)$. A similar argument is implicit in [@BohmanKeevash2010H].
\[lem:bound:manypaths\] For every $1 \leq j \leq \ell-2$ we have $|M^{(j)}(A)| \leq (np)^{\ell-2-j}n^{2\ell\tau{\varepsilon}}$.
Set $H^{(0)}(A) = N^{(\ell-3)}(A,R)$, and for every $j \geq 1$ we let $H^{(j)}(A)$ contain all $v \in [n]$ with $|\Gamma(v) \cap H^{(j-1)}(A) | \geq np n^{-2\tau{\varepsilon}}$. First, we claim that for all $1 \leq j \leq \ell-2$ we have $$\label{eq:lem:bound:manypaths:HP:inclusion}
M^{(j)}(A) \subseteq H^{(j)}(A) .$$ Since $\tau \geq 2\ell$ by , it clearly suffices to show that for all $1 \leq j \leq \ell-2$, for every $v \notin H^{(j)}(A)$ we have $|W^{(j)}(v,A)| \leq j (npn^{{\varepsilon}})^{j} n^{-2\tau{\varepsilon}}$. We proceed by induction on $j$. For the base case $j=1$ the claim is trivial, since $H^{(1)}(A)$ contains all vertices $v \in [n]$ with $|\Gamma(v) \cap N^{(\ell-3)}(A,R)| \geq np n^{-2\tau{\varepsilon}}$. Turning to $j \geq 2$, fix $v \notin H^{(j)}(A)$. By distinguishing between the neighbours of $v$ inside and outside of $H^{(j-1)}(A)$, using the induction hypothesis and that the degree is bounded by $npn^{{\varepsilon}}$, we obtain $$|W^{(j)}(v,A)| \leq np n^{-2\tau{\varepsilon}} \cdot (npn^{{\varepsilon}})^{j-1} + np n^{{\varepsilon}} \cdot (j-1) (npn^{{\varepsilon}})^{j-1} n^{-2\tau{\varepsilon}} \leq j (npn^{{\varepsilon}})^{j} n^{-2\tau{\varepsilon}} ,$$ which, as explained, establishes .
To finish the proof, again using $\tau \geq 2\ell$, it suffices to show that for all $0 \leq j \leq \ell-2$ we have $$\label{eq:lem:bound:manypaths:H:bound}
|H^{(j)}(A)| \leq (np)^{\ell-2-j}n^{(2j\tau+\ell+j){\varepsilon}} .$$ As before, we proceed by induction on $j$. Using $|A| \leq k \leq npn^{{\varepsilon}}$ and that the degree is bounded by $npn^{{\varepsilon}}$, we establish the base case $j=0$ by observing that $|H^{(0)}(A)| \leq |\Gamma^{(\ell-3)}(A)| \leq (npn^{{\varepsilon}})^{\ell-2}$. Suppose $j \geq 1$. Recall that $(np)^{\ell-2} = p^{-1}$. Since ${{\mathcal L}}_i$ holds, using the induction hypothesis we obtain $$|H^{(j)}(A)| \leq 16{\varepsilon}^{-1} (np)^{\ell-2-j}n^{(2j\tau+\ell+j-1){\varepsilon}} \leq (np)^{\ell-2-j}n^{(2j\tau+\ell+j){\varepsilon}} ,$$ completing the proof.
With Lemma \[lem:bound:manypaths\] in hand, combing – with $|B| = k \leq npn^{{\varepsilon}}$ as well as $|I_B| \leq n^{2\vartheta{\varepsilon}}$, and then using , as well as $\ell \geq 4$, $np=n^{1/(\ell-1)}$ and $(np)^2 \leq (np)^{\ell-2} = p^{-1}$, we deduce that $$|L_{\Sigma^*}(i)| \leq npn^{{\varepsilon}} \cdot n^{2\ell\tau{\varepsilon}} + n^{2\vartheta{\varepsilon}} \cdot npn^{2\ell\tau{\varepsilon}} \leq npn^{5\vartheta{\varepsilon}} < (np)^{2} n^{-1/(2\ell)} \leq p^{-1} n^{-1/(2\ell)} ,$$ which establishes $\neg{{{\mathcal B}}_{{2,i}}(\Sigma^*)}$.
Few tuples are ignored for $\Sigma^*$ {#sec:good_configuration:few_ignored}
-------------------------------------
In this section we estimate the size of $T_{\Sigma^*,\ell-3}(i) \setminus Z_{\Sigma^*,\ell-3}(i)$. Let $Q_{\Sigma^*}(i)$ contain all pairs $(w_1, w_\ell) \in B \times N^{(\ell-3)}(A,R)$ for which there exists a path $w_1 \cdots w_{\ell}$ with $w_2 \in I_B \cup M^{(\ell-2)}(A)$. We claim that $$\label{eq:bad:inequality}
|T_{\Sigma^*,\ell-3}(i) \setminus Z_{\Sigma^*,\ell-3}(i)| \leq |Q_{\Sigma^*}(i)| .$$ Every tuple $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma^*,\ell-3}(i) \setminus Z_{\Sigma^*,\ell-3}(i)$ was ignored in one of the first $i$ steps because (R2) failed. Recall that $C_{x,y,\Sigma^*}(i,j)$ contains all pairs $bw \in B \times N^{(\ell-3)}(A,R)$ for which there exist disjoint paths $b=w_1 \cdots w_{j}=x$ and $y=w_{j+1} \cdots w_{\ell}=w$ in $G(i)$. Observe that for every ignored tuple there exists $i' < i$, distinct $x,y \in [n]$ and $j \in [\ell-1]$ with $e_{i'+1} =xy$, $f_{\ell-2} \in C_{x,y,\Sigma}(i',j)$ and $|C_{x,y,\Sigma}(i',j)| > p^{-1} n^{-30\ell{\varepsilon}}$. So, since $e_{i'+1} = xy$ was added, for every such tuple there exists a path $v_{\ell-2}=w_1 \cdots w_{j}w_{j+1} \cdots w_{\ell}=v_{\ell-3}$ with $w_{j}=x$ and $w_{j+1}=y$ in $G(i'+1) \subseteq G(i)$. Note that by monotonicity we have $C_{x,y,\Sigma^*}(i',j) \subseteq C_{x,y,\Sigma^*}(i,j)$, and therefore all such ‘bad’ pairs $xy$ satisfy $|C_{x,y,\Sigma^*}(i,j)| > p^{-1} n^{-30\ell{\varepsilon}}$. By the findings of Section \[sec:good\_configuration:bad:1\] it thus suffices to consider $C_{x,y,\Sigma^*}(i,j)$ for $xy \in L_{\Sigma^*}(i,j)$ with $j \in \{1,2\}$, since for all others holds. Now, using and , it is not difficult to see that the corresponding paths $v_{\ell-2}=w_1 \cdots w_{\ell}=v_{\ell-3}$ satisfy $w_1 \in B$, $w_2 \in I_B \cup M^{(\ell-2)}(A)$ and $w_{\ell} \in N^{(\ell-3)}(A,R)$. Putting things together, the extension property ${{\mathcal U}}_T$ (cf. Lemma \[lemma:extension:property\]) implies , since every $(v_0, \ldots, v_{\ell-2}) \in T_{\Sigma^*,\ell-3}(i) \setminus Z_{\Sigma^*,\ell-3}(i)$ is uniquely determined by the pair $f_{\ell-2}=v_{\ell-3}v_{\ell-2}$.
Let $Q_{\Sigma^*,I}(i)$ and $Q_{\Sigma^*,M}(i)$ contain all pairs $(w_1,w_\ell) \in Q_{\Sigma^*}(i)$ where at least one corresponding path $w_1 \cdots w_{\ell}$ satisfies $w_2 \in I_B$ and $w_2 \in M^{(\ell-2)}(A) \setminus I_B$, respectively. Now, using and , to establish , it suffices to prove, say, $$\label{eq:bad:paths:estimate}
\max\{|Q_{\Sigma^*,I}(i)|,|Q_{\Sigma^*,M}(i)|\} \leq (np)^{\ell-1}n^{-15{\varepsilon}} .$$ Using Lemma \[lem:bound\_paths\_IB\], $|I_B| \leq n^{2\vartheta{\varepsilon}}$ and that the degree is at most $npn^{{\varepsilon}}$, we obtain, with room to spare, $$\label{eq:bad:paths:estimate:small:Q1}
|Q_{\Sigma^*,I}(i)| \leq npn^{{\varepsilon}} \cdot |I_B| \cdot (np)^{\ell-3} n^{15 \ell{\varepsilon}} \leq (np)^{\ell-2} n^{(15 \ell + 2\vartheta + 1){\varepsilon}} \leq (np)^{\ell-1}n^{-15{\varepsilon}} .$$ Turning to $Q_{\Sigma^*,M}(i)$, note that for every $w_2 \in M^{(\ell-2)}(A) \setminus I_B$ we have $|\Gamma(w_2) \cap B| \leq npn^{-\vartheta{\varepsilon}}$ by . With a similar argument as above, using Lemma \[lem:bound:manypaths\], i.e., $|M^{(\ell-2)}(A)| \leq n^{2\ell\tau{\varepsilon}}$, we see that $$\label{eq:bad:paths:estimate:small:Q2}
|Q_{\Sigma^*,M}(i)| \leq npn^{-\vartheta{\varepsilon}} \cdot |M^{(\ell-2)}(A)| \cdot (npn^{{\varepsilon}})^{\ell-2} \leq (np)^{\ell-1} n^{(2\ell\tau+\ell-\vartheta){\varepsilon}} \leq (np)^{\ell-1}n^{-15{\varepsilon}} ,$$ where the last inequality follows from , i.e., $\vartheta = 20\ell\tau$. This establishes , which, as explained, completes the proof of Lemma \[lem:dem:config\].
I would like to thank my supervisor Oliver Riordan for many stimulating discussions and helpful comments on an earlier version of this paper. Part of this research was done while visiting the University of Memphis, and I am grateful for the hospitality and great working conditions. Finally, I would also like to thank the anonymous referees for several suggestions improving the presentation of the paper.
[0.4]{}
[10]{}
N. Alon, M. Krivelevich, and B. Sudakov. Coloring graphs with sparse neighborhoods. (1999), 73–82.
T. Bohman. The triangle-free process. (2009), 1653–1677.
T. Bohman and P. Keevash. The early evolution of the [$H$]{}-free process. (2010), 291–336.
B. Bollobás. Personal communication (2010).
B. Bollob[á]{}s. . Cambridge University Press, Cambridge, [S]{}econd edition, 2001.
B. Bollobás and O. Riordan. Constrained graph processes. (2000), \#R18.
F. Chung and R. Graham. . A K Peters, Ltd., Massachusetts, 1998.
R. Durrett. . Cambridge University Press, Cambridge, 2007.
P. Erdős, S. Suen, and P. Winkler. On the size of a random maximal graph. (1995), 309–318.
P. Erd[ő]{}s and A. R[é]{}nyi. On random graphs. [I]{}. (1959), 290–297.
S. Gerke and T. Makai. No dense subgraphs appear in the triangle-free graph process. (2011), \#P168.
S. Janson, T. [Ł]{}uczak, and A. Ruci[ń]{}ski. . Wiley-Interscience, New York, 2000.
S. Janson and A. Ruciński. The deletion method for upper tail estimates. (2004), 615–640.
J.H. Kim and V.H. Vu. Divide and conquer martingales and the number of triangles in a random graph. (2004), 166–174.
J.H. Kim. The [R]{}amsey number [$R(3,t)$]{} has order of magnitude $t^2/\log t$. (1995), 173–207.
D. Osthus and A. Taraz. Random maximal [$H$]{}-free graphs. (2001), 61–82.
M.E. Picollelli. The diamond-free process. Preprint, 2010. [` \simarXiv:1010.5207`]{}.
M.E. Picollelli. The final size of the [$C_4$]{}-free process. (2011), 939–955.
M.E. Picollelli. The final size of the [$C_\ell$]{}-free process. Available at `http://sites.google.com/site/mepicollelli/`.
V. R[ö]{}dl and A. Ruci[ń]{}ski. Threshold functions for [R]{}amsey properties. (1995), 917–942.
A. Ruci[ń]{}ski and N.C. Wormald. Random graph processes with degree restrictions. (1992), 169–180.
J. Spencer. Maximal triangle-free graphs and [R]{}amsey [$R(3,t)$]{}. Unpublished manuscript, 1995. `http://cs.nyu.edu/spencer/papers/ramsey3k.pdf`.
L. Warnke. Dense subgraphs in the [$H$]{}-free process. (2011), 2703–2707.
L. Warnke. When does the [$K_4$]{}-free process stop? , to appear. [` \simarXiv:1007.3037`]{}.
G. Wolfovitz. Lower bounds for the size of random maximal [$H$]{}-free graphs. (2009), \#R16.
G. Wolfovitz. The [$K_4$]{}-free process. Preprint, 2010. [` \simarXiv:1008.4044`]{}.
G. Wolfovitz. Triangle-free subgraphs in the triangle-free process. , [**39**]{} (2011), 539–-543.
N.C. Wormald. Differential equations for random processes and random graphs. , [**5**]{} (1995), 1217–1235.
N.C. Wormald. The differential equation method for random graph processes and greedy algorithms. In [*Lectures on approximation and randomized algorithms*]{}, pages 73–155. PWN, Warsaw, 1999.
[^1]: As usual, we say that an event holds *with high probability*, or *whp*, if it holds with probability $1-o(1)$ as $n\to\infty$.
|
---
abstract: 'Following the traditional naming of “eruptive flare" and “confined flares" but not implying a causal relationship between flare and coronal mass ejection (CME), we refer to the two kinds of large energetic phenomena occurring in the solar atmosphere as eruptive event and confined event, respectively: the former type refers to flares with associated CMEs, while the later type refers to flares without associated CMEs. We find that about 90% of X-class flares, the highest class in flare intensity size, are eruptive, but the rest 10% confined. To probe the question why the largest energy release in the solar corona could be either eruptive or confined, we have made a comparative study by carefully investigating 4 X-class events in each of the two types with a focus on the differences in their magnetic properties. Both sets of events are selected to have very similar intensity (X1.0 to X3.6) and duration (rise time less than 13 minutes and decaying time less than 9 minutes) in soft X-ray observations, in order to reduce the bias of flare size on CME occurrence. We find no difference in the total magnetic flux of the photospheric source regions for the two sets of events. However, we find that the occurrence of eruption (or confinement) is sensitive to the displacement of the location of the energy release, which is defined as the distance between the flare site and the flux-weighted magnetic center of the source active region. The displacement is 6 to 17 Mm for confined events, but is as large as 22 to 37 Mm for eruptive events, compared to the typical size of about 70 Mm for active regions studied. In other words, confined events occur closer to the magnetic center while the eruptive events tend to occur closer to the edge of active regions. Further, we have used potential-field source-surface model (PFSS) to infer the 3-D coronal magnetic field above source active regions. For each event, we calculate the coronal flux ratio of low corona ($<$ 1.1 $R_\odot$) to high corona ($\geq$ 1.1 $R_\odot$). We find that the confined events have a lower coronal flux ratio ($< 5.7$), while the eruptive events have a higher flux ratio($> 7.1$). These results imply that a stronger overlying arcade field may prevent energy release in the low corona from being eruptive, resulting in flares without CMEs. A CME is more likely to occur if the overlying arcade field is weaker.'
author:
- 'Yuming Wang, and Jie Zhang'
title: 'A Comparative Study Between Eruptive X-class Flares Associated with Coronal Mass Ejections and Confined X-class Flares'
---
Introduction
============
Coronal mass ejections (CMEs) and flares are known to be the two most energetic phenomena that occur in the atmosphere of the Sun, and have profound effects on the physical environment in the geo-space environment and human technological systems. In this paper, we intend to understand the physical origins of CMEs and flares by comparatively studying two different kinds of energetic phenomena, both of which have almost identical flares, but one is associated with CMEs and the other not. In the past when lacking direct CME observations, these two kinds of phenomena have been called as eruptive flares and confined flares, respectively [e.g., @Svestka_Cliver_1992]. An eruptive flare (also called a dynamic flare) is usually manifested as having two-ribbons and post-flare loops in H$\alpha$ imaging observations and long duration (e.g., tens of minutes or hours) in soft X-ray, while a confined flare occurs in a compact region and last for only a short period (e.g., minutes). Following this convention but trying not to imply a causal relation between flare and CME, hereafter we refer the flare event associated with an observed CME as an “eruptive event", and the flare event not associated with a CME as a “confined event".
It has been suggested based on observations that CMEs and flares are the two different phenomena of the same energy release process in the corona [e.g., @Harrison_1995; @Zhang_etal_2001a; @Harrison_2003]. They do not drive one another but are closely related. In particular, @Zhang_etal_2001a and @Zhang_etal_2004 showed that the fast acceleration of CMEs in the inner corona coincides very well in time with the rise phase (or energy release phase) of the corresponding soft X-ray flares, strongly implying that both phenomena are driven by the same process at the same time, possibly by magnetic field reconnections. However, this implication raises another important question of under what circumstances the energy release process in the corona leads to an eruption, and under what circumstances it remains to be confined during the process. An answer to this question shall shed the light on the origin of flares as well as CMEs.
The occurrence rate of eruptive events depends on the intensity and duration of flares. A statistical study performed by @Kahler_etal_1989 showed that the flares with longer duration tend to be eruptive, while more impulsive flares tend to be confined. By using the CME data from Solar Maximum Mission (SMM) and the flare data from GOES satellites during 1986 – 1987, @Harrison_1995 found that the association ratio of flares with CMEs increases from about 7% to 100% as the flare class increases from B-class to X-class, and from about 6% to 50% as the duration of flares increases from about 1 to 6 hours. @Andrews_2003 examined 229 M and X-class X-ray flares during 1996 – 1999, and found that the CME-association rate, or eruptive rate is 55% for M-class flares and 100% for X-class flares. With a much lager sample of 1301 X-ray flares, @Yashiro_etal_2005 obtained a similar result that the eruptive rates of C, M, and X-class flares are $16-25$%, $42-55$%, and $90-92$%, respectively. @Yashiro_etal_2005 work showed that a flare is not necessarily associated with a CME even if it is as intense as an X-class flare. Such kind of confined but extremely energetic events were also reported by @Feynman_Hundhausen_1994 and @Green_etal_2002.
The studies mentioned above indicate the probability of CME occurrence for a given flare. On the other hand, there is also a probability of flare occurrence for a given CME. There are CMEs that may not be necessarily associated with any noticeable X-ray flares. @Zhang_etal_2004 reported an extremely gradually accelerated slow CME without flare association, implying that the non-flare eruptive event tends to be slowly driven. By Combining the corona data from SMM and 6-hr soft X-ray data from GOES satellite, @St_Cyr_Webb_1991 reported that about 48% frontside CMEs were associated with X-ray events near the solar minimum of solar cycle 21. Based on SOHO observations, @Wang_etal_2002a studied 132 frontside halo CMEs, and found that the association rate of CMEs with X-ray flares greater than C-class increased from 55% at solar minimum to 80% near solar maximum. With 197 halo CMEs identified during 1997 – 2001, @Zhou_etal_2003 concluded that 88% CMEs were associated with EUV brightenings.
Some attempts have been made to explain the confinement or the eruptiveness of solar energetic events in the context of the configuration of coronal magnetic field. @Green_etal_2001 analyzed the 2000 September 30 confined event utilizing multi-wavelength data, and suggested that the event involve magnetic reconnection of two closed loops to form two newly closed loops without the opening of the involved magnetic structure. @Nindos_Andrews_2004 made a statistical study of the role of magnetic helicity in eruption rate. They found that the coronal helicity of active regions producing confined events tends to be smaller than the coronal helicity of those producing eruptive events.
In this paper, we address the eruptive-confinement issue of solar energetic events with the approach of a comparative study. We focus on the most energetic confined events that produce X-class soft X-ray flares but without CMEs. While the majority of X-class flares are eruptive, a small percentage (about 10%) of them are confined. The magnetic properties of these confined events shall be more outstanding than those less-energetic confined events. For making an effective comparison, we select eruptive events with X-ray properties, in terms of intensity and duration, very similar to those selected confined events. The differences on magnetic configuration between these two sets of events shall most likely reveal the true causes of eruption or confinement. How to select events and the basic properties of these events are described in section 2. Detailed comparative analyses of the two sets of events are given in section \[sec\_photosphere\] and \[sec\_corona\], which are on the properties of the photospheric magnetic field distribution and the extrapolated coronal magnetic field distribution, respectively. In section \[sec\_summary\], we summarize the paper.
Selection of Events and Observations {#sec_selection}
====================================
Confined Events: X-class Flares without CMEs
--------------------------------------------
From 1996 to 2004, there are 104 X-class soft X-ray flares reported by the NOAA (National Oceanic and Atmosphere Administration) Space Environment Center (SEC). Flares are observed by Geosynchronous Earth Observing Satellites (GOES), which record in high temporal resolution (every 3 seconds) the disk-integrated soft X-ray flux in two pass-bands: 1.0 – 8.0 Åand 0.5 – 4.0 Å, respectively. The flare catalog provides the peak intensity, beginning time, peak time, and ending time of flares. Based on peak intensity, flares are classified into five categories: A, B, C, M and X in the order of increasing strength. An X-class flare, in the strongest category, is defined by the peak flux in the 1.0 – 8.0 Å band exceeding $10^{-4}$ Wm$^{-2}$.
To find out whether a flare is associated with a CME or not, we make use of both the CME observations by Large Angle and Spectrometric Coronagraph (LASCO, [@Brueckner_etal_1995]) and coronal disk observations by Extreme Ultraviolet Imaging Telescope (EIT, [@Delaboudiniere_etal_1995]); both instruments are on-board the Solar and Heliospheric Observatory (SOHO) spacecraft. The search process started with the CME catalog[^1] [@Yashiro_etal_2004] for an initial quick look. A flare becomes a candidate of confined type if there is no any CME whose extrapolated onset time is within the 60-minute time window centered at the fare onset time. The onset time of a CME is calculated by linearly extrapolating the height-time measurement in the outer corona back to the surface of the Sun, which shall give the first-order approximation of the true onset time of CME. Further, we visually examine the sequence of the LASCO and EIT images around the flare time to verify that indeed there is no CME associated with the flare studied. One common property of these confined events is the lack of EUV dimming in EIT images, even though they show strong compact brightenings in EIT images. Following the compact brightening, there is no corresponding CME feature appearing in subsequent LASCO images. This scenario is in sharp contrast to that of an eruptive event, in which an EIT dimming accompanies the brightening, and within a few frames, a distinct CME feature appears in appropriate position angle in LASCO images. After applying this process on all X-class flares, we find 11 events are confined; they are listed in Table \[tb\_flares\_list\]. We notice that event 7 to 11 occurred within three days between July 15 - 17, 2004, and they all originated from the same solar active region (NOAA AR10649).
Among the 11 confined X-class flares from 1996 to 2004, the first four events have been reported earlier by @Yashiro_etal_2005.The third one has also been reported and studied by @Green_etal_2002. During 1996 – 2004, there are in total 104 X-class flares. Thus the percentage of confined X-class flares is about $10\%$. As shown in Table \[tb\_flares\_list\], all these confined flares are impulsive. Their rise time does not exceed 13 minutes except for event 8 (23 minutes). The decay time does not exceed 10 minutes for all the 11 events. The rise and decay times are derived from the begin, maximum and end times of flares, which are defined and compiled by NOAA/SEC [^2]. The peak intensity of all these events was less than X2.0 level except for event 10 (X3.6). Any event stronger than X3.6 is found always associated with a CME.
Out of the 11 confined events, we are able to select 4 of them suitable for further in-depth analysis. These events are numbered as 4, 5, 6 and 11, respectively. They are suitable because (1) the flare is isolated, which means that there is no other flare immediately preceding and following that flare, and (2) there is no other coronal dimming or CME eruption in the vicinity of the flare region within a certain period. Events 1 and 3 are not selected, because they mixed up with a flare-CME pair from the same source regions. In the presence of an eruptive flare immediately preceding or following a confined event, we are not certain how well the confined event is related with the earlier or later eruptive one. In order to make our analysis as “clean" as possible, such events are discarded. Event 2 is also excluded because its source region is right behind the western limb and hence no timely magnetogram data is available. Events 7 through 11 were all from the same active region. By over-plotting EIT images showing flare locations on the MDI magnetogram images, we find that these flares essentially occurred at the same location within the active region. Thus we choose only the last event representing all the five events. The four confined events selected for further analysis are labelled by $C_1$ through $C_4$ in the second column of Table \[tb\_flares\_list\].
Eruptive Events: X-class Flares with CMEs
-----------------------------------------
For making the comparative study with the 4 confined events mentioned above, a set of four eruptive X-class flares are selected. These eruptive flares are chosen to have similar properties in X-rays as those confined events:(1) their rise time and decay time are less than 13 minutes, (2) their intensities are between X1.0 to X2.0. Further, their locations are within $60^\circ$ from solar central meridian in longitude in order to reduce the projection effect in magnetograms. These flares, which are relatively impulsive, are indeed associated with CMEs, as shown in LASCO images. The four events are also listed in Table \[tb\_flares\_list\] labelled as $E_1$ through $E_4$, respectively.
An overview of the two sets of events is given in Figure \[fg\_flare\_overview\]. The upper panels show the four confined events and the lower panels show the four eruptive events. For each event we show the GOES soft X-ray flux profile (all in a 2-hour interval), running-difference EIT image and running-difference LASCO image in the top, middle and low panels, respectively. Apparently, the temporal profiles of GOES soft X-ray fluxes exhibit no noticeable difference between the two sets of events, due to the constraint in our selection of events. Moreover, as seen in the EIT images, the two sets of events are all associated with compact coronal brightening indicating the occurrence of flares. However, for eruptive flares, the accompanying CMEs are clearly seen in those LASCO images. In contrast, there is no apparent brightening CME feature seen in LASCO images for those confined events (only one image is shown here to represent the observed sequence of images, which all indicate a non-disturbed corona). For eruptive events, the speeds and angular widths of CMEs are also listed in the Table \[tb\_flares\_list\].
Magnetic Properties in the Photosphere {#sec_photosphere}
======================================
Flare Location and Active Region Morphology
-------------------------------------------
To explore what physical factors lead the two similar sets of flares, all strong and impulsive, to have difference in CME production, we first study the magnetic properties of their surface source regions. SOHO/MDI provides the observations of photospheric magnetic field (the component along the line of sight) every 96 minutes. The spatial resolution of MDI magnetograms is about 4 arcsec with a plate scale of 2 arcsec per pixel, at which detailed magnetic features across the source active regions are reasonably resolved. To reduce the projection effect of line-of-sight magnetic field, we have only chosen the events within $60^\circ$ from solar central meridian.
For each event, we determine the location of the flare seen in EIT relative to the magnetic features seen in MDI. We first align the MDI image with the corresponding EIT image. The difference in the timing between MDI and EIT images have been taken into account. Figure \[fg\_pmdieit\_example\] illustrates the alignment for the 2001 June 23 event. The soft X-ray flare began at 04:02 UT and peaked at 04:09 UT. In EIT 195 Å image showing the flare was taken at 04:11 UT. The nearest full disk MDI image prior to the flare was taken at 03:11 UT. The MDI magnetogram is rotated to fit the EIT time, and then superimposed in contours onto the EIT image. In the right panel of Figure \[fg\_pmdieit\_example\], we display the aligned images; only the region of interest is shown. Using this method, we are able to determine the location of flares, which is just above the neutral lines seen in the magnetogram.
In Figure \[fg\_segmdi\_noncme\] and \[fg\_segmdi\_cme\], we show the magnetogram images for the four confined events and the four eruptive events, respectively. The flare sites, or bright patches seen in EIT, are marked by red asterisks in the images. The magnetogram images have been re-mapped onto the Carrington coordinate, which reduces the spherical projection effect of the image area. The $x$-axis is Carrington longitude in units of degree, and the $y$-axis is the sine of latitude. The area of the images shown in Figure \[fg\_segmdi\_noncme\] and \[fg\_segmdi\_cme\] are all $30^\circ\times30^\circ$ squares, which usually cover the entire active regions producing the CMEs/flares of interest. To highlight the magnetic features, the displayed images have been segmented into three different levels: strong positive magnetic field ($\geq 50$ Gauss) indicated by white color, strong negative magnetic field ($\leq -50$ Gauss) indicated by black color, and weak field (from $-50$ to 50 Gauss) indicated by gray color. Note that the noise level of a MDI magnetogram image is typically at about 10 Gauss. As shown in the figures, an active region is naturally segmented into many individual pieces. Those pieces with magnetic flux larger than $10^{13}$ wb are labelled by a letter with a number in the neighbouring bracket indicating the magnetic flux in units of $10^{13}$ wb.
Results
-------
We find that there is no apparent difference in term of total magnetic flux of the source region between the confined events and eruptive events. The total magnetic flux, combining both positive and negative flux, are listed in Table \[tb\_photosphere\]. The total flux for confined events varies from about 5 to 36$\times10^{13}$ wb, while for eruptive events it varies from about 11 to 24$\times10^{13}$ wb.
However, there is a noticeable pattern that the confined flares all originated in a location relatively closer to the center of the host active region. Figure \[fg\_segmdi\_noncme\]a shows the confined event of 2001 June 23. There are three relatively large magnetic pieces labelled by ‘A’, ‘B’ and ‘C’. The flare site is surrounded by the pieces ‘A’ and ‘B’. Figure \[fg\_segmdi\_noncme\]b shows the 2003 August 9 confined event. The flare location is associated with three small negative patches (marked by the red asterisks) embedded in a large positive piece ‘A’. Figure \[fg\_segmdi\_noncme\]c shows the 2004 February 26 confined event. The flare occurred just above the neutral line between thye large positive piece ‘A’ and the large negative piece ‘B’. Figure \[fg\_segmdi\_noncme\]d shows the 2004 July 17 confined event. The flare was located in a complex active region with a large number of sunspots. It occurred right at the boundary between pieces ‘C’ and ‘F’. From the view of the entire active region, pieces ‘C’ and ‘F’ were further enclosed by two much larger and stronger pieces ‘A’ and ‘D’ whose fluxes were about 10 times larger.
For those eruptive flares, on the other hand, the flare sites were all relatively further from the center of the magnetic flux distribution. In other words, they were closer to the edge of hosting active regions. Figure \[fg\_segmdi\_cme\]a shows the 1998 May 2 event. The strongest pair of magnetic pieces are ‘A’ and ‘D’, but the eruptive flare occurred at the neutral line between pieces ‘D’ and ‘C’, which was the smallest amongst the 4 labelled pieces in the active region. Figure \[fg\_segmdi\_cme\]b shows the 2000 March 2 event. Similarly, the strongest pair of pieces were ‘A’ and ‘C’, but the flare was from the neutral line between pieces ‘C’ and ‘B’, which was the smallest labelled piece. Figure \[fg\_segmdi\_cme\]c shows the 2000 November 24 event. The flare occurred at the outer edge of the strongest piece ‘A’, which was neighbored by a very small region with negative flux. Figure \[fg\_segmdi\_cme\]d shows the 2004 October 30 event. The flare site was also close to the edge of the entire active region.
To quantify this observation of different displacements of flare locations, we here introduce a flare displacement parameter, which is defined by the surface distance between the flare site and the weighted center of the magnetic flux distribution of the host active region, or center of magnetic flux (COM) for short. The COM might be the place that has the most overlying magnetic flux. The COM is calculated based on the re-mapped $30^\circ\times30^\circ$ MDI images (without segmentation). It is a point, across which any line can split the magnetogram into two flux-balanced halves, and can be formulated as $x_c=\frac{\sum_iF_i*x_i}{\sum_iF_i}$ and $y_c=\frac{\sum_iF_i*y_i}{\sum_iF_i}$. The COM of these events have been marked by the diamonds in Figure \[fg\_segmdi\_noncme\] and \[fg\_segmdi\_cme\]. With known COM, it is easy to derive the displacement parameter, which is listed in Table \[tb\_photosphere\]. Consistent with the earlier discussions, it is found that the displacement parameters for the four confined events are all smaller than 17 Mm, while for the four eruptive events, they are all larger than 22 Mm.
We now consider possible errors in calculating the displacement parameter. The error main arises from the uncertainty in the recorded weak magnetic field around the active regions. However, in the selected region of study that is $30^\circ\times30^\circ$ across, the highlighted white and black pieces contain about 99% of the total magnetic flux in the region. Therefore the uncertainty of the flux is expected to be at the order of 1%. Considering the formula of the coordinates of COM given in the last paragraph and assuming a typical scale of 100 Mm, the error of the calculated distance is about 1 Mm. Further, considering the spatial resolution of MDI of $\sim1$ Mm (varing from $\sim0.7$ Mm at central meridian to $\sim1.4$ Mm at longitude of $\pm60^\circ$), the overall uncertainty should be about $\pm2$ Mm. With these consideration, the displacement parameters and their uncertainties are plotted in Figure \[fg\_ratio\_distance\]. The confined events are indicated by diamonds, and the eruptive events are indicated by asterisks. The vertical dashed line, which corresponds to a displacement of about 19.5 Mm, effectively separates the two sets of events.
Magnetic Properties in the Corona {#sec_corona}
=================================
Method
------
Having studied the magnetic field distributions in the photosphere, we further investigate into magnetic field distributions in the 3-D corona. The magnetic field configuration in the corona shall ultimately determine the eruption/confinement since the energy releases occurs in the corona. There is so far no direct observations of coronal magnetic fields. We have to utilize certain models to calculate the coronal magnetic field based on observed photospheric boundary. In this paper, we apply the commonly used potential-field source-surface (PFSS) model [e.g., @Schatten_etal_1969; @Hoeksema_etal_1982]. The PFSS model is thought to be a useful first-order approximation to the global magnetic field of the solar corona. Nevertheless, we realize that the current-carrying core fields, which are low-lying and near the magnetic neutral line, are far from the potential field approximation; these core fields are likely to be the driving source of any energy release in the corona. Therefore, the usage of PFSS model in this study is limited to calculate the total flux of the overlying large scale coronal field, which is believed to be closer to a potential approximation. These overlying fields are thought to constrain the low-lying field from eruption.
A modified MDI magnetic field synoptic chart is used as input to our PFSS model. The high-resolution MDI magnetic field synoptic chart[^3] is created by interpolating data to disk-center resolution, resulting in a $3600\times1080$ pixel map. The X and Y axis are linear in Carrington longitude ($0.1^\circ$ intervals) and sine latitude, respectively. This high resolution is useful in creating detailed coronal magnetic field above the source regions of interest. Since an MDI synoptic chart is created from the magnetogram images over a $\sim27$-day solar rotation, the synoptic chart does not exactly represent the photospheric magnetic field in the region of interest at the flare/CME time. The details of the source region may be different because of the evolution of photospheric magnetic field. To mitigate this problem, we use the MDI daily magnetogram to update the original MDI synoptic chart. The process is to re-map the snapshot magnetogram image prior to the flare occurrence to the Carrington grid, and then slice out the region of interest, $30^\circ$ in longitude and $60^\circ$ in latitude. This sliced region, to replace the corresponding portion in the original synoptic chart.
Since a PFSS model makes use of the spheric harmonic series expansion, we realize that a high-resolution data requires a high order expansion in order to have a consistent result. We calculate the spheric harmonic coefficients to as high as 225 order for the input $3600\times1080$ boundary image. We find that, at this order, we can get the best match between the calculated photospheric magnetic field and the input synoptic chart. The mean value of the difference between them is less than 0.5 Gauss, and the standard deviation is generally $\lsim 15$ Gauss for solar minimum and $\lsim 25$ Gauss for solar maximum, which are comparable to the noise level of MDI magnetograms. That means, we can effectively reproduce the observed photospheric magnetic field with the 225-order PFSS model. With spheric harmonic coefficients known, it is relatively straightforward to calculate the magnetic field in the 3-D volume of the corona.
Results
-------
Figure \[fg\_mag\_n\_20040717\] and \[fg\_mag\_c\_20041030\] show the extrapolated coronal magnetic field lines for one confined event (2004 July 17) and one eruptive event (2004 October 30), respectively. In each figure, the left panel shows field lines viewed from top, and the right panel shows field lines viewed from the side by rotating the left panel view of $60$ degrees into the paper. The green-yellow colors denote the closed field lines, with green indicating the loop part of outward magnetic field (positive magnetic polarity at the footpoints) and yellow the inward (negative magnetic polarity at the footpoints), and the blue color indicates the open field lines. The two examples show that the location of the confined flare, which is near the center of the active region, is covered by a large tuft of overlying magnetic loop arcades, while the location of the eruptive flare, which is near the edge of the active region, has relatively few directly overlying loop arcade. In particular, for the eruptive event, the nearby positive and negative magnetic field lines seem to connect divergently with other regions, instead of forming a loop arcade of its own.
To quantify the strength of the overlying field, we calculate the total magnetic flux cross the plane with the $x$ direction extending along the neutral line and the $y$ direction vertically along the radial direction. The thick blue lines on the photospheric surface in Figure \[fg\_segmdi\_noncme\] and \[fg\_segmdi\_cme\] indicate the neutral lines used in the calculation. The length of the neutral lines is determined as it encompasses the major part of the eruption region. The overlying magnetic field flux then is normalized to the length of the neutral line. Thus obtained normalized overlying magnetic flux is a better quantity to be used for comparison between different events, because this parameter is not sensitive to the exact length of the neutral lines selected, which may vary significantly from event to event.
Such calculated magnetic fluxes for the 8 events are listed in Table \[tb\_flux\]. In calculating the flux, we do not consider that the crossing direction of the field lines over the neutral line is from one side to the other or opposite. The relative uncertainty of the calculated magnetic flux can be estimated as $\sigma_B/B_0$ where $\sigma_B$ is the uncertainty of calculated magnetic field strength in the corona and $B_0$ is the magnetic field strength in the active regions at photosphere. Considering $\sigma_B$ is about 15 to 25 Gauss, the standard deviation mentioned before, and $B_0$ is usually hundreds of Gauss, we infer that the uncertainty of overlying magnetic flux is about 10%. The estimate should be true in case that the coronal magnetic field is correctly obtained by our extrapolation method. If the extrapolated field largely deviates from the realistic status, the uncertainty may probably be slightly different.
The total overlying flux, $F_{total}$, in the height range from 1.0 to 1.5 $R_\odot$, is given in the third column. It seems that there is no systematic difference between the confined events and eruptive events. The value for the confined events varies from 0.40 to 1.27 $\times 10^{10}$ wb/Mm, and that for the eruptive events from 0.73 to 1.34 $\times10^{10}$ wb/Mm.
We further calculate the flux in two different height ranges, the lower flux from 1.0 to 1.1 $R_\odot$ and the higher flux from 1.1 to 1.5 $R_\odot$. A common accepted scenario is that the lower flux shall correspond to the inner sheared core field (or fully-fledged flux rope if filament present) that tends to move out, while the outer flux is the large scale overlying arcade that tends to constrain the inner flux from eruption. Note that the chosen of 1.1 $R_\odot$, which corresponds to a height of about 70 Mm above the surface, is rather arbitrary. However, slightly changing this number will not affect any overall results that will be reached. The magnetic flux in the low corona, $F_{low}$, and in the high corona, $F_{high}$, are listed in the fourth and fifth columns of Table \[tb\_flux\], respectively.
There is a trend that the low-corona flux for the eruptive events is generally larger than that for the confined events. Three out of the four eruptive events have their low-corona overlying flux more than 1.0 $\times10^{10}$ wb/Mm, while three out of the four confined events have the flux less than 1.0 $\times10^{10}$ wb/Mm. On the other hand, the high-corona flux for the eruptive events seems smaller than that for the confined events. Three confined events have their high-corona flux more than 0.15 $\times10^{10}$ wb/Mm, while all four eruptive events have the flux less than 0.15 $\times10^{10}$ wb/Mm.
We further calculate the flux ratio parameter, which is defined as $R=F_{low}/F_{high}$. This quantity is independent of the normalization. It may serve as an index of how weak the constraint on the inner eruptive field is. Interestingly, the flux ratios for the two sets of events fall into two distinct groups. For the confined events, $R$ varies from 1.59 to 5.68, while for the eruptive events, the value of $R$ is larger, from 7.11 to 10.17. The value of 6.5 may be used as a boundary separating the two sets of events. This value probably implies a threshold for confinement or eruptiveness. This is to say, if the flux ratio is less than 6.5, a flare is likely to be confined, otherwise eruptive. The higher the ratio, the higher the possibility of a coronal energy release being eruptive.
Summary and Discussions {#sec_summary}
=======================
In summary, among the 104 X-class flares occurred during 1996 – 2004, we found a total of 11 ($\sim10\%$) are confined flares without associated CMEs, and all the others ($\sim90\%$) are eruptive flares associated with CMEs. Four suitable confined flares are selected to make a comparative study with four eruptive flares, which are similar in X-ray intensity and duration as those confined events. We have carefully studied the magnetic properties of these events both in the photosphere and in the corona. The following results are obtained:
\(1) In the photosphere, we can not find a difference of the total magnetic fluxes in the surface source regions between the two sets of events. However, there is an apparent difference in the displacement parameter, which is defined as the surface distance between the flare site and the center of magnetic flux distribution. For the confined events, the displacement is from 6 to 17 Mm, while for those eruptive events it is from 22 to 37 Mm. This result implies that the energy release occurring in the center of an active region is more difficult to have a complete open eruption, resulting in a flare without CME. On the other hand, the energy release occurring away from the magnetic center has a higher probability to have an eruption, resulting in both flares and CMEs. Whether an eruption could occur or not may be strongly constrained by the overlying large scale coronal magnetic field. The overlying coronal magnetic field shall be strongest and also longest along the vertical direction over the center of an active region. On the other hand, the overlying constraining field shall be weaker if the source is away from the center. This scenario is further supported by our study of coronal magnetic field.
\(2) Calculation of coronal magnetic field shows that the flux ratio of the magnetic flux in the low corona to that in the high corona is systematically larger for the eruptive events than that for the confined events. The magnetic flux ratio for the confined events varies from 1.6 to 5.7, while the ratio for the eruptive events from 7.1 to 10.2. However, there is no evident difference between the two sets of events in the total magnetic flux straddling over neutral lines, and there is only a weak trend indicating a systematic difference in the low- and high-corona magnetic fluxes. This low-to-high corona magnetic flux ratio serves as a proxy of the strength of the inner core magnetic field, which may play an erupting role, relative to the strength of the overlying large scale coronal magnetic field, which may play a constraining role to prevent eruption. The lower this ratio, the more difficult the energy release in the low corona can be eruptive.
There is variety of theoretical models on the initiation mechanism of CMEs and the energy release of flares [e.g., @Sturrock_1989; @Chen_1989; @vanBallegooijen_Martens_1989; @Forbes_Isenberg_1991; @Moore_Roumeliotis_1992; @Low_Smith_1993; @Mikic_Linker_1994; @Antiochos_etal_1999; @Lin_Forbes_2000]. These models differ in pre-eruption magnetic configurations, trigger processes, or where magnetic reconnection occurs. Nevertheless, in almost all these models, the magnetic configuration involves two magnetic regimes, one is the core field in the inner corona close to the neutral line, the other is the large scale overlying field or background field. The core field is treated as highly sheared or as a fully-fledged flux rope; in either case, the core field stores free energy for release. On the other hand, the overlying field is regarded as potential and considered to be main constraining force to prevent the underlying core field from eruption or escaping. @Torok_Kliem_2005 and @Kliem_Torok_2006 recently pointed out that the decrease of the overlying field with height is a main factor in deciding whether the kink-instability (in their twist flux rope model) leads to a confined event or a CME. On the other hand, @Mandrini_etal_2005 reported the smallest CME event ever observed by 2005, in which the CME originated from the smallest source region, a tiny dipole, and developed into the smallest magnetic cloud. They suggested that the ejections of tiny flux ropes are possible. Therefore, it is reasonable to argue that, whether an energy release in the corona is eruptive or confined, is sensitive to the balance between the inner core field and the outer overlying field. Our observational results seem to be consistent with this scenario.
This study is only a preliminary step to investigate the confinement and/or eruptiveness of solar flares, or coronal energy releases in general. However, it demonstrates that the distribution of magnetic field both in the photosphere and in the corona may effectively provide the clue of the possible nature of an energetic event: whether a flare, a CME or both. To further evaluate the effectiveness of this methodology, a more robust study involving more events is needed.
[xx]{}
Andrews, M. D.: 2003, A search for [CMEs]{} associated with big flares, [*Sol. Phys.*]{} [**218**]{}, 261–279.
Antiochos, S. K., DeVore, C. R. and Klimchuk, J. A.: 1999, A model for solar coronal mass ejections, [*Astrophys. J.*]{} [**510**]{}, 485–493.
Brueckner, G. E., Howard, R. A., Koomen, M. J., Korendyke, C. M., Michels, D. J., Moses, J. D., Socker, D. G., Dere, K. P., Lamy, P. L., Llebaria, A., Bout, M. V., Schwenn, R., Simnett, G. M., Bedford, D. K. and Eyles, C. J.: 1995, The large angle spectroscopic coronagraph ([LASCO]{}), [*Sol. Phys.*]{} [**162**]{}, 357–402.
Chen, J.: 1989, Effects of toroidal forces in current loops embedded in a background plasma, [*Astrophys. J.*]{} [**338**]{}, 453–470.
Delaboudiniere, J.-P., Artzner, G. E., Brunaud, J. and et al.: 1995, [EIT]{}: Extreme-ultraviolet imaging telescope for the [SOHO]{} mission, [*Sol. Phys.*]{} [**162**]{}, 291–312.
Feynman, J. and Hundhausen, A. J.: 1994, Coronal mass ejections and major solar flares: The great active center of march 1989, [*J. Geophys. Res.*]{} [ **99(A5)**]{}, 8451–8464.
Forbes, T. G. and Isenberg, P. A.: 1991, A catastrophe mechanism for coronal mass ejections, [*Astrophys. J.*]{} [**373**]{}, 294–307.
Green, L. M., Harra, L. K., Matthews, S. A. and Culhane, J. L.: 2001, Coronal mass ejections and their association to active region flaring, [*Sol. Phys.*]{} [**200**]{}, 189–202.
Green, L. M., Matthews, S. A., [van Driel-Gesztelyi]{}, L., Harra, L. K. and Culhane, J. L.: 2002, Multi-wavelength observations of an x-class flare without a coronal mass ejection, [*Sol. Phys.*]{} [**205**]{}, 325–339.
Harrison, R. A.: 1995, The nature of solar flares associated with coronal mass ejection, [*Astron. & Astrophys.*]{} [**304**]{}, 585–594.
Harrison, R. A.: 2003, Soho observations relating to the association between flares and coronal mass ejections, [*Adv. Space Res.*]{} [ **32(12)**]{}, 2425–2437.
Hoeksema, J. T., Wilcox, J. M. and Scherrer, P. H.: 1982, Structure of the heliospheric current sheet in the early portion of sunspot cycle 21, [*J. Geophys. Res.*]{} [**87**]{}, 10331–10338.
Kahler, S. W., [Sheeley, Jr.]{}, N. R. and Liggett, M.: 1989, Coronal mass ejections and associated [X-ray]{} flare durations, [*Astrophys. J.*]{} [ **344**]{}, 1026–1033.
Kliem, B., and Torok, T.: 2006, Torus Instability, [*Phys. Rev. Lett.*]{} [ **96**]{}, 255002.
Lin, J. and Forbes, T. G.: 2000, Effects of reconnection on the coronal mass ejection process, [*J. Geophys. Res.*]{} [**105(A2)**]{}, 2375–2392.
Low, B. C. and Smith, D. F.: 1993, The free energies of partially open coronal magnetic fields, [*Astrophys. J.*]{} [**410**]{}, 412–425.
Mandrini, C. H., Pohjolainen, S., Dasso, S., Green, L. M., Demoulin, P., van Driel-Gesztelyi, L., Copperwheat, C. and Foley, C.: 2005, Interplanetary flux rope ejected from an X-ray bright point: The smallest magnetic cloud source-region ever observed, [*Astron. & Astrophys.*]{} [**434**]{}, 725–740.
Mikic, Z. and Linker, J. A.: 1994, Disruption of coronal magnetic field arcades, [*Astrophys. J.*]{} [**430**]{}, 898–912.
Moore, R. L. and Roumeliotis, G.: 1992, Triggering of eruptive flares - destabilization of the preflare magnetic field configuration, [*in*]{} Z. Svestka, B. V. Jackson and M. Machado (eds), [*Eruptive Solar Flares*]{}, Proceedings of IAU Colloquium No. 113, Springer-Verlag, New York, pp. 107–121.
Nindos, A. and Andrews, M. D.: 2004, The association of big flares and coronal mass ejections: What is the role of magnetic helicity?, [*Astrophys. J.*]{} [**616**]{}, L175–L178.
Schatten, K. H., Wilcox, J. M. and Ness, N. F.: 1969, A model of interplanetary and coronal magnetic fields, [*Sol. Phys.*]{} [**6**]{}, 442–455.
, O. C. and Webb, D. F.: 1991, Activity associated with coronal mass ejections at solar minimum - [SMM]{} observations from 1984-1986, [*Sol. Phys.*]{} [**136**]{}, 379–394.
Sturrock, P. A.: 1989, The role of eruption in solar flares, [*Sol. Phys.*]{} [**121**]{}, 387–397.
Svestka, Z. and Cliver, E. W.: 1992, History and basic characteristics of eruptive flares, [*in*]{} Z. Svestka, B. V. Jackson and M. E. Machado (eds), [*Eruptive Solar Flares*]{}, Proceedings of IAU Colloquium No. 113, Springer-Verlag, New York, pp. 1–11.
Torok, T. and Kliem, B.: 2005, Confined and ejective eruptions of kink-unstable flux ropes, [*Astrophys. J.*]{} [**630**]{}, L97–L100.
, A. A. and Martens, P. C. H.: 1989, Formation and eruption of solar prominences, [*Astrophys. J.*]{} [**343**]{}, 971–984.
Wang, Y. M., Ye, P. Z., Wang, S., Zhou, G. P. and Wang, J. X.: 2002, A statistical study on the geoeffectiveness of earth-directed coronal mass ejections from [March]{} 1997 to [December]{} 2000, [*J. Geophys. Res.*]{} [ **107(A11)**]{}, 1340, doi:10.1029/2002JA009244.
Yashiro, S., Gopalswamy, N., Akiyama, S., Michalek, G. and Howard, R. A.: 2005, Visibility of coronal mass ejections as a function of flare location and intensity, [*J. Geophys. Res.*]{} [**110(A12)**]{}, A12S05.
Yashiro, S., Gopalswamy, N., Michalek, G., [St. Cyr]{}, O. C., Plunkett, S. P., Rich, N. B. and Howard, R. A.: 2004, A catalog of white light coronal mass ejections observed by the soho spacecraft, [*J. Geophys. Res.*]{} [ **109(A7)**]{}, A07105.
Zhang, J., Dere, K. P., Howard, R. A., Kundu, M. R. and White, S. M.: 2001, On the temporal relationship between coronal mass ejections and flares, [ *Astrophys. J.*]{} [**559**]{}, 452–462.
Zhang, J., Dere, K. P., Howard, R. A. and Vourlidas, A.: 2004, A study of the kinematic evolution of coronal mass ejections, [*Astrophys. J.*]{} [ **604**]{}, 420–432.
Zhou, G., Wang, J. and Cao, Z.: 2003, Correlation between halo coronal mass ejections and solar surface activity, [*Astron. & Astrophys.*]{} [ **397**]{}, 1057.
[lcccccccccp[100pt]{}]{} \# &Label &Date &Begin &$T_R^a$ &$T_D^b$ &Class &Location &NOAA &CME$^c$ &Comment\
& & &UT &min &min & & &AR &V(km/s)/Width &\
\
1 & &2000/06/06 &13:30 &9.0 &7.0 &X1.1 &N20E18 &9026 &- &Contained by a preceding and a following M-class flares (Y)\
2 & &2000/09/30 &23:13 &8.0 &7.0 &X1.2 &N07W91 &9169 &- &Limb event (G, Y)\
3 & &2001/04/02 &10:04 &10.0 &6.0 &X1.4 &N17W60 &9393 &- &Contained by a preceding eruptive flare (Y)\
4 &$C_1$ &2001/06/23 &04:02 &6.0 &3.0 &X1.2 &N10E23 &9511 &- &(Y)\
5 &$C_2$ &2003/06/09 &21:31 &8.0 &4.0 &X1.7 &N12W33 &10374 &- &\
6 &$C_3$ &2004/02/26 &01:50 &13.0 &7.0 &X1.1 &N14W14 &10564 &- &\
7 & &2004/07/15 &18:15 &9.0 &4.0 &X1.6 &S11E45 &10649 &- &\
8 & &2004/07/16 &01:43 &23.0 &6.0 &X1.3 &S11E41 &10649 &- &\
9 & &2004/07/16 &10:32 &9.0 &5.0 &X1.1 &S10E36 &10649 &- &\
10 & &2004/07/16 &13:49 &6.0 &6.0 &X3.6 &S10E35 &10649 &- &\
11&$C_4$ &2004/07/17 &07:51 &6.0 &2.0 &X1.0 &S11E24 &10649 &- &Event 7–11 all from the same AR\
\
1 &$E_1$ &1998/05/02 &13:31 &11.0 &9.0 &X1.1 &S15W15 &8210 &936/halo &\
2 &$E_2$ &2000/03/02 &08:20 &8.0 &3.0 &X1.1 &S18W54 &8882 &776/62$^\circ$ &\
3 &$E_3$ &2000/11/24 &04:55 &7.0 &6.0 &X2.0 &N19W05 &9236 &1289/halo &\
4 &$E_4$ &2004/10/30 &11:38 &8.0 &4.0 &X1.2 &N13W25 &10691 &427/halo &\
\
$^a$ Rise time of flares.\
$^b$ Decay time of flares.\
$^c$ Apparent speed and angular width of CMEs. Adopted from the online GSFC-NRL-CUA CME catalog.\
G and Y in comment column mean that the corresponding events have been reported by @Green_etal_2002 and @Yashiro_etal_2005, respectively.
[lccc]{} Event &Date &Flux$^a$ &Distance$^b$\
& &$10^{13}$ wb &Mm\
\
$C_1$ &2001/06/23 &5 &6\
$C_2$ &2003/06/09 &36 &17\
$C_3$ &2004/02/26 &23 &8\
$C_4$&2004/07/17 &34 &10\
\
$E_1$ &1998/05/02 &17 &22\
$E_2$ &2000/03/02 &24 &33\
$E_3$ &2000/11/24 &18 &37\
$E_4$ &2004/10/30 &11 &29\
\
$^a$ Total magnetic flux in active regions measured in MDI magnetogram.\
$^b$ Surface distance between the flare site and the COM of the associated active region.
[lccccc]{} Event &Date &$F_{total}$ &$F_{low}$ &$F_{high}$ &Ratio$=\frac{F_{low}}{F_{high}}$\
& &$10^{10}$ wb/Mm &$10^{10}$ wb/Mm &$10^{10}$ wb/Mm &\
\
$C_1$ &2001/06/23 &0.40 &0.34 &0.06 &5.67\
$C_2$ &2003/06/09 &0.83 &0.61 &0.22 &2.77\
$C_3$ &2004/02/26 &1.27 &1.08 &0.19 &5.68\
$C_4$ &2004/07/17 &1.19 &0.73 &0.46 &1.59\
\
$E_1$ &1998/05/02 &1.34 &1.22 &0.12 &10.17\
$E_2$ &2000/03/02 &1.17 &1.06 &0.11 & 9.64\
$E_3$ &2000/11/24 &1.14 &1.03 &0.11 & 9.36\
$E_4$ &2004/10/30 &0.73 &0.64 &0.09 & 7.11\
![An overview of the four confined ($C_1 - C_4$, upper panels) and four eruptive ($E_1 - E_2$, lower panels) flares. For each events, we display its GOES X-ray flux profile (spanning 2 hours), running-difference images of EIT 195Å and LASCO/C2 in the three sub-panels from top to bottom.[]{data-label="fg_flare_overview"}](f1.eps){width="0.8\hsize"}
![An example showing flare and its source region. The left image is a full disk MDI magnetogram taken before the flare onset. The right image shows the EIT image in green-white false colors; the white patch at the center denotes the flare location. The superimposed contours show the magnetogram, with yellow the positive and blue the negative field.[]{data-label="fg_pmdieit_example"}](f2.ps){width="\hsize"}
![Segmented MDI magnetograms for the four confined events, in which strong positive ($\geq 50$ Gauss) and negative ($\leq -50$ Gauss) magnetic fields are highlighted as white and black colors, respectively. Red asterisk symbols indicate the flare sites, the red diamond symbols indicate the COM of the active regions, and the blue lines denote the neutral lines over which the flares occurred. See text for more details.[]{data-label="fg_segmdi_noncme"}](f3.eps){width="\hsize"}
![Segmented MDI magnetograms for the four eruptive events. See the caption in Figure \[fg\_segmdi\_noncme\].[]{data-label="fg_segmdi_cme"}](f4.eps){width="\hsize"}
![Calculated coronal magnetic field of one confined event. The closed field lines are denoted by the green & yellow colors, corresponding to the outward and inward direction respectively, and the open field lines are denoted by the blue color. The left image is of a top view while the right image is of a side view.[]{data-label="fg_mag_n_20040717"}](f5.eps){width="\hsize"}
![An example of eruptive events showing the extrapolated magnetic field above the active region.[]{data-label="fg_mag_c_20041030"}](f6.eps){width="\hsize"}
![The scattering plot showing magnetic properties of both confined events (diamond symbols) and the eruptive events (asterisk symbols). The $x$-axis denotes the distance between the flare site and the center of magnetic flux distribution (COM) of the active region, and the $y$-axis denotes the ratio of magnetic flux in the low to high corona above the neutral line.[]{data-label="fg_ratio_distance"}](f7.ps){width="\hsize"}
[^1]: the NRL-GSFC-CUA CME catalog at http://cdaw.gsfc.nasa.gov/CME\_list/
[^2]: http://www.sec.noaa.gov/ftpdir/indices/events/README
[^3]: http://soi.stanford.edu/magnetic/index6.html
|
---
author:
- 'L. Sidoli'
- 'S. Mereghetti'
- 'F. Favata'
- 'T. Oosterbroek'
- 'A.N. Parmar'
date: 'Received 12 April 2006; Accepted: 6 June 2006'
title: 'Unveiling the nature of the highly absorbed X–ray source SAX J1748.2-2808 with XMM-Newton'
---
Introduction {#sect:intro}
============
is an X–ray source discovered with the Narrow Field Instruments on-board during a survey of the Galactic Center region (hereafter GC) performed in September 1997 (Sidoli [@s:00], Sidoli et al. [@s:01]). The spectrum was severely absorbed and displayed an intense Fe K emission line. The spectrum was poorly constrained, making both thermal and non-thermal nature for the X–ray emission possible. The source, unresolved at the angular resolution of the MECS instrument (FWHM $\sim$1$'$), is located in the direction of the giant molecular cloud Sgr D.
At the time of its discovery, the nature of was uncertain and its intense Fe K line emission, together with its highly absorbed spectrum, made it a unique object in the GC region which could well represent the bright tail of a distribution of similar unresolved objects significantly contributing to the diffuse Fe line emission (at 6.7 keV) from the galactic ridge (Koyama et al. [@k:89]; Ebisawa et al. [@e:01]).
Interestingly, displays properties very similar to a class of sources subsequently discovered with the INTEGRAL satellite (see e.g., Walter et al. [@w:04], and references therein): these objects show strong photoelectric absorption, hard 2–10 keV spectra, and often display intense Fe line emission. Most of them also show X–ray pulsations, thus indicating that they are likely high mass X–ray binaries embedded in a local absorbing gas.
Here we report the results of an observation the region of sky around , performed with the main goal of unveiling the nature of this intriguing source.
{height="8cm"}
Observations {#sect:obs}
============
The XMM-Newton Observatory (Jansen et al. [@ja:01]) includes three 1500 cm$^2$ X–ray telescopes each with an European Photon Imaging Camera (EPIC) at the focus, composed of and one pn (Strüder et al. [@st:01]) two MOS CCD detectors (Turner et al. [@t:01]). The field was observed with on 2005 February 26-27 for about 50 ks.
Data were processed using version 6.1 of the Science Analysis Software (SAS). Known hot, or flickering, pixels and electronic noise were rejected using the SAS. A further severe cleaning was necessary because of the presence of several soft proton flares. After rejecting the time intervals where the flares were present, the net good exposure times reduced to about 32.3 ks for the MOS1 and the MOS2, and to 13.3 for the pn. Cleaned MOS1 (with pattern selection from 0 to 12), MOS2 (pattern selection from 0 to 12) and pn (patterns from 0 to 4) events files were extracted, and used for the subsequent analysis: a source detection analysis and a spectral analysis of the brightest sources.
Spectra were rebinned such that at least 20 counts per bin were present and such that the energy resolution was not over-sampled by more than a factor 3. Free relative normalizations between the MOS1, MOS2 and pn instruments were included. The background spectra were extracted from source free regions of the same observation. All spectral uncertainties and upper-limits are given at 90% confidence for one interesting parameter.
: analysis and results
======================
A close-up view of the combined EPIC 2–10 keV image is shown in Fig. \[fig:resolved\], together with an optical image of the same field. resolved into two sources, a brighter (here called “main” source) and a fainter one.
In order to minimize contamination from the nearby fainter source, we extracted source counts centered on the “main” source, from a circular region of 20 radius, for the MOS1, MOS2 and pn detectors separately.
The fit of the spectrum with an absorbed power-law ($\chi^2$=41.5 with 35 degrees of freedom, d.o.f.) resulted in structured residuals around 6–7 keV, confirming the presence of the Fe-K line already observed with . Adding a Gaussian line to the power-law model resulted in a significantly better fit ($\chi$/d.o.f.=24.9/32), with a broad line ($\sigma$=0.43$^{+0.33}_{-0.20}$ keV) centered in the range 6.4–6.8 keV (see Figs. \[fig:cont\_pow\] and \[fig:mainspec\], and Table \[tab:mainspec\] for the resulting parameters). We next tried with other simple models, such as a hot plasma model ([mekal]{} in [xspec]{}), a thermal bremsstrahlung spectrum, and a blackbody model (see Table \[tab:mainspec\] for the spectral analysis results).
[llllll]{} Model & Column density & Parameter & $\chi^2$/dof &Flux (2–10 keV) & $L$ (2–10 keV)\
&($10^{22}$ cm$^{-2}$)& & &($10^{-13}$ erg cm$^{-2}$ s$^{-1}$ ) & ($10^{34}$ erg s$^{-1}$ )\
Power law & $16.5^{+6.0}_{-4.5} $ & $\Gamma=1.3^{+0.6}_{-0.3}$ & $41.5/35$ & $6.8$ & 1.1\
Power law + line & $14.3^{+6.0}_{-4.0} $ & $\Gamma=1.4^{+0.4}_{-0.5}$ & $24.9/32$ & $6.6$ & 1.0\
& & E$_{line}$=$6.6^{+0.2}_{-0.2}$ & & &\
& & $\sigma$=0.43$^{+0.33}_{-0.20}$ & & &\
& & EW=$0.78^{+0.62}_{-0.38}$ & & &\
& & I$_{line}$=$9.0^{+7.0}_{-4.1}$ & & &\
[mekal]{} & $16.6^{+2.7}_{-2.7}$ & T$_{\rm M}=30^{+50}_{-15}$ keV & $38.1/35$ & $6.7$ & 1.1\
Bremsstrahlung & $16.8^{+4.8}_{-3.2}$ &T$_{\rm br}> 12$ keV & $41.3/35$ & $6.8$ & 1.1\
Black body & $10.4^{+3.9}_{-3.2}$ &T$_{\rm bb}=2.2^{+0.4}_{-0.3}$ keV & $37.5/35$ & $6.7$ & 0.8\
& & R$_{\rm bb}$=$0.07^{+0.02}_{-0.02}$ km & & &\
The second fainter source resolved with is too weak to allow a meaningful spectral analysis.
It is reasonable to assume that the emission was mostly contributed by the brighter, “main”, source (and in the following we will call it ). The fainter source probably contributed a fraction of the measured flux from the iron line in the spectrum. In order to allow a proper comparison with the observation, we extracted a combined spectrum from both the main and the secondary fainter source. The residuals to the fit with an absorbed power-law again clearly show an excess around 6.5–6.7 keV (see Fig. \[fig:unres\_res\]), requiring the addition of a Gaussian emission line. The line centroid is 6.6$^{+0.2}_{-0.1}$ keV, the normalization is (8$^{+9}_{-3}$) $\times$10$^{-6}$ photons cm$^{-2}$ s$^{-1}$ and the equivalent width is 400$^{+500} _{-150}$ eV. The absorbing column density is (13$^{+7} _{-3}$) $\times$10$^{22}$ cm$^{-2}$, while the photon index, $\Gamma$, is 1.2 $^{+0.9}_{-0.3}$ (which is harder than the power-law fit to the “main” source alone, likely due to the hard emission contributed by the fainter source). The observed flux is 9$\times$10$^{-13}$ erg cm$^{-2}$ s$^{-1}$ (2–10 keV). We then re-extracted the MECS spectrum, and deconvolved it with this best fit model for the main plus faint sources emission. The result is shown in Fig. \[fig:sax\] and demonstrates that there is no evidence for dramatic changes in the flux level and spectral shape.
A source catalog of the field {#sect:det}
==============================
Examination of the EPIC images of the field in different energy ranges shows a region rich in faint X–ray sources. We performed a detection analysis in order to obtain a source catalog of this region, using the source detection procedure described in Baldi et al. ([@b:02]). All the source detection chain uses SAS version 6.1 tasks.
The cleaned events files were used to produce MOS1, MOS2 and pn images and exposure maps (which also include the vignetting effect) in 4 energy ranges: 0.5–2 keV (soft band, hereafter “S”), 2–5 keV (medium band, “M”), 5–10 keV (hard band, “H”), and 0.5–10 keV (total band). For each energy band independently, the MOS1, MOS2 and pn images (and the corresponding instrumental exposure maps) were then merged in order to get a higher signal-to-noise ratio in the source detection. After the production of a detector mask (with the task [*emask*]{}), a source detection in local mode was performed in each energy band separately with the task [*eboxdetect*]{} to produce a preliminary list of sources using a sliding box technique. Then, with task [*esplinemap*]{}, all the sources in the list were removed from the image and the resulting source–free image was fitted with a cubic spline function in order to create a background map for each energy band. Then, a second run with [*eboxdetect*]{} in map mode was made, this time using the background maps produced before. Lastly, the final source positions, together with the EPIC combined (MOS1+MOS2+pn) count rates for each source in each energy range, were calculated using the task [*emldetect*]{}, which performs maximum likelihood fits to the source spatial count distribution in all energy bands.
This detection procedure resulted in 31 sources, reported in Table \[tab:cat\]. Most of them have been detected in the medium and hard energy ranges, while few sources were detected only in the soft band (0.5-2 keV), very likely foreground stars.
The observed fluxes have been calculated from the combined EPIC count rate, $cr$, and a total conversion factor $f$ (flux=$f$ $\cdot$ $cr$) derived as in Baldi et al. ([@b:02]):
$$\frac{T_{tot}}{f}=\frac{T_{MOS1}}{f_{MOS1}}+\frac{T_{MOS2}}{f_{MOS2}}+\frac{T_{pn}}{f_{pn}}\:,$$ where $T_{tot}$ is the sum of the exposure times from the three instruments, $T_{tot}$=$(T_{MOS1}+T_{MOS2}+T_{pn}$), and $f_{MOS1}$, $f_{MOS2}$ and $f_{pn}$ are the single count-rate-to-flux conversion factors.
The sources listed in our catalog show a distribution of hardness ratios, ranging from sources detected only in the soft band (likely foreground stars), up to extremely hard sources, detected only above 5 keV (see Fig. \[fig:hr\]). In order to account for the different source hardness, we derived three different total conversion factors $f$, assuming three kinds of “typical” spectra: for sources detected only in the “S” band, we considered a blackbody spectrum with temperature 200 eV and a column density of $10^{21}$ cm$^{-2}$ (and the derived flux is limited to the 0.5–2 keV energy range); for sources detected only in the “H” band, we have chosen a power-law spectrum with a photon index $\Gamma$=2 and an absorbing column density of $10^{24}$ cm$^{-2}$ (flux derived in the 5–10 keV energy range); a power-law spectrum with a photon index $\Gamma$=2 absorbed with =2$\times$$10^{23}$ cm$^{-2}$ has been considered for all other sources (flux derived in the 2–10 keV range).
The absorbed fluxes calculated in this way are reported in Table \[tab:cat\] for the faint sources, while for the brightest ones, for which a reliable spectrum can be obtained, the flux measured from the spectral analysis is reported.
[llll]{} Src ID & & Parameter & Flux\
&($10^{22}$ cm$^{-2}$)& & (2–10 keV)\
1 & $1.3^{+0.3}_{-0.3} $ & $\Gamma = 1.7^{+0.2}_{-0.1}$ & 5\
2 & $8^{+2}_{-2} $ & $\Gamma = 2.0^{+0.4}_{-0.6}$ & 20\
4 & $15^{+5}_{-6} $ & $\Gamma = 1.6^{+0.9}_{-0.9}$ & 30\
5 & $0.7^{+0.8}_{-0.4} $ & $\Gamma = 1.8^{+0.8}_{-0.6}$ & 2\
6 & $<0.2$ & T$_{\rm bb}$=$0.2\pm{0.1}$ & 0.5 (0.5–2 keV)\
7 & $1.6^{+1.3}_{-0.7} $ & $\Gamma=2.1^{+1.2}_{-0.6}$ & 1.4\
A search in the SIMBAD database resulted in two HD stars positionally coincident with two soft X-ray sources (see Table \[tab:cat\]): HD 316290, located 31 from source 11, and HD 161824, at 1$''$ from the soft X–ray source 15. A third star, Tyc2 929, is located within 12 from the soft X–ray source 28. In Table \[tab:stars\] we report the sky positions and B, V magnitudes of these 3 optical counterparts, together with the associated ratios between X–ray and optical flux (Maccacaro et al. 1988), which strengthen the physical association with the soft X–ray sources.
[llllllll]{} Star & R.A. (J2000) & Dec. (J2000) & B & V & Sp. type & Associated & log(f$_{X}$/f$_{V}$)\
& & & (mag) & (mag) & & X–ray source ID &\
HD 316290 & 267.203000 & $-$28.016972 & 10.25 & 9.76 & F8 & 11 & $-4.3$\
HD 161824 & 267.212250 & $-$28.246139 & 9.63 & 8.33 & K1/K2III & 15 & $-4.5$\
Tyc2 929 & 267.045025 & $-$28.308665 & 12.27 & 10.55 & & 28 & $-4.8$\
In the search for counterparts of the X–ray sources at other wavelengths, we have conservatively assumed a circular uncertainty region with a radius of 4$''$. The results from a search in the SIMBAD database are reported in Table \[tab:cat\] (last column) while a cross-correlation with the 2MASS All-Sky Catalog of Point Sources (Cutri et al. [@c:03]) resulted into 68% X–ray matches with 2MASS counterparts.
Within the 14$'$ radius of the X–ray field of view, the 2MASS catalog (Cutri et al. 2003) list 26,413 stars, translating into a surface density $\mu\sim1.19$ $\times$ 10$^{-2}$ sources arcsec$^{-2}$. This corresponds to 0.6 sources within each error region of 4$''$ radius. Therefore it is likely that a large number of the 2MASS counterparts are just random coincidences, as suggested also by the fact that several X–ray sources are positionally associated with more than one infrared counterpart. At the spatial resolution of XMM-Newton, stellar confusion in the Galactic plane prevents from unambiguously associating infrared sources with the X–ray ones. Thus, we will not discuss further the possible association with 2MASS sources. On the other hand, for the two brightest HD stars, we estimate a probability of chance coincidence around 0.02%. This, together with the measured log(f$_{X}$/f$_{V}$), confirm the real association of the brightest stars with the X–ray sources.
Among the brightest X–ray sources, for which a spectral analysis has been possible, few can be firmly identified with known objects. Source 2 in Table \[tab:cat\] is a transient discovered with EPIC in an XMM-Newton observation performed on 12 March 2003, pointed on the composite SNR G0.9+0.1 (Sidoli & Mereghetti 2003; Sidoli et al. 2004). The observed flux during the discovery observation was 3.7$\times$10$^{-12}$ erg cm$^{-2}$ s$^{-1}$ (2–10 keV), almost a factor of 2 higher than that measured in February 2005, while the spectral parameters remained constant, within the uncertainties. The detection of the transient in 2005 could indicate that we are observing a second outburst from the source, or that the source is still in outburst since 2003. Source 4 is the pulsar wind nebula in the supernova remnant G0.9+0.1 (Mereghetti et al. [@msi:98]). The spectral parameters are similar to those measured during previous observations with (Porquet et al. 2003; Sidoli et al. 2004). Source 6 is positionally coincident with source 80 in the ROSAT catalog of the GC sources (Sidoli et al. 2001b). Compared with the ROSAT observation, it displays variability.
Five faint X–ray sources have been detected only above 5 keV. While it is somehow expected that the faintest sources be more distant and absorbed, on the other hand it is remarkable that they are observed only above 5 keV, meaning that their spectrum is both truly hard and absorbed. Few of these faint hard sources could be background AGNs. From the Log(N)–Log(S) measured in the energy range 5–10 keV (Baldi et al. 2002) with , the expected number of hard sources with flux larger than 5$\times$10$^{-14}$ erg cm$^{-2}$ s$^{-1}$ is 5–11 sources deg$^{-2}$, which translates into 1–2 extragalactic hard X–ray sources in the field of view. The remaining sources are probably CVs located close to the GC distance (their fluxes translate into luminosities in the range $\sim$10$^{32}$–10$^{33}$ erg s$^{-1}$).
Discussion and Conclusions {#sect:discussion}
==========================
In Sidoli et al. ([@s:01]) we reported the discovery of a new X–ray source in the direction of the Sgr D region, . Our new observation allows to resolve it into two sources (sources 3 and 12 in Table \[tab:cat\]), with a brighter “main” source contributing almost 80% of the source flux in the 2–10 keV energy range.
The fainter source is harder (detected only above 5 keV) than the “main” one. A possible optical counterpart is the star 0600-28834001 of the USNO-A2.0 catalog (B=18.1, R=13.4), which is listed as \[RHI84\]10-672 in the Raharto et al. (1984) catalog of M-type stars. The derivation of log(f$_{X}$/f$_{V}$) is highly uncertain, but assuming, e.g., a blackbody emission at kT$\sim$1 keV, absorbed with a column density of 10$^{24}$ cm$^{-2}$, the 5–10 keV flux translates into a 0.3-3.5 keV flux $\sim$4$\times$10$^{-11}$ erg cm$^{-2}$s$^{-1}$ (corrected for the absorption), and to a log(f$_{X}$/f$_{V}$)$\sim$1.8, clearly not stellar. Thus, the hardness of the X–ray emission excludes a coronal emission for the fainter source.
The refined sky position of the brighter source allows to reject all the possible associations discussed in Sidoli et al. ([@s:01]). The spectrum was affected by a high interstellar absorption, $N_{\rm H}$$\sim$$10^{23}$ cm$^{-2}$, suggesting that the source is probably located at the GC distance (in this case the luminosity in the 2–10 keV energy band is $\sim$10$^{34}$ erg s$^{-1}$). A strong Fe K line was present (with a line centroid of $6.62\pm0.30$ keV), and a good fit was obtained both with a power-law plus a Gaussian line, and with a hot thermal plasma model with a temperature, kT, of $6^{+35} _{-4}$ keV. Thus, the spectrum was consistent with both thermal and non-thermal models.
The significantly better statistics of the spectrum and the smaller uncertainties in the spectral slope, favor a non-thermal nature for the X–ray emission of the “main” source. A hard power-law ($\Gamma \sim$ 1.4) is a good fit to the data, with an iron line and a high photoelectric absorption (= 10–20 $\times$10$^{22}$ cm$^{-2}$). The absorption is probably not intrinsic, since the source is located within about 1 degree from the direction of the Galactic center. Thermal models do not fit the X–ray spectrum as well, and result in very high temperatures (for example, a thermal plasma should be hotter than 15 keV). Among the thermal models tried, the blackbody is the best in fitting the spectrum, but results in a high temperature ($\sim$2 keV) and in an emitting region of less then 0.1 km at the galactic center distance. Thus, the X–ray spectral shape favors a non-thermal nature for the X–ray emission.
The X–ray emission appears to be stable; the “main” source has been detected at large off-axis angle during two previous observations performed in September 2000 and March 2003 (both pointed at the SNR G0.9+0.1). In both occasions, did not show evidence for any strong flux variability. Moreover, the total emission from “main” plus “faint” source, is compatible with that observed with in 1997 (see Fig. \[fig:sax\]).
These properties are suggestive of three possibilities: a binary system containing a compact object, a background AGN, or reflection from a molecular cloud core (e.g., similar to the X–rays emission and fluorescent iron line produced from the molecular cloud Sgr B2; Revnivtsev et al. [@r:04]). This third possibility, already discussed in Sidoli et al. [@s:01], seems now unlikely, based on the high spatial resolution of the observation. The compact cores contained in the giant molecular cloud Sgr B2, for example, are about 1 pc in size (Lis & Goldsmith [@lg:91]), while the spatial resolution (FWHM$\sim$6$''$) allows us to exclude a source with a size larger than $\sim$0.25 pc at a distance of 8.5 kpc.
The shape of the X–ray spectrum and the parameters of the iron line are consistent with a background AGN. It should be a nearby object, since the Fe line is not red-shifted. Assuming a distance of 5 Mpc, the 2–10 keV unabsorbed flux corresponds to a luminosity of $\sim$3.4$\times$10$^{39}$ erg s$^{-1}$, which is quite low, but still compatible with a low-luminosity Seyfert galaxy (Terashima et al. [@t:02]). Note that no radio counterpart is present in the NED catalogue within 30$''$ of the X–ray position, and does not show evidence for X–ray variability on timescales of years, while X–ray temporal variability and presence of radio emission are typical properties of AGNs.
The X–ray spectral properties of are reminiscent of the soft gamma-ray sources discovered with the INTEGRAL satellite (see e.g., Kuulkers [@k:05] for a review). Several of their X–ray counterparts display hard and heavily absorbed spectra, together with intense fluorescent Fe line emission, indicative of dense gaseous envelopes around the compact object, illuminated by the central source. In few of them, the association with OB optical counterparts and the detection of X–ray pulsations, suggest that they are highly absorbed HMXRBs, not detected in previous surveys at soft X–rays. The derived luminosity of these INTEGRAL sources is around 10$^{36}$ erg s$^{-1}$, although there is a large uncertainty in the distance estimates, and the true luminosity could be much less than this. lies in the direction of SgrD molecular cloud, near to SgrB2, which is an important site of star formation, so it is not unlikely that is indeed a HMXRB. The low X–ray luminosity suggests that it belongs to a class of massive X–ray binaries with low persistent emission (in the range 10$^{34}$–10$^{35}$ erg s$^{-1}$), wind-accreting and with no outbursts (e.g., 4U 2206+54, Masetti et al. [@m:04]). On the other hand, these sources typically show temporal variability on different timescales (sometimes with flares), which has not been observed in (perhaps because of the poor coverage). However, wind-fed HMXRBs are usually quite stable X–ray emitters on long timescales (months or years). Low luminosity wind-accreting neutron stars has been predicted by Pfahl et al. (2002), who proposed that most of the faint sources detected in the $Chandra$ survey of the GC (Wang et al. 2002) could be of this kind. A search for hard unidentified sources from ROSAT PSPC observations seems to confirm that a new class of fainter wind-fed X–ray binaries exists in our Galaxy (Suchkov & Hanisch 2004).
Other kinds of galactic X–ray binaries, containing neutron stars or black-holes, seem to be unlikely; the luminosity ($\sim$10$^{34}$ erg s$^{-1}$) suggests an object in quiescence: but low-mass X-ray binaries (LMXRBs) in quiescence (soft X–ray transients) typically have much softer spectra (blackbody temperatures $\sim$0.1–0.3 keV; e.g., Verbunt & Lewin [@vl:04]), while black-hole X–ray novae in quiescence have much lower luminosities (Kong et al. [@k:02]).
In conclusion, among the different hypotheses discussed above, the spectral shape (hard, non–thermal), X–ray luminosity, the presence of Fe line emission, seem to favor a low luminosity HMXRB.
Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA member states and the USA (NASA). We thank Giovanna Giardino, Nicola La Palombara and Silvano Molendi for useful discussions. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The data analysis is supported by the Italian Space Agency (ASI), through contract ASI/INAF I/023/05/0.
Baldi, A., Molendi, S., Comastri, A., et al. 2002, ApJ, 564, 190B
Cutri, R.M., Skrutskie, M.F., Van Dyk, S. et al. 2003, University of Massachusetts and Infrared Processing and Analysis Center, (IPAC/California Institute of Technology)
Ebisawa, K., Maeda, Y., Kaneda, H., et al. 2001, Science, 293, 1633
Hog, E., Fabricius, C., Makarov, V.V., et al. 2000, A&A, 355, L27
Jansen, F., Lumb, D., Altieri, B., et al. 2001, A&A, 365, L1
Koyama, K., Awaki, H., Kunieda, H., et al. 1989, Nature 339, 60
Kong, A.K.H., McClintock, J.E., Garcia, M.R., et al. 2002, ApJ, 570, 277
Kuulkers, E., in “Interacting Binaries: Accretion, Evolution and Outcomes", Eds. L.A. Antonelli, et al., Proc. of the Interacting Binaries Meeting of Cefalu, Italy, July 2004, AIP, in press, astro-ph/0504625
Lis, D. C., Goldsmith, P. F. 1991, ApJ 286, 232
Maccacaro, T., Gioia, M.I., Wolter, A., et al. 1988, ApJ 326, 680
Masetti, N., Dal Fiume, D., Amati, L., et al. 2004, A&A, 423, 311
Mereghetti, S., Sidoli, L., Israel, G. L., 1998, A&A, 331, L77
Pfahl, E., Rappaport, S., Podsiadlowski, Ph., 2002, ApJ, 571, L37
Porquet, D., Decourchelle, A., Warwick, R. S., 2003, A&A, 401, 197
Raharto, M., Hamajima, K., Ichikawa, T., et al. 1984, Ann. Tokyo Astron. Obs., 19, 469
Revnivtsev, M.G., Churazov, E.M., Sazonov, S.Yu., et al. 2004, A&A, 425, L49
Sidoli, L., 2000, PhD Thesis, Univ. of Milan, Italy
Sidoli, L., Mereghetti, S., Treves, A., et al. 2001, A&A, 372, 651
Sidoli, L., Belloni, T., Mereghetti, S., 2001, A&A, 368, 835
Sidoli, L., Mereghetti, S., 2003, ATel 147
Sidoli, L., Bocchino, F., Mereghetti, S., et al. 2004, MmSAI, 75, 507
Strüder, L., Briel, U., Dennerl, K., et al. 2001, A&A, 365, L18
Suchkov, A.A., Hanish, R.J., 2004, ApJ, 612, 437
Terashima, Y., Iyomoto, N., Ho, L.C., et al. 2002, ApJ Suppl, 139, 1
Turner, M. J. L., Abbey, A., Arnaud, M., et al. 2001, A&A, 365, L27
Verbunt, F., Lewin, W.H.G., 2005, to appear in “Compact Stellar X–ray Sources", eds. W.H.G. Lewin and M. van der Klis, Cambridge University Press, astro-ph/0404136
Walter, R., Courvoisier, T. J.-L., Foschini, L., et al. 2004, Proc. of the V INTEGRAL Workshop, 16-20 February 2004, Munich, Germany. V. Schoenfelder, G. Lichti, & C. Winkler. ESA SP-552, Noordwijk: ESA Publication Division, ISBN 92-9092-863-8, p. 417-422.
Wang, Q.D., Gotthelf, E.V., Lang, C.C., 2002 Nature, 415, 148
-------- ------------ ------------ ----------------------- ---------------------- ---------------------- ----------------------- ----------------------- ---------- ---------------------------------
Source R.A. Dec. S M H HR1 HR2 Flux Notes$^{a}$
ID (J2000) (J2000)
1 267.200047 -28.211022 15.14$\pm{ 0.67}$ 23.76$\pm{ 0.92}$ 5.99$\pm{ 0.47}$ 0.22$\pm{ 0.03}$ -0.60$\pm{ 0.03} $ 5
2 266.817367 -28.180065 2.38$\pm{ 0.49}$ 66.02$\pm{ 2.27}$ 29.30$\pm{ 1.44}$ 0.93$\pm{ 0.01}$ -0.39$\pm{ 0.03} $ 20 Transient (1)
3 267.070445 -28.130656 $-$ 5.60$\pm{ 0.38}$ 10.87$\pm{ 0.46}$ 1.0 0.32$\pm{ 0.04} $ 6.8 “main” source
4 266.845425 -28.151610 $-$ 20.60$\pm{ 1.28}$ 19.03$\pm{ 1.14}$ 1.0 -0.04$\pm{ 0.04}$ 30 G0.9+0.1 PWN
5 267.242974 -28.239226 7.55$\pm{ 0.59}$ 6.82$\pm{ 0.67}$ 2.38$\pm{ 0.36}$ -0.05$\pm{ 0.06}$ -0.48$\pm{ 0.07}$ 2
6 266.878546 -28.229876 16.66$\pm{ 0.99}$ 0.84$\pm{ 0.47}$ $-$ -0.90$\pm{ 0.05}$ -1.0 0.4 SBM80 (2); variable
7 267.149164 -27.938457 5.16$\pm{ 0.53}$ 5.53$\pm{ 0.66}$ 1.31$\pm{ 0.30}$ 0.03$\pm{ 0.08}$ -0.62$\pm{ 0.08}$ 1.4
8 267.017602 -28.245968 0.47$\pm{ 0.23}$ 3.11$\pm{ 0.45}$ 5.98$\pm{ 0.49}$ 0.74$\pm{ 0.12}$ 0.32$\pm{ 0.07}$ 4
9 267.058541 -28.272493 $-$ 3.37$\pm{ 0.51}$ 5.39$\pm{ 0.52}$ 1.0 0.23$\pm{ 0.08}$ 4
10 267.245308 -28.188918 4.45$\pm{ 0.42}$ 0.96$\pm{ 0.30}$ $-$ -0.65$\pm{ 0.10}$ -0.62$\pm{ 0.26}$ 0.5
11 267.202258 -28.016414 7.89$\pm{ 0.58}$ $-$ $-$ -1.0 $-$ 0.2 (S) HD 316290
12 267.081919 -28.124028 $-$ 0.49$\pm{ 0.21}$ 3.15$\pm{ 0.31}$ 1.0 0.73$\pm{ 0.10}$ 2 (H) faint source near ;
\[RHI84\] 10-672;sp.type M6 (3)
13 267.071020 -28.262540 3.17$\pm{ 0.40}$ 1.70$\pm{ 0.43}$ $-$ -0.30$\pm{ 0.13}$ -1.0 0.8
14 267.118308 -28.158466 1.45$\pm{ 0.22}$ 0.85$\pm{ 0.22}$ $-$ -0.26$\pm{ 0.14}$ -1.0 0.4
15 267.212103 -28.245925 5.32$\pm{ 0.49}$ $-$ $-$ -1.0 $-$ 0.2 (S) HD 161824
16 267.162129 -28.176110 0.31$\pm{ 0.15}$ 0.52$\pm{ 0.23}$ 0.78$\pm{ 0.18}$ 0.25$\pm{ 0.31}$ 0.20$\pm{ 0.24}$ 0.5
17 266.943950 -28.150689 $-$ $-$ 1.33$\pm{ 0.26}$ $-$ 1.0 0.8 (H)
18 266.884630 -28.148614 $-$ $-$ 2.17$\pm{ 0.39}$ $-$ 1.0 1.4 (H)
19 267.215469 -28.180396 1.27$\pm{ 0.23}$ $-$ $-$ -1.0 $-$ 0.04 (S)
20 267.170685 -28.026434 $-$ 0.58$\pm{ 0.29}$ 0.69$\pm{ 0.21}$ 1.0 0.091$\pm{ 0.29}$ 0.5
21 267.144386 -27.926348 1.64$\pm{ 0.35}$ 0.98$\pm{ 0.48}$ $-$ -0.25$\pm{ 0.25}$ -0.56$\pm{ 0.40}$ 0.4
22 267.021045 -28.117880 $-$ $-$ 1.69$\pm{ 0.22}$ $-$ 1.0 1.1 (H)
23 266.951207 -28.065710 $-$ $-$ 1.83$\pm{ 0.27}$ $-$ 1.0 1.2 (H)
24 267.129677 -28.197335 $-$ $-$ 0.87$\pm{ 0.19}$ $-$ 1.0 0.5 (H)
25 266.982671 -28.052198 1.85$\pm{ 0.27}$ $-$ $-$ -1.0 $-$ 0.06 (S)
26 267.103188 -28.222073 $-$ $-$ 0.87$\pm{ 0.29}$ $-$ 1.0 0.5 (H)
27 267.169872 -28.306960 1.18$\pm{ 0.32}$ $-$ $-$ -1.0 $-$ 0.04 (S)
28 267.044643 -28.308560 3.01$\pm{ 0.40}$ $-$ $-$ -1.0 $-$ 0.09 (S) TYC2 929 (4)
29 267.056520 -28.045786 $-$ $-$ 0.86$\pm{ 0.17}$ $-$ 1.0 0.5 (H)
30 267.057854 -28.296586 1.61$\pm{ 0.31}$ $-$ $-$ -1.0 $-$ 0.05 (S)
31 267.303578 -28.077256 1.42$\pm{ 0.26}$ $-$ $-$ -1.0 $-$ 0.05 (S)
-------- ------------ ------------ ----------------------- ---------------------- ---------------------- ----------------------- ----------------------- ---------- ---------------------------------
$^{a}$[Numbers in parentheses are the following references: (1) Sidoli & Mereghetti [@s:03]; (2) Sidoli, Mereghetti & Belloni [@sbm:01]; (3) Raharto et al., [@r:84]; (4) Hog et al. [@h:00] ]{}\
|
---
abstract: 'We have explored the impact of sterile neutrino dark matter on core-collapse supernova explosions. We have included oscillations between electron neutrinos or mixed $\mu,\tau$ neutrinos and right-handed sterile neutrinos into a supernova model. We have chosen sterile neutrino masses and mixing angles that are consistent with sterile neutrino dark matter candidates as indicated by recent x-ray flux measurements. Using these simulations, we have explored the impact of sterile neutrinos on the core bounce and shock reheating. We find that, for ranges of sterile neutrino mass and mixing angle consistent with most dark matter constraints, the shock energy can be significantly enhanced and even a model that does not explode can be made to explode. In addition, we have found that the presence of a sterile neutrino may lead to detectable changes in the observed neutrino luminosities.'
address:
- |
Department of Physics, University of Notre Dame,\
Notre Dame, IN 46556, USA\
$^*$E-mail: gmathews@nd.edu
- |
National Astronomical Observatory of Japan\
Mitaka, Tokyo 181-8599, Japan
author:
- 'Grant J. Mathews$^*$ and MacKenzie Warren'
- Jun Hidaka and Toshitaka Kajino
title: 'Sterile neutrino dark matter and core-collapse supernovae'
---
Introduction
============
Sterile neutrinos, a proposed fourth neutrino flavor, are one viable dark matter candidate.[@abazajian2001b] A sterile neutrino is an electroweak singlet and is thus consistent with limits from the [*LEP*]{} measurement [@LEP2006] of the width of the $Z^{0}$ gauge boson. The standard model does not provide any predictions about this proposed particle, but bounds can be placed using astronomy, cosmology and supernovae[@abazajian2001a; @abazajian2001b; @boyarsky2006a; @boyarsky2006b; @boyarsky2007; @boyarsky2009b; @boyarsky2009c; @boyarsky2014; @bulbul2014; @chan2014] on the mass and mixing angle parameter space. If sterile neutrino dark matter can radiatively decay,[@pal1982] x-ray observations[@boyarsky2006b; @boyarsky2007; @boyarsky2006a; @boyarsky2009b; @boyarsky2009c; @boyarsky2014; @bulbul2014] from galaxies and galaxy clusters can also be used to place bounds, i.e $1\text{ keV} < m_{s}< 18 \text{ keV}$ and $\sin^{2} 2 \theta_{s} \lesssim1.93 \times 10^{-5} \left({m_{s}}/{\text{keV}}\right)^{-5.04}$.
The explosion mechanism of CCSNe remains an outstanding problem in physics as well. Spherically symmetric models do not easily explode and two-dimensional and three-dimensional models that do explode often have too little energy to match observations.[@janka2012] The problem remains that, although a shock forms successfully and propagates outward in mass, it loses energy to the photodissociation of heavy nuclei and becomes a standing accretion shock.
However, even one dimensional models explode in simulations with enhanced neutrino fluxes, either from convection below the neutrinosphere [@wilson1988; @wilson1993; @book] or a QCD phase transition.[@fischer2011] Along this vein, we have explored the resonant mixing between sterile and electron neutrinos (or antineutrinos) to increase the early neutrino luminosity and revitalize the shock.[@hidaka2006; @hidaka2007; @warren2014; @warren2016]
Hidaka and Fuller [@hidaka2006; @hidaka2007] were the first to propose that sterile neutrinos could serve as an efficient neutrino energy transport mechanism in the supernova core. They used a one zone collapse calculation to study the resonant oscillations of a sterile neutrino with the mass and mixing angle of a warm dark matter candidate. They found that the resonant mixing of electron and sterile neutrinos can serve as an efficient method of transporting neutrino energy from the protoneutron star core, where high energy neutrinos are trapped, to the stalled shock to assist in neutrino reheating. This mechanism is highly sensitive to the feedback between neutrino oscillations and the local composition, energy transport, and hydrodynamics and warranted detailed numerical studies.[@warren2014; @warren2016]
In Refs. & coherent active-sterile neutrino oscillations were studied using the University of Notre Dame-Lawrence Livermore National Laboratory (UND/LLNL) code,[@book; @bowers1982] a spherically symmetric general relativistic supernova model with detailed neutrino transport and hydrodynamics. The impact on shock reheating of sterile neutrinos with masses and mixing angles consistent with dark matter constraints was studied. Sterile neutrino dark matter candidates can enhance the shock energy and lead to a successful explosion, even in a simulation that would not otherwise explode.[@warren2014]
Matter-enhanced Neutrino Oscillations\[sec:osc\]
================================================
With the inclusion of a sterile neutrino, the full neutrino mixing problem requires a complete understanding of all mass differences and mixing angles in the $4\times 4$ mixing matrix. However, for this work, only two neutrino mixing between electron neutrinos $\nu_{e}$ and sterile neutrino $\nu_{s}$ (or their antiparticles) have been considered. This is sufficient for exploring the impact on the explosion energy since electron neutrinos and antineutrinos dominate in the gain region during shock reheating.
Matter-enhanced neutrino oscillations in supernovae occur via the Mikheyev-Smirnov-Wolfenstein (MSW) effect.[@mikheyev1985; @wolfenstein1978] As neutrinos propagate through matter, they experience an effective potential from charged and neutral current interactions due to forward scattering on baryonic matter, electrons, and other neutrinos. Each neutrino flavor will experience a different potential because $\nu_{e}$ experience both charged and neutral current interactions whereas $\nu_{s}$ do not experience any weak interactions. This can induce a coherent effect where maximum mixing is possible, even for a small vacuum mixing angle, when the phase arising from the potential difference cancels the phase due to the mass difference. The forward scattering potential experienced by electron neutrinos is $$V(r) = \frac{3 \sqrt{2}}{2} G_{F} n_{B} \left(Y_{e} + \frac{4}{3} Y_{\nu_{e}} + \frac{2}{3} Y_{\nu_{\tau}} -1\right)~,$$ where $G_{F}$ is the fermi coupling constant, $n_{B}$ is the baryon number density, and $Y_{i}$ is the number fraction of species $i$. The antineutrino species will experience forward scattering potentials with the opposite sign. In the supernova environment, one can assume $Y_{\nu_{\mu}} = Y_{\nu_{\tau}} = 0$, since $\nu_{\mu}$ and $\nu_{\tau}$ neutrinos and antineutrinos are produced via pair emission processes and thus occur in equal numbers.
This results in an “effective” in-medium mixing angle,[@warren2014; @warren2016] $$\sin^{2} 2 \theta_{M} (r) = \frac{\Delta^{2} \sin^{2} 2 \theta_{s}}{(\Delta \cos 2 \theta_{s} - V(r))^{2} + \Delta^{2} \sin^{2} 2 \theta_{s}}~.$$ From this expression, it is simple to see that one can achieve maximal mixing in matter, even for small vacuum mixing angles. Such resonant mixing will occur for neutrinos with the energy where $\sin^{2} 2 \theta_{M} = 1$, $$E_{res} = \frac{\Delta m^{2}}{2 V(r)} \cos 2 \theta_{s}~.$$
In this work, only coherent and adiabatic oscillations are considered.[@warren2014; @warren2016] This ensures that all electron neutrinos of the given flavor with the resonance energy $E_{res}$ will oscillate to sterile neutrinos, and vice versa.[@mikheyev1985; @wolfenstein1978] It is probable that incoherent, scattering-induced oscillations will be significant in the high matter densities of the central core, but this will not be a dominant effect and will be left to future work.
Results\[sec:results\]
======================
A model that can successfully explode was used as a baseline for comparison. The UND/LLNL supernova model[@bowers1982; @book] is a spherically symmetric, general relativistic hydrodynamic supernova model. We have used the $20 \text{ M}_{\odot}$ progenitor from Woosley & Weaver [@woosley1995].
Figure \[fig:parameter\] shows the enhancement to the explosion energy in a simulation with a sterile neutrino compared to a simulation without a sterile neutrino. Sterile neutrino masses were considered in the range $1 \text{ keV} < m_{s} < 10 \text{ keV}$ and mixing angles in the range of $10^{-11} < \sin^{2} 2\theta_{s} < 10^{-2}$, which includes the region that corresponds with dark matter candidates. The shaded regions show the parameter space allowed for sterile dark matter by recent x-ray flux measurements.[@abazajian2001a; @boyarsky2006a; @boyarsky2006b; @boyarsky2007; @boyarsky2009c] There is a large region of the parameter space that both enhances the explosion energy of core-collapse supernovae and is satisfies constraints on sterile neutrino dark matter.
![Sterile neutrino mass $m_{s}$ and mixing angle $\sin^{2} 2 \theta_{s}$ parameter space. The region above the solid line enhances the supernova explosion energy by $1.01\times$ compared to a simulation without a sterile neutrino. The region above the dashed line enhances the explosion energy by $1.1\times$. The dark gray shaded region shows the parameter space allowed by x-ray flux measurements if sterile neutrinos make 100% of the observed dark matter mass.[@abazajian2001a; @boyarsky2006a; @boyarsky2006b; @boyarsky2007; @boyarsky2009c] The medium gray region is for 10% of the observed dark matter mass and the light gray region is for 1% of the observed dark matter mass. The solid black square shows the most recent best fit point from the x-ray flux from Boyarsky et al[@boyarsky2014] and Bulbul et al.[@bulbul2014] Figure from Warren et al.[@warren2016][]{data-label="fig:parameter"}](eparameter.eps){width="3.0in"}
To illustrate how the explosion energy enhancement occurs, consider a single choice of sterile neutrino mass $m_{s} = 1$ keV and mixing angle $\sin^{2} 2\theta_{s} = 10^{-5}$. Any mass and mixing angle that causes an enhancement will show similar behavior.
The enhancement to the kinetic energy isn’t evident until $\sim0.2$s post-bounce, when neutrino reheating becomes important. By 1s post-bounce, the explosion energy is enhanced by a factor of $2.2\times$ compared to the simulation without a sterile neutrino.
This dramatic enhancement to the kinetic energy is due to increased neutrino heating in the simulation with a sterile neutrino. The presence of a sterile neutrino enhances the luminosities of all neutrino and antineutrino species considered here. The enhancement to the neutrino luminosities does not become significant until $\sim 0.1$s post-bounce, which is when oscillations to a sterile neutrino become important[@warren2014]. The luminosities of all neutrino and antineutrino species are enhanced by about $2\times$ until 0.4s post-bounce, which corresponds to the enhancement in the kinetic energy. The increased neutrino luminosities emerging from the protoneutron star increase the rate of neutrino heating in the gain region and lead to the enhanced explosion energy.
Although oscillations are only allowed between electron neutrinos $\nu_{e}$ and sterile neutrinos $\nu_{s}$ (and their antiparticles), enhancements are seen in the luminosities of all neutrino species. This is because the oscillations $\nu_{e} \leftrightarrow \nu_{s}$ leads to a “double reheating” scenario: the $\nu_{e} \leftrightarrow \nu_{s}$ oscillations cause additional heating near the location of the neutrinosphere, which causes increased neutrino cooling in all flavors, and finally these enhanced neutrino luminosities reheat the stalled shock.
In the simulation with a sterile neutrino, the neutrinosphere radius increases by about $1.4\times$ between 0.1s and 0.4s post-bounce, which corresponds with the increased neutrino luminosities. The neutrinosphere radius is increased because the oscillations heat the protoneutron star surface by depositing energy at the location of the $\nu_{s} \rightarrow \nu_{e}$ resonance. However, this additional heating of the protoneutron star surface does not increase the temperature of the neutrinosphere, but instead causes it to expand outward. Although the location of the neutrinosphere is increased by $\sim1.4\times$, the temperature at the neutrinosphere is roughly the same in both simulations. Thus the larger neutrinosphere radius leads to enhanced emission of neutrinos, but the neutrinos in both simulations have roughly the same characteristic temperature.
Conclusions\[sec:conc\]
=======================
Recent observations of galaxies and galaxy clusters indicate an unidentified emission line at $\sim3.5$keV. This line may be due to the radiative decay of sterile neutrino dark matter with $m_{s} \approx 7$keV. If this is the case, bounds can be placed on the sterile neutrino mass and mixing angle from the observed photon energies and fluxes. Further observations are needed to confirm the presence of this line in additional dark matter dominated environments, such as dwarf spheroidal galaxies.
For oscillations between an electron neutrino and sterile neutrino, a large region of the sterile neutrino mass and mixing angle parameter space that is allowed by these observations leads to an enhancement of the explosion energy in core-collapse supernovae. The enhancement is due to increased neutrino heating in the gain region caused by increased neutrino luminosities of all neutrino and antineutrino flavors. The neutrino luminosities are enhanced due to a “double reheating” mechanism in the protoneutron star, where the surface of the protoneutron star is heated due to the oscillations between electron and sterile neutrinos, and in turn, the heating of the protoneutron star increases the luminosities of all neutrino and antineutrino flavors.
Acknowledgments {#acknowledgments .unnumbered}
===============
Work at the University of Notre Dame supported by the U.S. Department of Energy under Nuclear Theory Grant DE-FG02-95-ER40934. One of the authors (MW) is also supported by U.S. National Science Foundation through the Joint Institute for Nuclear Astrophysics (JINA) Frontier center. This work was also supported in part by Grants-in-Aid for Scientific Research of the JSPS (20105004, 24340060).
[0]{}
K. Abazajian, G.M. Fuller, and M. Patel, [*Phys.Rev.D*]{} [**64**]{}, 023501 (2001).
, [DELPHI]{}, [L3]{}, [OPAL]{}, [SLD Collaborations]{}, [LEP Electroweak Working Group]{},, and [SLD Electroweak and Heavy Flavor Groups]{}, [*Phys. Rep.*]{} [**427**]{}, 257 (2006).
K. Abazajian, G.M. Fuller, and W.H. Tucker, [*Ap.J.*]{} [**562**]{}, 593 (2001).
A. Boyarsky and A. Neronov and O. Ruchayskiy and M. Shaposhnikov and I. Tkachev, [*Phys.Rev.Lett.*]{} [**97**]{}, 261302 (2006).
A. Boyarsky, A. Neronov, O. Ruchayskiy, and M. Shaposhnikov, [*Phys.Rev.D*]{} [**74**]{}, 103506 (2006).
A. Boyarsky, J. Nevalainen, and O. Ruchayskiy, [*A&A*]{} [**471**]{} 51 (2007).
A. Boyarsky, J. Lesgourgues, O. Ruchayskiy, and M. Viel, [*JCAP*]{} [**05**]{} 012, (2009).
A. Boyarsky, O. Ruchayskiy, and D. Iakubovskyi, [*JCAP*]{} [**03**]{}, 005 (2009).
A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, and J. Franse. [*Phys.Rev.Lett.*]{} [**113**]{}, 251301 (2014).
E. Bulbul, M. [Markevitch]{}, A. [Foster]{}, R.K. [Smith]{}, M. [Loewenstein]{}, and S.W. [Randall]{}, [*Ap.J.*]{} [**789**]{}, 1 (2014).
M.H. Chan and R. Ehrlich, [*Astro.Space Sci*]{} [**349**]{}, 407 (2014).
P.B. Pal and L. Wolfenstein, [*Phys.Rev.D*]{} [**25**]{}, 766 (1982).
H.-Th. Janka, [*Annu.Rev.Nucl.Part.Sci.*]{} [**62**]{}, 1 (2012).
J.R. Wilson and R.W. Mayle, [*Phys.Rep.*]{} [**163**]{}, 63 (1988).
J.R. Wilson and R.W. Mayle, [*Phys.Rep.*]{} [**227**]{}, 97 (1993).
J.R. Wilson and G. Mathews, [*Relativistic Numerical Hydrodynamics*]{}, (Cambridge University Press, 2003).
T. Fischer, I. Sagert, G. Pagliara, M. Hempel, J. Schaffner-Bielich, T. Rauscher, F.-K. Thielemann, R. Kaeppeli, G. Martínez-Pinedo, and M. Liebendörfer, [*Ap.J.Supp.*]{} [**194**]{}, 39 (2011).
J. Hidaka and G.M. Fuller, [*Phys.Rev.D*]{} [**74**]{}, 125015 (2006).
J. Hidaka and G.M. Fuller, [*Phys.Rev.D*]{} [**76**]{}, 083516 (2007).
M. Warren, M. Meixner, G.J. Meixner, J. Hidaka, and T. Kajino, [*Phys.Rev.D*]{} [**90**]{}, 103007 (2014).
M. Warren, G.J. Mathews, M. Meixner, J. Hidaka, and T. Kajino. [*MPLA*]{} Submitted. arXiv:1603.05503 (2016).
R.L. Bowers and J.R. Wilson, [*Ap.J.Supp*]{} [**50**]{}, 115 (1982).
S. Mikheyev and A. Smirnov, [*Yad.Fiz.*]{} [**42**]{}, 1441 (1985).
L. Wolfenstein, [*Phys.Rev.D*]{} [**17**]{}, 2634 (1978).
S.E. Woosley and T.A. Weaver, [*Ap.J.Supp.*]{} [**101**]{}, 181 (1995).
M. Herant, W. Benz, W.R. Hix, C.L. Fryer, and S.A. Colgate, [*Ap.J.*]{} [**435**]{}, 339 (1994).
A. Burrows, J. Hayes, and B. Fryxell, [*Ap.J.*]{} [**450**]{}, 830 (1995).
H.-Th. Janka and E. Mueller, [*Astron. Astrophys.*]{} [**306**]{}, 167 (1996).
J.W. Murphy, J.C. Dolence, and A. Burrows, [*Ap.J.*]{} [**771**]{}, 52 (2013).
J. Blondin, A. Mezzacappa, and C. DeMarino, [*Ap.J.*]{} [**584**]{}, 2 (2003).
J. Blondin and A. Mezzacappa, [*Ap.J.*]{} [**642**]{}, 1 (2006)
L. Scheck, H.-Th. Janka, T. Foglizzo, and K. Kifonidis, [*Astr. Astrophys.*]{} [**477**]{}, 3 (2008).
|
---
author:
- Serge Dumont$^1$
title: 'On enhanced descend algorithms for solving frictional multi-contact problems : applications to the Discrete Element Method'
---
${^{1}}$ LAMFA, Université de Picardie Jules Verne - CNRS UMR 7352, 33, rue Saint-Leu,\
80 000 Amiens, France.\
e-mail : serge.dumontu-picardie.fr\
[**Abstract**]{} In this article, we present various numerical methods to solve multi-contact problems within the Non-Smooth Discrete Element Method. The techniques considered to solve the frictional unilateral conditions are based both on the bi-potential theory introduced by de Saxcé et al. [@dSF91] and the Augmented Lagrangian theory introduced by Alart et al. [@AC91]. Following the ideas of Feng et al. [@FJCM05], a new Newton method is developed to improve these classical algorithms and numerical experiments are presented to show that these methods are faster than the previous ones and provides results with a better quality.
[**Key words:**]{} Granular materials, Contact mechanics, Newton algorithms, Bi-potential, Augmented Lagrangian.
Introduction
============
This is a first draft of a paper that will be submitted in a near future.
The paper is organized as follow: in the next part, we present the equations to be solved for the Discrete Element Method, and the frictional contact law considered. In the third part, we first present two classical methods to numerically solve the full problem, the first one based on the bi-potential theory, and the second one on the Augmented Lagrangian theory. Then, we show how these methods can be enhanced using an appropriate Newton method. The last part on this article is devoted to the numerical experiments in order to show the main properties of these algorithms.
Problem Setting
===============
The equations of motion of a multi-contact system
-------------------------------------------------
Classically (see for example [@J99; @JM92; @M88]), the motion of a multi-contact system is described using a global generalized coordinate $\bf q$ (for $N_p$ particles, ${\bf q}\in {\ensuremath{\mathbb R}}^{\tilde d \times N_p}$, where $\tilde d=6$ for a 3D problem and $\tilde d=3$ for a 2D problem). Due to the possible shocks between particles, the equations of motion has to be formulated in term of differential measure equation: $$\label{eqmotion}
{\mathbb M} d{\dot{\bf{q}}}+ {\bf F}^{int}(t,{\bf q},{\dot{\bf{q}}})dt = {\bf F}^{ext}(t,{\bf q},{\dot{\bf{q}}})dt + d{R}$$
where
- $\mathbb M$ represents the generalized mass matrix;
- $ {\bf F}^{int}$ and $ {\bf F}^{ext}$ represent the internal and external forces respectively;
- $d{\bf R}$ is a non-negative real measure, representing the reaction forces and impulses between particles in contact.
For the sake of simplicity and without lost of generality, only the external forces are considered in the following. The internal forces are neglected because the general case can be easily derived through a linearizing procedure.
Then, for the numerics, the equation (\[eqmotion\]) is integrated on each time interval $[t_k,t_{k+1}]$, and approximated using a $\theta$-method with $\theta\in]\frac12,1]$ for stability reason (see [@RA04]).
Therefore, the classical approximation of equation (\[eqmotion\]) yields $$\label{eqdiscr1}
\left\{\begin{array}{l}
\mathbb{M}({\dot{\bf{q}}}_{k+1}-{\dot{\bf{q}}}_n) = \Delta t(\theta {\bf F}_{k+1}+(1-\theta){\bf F}_k)+{\bf R}_{k+1} \\
{\bf q}_{k+1}={\bf q}_k+\Delta t \theta {\dot{\bf{q}}}_{k+1}+\Delta t(1-\theta){\dot{\bf{q}}}_k
\end{array}
\right.$$
We will denote ${\dot{\bf{q}}}_k^{free}={\dot{\bf{q}}}_k + \mathbb{M}^{-1}\Delta t(\theta {\bf F}_{k+1}+(1-\theta){\bf F}_k)$ the free velocity (velocity when the contact forces vanish). Then, the first equation in (\[eqdiscr1\]) becomes $${\dot{\bf{q}}}_{k+1}={\dot{\bf{q}}}_k^{free}+ \mathbb{M}^{-1}{\bf R}_{k+1}.$$
In order to write the contact law, for a contact $c$ between two particles ($1\leq c\leq N_c$, where $N_c$ is is the total number of contact), we define the local-global mapping
$$\left\{\begin{array}{l}
{\bf u}^c=P^*({\bf q},c){\dot{\bf{q}}}\\
{\bf R}=P({\bf q},c){\textbf r}^c
\end{array}\right.$$
where ${\bf u}^c$ is the local relative velocity between the two bodies in contact and ${\textbf r}^c$ is the local contact forces (${\bf u}^c,{\textbf r}^c\in {\ensuremath{\mathbb R}}^d$ where $d$ is the dimension of the problem, and $P^*$ is the transpose of matrix $P$). We also denote ${\mathbb P}({\bf q})$ the total-global mapping, for ${\bf u}$ and ${\textbf r}$ in ${\ensuremath{\mathbb R}}^{d\times N_c}$ (vectors composed of all relative velocity and contact forces respectively): $$\label{globalmap}
\left\{\begin{array}{l}
{\bf u}={\mathbb P}^*({\bf q}){\dot{\bf{q}}}\\
{\bf R}={\mathbb P}({\bf q}){\textbf r}\end{array}\right.$$
In the discretization, a prediction of $\bf q$ is computed to estimate the mapping ${\mathbb P}({\bf q})$ (see equations (\[globalmapapp1\]) and (\[globalmapapp2\]) in the following).
Using the equations (\[eqdiscr1\]) and (\[globalmap\]), the discretization of the motion of a multi-contact system, with frictional contact between particles can be written:
$$\label{eqdiscr2}
\left\{\begin{array}{l}
{\tilde{{{\bf u}}}}_{k+1}= {\tilde{{{\bf u}}}}_k^{free} +{\mathbb W}{\textbf r}_{k+1} \\
law_c({\tilde{{{\bf u}}}}^c_{k+1},{\textbf r}^c_{k+1})=\mbox{.true.}\qquad \forall c\in \{1,2,...,N_c\}
\end{array}
\right.$$
where ${\mathbb W}={\mathbb P}^* {\mathbb M}^{-1} {\mathbb P}$ is the Delassus operator, and ${\tilde{{{\bf u}}}}_k^{free}={\mathbb P}^*{\dot{\bf{q}}}_k^{free}$ is the relative free velocity. Notice that a Newton impact law is also considered (see [@M88] and equation (\[newtonlaw\]) in the following), that modify ${\bf u}_k$ and ${\bf u}^{free}_k$ by ${\tilde{{{\bf u}}}}_k$ and ${\tilde{{{\bf u}}}}^{free}_k$ respectively.
The second equation in (\[eqdiscr2\]) is the implicit frictional contact law that is in our case the classical Signorini condition and Coulomb’s friction law.
The frictional contact law
--------------------------
In the local coordinates system defined by the local normal vector ${{\bf n}}$ and the tangential vector ${\bf t}\perp{{\bf n}}$, any element ${\bf u}$ and ${\textbf r}$ can be uniquely decomposed as ${\bf u}=u_n{{\bf n}}+{\bf u}_t$ and ${\textbf r}={r_n}{{\bf n}}+{\textbf r}_t$ respectively. In these coordinates, the unilateral contact law can be stated using the Signorini’s conditions (see figure \[signorini\] for a graphical representation): $$u_n\geq 0,\quad r_n\geq 0,\quad u_nr_n=0.$$
On the other hand, the Coulomb’s law of friction can be stated using the algorithmic form (see figure \[coulomb\] for a graphical representation): $$\left[\begin{array}{lll}
\mbox{If } {r_n}=0\quad &\mbox{ then } u_n\geq 0 & \mbox{! No contact}\\
& & \\
\mbox{Else if } {r_n}>0 \mbox{ and } \|{\textbf r}_t\|<\mu {r_n}& \mbox{ then } {\bf u}=0 & \mbox{! Sticking}\\
& & \\
\mbox{Else } {r_n}>0 \mbox{ and } \|{\textbf r}_t\|=\mu {r_n}& \mbox{ then } \exists \lambda \geq 0 \mbox{ such that } {\bf u}_t=\lambda \frac{{\textbf r}_t}{\|{\textbf r}_t\|} & \mbox{! Sliding}
\end{array}\right.$$
For a given friction coefficient $\mu$, let $K_\mu$ be the isotropic Coulomb’s cone, which defines the set of admissible forces (see figure \[coulombcone\]): $$K_\mu=\left\{ {\textbf r}={r_n}{{\bf n}}+{\textbf r}_t :\ \|{\textbf r}_t\|-\mu {r_n}\leq 0 \right\}$$
![The Coulomb’s cone[]{data-label="coulombcone"}](coulomb.pdf){width="6.0cm"}
The previous law can be also written: $$\left[\begin{array}{lll}
\mbox{If } {r_n}=0\quad &\mbox{ then } u_n\geq 0 & \mbox{! No contact}\\
& & \\
\mbox{Else if } {\textbf r}\in I(K_\mu) & \mbox{ then } {\bf u}=0 & \mbox{! Sticking}\\
& & \\
\mbox{Else } {r_n}>0 \mbox{ and } {\textbf r}\in B(K_\mu) & \mbox{ then } \exists \lambda \geq 0 \mbox{ such that } {\bf u}_t=\lambda \frac{{\textbf r}_t}{\|{\textbf r}_t\|} & \mbox{! Sliding}
\end{array}\right.$$
where $I(K_\mu)$ and $B(K_\mu)$ are respectively the interior and the boundary of the cone $K_\mu$.
Numerical Resolution of the contact/friction problems
=====================================================
We will describe in this section the numerical algorithms that will be considered in the following. Generally, to solve the problem (\[eqdiscr2\]), the numerical algorithms considered are based on two levels: the global level where the equations of motion are solved, and the local level devoted to the resolution of the contact law.
Resolution of the global problem : the Non Linear Gauss Seidel Method (NLGS)
-----------------------------------------------------------------------------
In this paragraph, we describe the algorithm used at the global level to solve the problem (\[eqdiscr2\]). Following the ideas of Jean and Moreau [@J99; @M88], we use the non-linear Gauss-Seidel algorithm which is the most commonly used. It consists in considering successively each contact until the convergence. The numerical criterion used to state the convergence will be studied latter in the paper.
This method is intrinsically sequential but it is possible to used a simple multi-threading technique which consists in splitting the contact loop into several threads. This method has been studied in [@RDA04] in the case where the local algorithm is based on the Augmented Lagrangian method.
Notice that it is also possible to consider at this stage more sophisticated method such as a conjugate gradient type method (see for example [@RA04]).
The standard bi-potential based method (SBP)
---------------------------------------------
In this paragraph, we provide a first method to solve the contact problem, at the local level (contact point between two particles). The method is based on the notion of bi-potential, introduced by de Saxcé et al. [@dSF91].
Using the bi-potential framework, it can be shown (see for example [@dSF91; @F00; @Fortin2005; @S06]) that a couple $({\bf u},{\textbf r})$ verifies the Signorini-Coulomb contact rules if $$\label{SCrules}
b_c({\bf v},{\bf s})+{\bf v}\cdot {\bf s}\geq b_c({\bf u},{\textbf r})+{\bf u}\cdot {\textbf r}=0\qquad \forall {\bf v},{\bf s}$$
where $b_c$ is the bi-potential $$b_c({-\bf u},{\textbf r})=\Psi_{{\ensuremath{\mathbb R}}^+}(u_n) +\Psi_{K_\mu}({\textbf r}) +\mu {r_n}\|{\bf }u_t\|$$
and $\Psi_C$ stands for the indicatrix function of the set $C$: $\Psi_C(x)=0$ if $x\in C$, $\Psi_C(x)=+\infty$ if $x\notin C$.
Consequently, the contact law can be written in a compact form of an implicit subnormality rule (or a differential inclusion rule): $$-{\bf u} \in \partial_{\textbf r}b_c(-{\bf u},{\textbf r}).$$
Then, for a contact $c$, at a NLGS iteration $i$, knowing the relative velocity ${\tilde{{{\bf u}}}}^{c,i}$, the algorithm to compute ${\textbf r}^{c,i+1}$ from ${\textbf r}^{c,i}$ is based on the minimization of the bi-potential (see for exemple [@F00], page 51), using the inequality:
$$\label{minbipo}
b_c(-{\tilde{{{\bf u}}}}^{c,i},{\textbf r})+{\tilde{{{\bf u}}}}^{c,i}\cdot{\textbf r}\geq b_c(-{\tilde{{{\bf u}}}}^{c,i},{\textbf r}^{c,i+1})+{\tilde{{{\bf u}}}}^{k,i}\cdot{\textbf r}^{c,i+1}\qquad \forall {\textbf r}\in K_\mu$$
or $g({\textbf r})\geq g({\textbf r}^{c,i+1})$, $\forall {\textbf r}\in K_\mu$, if we denote $$\label{defg}
g({\textbf r})=\Psi_{{\ensuremath{\mathbb R}}^+}({\tilde{{{u}}}_n}^{c,i})+\Psi_{K_\mu}({\textbf r})+\mu {r_n}\|{\tilde{{{\bf u}}}}^{c,i}_t \|+{\tilde{{{\bf u}}}}^{c,i}\cdot{\textbf r}.$$
The minimization of (\[minbipo\]) is classically realized using a projected gradient projection (Uzawa method) without considering the singular term $\Psi_{{\ensuremath{\mathbb R}}^+}({\tilde{{{u}}}_n}^{c,i})$. This minimization can also be viewed as the proximal point of the augmented force ${\textbf r}-\rho {\tilde{{{\bf u}}}}$, with respect to the function ${\textbf r}\mapsto \rho b_c(-{\tilde{{{\bf u}}}},{\textbf r})$ (see for example [@dSF91; @F00; @Fortin2005]): $${\textbf r}=prox({\textbf r}-\rho{\tilde{{{\bf u}}}},\rho b_c(-{\tilde{{{\bf u}}}},{\textbf r})).$$
More precisely, the Uzawa method leads to compute the augmented force ${{\mathbf \tau}}^{c,i+1}={\textbf r}^{c,i}-\rho \nabla \tilde{g} ({\textbf r}^{c,i})$, where $\tilde{g}$ is the differential part of $g$: $$\nabla \tilde{g}({\textbf r}^{c,i})=\nabla_{\textbf r}(\mu {r_n}\|{\tilde{{{\bf u}}}}^{c,i}_t \|+{\tilde{{{\bf u}}}}^{c,i}\cdot{\textbf r})=\mu \|{\tilde{{{\bf u}}}}^{c,i}_t \| {{\bf n}}+ {\tilde{{{\bf u}}}}^{c,i},$$ and to consider the force at next step as a projection of the augmented force onto the set of admissible force ${\textbf r}^{c,i+1}=proj({{\mathbf \tau}}^{c,i+1},K_\mu)$, that provides equations (\[uzawa1\]) and (\[uzawa2\]) in the resolution algorithm of the global problem. The $proj({{\mathbf \tau}}^{c,i+1},K_\mu)$ stands for the orthogonal projection over the convex $K_\mu$, that can be computed exactly (see [@F00]).
This algorithm will be referred as the SBP (Standard Bi-Potential) method above and throughout.
For a sake of simplicity, we denote hereafter the descent direction $${{\bf D}^{c,i}}=\mu \|{\tilde{{{\bf u}}}}^{c,i}_t \| {{\bf n}}+ {\tilde{{{\bf u}}}}^{c,i}.$$
A first improvement of this method could be to compute the optimal step $\rho^{c,i}$. To do so, we have to minimize
$$\label{minrho}
\rho\mapsto g({\textbf r}^{c,i}-\rho {{\bf D}^{c,i}}),$$
or, more precisely, $$\label{minrho1}
\begin{array}{ll}
\rho\mapsto & \ \ \ \Psi_{{\ensuremath{\mathbb R}}^+}({\tilde{{{u}}}_n}^{c,i})+\Psi_{K_\mu}({\textbf r}^{c,i}-\rho {{\bf D}^{c,i}})+\mu ({r_n}^{c,i}-\rho {{\bf D}^{c,i}}\cdot {{\bf n}}) \|{\tilde{{{\bf u}}}}^{c,i}_t \|+{\tilde{{{\bf u}}}}^{c,i}\cdot({\textbf r}^{c,i}-\rho {{\bf D}^{c,i}})\\
& =\Psi_{{\ensuremath{\mathbb R}}^+}({\tilde{{{u}}}_n}^{c,i})+\Psi_{K_\mu}({\textbf r}^{c,i}-\rho {{\bf D}^{c,i}})-\rho {{\bf D}^{c,i}}\cdot(\mu \|{\tilde{{{\bf u}}}}^{c,i}_t \| {{\bf n}}+ {\tilde{{{\bf u}}}}^{c,i})+Cte\\
& = \Psi_{{\ensuremath{\mathbb R}}^+}({\tilde{{{u}}}_n}^{c,i})+\Psi_{K_\mu}({\textbf r}^{c,i}-\rho {{\bf D}^{c,i}})-\rho \|{{\bf D}^{c,i}}\|^2+Cte.\\
\end{array}$$ We can observe that this method do not permit to choose an optimal parameter $\rho$ since $g$, as a function of $\rho$, is linear, excepted in the case where $ {{\bf D}^{c,i}}\notin K_\mu$. A solution could be to modify the function $g$, for example by replacing ${\tilde{{{\bf u}}}}^{c,i}$ by a prediction of ${\tilde{{{\bf u}}}}^{c,i+1}$ using the equations of the dynamics. Unfortunately, this method do not provides good numerical results.
Then, the standard bi-potential based algorithm (SBP) can be written (see [@S06] for example):
- Loop on the step time $k$
- Prediction of a position (for the computation of the local-global mapping):\
$$\label{globalmapapp1}
{\bf q}_{k+\frac12}={\bf q}_k+\frac{\Delta t}{2}{\dot{\bf{q}}}_k;$$
- Initialization of the motion: ${\dot{\bf{q}}}^{0}_{k+1}={\dot{\bf{q}}}_{k}^{free}$(initialization of the contact forces with ${\bf R}=0$).
- Loop on $i\geq 0$ (NLGS), until convergence
- Loop on the contacts $c$:
- Computation of the local-global mapping $$\label{globalmapapp2}
\dot{\bf u}^-=P^*({\bf q}_{k+\frac12},c){\dot{\bf{q}}}_k\ ;\qquad\dot{\bf u }^{c,+i}=P^t({\bf q}_{k+\frac12},c){\dot{\bf{q}}}^{i}_{k+1}$$
- Newton shock law $$\label{newtonlaw}
{\tilde{{{u}}}_n}^{c,i}=\frac{u^{c,+i}_n+e_nu^-_n}{1+e_n}\ ;\qquad {\tilde{{{\bf u}}}}^{c,i}_t=\frac{{\bf u}^{c,+i}_t+e_n{\bf u}^-_t}{1+e_t}$$
- Prediction of the reaction: $$\label{uzawa1}
{{\mathbf \tau}}^{c,i+1}= {\textbf r}^{c,i} -\rho \left[ {\tilde{{{\bf u}}}}^{c,i}_t+( {\tilde{{{u}}}_n}^{c,i}+\mu \|{\tilde{{{\bf u}}}}^{c,i}_t \|){{\bf n}}\right]$$
- Correction of the reaction: $$\label{uzawa2}
{\textbf r}^{c,i+1}=proj({{\mathbf \tau}}^{c,i+1},K_{\mu})$$
- Actualization of the generalized displacement: $$\label{uzawa3}
\displaystyle {\dot{\bf{q}}}^{i+1}_{k+1}={\dot{\bf{q}}}_{k}^{free}+{\mathbb M}^{-1}(\sum_{\alpha\leq c}P({\bf q}_{k+\frac12},\alpha){\textbf r}^{\alpha,i+1}
+\sum_{ \alpha>c}P({\bf q}_{k+\frac12},\alpha){\textbf r}^{\alpha,i})$$
- End of the loop on contacts $c$.\
- End of the loop on $i$ of NLGS when the convergence is reached: ${\dot{\bf{q}}}_{k+1}={\dot{\bf{q}}}_{k+1}^{i+1}$
- Actualization of the generalized displacements: ${\bf q}_{k+1}={\bf q}_{k+\frac12}+\frac{\Delta t}{2}{\dot{\bf{q}}}_{k+1}$
- End of the loop on the step time $k$.
Notice that only one iteration of the Uzawa algorithm at the local level is considered. Various previous studies (see for example [@JF08]) show that there is no significant improvement of the method if several iterations of the Uzawa algorithm are considered at this stage.
Newton method and enhanced bi-potential method (EBP)
----------------------------------------------------
We introduce in this section a Newton method in order to speed up the convergence of the computation of the solution. This method has been already used, especially in the case of the augmented lagrangian method developed by Alart et al. [@AC91], and the ideas presented in this article follows those of Feng et al. [@FJCM05] and have been adapted to the problem of the discrete element method. The main idea of this technique is to find the solution of the optimization problem, not as a minimum of a functional, but rather as a zero of a function, using the Euler equation of the problem. Then a standard Newton method can be developed to solve this Euler equation.
The technique is first described in the case of the bi-potential framework, and will adapted to the augmented lagrangian method farther.
We recall that the local problem that has to be solved, for each contact $c$ can be written $$\label{locprob1}
\left\{\begin{array}{l}
\displaystyle {\tilde{{{\bf u}}}}^{c}_{k+1}={\tilde{{{\bf u}}}}^{c,free}_k+\sum_{\alpha=1}^{N_c} W_{c\alpha} {\textbf r}^\alpha
\ \\
{\textbf r}^c=proj({{\mathbf \tau}}^c,K_\mu)
\end{array}\qquad \forall c=1,\ ...,\ N_c
\right.$$
where ${{\mathbf \tau}}^c={\textbf r}^c-\rho(\mu\|{\tilde{{{\bf u}}}}_t^c\|{{\bf n}}+{\tilde{{{\bf u}}}})$ is the augmented reaction (see \[uzawa1\]), and $W_{c\alpha}=P^*({\bf q}_{k+\frac12,c}){\mathbb M}^{-1}P({\bf q}_{k+\frac12,\alpha})$ is the local Delassus operator.
This problem can be written equivalently $$\label{locprob2}
\left\{\begin{array}{l}
\displaystyle {\tilde{{{\bf u}}}}^c_{k+1}-{\tilde{{{\bf u}}}}^{c,free}_{k}-\sum_{\alpha=1}^{N_c} W_{c\alpha} {\textbf r}^\alpha=0 \\
\ \\
{\textbf r}^c-proj({{\mathbf \tau}}^c,K_\mu)=0
\end{array}\right.\qquad \forall c=1,\ ...,\ N_c$$
Reminding now that we want to use a Newton algorithm to solve theses equations inside the Non Linear Gauss Seidel loop on the variable $i$, we define now, for each contact $c=1,\ ...,N_c$, the function $$f^i_c(\chi)=
\left(\begin{array}{c}
\displaystyle{\tilde{{{\bf u}}}}^{c,i}-{\tilde{{{\bf u}}}}^{c,free}_{k}-\sum_{\alpha=1}^{N_c} W_{c\alpha} {\textbf r}^{\alpha,i} \\
\ \\
{{\bf Z}}^{c,i}
\end{array}\right)$$
where :
- the vector ${{\bf Z}}^c$ is the error on the prediction of the reaction $$\label{errorpred}
{{\bf Z}}^{c,i}({\textbf r}^{c,i},{\tilde{{{\bf u}}}}^{c,i})={\textbf r}^{c,i}-proj({{\mathbf \tau}}^{c,i},K_\mu),$$
- $\chi_c=({\textbf r}^{c,i},{\tilde{{{\bf u}}}}^{c,i})^t$,
- $\chi=(\chi_1,\chi_2,...,\chi_{N_c})^t$
The first equality in the relation $f(\chi)=0$ is the equation of motion for the bodies in contact, and the second relation is the frictional Coulomb law between the bodies in contact, written within the bipotential framework.
Then we have to write a Newton algorithm to solve the problem $f(\chi)=0$. This algorithm can be written, for a contact $c$, by substituting equations (\[uzawa1\]) and (\[uzawa2\]) in algorithm (SBP) by the followings:
- Initialization: $$\chi^0_c=\left( {\textbf r}^0={\textbf r}^{c,i},\ {{\bf v}}^0={\tilde{{{\bf u}}}}^{c,i} \right)^t,\quad \ell=0$$
- Loop on $\ell$, until convergence:
- ${{\mathbf \tau}}^c_\ell={\textbf r}^\ell-\rho(\mu\|{{\bf v}}^\ell_t\|{{\bf n}}+{{\bf v}}^\ell)$
- Resolution: $$\label{eqnewt}
\left[ \frac{\partial f_c}{\partial \chi^c}(\chi^\ell)\right] \Delta \chi_c= -f_c(\chi^\ell)$$
- Actualization: $\chi_c^{\ell+1}=\chi_c^\ell+\Delta \chi_c$
- End of the loop on $\ell$ until convergence, ${\tilde{{{\bf u}}}}^{c,i+1}={{\bf v}}^{\ell}$ and ${\textbf r}^{c,i+1}={\textbf r}^\ell$.
This algorithm needs more than one iteration at each Non Linear Gauss Seidel iteration to be efficient. As a consequence and compared to the Uzawa algorithm, the solution in the Newton algorithm is controlled by both the local (iteration $\ell$) and global convergence criteria (iteration $i$, see [@FJCM05; @JF08]).
The local convergence criterion for the Newton algorithm is defined by: $$\label{er_newt}
\varepsilon^c_{Newt}(\chi_\ell)=\|{{\bf v}}^\ell-{\bf u}^{c,free}_k-W{\textbf r}^\ell\|+\|{\textbf r}^\ell-proj({\textbf r}^\ell,K_\mu)\|$$ This criterion measure $f_c(\chi_\ell)$ that has to be sufficiently small.
The matrix $\left[ \frac{\partial f_c}{\partial \chi_c}(\chi)\right]$ represents the tangential matrix of the local equations for the contact $c$. This matrix is of dimension $6\times6$ for a 3 dimensional problem, and $3\times 3$ for a 2 dimensional problem. For a 3 dimensional problem, the general form of this matrix is the following: $$\left[ \frac{\partial f_c}{\partial \chi_c}(\chi)\right]
=
\left[\begin{array}{cc}
-W & Id_{3\times3} \\
A_c & B_c\\
\end{array}\right]$$ where $$A_c=\left[ \frac{\partial Z_c}{\partial {r_n}}\left| \frac{\partial Z_c}{\partial r_{t_1}}\right| \frac{\partial Z_c}{\partial r_{t_2}}\right]
\qquad
B_c=\left[ \frac{\partial Z_c}{\partial v_n}\left| \frac{\partial Z_c}{\partial v_{t_1}}\right| \frac{\partial Z_c}{\partial v_{t_2}}\right]$$
The matrices $A_c$ and $B_c$ takes different forms according to the contact status:
- First case: sliding contact.
In that case, we have $$\mu \|{{\mathbf \tau}}_t\|\geq-\tau_n \qquad \|{{\mathbf \tau}}_t\|\geq \mu \tau_n$$ then $$Proj({{\mathbf \tau}},K_\mu)=\tau - \left(\frac{\|{{\mathbf \tau}}_t\|-\mu \tau_n}{1+\mu^2} \right) \left(\frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|}-\mu {{\bf n}}\right)$$ and $${\bf Z}_c=\rho(\mu\|{{\bf v}}^k_t\|{{\bf n}}+{{\bf v}}^k)+ \left(\frac{\|{{\mathbf \tau}}_t\|-\mu \tau_n}{1+\mu^2} \right) \left(\frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|}-\mu {{\bf n}}\right)$$
The computation of the derivatives of ${\bf Z}_c$ provides the matrices $A_c$ and $B_c$:
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial {r_n}} = -\frac{\mu}{1+\mu^2} \left( \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|} -\mu {{\bf n}}\right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial r_{t_1}} = \frac{\tau_{t_1}}{(1+\mu^2)\|\tau_t\|}\left( \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|} -\mu {{\bf n}}\right)+
\frac{\|{{\mathbf \tau}}_t\|-\mu{{\mathbf \tau}}_n}{1+\mu^2}\left(\frac{{\bf t}_1}{\|{{\mathbf \tau}}_t\|}-\frac{\tau_{t_1}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t \right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial r_{t_2}} = \frac{\tau_{t_2}}{(1+\mu^2)\|\tau_t\|}\left( \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|} -\mu {{\bf n}}\right )+
\frac{\|{{\mathbf \tau}}_t\|-\mu{{\mathbf \tau}}_n}{1+\mu^2}\left(\frac{{\bf t}_2}{\|{{\mathbf \tau}}_t\|}-\frac{\tau_{t_2}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t \right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_n} = \rho {{\bf n}}+\frac{\rho\mu}{1+\mu^2}\left(\frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|}-\mu {{\bf n}}\right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_{t_1}} = \rho\left( {\bf t}_1+\mu\frac{ v_{t_1}}{\|{{\bf v}}_t\|}{{\bf n}}\right)-\frac{\rho}{1+\mu^2}\left(
\begin{array}{l}
\left(\frac{\tau_{t_1}}{\|{{\mathbf \tau}}_t\|}-\frac{\mu^2v_{t_1}}{\|{{\bf v}}_t\|}\right)\left( \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|} -\mu {{\bf n}}\right )+\\
\qquad\quad(\|{{\mathbf \tau}}_t\|-\mu \tau_n)\left( \frac{{\bf t}_1}{\|{{\mathbf \tau}}_t\|}-\frac{\tau_{t_1}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t\right)
\end{array}\right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_{t_2}} = \rho\left( {\bf t}_2+\mu \frac{v_{t_2}}{\|{{\bf v}}_t\|}{{\bf n}}\right)-\frac{\rho}{1+\mu^2}\left(
\begin{array}{l}
\left(\frac{\tau_{t_2}}{\|{{\mathbf \tau}}_t\|}-\frac{\mu^2v_{t_2}}{\|{{\bf v}}_t\|}\right)\left( \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|} -\mu {{\bf n}}\right )+\\
\qquad\quad(\|{{\mathbf \tau}}_t\|-\mu \tau_n)\left( \frac{{\bf t}_2}{\|{{\mathbf \tau}}_t\|}-\frac{\tau_{{\bf t}_2}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t\right)\end{array}\right)$
For a 2D problem, these computations yields:
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial {r_n}} = \frac{\mu}{1+\mu^2} \left( \mu {{\bf n}}- \theta_r {\bf t} \right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial r_{t}} = \frac{1}{(1+\mu^2)}\left( -\mu \theta_r {{\bf n}}+ {\bf t} \right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_n} = \frac{\rho}{1+\mu^2}\left({{\bf n}}+ \mu \theta_r {\bf t}\right)$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_{t}} =\frac{\rho\mu}{1+\mu^2}\Bigl((\theta_v+\theta_r){{\bf n}}+ \mu(1-\theta_r\theta_v){\bf t} \Bigr)$
where $\theta_v=\mbox{sign}({{\bf v}}_t)$ and $\theta_r=\mbox{sign}({{\mathbf \tau}}_t)$.
- Second case: sticking contact.
In that case, we have $$\mu \|\tau_t\|\geq -\tau_n \qquad \|\tau_t\|< \mu \tau_n$$ then
$${\bf Z}_c=\rho (\mu\|{{\bf v}}^k_t\|{{\bf n}}+{{\bf v}}^k)$$ and the computation of the derivatives of ${\bf Z}_c$ reads:
- $A_c=0_{3\times 3}$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_n} =\rho {{\bf n}}$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_{t_1}} = \rho\mu \frac{v_{t_1}}{\|{{\bf v}}_t\|}{{\bf n}}+ \rho {\bf t}_1$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_{t_2}} = \rho\mu \frac{v_{t_2}}{\|{{\bf v}}_t\|}{{\bf n}}+ \rho {\bf t}_2$
For a 2D problem, these computations leads to:
- $A_c=0_{2\times 2}$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_n} = \rho {{\bf n}}$
- $\displaystyle \frac{\partial {\bf Z}_c}{\partial v_{t}} =\rho\mu \theta_v {{\bf n}}+\rho {\bf t}$
- Third case: no contact.
In that case, the matrices $A_c= \mbox{Id}_{3\times 3}$ and $B_c$ vanishes, and $\chi^{\ell+1}_c=\left\{\begin{array}{c}
0\\ {{\bf v}}^k\end{array}
\right\}$
Resolution of the linear system
-------------------------------
Generally, the drawback of a Newton is the computational cost of the linear system to be solved at each iteration. Here, the particular form of the tangent matrix allows the use of a condensation technique. More precisely, the linear system to be solved can be written: $$\left[\begin{array}{cc}
-W & Id_{3\times3} \\
A_c & B_c\\
\end{array}\right]
\left(\begin{array}{c}
\delta {\textbf r}\\ \delta {{\bf v}}\end{array}\right) =
\left(\begin{array}{c}
\bf
-f\\ \bf -g \end{array}\right).$$
The first equation yields $\delta {{\bf v}}=-{\bf f}+W\delta {\textbf r}$, and introducing this equality is the second equation leads to solve the linear system $$(A_c+B_cW)\delta {\textbf r}= -{\bf g}+B_c {\bf f}.$$ This properties halves the size of the linear system to be solved.
A drawback of the bi-potential framework is that, due to is specificity, it is rather difficult to consider fully coupled problems, where the contact law and another phenomena, such as electricity or thermic effects are strongly coupled. The other method presented in this paper has a better property from this point of view because it is based on a more standard mathematical background in the theory of optimization.
Newton method and enhanced augmented lagrangian method, (SAL) and (EAL)
-----------------------------------------------------------------------
In [@AC91], Alart et al. propose another method to solve the frictional contact problem. This method has been also used with various improvement (parallelization, conjugate gradient method for example) to solve multi-contact problems [@RA04; @RA04b; @RDA04; @RAD05; @RBDA05]. Even if the coupled frictional contact problem is not an optimization problem anymore, it is always possible to formally formulate a “quasi“- optimization problem, for which the constraint set depends on the normal components of the solution as a parameter. The solution is then searched as a saddle point of a ”quasi" augmented Lagrangian of the problem. More precisely, the global problem on all unknowns that has to be solved at each time step (in place of equation (\[locprob1\])) has the following form: $$\label{globprob1}
\left\{\begin{array}{l}
{\bf u}={\bf u}^{free} + {\ensuremath{\mathbb W}}{\textbf r}\\
\ \\
{\textbf r}\geq 0,\ {\bf u}\geq 0,\ {\textbf r}\cdot {\bf u}=0.
\end{array}
\right.$$
In order to solve this problem, for a given ${\textbf r}\in {\ensuremath{\mathbb R}}^{3\times N_c}$, one can define the cartesian product of infinite half cylinder with section equal to the ball ${\cal B}(0,\mu r_c)$ of radius $\mu r_c$ by: $${\cal C}(\mu {\textbf r})=\prod_{c=1}^{N_c}{\ensuremath{\mathbb R}}^+\times {\cal B}(0,\mu r_c)$$ and then, the granular type frictional contact problem is given by $$\label{AL}
{\textbf r}\in \mbox{argmin}_{{\textbf r}\in {\cal C}(\mu {\textbf r})} \frac12 {\textbf r}\cdot {\ensuremath{\mathbb W}}{\textbf r}+{\bf u}^{free}\cdot {\textbf r}= \mbox{argmin}_{{\textbf r}\in {\cal C}(\mu {\textbf r})} J({\textbf r}),$$ and the projected gradient method the minimize this problem reads (for each iteration $i$ of the NLGS algorithm): $$\label{gradientAL}
{\textbf r}^{i+1}=proj({\textbf r}^i-\rho({\bf u}^{free}+{\mathbb W}{\textbf r}^i),\ {\cal C}(\mu{{\textbf r}^{i+1}})),$$ or ${\textbf r}^{i+1}=proj({{\mathbf \tau}}^{i+1}, \ {\cal C}(\mu{{\textbf r}^{i+1}}))$, with ${{\mathbf \tau}}^{i+1}={\textbf r}^i-\rho{\bf u}^i$, ${\bf u}^i={\bf u}^{free}+{\mathbb W}{\textbf r}^i$. This algorithm will be referred to hereinafter as the SAL (Simple Augmented Lagrangian) method.
Notice that this method is very closed to the SBP method. More precisely, for a contact $c$, only the descent direction ${\tilde{{{\bf u}}}}^{c,i}+\mu \|{\tilde{{{\bf u}}}}^{c,i}_t \|{{\bf n}}$ in (\[uzawa1\]) is replaced by ${\tilde{{{\bf u}}}}^{c,i}$ and the projection ${\textbf r}^{c,i+1}=proj({{\mathbf \tau}}^{c,i+1},K_{\mu})$ in (\[uzawa3\]) is replaced by $$\left\{\begin{array}{l}
{r_n}^{c,i+1}=max(0,\tau_n^{c,i+1})\\
\ \\
{\textbf r}_t^{c,i+1}=\frac{{{\mathbf \tau}}_t^{c,i+1}}{\|{{\mathbf \tau}}_t^{c,i+1}\|}\mu {r_n}^{c,i+1}.
\end{array}
\right.$$
On the contrary, it is possible to see the algorithm developed from the bi-potential formalism as a slight modification of the algorithm above. Indeed, it is only necessary to change the set ${\cal C}({\textbf r})$ by $\displaystyle {\cal K}=\prod_{c=1}^{N_c}K_\mu$, and to change the descent direction ${\tilde{{{\bf u}}}}^{c,i}$ by ${\tilde{{{\bf u}}}}^{c,i}+\mu \|{\tilde{{{\bf u}}}}^{c,i}_t \|{{\bf n}}$ which remains a descent direction for the SAL method, since $$\nabla J({\textbf r}^{c,i+1})\cdot {{\bf D}^{c,i}}=-\|{\tilde{{{\bf u}}}}^{c,i}\|^2-{\tilde{{{\bf u}}}}^{c,i}\cdot(\mu \|{\tilde{{{\bf u}}}}^{c,i}_t\|){{\bf n}}=-\|{\tilde{{{\bf u}}}}^{c,i}\|^2-\mu u^{c,i}_n\|{\tilde{{{\bf u}}}}^{c,i}_t\|$$ which is negative since $\mu\in[0,1]$.
Then, acting by analogy, we can develop a Newton method to find the minimum of $J$ by seeking the solution as a zero of the function $\tilde{f}(\chi)$ where, for a contact $c$ $$\tilde{f}_c(\chi)=
\left(\begin{array}{c}
\displaystyle{\tilde{{{\bf u}}}}^c_{k+1}-{\tilde{{{\bf u}}}}^{c,free}_{k}-\sum_{\alpha=1}^{N_c} W_{c\alpha} {\textbf r}^\alpha \\
\ \\
{\bf \tilde{Z}}^c
\end{array}\right),$$ the vector ${\bf \tilde{Z}}^c$ is the error on the prediction of the reaction $$\label{predbipo}
{\bf \tilde{Z}}^c({\textbf r}^c,{\tilde{{{\bf u}}}}^c_{k+1})={\textbf r}^c-proj({{\mathbf \tau}}^c_{k+1},{\cal C}_c(\mu{{\mathbf \tau}}^c_{k+1})),$$
and the set ${\cal C}_c(\mu{\textbf r}^c)$ is the set of admissible forces ${\cal C}_c(\mu{\textbf r}^c)={\ensuremath{\mathbb R}}^+\times {\cal B}(0,r_c)$. This method will be refered as the EAL (Enhanced Augmented Lagrangian) method hereafter.
Then, as bellow, we have three cases in the computation of the tangent matrix $\left[\frac{\partial \tilde{f}}{\partial \chi^c}(\chi^\ell)\right]$:
- First case: sliding contact ($\tau_n>0,\ {{\mathbf \tau}}_t\geq \mu \tau_n$)
We have: $proj({{\mathbf \tau}}^c,{\cal C}_c(\mu{{\mathbf \tau}}^c))=\tau_n{{\bf n}}+ \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|}\mu\tau_n {\bf t}$ and $ \tilde{\bf Z}_c=\rho v_n{{\bf n}}-\frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|}\mu\tau_n +{\textbf r}_t$.
The computation of the derivatives of $ \tilde{\bf Z}_c$ provides the matrices $A_c$ and $B_c$:
- $\displaystyle \frac{\partial \tilde{\bf Z}_c}{\partial {r_n}} = -\mu \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|}$
- $\displaystyle \frac{\partial \tilde{\bf Z}_c}{\partial r_{t_1}} = {\bf t}_1-\mu \tau_n\left( \frac{{\bf t}_1}{\|{{\mathbf \tau}}_t\|}-\frac{{{\mathbf \tau}}_{t_1}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t \right)$
- $\displaystyle \frac{\partial \tilde{\bf Z}_c}{\partial r_{t_2}} = {\bf t}_2-\mu \tau_n\left( \frac{{\bf t}_2}{\|{{\mathbf \tau}}_t\|}-\frac{{{\mathbf \tau}}_{t_2}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t \right) $
- $\displaystyle \frac{\partial \tilde{\bf Z}_c}{\partial v_n} = \rho \left({{\bf n}}+\mu \frac{{{\mathbf \tau}}_t}{\|{{\mathbf \tau}}_t\|}\right)$
- $\displaystyle \frac{\partial \tilde{\bf Z}_c}{\partial v_{t_1}} = -\rho \mu \tau_n \left( \frac{{\bf t}_1}{\|{{\mathbf \tau}}_t\|}-\frac{{{\mathbf \tau}}_{t_1}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t \right) $
- $\displaystyle \frac{\partial \tilde{\bf Z}_c}{\partial v_{t_2}} = -\rho \mu \tau_n \left( \frac{{\bf t}_2}{\|{{\mathbf \tau}}_t\|}-\frac{{{\mathbf \tau}}_{t_2}}{\|{{\mathbf \tau}}_t\|^3}{{\mathbf \tau}}_t \right)$
For a two dimensional problem, these computations yields $$A_c=\left(\begin{array}{cc}
0 & 0 \\
-\mu \theta_r & 1\\
\end{array}\right)\qquad
B_c=\left(\begin{array}{cc}
1 & 0\\
\mu \theta_r & 0\\
\end{array}\right).$$
- Second case: sticking contact ($\tau_n>0$, ${{\mathbf \tau}}_t<\mu \tau_n$)
$proj({{\mathbf \tau}}^c,{\cal C}_c(\mu{{\mathbf \tau}}^c))={{\mathbf \tau}}^c$ and the computation of the derivatives of ${\bf Z}_c$ reads
- $A_c=0_{3\times 3}$
- $B_c=\rho \mbox{Id}_{3\times 3}$
- Third case: no contact (${{\mathbf \tau}}_n\leq0$)
$proj({{\mathbf \tau}}^c,{\cal C}_c(\mu{{\mathbf \tau}}^c))=0$, then the matrices $A_c= \mbox{Id}_{3\times 3}$ and $B_c$ vanishes, and $\chi^{\ell+1}_c=\left\{\begin{array}{c}
0\\ {{\bf v}}^k\end{array}
\right\}$
The global stopping (convergence) criterion
-------------------------------------------
We present in this paragraph the convergence criterion on the global non linear Gauss-Seidel iterations. This criterion, developed from that proposed in [@FHdS02] has been extended in the case of the Newton and bi-potential (EBP) method, where some term are naturally vanishing in the original Uzawa and bi-potential (SBP) method. This criterion $\varepsilon_{glob}$ has been written in such a way that if the solution verify that $\varepsilon_{glob}$ is sufficiently small, then this solution has good properties on the equation of motion and Signorini Coulomb contact law. Consequently, this criterion stays valid for the methods developed with the augmented lagrangian (SAL and EAL methods).
This criterion can be stated: $$\label{estimator}
\varepsilon_{glob}=\frac{1}{N_c}\sum_{c=1}^{N_c}\left[\varepsilon^c_{motion}+ \varepsilon^c_{proj}+\varepsilon_{b_c}+\varepsilon^c_{pen}
\right]$$
where:
- $\varepsilon^c_{motion}=\|{\tilde{{{\bf u}}}}^c-{\tilde{{{\bf u}}}}_m^c\|$ where ${\tilde{{{\bf u}}}}^c_m={\tilde{{{\bf u}}}}^{c,i}+\sum_{\alpha=1}^{N_c} W_{c\alpha} {\textbf r}^\alpha$, so $\varepsilon_{motion}$ measures the error on the equation of motion (see equation (\[locprob1\]), this term vanishes for the SBP and SAL method);
- $\varepsilon^c_{proj}=\sqrt{\|{\textbf r}^c-proj({\textbf r}^c,{K_\mu})\|^2}$ is the error for the projection on the Coulomb cone (vanishing for the SBP method);
- $\varepsilon_{b_c}=\Bigl|{\tilde{{{\bf u}}}}^c\cdot {\textbf r}^c+\mu {r_n}^c \|{\tilde{{{\bf u}}}}^c_t\| \Bigr|$ is the absolute value of the bi-potential that has to vanish if and only if the couple $({\tilde{{{\bf u}}}}^c,{\textbf r}^c)$ verifies the Signorini Coulomb contact law (see formula \[SCrules\]);
- $\varepsilon^c_{pen}=-\min(0,{\tilde{{{u}}}_n}^c)$ is the value of the penetration.
One can notice that is absolutely necessary to verify in the criterion that there is no penetration, because nothing in the presented algorithm ensures that is condition is verify at the end of the loop. Moreover, if this condition is not satisfied, the rest of bi-potential can be negative or equal to zero, even if the couple $({\tilde{{{\bf u}}}},{\textbf r})$ is not a solution.
Numerical results
=================
We present in the section three numerical examples with an increasing complexity.
In these computations, the descent parameter $\rho$ is taken in such a way that the result is optimal, in terms of time computing. Denoting $\bar \rho= \frac{m_im_j}{m_i+m_j}\frac1{\Delta t}$, for the SBP and the SAL methods, we have chosen $\rho=0.6 \bar \rho$, whereas for the EBP and the EAL methods, it is better the take $\rho=\bar \rho$. We recall that it has been show that, for the the bi-potential method (see for example [@FJCM05]) and the augmented lagrangian method (see for example [@RA04]), the parameter $\rho$ has to verify $\rho< 2\bar \rho$ in order to ensure the convergence. Generally, for these two methods, the convergence is very sensitive on this parameter. We will show in the last paragraph of this study that for the EBP method, the parameter $\rho$ can be taken in a large range around the value $\bar \rho$ without changing dramatically the convergence of the method.
At each iteration of the NLGS algorithm, the Newton algorithm is stopped either if the convergence is obtained ($\varepsilon_{Newt}^c\leq 10^{-5}$), or if the number of iteration of the Newton algorithm reached 100 when there is no convergence.
Ball sliding on a plane
-----------------------
In this first example, we consider a ball placed on a table with an initial horizontal velocity equal to 1.5 m$\cdot$ s$^{-1}$. The ray of the ball is equal to $5\cdot 10^{-3}$ m, and the friction coefficient between wall and ball is equal to $\mu=0.7$. The time step of discretization is equal to $10^{-4}$ s. In this experiment, the ball first slides on the table, and then the ball rolls without sliding. The global stoping criterion is equal to $\varepsilon_{glob}=10^{-10}$.
![Example 1 – A ball is launched with an initial horizontal velocity (left). First, the ball slides. Then, the ball rolls without slipping (right).[]{data-label="fig1"}](gliss1.pdf "fig:"){width="6.0cm"}![Example 1 – A ball is launched with an initial horizontal velocity (left). First, the ball slides. Then, the ball rolls without slipping (right).[]{data-label="fig1"}](gliss2.pdf "fig:"){width="6.0cm"}
-------- ------------------ ---------------------- --------------
Method Number of Error Total
NLGS iterations $\varepsilon_{glob}$ CPU time (s)
(last time step) (last time step)
SBP 18 $0.384\cdot10^{-10}$ 9.44
SAL 18 $0.384\cdot10^{-10}$ 9.28
EBP 1 0 8.83
EAL 1 $0.175\cdot10^{-13}$ 8.78
-------- ------------------ ---------------------- --------------
: Comparison of the results obtained by the four methods on the first example (after the 2000th time step).[]{data-label="Tab:tab1"}
![Example 1 – Convergence for the standard bi-potential based method, 5$^{th}$ iteration[]{data-label="conv1-1"}](sbp-gliss.pdf){width="12.0cm"}
![Example 1 – Convergence for the standard augmented lagrangian method, 5$^{th}$ iteration[]{data-label="conv1-2"}](sal-gliss.pdf){width="12.0cm"}
![Example 1 – Convergence for the Newton and bi-potential method, 5$^{th}$ iteration[]{data-label="conv1-3"}](ebp-gliss.pdf){width="12.0cm"}
We can observe from these numerical results that the error coming from the projection is very small for the four methods. The Standard Bi-Potential (SBP) method and the Standard Augmented Lagrangian method (SAL) give very closed results, both in term of quality (see figures \[conv1-1\] and \[conv1-2\]) and in term of time computing (see table \[Tab:tab1\]). Nevertheless, we can notice that the time computing is smaller with the SAL method, because there is less computations at each iteration (no term such as $\|{\tilde{{{\bf u}}}}_t\|$ and projection easier to compute for example). The Enhanced Bi-Potential method provides better results, both in term of quality (see figure \[conv1-3\]) and in term of time computing (6.5% better). The Enhanced Augmented Lagrangian method converges after the first Non Linear Gauss Seidel iteration for every time steps, and consequently, this is the faster method on this example (7% faster than the SBP method).
Sedimentation of 4 balls in a box
---------------------------------
In this second experiment, we consider the sedimentation of 4 balls of radius ranging from $4\cdot10^{-4}$ m to $5\cdot10^{-4}$ m. For the computations, the time step of discretization is equal to $\Delta t= 10^{-4}$ s., and the Non linear Gauss-Seidel loop is stopped either if the the global stopping criterion on the NLGS method is equal to $\varepsilon_{glob}=10^{-10}$, or after 5000 iterations if there is no convergence (this case never occurs in this experiment). The friction coefficient between the balls and between the balls and the walls is equal to $\mu=0.3$.
![Example 2 – Sedimentation of four balls under the gravity effect.[]{data-label="fig2"}](b4-1.pdf "fig:"){width="6.0cm"}![Example 2 – Sedimentation of four balls under the gravity effect.[]{data-label="fig2"}](b4-2.pdf "fig:"){width="6.0cm"}
-------- ------------------ ---------------------- ---------------------- --------------
Method Number of Error Maximal Total
NLGS iterations $\varepsilon_{glob}$ penetration CPU time (s)
(last time step) (last time step) (last time step)
SBP 305 $0.949\cdot10^{-12}$ $0.310\cdot10^{-11}$ 2.92
SAL 301 $0.980\cdot10^{-12}$ $0.340\cdot10^{-11}$ 2.87
EBP 161 $0.635\cdot10^{-12}$ $0.641\cdot10^{-12}$ 2.59
EAL 158 $0.973\cdot10^{-12}$ $0.208\cdot10^{-19}$ 2.43
-------- ------------------ ---------------------- ---------------------- --------------
: Comparison of the results obtained by the four methods on the second example (after the 1000th time step)\[Tab:tab2\]
![Example 2 – Convergence of the non-linear Gauss-Seidel iterations for the standard bi-potential based method (1000th time step) The two last curves overlaps, showing that the global error is governed by the error of penetration.[]{data-label="conv2-1"}](sbp-essai.pdf){width="12.0cm"}
![Example 2 – Convergence of the non-linear Gauss-Seidel iterations for the standard augmented lagrangian method (1000th time step). The two last curves overlaps, showing that the global error is governed by the error of penetration.[]{data-label="conv2-2"}](sal-essai.pdf){width="12.0cm"}
![Example 2 – Convergence of the non-linear Gauss-Seidel iterations for the Newton and bi-potential method (1000th time step). The two last curves overlaps, that shows that the global error is governed by the error on the equations of motion.[]{data-label="conv2-3"}](ebp-essai.pdf){width="12.0cm"}
![Example 2 – Convergence of the non-linear Gauss-Seidel iterations for the Newton and Augmented Lagrangian method (1000th time step). The two last curves overlaps, and the other ones does not appear on the figure because the corresponding errors are lower than $10^{-16}$.[]{data-label="conv2-4"}](eal-essai.pdf){width="12.0cm"}
Like in the previous simulation,, the SBP and the SAL method methods provide very similar results (see figures \[conv2-1\] and \[conv2-2\]). For these two methods, we can notice that here the global error is essentially due to the penetrations. The SAL method is 2% faster than the SBP method (table \[Tab:tab2\]).
Results obtained by the EBP method are better (figure \[conv2-3\]), and here the overall error is governed by the error on the equations of the motion. The EBP method is 11% faster than the SBP method, and the penetration is 5 times smaller. In this example the EAL is the faster method (16,8% faster than the SBP method), and the penetration is very small (see figure \[conv2-4\]).
Sedimentation of 500 balls
--------------------------
In this example, we consider the sedimentation of 500 balls (see figure \[fig3\]) of radii ranging from $2.5\cdot10^{-4}$ m to $5\cdot10^{-4}$ m, the time step of discretization is equal to $\Delta t=5\cdot 10^{-5}$ s, and the Non linear Gauss-Seidel loop is stopped if the global estimator (\[estimator\]) verifies $\varepsilon_{glob}\leq 10^{-12}$ or after after 5000 iterations if there is no convergence. The friction coefficient between the balls and between the balls and the walls is equal to $\mu=0.3$.
![Example 3 – Zoom on balls falling under the gravity effect. Initial configuration on the left, final configuration on the right.[]{data-label="fig3"}](b500-1.pdf "fig:"){width="6.0cm"}![Example 3 – Zoom on balls falling under the gravity effect. Initial configuration on the left, final configuration on the right.[]{data-label="fig3"}](b500-2.pdf "fig:"){width="6.0cm"}
The results in table \[Tab:tab3-1\] are obtained after 1000 time steps.
-------- ------------------ ---------------------- ----------------------- --------------
Method Number of Error Maximal Total
NLGS iterations $\varepsilon_{glob}$ penetration CPU time (s)
(last time step) (last time step) (last time step)
SBP 5000 $0.119\cdot 10^{-6}$ $0.213\cdot 10^{-5}$ 1092.95
SAL 5000 $0.135\cdot 10^{-6}$ $0.533\cdot 10^{-5}$ 973.31
EBP 5000 $0.156\cdot 10^{-6}$ $0.286\cdot 10^{-6}$ 854.31
EAL 5000 $0.101\cdot 10^{-6}$ $0.390\cdot 10^{-17}$ 916.65
-------- ------------------ ---------------------- ----------------------- --------------
: Comparison of the results obtained by the four methods on the third example (after the 1000th iteration, $N_{max}=5000$ iterations)\[Tab:tab3-1\]
![Example 3 – Convergence of the non-linear Gauss-Seidel iterations for the standard bi-potential based method (1000th time step). The two last curves collapse.[]{data-label="conv3-1"}](sbp-sedi.pdf){width="12.0cm"}
![Example 3 – Convergence of the non-linear Gauss-Seidel iterations for the standard augmented lagrangian method (1000th time step). The two last curves collapse.[]{data-label="conv3-2"}](sal-sedi.pdf){width="12.0cm"}
![Example 3 – Convergence of the non-linear Gauss-Seidel iterations for the Newton and bi-potential method (1000th time step). The two last curves collapse.[]{data-label="conv3-3"}](ebp-sedi.pdf){width="12.0cm"}
![Example 3 – Convergence of the non-linear Gauss-Seidel iterations for the Newton and Augmented Lagragian method (1000th time step). The two last curves collapse.[]{data-label="conv3-4"}](eal-sedi.pdf){width="12.0cm"}
In this example, the difference between methods SBP and SAL on the one hand, and the method EBP and EAL on the other hand is larger (see table \[Tab:tab3-1\]). We can notice that the SAL method is 10.95% faster than the SBP method, and the EBP is 21.83% faster than the SBP method. Here, the EAL method is no longer the faster one, but the penetration is very small. Again, for the two first methods the global error is essentially due to the penetrations whereas the two last methods, the error is essentially due to the failure to follow precisely the equations of motion.
Discussion on the descent parameter $\rho$
------------------------------------------
We consider again the third example solved by the Newton and bi-potential method ($\varepsilon_{tot}=10^{-8}$, maximal number of iterations of Newton method equal to 100, $\varepsilon_{Newt}=10^{-5}$, $500^{th}$ time step). Here, we take $\bar \rho=\frac{m_im_j}{m_i+m_j}\frac{1}{\Delta t}$, and we consider $\rho=\alpha\bar \rho$, for various values of $\alpha$.
----------- ------------------ ---------------------- -----------
$\alpha$ Number of NLGS Maximal Total CPU
iterations penetration time (s)
(last time step) (last time step)
5 652 $0.110\cdot 10^{-6}$ 65.08
2 414 $0.177\cdot 10^{-6}$ 55.86
1 750 $0.149\cdot 10^{-6}$ 53.95
$\frac12$ 812 $0.634\cdot 10^{-6}$ 78.43
$\frac15$ 667 $0.219\cdot 10^{-5}$ 176.14
----------- ------------------ ---------------------- -----------
: Comparison of the results obtained for various values of $\rho=\alpha \bar \rho$ on the third example (after the 500th iteration, $N_{max}=5000$ iterations)\[Tab:tab4-1\]
These results show one of the main advantage of the EBP method. Indeed, one can notice that in table \[Tab:tab4-1\], the CPU time and the quality of the solution are very similar if $\alpha$ is equal to 1 or 2. Even if $\alpha$ is equal to five, the convergence is not to damaged. In that case, one remain the the SBP and the SAL are no longer convergent. If the parameter $\alpha$ is small, the method converges but the convergence rate is very small. One can notice that the EAL method is much more sensitive about the parameter $\alpha$, essentially in the convergence of the Newton method.
Conclusion
==========
The results presented show that, using an appropriate Newton method, it is possible to improve the computational time over that 20% compared to the standard methods. Moreover, one principal drawback of that type of methods, that is the dependance of the results on the parameter $\rho$ does not exist anymore.
In the future, this method will be extended to the case of a contact law with adhesion. This improvement will be realized in a near future.
[**Acknowledgment:**]{} This work was supported by the french CNRS grant “PEPS Opticontact”, and by the project “TRIBAL” supported by the Picardy region and the European funds FEDER.
[99]{} P. Alart and A. Curnier. A mixed formulation for frictional contact problems prone to Newton like solution method. Computer Methods in Applied Mechanics and Engineering, 92(3):353–375, 1991.
G. De Saxcé, Z.-Q. Feng. New inequality and functional for contact with friction: the implicit standard material approach. Mechanics of Structures and Machines, 19:301-325, 1991.
Z.-Q. Feng, P. Joli, J.-M. Cros, B. Magnain, The bi-potential method applied to the modeling of dynamics problems with friction. Math. Comput., 36:375-383 (2005).
J. Fortin, Simulation numérique de la dynamique des systèmes multicorps appliquée aux milieux granulaires. PhD Thesis (en french), University of Lille 1, 2000.
J. Fortin, M. Hjiaj, G. de Saxcé, [*An improved discrete element method based on a variational formulation of the contact law*]{}, Computers and Geotechnics, 29 (2002), 609-640.
J. Fortin, O. Millet, G. de Saxcé, [*Numerical Simulation of Granular Materials by an improved Discrete Element Method*]{}, Int. J. for Num. Methods in Engineering, no. 62, pp. 639-663, (2005).
M. Jean and J. J. Moreau. Unilaterality and dry friction in the dynamics of rigid bodies collection. In A. Curnier, editor, Contact Mechanics International Symposium, pages 31– 48. Presses Polytechniques et Universitaires Romanes, 1992.
M. Jean. The non smooth contact dynamics method. Compt. Methods Appl. Math. Engrg., 177:235–257, 1999.
P. Joli, Z.-Q. Feng. Uzawa and Newton algorithms to solve frictional contact problems within the bi-potential framework. Int. J. for Num. Methods in Engineering, no. 73, pp. 317-330, (2008).
F. Jourdan, P. Alart, M. Jean. A Gauss-Seidel like algorithm to solve frictional contact problems. Computer Methods in Applied Mechanics an Engineering, 155:31-47 (1998).
J. J. Moreau. Unilateral contact and dry friction in finite freedom dynamics. In J.J.Moreau and eds. P.-D. Panagiotopoulos, editors, Non Smooth Mechanics and Applications, CISM Courses and Lectures, volume 302 (Springer-Verlag,Wien, New York), pages 1–82, 1988.
J. J. Moreau. Some numerical methods in multibody dynamics: application to granular materials. Eur. J. Mech. A Solids, 13(4-suppl.):93–114, 1994.
J. J. Moreau. Numerical aspect of sweeping process. Compt. Methods Appl. Math. Engrg., 177:329–349, 1999.
M. Renouf and P. Alart. Conjugate gradient type algorithms for frictional multicontact problems: applications to granular materials. Comput. Methods Appl. Mech. Engrg., 194(18-20):2019–2041, 2004.
M. Renouf and P. Alart. Gradient type algorithms for 2d/3d frictionless/frictional multicontact problems. In K. Majava P. Neittaanmaki, T. Rossi, O.Pironneau (eds.) R. Owen, and M. Mikkola (assoc. eds.), editors, ECCOMAS 2004, Jyvaskyla, 24–28 July 2004.
M. Renouf, F. Dubois, and P. Alart. A parallel version of the Non Smooth Contact Dynamics algorithm applied to the simulation of granular media. J. Comput. Appl. Math., 168:375–38, 2004.
M. Renouf, V. Acary, and G. Dumont. Comparison of algorithms for collisions, contact and friction in view of real-time applications. In Multibody Dynamics 2005 proceedings, International Conference on Advances in Computational Multibody, Madrid, 21-24 June 2005.
M. Renouf, D. Bonamy, F. Dubois, and P. Alart. Numerical simulation of two-dimensional steady granular flows in rotating drum: On surface flow rheology. Phys. Fluids, 17:103303, 2005.
R.T. Rockafellar. Convex Analysis, Vol. 28, Princeton Math. Series, Princeton Univ. Press, 1970.
N. A. Ramaniraka, Thermomémcanique des contacts entre deux solides déformables, EPFL PhD no. 1651 (1991), Ecole Polytechnique Fédérale de Lausanne.
I. Sanni. Modélisation et simulation bi et tri-dimensionnelles de la dynamique unilatérale des systèmes multi-corps de grandes tailles: application aux milieux granulaires. PhD Thesis (en french), University of Picardy, 2006.
|
---
abstract: 'The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in $\mathbb P^4$ whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme.'
address:
- 'Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720'
- 'Department of Mathematics, Texas A&M University, College Station, TX 77843'
author:
- Diane Maclagan
- 'Rekha R. Thomas'
title: Combinatorics of the toric Hilbert scheme
---
Introduction
============
Let $A = [a_1 \cdots a_n]$ be a $d \times n$ integer matrix of rank $d$ such that $ker(A) \cap \mathbb N^n = \{ 0 \}$ and let ${\mathbb N
A} := \{\sum_{i=1}^{n} m_ia_i \, : \, m_i \in {\mathbb N} \}
\subseteq {\mathbb Z^d}$ be the non-negative integer span of the columns $a_1, \ldots, a_n$ of $A$. The symbol ${\mathbb N}$ denotes the set of natural numbers including zero. Consider the ${\mathbb Z^d}$-graded polynomial ring $S :=
k[x_1,\ldots,x_n]$ over a field $k$ with $\deg x_i := a_i$ for all $i$ and an ideal $I \subseteq S$ that is homogeneous with respect to the grading by $\mathbb N A$, which we call [ *$A$-homogeneous*]{}. The $k$-algebra $R=S/I$ is called an $A$-[ *graded algebra*]{} if its Hilbert function $H_R(b):=\dim_k (R_b)$ is: $$H_R(b) = \left\{ \begin{array}{lll} 1 & \mathrm{ if }\, \, b\in
\mathbb N A & \\ 0 & \mathrm{ otherwise} & \\
\end{array} \right.$$ The presentation ideal $I$ is called an $A$-[*graded ideal*]{} and if $I$ is generated by monomials it is called a [*monomial $A$-graded ideal*]{}.
$A$-graded algebras were introduced by Arnold [@Arnold] who investigated matrices of the form $A = [1 \,\, p \,\, q]$ where $p$ and $q$ are positive integers. A complete classification of all $A$-graded algebras arising from one by three matrices can be found in [@Arnold], [@Korkina] and [@KPR]. The generalization to $d$ by $n$ matrices is due to Sturmfels [@berndpreprint]. The canonical example of an $A$-graded ideal is the [*toric ideal*]{} of $A$, denoted as $I_A$. [*Initial ideals*]{} of $I_A$ [@GBCP] are also $A$-graded.
In [@berndpreprint], Sturmfels constructed a parameter space whose points are in bijection with the distinct $A$-graded ideals in $S$. This variety is the underlying reduced scheme of the [*toric Hilbert scheme*]{} of $A$, denoted as $H_A$, which has been defined recently by Peeva and Stillman [@PS2], [@PS1]. The classical Hilbert scheme parameterizes all homogeneous, saturated ideals in $S$ with a fixed Hilbert polynomial, where $S$ is graded by total degree. However, unlike classical Hilbert schemes which are known to be connected [@Hartshorne], it is unknown whether toric Hilbert schemes are connected. Several of the techniques applied to classical Hilbert schemes cannot be used in the toric situation. In particular, the multigraded Hilbert function used to define $A$-graded ideals is not preserved under a change of coordinates. See [@PS1] for further discussions. The only cases in which $H_A$ is known to be connected are when $A$ has corank one (i.e. $n-d=1$) or two. In the former case the connectivity is trivial, and in the latter it follows from results in [@GP].
In Section 2 we define a graph on all the monomial $A$-graded ideals in $S$, called the [*flip graph*]{} of $A$, by defining an adjacency relation among these ideals. This generalizes the notion of adjacency between two monomial initial ideals of the toric ideal $I_A$, given by the edges of the [*state polytope*]{} of $I_A$ [@ST]. Our main result in Section 3 reduces the connectivity of the toric Hilbert scheme to a combinatorial problem.
[**Theorem 3.1.**]{} The toric Hilbert scheme $H_A$ is connected if and only if the flip graph of $A$ is connected.
The flip graph of $A$ provides information on the structure of $H_A$. In Section 4 we use these graphs to prove that two by five matrices can have toric Hilbert schemes of arbitrarily high dimension. The projective toric variety of such a matrix is a curve in $\mathbb P^4$.
[**Theorem 4.1.**]{} For each $j \in {\mathbb N} \backslash
\{0\}$, there exists a two by five matrix $A(j)$ such that its toric Hilbert scheme $H_{A(j)}$ has an irreducible component of dimension at least $j$.
In Section 5 we relate the flip graph of $A$ to the [*Baues graph*]{} of $\mathcal A$ which is a graph on all the triangulations of the point configuration ${\mathcal A} := \{ a_1, \ldots, a_n \} \subset {\mathbb
Z^d}$ consisting of the columns of $A$. The edges of the Baues graph are given by [*bistellar flips*]{}. This graph and its relatives have been studied extensively in discrete geometry [@reiner]. Sturmfels proved that the radical of a monomial $A$-graded ideal $I$ is the [*Stanley-Reisner*]{} ideal of a triangulation of $\mathcal A$, which we denote as $\Delta(rad(I))$ (see Theorem 4.1 in [@berndpreprint] or Theorem 10.10 in [@GBCP]). This gives a map from the vertices of the flip graph into the vertices of the Baues graph. We extend this map to the edges of the flip graph.
[**Theorem 5.2.**]{} If $I$ and $I'$ are adjacent monomial $A$-graded ideals in the flip graph of $A$, then either they have the same radical and hence $\Delta(rad(I)) = \Delta(rad(I'))$ or $\Delta(rad(I))$ differs from $\Delta(rad(I'))$ by a bistellar flip.
Recently Santos [@Santos] constructed a configuration ${\mathcal A}$ with a disconnected Baues graph, settling the [*generalized Baues problem*]{} (see [@reiner] for a survey). Although his example does not immediately give a disconnected flip graph, it strongly supports the possibility of one. In Section 6 we explain this connection and provide results that point toward a disconnected flip graph and hence, by Theorem 3.1, a disconnected toric Hilbert scheme.
The Flip Graph of $A$
=====================
In this section we define an adjacency relation on all monomial $A$-graded ideals which, in turn, defines the flip graph of $A$. This graph is the main combinatorial object and tool in this paper. We first recall the definition of an $A$-graded ideal.
\[agi\] Let $A = [a_1 \cdots a_n] \in {\mathbb Z^{d \times n}}$ be a matrix of rank $d$ such that $ker(A) \cap \mathbb N^n = \{ 0 \}$ and let ${\mathbb N A} := \{ \sum_{i=1}^n m_ia_i \, : \, m_i \in {\mathbb N}
\}$. An ideal $I$ in $S=k[x_1,\ldots,x_n]$ with $\deg x_i = a_i$ is called an $A$-graded ideal if $I$ is $A$-homogeneous and $R = S/I$ has the ${\mathbb Z^d}$-graded Hilbert function: $$H_R(b) := \dim_k (R_b) = \left\{ \begin{array}{lll} 1 & \mathrm
{ if }\, \, b\in \mathbb N A & \\ 0 & \mathrm{ otherwise} & \\
\end{array} \right.$$
The canonical example of an $A$-graded ideal is the toric ideal $I_A$ which is the kernel of the ring homomorphism $\phi : S \rightarrow
k[t_1^{\pm}, \ldots, t_d^{\pm}]$ given by $x_j \mapsto t^{a_j}$. See [@GBCP] for more information. To see that $I_A$ is $A$-graded, recall that $I_A = \langle x^u - x^v \, : \, Au = Av, \, u,v \in
{\mathbb N^n} \rangle$, and is hence $A$-homogeneous. For each $b \in
{\mathbb N A}$, any two monomials $x^u$ and $x^v$ in $S$ of $A$-degree $b$ (i.e. with $Au = Av = b$) are $k$-linearly dependent modulo $I_A$ making $\dim_k((S/I_A)_b) = 1$. If $b \in {\mathbb Z^d} \backslash
{\mathbb N A}$, $(I_A)_b$ is empty.
Given a [*weight vector*]{} $w \in {\mathbb N^n}$, the initial ideal of an ideal $I \subseteq S$ with respect to $w$ is the ideal $in_w(I) := \langle in_w(f) : f \in I \rangle$ where $in_w(f)$ is the sum of all terms in $f$ of maximal $w$-weight. Our assumption that $ker(A) \cap {\mathbb N}^n = \{0\}$ implies that there is a strictly positive integer vector ${w'}$ in the row space of $A$. Using the binomial description of $I_A$ given above, we then see that $I_A$ is homogeneous with respect to the grading $deg(x_i) =
w'_i$. Hence, the [*Gröbner fan*]{} of $I_A$ covers ${\mathbb R^n}$ and each cell in this fan contains a non-zero non-negative integer vector in its relative interior (see Proposition 1.12 in [@GBCP]). Therefore, for any weight vector $w \in {\mathbb Z^n}$, the initial ideal $in_{w}(I_A)$ is well defined as it coincides with $in_{\bar w}(I_A)$ where $\bar w$ is a non-negative integer vector in the relative interior of the Gröbner cone of $w$. Since the Hilbert function is preserved when passing from an ideal to one of its initial ideals, all initial ideals of $I_A$ are also $A$-graded.
If $M$ is a monomial $A$-graded ideal, then for each $b
\in {\mathbb N A}$ there is a unique monomial of degree $b$ that does not lie in $M$ and is hence a [*standard*]{} monomial of $M$. Definition \[agi\] implies that all $A$-graded ideals are generated by $A$-homogeneous [*binomials*]{} (polynomials with at most two terms) since any two monomials of the same $A$-degree have to be $k$-linearly dependent modulo an $A$-graded ideal.
There is a natural action of the algebraic torus $(k^*)^n$ on $S$ given by $\lambda \cdot x_i=\lambda_ix_i$ for $\lambda \in (k^*)^n$.
An $A$-graded ideal is said to be [*coherent*]{} if it is of the form $\lambda \cdot in_w(I_A)$ for some $\lambda \in (k^*)^n$ and $w \in
{\mathbb Z}^n$.
We recall the definition of the [*Graver basis*]{} of $A$ [@GBCP]. For $u,v \in {\mathbb N^n}$ we write $u < v$ if for each $i=1,\ldots,n$, $u_i \leq v_i$ and $u \neq v$.
A binomial $x^u-x^v$ with $Au=Av$ is a [*Graver binomial*]{} if there do not exist $u^{\prime}, \, \, v^{\prime} \in \mathbb N^n$ with $Au^{\prime}=Av^{\prime}$ and $u^{\prime} < u$, $v^{\prime}<v$. The collection of all Graver binomials is called the [*Graver basis*]{}, $Gr_A$.
The following lemma is a strengthening of Lemma 10.5 in [@GBCP] and was also independently discovered by Peeva and Stillman ([@PS1 Proposition 2.2]). The universal Gröbner basis of an ideal is the union of all the finitely many reduced Gröbner bases of the ideal.
\[gensaregraver\] Let $I$ be an $A$-graded ideal, and let $\mathcal
G=\{x^{a_1}-c_1x^{b_1},\dots,\\ x^{a_k}-c_kx^{b_k} \}$ be the universal Gröbner basis of $I$. Here the $c_i$ may be zero and for each binomial, $x^{a_i}$ and $x^{b_i}$ are not both in $I$. If $c_i=0$, choose $b_i$ so that $Aa_i=Ab_i$ and $x^{b_i} \not \in I$. Then for all $i$, $x^{a_i} - x^{b_i}$ is a Graver binomial. Hence, every minimal generator of $I$ is of this form.
If $x^{a_i}-c_ix^{b_i} \in \mathcal G$, then there is some term order $\prec$ such that one of $x^{a_i}$ and $x^{b_i}$ is a minimal generator of $in_{\prec}(I)$, and the other is standard for $in_{\prec}(I)$. Since $in_{\prec}(I)$ is also $A$-graded, it suffices to prove the lemma for monomial $A$-graded ideals, where $c_i=0$ for all $i$.
Suppose there exist an $i$ such that $x^{a_i}-x^{b_i}$ is not a Graver binomial. Then there exists $u,v \in \mathbb N^n$ with $Au=Av$ such that $u<a_i$ and $v < b_i$. Since $I$ is $A$-graded, one of $x^u$ or $x^v$ is in $I$. If we have $x^u \in I$ then $x^{a_i}$ would not be a minimal generator of $I$, and if $x^v \in I$ then $x^{b_i}$ would not be standard. Therefore, $x^{a_i}-x^{b_i}$ is a Graver binomial for all $i$.
An $A$-homogeneous ideal $I$ in $S$ is [*weakly $A$-graded*]{} if $H_{S/I}(b) \in \{0,1 \}$ for all $b \in \mathbb Z^d$, and $H_{S/I}(b)=1 \Rightarrow b \in \mathbb N A$.
\[UGB\] Let $I$ be an ideal which contains a binomial of the form $x^a-cx^b$ for every Graver binomial $x^a-x^b$. Then $I$ is weakly $A$-graded.
It suffices to prove that $M=in_{\prec}(I)$ is weakly $A$-graded, where $\prec$ is any term order, since $in_{\prec}(I)$ has the same Hilbert series as $I$. If $x^a-x^b$ is a Graver binomial, then since there is some $c$ with $x^a-cx^b \in I$, one of $x^a$ and $x^b$ lies in $M$. Let $x^u$ and $x^v$ be two monomials of degree $b$, and let $x^a-x^b$ be a Graver binomial with $x^a |x^u$ and $x^b
|x^u$. Since one of $x^a$ and $x^b$ lies in $M$, one of $x^u$ and $x^v$ lies in $M$. It thus follows that there is at most one standard monomial of $M$ in each degree $b$, and so $M$ is weakly $A$-graded.
We now define a “flipping” procedure on a monomial $A$-graded ideal which transforms this ideal into an “adjacent” monomial $A$-graded ideal. The idea is motivated by a similar procedure for toric initial ideals which we describe briefly.
The distinct monomial initial ideals of $I_A$ are in bijection with the vertices of the state polytope of $I_A$, an $(n-d)$-dimensional polytope in ${\mathbb R}^n$ [@ST]. Two initial ideals are said to be adjacent if they are indexed by adjacent vertices of the state polytope. The edges of the state polytope are labeled by the binomials in the universal Gröbner basis of $I_A$, $UGB_A \subseteq Gr_A$.
Suppose $I$ and $I'$ are two adjacent monomial initial ideals of $I_A$ connected by the edge $x^a-x^b$. The closure of the outer normal cone at the vertex $I$ (respectively $I'$) is the [*Gröbner cone*]{} $K$ (respectively $K'$) of $I$ (respectively $I'$), the interior of which contains all the weight vectors $w$ such that $in_w(I_A) = I$ (respectively $in_w(I_A) = I'$). The linear span of the common facet of $K$ and $K'$ is the hyperplane $\{u \in {\mathbb R^n} \, :\, (a-b)
\cdot u = 0 \}$. When $w$ is in the interior of $K$, $in_w(x^a-x^b) =
x^a$, $x^a$ is a minimal generator of $I$ and $x^b \not \in I$, and when $w$ is in the interior of $K'$, $in_w(x^a-x^b) = x^b$, $x^b$ is a minimal generator of $I'$ and $x^a \not \in I'$. For a $w$ in the relative interior of the common facet of $K$ and $K'$, $in_w(x^a-x^b)
= x^a-x^b$. Hence passing from $I$ to $I'$ involves “flipping” the orientation of the binomial $x^a-x^b$. No other binomial in $UGB_A$ changes orientation during this passage. See [@HuT] for details. We extend this notion of “flip” to all monomial $A$-graded ideals.
Let $I$ be a monomial $A$-graded ideal and $x^a-x^b$ a Graver binomial with $x^a$ a minimal generator of $I$ and $x^b \not \in I$. We define $I_{flip}$, the result of flipping over this binomial, to be $$I_{flip} := \langle x^c | \exists \, \, d : x^c-x^d \in Gr_A, x^c
\in I, x^d \not \in I, c \neq a \rangle + \langle x^b \rangle.$$
\[weaklyA\] The ideal $I_{flip}$ is weakly $A$-graded.
Let $x^{\alpha}-x^{\beta}$ be a Graver binomial, with $x^{\alpha}
\in I$. By Lemma \[UGB\] it suffices to show that either $x^{\alpha} \in I_{flip}$, or $x^{\beta} \in I_{flip}$. Since $x^{\alpha} \in I$, there is some (possibly identical) Graver binomial $x^{\alpha^{\prime}}-x^{\beta^{\prime}}$ with $x^{\alpha^{\prime}}$ a minimal generator of $I$, and $x^{\beta^{\prime}} \not \in I$, and $x^{\alpha^{\prime}} |x^{\alpha}$. If $\alpha^{\prime} \neq a$, then $x^{\alpha^{\prime}} \in I_{flip}$, and so $x^{\alpha}\in I_{flip}$. If $\alpha^{\prime}=a$, then $\beta^{\prime}=b$, so $x^{\beta} \in
I_{flip}$.
As defined above, to construct $I_{flip}$ requires knowledge of the entire Graver basis. However, the local change algorithm in [@HuT] can be used to construct $I_{flip}$.
\[localflip\] The ideal $I_{flip}$ is the initial ideal with respect to $x^a \prec
x^b$ of $W_{a-b}=\langle x^c | c \neq a, x^c$ is a minimal generator of $I \rangle + \langle x^b-x^a \rangle$.
We note first that this initial ideal is well-defined. The only non-trivial S-pairs formed during its construction are those of a monomial with $x^b-x^a$, in which case the result is a monomial multiple of $x^a$, so there is never any question of what the leading term of a polynomial is. This means that $I_{flip}$ is in fact the initial ideal of $W_{a-b}$ with respect to [*any*]{} term order in which $x^a \prec x^b$. We call $W_{a-b}$ a [*wall ideal*]{} since in the coherent situation, it is the initial ideal of any weight vector in the relative interior of the common facet/wall between the Gröbner cones of $I$ and $I_{flip}$ [@HuT].
Let $K$ be the initial ideal of $W_{a-b}$ with respect to $x^a \prec
x^b$. We first show the containment $K \subseteq I_{flip}$. Let $x^c$ be a minimal generator of $K$. If $x^c=x^b$, or $x^c$ is a minimal generator of $I$ other than $x^a$, then $x^c \in I_{flip}$. So we need only consider the case that $c=ra+g$, where $r > 0$ and $a,b \not \leq g$, as this is the only other form minimal generators of $K$ can have. In order to show that $x^c$ is in $I_{flip}$, it suffices to show that $x^c-x^d$ is a Graver binomial, where $x^d$ is the unique standard monomial of $I$ of the same $A$-degree as $x^c$.
Suppose $x^c-x^d$ is not a Graver binomial, so we can write $c=\sum u_i + g^{\prime}$, $d=\sum v_i +
g^{\prime}$, where for each $i$, $x^{u_i}-x^{v_i}$ is a Graver binomial. Since $x^d \not \in I$, we must have $x^{u_i} \in I$ and $x^{v_i} \not \in I$ for all $i$. If $u_i \neq a$ for some $i$, this would mean that $x^{u_i}$, and hence $x^c$, was in $I_{flip}$. We can thus reduce to the case where $g'=g$ and $v_i = b$ for all $i$, and so $d=rb+g$. Now since $x^c$ is a minimal generator of $K$, there must be some minimal generator, $x^{\alpha}$, of $I$ for which the result of the reduction of the S-pair of $x^{\alpha}$ and $x^b-x^a$ is $x^c$. The only binomial that can be used in the reduction is $x^b-x^a$, and hence there exists $l,m \geq 0$ such that $l+m=r$ and $x^{la+mb+g}$ is the least common multiple of $x^b$ and $x^{\alpha}$. If $l \neq 0$, then $x^a | x^{la+mb+g} | x^{\alpha+b}$. Since $x^a$ and $x^b$ have no common variables, we get that $x^a | x^{\alpha}$ which contradicts $x^{\alpha}$ being a minimal generator of $I$. So we must have $l=0$ and $x^{rb+g}$ is the least common multiple of $x^b$ and $x^{\alpha}$. But this implies that $x^{rb+g}=x^d$ is a multiple of $x^{\alpha}$ and hence in $I$, which is a contradiction. Therefore, this case cannot occur and we conclude that $K \subseteq I_{flip}$.
We now show the reverse inclusion. Suppose $x^c$ is a minimal generator of $I_{flip}$ not equal to $x^b$, and $x^c-x^d$ is the corresponding Graver binomial with $x^d \not \in I$. We may assume that $x^c$ is a multiple of $x^a$, as otherwise it is a generator of $W_{a-b}$, and thus in $K$ automatically. Write $c=ra+\gamma$, where $a \not \leq \gamma$. Suppose that $x^{rb+\gamma} \not \in I$. Then $d=rb+\gamma$, so we must have $\gamma=0$ and $r=1$ to preserve $x^c-x^d$ being a Graver binomial. But then $c=a$, contradicting $x^c$ being a minimal generator of $I_{flip}$. Thus $x^{rb+\gamma} \in I$, and so there is some $\alpha \neq a$ with $x^{\alpha}$ a minimal generator of $I$ such that $\alpha \leq rb+\gamma$. This means that $x^{rb+\gamma} \in
W_{a-b}$, and so $x^{ra+\gamma}=x^c\in W_{a-b}$ because $x^b-x^a \in
W_{a-b}$. Any monomial in $W_{a-b}$ is in $K$, so we conclude that $x^c \in K$.
We say that a binomial $x^{a} -x^{b}$ in the Graver basis is [*flippable*]{} for a monomial $A$-graded ideal $I$ if $x^a$ is a minimal generator of $I$, $x^b \not \in I$ and the ideal $I_{flip}$ obtained by flipping $I$ over $x^a-x^b$ is again a monomial $A$-graded ideal.
We now give a characterization of when a binomial is flippable.
\[flipcrit\] Let $I$ be a monomial $A$-graded ideal, and $x^a-x^b$ a Graver binomial. Then $x^a-x^b$ is flippable for $I$ if and only if $I$ is the initial ideal with respect to $x^b \prec x^a$ of the wall ideal $W_{a-b}=\langle x^c | c \neq a, x^c$ is a minimal generator of $I
\rangle + \langle x^a-x^b \rangle$.
Since $W_{a-b}$ is $A$-homogeneous, $I$ is the initial ideal of $W_{a-b}$ if and only if $W_{a-b}$ is an $A$-graded ideal. But by Lemma \[localflip\] $I_{flip}$ is an initial ideal of $W_{a-b}$, so is $A$-graded exactly when $W_{a-b}$ is.
The [*flip graph*]{} of $A$ has as its vertices all the monomial $A$-graded ideals in $S$. There is an edge labeled by the Graver binomial $x^a-x^b$ between two vertices $I$ and $I'$, if $I'$ can be obtained from $I$ by flipping over $x^a-x^b$.
The edge graph of the state polytope of $I_A$ is a subgraph of the flip graph of $A$. Since the state polytope of $I_A$ is $(n-d)$-dimensional, this subgraph is $(n-d)$-connected and so every vertex in this subgraph has valency at least $n-d$.
Let $Flips_A$ denote the set of binomials labeling the edges of the flip graph of $A$. Since the edges of the state polytope of $I_A$ are labeled by the elements in $UGB_A$, we have $UGB_A \subseteq Flips_A \subseteq Gr_A$.
\(i) Gasharov and Peeva [@GP] proved that all monomial $A$-graded ideals of corank two matrices are coherent. Hence, in this case, the flip graph of $A$ is precisely the edge graph of the state polytope of $I_A$, which is a polygon since $n-d=2$, and $UGB_A = Flips_A$. However, even in this case, $Flips_A$ may be properly contained in $Gr_A$: for $A = [1\,\,3\,\,7]$, $UGB_A = Flips_A = \{a^2c-b^3, a^3-b, ac^2-b^5,
b^7-c^3, c-a^7, ab^2-c \}$ while $Gr_A = Flips_A \cup
\{a^4b-c\}$.\
(ii) For $A = [1 \,\,3\,\,4]$, $UGB_A = Flips_A = Gr_A =
\{ac^2-b^3, a^2c-b^2, b^4-c^3, b-a^3, ab-c, a^4-c\}$.\
(iii) For $A = [3\,\,4\,\,5\,\,13\,\,14]$, $UGB_A \subsetneq Flips_A
\subsetneq Gr_A$. In this case, $Flips_A \backslash UGB_A = \{
a^2bcd-e^2\}$ while $Gr_A \backslash Flips_A = \{d^4-bc^4e^2,
ad^3-bc^2e^2, e^3-b^6cd, b^3cd^3-e^4, e^3- a^2c^2d^2, e^2-ab^5c,
e^3-ab^2cd^2, e^3-a^4bd^2 \}$.
For fixed $A$, let $S_A$ be the intersection of all the monomial $A$-graded ideals in $S$ and let $P_A := \langle x^ax^b :
x^a-x^b \in Gr_A \rangle$. Then $P_A$ is contained, sometimes strictly, in $S_A$ since for each Graver binomial $x^a-x^b$, at least one of $x^a$ or $x^b$ belongs to each monomial $A$-graded ideal.
If $x^a-x^b \in Gr_A$ has at least one of $x^a$ or $x^b$ in $P_A$, then $x^a-x^b \in Gr_A \backslash
Flips_A$. The converse is false.
Suppose $x^a - x^b$ is a flippable binomial for a monomial $A$-graded ideal $M$ such that $x^a \in M$ and $x^b \not \in M$. If $x^a \in P_A \subseteq
S_A$ then $x^a \in M_{flip}$ and if $x^b \in P_A \subseteq S_A$ then $x^b \in M$ both of which are contradictions. To see that the converse is false, consider $$A = \left(
\begin{array}{cccccc}
2 & 1 & 0 & 1 & 0 & 0\\
0 & 1 & 2 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 2 \end{array} \right)$$ which has 29 monomial $A$-graded ideals all of which are coherent. The binomial $x_1x_4x_6-x_2x_3x_5 \in Gr_A \backslash Flips_A$, but neither $x_1x_4x_6$ nor $x_2x_3x_5$ lies in $P_A = \langle
x_1x_2^2x_4, x_1x_3^2x_6, x_4x_5^2x_6, x_1x_2x_3x_5, x_2x_3x_4x_5,
x_2x_3x_5x_6, \\
x_1x_2^2x_5^2x_6, x_2^2x_3^2x_4x_6, x_1x_3^2x_4x_5^2,
x_1x_2x_3x_4x_5x_6
\rangle$.
Connection to the toric Hilbert Scheme
======================================
In this section we explain the relevance of flips for the toric Hilbert scheme $H_A$. We begin by describing the toric Hilbert scheme.
A parameter space for the set of $A$-graded ideals was first described by Sturmfels [@berndpreprint]. Peeva and Stillman improved on this construction by producing the toric Hilbert scheme of $A$ [@PS2], [@PS1], which they show satisfies an important universal property. It is a version of their equations we explain below.
A degree $b \in \mathbb N A$ is a Graver degree if there is some Graver binomial $x^{\alpha}-x^{\beta}$ with $A\alpha=A\beta=b$. We denote by $b_1,\dots,b_N$ the Graver degrees and by $m_i$ the number of monomials of degree $b_i$. Let $$X=\mathbb P^{m_1-1} \times \mathbb P^{m_2-1} \times \dots \times
\mathbb P^{m_N-1}.$$
We now describe $H_A$ as a subscheme of $X$. The coordinates of each $\mathbb P^{m_i-1}$ can be labeled by the monomials of degree $b_i$ as $\{\xi_u : Au=b_i \}$. A point $p\in X$ corresponds to a weakly $A$-graded ideal $I_p$ by the following procedure: For each pair $x^u, x^v$ of degree $b_i$, we place the binomial $\xi_v x^u-\xi_u x^v$ in $I_p$. For each Graver binomial $x^{\alpha}-x^{\beta}$ there thus is a binomial of the form $x^{\alpha}-cx^{\beta}$ in the resulting ideal, where $c$ may be zero. This is immediate except in the case that $\xi_u=\xi_v=0$. In that case, choose $w$ with $Aw=A\alpha$ such that $\xi_w \neq 0$. Then the binomial $\xi_wx^u-\xi_ux^w \in I_p$, so $x^u \in I_p$, and so $x^u-0\cdot x^v$ is the required binomial. Lemma \[UGB\] now implies that $I_p$ is weakly $A$-graded.
We note that the toric ideal $I_A$ corresponds to the point in $X$ with $\xi_u=1$ for all $u$. A monomial $A$-graded ideal corresponds to a point in $X$ where for each $1 \leq i \leq N$ there is exactly one $\xi_{u_i}=1$, and $\xi_v=0$ if $v\neq u_i$ for some $i$. In general, if $Au=b_i$, then $x^u \in I_p$ exactly if $\xi_u=0$.
We now give equations for $H_A$, which guarantee that the resulting ideals $I_p$ are in fact $A$-graded. Let $B
\subset \mathbb NA$ be a finite collection of degrees such that if a weakly $A$-graded ideal generated in Graver degrees is $A$-graded in every degree in $B$, then it is $A$-graded. We know that such a $B$ exists because of bounds given by Sturmfels [@berndpreprint] and Peeva and Stillman [@PS2].
For each $b \in B$ we construct the matrix $M_b$ whose $d_b$ rows are labeled by the monomials of degree $b$. The $n_b$ columns of $M_b$ are labeled by pairs $x^u, x^v$ of degree $b$ such that there is some Graver binomial $x^{\alpha}-x^{\beta}$ such that $u=v-\alpha+\beta$. The corresponding column consists of $\xi_{\alpha}$ in the $x^u$ row, $-\xi_{\beta}$ in the $x^v$ row, and zeroes elsewhere.
The global equations for $H_A$ are now given by the maximal minors of $M_b$ for every $b\in B$. To see that these equations guarantee that $I_p$ is $A$-graded, note that if $I_p$ is not $A$-graded, there is some degree $b \in B$ with all monomials of degree $b$ contained in $I_p$. Now homogeneous polynomials of degree $b$ are in one-to-one correspondence with vectors in $k^{d_b}$. The bijection takes the basis vector with a one in the row corresponding to $x^u$ and zeroes elsewhere to $x^u$, and is defined on other vectors by linear extension. Homogeneous polynomials of degree $b$ contained in $I_p$ are those corresponding to the image of the map $\sigma: k^{n_b}
\rightarrow k^{d_b}$ given by $\sigma: x \mapsto M_bx$. Thus if all monomials of degree $b$ are in $I_p$, $M_b$ must have full rank, which means that there is a maximal minor which does not vanish.
While these equations for $H_A$ are not binomial, it follows from the work of Peeva and Stillman [@PS2] that each irreducible component of the scheme is given by binomial equations. The work of Eisenbud and Sturmfels on binomial ideals [@ES] now implies that the radical of the ideal defining each component is also a binomial ideal, and so the reduced structure on each irreducible component is a toric variety. We denote by $\tilde{H}_A$ the underlying reduced scheme of $H_A$.
The main result of this section is:
\[conniffconn\] The toric Hilbert scheme $H_A$ is connected if and only if the flip graph of $A$ is connected.
The remainder of this section builds up to the proof of Theorem \[conniffconn\]. By the support of a point $v \in \mathbb A^n$ we mean $supp(v):=\{i : v_i \neq 0 \}$. In what follows we assume some familiarity with toric varieties, such as that given in [@Ewald] or [@Fulton].
Corollary 2.6 of [@ES] says that every prime binomial ideal determines a (not necessarily normal) toric variety. The next lemma gives a property of such varieties. When $Q$ is a prime ideal of $S$ we denote by $V(Q)$ the the zero set of $Q$ in $\mathbb A^n$.
\[torussupport\] Consider the point configuration $\{p_1,\dots,p_n\} \subseteq \mathbb
Z^d$ and its toric ideal $Q=ker(\phi:k[x_1,\ldots,x_n] \rightarrow
k[t^{p_1},\dots,t^{p_n}])$ which is a prime binomial ideal. Let $v_1$ and $v_2$ be two points in $V(Q)\subseteq \mathbb A^n$. Then $v_1$ and $v_2$ lie in the same torus orbit of $V(Q)$ if and only if they have the same support.
The dense torus in $V(Q)$ is $V(Q) \cap (k^*)^n$, and the action of this torus on $V(Q)$ is by coordinate-wise multiplication. It thus follows that if $v_1$ and $v_2$ are in the same torus orbit, they have the same support.
Suppose $v_1,v_2 \in V(Q)$ have the same support. If this support is the entire set $\{1,\dots,n\}$, then define $u_i=(v_1)_i/(v_2)_i$. Then if $x^a-x^b$ is a binomial in $Q$, $u^a-u^b=\frac{v_1^a}{v_2^a}-\frac{v_1^b}{v_2^b}=
\frac{1}{v_2^av_2^b}(v_1^av_2^b-v_1^bv_2^a)=0$, so $u$ is in $V(Q) \cap (k^*)^n$, and so $v_1$ and $v_2$ are in the same torus orbit.
Suppose now that $v_1$ and $v_2$ have the same support $\tau
\subsetneq \{1,\dots,n\}$. Since $v_1$ and $v_2$ are in $V(Q)$, this means that there is no binomial in $Q$ of the form $x^a-x^b$ where $supp(a) \subseteq \tau$ and $supp(b) \not \subseteq \tau$. This is because if such a binomial were in $Q$, we will have $v_i^b=0$ for $i=1,2$, and $v_i^a \neq 0$ for $i=1,2$, which contradicts $v_1,v_2
\in V(Q)$. This means that there is no affine dependency between $\{p_i : i \in \tau \}$ and $\{p_i : i \not \in \tau \}$. But this implies that $conv(p_i : i \in \tau)$ is a face of $conv(p_i : 1 \leq
i \leq n)$, and if $p_j \in conv(p_i : i \in \tau)$, then $j \in
\tau$. This means that $v_1$ and $v_2$ lie in an invariant toric subvariety, and so by a similar argument to above are torus isomorphic.
The action of $(k^*)^n$ on $A$-graded ideals gives an action of $(k^*)^n$ on $\tilde{H}_A$. The $n$-torus acts by mapping $v\in
\tilde{H}_A$ to $t \cdot v$ via the map $(t \cdot v)_u=t^uv_u$.We will refer to this action as the $n$-torus action. There is also a torus action on a point for every irreducible component of the reduced toric Hilbert scheme it belongs to. We will refer to these actions as the ambient torus actions. We note that these torus actions are usually different from the $n$-torus action, as each of the finitely many irreducible components of $H_A$ has only finitely many ambient torus orbits, but there can be an infinite number of $n$-torus orbits. An example of this situation is given in Theorem 10.4 of [@GBCP]. The $n$-torus orbit is, however, contained inside all ambient torus orbits.
\[toruscontainment\] Let $v$ be a point on $\tilde{H}_A$. Then the $n$-torus orbit of $v$ is contained in any ambient torus orbit of $v$.
It is straightforward to see that $t \cdot v$ lies in every irreducible component of $\tilde{H}_A$ in which $v$ does (this follows from the fact that $S[l]/(lI_v+(1-l)(I_{t\cdot v}))$ is a flat $k[l]$ module). All points in the $n$-torus orbit of $v$ have the same support, and thus lie on the same ambient torus orbit by Lemma \[torussupport\].
Fix an irreducible component $V$ of $\tilde{H}_A$. Since $V$ is a projective toric variety, there is a polytope $P$ corresponding to $V$. An ambient torus orbit of a point $v \in \tilde{H}_A$ corresponds to a face of $P$. In the case of the coherent component, this polytope is the state polytope of $I_A$. Over the course of the next three lemmas, we show that the edges of $P$ correspond exactly to flips.
\[verticesaremono\] Vertices of $P$ correspond exactly to the monomial $A$-graded ideals in V.
Let $I$ be the ideal corresponding to a vertex $p$ of $P$. The orbit of $I$ under the ambient torus corresponding to $P$ is just the ideal $I$. By Corollary \[toruscontainment\] the $n$-torus orbit of $I$ is contained in any ambient torus orbit, so $I$ is $n$-torus fixed as well, and thus is a monomial ideal.
For the other implication, let $I$ be a monomial $A$-graded ideal corresponding to a point $v$ in $V$. As a point in $X$, $v$ is invariant under any scaling of its coordinates in any fashion, and thus is invariant under any ambient torus action. It thus corresponds to a vertex of $P$.
\[twoinismeansflip\] Let $I$ be an $A$-graded ideal. If $I$ has exactly two initial ideals, then $I$ is $n$-torus isomorphic to an ideal of the form $J=\langle x^a-x^b, x^{c_1}, \ldots, x^{c_r} \rangle$.
Let $M_1$ and $M_2$ be the two initial ideals of $I$, and let $\mathcal G$ be the universal Gröbner basis of $I$. The set $\mathcal G$ contains a reduced Gröbner basis for $I$ with respect to a term order for which $M_1$ is the initial ideal, and so there exist binomials $x^a-cx^b \in \mathcal G$ with $c \neq 0$ for which $x^a$ is a minimal generator of $M_1$, $x^b \not \in M_1$. Suppose for all such binomials we have $x^a \in M_2$. Then $M_1 \subseteq
M_2$ is an inclusion of distinct monomial $A$-graded ideals, which is impossible. So we conclude that there is some binomial $x^{a_1}-c_1x^{b_1}\in \mathcal G$ with $c_1 \neq 0$, $x^{a_1} \in M_1
\setminus M_2$ and $x^{b_1} \in M_2 \setminus M_1$.
Suppose there is some other binomial $x^{a_2}-c_2x^{b_2} \in \mathcal
G$ with $c_2 \neq 0$. Without loss of generality we may assume that $x^{a_2} \in M_1$ and $x^{b_2} \not \in M_1$. We note that $(a_1-b_1)
\neq (a_2-b_2)$, as by Lemma \[UGB\] the two binomials $x^{a_1}-x^{b_1}$ and $x^{a_2}-x^{b_2}$ are Graver binomials, and they must be distinct since $\mathcal G$ is the universal Gröbner basis of $I$. We can thus find a supporting hyperplane for $pos(a_1-b_1,b_2-a_2)$, which intersects the cone only at the origin. This implies the existence of a vector $w$ which satisfies $w \cdot
(a_1-b_1) >0$ and $w \cdot (b_2-a_2) > 0$. Let $M=in_w(I)$. Then $x^{a_1} \in M$, and $x^{b_2} \in M$, so $M \neq M_1$, and $M \neq
M_2$. This means that $I$ has a third initial ideal, which contradicts our assumption, and so we conclude that $x^{a_1}-c_1x^{b_1}$ is the only binomial in $\mathcal G$.
Pick $i\in supp(b_1)$. Define $\lambda_i=\frac{1}{c_1}$, and $\lambda_j=1$ for $j \neq i$. Then $\lambda I$ is in the desired form.
\[edgesareflips\] Let $M_1$ and $M_2$ be monomial $A$-graded ideals corresponding to vertices $p_1$ and $p_2$ of $P$. $M_1$ and $M_2$ are connected by a single flip if and only if there is an edge $e$ of $P$ connecting $p_1$ and $p_2$.
Suppose $p_1$ and $p_2$ are connected by an edge $e$. Let $I$ be the ideal corresponding to a point $p$ in the relative interior of $e$. By Corollary \[toruscontainment\] the $n$-torus closure of $p$ is contained in $e$. Thus $I$ has at most two initial ideals. If $I$ had only one initial ideal, it would be a monomial ideal and thus corresponds to a vertex of $P$, by Lemma \[verticesaremono\]. We thus conclude that $I$ has exactly two initial ideals, $M_1$ and $M_2$, corresponding to $p_1$ and $p_2$ respectively. Now by Lemma \[twoinismeansflip\] $I$ is $n$-torus isomorphic to $J= \langle
x^a-x^b, x^{c_1}, \ldots, x^{c_r} \rangle$, where $x^a \in M_1
\setminus M_2$ and $x^b \in M_2 \setminus M_1$. Since $J$ is $A$-graded, $x^a-x^b$ is a Graver binomial. Because $J$ has initial ideals $M_1$ and $M_2$, it is their wall ideal $W_{a-b}$, and so $M_1$ and $M_2$ are connected by a flip over $x^a-x^b$.
Conversely, suppose $M_1$ and $M_2$ are connected by a single flip. Then there is an ideal $W_{a-b}=\langle x^a-x^b, x^{c_1}, \ldots,
x^{c_r} \rangle$ which has as its two initial ideals $M_1$ and $M_2$. Let $J$ be an $A$-graded ideal which is isomorphic to $W_{a-b}$ under the ambient torus corresponding to $P$. Let $x^d$ be a minimal generator of $M_1$, with $d \neq a$, and $x^d-x^e$ the corresponding Graver binomial with $x^e \not \in M_1$. Then $x^d \in W_{a-b}$, and thus $x^d \in J$, as the ambient torus action preserves the monomials in an $A$-graded ideal. So $J$ contains all minimal generators of $M_1$ and $M_2$ except $x^a$ and $x^b$. Suppose $J$ has a minimal generator $x^{\alpha}-cx^{\beta}$, where $x^{\alpha}-x^{\beta}$ is a Graver binomial, $x^{\alpha},x^{\beta} \not \in J$, and $\alpha, \beta
\neq a,b$. Without loss of generality we may assume that $x^{\alpha}
\in M_1$. If $x^{\beta} \not \in M_1$ then $x^{\alpha} \in W_{a-b}$ by the definition of $W_{a-b}$, and thus also $x^{\alpha} \in J$. We thus conclude that $x^{\beta} \in M_1$. But this means there exist $\alpha^{\prime} \leq \alpha$, $\beta^{\prime} \leq \beta$, such that $x^{\alpha^{\prime}}$ and $x^{\beta^{\prime}}$ are minimal generators of $M_1$. Since $x^{\alpha}$ and $x^{\beta}$ have disjoint support, we cannot have $\alpha^{\prime}=\beta^{\prime}=a$, so at least one of $x^{\alpha^{\prime}}$ and $x^{\beta^{\prime}}$ is in $W_{a-b}$. But this means at least one of $x^{\alpha}$ and $x^{\beta}$ is in $J$, giving a contradiction. Hence the only binomial minimal generator of $J$ is of the form $x^a-c^{\prime}x^b$, so as in the proof of Lemma \[twoinismeansflip\] $J$ is $n$-torus isomorphic to $W_{a-b}$. We thus see that all ambient torus closures of $W_{a-b}$ are the same as the $n$-torus closure, and so $p_1$ and $p_2$ are connected by an edge.
It suffices to show that the reduced scheme $\tilde{H}_A$ is connected if and only if the flip graph of $A$ is connected. Since passing to an initial ideal is a flat deformation, each irreducible component contains a monomial $A$-graded ideal. It thus suffices to show that all monomial $A$-graded ideals lie in the same connected component of $\tilde{H}_A$ if and only if the flip graph is connected. The “if” direction follows from the fact that if $I_1$ and $I_2$ are connected by a single flip, then they are both initial ideals of a single wall ideal $W_{a-b}$, and so lie in the same connected component of $\tilde{H}_A$. The “only-if” direction follows from Lemmas \[verticesaremono\] and \[edgesareflips\], which imply that the flip graph restricted to an irreducible component of $\tilde{H}_A$ is the edge skeleton of a polytope whose vertices are the monomial $A$-graded ideals in that component, and so is connected. As the intersection of two irreducible components of $\tilde{H}_A$ contains a monomial $A$-graded ideal by Gröbner deformation, this means that if $\tilde{H}_A$ is connected, the flip graph of $A$ is connected.
Toric Hilbert Schemes of Arbitrarily High Dimension from Curves in $\mathbb P^4$
================================================================================
In this section we exhibit toric Hilbert schemes of arbitrarily high dimensions for which the associated toric varieties are curves in ${\mathbb P^4}$. When $A$ has corank one, its Graver basis consists of precisely one binomial $x^a-x^b$, and the flip graph of $A$ has only the two vertices $I= \langle x^a \rangle$ and $I' = \langle
x^b \rangle$ which are connected by the flip $x^a-x^b$. Hence $H_A$ is one-dimensional and connected. All $A$-graded ideals of a corank two matrix are coherent [@GP] which implies that the flip graph of $A$ is connected since it coincides with the edge graph of the state polytope of $I_A$. In this case, $H_A$ has exactly one irreducible component which is two dimensional and smooth [@PS1]. The toric Hilbert scheme of a corank three matrix is at least three dimensional since the irreducible component containing the coherent $A$-graded ideals has dimension three. In contrast to the results in coranks one and two, Theorem \[highdim\] gives a family of two by five matrices of corank three whose toric Hilbert schemes can have arbitrarily high dimensions. The projective toric variety of each matrix in the family is a curve in ${\mathbb P^4}$. Note that both the corank $n-d$ and the number of columns $n$ are fixed for these matrices.
\[highdim\] For each $j \in {\mathbb N} \backslash \{0\}$, the toric Hilbert scheme $H_{A(j)}$ of $$A(j) = \left ( \begin{array}{ccccc} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 &
3+3j & 4+3j & 6+3j \end{array} \right )$$ has an irreducible component of dimension at least $j$.
These matrices were motivated by Example 5.11 in [@ST], and the theorem was inspired by computer experiments on their flip graphs. We first define the following monomial ideals and sets of binomials that will be used in the proof of Theorem \[highdim\]. For each $j \in {\mathbb N} \backslash \{0\}$, let\
$\begin{array}{ll}
P_j = \langle c^2e, bc, a^2e, ace, ae^{j+2} \rangle,&
R_j = \langle a^5c^j, a^8c^{j-1}, \ldots, a^{5+3(j-1)}c \rangle,\\
Q_j = \langle be^{j+1}, a^2c^{j+1}, b^4e^j, c^{j+2} \rangle,&
S_j = \langle b^7e^{j-1}, b^{10}e^{j-2}, \ldots, b^{7+3(j-1)} \rangle
\end{array}$
and
$\begin{array}{l}
{\mathcal P}_j = \{ c^2e-d^3, bc-ad, a^2e-b^2d, ace-bd^2,
ae^{j+2}-c^jd^3 \},\\
{\mathcal Q}_j = \{be^{j+1}-c^{j+1}d,
a^2c^{j+1}-b^3e^j, b^4e^j-a^3c^jd, c^{j+2}-ae^{j+1}\},\\
{\mathcal R}_j = \{a^{5+3t}c^{j-t} - b^{6+3t}e^{(j-1)-t}, \, \,
t=0,1,\ldots, j-1 \},\\
{\mathcal S}_j = \{ b^{7+3t}e^{(j-1)-t}-a^{6+3t}c^{(j-1)-t}d,
\,\,t=0,1,\ldots, j-1 \}.
\end{array}$\
\[gbasis\] The ideal $M_j = P_j + Q_j + R_j + S_j$ is the initial ideal of $I_{A(j)}$ with respect to the weight vector $w = (1,1,2,0,2)$.
By computing the $A(j)$-degree of both terms in each binomial of ${\mathcal G}_j := {\mathcal P}_j \cup {\mathcal Q}_j \cup
{\mathcal R}_j \cup {\mathcal S}_j$, it can be seen that ${\mathcal
G}_j$ is a subset of $I_{A(j)}$. It can also be checked that for each binomial in ${\mathcal G}_j$, the positive term is the leading term with respect to $w = (1,1,2,0,2)$. Hence $M_j = \langle in_{w}(g) : g \in
{\mathcal G}_j \rangle$ is contained in the initial ideal of $I_{A(j)}$ with respect to $w$ and no generator of $M_j$ is redundant. The monomial ideal $M_j$ will equal $in_{w}(I_{A(j)})$ if ${\mathcal G}_j$ is the reduced Gröbner basis of $I_{A(j)}$ with respect to $w$. Consider the elimination order $x,y \succ a,b,c,d,e$ refined by the graded reverse lexicographic order $x > y$ on the first block of variables and the weight vector $w$ on the second block of variables. Then the reduced Gröbner basis of $I_{A(j)}$ with respect to $w$ is the intersection of the reduced Gröbner basis of $$J(j) := \langle
a-x, b-xy, c-xy^{3+3j}, d-xy^{4+3j}, e-xy^{6+3j} \rangle$$ with respect to $\succ$ with $k[a,b,c,d,e]$ (see Algorithm 4.5 in [@GBCP]). By a laborious check it can be shown that the reduced Gröbner basis of $J(j)$ with respect to $\succ$ is\
$\begin{array}{l}
{\mathcal G}_j \cup \{x-a, ya-b, ybd-ae, yc-d, yd^2-ce, y^2d-e,\\
yb^{3t+2}e^{j-t}-a^{3t+1}c^{(j+1)-t},\, t=0,\ldots,j,\\
y^2b^{3t+1}e^{j-t}-a^{3t}c^{(j+1)-t}, \, t=0,\ldots,j, \\
y^{3l}b^{3t}e^{p_l-t} - a^{3t-1}c^{p_l-t+1},\,\,l=1,\ldots,j,
\,\,t=1,\ldots, p_l:=(j+1)-l,\\
y^{3l+1}b^{3t-1}e^{p_l-t} - a^{3t-2}c^{p_l-t+1},\,\,l=1,\ldots,j,\,\,
t=1,\ldots, p_l,\\
y^{3l+2}b^{3t-2}e^{p_l-t} - a^{3t-3}c^{p_l-t+1},\,\,l=1,\ldots,j,\,\,
t=1,\ldots, p_l \}.
\end{array}$\
\[highval\] For each $j \in {\mathbb N} \backslash \{0\}$ the monomial $A(j)$-graded ideal $M_j$ from Lemma \[gbasis\] has exactly $2j+4$ flippable binomials.
We will show that the binomials in ${\mathcal Q_j} \cup
{\mathcal R}_j \cup {\mathcal S}_j$ are flippable for $M_j$ while those in ${\mathcal P_j}$ are not. In order to show that a binomial $x^a-x^b$ is flippable for $M_j$ we need to show that every $S$-polynomial (monomial in our case) formed from the binomial $x^a-x^b$ (with $x^a$ as leading term) and a minimal generator $x^c$ of $M_j$ different from $x^a$ reduces to zero modulo $W_{a-b} = \langle
x^c : c \neq a,\, x^c$ a minimal generator of $M_j \rangle +
\langle x^a-x^b \rangle$.
We first consider ${\mathcal R_j}$. A binomial $a^{5+3t}c^{j-t} - b^{6+3t}e^{(j-1)-t}$ in ${\mathcal R}_j$ can form a non-trivial $S$-pair ($S$-monomial) with (i) $c^2e$, (ii) $bc$, (iii) $a^2e$, (iv) $ace$, (v) $ae^{j+2}$, (vi) $a^2c^{j+1}$, (vii) $c^{j+2}$ and (viii) a monomial $a^{5+3l}c^{j-l}$ from $R_j$ such that $t \neq l$. The remaining generators of $M_j$ (except $a^{5+3t}c^{j-t}$ itself) are relatively prime to $a^{5+3t}c^{j-t}$ and so the $S$-pairs formed reduce to zero by Buchberger’s first criterion. We consider each case separately.
\(i) The $S$-monomials formed from $c^2e$ and $a^{5+3t}c^{j-t}-
b^{6+3t}e^{(j-1)-t}$ are $b^{6+3t}c^pe^{j-t}$, $0 \leq t \leq j-1$, where $p=1$ if $j-t = 1$ and $p=0$ if $j-t >
1$.\
(a) If $t=0$, $b^6c^pe^j$ is a multiple of $b^4e^j \in Q_j$.\
(b) If $t > 0$, $b^{6+3t}c^pe^{j-t}$ reduces to zero modulo $b^{7+3(t-1)}e^{j-t} \in S_j$.
\(ii) The $S$-monomials formed from $bc$ are $b^{7+3t}e^{(j-1)-t}$, $0 \leq t \leq j-1$ all of which lie in $S_j$ and hence reduce to zero modulo $W_{a-b}$.
\(iii) The $S$-monomials between $a^2e$ and $a^{5+3t}c^{j-t}-
b^{6+3t}e^{(j-1)-t}$ are $b^{6+3t}e^{j-t}$ for $0 \leq t \leq j-1$. If $t=0$, $b^6e^j$ is a multiple of $b^4e^j \in Q_j$, and if $t > 0$ then $b^{6+3t}e^{j-t}$ is divisible by $b^{7+3(t-1)}e^{j-t}
\in S_j$.
\(iv) The $S$-monomials from $ace$ are $b^{6+3t}e^{j-t}$ for $0 \leq t \leq j-1$, all of which reduce to zero as in (iii).
\(v) The monomial $ae^{j+2}$ gives $b^{6+3t}e^{2j+1-t}$ for $0 \leq t \leq j-1$, all of which reduce to zero modulo $be^{j+1} \in Q_j$.
\(vi) From $a^2c^{j+1}$ we get $b^{6+3t}c^{t+1}e^{(j-1)-t}$, $0 \leq t \leq j-1$, all of which are multiples of $bc \in P_j$.
\(vii) The $S$-monomials from $c^{j+2}$ are $b^{6+3t}c^{t+2}e^{(j-1)-t}$ which are also multiples of $bc \in P_j$ for $0 \leq t \leq j-1$.
\(viii) For this last case, suppose first that $l < t \in
\{0,1,2,\ldots,j-1\}$. Then $lcm(a^{5+3l}c^{j-l},a^{5+3t}c^{j-t}) = a^{5+3t}c^{j-l}$ and the $S$-monomial between $a^{5+3l}c^{j-l}$ and $a^{5+3t}c^{j-t}-
b^{6+3t}e^{(j-1)-t}$ is $b^{6+3t}c^{t-l}e^{(j-1)-t}$ which is a multiple of $bc \in P_j$. If $l > t$, then the $S$-monomial is $a^{3(l-t)}b^{6+3t}e^{(j-1)-t}$ which is divisible by $a^2e \in P_j$ since $t < l \leq j-1$ and hence $t < j-1$.
Similarly, one can check that the binomials in ${\mathcal Q_j} \cup {\mathcal S}_j$ are all flippable for $M_j$, which shows that $M_j$ has at least $2j+4$ flippable binomials. To finish the proof, we argue that no binomial in ${\mathcal P_j}$ is flippable for $M_j$.
\(i) The $S$-binomial between $c^2e-d^3 \in {\mathcal P_j}$ and $bc \in
P_j$ is $bd^3$ which is not divisible by any generator of $M_j$.
\(ii) The binomials $bc-ad,\,a^2e-b^2d$ and $ace-bd^2 \in {\mathcal P_j}$ form the $S$-binomials $ade^{j+1}, \, b^3de^j$ and $b^2d^2e^j$ respectively with $be^{j+1} \in Q_j$. None of them can be divided by a minimal generator of $M_j$.
\(iii) The $S$-binomial of $ae^{j+2}-c^jd^3 \in {\mathcal Q_j}$ and $a^2e \in P_j$ is $ac^jd^3$ which does not lie in $M_j$.
Hence $M_j$ has exactly $2j+4$ flippable binomials.
The same proof as in Lemma \[highval\] shows that the generators of ${\mathcal I}(\mu_0, \ldots, \mu_{j-1}) := P_j + Q_j
+ \langle a^{5+3t}c^{j-t}-\mu_tb^{6+3t}e^{j-1-t}\,,t=0,\ldots,j-1 \rangle
+ S_j$ form a Gröbner basis with respect to $w = (1,1,2,0,2)$ with initial ideal $M_j$, for every choice of scalars $\mu_0,\ldots,\mu_{j-1}$ from the underlying field $k$. Lemma \[highval\] proved this claim for the case $\mu_i = 1$, for an $0 \leq i \leq j-1$ and $\mu_j = 0$ for all $j \neq i$. Since $M_j$ is $A(j)$-graded, the $A(j)$-homogeneous ideal ${\mathcal
I}(\mu_0,\ldots,\mu_{j-1})$ is also $A(j)$-graded for every choice of scalars $\mu_0,\ldots,\mu_{j-1}$. Hence there is an injective polynomial map from ${\mathbb A_k^j} \rightarrow H_{A(j)}$, such that $(\mu_0,
\ldots, \mu_{j-1})$ maps to the point on $H_{A(j)}$ corresponding (uniquely) to ${\mathcal I}(\mu_0, \ldots, \mu_{j-1})$. Since ${\mathbb A_k^j}$ is irreducible, the image of this map lies entirely in one irreducible component of the toric Hilbert scheme $H_{A(j)}$ and the dimension of this component is at least $dim({\mathbb A_k^j})
= j$.
In [@ST] it was conjectured that the maximum valency of a vertex in the state polytope of $I_A$ is bounded above by a function in just the corank of $A$. As a particular case, it was also conjectured that if $A$ is of corank three, then every vertex in the state polytope of $I_A$ has at most four neighbors. This latter conjecture was recently disproved by Hoşten and Maclagan [@HuT] who have found vertices with up to six neighbors. Lemma \[highval\] shows that even in corank three, a vertex in the flip graph of $A$ can have arbitrarily many neighbors.
Connection to the Baues Problem
===============================
In this section we elaborate a connection between $A$-graded ideals and the Baues problem for triangulations. A good reference for all forms of the Baues problem is [@reiner].
A triangulation of a point configuration $\mathcal A=
\{a_1,\ldots,a_n\}\subseteq \mathbb R^d$ is a geometric simplicial complex covering $\text{conv}(a_1,\ldots,a_n)$ with the vertices of each simplex being a subset of $\mathcal A$. Each simplex $\sigma$ is indexed by the set $\{i: a_i \text{ is a vertex of }\sigma\}$.
A basic operation on triangulations of a point configuration is the [*bistellar flip*]{}. The two basic types of non-degenerate bistellar flips in the plane are shown in Figure \[flipeg\].
Intuitively, a bistellar flip should be thought of as gluing in a higher dimensional simplex, and then turning that simplex over and viewing it from the other side. This can be seen most clearly in the second example in Figure \[flipeg\], which can be viewed as the top and bottom of a tetrahedron. The first example can also be thought of as two opposite views of a tetrahedron.
More formally, a bistellar flip interchanges the two different triangulations of a [*circuit*]{} (minimal affine dependence) of $\mathcal A$. Let $t$ be a circuit of the configuration $\mathcal A$, and $T=\{i : t_i \neq 0\}$ be its support. We denote by $T^+$ the set $\{ i : t_i >0 \}$ and by $T^-$ the set $\{ i : t_i <0 \}$. There are exactly two triangulations of $C=conv(a_i : i \in T)$. The first, $C^+$, has $|T^+|$ simplices, which are the simplices indexed by the sets $\{ T \setminus \{i\} : i \in T^+\}$. The second, $C^-$, has $|T^-|$ simplices, which are the sets in $\{ T \setminus \{i\} : i \in
T^-\}$. The unique minimal non-face of $C^+$ ($C^-$) is $T^+$ ($T^-$). If $C$ is $d$-dimensional, and one of $C^+$ and $C^-$ is a subcomplex of the triangulation $\Delta$, then a bistellar flip over the circuit $t$ involves replacing the subcomplex $C^+$ by $C^-$ or vice versa.
If $C$ is lower dimensional, we impose an additional condition for $t$ to be flippable. By the [*link*]{} of a simplex $\sigma$ in a simplicial complex $\Delta$ we mean the collection of simplices $\{
\tau : \tau \cap \sigma=\emptyset, \tau \cup \sigma \in \Delta \}$. We say $t$ is flippable if $C^+$ (or $C^-$) is a subcomplex of $\Delta$, [*and*]{} the link in $\Delta$ of every maximal simplex of $C^+$ (respectively $C^-$) is the same subcomplex $L$. This second condition is trivially satisfied if $C$ is $d$-dimensional, as the link of every maximal simplex is the empty set. A bistellar flip over the circuit $t$ from $C^+$ to $C^-$ then involves replacing the simplices $\{ l \cup \sigma : \sigma \in C^+, l\in L \}$ by the simplices $\{ l \cup \tau : \tau \in C^-,l \in L\}$.
Examples of bistellar flips are shown in Figure \[flipseg2\].
We can form a graph, called the [*Baues graph*]{}, on the set of all triangulations of a point configuration, with an edge connecting two triangulations when they differ by a bistellar flip. Figure \[flipseg2\] is a subgraph of the Baues graph for a particular collection of six points in the plane. An obvious question to ask is whether the Baues graph is connected. Santos recently answered this question negatively [@Santos], constructing a configuration of 324 points in $\mathbb R^6$ which has a disconnected Baues graph.
The rest of this section will relate the Baues graph to the flip graph and the toric Hilbert scheme. The connection is through the following lemma, which is a special case of Theorem 10.10 in [@GBCP]. It links monomial $A$-graded ideals and triangulations of $\mathcal A$, where $A$ is the matrix whose columns are the points in $\mathcal A$, with an additional row of ones added. We will denote both the $i$th row of $A$ and the $i$th point of $\mathcal A$ by $a_i$. We adopt the notational convention that if $\sigma \subseteq
\{1,\dots,n\}$ is a set, then $x^{\sigma}=\prod_{i \in \sigma} x_i$. The [*Stanley-Reisner ideal*]{} (see [@Stanley]) $I(\Delta)$ of a simplicial complex $\Delta$ is the ideal generated by the monomials $x^{\sigma}$ where the sets $\sigma$ are the minimal non-faces of $\Delta$. Similarly, every squarefree monomial ideal $I$ in $S$ defines a unique simplicial complex $\Delta(I)$ on $\{1,\dots,n\}$.
\[triangs\] [@GBCP Theorem 10.10] Let $I$ be a monomial $A$-graded ideal. Then $\Delta(rad(I))$, the simplicial complex associated to $rad(I)$ via the Stanley-Reisner correspondence, is a triangulation of $\mathcal A$.$\Box$
We can now state the main theorem of this section.
\[flipmeansbistellar\] Let $I$ be a monomial $A$-graded ideal and $x^a-x^b$ a flippable binomial for $I$. Then either $\Delta(rad(I))=\Delta(rad(I_{flip}))$, or they differ by a bistellar flip.
The proof will be developed through the following series of lemmas. We need to show that if $I_1$ and $I_2$ are monomial $A$-graded ideals which differ by a flip, then either the radicals are the same, or $\Delta(rad(I_1))$ and $\Delta(rad(I_2))$ differ by a bistellar flip. This involves showing:
1. $t=a-b$ is a circuit of ${\mathcal A}$ (Lemma \[radsthesame\]).
2. $C^+$ is a subcomplex of $\Delta(rad(I))$ with the link of all maximal simplices of $C^+$ the same (Lemma \[ksubcomplex\]).
3. $\Delta(rad(I_{flip}))$ differs from $\Delta(rad(I))$ exactly by replacing $C^+$ and its link by $C^-$ and corresponding link.
By a circuit of $A$ we mean a binomial $x^a-x^b$ such that $a-b$ is a circuit of $\mathcal A$. Note that all circuits are Graver binomials.
\[radsthesame\] Let $I$ be a monomial $A$-graded ideal, with $x^a-x^b$ a flippable binomial with $x^a
\in I$. Then $x^b \in rad(I) \Leftrightarrow rad(I)=rad(I_{flip})$. If $x^b \not \in rad(I)$, then $x^a-x^b$ is a circuit of $A$.
The implication $\Leftarrow$ is immediate in the first statement so we need only show that $x^b \in rad(I)$ implies $rad(I)=rad(I_{flip})$. Suppose $x^b \in rad(I)$. Let $x^c$ be a minimal generator of $I_{flip}$. Then either $x^c$ is a minimal generator of $I$, $c=b$, or $c=a+g$ for some $g$. In each case $x^c \in rad(I)$, so $rad(I_{flip}) \subseteq rad(I)$. If the containment is proper, Lemma \[triangs\] gives a proper containment of Stanley-Reisner ideals of triangulations of $\mathcal A$, which is not possible. So we conclude $rad(I)=rad(I_{flip})$.
For the second statement, suppose $x^a-x^b$ is not a circuit. Then there exists a circuit $x^c-x^d$ with $supp(c) \subseteq supp(a)$, and $supp(d) \subseteq supp(b)$ where at least one of these inclusions is proper. Since $x^b \not \in rad(I)$, we must have $x^{supp(d)} \not
\in rad(I)$, and thus $x^d \not \in I$. This implies $x^c \in I$, and so, since we know $c \neq a$, $x^c \in I_{flip}$. This means $x^{supp(c)} \in rad(I_{flip})$, and so $x^a \in rad(I_{flip})$. But this means, as above, that $rad(I_{flip})=rad(I)$, which in turn implies that $x^b \in rad(I)$, contradicting the hypothesis.
Let $I$ be a monomial $A$-graded ideal, with $x^a -x^b$ flippable, where $x^a \in I$, $x^b \not \in rad(I)$. Let $t=a-b$, and $T=supp(t)$. By Lemma \[radsthesame\] we know that $t$ is a circuit, so we can consider the triangulation $C^+=\{T \setminus \{i\} : i
\in T^+ \}$ of $C$.
\[ksubcomplex\] Let $I$, $x^a-x^b$, $t$, and $C^+$ be as above. Then $C^+$ is a subcomplex of $\Delta=\Delta(rad(I))$, and there is a subcomplex of $\Delta$ which is the common link of all maximal simplices of $C^+$.
$T^+$ is the only minimal non-face of $C^+$, so to show that $C^+$ is a subcomplex of $\Delta$, we need to show that $x^{T^+}$ is the only minimal generator of $rad(I)$ with support in $T$.
Suppose $x^c$ is a minimal generator of $rad(I)$, with $supp(c)
\subseteq T$. Then there is some $l \geq 1$ such that $x^{lc} \in I$. Write $c=a^{\prime}+b^{\prime}$, where $supp(a^{\prime}) \subseteq
supp(a)$, and $supp(b^{\prime}) \subseteq supp(b)$. If $supp(a^{\prime}) \neq supp(a)$, then $x^a$ does not divide $x^{lc}$ and so $x^{lc}$ is in the wall ideal $W_{a-b}$. We can choose $\delta$ with $supp(\delta) \subseteq supp(b)$ so that $lc+
\delta=mb+a^{\prime}$ for some $m \geq 1$. Since $x^{mb+a^{\prime}}=x^{lc+\delta} \in W_{a-b}$, it follows that $x^{ma+a^{\prime}} \in W_{a-b}$, because $x^a-x^b \in W_{a-b}$. So $x^{ ma+a^{\prime}} \in I_{flip}$, and there is thus some $p \geq m+1$ such that $x^{pa} \in I_{flip}$. This implies that $x^a \in
rad(I_{flip})$. But, by Lemma \[radsthesame\], this means that $rad(I_{flip})=rad(I)$, which in turn implies that $x^b \in rad(I)$, contradicting our hypothesis. So $supp(a^{\prime}) =
supp(a)=T^+$, and thus $x^{T^+} | x^c$. This shows that $x^{T^+}$ is the only minimal generator of $rad(I)$ with support in $T$, as required. From this we conclude that $C^+$ is a subcomplex of $\Delta$.
We now show that every maximal simplex $\sigma \in C^+ \subseteq
\Delta$ has the same link. We do this by showing that any simplex not in the link of one maximal simplex of $C^+$ is not in the link of any other maximal simplex of $C^+$.
Suppose $\sigma \subseteq \{1,\dots,n\}$ is not a simplex in the link of a maximal simplex $\gamma$ of $C^+ \subseteq \Delta$, where $\gamma=T \setminus \{p\}$ for some $p \in T^+$ and $\sigma \cap T =
\emptyset$. Then $x^{\sigma \cup \gamma} \in rad(I)$, because $\sigma \cup \gamma$ is not a face of $\Delta$, and so there exists $l \geq 1$, and $x^{\alpha}$ a minimal generator of $I$ with $\alpha
\neq a$, such that $x^{\alpha} | (x^{\sigma \cup \gamma})^l$. Write $\alpha=a^{\prime}+b^{\prime}+\sigma^{\prime}$, where $supp(a^{\prime}) \subsetneq supp(a)$, $supp(b^{\prime}) \subseteq
supp(b)$, and $supp(\sigma^{\prime}) \subseteq \sigma$. Choose $\delta$ with $supp(\delta) \subseteq supp(a)$ such that $\alpha+\delta=ma+b^{\prime}+\sigma^{\prime}$ for some $m \geq 0$. Then because $x^{\alpha} \in W_{a-b}$, we have $x^{\alpha+\delta} \in
W_{a-b}$, and so $x^{mb+b^{\prime}+\sigma^{\prime}}$ is in $W_{a-b}$ and thus in $I$. So $x^{supp(b)\cup supp(\sigma^{\prime})} \in rad(I)$. Let $\tau$ be another maximal simplex of $C^+$, so $\tau=(\gamma \cup \{p\})
\setminus \{p^{\prime}\}$ for some $p^{\prime} \in T^+$. Then $supp(b)
\cup supp(\sigma^{\prime}) \subseteq \tau \cup \sigma$, and so $x^{\tau \cup \sigma} \in rad(I)$, and thus $\sigma$ is not in the link of $\tau$ in $\Delta$. This shows that every maximal simplex $\sigma
\in C^+ \subseteq \Delta$ has the same link, as required.
If $x^b \in rad(I)$ then $rad(I)=rad(I_{flip})$ by Lemma \[radsthesame\], and so $\Delta(rad(I))=\Delta(rad(I_{flip}))$.
Suppose $x^b \not \in rad(I)$. Then Lemma \[radsthesame\] implies that $t=a-b$ is a circuit of $\mathcal A$. By Lemma \[ksubcomplex\] $C^+$ is a subcomplex of $\Delta(rad(I))$ with each maximal simplex of $C^+$ having the same link in $\Delta(rad(I))$. It remains to show that $\Delta(rad(I_{flip}))$ is the result of performing a bistellar flip on $\Delta(rad(I))$.
Let $\Delta^{\prime}$ be the result of performing the bistellar flip on $\Delta(rad(I))$ over $t$, and let $M$ be the Stanley-Reisner ideal of $\Delta^{\prime}$.
We claim that M is the squarefree monomial ideal generated by $x^{supp(b)}$, all the generators of $rad(I)$ except $x^{supp(a)}$, and also all monomials of the form $x^{\sigma}$, such that $supp(a) \subseteq
\sigma$, and $\sigma \setminus (T \cap \sigma)$ is not in the link of the maximal simplices of $C^+$. Let $\alpha \subseteq
\{1,\dots,n\}$. Then $\alpha$ is a face of $\Delta^{\prime}$ exactly when either $\alpha$ is a face of $\Delta$ and $T^- \not \subseteq
\alpha$, or $\alpha= T^+ \cup \tau \cup \gamma$, where where $\tau
\subsetneq
T^-$, and $\gamma$ is in the link of the maximal simplices of $C^+$. This means that $\beta \subseteq \{1,\dots,n\}$ is not a face of $\Delta^{\prime}$ exactly when either $T^- \subseteq \beta$, or $\beta$ is not a face of $\Delta$ and $\beta \neq T^+ \cup \tau \cup
\gamma$ for any $\tau \subsetneq T^-$ and $\gamma$ in the link of the maximal simplices of $C^+$. This proves the claim.
We now show that $rad(I_{flip}) \subseteq M$. Let $x^{\alpha}$ be a minimal generator of $I_{flip}$ such that $x^{supp(\alpha)}$ is a minimal generator of $rad(I_{flip})$. If $x^{\alpha}$ is also a minimal generator of $I$, then $x^{supp(\alpha)}$ is in the square free ideal generated by all the generators of $rad(I)$ except $x^{supp(a)}$, so $x^{supp(\alpha)} \in M$. Since $x^{supp(b)} \in
M$, the only case left to consider is $\alpha=a+g$ for some $g \neq 0$ with $b \not \leq g$. Write $g=a^{\prime}+b^{\prime}+\gamma$, where $supp(a^{\prime}) \subseteq supp(a)$, $supp(b^{\prime}) \subsetneq
supp(b)$, and $supp(\gamma) \cap T = \emptyset$. Choose $\delta$ so that $\delta+a^{\prime}=la+\tilde{a}$ for some $l \geq 0$, where $supp(\tilde{a})=T^+\setminus \{p\}$ for some $p \in T^+$. Since $x^{\alpha}$ is a minimal generator of $I_{flip}$ different from $x^a$, it is in $W_{a-b}$. It thus follows that $x^{a+g+\delta} \in
W_{a-b}$, and so, because $x^a-x^b \in W_{a-b}$, we have $x^{(l+1)b+\tilde{a}+b^{\prime}+\gamma} \in W_{a-b}$ and thus in $I$. Since $supp((l+1)b+\tilde{a}+b^{\prime})=T \setminus \{p\}$, $x^{(T\setminus \{p\}) \cup supp(\gamma)} \in rad(I)$ and thus $supp(\gamma)$ is not in the link of the maximal simplices of $C^+$. Because $supp(\gamma)=supp(\alpha) \setminus T$, this means $x^{\alpha} \in M$, and therefore $rad(I_{flip}) \subseteq M$.
Now because $\Delta(rad(I_{flip}))$ and $\Delta^{\prime}$ are both triangulations of $\mathcal A$, this inclusion cannot be proper. So $M=rad(I_{flip})$, and thus $\Delta(rad(I_{flip}))$ is the result of performing a bistellar flip on $\Delta(rad(I))$.
Toward a disconnected toric Hilbert scheme
==========================================
We conclude with some results that suggest the existence of a toric Hilbert scheme. As mentioned earlier, Santos [@Santos] has recently constructed a six dimensional point configuration with 324 points for which there is a triangulation that admits no bistellar flips. Hence this configuration has a disconnected Baues graph. By the results in [@berndpreprint] and the previous section, every monomial $A$-graded ideal $I$ is supported on a triangulation of $\mathcal A$ via the correspondence $I \mapsto
\Delta(rad(I))$, and if two monomial $A$-graded ideals are adjacent in the flip graph of $A$, then either they are supported on the same triangulation or on two triangulations that are adjacent in the Baues graph of $A$. Just as for monomial $A$-graded ideals, there is a notion of coherence for triangulations of ${\mathcal A}$. Every [*coherent*]{} triangulation of $\mathcal A$ (often called a [*regular*]{} triangulation in the literature) supports at least one monomial $A$-graded ideal, and at least one of these ideals is coherent (see Chapter 8 in [@GBCP]). On the other hand, Peeva has shown that if a triangulation of $\mathcal A$ is [*non-coherent*]{}/[*non-regular*]{} then there may be no monomial $A$-graded ideal whose radical is the Stanley-Reisner ideal of this triangulation (see Theorem 10.3 in [@GBCP]). Hence in order for Santos’ example to lift to an example of a disconnected toric Hilbert scheme, it suffices to show that there is a monomial $A$-graded ideal whose radical is the Stanley-Reisner ideal of his isolated (non-regular) triangulation. A straightforward search for such a monomial $A$-graded ideal from his $6 \times 324$ matrix is, however, a daunting computational endeavor. Nonetheless, Santos’ disconnected Baues graph seems to be evidence in favor of a disconnected flip graph.
Recall that every coherent monomial $A$-graded ideal has at least $n-d$ neighbors in the flip graph of $A$. We say that a monomial $A$-graded ideal is [*flip deficient*]{} if its valency in the flip graph of $A$ is strictly less than $n-d$. All flip deficient monomial $A$-graded ideals are necessarily non-coherent. Before Santos constructed an isolated triangulation, discrete geometers provided several examples of flip deficient triangulations (triangulations with valency less that $n-d$ in the Baues graph) as evidence in support of the existence of a disconnected Baues graph. We provide examples of flip deficient monomial $A$-graded ideals.
For each matrix $A(n) := [1 \,\,2\,\,3\,\,7\,\,8\,\,9\,\,a_7 \cdots
a_n]$ with $a_i \in {\mathbb N}$ and $9 < a_7 < \cdots < a_n$, there is a monomial $A(n)$-graded ideal with at most $n-3 < n-1 =
corank(A(n))$ flips.
For the matrix $A = [1\,\,2\,\,3\,\,7\,\,8\,\,9]$, the monomial ideal $J = \langle x_1x_5, x_2x_4, x_1x_4, x_1x_2, x_4x_6, x_2x_6, x_1x_6,
x_3x_4, x_2^2x_3, x_1x_3, x_2x_5^2, x_2^2x_5, x_1^2,\\ x_3^2, x_2^4,
x_3x_5^3, x_4^2x_5^2, x_4^3, x_5^6, x_4x_5^4
\rangle$ is $A$-graded. The flippable binomials of $J$ are $x_5^6-x_3x_6^5, \, x_2x_6-x_3x_5$ and $x_3^2-x_2^3$. In this example, there are 2910 monomial $A$-graded ideals in total and the flip graph of $A$ is connected.
Consider the monomial ideal $J' = J + \langle x_7, \ldots, x_n \rangle
\subseteq k[x_1,\ldots,x_n]$ and a degree $b \in {\mathbb N}A(n) =
{\mathbb N}A = {\mathbb N}$. All the monomials in $k[x_1,\ldots, x_n]$ of $A(n)$-degree $b$ that are divisible by at least one of $x_7,
\ldots, x_n$ are in $J'$ by construction. Among the monomials in $k[x_1, \ldots, x_6]$ of degree $b$ (there is at least one such since $b \in {\mathbb N}A$), there is precisely one that is not in $J$ and hence not in $J'$ and hence $J'$ is $A(n)$-graded. If $x^a - x^b \in
k[x_1, \ldots, x_6]$ is flippable for $J'$ then $in_{x^a \succ x^b}(
\langle x^a - x^b \rangle + \langle x^c : x^c$ minimal generator of $J$, $c \neq a \rangle + \langle x_7, \ldots, x_n \rangle)$ = $J'$. The only non-trivial $S$-pairs that are produced during this calculation are those between $x^a-x^b$ and a monomial minimal generator $x^c$ of $J$. Since the resulting initial ideal equals $J'$, it follows that $in_{x^a \succ x^b} ( \langle x^a - x^b \rangle +
\langle x^c : x^c$ minimal generator of $J$, $c \neq a \rangle) = J$ and hence $x^a - x^b$ is flippable for $J$. So $x^a - x^b$ must be one of the three flippable binomials of $J$. Additionally, each of the minimal generators $x_7, \ldots, x_n$ of $J'$ provides a flippable binomial and hence $J'$ has $3+(n-6) = n-3$ flippable binomials.
We have not found matrices of corank three with flip deficiency in our experiments. However, flip deficiency occurs in corank four. Consider $A = [3\,\,6\,\,8\,\,10\,\,15]$ and its monomial $A$-graded ideal $$\langle ae, bd, ab^2, be, a^2, d^2, e^2, b^3, abc^2 \rangle.$$ The neighboring monomial $A$-graded ideals are:\
$\langle ae, bd, ab^2, be, de, a^2, d^2, e^2, b^3 \rangle$ from $de-abc^2$,\
$\langle ae, bd, ab^2, b^2e, a^2, d^2, e^2, b^3, acd, abc^2 \rangle$ from $acd-be$, and\
$\langle ae, bd, ab^2, ad^2, be, b^2c, a^2, d^3, e^2, b^3, abc^2, d^2e
\rangle$ from $b^2c-d^2$.
The above computations were made using two different programs. Starting with a monomial initial ideal of the toric ideal $I_A$ one can compute all monomial $A$-graded ideals in the same connected component as this initial ideal by using the results in Section 2 to calculate all the neighbors of a monomial $A$-graded ideal. This computation can be done using the program TiGERS [@HuT] with the command [tigers -iAe filename]{} where [filename]{} is the standard input file for [TiGERS]{} with the data of the matrix $A$. In order to find all monomial $A$-graded ideals, we resort to a second program (available from the authors) that first computes the Graver basis of $A$ and then systematically constructs weakly $A$-graded monomial ideals by choosing one monomial from each Graver binomial to be in the ideal (cf. Lemma \[UGB\]). The program then compares the Hilbert series of each such ideal against that of an initial ideal of $I_A$ to decide if it is A-graded. Comparing the total number of ideals produced by the two programs gives a convenient way to decide if the flip graph is connected.
We conclude with an algorithmic issue concerning the enumeration of all $A$-graded monomial ideals in the same connected component as a fixed one. The main program in TiGERS enumerates the vertices of the state polytope of $I_A$ by using the [*reverse search*]{} strategy of Avis and Fukuda [@AF], which requires only the current vertex to be stored at any given time. The input to the program is any one monomial initial ideal of $I_A$ from which the program reconstructs all the others without needing to consult the list of ideals it has already found. An essential requirement of this algorithm is a method by which the input ideal can be distinguished from any other monomial initial ideal of $I_A$ by considering only the edges of the state polytope. This is done in TiGERS as follows:
Suppose $M_1$ and $M_2$ are two monomial initial ideals of $I_A$ induced by the weight vectors $w_1$ and $w_2$ respectively. Let ${\mathcal G}_1$ and ${\mathcal G}_2$ be the corresponding reduced Gröbner bases of $I_A$. Then for each facet binomial $x^a-x^b$ in ${\mathcal G}_1$ we have $w_1 \cdot (a-b) >
0$ and for each facet binomial $x^{\alpha}-x^{\beta} \in {\mathcal
G}_2$ we have $w_2 \cdot (\alpha-\beta) > 0$. The reduced Gröbner bases ${\mathcal G}_1$ and ${\mathcal G}_2$ coincide if and only if each facet binomial $x^{\alpha}-x^{\beta}$ of ${\mathcal G}_2$ satisfies the inequality $w_1 \cdot (\alpha - \beta) > 0$. Suppose the input is a fixed initial ideal of $I_A$. By the previous observation, every other monomial initial ideal of $I_A$ will have a mismarked facet binomial with respect to this term order and hence can be distinguished from the input ideal. The following example shows that monomial $A$-graded ideals cannot be distinguished by checking the orientation of their flippable binomials.
Consider $A=[3\,\,4\,\,5\,\,13\,\,14]$ and its non-coherent monomial $A$-graded ideal $$M = \langle cd^5, c^2e^3, be, d^9, b^2, c^3, a^6, bd, ae^2, ad^3,
ac^2, a^2d, a^2b, bc, a^3e, a^3c \rangle.$$ The flippable binomials of $M$ are $ae^2-cd^2$, $c^3-a^5$ and $d^9-ce^8$. With respect to the weight vector $w =
(0,0,1,20,22)$, each of these flippable binomials has its positive term as leading term and hence $M$ cannot be distinguished from $in_{w}(I_A)$ by checking whether its flippable binomials are mismarked with respect to $w$.
Acknowledgments
===============
We would like to thank Bernd Sturmfels for helpful conversations.
[10]{}
V. I. Arnold. ${A}$-graded algebras and continued fractions. , 42(7):993–1000, 1989.
D. Avis and K. Fukuda. A basis enumeration algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. , 8:295–313, 1992.
D. Eisenbud and B. Sturmfels. Binomial ideals. , 84(1):1–45, 1996.
G. Ewald. . Springer-Verlag, New York, 1996.
W. Fulton. . Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry.
V. Gasharov and I. Peeva. Deformations of codimension 2 toric varieties. Preprint. Available at [ http://math.cornell.edu/$\sim$irena/abstracts]{}.
R. Hartshorne. Connectedness of the [H]{}ilbert scheme. , 29:261–304, 1966.
B. Huber and R.R. Thomas. Computing [G]{}r[ö]{}bner fans of toric ideals. 1999. To appear in Experimental Mathematics. Software : [TiGERS]{} available from [http://www.math.tamu.edu/$\sim$rekha/programs.html]{}.
E. Korkina. Classification of ${A}$-graded algebras with $3$ generators. , 3(1):27–40, 1992.
E. Korkina, G. Post, and M. Roelofs. Classification of generalized [A]{}-graded algebras with 3 generators. , 119:267–287, 1995.
I. Peeva and M. Stillman. Local equations of the toric [H]{}ilbert scheme. Preprint, 1999.
I. Peeva and M. Stillman. Toric [H]{}ilbert schemes. Preprint, 1999. Available at [ http://math.cornell.edu/$\sim$irena/publications.html]{}.
V. Reiner. The generalised [B]{}aues problem. In L. Billera, A. Bj[ö]{}rner, C. Greene, R. Simion, and R. Stanley, editors, [*New Perspectives in Algebraic Combinatorics*]{}. Cambridge University Press, 1999.
F. Santos. A point configuration whose space of triangulations is disconnected. Preprint, 1999.
R.P. Stanley. . Birkhäuser Boston Inc., Boston, MA, second edition, 1996.
B. Sturmfels. The geometry of ${A}$-graded algebras. (alg-geom/9410032).
B. Sturmfels. . American Mathematical Society, Providence, RI, 1996.
B. Sturmfels and R.R. Thomas. Variation of cost functions in integer programming. , 77(3, Ser. A):357–387, 1997.
|
---
abstract: 'Artistic style transfer is the problem of synthesizing an image with content similar to a given image and style similar to another. Although recent feed-forward neural networks can generate stylized images in real-time, these models produce a single stylization given a pair of style/content images, and the user doesn’t have control over the synthesized output. Moreover, the style transfer depends on the hyper-parameters of the model with varying “optimum" for different input images. Therefore, if the stylized output is not appealing to the user, she/he has to try multiple models or retrain one with different hyper-parameters to get a favorite stylization. In this paper, we address these issues by proposing a novel method which allows adjustment of crucial hyper-parameters, after the training and in real-time, through a set of manually adjustable parameters. These parameters enable the user to modify the synthesized outputs from the same pair of style/content images, in search of a favorite stylized image. Our quantitative and qualitative experiments indicate how adjusting these parameters is comparable to retraining the model with different hyper-parameters. We also demonstrate how these parameters can be randomized to generate results which are diverse but still very similar in style and content.'
author:
- Mohammad Babaeizadeh
- Golnaz Ghiasi
bibliography:
- 'egbib.bib'
title: 'Adjustable Real-time Style Transfer'
---
Introduction
============
Style transfer is a long-standing problem in computer vision with the goal of synthesizing new images by combining the *content* of one image with the *style* of another [@efros2001image; @hertzmann1998painterly; @ashikhmin2001synthesizing]. Recently, neural style transfer techniques [@gatys2015texture; @gatys2016image; @johnson2016perceptual; @ghiasi2017exploring; @li2018closed; @li2017universal] showed that the correlation between the features extracted from the trained deep neural networks is quite effective on capturing the visual styles and content that can be used for generating images *similar* in style and content. However, since the definition of similarity is inherently vague, the objective of style transfer is not well defined [@dumoulin2017learned] and one can imagine multiple stylized images from the same pair of content/style images.
Existing real-time style transfer methods generate only one stylization for a given content/style pair and while the stylizations of different methods usually look distinct [@sanakoyeu2018style; @huang2017arbitrary], it is not possible to say that one stylization is better in all contexts since people react differently to images based on their background and situation. Hence, to get favored stylizations users must try different methods that is not satisfactory. It is more desirable to have a single model which can generate *diverse* results, but still *similar* in style and content, in real-time, by adjusting some input parameters.
One other issue with the current methods is their high sensitivity to the hyper-parameters. More specifically, current real-time style transfer methods minimize a weighted sum of losses from different layers of a pre-trained image classification model [@johnson2016perceptual; @huang2017arbitrary] (check Sec 3 for details) and different weight sets can result into very different styles (Figure \[fig:fig\_compare\]). However, one can only observe the effect of these weights in the final stylization by fully retraining the model with the new set of weights. Considering the fact that the “optimal" set of weights can be different for any pair of style/content (Figure \[fig:fig\_weights\]) and also the fact that this “optimal" truly doesn’t exist (since the goodness of the output is a personal choice) retraining the models over and over until the desired result is generated is not practical.
The primary goal of this paper is to address these issues by providing a novel mechanism which allows for adjustment of the stylized image, in ***real-time*** and ***after*** training. To achieve this, we use an auxiliary network which accepts additional parameters as inputs and changes the style transfer process by adjusting the weights between multiple losses. We show that changing these parameters at inference time results to stylizations similar to the ones achievable by retraining the model with different hyper-parameters. We also show that a random selection of these parameters at run-time can generate a random stylization. These solutions, enable the end user to be in full control of how the stylized image is being formed as well as having the capability of generating multiple stochastic stylized images from a fixed pair of style/content. The stochastic nature of our proposed method is most apparent when viewing the transition between random generations. Therefore, we highly encourage the reader to check the project website to view the generated stylizations.
Related Work
============
The strength of deep networks in style transfer was first demonstrated by Gatys et al. [@gatys2016image]. While this method generates impressive results, it is too slow for real-time applications due to its optimization loop. Follow up works speed up this process by training feed-forward networks that can transfer style of a single style image [@johnson2016perceptual; @ulyanov2016texture] or multiple styles [@dumoulin2017learned]. Other works introduced real-time methods to transfer style of arbitrary style image to an arbitrary content image [@ghiasi2017exploring; @huang2017arbitrary]. These methods can generate different stylizations from different style images; however, they only produce one stylization for a single pair of content/style image which is different from our proposed method.
Generating diverse results have been studied in multiple domains such as colorizations [@deshpande2017learning; @cao2017unsupervised], image synthesis [@chen2017photographic], video prediction [@babaeizadeh2017stochastic; @lee2018stochastic], and domain transfer [@huang2018multimodal; @zhang2018xogan]. Domain transfer is the most similar problem to the style transfer. Although we can generate multiple outputs from a given input image [@huang2018multimodal], we need a collection of target or style images for training. Therefore we can not use it when we do not have a collection of similar styles.
Style loss function is a crucial part of style transfer which affects the output stylization significantly. The most common style loss is Gram matrix which computes the second-order statistics of the feature activations [@gatys2016image], however many alternative losses have been introduced to measure distances between feature statistics of the style and stylized images such as correlation alignment loss [@peng2018synthetic], histogram loss [@risser2017stable], and MMD loss [@li2017demystifying]. More recent work [@liu2017depth] has used depth similarity of style and stylized images as a part of the loss. We demonstrate the success of our method using only Gram matrix; however, our approach can be expanded to utilize other losses as well.
To the best of our knowledge, the closest work to this paper is [@ulyanov2017improved] in which the authors utilized Julesz ensemble to encourage diversity in stylizations explicitly. Although this method generates different stylizations, they are very similar in style, and they only differ in minor details. A qualitative comparison in Figure \[fig:eagle\] shows that our proposed method is more effective in diverse stylization.
![Effect of adjusting the style weight in style transfer network from [@johnson2016perceptual]. Each column demonstrates the result of a separate training with all $w_s^l$ set to the printed value. As can be seen, the “optimal" weight is different from one style image to another and there can be multiple “good" stylizations depending on ones’ personal choice. Check supplementary materials for more examples.[]{data-label="fig:fig_weights"}](fig_weights.pdf){width="1.0\columnwidth"}
Background
==========
Style transfer using deep networks
----------------------------------
Style transfer can be formulated as generating a stylized image ${\mathbf{p}}$ which its content is similar to a given content image ${\mathbf{c}}$ and its style is close to another given style image ${\mathbf{s}}$. $${\mathbf{p}}=\Psi({\mathbf{c}}, {\mathbf{s}})$$
The similarity in style can be vaguely defined as sharing the same spatial statistics in low-level features, while similarity in content is roughly having a close Euclidean distance in high-level features [@ghiasi2017exploring]. These features are typically extracted from a pre-trained image classification network, commonly VGG-19 [@simonyan2014very]. The main idea here is that the features obtained by the image classifier contain information about the content of the input image while the correlation between these features represents its style.
In order to increase the similarity between two images, Gatys et al. [@gatys2016image] minimize the following distances between their extracted features: $$\begin{aligned}
{\mathcal{L}}^l_c({\mathbf{p}}) &= \big|\big|\phi^l({\mathbf{p}})-\phi^l({\mathbf{s}})\big|\big|^2_2 \\
{\mathcal{L}}^l_s({\mathbf{p}}) &= \big|\big|G(\phi^l({\mathbf{p}}))-G(\phi^l({\mathbf{s}}))\big|\big|^2_F\end{aligned}$$ where $\phi^l({\mathbf{x}})$ is activation of a pre-trained classification network at layer $l$ given the input image ${\mathbf{x}}$, while ${\mathcal{L}}^l_c({\mathbf{p}})$ and ${\mathcal{L}}^l_s({\mathbf{p}})$ are content and style loss at layer $l$ respectively. $G(\phi^l({\mathbf{p}}))$ denotes the Gram matrix associated with $\phi^l({\mathbf{p}})$.
The total loss is calculated as a weighted sum of losses across a set of *content layers ${\mathit{C}}$* and *style layers ${\mathit{S}}$*: $$\begin{aligned}
{\mathcal{L}}_c({\mathbf{p}}) &= \sum_{l \in {\mathit{C}}} w^l_c{\mathcal{L}}^l_c({\mathbf{p}}) \text{\space and \space} {\mathcal{L}}_s({\mathbf{p}}) &= \sum_{l \in {\mathit{S}}} w^l_s{\mathcal{L}}^l_s({\mathbf{p}})
\label{eqn:losses}\end{aligned}$$ where $w^l_c$, $ w^l_s$ are hyper-parameters to adjust the contribution of each layer to the loss. Layers can be shared between ${\mathit{C}}$ and ${\mathit{S}}$. These hyper-parameters have to be manually fine tuned through try and error and usually vary for different style images (Figure \[fig:fig\_weights\]). Finally, the objective of style transfer can be defined as: $$\begin{aligned}
\min_{\mathbf{p}}\big({\mathcal{L}}_c({\mathbf{p}})+{\mathcal{L}}_s({\mathbf{p}})\big)
\label{eqn:objective}\end{aligned}$$ This objective can be minimized by iterative gradient-based optimization methods starting from an initial ${\mathbf{p}}$ which usually is random noise or the content image itself.
Real-time feed-forward style transfer
-------------------------------------
Solving the objective in Equation \[eqn:objective\] using an iterative method can be very slow and has to be repeated for any given pair of style/content image. A much faster method is to directly train a deep network $T$ which maps a given content image ${\mathbf{c}}$ to a stylized image ${\mathbf{p}}$ [@johnson2016perceptual]. $T$ is usually a feed-forward convolutional network (parameterized by $\theta$) with residual connections between down-sampling and up-sampling layers [@ruder2018artistic] and is trained on many content images using Equation \[eqn:objective\] as the loss function: $$\begin{aligned}
\min_\theta\big({\mathcal{L}}_c(T({\mathbf{c}}))+{\mathcal{L}}_s(T({\mathbf{c}}))\big) \end{aligned}$$ The style image is assumed to be fixed and therefore a different network should be trained per style image. However, for a fixed style image, this method can generate stylized images in real-time [@johnson2016perceptual]. Recent methods [@dumoulin2017learned; @ghiasi2017exploring; @huang2017arbitrary] introduced real-time style transfer methods for multiple styles. But, these methods still generate only one stylization for a pair of style and content images.
Proposed Method
===============
Problem Statement
-----------------
In this paper we address the following issues in real-time feed-forward style transfer methods:\
1. The output of these models is sensitive to the hyper-parameters $w^l_c$ and $ w^l_s$ and different weights significantly affect the generated stylized image as demonstrated in Figure \[fig:fig\_compare\]. Moreover, the “optimal" weights vary from one style image to another (Figure \[fig:fig\_weights\]) and therefore finding a good set of weights should be repeated for each style image. Please note that for each set of $w^l_c$ and $ w^l_s$ the model has to be fully retrained that limits the practicality of style transfer models.\
2. Current methods generate a *single* stylized image given a content/style pair. While the stylizations of different methods usually look very distinct [@sanakoyeu2018style], it is not possible to say which stylization is better for every context since it is a matter of personal taste. To get a favored stylization, users may need to try different methods or train a network with different hyper-parameters which is not satisfactory and, ideally, the user should have the capability of getting different stylizations in real-time.
We address these issues by conditioning the generated stylized image on additional input parameters where each parameter controls the share of the loss from a corresponding layer. This solves the problem (1) since one can adjust the contribution of each layer to adjust the final stylized result after the training and in real-time. Secondly, we address the problem (2) by randomizing these parameters which result in different stylizations.
Style transfer with adjustable loss
-----------------------------------
We enable the users to adjust $w^l_c$,$w^l_s$ without retraining the model by replacing them with input parameters and conditioning the generated style images on these parameters: $${\mathbf{p}}=\Psi({\mathbf{c}}, {\mathbf{s}}, {{\pmb{\alpha}}}_c, {{\pmb{\alpha}}}_s)$$ ${{\pmb{\alpha}}}_c$ and ${{\pmb{\alpha}}}_s$ are vectors of parameters where each element corresponds to a different layer in content layers ${\mathit{C}}$ and style layers ${\mathit{S}}$ respectively. $\alpha^l_c$ and $\alpha^l_s$ replace the hyper-parameters $w^l_c$ and $w^l_s$ in the objective Equation \[eqn:losses\]: $$\begin{aligned}
{\mathcal{L}}_c({\mathbf{p}}) &= \sum_{l \in {\mathit{C}}} \alpha^l_c{\mathcal{L}}^l_c({\mathbf{p}}) \text{\space and \space} {\mathcal{L}}_s({\mathbf{p}}) &= \sum_{l \in {\mathit{S}}} \alpha^l_s{\mathcal{L}}^l_s({\mathbf{p}})
\label{eqn:alphalosses}\end{aligned}$$ To learn the effect of ${{\pmb{\alpha}}}_c$ and ${{\pmb{\alpha}}}_s$ on the objective, we use a technique called *conditional instance normalization* [@ulyanovinstance]. This method transforms the activations of a layer $x$ in the feed-forward network $T$ to a normalized activation $z$ which is conditioned on additional inputs ${{\pmb{\alpha}}}=[{{\pmb{\alpha}}}_c, {{\pmb{\alpha}}}_s]$: $$\begin{aligned}
z=\gamma_{{\pmb{\alpha}}}\big(\frac{x-\mu}{\sigma}\big)+\beta_{{\pmb{\alpha}}}\label{eq:style_params}\end{aligned}$$ where $\mu$ and $\sigma$ are mean and standard deviation of activations at layer $x$ across spatial axes [@ghiasi2017exploring] and $\gamma_{{\pmb{\alpha}}},\beta_{{\pmb{\alpha}}}$ are the learned mean and standard deviation of this transformation. These parameters can be approximated using a second neural network which will be trained end-to-end with $T$: $$\begin{aligned}
\gamma_{{\pmb{\alpha}}},\beta_{{\pmb{\alpha}}}=\Lambda({{\pmb{\alpha}}}_c,{{\pmb{\alpha}}}_s)\end{aligned}$$ Since ${\mathcal{L}}^l$ can be very different in scale, one loss term may dominate the others which will fail the training. To balance the losses, we normalize them using their exponential moving average as a normalizing factor, i.e. each ${\mathcal{L}}^l$ will be normalized to: $$\begin{aligned}
{\mathcal{L}}^l({\mathbf{p}})=\frac{\sum_{i\in{\mathit{C}}\cup{\mathit{S}}}\overline{{\mathcal{L}}^i}({\mathbf{p}})}{\overline{{\mathcal{L}}^l}({\mathbf{p}})} * {\mathcal{L}}^l({\mathbf{p}})\end{aligned}$$ where $\overline{{\mathcal{L}}^l}({\mathbf{p}})$ is the exponential moving average of ${\mathcal{L}}^l({\mathbf{p}})$.
Experiments
===========
In this section, first we study the effect of adjusting the input parameters in our method. Then we demonstrate that we can use our method to generate random stylizations and finally, we compare our method with a few baselines in terms of generating random stylizations.
Implementation details
----------------------
We implemented $\Lambda$ as a multilayer fully connected neural network. We used the same architecture as [@johnson2016perceptual; @dumoulin2017learned; @ghiasi2017exploring] for $T$ and only increased number of residual blocks by 3 (look at supplementary materials for details) which improved stylization results. We trained $T$ and $\Lambda$ jointly by sampling random values for ${{\pmb{\alpha}}}$ from $U(0,1)$. We trained our model on ImageNet [@deng2009imagenet] as content images while using paintings from Kaggle Painter by Numbers [@Kaggle] and textures from Descibable Texture Dataset [@cimpoi14describing] as style images. We selected random images form ImageNet test set, MS-COCO [@lin2014microsoft] and faces from CelebA dataset [@liu2018large] as our content test images. Similar to [@ghiasi2017exploring; @dumoulin2017learned], we used the last feature set of $conv3$ as content layer ${\mathit{C}}$. We used last feature set of $conv2$, $conv3$ and $conv4$ layers from VGG-19 network as style layers ${\mathit{S}}$. Since there is only one content layer, we fix ${{\pmb{\alpha}}}_c=1$. Our implementation can process $47.5$ fps on a NVIDIA GeForce 1080, compared to $52.0$ for the base model without $\Lambda$ sub-network.
Effect of adjusting the input parameters
----------------------------------------
The primary goal of introducing the adjustable parameters ${{\pmb{\alpha}}}$ was to modify the loss of each separate layer manually. Qualitatively, this is demonstrable by increasing one of the input parameters from zero to one while fixing the rest of them to zero. Figure \[fig:gradient\] shows one example of such transition. Each row in this figure is corresponding to a different style layer, and therefore the stylizations at each row would be different. Notice how deeper layers stylize the image with *bigger* stylization elements from the style image but all of them still apply the coloring. We also visualize the effect of increasing two of the input parameters at the same time in Figure \[fig:fig\_results\_grad\]. However, these transitions are best demonstrated interactively which is accessible at the project website .
To quantitatively demonstrate the change in losses with adjustment of the input parameters, we rerun the same experiment of assigning a fixed value to all of the input parameters while gradually increasing one of them from zero to one, this time across 100 different content images. Then we calculate the median loss at each style loss layer ${\mathit{S}}$. As can be seen in Figure \[fig:fig\_loss\_changes\]-(top), increasing $\alpha_s^l$ decreases the measured loss corresponding to that parameter. To show the generalization of our method across style images, we trained 25 models with different style images and then measured median of the loss at any of the ${\mathit{S}}$ layers for 100 different content images (Figure \[fig:fig\_loss\_changes\])-(bottom). We exhibit the same drop trends as before which means the model can generate stylizations conditioned on the input parameters.
Finally, we verify that modifying the input parameters ${{\pmb{\alpha}}}_s$ generates visually similar stylizations to the retrained base model with different loss weights $w^l_s$. To do so, we train the base model [@johnson2016perceptual] multiple times with different $w_s^l$ and then compare the generated results with the output of our model when $\forall l\in{\mathit{S}}, \space \alpha_s^l=w^l_s$. Figure \[fig:fig\_compare\] demonstrates this comparison. Note how the proposed stylizations in test time and without retraining match the output of the base model.
Generating randomized stylizations
----------------------------------
One application of our proposed method is to generate multiple stylizations given a fixed pair of content/style image. To do so, we randomize ${{\pmb{\alpha}}}$ to generate randomized stylization (top row of Figure \[fig:noise\]). Changing values of ${{\pmb{\alpha}}}$ usually do not randomize the position of the “elements" of the style. We can enforce this kind of randomness by adding some noise with the small magnitude to the content image. For this purpose, we multiply the content image with a mask which is computed by applying an inverse Gaussian filter on a white image with a handful ($< 10$) random zeros. This masking can shadow sensitive parts of the image which will change the spatial locations of the “elements" of style. Middle row in Figure \[fig:noise\] demonstrates the effect of this randomization. Finally, we combine these two randomizations to maximizes the diversity of the output which is shown in the bottom row of Figure \[fig:noise\]. More randomized stylizations can be seen in Figure \[fig:fig\_results16\] and at .
### Comparison with other methods
To the best of our knowledge, generating diverse stylizations at real-time is only have been studied at [@ulyanov2017improved] before. In this section, we qualitatively compare our method with this baseline. Also, we compare our method with a simple baseline where we add noise to the style parameters.
The simplest baseline for getting diverse stylizations is to add noises to some parameters or the inputs of the style-transfer network. In the last section, we demonstrate that we can move the locations of elements of style by adding noise to the content input image. To answer the question that if we can get different stylizations by adding noise to the style input of the network, we utilize the model of [@dumoulin2017learned] which uses conditional instance normalization for transferring style. We train this model with only one style image and to get different stylizations, we add random noise to the style parameters ($\gamma_{{\pmb{\alpha}}}$ and $\beta_{{\pmb{\alpha}}}$ parameters of equation \[eq:style\_params\]) at run-time. The stylization results for this baseline are shown on the top row of Figure \[fig:eagle\]. While we get different stylizations by adding random noises, the stylizations are no longer similar to the input style image.
To enforce similar stylizations, we trained the same baseline while we add random noises at the training phase as well. The stylization results are shown in the second row of Figure \[fig:eagle\]. As it can be seen, adding noise at the training time makes the model robust to the noise and the stylization results are similar. This indicates that a loss term that encourages diversity is necessary.
We also compare the results of our model with StyleNet [@ulyanov2017improved]. As visible in Figure \[fig:eagle\], although StyleNet’s stylizations are different, they vary in minor details and all carry the same level of stylization elements. In contrast, our model synthesizes stylized images with varying levels of stylization and more randomization.
Conclusion
==========
Our main contribution in this paper is a novel method which allows adjustment of each loss layer’s contribution in feed-forward style transfer networks, in real-time and after training. This capability allows the users to adjust the stylized output to find the favorite stylization by changing input parameters and without retraining the stylization model. We also show how randomizing these parameters plus some noise added to the content image can result in very different stylizations from the same pair of style/content image.
Our method can be expanded in numerous ways e.g. applying it to multi-style transfer methods such as [@dumoulin2017learned; @ghiasi2017exploring], applying the same parametrization technique to randomize the correlation loss between *the features of each layer* and finally using different loss functions and pre-trained networks for computing the loss to randomize the outputs even further. One other interesting future direction is to apply the same “loss adjustment after training" technique for other classic computer vision and deep learning tasks. Style transfer is not the only task in which modifying the hyper-parameters can greatly affect the predicted results and it would be rather interesting to try this method for adjusting the hyper-parameters in similar problems.
Operation input dimensions output dimensions
------------------------------------- ------------------ ------------------------------------------------------ -- -- -- --
input parameters ${{\pmb{\alpha}}}$ $3$ $1000$
$10 \times $Dense $1000$ $1000$
Dense $1000$ $2 (\gamma_{{\pmb{\alpha}}},\beta_{{\pmb{\alpha}}})$
Optimizer
Training iterations
Batch size
Weight initialization
Operation Kernel size Stride Feature maps Padding Nonlinearity
--------------------------------------------------- ------------- -------- -------------- --------- -------------- --
[**Network**]{} – $256 \times 256 \times 3$ input
Convolution $9$ $1$ $32$ SAME ReLU
Convolution $3$ $2$ $64$ SAME ReLU
Convolution $3$ $2$ $128$ SAME ReLU
Residual block $128$
Residual block $128$
Residual block $128$
Residual block $128$
Residual block $128$
Residual block $128$
Residual block $128$
Upsampling $64$
Upsampling $32$
Convolution $9$ $1$ $3$ SAME Sigmoid
[**Residual block**]{} – $C$ feature maps
Convolution $3$ $1$ $C$ SAME ReLU
Convolution $3$ $1$ $C$ SAME Linear
[**Upsampling**]{} – $C$ feature maps
Convolution $3$ $1$ $C$ SAME ReLU
Normalization
Optimizer
Training iterations
Batch size
Weight initialization
{width="100.00000%"}
{width="100.00000%"}
{width="100.00000%"}
|
---
author:
- |
Omer Faruk Gulban\
Maastricht University
bibliography:
- 'references.bib'
title: The relation between color spaces and compositional data analysis demonstrated with magnetic resonance image processing applications
---
Introduction
============
Compositional data analysis can be applied to color images in the context of image processing and analysis. Color images are stored as triplets of non-negative integers representing the additive primary colors red, green and blue (referred as RGB channels). Once the color image is formed, the transformation from RGB to hue, saturation and intensity (HSI) coordinate system is an often used first step to improve the image visualization [@Smith1978; @Ledley1990; @Grasso1993] (for the mathematical expression of this transformation see Appendix \[RGB2HSI\]). HSI color space is commonly used in computer applications because of the commonly accepted property of its components relating to human color perception [@Joblove1978; @Levkowitz1993; @Pohl2016]. Hue relates to the dominant component of the primary colors (red, green, blue) in the mixture, saturation relates to the distance from an equal mixture (gray), and intensity relates to distance from total darkness (black to white). In this article, inspiration is drawn from RGB to HSI color space transformation to propose an analogous and more general method based on compositional data analysis. During this process, a simple vector decomposition is outlined to justify the use of compositional data analysis [@Aitchison1982; @Pawlowsky-Glahn2015] and the simplex space is leveraged to manipulate vector fields to perform image enhancement. The proposed method is demonstrated using a magnetic resonance imaging (MRI) dataset. The results are also visualized using a digital color photograph to provide a general intuition. Due to the interdisciplinary nature of this work a glossary is provided in Appendix \[glossary\] to accompany readers from different fields.
Materials and Methods
=====================
MRI data acquisition and preprocessing {#data_and_preproc}
--------------------------------------
Whole head T1 weighted (T1w), proton density weighted (PDw) and T2\* weighted (T2\*w) images at 0.7 mm isotropic volumetric cubic element (voxel) resolution were acquired in one male participant using a three dimensional magnetization prepared rapid acquisition gradient echo sequence with a 32-channel head coil (Nova Medical) on a 7 Tesla whole-body scanner (Siemens) in Maastricht Brain Imaging Center. These measurements reflect different intrinsic properties of the tissues, for instance T1w image shows the optimal contrast between white matter and gray matter, PDw image shows the density of the hydrogen atoms, T2\*w image shows the iron content. For the detailed report of the data acquisition parameters see Appendix \[mri\_parameters\]. For a review on ultra high field MRI (($\geq7$ Tesla)) see [@Ugurbil2014].
As a standard preprocessing step, a masking operation was performed based on the PDw image using FSL-BET software (version 2.1; [@Smith2002; @Smith2004]). The resulting volumetric mask was applied to all images in order to discard parts containing non-brain tissues (e.g. bones, muscles, skin, air). This masking step was necessary to reduce the overall processing time in the following analyses by reducing the total number of voxels from 26214400 ($320\times320\times256$) to $\sim4.5$ million (4543582 to be exact). Measurements (T1w, PDw and T2\*w) stored at each one of the $\sim4.5$ million voxels are considered as components of three separate scalar fields constituting a vector field when combined.
Barycentric decomposition {#methods_bary_decomp}
-------------------------
The image analysis illustrated in this paper starts from the following vector decomposition, where the real space of $n$ dimensions is indicated with $\mathbb{R}^n$ and the simplex space is indicated with $\mathbb{S}^n$ symbols: $$\begin{aligned}
\label{bary_decomp}
& \vec{v} = [v_1, v_2, \ldots, v_D] \text{, where } v_1, v_2, \ldots, v_D \in \mathbb{R}^D_{>0},\nonumber\\
& \vec{v} = \frac{1}{k} C(\vec{v}) s \text{, where } C(\vec{v}) \in \mathbb{S}^D \text{, and } s \in \mathbb{R}^1_{>0},\end{aligned}$$ where $\mathbb{R}^D_{>0}$ indicates positive real numbers, $k$ is an arbitrary scalar and the letter $C$ stands for the closure operation used in compositional data analysis [@Aitchison2002; @Pawlowsky-Glahn2015]: $$\label{closure}
C(\vec{v}) = k \frac{\vec{v}}{s}.$$ The letter $s$ is another scalar that is the sum of the vector components: $$\label{intensity}
s = \sum_{i=1}^D v_i.$$ Note that $k$ disappears from the expression in Eq. \[bary\_decomp\] and \[closure\] when selected as one. The decomposition (Eq. \[bary\_decomp\]) of the vector $\vec{v}$ into its barycentric coordinates ($C(\vec{v})$) and a scalar ($s$) allows the application of the compositional data analysis to the barycentric coordinates. Historically, the closure operation was used by August Ferdinand Mobius [@fauvel1993] and the resulting vector was interpreted as relating to the barycenter (center of mass) of a simplex, hence giving the name to the vector decomposition (Eq \[bary\_decomp\]) demonstrated here.
Compositional image analysis {#methods_coda}
----------------------------
In this section, operations performed on the the barycentric coordinates are laid out. These operations are mostly adapted from [@Pawlowsky-Glahn2015] to fit the context of the analyzed vector field. Tensor notation is used for explicit formulations:
Let $A$ be a tensor with three dimensions, $A_{x, y, z}$, where the coordinates $x, y, z$ of an element represents the spatial location. As mentioned the in Section \[data\_and\_preproc\], there are $\sim4.5$ million tensors (i.e. voxels) of interest in total.
Let the superscript of tensor $A$ indicate different MRI measurements acquired at every voxel; $A_{x,y,z}^{T1w}$, $A_{x,y,z}^{PDw}$, $A_{x,y,z}^{T2^*w}$. As the first step, all three tensors are vectorized (flattened): $$\label{coda_start}
\mathsf{vec}(A^t) = [a_{1,1,1},\ \hdots,\ a_{x,1,1}, a_{1,2,1},\ \hdots,\ a_{1,y,1}, a_{1,2,2},\ \hdots,\ a_{1,2,z}]^T.$$ $T$ stands for the transpose operator and $ t \in T1w, PDw, T2^*w$.
Concatenation denoted by $\Vert$ is used to matricize the vectorized tensors as the next step: $$V = \mathsf{vec}(A^{T1w})\ \Vert \ \mathsf{vec}(A^{PDw})\ \Vert \ \mathsf{vec}(A^{T2w}),$$ where number of rows of the matrix $V$ is $x \times y \times z$ ($n = \sim4.5$ million), and the number of columns is $3$. These operations are done for convenient indexing in the following equations.
Let $V$ indicate the set of voxels where the vector $v_i$ is the composition of T1w, PDw and T2\*w measurements: $$V = [v_i,\ \hdots,\ v_{n}] \text{ where } i \in [1, 2,\ \hdots,\ n] \text{ and } v_i = [v_{i}^{T1w}, v_{i}^{PDw}, v_{i}^{T2^*w}].$$ At this stage, $V \in \mathbb{R}^3$. Voxel-wise (i.e. row-wise) closure is applied to $V$ to acquire the barycentric coordinates of every composition: $$\label{barycentric_component}
X = C(V) = [C(v_i),\ \hdots,\ C(v_n)],\ X \in \mathbb{S}^3.$$ The set of compositions $X$ was centered by finding the *sample center* and *perturbing* each composition with the inverse of the sample center: $$\label{centering}
\hat{X} = X \oplus \mathbf{\mathsf{cen}(X)^{-1}},$$ where $\oplus$ denotes the perturbation operator (analogous to addition in real space): $$x \oplus y = C[x_{T1w}\mathbf{y}_1, x_{PDw}\mathbf{y}_2, x_{T2^*w}\mathbf{y}_3].$$ Multipliers of the components are indicated with $\mathbf{y}_1$, $\mathbf{y}_2$ and $\mathbf{y}_3$. The term $\mathsf{cen}(X)$ is a vector that stands for the component-wise (i.e. column-wise) geometric mean across all voxels: $$\mathsf{cen}(X) = [\mathbf{g}_{T1w}, \mathbf{g}_{PDw}, \mathbf{g}_{T2^*w}] = \left( \prod_{i=1}^{n} x_{ij} \right) ^ {1/n}, j = [T1w, PDw, T2^*w].$$ After centering, the data is standardized: $$\label{standardize}
\hat{\hat{X}} = \hat{X} \odot \mathsf{totvar}[X]^{-1/2},$$ where $\odot$ symbol stands for the power operator (analogous to scaling in real space). The exponent $p$ is applied to every component: $$x\odot p = C[x_{T1w}^p, x_{PDw}^p, x_{T2*w}^p],$$ and the total variance is computed by: $$\mathsf{totvar}[X] = \frac{1}{n} \sum_{i=1}^{n} d_a^2(x_i,\ \mathsf{cen}(X)),$$ where $d_a^2$ indicates squared Aitchison distance: $$d_a(x, y) = \sqrt{\frac{1}{2D} \sum_{j=1}^D \sum_{k=1}^D \left( \ln\frac{x_j}{x_k} - \ln\frac{y_j}{y_k} \right)^2}.$$ In the current example $D=3$ because of operating on a set of three part compositions $X$ (see Equation \[barycentric\_component\]).
At this stage some visual intuition could be gained with regards to the vector field (MR images) that is decomposed and processed via compositional data analysis methods. For illustration purposes, this is done by generating virtual image contrasts computed through metrics of simplex space (see Figure \[Fig2\]).
The norm in $S^3$ image in Figure \[Fig2\] is generated by computing Aitchison norm voxel-wise: $$\parallel X \parallel _{a} = [ \parallel x_i \parallel _a,\ \hdots,\ \parallel x_n \parallel _a].$$ The subscript $a$ stands for the vector norm defined in simplex space (analogous to Euclidean norm in real space). Following [@Pawlowsky-Glahn2015], Aitchison norm is defined as: $$\label{a_norm}
\parallel x \parallel _a = \sqrt{\frac{1}{2D} \sum_{j=1}^D \sum_{k=1}^D \left( \ln\frac{x_j}{x_k} \right)^2},$$ where $D = 3$ considering three different MRI measurements (T1w, PDw, T2\*w).
It should be noted that this norm image is analogous to saturation dimension in the aforementioned RGB to HSI color space transformation (see Appendix \[RGB2HSI\]).
The angular difference in $S^3$ image in Figure \[Fig2\] is the voxel-wise angular difference ($\angle$) between the set of compositional vectors ($X$) and a reference vector ($r$): $$X_\measuredangle = [\angle{x_i r},\ \hdots,\ \angle{x_n r}],$$ where $$\label{angular_diff}
\angle{x_ir} = \arccos \left( \frac{\langle x_i, r \rangle _a}{\parallel x_i \parallel _a \parallel r \parallel _a} \right).$$ $\parallel x \parallel _a$ stands for Aitchison norm of the vector $x$ as defined in Equation \[a\_norm\] and $\langle x, r \rangle _a$ stands for inner product in simplex space: $$\langle x, r \rangle _a = \frac{1}{2D} \sum_{j=1}^D \sum_{k=1}^D \ln\frac{x_j}{x_k} \ln\frac{r_j}{r_k}.$$ Here, $D = 3$ again and it should be noted that the choice of the reference vector $r$ is arbitrary. In this example the reference vector is selected as $r = [0.05, 0.9, 0.05]$. This angle difference image is similar to hue dimension in RGB to HSI color space transformation (see Appendix \[RGB2HSI\]).
Recognizing the similarity of norm and angular difference in simplex space to saturation and hue dimensions of HSI color space at this stage leads to the proposal of a color balance algorithm which is used together with centering (Eq. \[centering\]) and standardization (Eq. \[standardize\]) to generate the color image in Figure \[Fig3\] lower row: $$\label{truncate}
\mathsf{truncate}(x) =
\begin{cases}
x \odot \left( \dfrac{\parallel x \parallel _a}{\lambda} \right) &, \parallel x \parallel _a > \lambda \\
x &, \parallel x \parallel _a \leq \lambda
\end{cases}$$ This method is inspired from other simple color balance methods that makes use of the dynamic range of RGB channels of color images [@Limare2011]. Truncate function is useful when when there are a small amount of outliers compositions. In essence this functions pulls the compositions that are too far away from the center of the simplex space towards the center. In the current work, distance threshold was arbitrarily chosen $\lambda = 3$ based on visual inspection of the resulting color enhancement. It is conceivable that other operations based on percentiles can also be used to the make the decision less arbitrary (e.g. $99^{th}$ percentile of Aitchison norm distribution consisting of all vectors of $X$).
The ilr transformation [@Egozcue2003; @Pawlowsky-Glahn2015] was performed voxel-wise to acquire real space coordinates (ilr coordinates) of the set compositions to generate Figure \[Fig3\] right column: $$X_{\mathsf{ilr}} = [\mathsf{ilr}(x_i),\ \hdots,\ \mathsf{ilr}(x_n)],$$ where the ilr transformation is defined as: $$\label{ilr_transformation}
\mathsf{ilr}(x_i) = \ln(x_i) \cdot \mathbf{H}.$$ $\mathbf{H}$ indicates the Helmert sub-matrix, chosen by following [@Tsagris2011; @Lancaster1965]. In this case $\mathbf{H}$ consists of 3 rows and 2 columns and selected as the following: $$\label{helmert_matrix}
\mathbf{H} =
\begin{bmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\
0 & -\sqrt{\frac{2}{3}}
\end{bmatrix}
.$$
To explore the usefulness of the ilr coordinates and probe physically interpretable compositional characteristics, a real-time interactive visualization method is used with joint ilr-coordinates and image space representations of the data. This method is similar to using pre-defined modulation transfer functions for mapping the bins of a 2D histogram data representation to 3D image space used in volume rendering software [@Kniss2005]. Major brain tissues such as gray matter, white matter, cerebrospinal fluid arteries and sinuses are identified manually by the author and delineated in both ilr-coordinates and brain images using neuro-anatomical expertise.
All analysis steps demonstrated in this work are implemented in a free and open source Python package (available at https://github.com/ofgulban/compoda; [@compoda0pt3pt1]) using Numpy [@numpy2011], Scipy [@scipy2001], Matplotlib [@matplotlib2007], Nibabel [@nibabel2017] and scikit-image [@scikit-image] as auxiliary scientific libraries. The joint exploration of tissues in ilr-coordinates and image space is performed by using a specialized software developed by the author [@segmentator1pt3pt0].
Results
=======
![\[Fig1\]Visual demonstration of the similarity between RGB to HSI transformation and the corresponding analogous concepts laid out in Sections \[methods\_bary\_decomp\] and \[methods\_coda\]. In the first row the original image consisting of three channels and each of its channels can be seen. The second row shows hue, saturation and intensity components after RGB-HSI transformation \[\[RGB2HSI\]\]. The third row shows angular difference in simplex space (analogous to hue), norm in simplex space (analogous to saturation) and derived by using compositional data methods (Equations \[coda\_start\]-\[angular\_diff\] and the scalar component in real space (Equation \[intensity\]). Color maps of all scalar images range between the minimum and the maximum values of the corresponding image (darker shades are assigned to low values). The image is acquired from official SpaceX photo gallery (https://www.flickr.com/photos/spacex) licensed under CC0 1.0 public domain.](figure_1.pdf){width="\textwidth"}
Figure \[Fig1\] shows the similarity between HSI and the analogous metrics demonstrated in methods section. When comparing Fig. \[Fig1\] rows 2-3 column 1 to column 3, it can be seen that hue and angular difference in simplex space are both invariant to intensity gradient visible in the sky (brightness change from upper right corner to lower left). The same observation can also be made for the saturation and norm in simplex space images (compare Fig. \[Fig1\] rows 2-3 column 2 to column 3). These observations can be related to the scale invariance principle of compositional data analysis. By applying the barycentric decomposition (Eq.\[bary\_decomp\]) to color triplets in each pixel, scale information is separated, revealing compositional vectors invariant to the underlying multiplicative scalar field (i.e. brightness). These image contrasts based on compositional metrics are useful in object recognition tasks, see how the clouds are invisible in hue and angular difference images or the well-defined tip of the rocket in saturation and norm images.
![\[Fig2\]T1w, PDw and T2\*w MR images rendered together with angular difference in simplex space ( Eq. \[angular\_diff\]), norm in simplex space ( Eq. \[a\_norm\]) and scalar channel sum component ( Eq. \[intensity\]). Left panel shows the sagittal slices (y-z plane) and the right panel shows a coronal slices (x-z plane) of the 3D images. Arrows with one asterisk (\*) indicates the parts of the images that are too bright and arrows with two asterisks (\*\*) indicates the parts that are too dim as a result of data acquisition imperfections. The legends on the upper left corner in left panel shows the colors bars which are also valid for the images in right panel.](figure_2.png){width="\textwidth"}
Figure \[Fig2\] depicts two different slices of the 3D brain images visualized in gray scale depicting different image contrasts. It can be seen that the smooth, artefactual multiplicative scalar field (referred as bias field in MRI literature, see arrows with asterisks in Fig. \[Fig2\]) is separated from the compositional components (compare Fig. \[Fig2\] rows 1-2 column 2 with row 3 column 2 in both panels). This is similar to the separation of the brightness gradient in the sky in Fig. \[Fig1\] as mentioned in the previous paragraph. The compositional image contrasts can be considered as virtual contrasts which enhance the visual appearance by being invariant to certain artefactual features. It should be mentioned that other image contrasts can be generated by changing the reference vector in angular difference images (see Eq. \[angular\_diff\]) to highlight specific tissues. Similarly, the compositions can also be perturbed (see Eq.\[centering\]) differently to generate different contrasts in norm images. The generation of task-specific virtual contrasts could potentially be useful in medical imaging applications since the input channels and the virtual contrasts are physically interpretable.
For the physical interpretation of the compositional vectors, a joint representation of color images and ilr coordinates of compositional vectors are presented in Figure \[Fig3\]. The difference between the rows in left column demonstrates the effect of color balance (see Equations \[centering\] and \[standardize\]). It can be seen that the image contrast is enhanced to the level of easily recognizing major brain tissues by associating them to different colors. This is a more convenient visualization compared to inspecting T1w, PDw, T2\*w image contrasts individually considering that the color image contains information from all three inputs at the same time. Right column in Fig. \[Fig3\] shows the 2D histograms of the ilr coordinates (see Eq. \[ilr\_transformation\]) of the vector compositions. The effect of color balance becomes apparent when the compositions are interpreted with regards to the embedded RGB color cube primary axes (real space axes). For instance most of the clusters are on the left hand side of the center, close to the green arrow (PDw). This indicates that PDw measurements were initially dominating the compositions causing the green heavy color image. After the color balance, compositions spread more equally along the primary color axes, indicating that color contrast in the image will be richer (i.e. less dominated by a single component). The color balanced visualization is more useful for human observers in terms of intuitively recognizing the tissues. For instance the arteries have mostly reddish-white colors and the sinuses appear in green, which corresponds to the areas delineated for these tissues in ilr coordinates when the positions of clusters are considered relative to the embedded primary axes of RGB color cube. Although both of these are blood vessels, the difference between arteries and sinuses is meaningful because sinuses contain mostly deoxygenated hemoglobin, leading to a rapid decay of the MRI signal. In contrast, arteries contain oxygenated blood with slower MR signal decay. The difference in signal decay times of blood vessels effects T2\*w and PDw measurements separating the compositional properties in relation of T1w images. Similarly the compositional change from white matter to gray matter to cerebrospinal fluid can be seen as an approximately straight line which corresponds to the change from red-heavy color of white matter to cyan of cerebrospinal fluid. It can be seen that these tissues are not dispersed towards PDw or T2\*w axes, which can be intepreted as PDw and T2\*w measurements not revealing different compositions in relation to T1w measurements when white matter, gray matter and cerebrospinal fluid is considered.
![\[Fig3\]MRI measurements rendered as a color image. Red channel is assigned to T1w, green to PDw and blue to T2\*w measurements. Left column shows the 2D histograms of the corresponding ilr coordinates (Eq. \[ilr\_transformation\]). The projection of primary axes of RGB color cube (in $R^3$) to ilr coordinates are embedded to provide an intuitive reference for the characteristics of the compositions. The effect of centering and standardization inside the simplex (Eq. \[centering\], \[standardize\]) and the compositional truncation (Eq. \[truncate\]) is visible as color balance improvement in the rendered brain slice. The labels pointing to the tissues in brain image and circles in 2D ilr coordinate histograms shows the relation of the compositional characteristics with the coloration. The circles in 2D histograms are the edges of the 2D transfer functions used to manually probe the tissue-ilr coordinate relationship.](figure_3.pdf)
Discussion
==========
In this work, compositional data analysis methods are used to reformulate RGB-HSI color space transformation. It is visually demonstrated that the compositional metrics such as angular difference and norm in simplex space relate to hue and saturation concepts in color space literature. This reformulation of hue and saturation would be advantageous for having a well-principled framework when operating on images with n-dimensions. For instance, a potential future application would be to analyze MRI datasets with more than three types of measurements by adding cerebral blood volume measurements [@Uludag2017] or multi-echo echo planar imaging [@Poser2009] to T1, T2 and PD weighted measurements at ultra high fields. Another application area would be processing of multi-spectral images for image fusion purposes [@Pohl2016]. In cases where more than three measurements are acquired for each element in an image, dimensionality reduction methods such as principal component analysis in simplex space [@Wang2015] would become relevant to maximize the visualized information content of color images. This is the disadvantage of being limited to three primary additive colors for color image rendering. However the virtual image contrasts (scalar images) could still be explored without dimensionality reduction. For instance the norm in simplex space can still be straightforwardly computed for n-dimensional compositions or the angular difference images can be explored can be explored by selecting different reference vectors to track the positions of n-dimensional compositions on an n-sphere.
The disadvantages of compositional metrics presented here should also be mentioned. If the signal to noise ratio of the acquired images are low, compositional image contrasts would become less useful. For instance, the parts of images where only the measurement noise is recorded (e.g. thermal noise) the compositional metrics would carry no meaning. In such cases the angular difference and norm in simplex space would return enhanced noise patterns therefore the local brightness (i.e. intensity) should be taken into account before the compositional image analysis. The domain-specific exploration of the parameter spaces are also needed. For instance what is the extent of the usefulness of the compositional image processing when the noise properties are dissimilar between different types of measurements or when the measurements have different spatial scales (e.g. images with different resolutions). Further investigation is necessary to establish the relevance of the proposed application of compositional data analysis specifically to MRI data and generally to image processing and image fusion.
Acknowledgements
================
The data was acquired thanks to Federico De Martino, under the project supported by NWO VIDI grant 864-13-012. The author O.F.G. was also supported by the same grant. I thank Ingo Marquardt for language editing in the initial version of the manuscript and Alberto Cassese for the advice on mathematical notations in Section \[methods\_bary\_decomp\]. In addition, I wish to express my appreciation for the comments and suggestions of the anonymous reviewers which I believe have helped to improve the manuscript.
Appendix
========
Glossary
--------
**Barycentric coordinates**: Center of mass-centric coordinates. Used in relation to the center of mass of an n-simplex.
**Color balance**: Used to indicate centering and standardizing compositional vectors in this work.
**HSI**: A three dimensional space with named axes relating to human color perception: hue, saturation, intensity.
**MRI**: Magnetic resonance imaging.
**PDw**: Proton density weighted MRI signal acquisition. Indicates density of the hydrogen atoms in different tissues.
**RGB**: A three dimensional space with named axes relating to color cube: red, green, blue
**Simplex**: Generalized geometrical notion of triangles. For example, 0-simplex is a point, 1-simplex is a line segment, 2-simplex is a triangle, 3-simplex is a tetrahedron etc.
**T1w**: T1 weighted MRI signal acquisition. Optimized to give the highest image contrast between white matter and gray matter brain tissues.
**T2\*w**: T2\* weighted MRI signal acquisition. Optimized to give the highest image contrast related to iron concentration between brain tissues.
**Voxel**: Volumetric cubic element, in other words a three dimensional pixel.
RGB to HSI transformation {#RGB2HSI}
-------------------------
$$Intensity = I = \left(\frac{R}{\max(R,\ G,\ B)} + \frac{G}{\max(R,\ G,\ B)} + \frac{B}{\max(R,\ G,\ B)}\right) \div 3.$$
$$Saturation = S =
\begin{cases}
0 &, I=0 \\
1 - \dfrac{\min(R,\ G,\ B)}{I} &, I > 0.
\end{cases}$$
$$Hue = H =
\begin{cases}
0^\circ &, \Delta=0\\
60^\circ \times \left( \dfrac{G' - B'}{\Delta}\mod6 \right) &, C_{max} = R'\\
60^\circ \times \left( \dfrac{B' - R'}{\Delta} + 2 \right) &, C_{max} = G'\\
60^\circ \times \left( \dfrac{R' - G'}{\Delta} + 4 \right) &, C_{max} = B'
\end{cases}$$
$$\text{where } \Delta = \max(R,\ G,\ B) - \min(R,\ G,\ B)$$
$$\text{ and } R' = \frac{R}{\max(R,\ G,\ B)},\ G' = \frac{G}{\max(R,\ G,\ B)},\ B' = \frac{B}{\max(R,\ G,\ B)}.$$
MRI data acquisition parameters and ethics statement {#mri_parameters}
----------------------------------------------------
Whole head images were acquired using a three dimensional magnetization prepared rapid acquisition gradient echo (MPRAGE) sequence. The data consisted of a T1w image (repetition time \[TR\] $= 3100\ ms$; time to inversion \[TI\] $= 1500\ ms$ \[adiabatic non-selective inversion pulse\]; time echo \[TE\] $= 2.42\ ms$; flip angle $= 5^\circ$; generalized auto-calibrating partially parallel acquisitions \[GRAPPA\] = 3 [@Griswold2002]; field of view \[FOV\] $= 224 \times 224\ mm^2$; matrix size $= 320 \times 320$; 256 slices; 0.7 mm isotropic voxels; pixel bandwidth $= 182$ Hz/pixel; first phase encode direction anterior to posterior; second phase encode direction left to right), a PDw image (0.7 mm isotropic) with the same 3D-MPRAGE sequence but without the inversion pulse (TR $= 1380\ ms$; TE $= 2.42\ ms$; flip angle $ = 5^\circ$; GRAPPA $= 3$; FOV $= 224 \times 224\ mm$; matrix size $= 320 \times 320$; 256 slices; 0.7 mm isotropic voxels; pixel bandwidth $= 182$ Hz/pixel; first phase encode direction anterior to posterior; second phase encode direction left to right), and a T2\*w anatomical image using a modified MPRAGE sequence TR ($= 4910\ ms$; TE $= 16\ ms$; flip angle $= 5^\circ$; GRAPPA $= 3$; FOV $= 224 \times 224 mm$; matrix size $= 320 \times 320$; 256 slices; 0.7 mm isotropic voxels; pixel bandwidth $= 473$ Hz/pixel; first phase encode direction anterior to posterior; second phase encode direction left to right). Only magnitude images are stored after reconstruction.
The experimental procedures were approved by the ethics committee of the Faculty for Psychology and Neuroscience at Maastricht University, and were performed in accordance with the approved guidelines and the Declaration of Helsinki. Informed consent was obtained from the participant before conducting the data acquisition.
|
---
abstract: 'A single crystal of the Co$^{2+}-$based pyrochlore [[NaCaCo$_2$F$_7$]{}]{} was studied by inelastic neutron scattering. This frustrated magnet with quenched exchange disorder remains in a strongly correlated paramagnetic state down to one 60th of the Curie-Weiss temperature. Below $T_f = 2.4$ K, diffuse elastic scattering develops and comprises $30 \pm 10\%$ of the total magnetic scattering, as expected for $J_{\text{eff}} = 1/2$ moments frozen on a time scale that exceeds $\hbar/\delta E$=3.8 ps. The diffuse scattering is consistent with short range $XY$ antiferromagnetism with a correlation length of 16 Å. The momentum ($\boldsymbol{Q}$) dependence of the inelastic intensity indicates relaxing $XY$-like antiferromagnetic clusters at energies below $\sim$ 5.5 meV, and collinear antiferromagnetic fluctuations above this energy. The relevant $XY$ configurations form a continuous manifold of symmetry-related states. Contrary to well-known models that produce this continuous manifold, order-by-disorder does not select an ordered state in [[NaCaCo$_2$F$_7$]{}]{} despite evidence for weak ($\sim 12 $%) exchange disorder. Instead, [[NaCaCo$_2$F$_7$]{}]{} freezes into short range ordered clusters that span this manifold.'
author:
- 'K.A. Ross'
- 'J.W. Krizan'
- 'J.A. Rodriguez-Rivera'
- 'R.J. Cava'
- 'C.L. Broholm'
title: 'Static and dynamic $XY$-like short-range order in a frustrated magnet with exchange disorder'
---
Introduction
============
The spin liquid state of the Heisenberg antiferromagnet (HAFM) on the pyrochlore lattice supports fluctuations within an extensively degenerate ground state manifold consisting of correlated, yet disordered, spin configurations [@moessner1998low; @moessner1998properties; @lacroix2011introduction]. This beautiful state of matter arises from a perfect frustration of antiferromagnetic (AFM) interactions on the corner sharing tetrahedra that comprise the pyrochlore lattice. However, the spin liquid is extremely susceptible to small perturbations that can reduce the ground state degeneracy and lower the free energy. The manner in which the spin liquid is modified in real materials with deviations from ideal Heisenberg exchange is thus a rich field of study, with many possible outcomes depending on the relevant perturbations [@gardner2010magnetic]. In particular, the role of fluctuations in selecting subsets of the ground state manifold must often be considered. Thermal and quantum fluctuations that are softer for certain spin configurations can, in some cases, select long range ordered (LRO) states in a mechanism called order-by-disorder [@champion2003er; @zhitomirsky2012quantum; @savary2012order; @maryasin2014order; @wong2013ground; @mcclarty2014order].
Quenched disorder, in the form of vacancies or bond disorder (i.e., local variations in the strength of the spin-spin interactions), also produces order-by-disorder, as described in the pioneering work by Villain [@villain1980order] and later studied in detail by others [@henley1987ordering; @henley1989ordering; @maryasin2014order; @mcclarty2014order]. Quenched disorder can compete with thermal fluctuations to determine the ordered state. An important recent example is the $XY$ antiferromagnetic pyrochlore material Er$_2$Ti$_2$O$_7$. For the pseudospin $\frac{1}{2}$ model believed to be appropriate for this material, thermal and quantum order-by-disorder have been shown to select a non-coplanar LRO state [@zhitomirsky2012quantum; @savary2012order; @oitmaa2013phase], while quenched disorder is predicted to favor a coplanar LRO state in the same model [@maryasin2014order; @andreanov2015order]. The role of quenched disorder for the HAFM pyrochlore model has been studied in the past by including a distribution of exchange interactions spanning $\bar{J} \pm \Delta$ in the HAFM Hamiltonian, $ H = \sum_{ij} J_{ij}
\mathbf{S}_i\cdot \mathbf{S}_j$. In the limit of weak disorder, $\Delta << \bar{J}$, where $\bar{J}$ is the mean exchange interaction, the spins are expected to form locally collinear antiferromagnetic correlations and the system eventually freezes at a temperature $T_f \approx \Delta$ [@saunders2007spin; @andreanov2010spin; @bellier2001frustrated].
Here we report on the nature of static and dynamic spin correlations in the recently discovered pyrochlore material, [[NaCaCo$_2$F$_7$]{}]{}, which has been synthesized in single crystal form via the optical floating zone method [@krizan2014nacaco]. The inherent local disorder arising from the mixed-charge $A$-site (Na$^{+}$/Ca$^{2+}$) is expected to yield exchange disorder. The availability of large single crystals has allowed us to measure the full dynamic structure factor for [[NaCaCo$_2$F$_7$]{}]{} in the high symmetry \[$HHL$\] plane. Our measurements indicate [[NaCaCo$_2$F$_7$]{}]{} adheres to the expectations for the HAFM with weak exchange disorder in the high energy limit ($E > 5.5$ meV), but at low energies it displays frozen short range correlations and relaxational dynamics associated with an easy plane ($XY$-like) manifold.
In [[NaCaCo$_2$F$_7$]{}]{} (space group $Fd\bar{3}m$, $a$ =10.4056(2) Å at $T$ = 295 K [@krizan2014nacaco]) the $A$-site of the pyrochlore lattice is occupied by Na$^{+}$ and Ca$^{2+}$ ions with equal concentration in a disordered configuration. Thermodynamic magnetic properties of [[NaCaCo$_2$F$_7$]{}]{} evidence a spin-freezing transition at $T_f \approx 2.4$ K [@krizan2014nacaco]. The low freezing temperature indicates *weak* exchange disorder with $\Delta/\bar{J} \sim 0.12$ (assuming that $T_f \approx \Delta$, as in the models of Refs. and $\bar{J} \sim 20 $ K based on the Curie-Weiss temperature, $\theta_{CW}$ = -140 K, and assuming $S=3/2$). The effective moment determined from Curie-Weiss analysis at high temperatures is 6.1 $\mu_B$; this large effective moment obtained at high temperatures indicates significant thermal population of a $J_{\text{eff}}$ = 3/2 quartet at room temperature. Nonetheless, the change in entropy at low temperatures approaches $R\ln2$ [@krizan2014nacaco], suggesting a spin-orbit coupled ground state Kramers doublet with $J_{\text{eff}} = 1/2$, as is often relevant to Co$^{2+}$ materials [@maartense1977field; @regnault1977magnetic; @zhou2012successive; @kenzelmann2002order]. The degree of anisotropy of the effective moments in NaCaCo$_2$F$_7$ is not yet determined, but magnetization at $T=2$ K and 40 K is linear and isotropic up to $\mu_0 H = 9$ T (Appendix \[sec:magnetization\]) [@krizan2014nacaco].
Experimental Method
===================
We studied a 0.87 g single crystal of [[NaCaCo$_2$F$_7$]{}]{} using the MACS spectrometer at the NIST Center for Neutron Research [@rodriguez2008macs]. The dynamic structure factor, $S(\mathbf{Q},E)$, was measured in the \[$HHL$\] reciprocal lattice plane. Two configurations were used for spectroscopic measurements; for low energy transfer scans, neutrons with a final energy $E_f = 3.7$ meV were selected, and post-sample BeO filters were used to remove higher harmonic contamination and reject neutrons for which $E_f>3.7$ meV. For incident energies below (above) $E_i$ = 5.2 meV, a Be filter (open channel) preceded the sample. For energy transfers above $E=8.2$ meV, fixed $E_f = 5.0$ meV was used with post-sample Be filters and an open channel pre-sample. In all figures, measurements made in the various configurations are normalized to the same intensity units (counts per monitor units) using overlapping energy scans. The energy resolution was $\delta E$ = 0.17 meV at the elastic line for $E_f = 3.7$ meV and $\delta E$ = 0.34 meV for $E_f = 5.0$ meV.
Results
=======
Below $T_f$, the elastic magnetic scattering indicates static short range AFM spin correlations. The subtraction of 14 K from 1.7 K data reveals strong magnetic diffuse scattering (Fig. \[fig:fig1\] a). There are two components to the this diffuse scattering; diffuse Bragg spots (not resolution limited in $\boldsymbol{Q}$, as will be discussed below), in addition to diffuse scattering taking the shape of a “zig-zag” pattern underlying the peaks, i.e. the extended diffuse intensity is strongest along the lines connecting certain zone centers. This zig-zag diffuse pattern persists at finite energy transfers (Fig. \[fig:fig3\]), as will be discussed below.
The diffuse Bragg spots arise at the (111), (220), and (113) zone centers, but importantly *not* at (002) or (222) (see also Fig. \[fig:rawdata\] in Appendix \[sec:moreneut\]). These absences strongly constrain the frozen spin configuration. Although a short range structure based on collinear antiferromagnetic moments might be expected based on the weak disorder HAFM model, the absence of the (002) diffuse Bragg spot rules out this scenario. The observed magnetic peaks are instead consistent with $XY$ spin configurations, specifically those transforming as the $\Gamma_5$ irreducible representation (IR) of the tetrahedral point group $T_d$. The $\Gamma_5$ IR admits a continuous manifold of states parameterized by a single angular parameter, $\alpha$, which rotates the spin on each sublattice around its local $<111>$ axis (Appendix \[sec:gamma5\])[@champion2003er; @zhitomirsky2014nature].
At 1.7K, the elastic peak near $(11\bar{1})$ (Fig. \[fig:fig2\] a)) can be described by the sum of a sharp Gaussian and a broad Lorentzian component. The Gaussian persists at all measured temperatures; this nuclear Bragg peak is a measure of the instrumental $\boldsymbol{Q}$-resolution. The Lorentzian magnetic component gradually develops upon cooling from 14 K (Fig. \[fig:fig2\]b), while the Full Width at Half Maximum (FWHM) decreases. This behavior is reminiscent of critical scattering preceding a transition to an ordered state. From this perspective, the transition in [[NaCaCo$_2$F$_7$]{}]{} may be thought of as being preempted by freezing. Below $T_f$, the FWHM of the Lorentzian saturates at 0.12(1) Å$^{-1}$, implying a correlation length of 16.1(1) Å for the short range magnetic order.
Inelastic scattering is also readily observed in [[NaCaCo$_2$F$_7$]{}]{} (Fig. \[fig:fig4\]). Using the total moment sum rule for magnetic neutron scattering, we find that the ratio of elastic magnetic to total magnetic scattering is $r = 0.3(1)$ in the measured region of the $[HHL]$ plane. This is consistent with the ratio expected for a fully frozen (static) configuration arising from moments with $J_{\text{eff}}=\frac{1}{2}$ (for which $r = \frac{{J_{\text{eff}}}^2}{J_{\text{eff}}(J_{\text{eff}}+1)} = \frac{1}{3}$) but it is half of what is expected for bare $S=\frac{3}{2}$ ($r =$ 0.6).
To interpret the inelastic magnetic neutron scattering we examine $\boldsymbol{Q}-E$ slices (Fig. \[fig:fig4\] a), the energy dependence of constant-$\boldsymbol{Q}$ cuts (Fig. \[fig:fig4\] b), as well as the $\boldsymbol{Q}$-dependence of constant-$E$ slices (Fig. \[fig:fig3\]). The latter reflect the spatial Fourier transform of spin correlations with a characteristic fluctuation frequency of $\omega = E/\hbar$. From the $\boldsymbol{Q}-E$ slices of Fig. \[fig:fig4\] a), the spectrum of magnetic scattering at the diffuse Bragg spots is seen to be gapless, while the spectrum at systematically absent magnetic Bragg spots (e.g. $(00\bar{2})$) is gapped and strongly damped.
The constant-$E$ slices reveal that the zig-zag structure of diffuse scattering, which is present in addition to the diffuse Bragg spots on the elastic line, persists to finite energy transfers (Fig. \[fig:fig3\]b)). For a more detailed analysis of these distinct spectra, Fig. \[fig:fig4\] b) shows constant-$\boldsymbol{Q}$ cuts at $T$= 1.7 K at two locations in the $[HHL]$ plane. For $\boldsymbol{Q}=(00\bar{2})$ where there is no elastic magnetic peak (orange symbols in Fig. \[fig:fig4\] b), the data can be fit to a Damped Harmonic Oscillator (DHO) spectral function with an excitation energy of 5.5(1) meV and a damping coefficient 8.7(4) meV (i.e., an overdamped mode, see Appendix \[sec:fits\]); we later identify this mode with collinear excitations out of the easy plane manifold. In contrast, the inelastic spectra at the diffuse Bragg positions (blue symbols in Fig \[fig:fig4\]b) can be fit to the sum of a Lorentzian relaxation function at $E=0$ meV with a HWHM of 0.33(1) meV and a DHO with central energy and damping fixed to the values extracted at $(00\bar{2})$ (Appendix \[sec:fits\]). This decomposition of the line shape of the spectrum at $(\bar{1}\bar{1}\bar{1})$ is detailed in the right inset of Fig. \[fig:fig4\]b).
The zig-zag pattern formed in the $[HHL]$ plane by the quasi-elastic scattering (Fig.\[fig:fig3\] b)) can be associated with low energy states related by easy plane spin rotations that span the $\Gamma_5$ manifold. To establish this, we first compared the constant energy slice at 1.25 meV to the calculated neutron scattering intensity from a spatial average of independent $XY$ tetrahedra (Appendix \[sec:moreneut\]). In this approximation, each tetrahedron supports one choice from the continuous $\Gamma_5$ manifold. This independent $XY$ tetrahedra model captures the lack of intensity near the $(00\bar{2})$ position and the general zig-zag shape of the diffuse scattering. A better agreement is obtained, however, when collinear AFM spin components are added to each $XY$ tetrahedron on the level of $\sim$ 50% (Fig. \[fig:fig3\] e)). This indicates that while the frozen state is $XY$-like, excitations for energy transfers beyond the freezing temperature involve both in- and out-of plane spin components. In particular, the DHO mode at 5.5 meV arises from out of plane excitations while the quasi-elastic component is associated with easy plane excitations. The inset to Fig. \[fig:fig4\] b) shows contributions from both types of excitations at 1.25 meV. Beyond 4 meV, the constant energy slices (Fig. \[fig:fig3\] c)) are reproduced by fully collinear AFM configurations on independent tetrahedra (Fig. \[fig:fig3\] f)). An important shortcoming of this simple model is the independent tetrahedra approximation. Specifically, the widths of the measured diffuse scattering for the slow spin fluctuations at 1.25 meV is significantly sharper in $Q$-space than predicted (Appendix \[sec:moreneut\]), and this is then evidence for inter-tetrahedron correlations. The size of the correlated region inferred from fitting raw data is 7.9 Å, which may be compared to the 3.65 Å side length of the tetrahedron. Even at $T=14$ K the inverse correlation length of the inelastic scattering is approximately half of that expected for independent tetrahedra.
Discussion
==========
The above analysis of the inelastic spectrum at $T=$ 1.7 K suggests there are two types of dynamics in this short-range correlated system. The first is a relaxing process that is qualitatively consistent with local rotations of $XY$ spin clusters through the continuous $\Gamma_5$ manifold with a relaxation time of $\tau_{XY} = 2.02$ ps. The second is a short-lived ($\tau_{H} = 0.15$ ps) inelastic mode at 5.5 meV, which is qualitatively consistent with collinear antiferromagnetic tetrahedral fluctuations, as might be expected from the weak disorder HAFM.
The $XY$ spin configurations relevant for [[NaCaCo$_2$F$_7$]{}]{} at low energies are already well-studied in the pyrochlore literature. As shown in Appendix \[sec:gamma5\], the $\Gamma_5$ IR can be decomposed into two basis vectors which have commonly been called $\psi_2$ (non-coplanar) and $\psi_3$ (coplanar) (Fig. \[fig:fig1\] b). LRO states based on the $\Gamma_5$ manifold are known to be selected in the HAFM model upon inclusion of “indirect” Dzyaloshinskii-Moriya (DM) interactions [@elhajal2005ordering], despite an accidental ground state degeneracy admitting all values of $\alpha$ at the mean field level. The same continuously degenerate manifold is also present at the mean field level for the $XY$ AFM pyrochlore model [@champion2004soft; @mcclarty2014order], and the $XY$-like anisotropic exchange model proposed for Er$_2$Ti$_2$O$_7$ [@zhitomirsky2012quantum; @savary2012order; @maryasin2014order; @andreanov2015order]. In all cases, the $\Gamma_5$ degeneracy is lifted by *disorder*, and a LRO state is selected. The “disorder” can arise from thermal or quantum fluctuations, or quenched exchange disorder. However, in [[NaCaCo$_2$F$_7$]{}]{}, despite a clear mechanism for weak bond disorder, the $\Gamma_5$ degeneracy is retained and explored by the system on short length scales and long time scales. In this case, exchange disorder does *not* lead to spin order, but instead to a a frozen spin configuration that appears to span the continuous $\Gamma_5$ manifold.
The microscopic reason for the stabilization of $XY$ spin configurations in [[NaCaCo$_2$F$_7$]{}]{} is not yet certain. However, the spin orbit coupled $J_{\text{eff}} = 1/2$ state expected for Co$^{2+}$ in a distorted octahedral environment could lead to $XY$ anisotropy, either in the $g$-tensor or the exchange interactions, or both. The central energy of the damped inelastic mode (5.5 meV) may be a measure of the strength of the anisotropy. This should be investigated in the future through measurements of single-ion energy levels of [[NaCaCo$_2$F$_7$]{}]{}. Additional open questions, aside from quantifying the single ion anisotropy, include whether orbital and lattice degrees of freedom are relevant to [[NaCaCo$_2$F$_7$]{}]{} as in the related spinel compound GeCo$_2$O$_4$ [@tomiyasu2011molecular].
Conclusions
===========
In summary, [[NaCaCo$_2$F$_7$]{}]{} is the first example of a new class of pyrochlore single crystals based on a structurally ordered magnetic $3d$ transition metal site in a varying local environment created by a disordered non-magnetic site [@krizan2015single; @krizan2015nacani2f7]. The disordered environment leads to weak disorder in the strong AFM interactions in [[NaCaCo$_2$F$_7$]{}]{} ($\theta_{CW}$ = -140 K), and ultimately a low temperature freezing transition at $T_f$ = 2.4 K. We have observed $XY$ spin configurations forming a short range ordered state below $T_f$ with a correlation length of $\xi = 16$ Å. The low energy fluctuations away from this frozen state are gapless to within the energy resolution of our measurement (0.17 meV) and take on a distinctive diffuse pattern that suggests relaxation through a continuous manifold of local $XY$ states. At higher energies, a strongly damped mode at 5.5 meV dominates the spectrum. The associated Q-dependence of the scattering intensity isconsistent with collinear antiferromagnetic tetrahedral fluctuations, indicating an $XY$ anisotropy barrier of $\sim$ 64 K.
The continuous manifold of $XY$ spin configurations present in [[NaCaCo$_2$F$_7$]{}]{} is known to collapse to an ordered state via order-by-disorder in models relevant to Er$_2$Ti$_2$O$_7$ as well as by DM interactions in the pyrochlore HAFM. However, unlike the aforementioned theoretical predictions, quenched exchange disorder in [[NaCaCo$_2$F$_7$]{}]{} does *not* lead to the selection of an ordered state, but instead a quasi-static disordered state. Apart from the low energy fluctuations that appear to span the $\Gamma_5$ manifold, a prominent out of plane damped mode is observed with the same local structure as predicted for the Heisenberg model with weak exchange disorder. An intriguing aspect of [[NaCaCo$_2$F$_7$]{}]{} is the potential for ice-like correlations on the Na$^{+}$, Ca$^{2+}$ disordered sublattice. Such correlated disorder might be necessary to explain why [[NaCaCo$_2$F$_7$]{}]{} fails to develop long range order.
The authors gratefully acknowledge enlightening discussions with O. Tchernyshyov, J.T. Chalker, and J.W. Lynn. KAR acknowledges the hospitality of Colorado State University during the writing of this manuscript, and the use of the SPINDIFF software package [@paddison2013spinvert]. The bulk of the work was supported by the US Department of Energy, office of Basic Energy Sciences, Division of Material Sciences and Engineering under grant DE-FG02-08ER46544. In particular this included the crystal growth activities and neutron scattering experiments. This work utilized facilities supported in part by the National Science Foundation under Agreement No. DMR-0944772. KAR was partially supported by NSERC of Canada.
Definition of states in the $\Gamma_5$ manifold {#sec:gamma5}
===============================================
The sublattices of the pyrochlore lattice are described by the following fractional coordinates:
$$\begin{aligned}
&&\mathbf{d}_0=\left(\frac{3}{8}, \frac{3}{8}, \frac{3}{8}\right),\quad \mathbf{d}_1=\left(\frac{3}{8}, \frac{1}{8}, \frac{1}{8}\right),\\
&&\mathbf{d}_2=\left(\frac{1}{8},\frac{3}{8},\frac{1}{8}\right),\quad\mathbf{d}_3=\left(\frac{1}{8}, \frac{1}{8}, \frac{3}{8}\right).\end{aligned}$$
The moments (pseudovectors) forming the $\psi_2$ and $\psi_3$ bases of the $\Gamma_5$ representation are assigned to these sublattices as:
$$\vec{\psi_2}
\left\{\begin{array}{l}
\mathbf{\hat{s}}_0=(1,1,\bar{2})/\sqrt{6}\\
\mathbf{\hat{s}}_1=(1,\bar{1}, 2)/\sqrt{6}\\
\mathbf{\hat{s}}_2=(\bar{1},1, 2)/\sqrt{6}\\
\mathbf{\hat{s}}_3=(\bar{1},\bar{1},\bar{2})/\sqrt{6},
\end{array}\right.,
\quad
\vec{\psi_3}
\left\{\begin{array}{l}
\mathbf{\hat{s}}_0=(1,\bar{1}, 0)/\sqrt{2}\\
\mathbf{\hat{s}}_1=(1, 1, 0)/\sqrt{2}\\
\mathbf{\hat{s}}_2=(\bar{1},\bar{1},0)/\sqrt{2}\\
\mathbf{\hat{s}}_3=(\bar{1},1,0)/\sqrt{2}
\end{array}\right.,
\label{eqn:spins}$$
A general tetrahedral state with the symmetry of $\Gamma_5$ can be written as a linear combination of these sets, $$\vec{\chi}(\alpha) = \cos{\alpha} \cdot \vec{\psi_2} + \sin{\alpha} \cdot \vec{\psi_3}
\label{eqn:alpha}$$
Assigning each “up” tetrahedron in the pyrochlore lattice a state $\vec{\chi}$ with a random value of $\alpha$ constitutes the independent tetrahedron $XY$ AFM state that is modeled in Figure \[fig:fullcomparison\] e). $\vec{\chi}(\alpha)$ spans a continuously deformable manifold of $XY$ states. These are the relevant ground states at the mean field level for the case of Er$_2$Ti$_2$O$_7$ [@champion2003er; @zhitomirsky2012quantum; @savary2012order; @maryasin2014order; @wong2013ground; @mcclarty2014order] or the HAFM model with “indirect” DM interactions [@elhajal2005ordering].
Details of constant-$\boldsymbol{Q}$ lineshapes {#sec:fits}
===============================================
Figure 3 in the main text presents fits to constant-$\boldsymbol{Q}$ cuts, i.e. $S(E)$. The fits to $S(E)$ include a relaxing diffusive component, $S_{XY}(E)$, and a damped harmonic oscillator, $S_{H}(E)$. These have the well-known forms [@lovesey1984theorych5; @lovesey1984theoryB], $$S_{XY}(E) = \frac{A_{XY}E(1+n(E))}{\pi}\frac{\Gamma_{XY}}{E^2 + \Gamma_{XY}^2},
\label{eqn:Sxy}$$ and, $$\begin{aligned}
S_{H}(E) = A_H(1+n(E)) \times \frac{2\Gamma_H E}{(E^2-E_c^2)^2 + (2\Gamma_H E)^2}
\label{eqn:Sh}\end{aligned}$$
where $\Gamma_{XY}$ is the HWHM of the diffusing component, and 2$\Gamma_{H}$ is the damping parameter of the DHO component. The relaxation rates are then given by $\tau_{XY}$ = 1/$\Gamma_{XY}$ and $\tau_{H}$ = 1/$\Gamma_{H}$ (with $\Gamma$’s expressed in units of frequency). $A_{XY}$ and $A_{H}$ are scale factors in arbitrary units. $n(E)$ is the Bose-Einstein population factor, $n(E) = (\exp({E/k_B T}) - 1)^{-1}$. $E_c/\hbar$ is the frequency of the DHO mode.
-- ---------- --------------------- ---------- ------------------ -------------
$A_{XY}$ $\Gamma_{XY}$ (meV) $A_H$ $\Gamma_H$ (meV) $E_c$ (meV)
– – 1157(52) 4.3(2) 5.5(1)
109(2) 0.33(1) 625(17) 4.3 5.5
-- ---------- --------------------- ---------- ------------------ -------------
: Parameters for fits of constant energy scans at $\boldsymbol{Q}$ = $(00\bar{2})$ and $\boldsymbol{Q}$ = $(\bar{1}\bar{1}\bar{1})$ to Eqns. \[eqn:Sxy\] and \[eqn:Sh\].
Supporting Neutron Scattering Data {#sec:moreneut}
==================================
Here we present additional information supporting the conclusions from the main text. Figure \[fig:fullcomparison\] shows a more detailed comparison to three choices of models; short range ordered states with 16 Å correlation lengths, and single tetrahedron states with $XY$ or locally collinear character, or a mixture of these. Fig. \[fig:widths\] a) shows the widths of diffuse features as compared to the independent tetrahedra model, as well as both lower temperature (100 mK) and higher temperature (14 K) inelastic scans. The low energy diffuse inelastic scattering corresponds to a correlated region ($\sim 8$ Å) much larger than a single tetrahedron ($3.5$ Å), at all temperatures measured, from 100 mK to 14 K. Figure \[fig:rawdata\] a) shows an elastic scan taken with $E_i = E_f$ = 13.5 meV at $T = 1.7$ K (after subtracting 14 K data), which reveals diffuse magnetic scattering throughout a larger range of $\mathbf{Q}$. Note in particular the absence of diffuse scattering at (222). Figure \[fig:widths\] b) also shows that the inelastic scattering takes on the same pattern in the thermal spin liquid phase (14 K) as it does in the frozen phase (100 mK) (“empty can” background subtractions made in both panels). In Fig. \[fig:rawdata\] b and c) we show raw elastic scattering data (no subtraction) at $T = 1.7$ K and $T = 14$ K.
Magnetization Data {#sec:magnetization}
==================
In order to investigate the possibility of an anisotropic $g$-tensor in [[NaCaCo$_2$F$_7$]{}]{}, magnetization measurements were performed at temperatures above the freezing transition ($T > 2.4$ K), using the extraction magnetometry technique in a commercial physical properties measurement system. Measurements with the field applied along three different crystallographic axes were compared. The data taken at $T= 40$ K are shown in Fig. \[fig:mag\] a) and b). Only slight deviations from isotropic behavior are observed at the highest field strengths ($\sim$ 9 T), and these could easily be due to demagnetization effects for crystals having slightly different shapes for the different field orientations. Furthermore, the deviation from isotropic magnetization does not correspond to the expected hierarchy for either $XY$-like ($|M_{(110)}| > |M_{(111)}| > |M_{(100)}|$) or Ising-like ($|M_{(100)}| > |M_{(111)}| > |M_{(110)}|$) $g$-tensors. For example, the magnetization of an ideal pyrochlore paramagnet with an $XY$-like $g$-tensor is shown in Fig. \[fig:mag\] c), using the equation, $$M_{\mathbf{d}}(H,T) = g_{\mathbf{d}} J \mu_B B_J(g_{\mathbf{d}} \mu_B J H / k_B T),
\label{eqn:brillouin}$$
where $H$ is the applied magnetic field, $g_{\mathbf{d}}$ is the average projection of the $g$-tensor onto the field direction (averaged over the four sites on the tetrahedron), $J$ is the effective angular momentum, here taken to be $1/2$ since we may assume a spin-orbit coupled Kramers doublet ground state for Co$^{2+}$, and $B_J$ is the Brillouin function. In Fig. \[fig:mag\] c) an $XY$-like $g$-tensor was assumed, with $g_{xy} = 3.6$ and $g_{z} = 3.0$. Although this equation is not expected to be valid at $T= 40$ K for [[NaCaCo$_2$F$_7$]{}]{}, since $T<\theta_{CW}$, one may expect the same hierarchy of magnetization strengths to be observed, even in such a correlated paramagnetic regime. Thus, at least to within the demagnetization effects in these measurements, the $g$-tensor anisotropy in [[NaCaCo$_2$F$_7$]{}]{} is shown to be small on average.
[33]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [**]{}, Vol. (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [ ()]{} @noop [****, ()]{} “,” (, ) Chap. , p. “,” (, ) Chap. , pp.
|
---
author:
- Florian Niedermann
- and Robert Schneider
bibliography:
- 'SLED\_BLF\_II.bib'
title: 'SLED Phenomenology: Curvature vs. Volume'
---
Introduction and Summary
========================
The SLED model [@Aghababaie:2003wz] provides a promising candidate for addressing the cosmological constant (CC) problem [@Weinberg:1988cp]. The main motivation is that for a codimension-two brane, the 4D CC only curves the transverse extra-space into a cone, while the on-brane geometry stays flat. However, it was realized from the very beginning [@Aghababaie:2003wz] that for compact extra dimensions this comes at the price of yet another tuning relation, stemming from the flux quantization condition, which in turn is required to stabilize the compact extra space. Alternatively, from a 4D point of view, the problem can be formulated as saying that it is simply the classical scale invariance (SI) of this theory which leads to a flat brane geometry, in which case Weinberg’s general no-go argument [@Weinberg:1988cp] applies.
To circumvent this problem, a brane-localized flux (BLF) term was later included [@Burgess:2011va; @Burgess:2011mt]; the idea was that if this term breaks SI, then it is in principle possible that the dilaton dynamically adjusts such that flux quantization is fulfilled, thereby avoiding the tuning relation (or runaway solutions). However, it was recently shown [@Niedermann:2015via] (and also confirmed in a specific UV model [@Burgess:2015gba]) that only SI brane couplings—including the BLF term—ensure a flat brane geometry. But then, it does not alter the tuning (or runaway) problem either, and we are basically back at square one.
However, the mere fact that the 4D curvature is zero in the SI case does not immediately rule out the model as a potential solution to the CC problem. It might still be possible to achieve a *nonzero but small* (compared to standard model loop contributions) curvature in a phenomenologically viable and technically natural way by breaking SI on the brane. The main purpose of this companion paper to [@Niedermann:2015via] is to investigate this remaining question in detail.
The starting point of our analysis is the effective theory that is obtained after solving for the Maxwell field in a 4D maximally symmetric configuration, and adding a counter-term to dispose of divergences which generically arise due to the BLF, as discussed in [@Niedermann:2015via]. The goal here is to explicitly solve the resulting Einstein-dilaton system for given model parameters and couplings. Explicitly, we will focus on a SI breaking brane tension. Since the standard model sector breaks SI on the brane, this term should be included in a realistic setup, and its size will be set by loop contributions of the brane matter fields. Furthermore, we will endow the brane with a finite thickness in order to avoid potential divergences. This should not be viewed as a mere technical regularization, but rather as another physically unavoidable feature: A realistic brane has to come with some microscopic thickness, which would ultimately be determined by an underlying UV model. We will find that both sources—the non SI tension and the brane width—contribute to the 4D curvature independently, and discuss them in detail.
To endow the brane with a thickness, we choose in Sec. \[sec:ring\_reg\] a convenient and well-known technique [see e.g. @Peloso:2006cq; @Burgess:2008yx] that replaces the infinitely thin brane by a ring of finite proper circumference $ \ell $. Most importantly, we expect the low energy questions we are going to ask to be insensitive to this microscopic choice. This setup only admits static solutions if there is some additional mechanism that prevents the ring from collapsing. Effectively, this boils down to adding an angular pressure component $ p_\theta $, the size of which can be inferred from the junction conditions across the brane. This allows us to generalize a previously derived formula for the 4D curvature to the regularized setup, thereby enabling us to study the tuning issue and the phenomenological viability of the model. Prior to that, we check in Sec. \[sec:delta\_limit\] whether our result are consistent with the delta-analysis in [@Niedermann:2015via]: We find that the delta-results are all recovered in the thin brane limit if and only if $ p_\theta \to 0 $. Since for an infinitely thin object there is no direction this pressure could act in, this is a reasonable physical assumption.[^1] Here, it will also be shown to be true for the case of exponential dilaton-brane couplings as introduced in Sec. \[sec:near\_SI\]. These couplings model the SI breaking and are of particular interest with respect to the CC problem as they allow to be close to SI without the need of tuning the coefficients small.
A discussion of the model’s phenomenological status is given in Sec. \[sec:phen\], leading to an unambiguous conclusion: [*Without tuning certain model parameters to be small compared to the bulk Planck scale, it is not possible to comply with both the observed value of the Hubble parameter as well as constraints on the size of the extra dimensions.*]{} This negative conclusion applies to both the SI breaking tension and the finite brane width effects independently.
This—so far analytical—verdict is based on several assumptions that are all confirmed by explicitly solving the brane-bulk system in Sec. \[sec:num\_results\]. To that end, the full set of field equations for a 4D maximally symmetric ansatz is integrated numerically, as explained in Sec. \[sec:num\_alg\]. Special attention is given to imposing the required regularity conditions at *both* axes of the compact space, because only then are all integration constants uniquely determined. The results and physical implications, both for SI and non SI dilaton-brane couplings, are discussed in Secs. \[sec:SI\] and \[sec:non\_SI\], respectively. We find that in both cases an acceptably small 4D curvature is typically only achieved by tuning the (dilaton independent part of the) brane tension, but that this tuning can indeed be alleviated for certain brane-dilaton couplings. However, we also confirm the analytic prediction, so that in either case the extra dimensions are way too large to be phenomenologically viable.
Let us note that the same model was recently analyzed in [@Burgess:2015lda] in a dimensionally reduced, effective 4D theory. Our present work instead solves the full 6D bulk-brane field equations, thus providing an alternative and complementary approach. While confirming the result of [@Burgess:2015lda] that a large extra space volume can be achieved for certain parameters without the need for putting in large hierarchies by hand, we are also able to go one step further and uncover the tuning that is always needed to get *both* the 4D curvature *and* the volume within their observational bounds. Our conclusions are summarized in Sec. \[sec:concl\].
Delta Brane Setup {#sec:thin_brane}
=================
Review
------
We first provide a brief review of the thin brane setup. The reader familiar with the corresponding discussion in our companion paper [@Niedermann:2015via] should feel free to skip this section.
The field content of the SLED model comprises the 6D metric $g_{AB}$, a Maxwell field $A_{B}$, which stabilizes the compact bulk dimensions, and the dilaton $\phi$, which renders the bulk theory SI. The corresponding action reads [@Burgess:2011mt] $$\label{eq:action}
S = S_\mathrm{bulk} + S_\mathrm{branes} \,,$$ where the bulk part is[^2] $$\label{eq:action_bulk}
S_\mathrm{bulk} = - \int {\mathrm{d}}^6 X \sqrt{-g}\,\left\{\frac{1}{2\kappa^2}\left[ R + (\partial_M\phi)(\partial^M\phi) \right] + \frac{1}{4} {\mathrm{e}}^{-\phi}F_{MN} F^{MN} + \frac{2e^2}{\kappa^4}{\mathrm{e}}^{\phi}\right\} ,$$ with $\kappa$ and $e$ the gravitational and U(1) coupling constants, respectively. The 6D Ricci scalar $R$ is built from the 6D metric $g_{AB}$, and $ F \equiv {\mathrm{d}}A $. The brane contributions are $$\label{eq:action_brane}
S_\mathrm{branes} = - \sum_b \int {\mathrm{d}}^4 x \sqrt{-g_4} \left\{ \mathcal{T}_b(\phi)-\frac{1}{2} \mathcal{A}_b(\phi) \epsilon_{mn} F^{mn} \right\} ,$$ where the index $b \in \{+,-\}$ runs over both branes situated at the north ($+$) and south ($-$) pole of the compact space, where the metric function $ B $ (see below) vanishes. The 4D brane tension is denoted by $\mathcal{T}_b(\phi)$. The second term, controlled by $\mathcal{A}_b(\phi)$, describes the brane localized flux (BLF). In general, both terms are allowed to have arbitrary dilaton dependences; in particular, the SI case corresponds to $\mathcal{T}_b(\phi)=\mathrm{const}$ and $\mathcal{A}_b(\phi) \propto {\mathrm{e}}^{-\phi}$.
In [@Niedermann:2015via] we investigated the theory under the assumption of 4D maximal symmetry and azimuthal symmetry in the bulk. This leads to the following general ansatz,
\[eq:ansatz\] $$\begin{aligned}
{\mathrm{d}}s^2 & = W^2(\rho) \,\hat g_{\mu\nu} {\mathrm{d}}x^{\mu} {\mathrm{d}}x^\nu + {\mathrm{d}}\rho^2 + B^2(\rho) {\mathrm{d}}\theta^2 \,, \label{eq:ansatz_met}\\
A &= A_\theta(\rho) {\mathrm{d}}\theta \,, \\
\phi & = \phi(\rho) \,, \label{eq:ansatz_phi}
\end{aligned}$$
where $\hat g_{\mu\nu}$ is 4D maximally symmetric and thus fully characterized by its (constant) 4D Ricci scalar $ \hat R $. With these symmetries, the Maxwell equations can be integrated analytically, yielding $$\label{eq:sol_F}
F_{\rho\theta} = {\mathrm{e}}^{\phi} B \left[\frac{Q}{W^4} + \sum_b \frac{\delta_b}{2\pi B} \mathcal{A}_b(\phi) \right] ,$$ where $ Q $ is an integration constant, and $ \delta_b $ is shorthand for the Dirac delta function $ \delta(\rho - \rho_b) $. In the case of a nonvanishing BLF, the second term leads to a divergence $\propto \delta(0)$ in the remaining equations of motion, which can be interpreted as a relict of treating the branes as point-like objects. We proposed a corresponding brane counter term which allowed to consistently dispose of this contribution. After this subtraction, the remaining field equations consist of the dilaton equation $$\label{eq:dilaton}
-\frac{1}{\kappa^2}\frac{1}{BW^4}\left( B W^4 \phi' \right)' = \frac{{\mathrm{e}}^{\phi}}{2}\left( \frac{Q^2}{W^8} - \frac{4e^2}{\kappa^4} \right)
- \sum_b \frac{\delta_b}{2\pi B} \left\{ \mathcal{T}^{\prime}_b(\phi) - \frac{Q}{W^4} {\mathrm{e}}^{\phi} \left[ \mathcal{A}^{\prime}_b(\phi) + \mathcal{A}_b(\phi) \right] \right\} \,,$$ and the $ {(\!\begin{smallmatrix}\mu\\#2\end{smallmatrix}\!)} $, $ {(\!\begin{smallmatrix}\rho\\#2\end{smallmatrix}\!)} $ and $ {(\!\begin{smallmatrix}\theta\\#2\end{smallmatrix}\!)} $ components of Einstein’s field equations,
\[eq:einstein\_expl\] $$\begin{aligned}
-\frac{1}{\kappa^2} \left( \frac{\hat R}{4 W^2} + 3 \frac{W''}{W} + \frac{B''}{B} + 3\frac{W'^2}{W^2} + 3 \frac{W'B'}{WB} + \frac{1}{2} \phi'^2 \right)
& = \frac{{\mathrm{e}}^\phi}{2} \left( \frac{Q^2}{W^8} + \frac{4e^2}{\kappa^4} \right) \nonumber\\
& \quad + \sum_b \frac{\delta_b}{2\pi B} \mathcal{T}_b(\phi)
\,, \label{eq:einstein_00}
\\
\frac{1}{\kappa^2} \left( \frac{\hat{R}}{2W^2} + 6\frac{W'^2}{W^2} + 4\frac{W'B'}{WB} - \frac{1}{2} \phi'^2 \right)
& = \frac{{\mathrm{e}}^{\phi}}{2} \left( \frac{Q^2}{W^8} - \frac{4e^2}{\kappa^4} \right) \,, \label{eq:einstein_rho}
\\
\frac{1}{\kappa^2} \left( \frac{\hat{R}}{2W^2} + 4\frac{W''}{W} + 6\frac{W'^2}{W^2} + \frac{1}{2} \phi'^2 \right)
& = \frac{{\mathrm{e}}^{\phi}}{2} \left( \frac{Q^2}{W^8} - \frac{4e^2}{\kappa^4} \right) \,. \label{eq:einstein_theta}
\end{aligned}$$
Integrating the dilaton equation over an infinitesimally small disc covering one of the axes yields the boundary condition for $\phi$. For $W$ and $B$ the same is achieved by taking appropriate combinations of the Einstein equations. Explicitly, one finds
\[eq:matching\_delta\] $$\begin{aligned}
\left[B \phi' \right]_{\rho=\rho_b} &= \frac{\kappa^2}{2\pi} {\mathcal{C}}_b \,, \label{eq:jump_delta_phi}\\
\left[B (W^4)' \right]_{\rho=\rho_b} &=0 \,,\\
[B']_{\rho=\rho_b} &= 1 - \frac{\kappa^2}{2\pi} \left [ \mathcal{T}_b(\phi) \right ]_{\rho=\rho_b} \,,
\end{aligned}$$
where we defined $$\label{def:SI_combination}
{\mathcal{C}}_b := \left\{ \mathcal{T}^{\prime}_b(\phi) - \frac{Q}{W^4} {\mathrm{e}}^{\phi} \left[\mathcal{A}^{\prime}_b(\phi) + \mathcal{A}_b(\phi)\right] \right\}_{\rho=\rho_b} \,,$$ which measures the brane coupling’s deviation from SI. Furthermore, integrating a suitable combination of the field equations over the whole compact extra space yields $$\label{eq:degrav_cond_delta}
V \hat R = 2 \kappa^2 \sum_b W_b^4 {\mathcal{C}}_b \,,$$ with the 2D volume defined as $$\label{def:volume}
V := 2\pi \int \!{\mathrm{d}}\rho\, BW^2 = \int {\mathrm{d}}^2 y \, \sqrt{g_2} \, W^2 \,.$$ Hence, the SI case ($ \mathcal{C}_b = 0 $) implies $ \hat{R} = 0 $.
Constraint {#sec:constraint}
----------
Let us now turn to a peculiarity [@Burgess:2015kda] of the delta setup which was not discussed in [@Niedermann:2015via]. Multiplying the constraint by $ B^2 $ and taking the limit $ \rho \to \rho_b $ yields (assuming that $ B^2 {\mathrm{e}}^\phi \to 0 $) $$\label{eq:brane_constraint}
\left\{ \frac{3}{8W^8} \left[ B (W^4)' \right]^2 + \frac{1}{W^4} \left[ B (W^4)' \right] \left[ B' \right] - \frac{1}{2} \left[ B \phi' \right]^2 \right\}_{\rho = \rho_b} = 0 \,.$$ The terms in square brackets are those which appear in the boundary conditions , and so we are lead to (assuming that $ \left [ \mathcal{T}_b(\phi) \right ]_{\rho=\rho_b} $ is finite, as it should be for physically relevant situations) $${\mathcal{C}}_b = 0 \,. \label{eq:SI_delta}$$ This is in clear contradiction to the SI breaking expectation $ {\mathcal{C}}_b \neq 0 $. In [@Burgess:2015kda], it was argued that this uncovers an inconsistency of the delta analysis; we will comment on this in more detail in Appendix \[ap:Cliff\]. Here, let us merely state the other possibility: that is in fact another prediction of the delta setup, saying that it is impossible to consistently break SI on a delta-brane, at least on-shell. In this work, we will explicitly verify that this option is indeed realized for a relevant class of couplings. More specifically, starting with exponential SI breaking couplings of the form $ {\mathcal{C}}_b \propto {\mathrm{e}}^{\gamma\phi_b} $ and a thick brane setup, we will find that $ \phi_b \to -\infty $ in the thin brane limit, thereby restoring $ {\mathcal{C}}_b \to 0 $.
At this point, let us also emphasize that the SI case is completely insensitive to this whole issue, because then is identically fulfilled. Thus, the important achievement of [@Niedermann:2015via], namely the first correct identification of those BLF couplings which unambiguously lead to $ \hat{R} = 0 $ (and the resulting tuning relation), remains unaffected.[^3]
However, also implies that the actual (nonzero) value of $ \hat{R} $ for broken SI cannot be inferred within the pure delta framework (which always[^4] predicts $ \hat{R} = 0 $), but requires studying a thick brane setup. This also has the advantage that potential singularities are regularized.
Thick Brane Setup {#sec:thick_brane}
=================
Ring Regularization {#sec:ring_reg}
-------------------
In order to avoid any singularities and potential ambiguities of the (non SI) delta brane setup, the authors in [@Burgess:2015nka; @Burgess:2015gba] introduced a specific UV model describing the brane as a vortex of finite width in extra space. We will instead use a different and technically simpler way of regularizing the system, in which the delta brane is replaced by a ring of circumference $\ell$ [@Peloso:2006cq; @Burgess:2008yx].[^5] We assume the microscopic details of the regularization to be irrelevant for the low energy questions we want to study.
Let us note that introducing the regularization scale $ \ell $ breaks SI. This, however, does not necessarily imply that the underlying UV theory (which would resolve the brane microscopically) breaks SI explicitly. Indeed, a SI mechanism could easily be built, in analogy to the flux stabilization which fixes the large size of the extra dimensions. In that case, the UV model parameters would not determine $ \ell $, but rather the SI combination $ \ell {\mathrm{e}}^{\phi_0/2} $.[^6] However, this does not change the fact that $ \ell $ has to take a specific value in order to comply with observations. For a SI UV model, this would correspond to a spontaneously broken SI; but the physical conclusions would be the same.
For simplicity, the brane at the south pole is chosen to be a pure tension brane without dilaton coupling, for which no regularization is required as it only leads to a conical defect of size $$\alpha_-=1-\frac{\kappa^2}{2\pi}\mathcal{T}_- \,.$$ The northern brane, which breaks SI, is regularized and now sits near the north pole at the coordinate position $ \rho_+ $, corresponding to a proper circumference $\ell \equiv 2 \pi B_+ >0$.[^7] The position of the (regular) axis at the north pole is denoted by $\rho_0$ $(<\rho_+)$. We can perform a shift of the $\rho$ coordinate such that $\rho_0=0$. Figure \[fig:rugby\_ball\] depicts the regularized bulk geometry for the exemplary parameter choice . The interior of the ring (red/dark) is almost flat, whereas the exterior (green/bright) has the usual rugby ball shape.
Since the delta function $ \delta_+ \equiv \delta(\rho - \rho_+) $ is now localized at the position of the finite width ring, the regularized equations of motion are then formally identical to those presented in Sec. \[sec:thin\_brane\], apart from one crucial further modification: In order to prevent the ring from collapsing, it is necessary to introduce an angular pressure component, i.e. to add the term $$\label{eq:add_p_theta}
\frac{\delta_+}{2\pi B} \, p_\theta$$ to the right hand side of the $ {(\!\begin{smallmatrix}\theta\\#2\end{smallmatrix}\!)} $ Einstein equation . A possible way of modeling such a stabilization microscopically was first given in [@Scherk:1978ta] and later also applied to the SLED model [@Burgess:2008yx]: The idea is to introduce a localized scalar field that winds around the compact brane dimension and is subject to nontrivial matching conditions. As a result, shrinking the extra dimensions causes the related field energy to increase, hence implying a stable configuration with finite ring size. By integrating out the scalar field, it was explicitly shown in [@Burgess:2008yx] that it contributes to the ($\phi$-dependent) tension on the brane and leads to a pressure in angular direction. The tension shift can be taken care of by an appropriate renormalization, and the whole stabilizing sector is then solely characterized by an angular pressure component $p_\theta$. Thus, without loss of generality, we will work with the renormalized theory. As argued in [@Burgess:2008yx], the value of $ p_\theta $ needed to stabilize the ring can be inferred from the Einstein equations.
The junction conditions across the brane can be readily derived and read[^8]
\[eq:matching\] $$\begin{aligned}
[B\phi']_{\rm disc} &= \frac{\kappa^2}{2\pi}\mathcal{C}_+\,, \label{eq:jump_phip}\\
4[B (\ln W)']_{\rm disc} &=\frac{\kappa^2}{2\pi}\,p_{\theta} \,, \label{eq:jump_Wp}\\
[B']_{\rm disc} &=-\frac{\kappa^2}{2\pi}\left[ \mathcal{T}_+(\phi)+\frac{3}{4}\,p_{\theta}\right]_{\rho=\rho_+}\,, \label{eq:jump_Bp}
\end{aligned}$$
where we introduced the notation $$[f]_{\rm disc}:=\lim\limits_{\epsilon\to 0}\left[f(\rho_+ + \epsilon) - f(\rho_+ - \epsilon)\right]\,,$$ for any function $ f(\rho) $. Furthermore, we have to impose appropriate boundary conditions at both axes. Since the north pole is regularized, the corresponding axis (at coordinate position $ \rho=0 $) is required to be elementary flat, i.e. $$\begin{aligned}
\label{eq:bdry_N}
\phi'_0=0\,, && W'_0=0\,, && B'_0=1\,,&& B_0=0\,.\end{aligned}$$ In general, the unregularized south pole (at coordinate position $ \rho=\rho_- $) features a conical singularity characterized by $$\begin{aligned}
\label{eq:bdry_S}
\phi'_-=0\,, && W'_-=0\,, && B'_-=-\alpha_-\,,&& B_-=0\,.\end{aligned}$$ Note that only three of the four boundary conditions at each axis are independent, due to the radial Einstein constraint . Let us now count the total number of integration constants: There are two second order and one first order equation, leading to a total of five a priori undetermined integration constants. In addition, there is one integration constant included in the metric ansatz , namely $\hat R$. All of them are fixed by imposing the six independent boundary conditions stated above. The closed system for $\phi$, $W$ and $B$ is thus given by the off-brane ($ \rho\neq\rho_b $) equations and , the junction conditions across the ring and the boundary conditions and at the north and south pole, respectively.
After fixing the above boundary conditions, we are left with a one-parameter family of solutions, parametrized by the Maxwell integration constant $ Q $. However, it cannot be chosen freely, because it contributes to the total flux $ \Phi_\mathrm{tot} := \int\!{\mathrm{d}}\rho\,{\mathrm{d}}\theta\, F_{\rho\theta} $, which is subject to the flux quantization condition [@Randjbar:1983; @Burgess:2011va], $$\label{eq:flux_quant}
\Phi_\mathrm{tot} = 2\pi Q \int \!{\mathrm{d}}\rho \, \frac{{\mathrm{e}}^{\phi} B}{W^4} + \left[ \mathcal{A}_+(\phi) {\mathrm{e}}^{\phi} \right]_{\rho=\rho_+} \stackrel{!}{=} \frac{2\pi n}{\tilde e} \qquad (n \in \mathds{N}) \,,$$ where in general the U(1) gauge coupling $\tilde e$ can be different from $e$.
4D Curvature
------------
The 4D curvature is crucial in studying the phenomenological viability of the model, so let us again derive its relation to the brane couplings, but now for the regularized model. Repeating the derivation that lead to in the thin brane setup, and taking into account , we now find $$\label{eq:R_hat_delta_2}
V \hat R = \kappa^2 \left( 2 \mathcal{C}_+ + p_\theta \right) \,.$$ We see that the regularized expression is only modified by the last term proportional to $p_{\theta}$. Next, we will also express $ p_\theta $ in terms of the brane couplings in the thin brane limit.
Angular Pressure and Delta Limit {#sec:delta_limit}
--------------------------------
The aim of this section is to explicitly check whether the above relations are compatible with the delta results of [@Niedermann:2015via], and to gain further intuition about the regularized system and its stabilization. This will in turn allow us to narrow down physically interesting dilaton couplings.
Whether the brane looks pointlike to a good approximation is determined by the hierarchy between brane and bulk size, i.e. by the dimensionless ratio $ \epsilon := \ell^2 / V $. Thus, the delta limit corresponds to $ \epsilon \to 0 $, and can be realized by letting $ \ell \to 0 $ and/or $ V \to \infty $. In this work, we will keep $ \ell $ fixed at a value not smaller than the bulk Planck length,[^9] and let $ V $ become large.
Let us first check whether the matching conditions are compatible with the delta results in the limit $\epsilon \to 0$. Since the geometry is close to flat space in the vicinity of the regularized axis, we assume[^10] $$\begin{aligned}
\label{eq:rho_deriv_in}
\lim_{\rho\nearrow\rho_+}\phi' = \mathcal{O}(\epsilon) \,, && \lim_{\rho\nearrow\rho_+}W' = \mathcal{O}(\epsilon)\,, && \lim_{\rho\nearrow\rho_+}B' = 1 + \mathcal{O}(\epsilon) \,.\end{aligned}$$ In that case, Eq. indeed reduces to the dilaton boundary condition as $ \epsilon \to 0 $. On the other hand, Eqs. and show that the boundary conditions for $W$ and $B$ are again modified by a term proportional to $p_{\theta}$. This was also observed in [@Burgess:2008yx].
At this point several remarks are in order:
- The delta results [@Niedermann:2015via] are recovered if and only if $\lim\limits_{\epsilon \to 0} p_\theta =0$.
- The occurrence of $p_\theta$ is expected, and a mere consequence of regularizing the setup as a ring. It has the clear physical interpretation as the angular pressure that is needed to stabilize the compact dimension.
- From a physical perspective, there is no understanding of an angular pressure for an infinitely thin object. As a result, we expect the pressure to vanish whenever there is a large hierarchy between the bulk size $V$ and the regularization scale $\ell$. This expectation is in accordance with the above observation that for $p_\theta \to 0$ all results of the delta analysis are recovered. Our present analysis allows to go beyond physical expectations and to explicitly take the thin brane limit.
- For the physically relevant class of exponential couplings (which admit a small 4D curvature and a large bulk volume), we will confirm the above expectation by showing $\lim\limits_{V \to \infty} p_\theta = 0$. This result also confirms the correctness of the delta approach in [@Niedermann:2015via] within this class of couplings. While it is possible to construct examples in which $ p_\theta \nrightarrow 0 $, these are typically plagued by some sort of pathology, like a runaway behavior or a diverging brane energy (cf. Sec. \[sec:near\_SI\]). Again, this is not very surprising, as there is no meaningful notion of a pointlike angular pressure.
- The authors of [@Burgess:2015kda] instead argued that $ p_\theta $ should be nonzero for SI breaking delta branes. We comment on this in Appendix \[ap:Cliff\].
We will now derive an expression for $p_\theta$ in terms of the dilaton coupling. This in turn enables us to identify and discuss those couplings that are compatible with the delta description. As we will see, these are also just the ones that lead to small $ \hat{R} $.
As pointed out in [@Burgess:2008yx], an expression for $p_\theta$ can be found by evaluating the radial Einstein constraint in the limit $\rho \searrow \rho_+$: $$\begin{gathered}
3\left( \kappa^2 p_{\theta} \right)^2 - 8 \left(2 \pi - \kappa^2 \mathcal{T}_+ \right) \kappa^2 p_\theta + 4 \kappa^4\, \mathcal{C}^2_+\\
-\epsilon\, 8 V \hat R + \epsilon\, 4 \kappa^2 V {\mathrm{e}}^{\phi_+} \left( Q^2 - \frac{4 e^2}{\kappa^4} \right) = \mathcal{O}(\epsilon)\,,\end{gathered}$$ where we used and to express the radial derivatives through the brane fields. The terms in the second line are suppressed by $\epsilon$ and can be neglected in the delta limit. Solving for $ p_\theta $, we find $$\begin{aligned}
\label{eq:ptheta1}
\kappa^2 p_\theta = \frac{4}{3}\left\{ \left(2\pi-\kappa^2 \mathcal{T}_+ \right)\pm \sqrt{ \left(2\pi-\kappa^2 \mathcal{T}_+ \right)^2-\frac{3}{4} \kappa^4\, \mathcal{C}^2_+}\right\} + \mathcal{O}(\epsilon)\,\end{aligned}$$ where the branch was chosen such that the delta result $ p_\theta = 0 $ is recovered for SI couplings in the limit $ \epsilon \to 0 $.[^11] For vanishing BLF this coincides with the result derived in [@Burgess:2008yx].
An important observation from the above equation is that for finite $\epsilon$ and SI couplings in general[^12] $p_\theta = \mathcal{O}(\epsilon)\neq 0$. The physical reason is that introducing a brane width in general requires a stabilizing angular pressure.
The requirement of being close to SI can be made more precise by defining a near SI regime according to $$\begin{aligned}
\label{eq:scale_invariant_limit}
\kappa^2 \mathcal{C}_+ \ll 1 \,.\end{aligned}$$ This in turn leads to an approximate expression for the stabilizing pressure, $$\begin{aligned}
\label{eq:ptheta2}
p_\theta = \frac{\kappa^2}{4\pi} \left(1-\frac{\kappa^2 \mathcal{T}_+}{2 \pi} \right)^{-1} \mathcal{C}^2_+ + \mathcal{O}(\epsilon) + \mathcal{O}(\mathcal{C}_+^4) \,.\end{aligned}$$ After inserting this into the formula for $\hat R$ in , we arrive at $$\label{eq:R_hat_delta_3}
\boxed{
V \hat R = 2 \kappa^2\,\mathcal{C}_+ + \frac{1}{4\pi} \left(1-\frac{\kappa^2 \mathcal{T}_+}{2 \pi} \right)^{-1} \kappa^4\,\mathcal{C}^2_+ +\mathcal{O}(\epsilon) + \mathcal{O}(\mathcal{C}_+^4)
}\,.$$ By comparing to its delta counterpart , we find two small corrections:
(i) a term quadratic in $\mathcal{C}_+$ and hence suppressed (in the near SI regime) relative to the leading linear term;
(ii) generic order $\epsilon$ contributions caused by the finite brane width.
Which of the two dominates depends on the details of the dilaton coupling. Later, we will find that both possibilities can be realized.
In summary, we have shown that the delta result for $\hat R$ receives two corrections which are small in the near SI regime (which we intend to study) and for a large hierarchy between the brane size and extra space volume.
Modeling Near Scale Invariance {#sec:near_SI}
------------------------------
As expected, the near SI regime is of superior phenomenological importance as it leads to parametrically small values of the 4D curvature due to . We look for a dilaton coupling which allows to keep the SI breaking effects small without introducing an a priori hierarchy of the coupling parameters. In principle, this can be realized by using exponential couplings [@Burgess:2015nka; @Burgess:2015lda], i.e.$$\begin{aligned}
\label{eq:BLFcoupling}
\mathcal{T}_+(\phi) = \lambda_+ + \tau\, {\mathrm{e}}^{\gamma\phi} &&\text{and} && \mathcal{A}_+(\phi) = \Phi_+ {\mathrm{e}}^{- \phi}\, ,\end{aligned}$$ with $\phi$-independent (and SI) tension $\lambda$ and constant parameters $ \gamma $, $ \tau $ and $\Phi_+$. For $ \tau $ and $ \gamma \neq 0$ the tension term breaks SI explicitly. We see that even for (a naturally) large $\tau$, the SI breaking given by $\mathcal{T}'_+$ becomes small when $\phi_+$ is sufficiently negative. This makes the exponential couplings interesting with respect to the CC problem.
By contrast, the BLF term preserves SI. Technically, we could have introduced the SI breaking also via the BLF term, which would lead to the same outcome.[^13] However, it should be noted that it is physically more imperative to include a SI breaking tension as we expect loops of localized brane matter, which in general breaks SI,[^14] to contribute to $\tau$. In other words, there is no obvious way of having $\tau$ small without imposing a fine-tuning. As a consequence, when looking for natural solutions, we have to consider a $\phi $-dependent tension with generic coefficient $\tau$. On the other hand, in the case of the BLF term, it depends on the details of the matter theory whether we expect loop corrections to $\Phi_+$. Following the discussion in [@Burgess:2015lda], if the matter fields are not coupled directly to the Maxwell sector, there might be a chance of keeping SI breaking contributions to $\mathcal{A}_+$ small. In any case, including a breaking via the BLF term would, due to , yield an additional contribution to $\hat R$ and, as we will see, would make it even more difficult to comply with the observational constraints.
With these couplings we find $$\label{eq:C_near_SI}
\mathcal{C}_+ = \tau \gamma\, {\mathrm{e}}^{\gamma\, \phi_+} \,,$$ leading to an angular pressure $$\begin{aligned}
\label{eq:ptheta3}
p_\theta = \frac{\kappa^2}{4\pi \alpha_+} \left(\tau \gamma {\mathrm{e}}^{\gamma \phi_+}\right)^2+ \mathcal{O}(\epsilon) + \mathcal{O}(\mathcal{C}_+^3)\,,\end{aligned}$$ where $ \alpha_+:= 1-\frac{\kappa^2}{2\pi}\lambda_+ $. The numerical analysis we conduct in this work (cf. Sec. \[sec:num\_results\]) will show emphatically that the volume obeys[^15] $$\begin{aligned}
\label{eq:volume_scaling}
V \propto {\mathrm{e}}^{-\phi_+} \,,\end{aligned}$$ hence implying $$\label{eq:p_theta_scaling}
p_\theta \propto
\begin{cases}
V^{-2\gamma} & \qquad (\text{for}\; 0 < \gamma <1/2) \\
V^{-1} & \qquad (\text{for}\; \gamma=0 \text{ or } \gamma > 1/2)
\end{cases} ,$$ asymptotically for $ V/\kappa \gg 1 $. The second line follows from the observation that for $\gamma>1/2$ the first expression in becomes sub-dominant compared to the $\mathcal{O}(\epsilon)$ contribution. The case $\gamma=0 $ is special as it corresponds to a SI coupling, where SI is only broken by the regularization. From it is clear that it is not continuously connected to $\gamma \neq 0 $ because the first term vanishes identically (irrespective of the value of $V$). In both cases, $\gamma=0 $ and $\gamma>1/2 $, the exponent saturates to the constant value $-1$.
The above formula allows us to discuss the consistency of the delta limit. We distinguish two cases:
1. For $\gamma \geq 0$, increasing the volume of the compact space leads to a decreasing angular pressure. In other words, when we make the hierarchy between transverse brane size and bulk volume large, the angular pressure tends to zero in accordance with the physical expectation. Moreover, in this limit the SI case is approached (since $ {\mathcal{C}}_+ \propto \gamma V^{-\gamma} \to 0 $), which renders the above approximations more and more accurate. As an aside, note that this observation, i.e. the concurrency of $p_\theta$ being small and having a small amount of SI breaking, is the loophole to the objections raised in [@Burgess:2015kda]. We discuss this more extensively in Appendix \[ap:Cliff\].
2. For $ \gamma < 0 $ the situation is different: If $\tau>0$, the system eventually hits a point (just before it becomes super-critical) where yields no real solution for $ p_\theta $ anymore, indicating a runaway behavior. Therefore, a discussion of that case requires the inclusion of a general time dependence of the fields which is beyond the scope of this work.
On the other hand, if $\tau<0$, there are static solutions for which $p_\theta$ grows as $V$ is increased due to . This is related to the observation that the system gets driven away from SI ($ {\mathcal{C}}_+ \to \infty $). As a result, the 4D curvature $\hat R$ cannot be kept under control for a phenomenologically large $V$ unless the coefficient $\tau$ is tuned to be extremely small. Moreover, the tension tends to $- \infty $ in this case which strongly questions the physical consistency of these solutions. So this case is not interesting, neither phenomenologically nor with respect to the tuning issue.
In summary, the exponential coupling with $\gamma \geq 0$ is of particular interest, as it allows to be close to SI, which is important to make the 4D curvature parametrically small. This is achieved by considering a sufficiently large bulk volume. Other types of couplings (including monomial and exponential ones with $ \gamma <0$) either lead to a runaway behavior or are incompatible with being close to SI (if the coefficient is not tuned to be small). The above discussion also shows that the physically relevant class of couplings is compatible with the delta description because $p_\theta$ (or any hidden metric dependence of the delta function as argued in [@Burgess:2015kda]) vanishes for $V \to \infty$.
Phenomenology {#sec:phen}
-------------
We have singled out the exponential tension-dilaton coupling as the phenomenologically relevant one, since its contribution to the 4D curvature can be made arbitrarily small. Let us now discuss whether this can lead to phenomenologically viable solutions.
At the present stage, there are two main phenomenological inputs the model has to comply with:
(1) In models with large extra dimensions the weakness of 4D gravity is a result of the large extra dimensions. This is possible because the 4D Planck mass is given, via dimensional reduction, by [@Burgess:2015lda] $$\begin{aligned}
\label{eq:Mp_phen}
{M_\mathrm{Pl}}^2 = \frac{V}{\kappa^2}\,.
\end{aligned}$$ Given present tests of the gravitational inverse square-law [@Kapner:2006si] (see [@Adelberger:2003zx] for a review), the upper bound on the size of the extra dimensions is of order of ten microns. Then, implies that the bulk gravity scale $\kappa^{-1/2}$ is not allowed to be significantly below $ \sim 10 \;\mathrm{TeV} $, which translates into the upper bound $$\begin{aligned}
\label{eq:V_phen}
\frac{V}{\kappa} \lesssim 10^{28} \,.
\end{aligned}$$
(2) The observed value of the 4D curvature measured in Planck units is notoriously small, viz. [@Ade:2013zuv] $$\begin{aligned}
\label{eq:R_phen}
\frac{\hat R}{{M_\mathrm{Pl}}^2} \sim 10^{-120}\,.
\end{aligned}$$
Let us now study whether the model is compatible with both requirements. For convenience, we will set $ \kappa = 1 $, i.e. here and henceforth dimensionful quantities are all measured in units of the bulk gravity scale.
We now make use of our central formula which permits to express the 4D curvature in terms of the extra space volume. Using , as well as , we then find that the leading contribution is $$\label{eq:R_scaling}
\frac{\hat R}{{M_\mathrm{Pl}}^2} = N_1 V^{-(2+\gamma)} + N_2 V^{-3} \,,
$$ where $ N_i $ are dimensionless coefficients, with $$\begin{aligned}
\label{eq:N12}
N_{1} \propto \gamma\tau && \text{and} && N_2 \propto \ell^2 \,.\end{aligned}$$ The unknown constants of proportionality are due to the unknown coefficients in and the $ \mathcal{O}(\epsilon) $ term in , respectively. For model parameters which do not contain a priori hierarchies among themselves, we expect them to be roughly $ \sim 1 $. While at this point it is merely a reasonable expectation, it will also be confirmed by the numerical solutions discussed in Sec. \[sec:num\_results\], which allow us to explicitly calculate these coefficients. The relation is one of the main results of this work. The two phenomenological bounds above then require $$N_1 \times 10^{-28(2+\gamma)} + N_2 \times 10^{-84} \lesssim 10^{-120}\,.$$ One way how this could in principle be fulfilled is by assuming a cancellation of the two terms. However, this would only be achieved by tuning the parameters $ \gamma $ and $ \tau $ very accurately. Therefore, we dismiss this possibility and demand both terms to fulfill the bound separately. From we know that the first term vanishes identically for a SI coupling ($ \gamma\tau = 0 $). If SI is broken, it could only comply with the bound without tuning $ N_1 $ (and thus $ \tau $) if $ \gamma \gtrsim 2.3 $.[^16] The second term, however, is more problematic: it implies that $ N_2 \lesssim 10^{-36} $. As expected from , and explicitly confirmed in Sec. \[sec:num\_results\], this could only be achieved by assuming the brane width $ \ell $ to be $ \sim 18 $ orders of magnitude smaller than the bulk Planck length. Not only would this again correspond to introducing an a priori hierarchy by hand, but also question the applicability of a classical analysis.
As a result, *if we do not allow the model parameters to be fine-tuned or to introduce large hierarchies, the model is ruled out phenomenologically*. Either the 4D curvature or the size of the extra dimensions would be too large to be phenomenologically viable.
Before concluding this sections, let us summarize the assumptions that went into this result:
- The interior profiles are close to their flat space estimates with corrections $\mathcal{O}(\epsilon)$, cf. Eq. .
- Motivated by the GGP result, the extra space volume is assumed to be proportional to $ {\mathrm{e}}^{-\phi_+}$, cf. Eq. .
- The coefficients in are of order unity.
They are all quite reasonable, and will indeed all be explicitly confirmed by our numerical analysis. Moreover, the numerical treatment will allow us to infer the amount of tuning (due to flux quantization) that is required to get a sufficiently small 4D curvature (albeit corresponding to a too large $ V $).
Numerical Results and Fine-Tuning {#sec:num_results}
=================================
In this section we present the results of our numerical studies of the regularized model and discuss their physical implications for the SLED scenario. We will first briefly sketch the numerical algorithm in Sec. \[sec:num\_alg\]. Next, in Sec. \[sec:SI\], we will discuss the simple case of SI brane couplings. In this case we know the exact analytic solutions for infinitely thin branes—the GGP solution, reviewed in [@Niedermann:2015via]—and so this provides a useful consistency check for our numerical solver. Finally, Sec. \[sec:non\_SI\] addresses the actual case of interest: a SI breaking tension. We derive the solutions of the full brane-bulk system without relying on any approximations, which in turn enables us to explicitly test (and confirm) the analytical approximations and results of the last section.
Numerical Algorithm and Parameters {#sec:num_alg}
----------------------------------
The goal is to determine the $ \rho $-profiles of the dilaton $ \phi $ and of the metric functions $ B $ and $ W $ for given model parameters. As explained above, this requires solving the bulk equations , , supplemented by the junction conditions and the boundary conditions , . We do so by starting at the north pole ($ \rho = 0 $) and integrating outward using the second order equations.[^17] Since the constraint is analytically conserved, it only needs to be imposed initially at $ \rho=0 $. For $ \rho > 0 $ it can then be used as a consistency check (or error estimator) of the numerical solution. At $ \rho=\rho_+ $, however, the constraint must be used once again, because it determines the stabilizing pressure $ p_\theta $. In other words, when the integration reaches $ \rho \nearrow \rho_+ $, the three junction conditions must be supplemented by the constraint (evaluated at $ \rho \searrow \rho_+ $) in order to determine the three exterior $ \rho $-derivatives and $ p_\theta $. Afterwards, the integration continues until $ B \to 0 $, defining the south pole $ \rho = \rho_- $.
Before the equations can actually be integrated in this way, we need to specify the three a priori unknown integration constants $ \phi_0 $, $ Q $ and $ \hat{R} $. In general, however, all of them are ultimately fixed via (the SI case is exceptional, see Sec. \[sec:SI\])
(i) flux quantization ,
(ii) regularity at the south pole, i.e., $ \phi'_- = 0 $,[^18]
(iii) the correct conical defect at the south pole, i.e., $ B'_- = -\alpha_- $.
Technically, this can be achieved by a standard shooting method: we choose some initial guesses for $ \phi_0 $, $ Q $ and $ \hat{R} $; after integrating the ODEs, the violations of (i)–(iii) can be computed, and finally be brought close to zero via an iterative root-finding algorithm.
In this way—and in agreement with the discussion in Sec. \[sec:ring\_reg\]—since there are no integration constants left (in the non SI case), we also see that the full solution is uniquely determined for a given set of model parameters. These consist of the bulk couplings $ \kappa = 1$ (in our present units), $ e $, the regularization width $ \rho_+ $, the brane couplings, parametrized by $ \alpha_\pm $, $\tau$, $\gamma$ and the BLF parameter $ \Phi_+ $, as well as the gauge coupling $ \tilde e $. Since the latter only enters via flux quantization , it is convenient to introduce the abbreviation $${\mathcal{N}}:= \frac{2\pi n}{\tilde e} \,,$$so that flux quantization simply reads $ \Phi_\mathrm{tot} = {\mathcal{N}}$.
Note that the solution would *not* be determined uniquely if, for instance, the boundary conditions ensuring regularity at the south pole were neglected. In this case, it would not be possible to numerically predict the value of $ \hat{R} $, since it could be chosen freely. Thus, in order to *compute* this quantity numerically, it is crucial to find complete, regular bulk solutions. To our knowledge, this is done here for the first time.[^19]
The main question is whether it is possible to find solutions for which $ \hat R $ is small enough and $ V $ is large enough to be phenomenological viable without fine-tuning, i.e. for generic values of the model parameters. For definiteness, and in order not to introduce any large hierarchies into the model by hand, we will choose the following parameters, $$\begin{aligned}
\label{eq:param}
e = 1\,, && \rho_+ = 1\,, && \Phi_+ = -0.6\,, && \tau = 0.9\times 2 \pi\,, &&\alpha_+= 0.9 && \text{and} && \alpha_- = 0.5 \,.\end{aligned}$$ (Somewhat different values would not change the main results, though.) The parameter $ {\mathcal{N}}$, determining the total flux, will be varied, and used as a dial to achieve different values of $ \hat{R} $ and $ V $.
![Complete numerical solutions of the coupled Einstein-dilaton system for the parameters and $ \gamma = 0.2 $. The axis at the north pole ($ \rho = 0 $) is regular ($ W'=\phi'=0 $) and elementary flat ($ B'=1 $), while the axis at the south pole is regular but has a defect angle corresponding to the unregularized pure tension brane ($ B'=-0.5 $); the regularized brane sits at $ \rho_+ = 1 $, and produces jumps in the $ \rho $-derivatives. The orange (light) and purple (dark) curves correspond to $ V=8\pi $ and $ V = 16\pi $, respectively (which were obtained for $ {\mathcal{N}}= -1.102 $ and $ {\mathcal{N}}= -0.885 $). The required 4D curvature was $ \hat{R} = 0.0571 $ and $ 0.0233 $, respectively. The constraint violation, i.e. the numerical deviation of from zero, was always smaller than $ 10^{-10} $ in this example, and the numerical error bars would not exceed the line widths in the plots. []{data-label="fig:profiles"}](figures/profiles.pdf){width="\textwidth"}
An exemplary numerical solution is shown in Fig. \[fig:profiles\], where the three functions $ B, W, \phi $, as well as their $ \rho $-derivatives are plotted, for $ \gamma = 0.2 $ and two different choices of $ {\mathcal{N}}$, leading to two different values of $ V $, as is evident from the profile of $ B $. Since we chose $ \alpha_+ \neq \alpha_- $, the solutions are warped—both $ W $ and $ \phi $ have nontrivial profiles.[^20]
Furthermore, one can already see that the profiles inside the regularized brane ($ \rho < \rho_+ $) become more trivial as $ V $ increases, as expected. This trend continues, and all functions and their derivatives at $ \rho \nearrow \rho_+ $ were always found to approach the corresponding values at the regular axis ($ \rho = 0 $) like $ V^{-1} $ for $ V \to \infty $, thereby confirming .
All of the $ \rho $-derivatives are discontinuous at the regularized brane ($ \rho = \rho_+ $), as required by the junction conditions . $ B' $ consistently approaches $ -\alpha_- = -0.5 $ at the south pole and, most importantly, both $ W' $ and $ \phi' $ vanish there, as required by regularity. By running the numerics similarly for different choices of $ \gamma $ and $ {\mathcal{N}}$, we can now systematically learn how these model parameters determine $ \hat{R} $ and $ V $.
Scale Invariant Couplings and Thick Branes {#sec:SI}
------------------------------------------
Let us first consider the case $ \tau = 0 $ corresponding to a SI tension $ \mathcal{T}_+ = 2\pi(1-\alpha_+) $. Incidentally, in this case the dilaton profile is regular, and so the solution can even be obtained for the idealized, infinitely thin brane, as already discussed in [@Niedermann:2015via]. It is given by the GGP solution [@Gibbons:2003di], for which $ \hat R = 0 $. In that case, the integral in the flux quantization condition can be performed explicitly, yielding $$\label{eq:flux_quant_GGP}
\frac{2 \pi}{e}\sqrt{\alpha_+\alpha_-} + \Phi_+ = {\mathcal{N}}\,.$$ The dilaton integration constant $ \phi_0 $ drops out of all equations due to SI, and thus the above counting of constants does not add up, resulting in the tuning relation among model parameters. If we chose parameters which do not fulfill this equation, there would not be a static solution, in accordance with the expected runaway behavior à la Weinberg [@Weinberg:1988cp]. In turn, the extra space volume $ V $, which turns out to be $ \propto {\mathrm{e}}^{-\phi_0} $ [@Niedermann:2015via], can be chosen freely. As a result, this model could have a phenomenologically viable volume (although a vanishing 4D curvature is not compatible with observations), but only at the price of a new fine-tuning. If SI is broken, things will change: on the one hand, $ \phi_0 $ will be fixed, and thus the tuning relation is expected to disappear. On the other hand, the volume $ V $ will also be determined, and $ \hat R $ is expected to be nonzero. The question then is if they can satisfy the phenomenological bounds presented in Sec. \[sec:phen\], and if so, whether this can be achieved without introducing yet another tuning.
Let us now present the numerical results for a regularized brane with $ \tau = 0 $ \[all other parameters as in \]. In that case SI is already broken by introducing a regularization scale $\ell$. Thus, the above discussion applies here as well: $ \phi_0 $ and $ V $ are fixed in terms of model parameters. Moreover, we expect $\hat R \neq 0$ due to $ \mathcal{O}(\epsilon)$ contributions caused by the finite brane width.[^21] However, if the thin brane limit is taken by letting $ V \to \infty $ (which can be achieved by adjusting $ {\mathcal{N}}$ appropriately), these effects should become suppressed, and we expect to recover the GGP solution with $ \hat{R} = 0 $. This is exactly what happens, as can be seen from Fig. \[fig:R0\]. Specifically, we find that $ \hat{R} \propto V^{-2} $ as $ V \to \infty $. Furthermore, the angular pressure $ p_\theta $ (not shown) is also nonvanishing, but goes to zero like $ V^{-1} $. These findings are in complete agreement with the analytic predictions , (with $ {\mathcal{C}}_+ = 0 $).
At the same time, the tuning relation is also violated, and the static solutions exist for any choice of parameters. But again this violation, $$\begin{aligned}
\delta\Phi := \Phi_\mathrm{GGP} - {\mathcal{N}}\,, && \text{with} && \Phi_\mathrm{GGP} := \frac{2 \pi}{e}\sqrt{\alpha_+\alpha_-} + \Phi_+ \,,\end{aligned}$$ vanishes (like $ V^{-1} $) as $ V \to \infty $, see Fig. \[fig:Flux0\].[^22]
In summary, we explicitly confirmed that introducing a regularization leads to $ \mathcal{O}(\epsilon) $ corrections of the GGP predictions ($ \hat R = 0 $, $ \Phi_\mathrm{GGP} = {\mathcal{N}}$, $ p_\theta = 0 $). In particular, this agrees with the analytic result of [@Niedermann:2015via] that $ \hat{R} = 0 $ is only guaranteed in the SI delta model (which is approached as $ \epsilon \to 0 $) via a tuning of model parameters ($ \Phi_\mathrm{GGP} = {\mathcal{N}}$). Furthermore, this simple example already shows that a stabilizing pressure $ p_\theta $ is necessary for a thick brane, but also that $ p_\theta \to 0 $ as $ \epsilon \to 0 $, allowing for a consistent delta description as in [@Niedermann:2015via].
But now we can even make a precise statement about the required tuning beyond the idealized delta brane limit. The phenomenological bound together with yields (recall that we are working in units in which $ \kappa = 1 $) $$\label{eq:RV_phen}
10^{-120} \stackrel{!}{\sim} \frac{\hat R}{V} \sim \delta\Phi^3 \,,$$ where the second estimate used (and extrapolated) our numerically inferred scaling relations (neglecting the $ \mathcal{O}(1) $ coefficients), cf. Fig. \[fig:SI\_coupling\]. Therefore, the parameter $ {\mathcal{N}}\equiv 2\pi n / \tilde e $ must be tuned close to $ \Phi_\mathrm{GGP} \equiv \frac{2 \pi}{e}\sqrt{\alpha_+\alpha_-} + \Phi_+ $ with a precision of $ \sim 10^{-40} $. This is clearly not better than the CC problem we started with. It is crucial to note that this can also directly be read as a tuning relation for the brane tension $ \lambda $, since $ \alpha_+ = 1 - \lambda / 2\pi $.
But—as already anticipated in Sec. \[sec:phen\]—there is also another problem regarding phenomenology, even if we allow for such a tuning: For $ \delta\Phi \sim 10^{-40} $, the extra space volume would be $ V \sim 10^{40} $, grossly violating the bound . Thus, by tuning $ \hat R $ small enough, we have at the same time tuned the extra space volume 12 orders of magnitude larger than allowed. Alternatively, if we require $ V $ to satisfy the observational bound , $ \hat{R} $ would still be 36 orders of magnitude larger than what is observed. Hence, as it stands, the model suffers not only from a tuning problem, but is not even phenomenologically viable.
This nicely agrees with the analytic discussion in Sec. \[sec:phen\]. Explicitly, we confirmed the relation (here for $ \gamma = 0 $), finding the coefficient $ N_2 = 3.16 $ for this specific set of parameters, i.e. $ e $, $ \rho_+ $, $ \Phi_+ $ and $ \alpha_\pm $ as given in . Now, since the resulting failure to get both $ \hat{R} $ and $ V $ within their phenomenological bounds is the central result of this work, it is worthwhile to discuss its robustness.
First, it should be noted that the main reason for this result can be traced back to the $ \mathcal{O}(\epsilon) $ contributions to the 4D curvature $ \hat{R} $, cf. Eq. , which are caused by endowing the brane with a finite width. Hence, they are unavoidable in a (realistic) thick brane setup; of course, we did our explicit calculations only in one particular regularization, but the standard EFT reasoning suggests that the qualitative answer would be the same for any other reasonable regularization.[^23] While there are additional contributions to $ \hat{R} $ if the dilaton couplings break SI, see Eq. , they can only make things worse (unless there were a miraculous cancellation—a possibility that we dismiss in the search of a natural solution to the CC problem). Again, this will be explicitly confirmed in the following section.
Next, we checked numerically that the scaling relation, as well as the order of magnitude of the coefficient $ N_2 $ do not change if different tensions (i.e. other generic values for $ \alpha_\pm $) are chosen. Furthermore, the parameters $ \Phi_+ $ and $ e $ have no influence on the result at all; this is obvious for the BLF $ \Phi_+ $, but also easily seen for the gauge coupling $ e $ as follows: For the SI couplings we are considering here, the full (regularized) equations of motion enjoy the exact symmetry $$\begin{aligned}
\label{eq:SI_symm}
e \mapsto a e \,, && Q \mapsto a Q \,, && {\mathrm{e}}^{\phi} \mapsto \frac{1}{a^2} {\mathrm{e}}^{\phi} \,,\end{aligned}$$ for any constant $ a $. Hence, after changing $ e $, the new solution is simply obtained from the old one by rescaling $ Q $ and $ {\mathrm{e}}^\phi $ appropriately. Since the metric is unaltered, this leaves $ \hat{R} $ and $ V $ unchanged.[^24]
Hence, the only parameter that could change things is $ \rho_+ $, determining the regularization scale $ \ell \approx 2\pi \rho_+ $, in accordance with the discussion below Eq. .
Non Scale Invariant Couplings {#sec:non_SI}
-----------------------------
We now turn to the case $ \tau \neq 0 $ (and $ \gamma > 0 $),[^25] where SI is broken explicitly via the tension term. The hope is to find values of $ \gamma $ for which no tuning is required in order to achieve a large volume and small curvature. As argued above, this suggests focusing on $ \gamma > 0 $, because then $ V \to \infty $ drives the model towards the SI case which in turn implies $ \hat R \to 0 $. While this case was already discussed in Sec. \[sec:phen\] under certain reasonable assumptions, the numerical analysis independently confirms the previous results and allows to quantify the amount of tuning necessary to get a viable 4D curvature.
\
Figure \[fig:non\_SI\] shows the numerical results for different values of $ \gamma > 0 $. Again, small $ \hat{R} $ and large $ V $ are generically realized for $ \delta \Phi \to 0 $, i.e. if $ \Phi_\mathrm{GGP}$ is tuned close to $ {\mathcal{N}}$. Evidently, both quantities again show a power law dependence on $ \delta\Phi $, with exponents which now depend on $ \gamma $. Empirically, we find the following laws, $$\label{eq:non_SI_scaling}
\hat{R} \propto
\begin{cases}
\delta\Phi^{1 + 1/\gamma} \\
\delta\Phi^{2}
\end{cases} , \qquad
V \propto
\begin{cases}
\delta\Phi^{-1/\gamma} & \qquad (\text{for}\; 0< \gamma <1) \\
\delta\Phi^{-1} & \qquad (\text{for}\; 1 < \gamma)
\end{cases} ,$$ as $ \delta\Phi \to 0 $. These are plotted in Figs. \[fig:R\] and \[fig:V\] as dashed lines, and evidently provide very good fits to the numerical data points. Note that the scalings for $ \gamma > 1 $ are the same as the ones obtained in the SI case $ \tau = 0 $. The transition to this generic scaling law occurs because for $ \gamma > 1 $ the finite width effects (which are independent of $ \gamma $) dominate, cf. Sec. \[sec:delta\_limit\]. Also note that combining the scaling relations for $ \hat{R} $ and $ V $ exactly reproduces the analytic prediction . For completeness, let us mention that the corresponding numerical coefficients for $ N_1 $ in , i.e. the ratios $ N_1 / (\gamma\tau) $, were found in the range $ \sim 2 $ to $ 6 $. Likewise, the scaling relations for $ p_\theta $, which are drawn as dashed lines in Fig. \[fig:pTheta\], again agree very well with the data. Finally, Fig. \[fig:phi\] shows the relation between the dilaton evaluated at the brane and the volume, confirming . With these results, we can now turn to the tuning question. For $ \gamma > 1 $, the discussion is exactly the same as for the SI case ($ \tau = 0 $) above, because the scaling relations are the same. But for $ \gamma < 1 $ there is a modification: Using the scaling relations , the phenomenological bound now implies $$10^{-120} \sim \delta\Phi^{1+2/\gamma} \,.$$ For $ \gamma \lesssim 1 $, $ \delta\Phi $ still has to be tuned tremendously close to zero; but for $ \gamma \ll 1 $, this is not the case anymore. Specifically, if we choose $ \gamma \approx 1/60 $ (which is not hierarchically small), this relation is already fulfilled if $ \delta\Phi \sim 0.1 $, i.e. without any fine-tuning of model parameters. So we find the remarkable result that the near-SI tension is capable of producing a small 4D curvature and a large volume (as compared to the fundamental bulk scale) without fine-tuning, although this was not possible for a SI tension ($ \tau = 0 $). At first sight, this looks very promising. However, on closer inspection, there is an even bigger problem with the volume bound than before, since $ \gamma \sim 1/60 $ and $ \delta\Phi \sim 0.1 $ now yields $ V \sim 10^{60} $, exceeding the bound by 32 orders of magnitude. In turn, if we chose $ \gamma \sim 1/28 $, so that the volume satisfies the bound for $ \delta\Phi \sim 0.1 $, then $ \hat{R} \sim 10^{-57} {M_\mathrm{Pl}}^2 $, which is $ 63 $ orders of magnitude larger than its observational bound.
In summary, while it is possible to get small $ \hat{R} $ and large $ V $ without tuning $ \Phi_\mathrm{GGP} $ extremely close to $ {\mathcal{N}}$, it is not possible for both of them to satisfy their phenomenological bounds, in accordance with the general discussion in Sec. \[sec:phen\].
Let us note that this possibility of getting a large volume without large parameter hierarchies was also recently observed in [@Burgess:2015lda], where the same model was studied in a dimensionally reduced, effective 4D theory. However, there it was also assumed that it would at the same time be possible to have $ \hat{R} $ within its bounds (possibly via some independent fine-tuning), so that the model could in this way at least address the electroweak hierarchy problem (albeit not the CC problem). Here we found that this is not possible, because $ \hat{R} $ and $ V $ are not independent, and so one cannot tune $ \hat{R} $ without at the same time ruining the value of $ V $.
Conclusion {#sec:concl}
==========
The main result of our preceding work [@Niedermann:2015via] was that the SLED model (with delta branes) only guarantees the existence of 4D flat solutions if the brane couplings respect the SI of the bulk theory, and that this comes at the price of a fine-tuning (or runaway), as expected [@Weinberg:1988cp]. Here, we took one step further and asked how large the 4D curvature $ \hat{R} $ is for SI breaking couplings and the (more realistic) case of a finite brane width not below the fundamental 6D Planck length.
Specifically, we worked with a regularization which replaces the delta brane by a ring of stabilized circumference $ \ell $, and considered a SI breaking tension term parametrized as $ \mathcal{T}_+ = \lambda + \tau\, {\mathrm{e}}^{\gamma\phi_+} $. This type of dilaton-brane coupling is particularly interesting with respect to the CC problem as it allows to be close to SI without assuming an unnaturally small coefficient $\tau$. We then followed two complementary routes:
First, we analytically derived a formula for $ \hat{R} $. Motivated by the GGP solution, the extra space volume was then assumed to be proportional to $ {\mathrm{e}}^{-\phi_+} $. This resulted in the rigid relation between $\hat R$ and the extra space volume $V$, consisting of two $ V $-dependent contributions to $ \hat{R} $ with unknown numerical constants of proportionality $ N_1 $ and $ N_2 $. They originate from the SI breaking dilaton coupling and the finite brane width, respectively. Provided that $ N_{1/2} \sim 1 $, we found that either $ \hat{R} $ or $ V $ exceeds its phenomenological bound (by 36 or 12 orders of magnitude, respectively).
Second, we solved the full bulk-brane field equations numerically. By enforcing the correct boundary conditions at both branes, we were able to calculate all observables, in particular $ \hat{R} $ and $ V $, for given model parameters. We thereby confirmed the analytically derived scaling relations without relying on any approximations and were able to explicitly compute the coefficients $ N_{1/2} $, indeed affirming $ N_{1/2} \sim 1 $. The only way to get $ N_1 \ll 1 $ would be to either require SI brane couplings—which would ruin solar system tests due to a fifth force [@Burgess:2015lda]—or to fine-tune (either $ \tau $ or $ \lambda $). As for $ N_2 $, the only caveat is provided by allowing the brane width $ \ell $ to be much ($ \sim 18 $ orders of magnitude) smaller than the bulk Planck scale. This, however, would confront us with the problem how such a hierarchy could arise naturally, and whether one would have to take quantum gravity effects into account.
Moreover, the numerical analysis admitted an extensive discussion of the tuning issue. To be precise, we calculated the amount of tuning necessary to realize a large hierarchy between the bulk scale and $ V $, as is phenomenologically required according to , with the following results:
- For SI couplings ($\tau=0$) a sufficiently large $V$ is only achieved by tuning the total flux (or, equivalently, the brane tension) close to the corresponding GGP value with a precision of $ \sim 10^{-28} $.
- If SI is broken explicitly by a $\phi$-dependent tension, it turns out that the tuning problem can in fact be avoided for near SI tension couplings $ \gamma \ll 1 $, in agreement with [@Burgess:2015lda]. However, the phenomenological problem still persists (and even gets worse). Explicitly, for $ \gamma \sim 1/28 $, which yields the required volume without tuning, $ \hat{R} $ would be 63 orders of magnitude above its measured value.
In summary, there are no phenomenologically viable solutions in the SLED model if the brane width is not smaller than the fundamental bulk Planck length. But even if this were allowed, the required SI breaking dilaton coupling of the brane fields would always lead to a way too large 4D curvature or extra space volume, unless some sort of fine-tuning is at work.
We thank Cliff Burgess, Ross Diener, Stefan Hofmann, Tehseen Rug and Matthew Williams for many helpful discussions. The work of FN was supported by TRR 33 “The Dark Universe”. The work of FN and RS was supported by the DFG cluster of excellence “Origin and Structure of the Universe”.
Validity of Delta-Analysis {#ap:Cliff}
==========================
The authors of [@Burgess:2015kda] critically assessed our preceding work [@Niedermann:2015via] based on a delta-analysis.[^26] Specifically, they argued that the unregularized approach did not take into account a hidden metric dependence of the delta-function of the form $$\begin{aligned}
\frac{\partial \delta^{(2)}(y)}{\partial g_{\theta\theta}} =: C \,\frac{\delta^{(2)}(y)}{B_+^2} \;,\end{aligned}$$ which would introduce an additional (localized) term in the $(\theta\theta)$-Einstein equation. In that case, the constant $C$ would be constrained by the radial Einstein equation in terms of the brane tension; specifically, we find[^27] $$\begin{aligned}
\label{eq:C2}
\mathcal{T}_+ C \simeq - \frac{\kappa^2}{8\pi} \frac{\mathcal{T}'^2_+}{\left(1-\frac{\kappa^2 \mathcal{T}_+}{2 \pi}\right)}\;,\end{aligned}$$ where higher order terms in $ \mathcal{T}'_+ $ were neglected.
The first important observation is that $ C $ vanishes for $ \mathcal{T}'_+ = 0 $. This shows that the concerns of [@Burgess:2015kda] do not apply to the SI case. So one of the central results of [@Niedermann:2015via], namely that $ \hat{R} = 0 $ for SI delta branes (and not for dilaton-independent couplings, as had been claimed previously [@Burgess:2011mt; @Burgess:2011va]), is insensitive to this issue.
But it also looks as if assuming $C=0$, as implicitly done in [@Niedermann:2015via], would be in conflict with the SI breaking case $ \mathcal{T}'_+ \neq 0 $. This was exactly the argument given in [@Burgess:2015kda]. However, there is a loophole to that reasoning: the right hand side of depends on $\phi$ evaluated at the position of the delta brane, so we cannot make any final statement without knowing its value. In particular, $\phi_+$ could be such that the right hand side vanishes in the case of an infinitely thin brane.
The intuitive explanation for $ C \neq 0 $ in [@Burgess:2015kda] was that a delta function should depend on the proper distance from the brane and thus implicitly on the off-brane metric. However, this picture is misleading since $ C $ is in fact not $ \partial\delta(y) / \partial g_{\rho\rho} $ (which vanishes!), but $ \partial\delta(y) / \partial g_{\theta\theta} $. Hence, in the parlance of [@Burgess:2015kda] $ C $ corresponds to the delta function’s knowledge about the azimuthal distance around a point. Equivalently, and more physically speaking, it is the azimuthal pressure of the point source. This is obvious after noticing that the introduction of $ C $ is formally equivalent to introducing $ p_\theta $ as we did in our ring-regularization, upon identifying $ \lim\limits_{\epsilon\to0} p_\theta \equiv -2 \mathcal{T}_+(\phi) C $. Either way, $ C \neq 0 $ seems to be rather unphysical.
While the analysis of [@Niedermann:2015via] is in line with the physical (but indeed more qualitative) argument that there is no well-defined notion of an angular pressure for an infinitely thin object, we think that a rigorous statement requires an explicit calculation of the right side of . Since $ \phi $ can generically diverge at the non SI delta brane, this can only be done by first introducing a regularization of (dimensionless) width $ \epsilon $ and then letting $ \epsilon \to 0 $. This was (admittedly) not done in [@Niedermann:2015via], but neither in [@Burgess:2015kda; @Burgess:2015nka; @Burgess:2015gba; @Burgess:2015lda]. But it was done in this work, and we were able to give an unambiguous answer: For the relevant case of an exponential dilaton coupling,[^28] $p_\theta \to 0$ in the delta limit (and thus $ C=0 $)—in accordance with our physical expectation. As a result, the old delta analysis correctly captures the physics of an exponential dilaton coupling.
However, it should be noted that whenever $ p_\theta \to 0 $, also $ \hat{R} \to 0 $, cf. and . As already mentioned in Sec. \[sec:constraint\], this was not realized in the delta-analysis [@Niedermann:2015via], where it would have translated to the impossibility of breaking SI on a delta brane. But this would only have given yet another reason for studying the (more realistic) regularized setup, as we now did. Nonetheless, it is true that the delta formula for $ \hat{R} $ gives the correct leading nonzero contributions that arise for a regularized, near SI brane, as discussed in Sec. \[sec:delta\_limit\].
Now, let us be more specific and explicitly evaluate . First, for all couplings studied, we verified numerically[^29] $$\begin{aligned}
\label{eq:phi_plus}
\phi_+ \to - \infty \qquad (\text{for} \quad V \to \infty)\;.\end{aligned}$$ We start with the physically relevant exponential coupling (as already discussed, this allows to be close to SI without tuning the coefficient). Then, Eq. implies a vanishing $C$ in the limit , hence proving that the loophole is realized.
We also considered monomial couplings; physically, they are less interesting as they either lead to a diverging negative or super-critical tension in the limit . Nevertheless, even in these cases, we find $C \to 0$. For concreteness, consider a linear coupling in $\phi$: In that case, it is easy to check that the denominator in diverges while the numerator is a constant, hence implying $C \to 0$ (albeit $ p_\theta \to \text{const} \neq 0 $, which we interpret as being caused by the pathological tension).
Of course, we could not check the validity of for all possible couplings and there might very well be more complicated ‘designed potentials’ with a different behavior. However, based on our previous findings we conjecture that these potentials either lead to a vanishing $C$ or again introduce some sort of pathology.
In summary, we agree with the formulas in [@Burgess:2015kda], yet we come to a different conclusion based on a simple loophole that applies for both exponential and linear couplings (and probably for a much broader class which was beyond the scope of the present work). Let us stress that rigorously proving this result required to solve the full bulk-brane system. In particular, to show the validity of , it would not suffice to consider only a single brane without demanding the second brane to be physically well-defined.
Finally, it should be emphasized that we do agree—as discussed in great detail in this work—that $ p_\theta $ must be included for a brane of finite width, and has important consequences for the 4D curvature. Since this is the physically more relevant case anyhow, the delta-limit question becomes somewhat irrelevant. Still, the important achievement of [@Niedermann:2015via], namely the first correct identification of those BLF couplings which lead to $ \hat{R} = 0 $ (and the worries it raises), remains unaffected.
[^1]: Nonetheless, it was recently disputed in [@Burgess:2015kda] and used as an argument against the trustworthiness of [@Niedermann:2015via]. We comment on this in Appendix \[ap:Cliff\].
[^2]: We use the same notation and conventions as in [@Niedermann:2015via].
[^3]: In fact, the whole analysis of [@Niedermann:2015via] could also be trivially adapted to the point of view of [@Burgess:2015kda] on the SI breaking case (by simply including an angular pressure $ p_\theta $), without changing any of the conclusions. It would only add another contribution $ \propto p_\theta $ to , which also only vanishes in the SI case. However, we regard an angular pressure for an infinitely thin object as unphysical, cf. Sec. \[sec:delta\_limit\] and Appendix \[ap:Cliff\].
[^4]: In the proposal of [@Burgess:2015kda] $ \hat{R} \neq 0 $ would still be possible for delta branes, but only at the price of allowing $ p_\theta \neq 0 $.
[^5]: Note that even though it is not obvious how the BLF term could be consistently adapted to the 5D brane in a covariant way at the level of the action, introducing the regularization after the Maxwell field has been solved for is straightforward. (In any case, the BLF term will in the end not be crucial for our main conclusions.)
[^6]: This is analogous to the SI GGP solutions [@Gibbons:2003di; @Niedermann:2015via], where not the extra space volume $ V $ is fixed, but only the combination $ V {\mathrm{e}}^{\phi_0} $.
[^7]: Here and henceforth, evaluation at $ \rho = \rho_0 $, $ \rho_+ $ and $ \rho_- $ will be denoted by subscripts “$ 0 $”, “$ + $” and “$ - $”, respectively.
[^8]: For convenience, here and throughout the rest of Sec. \[sec:thick\_brane\], we set $ W_+ = 1 $, which is always possible by a (rigid) rescaling of the 4D coordinates.
[^9]: Specifically, we will set $ \rho_+ = \sqrt{\kappa} $ in the numerical examples below, corresponding to $ \ell \approx 2 \pi \sqrt{\kappa} $.
[^10]: These assumptions were also verified numerically.
[^11]: Note that we only consider *subcritical* tensions $ \mathcal{T}_+ < 2\pi / \kappa^2 $.
[^12]: \[fn:no\_warp\]There is a special class of SI solutions with $W'=0$ (no warping), $Q=2e/\kappa^2$ and $\hat R=0$ for which $p_\theta=0$ as an exact result even for $\epsilon\neq 0$. Physically, these solutions correspond to the regularized rugby ball setup. However, with respect to the CC problem this class is of no interest as it requires to unacceptably tune the relative size of both tensions.
[^13]: In fact, we checked this explicitly. The reason is that the terms $\mathcal{T}'_+$ and $({\mathrm{e}}^{\phi}\mathcal{A}_+)'$ (which lead to SI breaking if nonvanishing) always occur in the combination , so technically it makes no difference which of the two mediates the SI breaking.
[^14]: A SI matter theory would lead to observational problems: As argued in [@Burgess:2015lda], this would imply a direct coupling between brane matter and $\phi $, corresponding to an additional (Brans-Dicke like) force of gravitational strength. This is clearly ruled out by solar system observations [@Will:2005va] unless a mechanism is included to shield the dilaton fluctuations inside the solar system. A complete study of this case is thus beyond the scope of our present work.
[^15]: In the special case of a scale invariant coupling ($\gamma=0$) and delta branes, this follows analytically from the GGP solutions [@Gibbons:2003di], see [@Niedermann:2015via].
[^16]: In Sec. \[sec:num\_results\], however, we will uncover yet another fine-tuning (imposed by flux quantization) which could only be avoided if $ \gamma \ll 1 $.
[^17]: We used two independent implementations: one in Python, using an explicit adaptive Runge-Kutta method, and one in Mathematica, using its “NDSolve” method. The corresponding results were found to agree within the numerical uncertainties.
[^18]: The corresponding regularity condition for $ W $ is not independent thanks to the constraint, i.e., $ W'_- = 0 $ automatically whenever $ \phi'_- = 0 $.
[^19]: Analytically, the regularity condition also implicitly entered the derivation of when integrating over the whole bulk. However, this equation for $ \hat{R} $ is not yet a prediction solely in terms of model parameters, since it still contains $ V $ and $ \phi_+ $, which are a priori unknown. We were only able to infer the explicit value of $ \hat{R} $ numerically.
[^20]: Note that here we chose the gauge $ W_0 = 1 $ for convenience.
[^21]: This is a qualitative difference to models with two *infinite* extra dimensions, where a regularized pure tension brane still has $ \hat{R} = 0 $ [@Kaloper:2007ap; @Eglseer:2015xla].
[^22]: Incidentally, it turns out that without warping, i.e. for $ \alpha_+ = \alpha_- $, the scalings are somewhat different: $ \hat{R} \propto V^{-3} $, $ \delta\Phi \propto V^{-2} $ and $ p_\theta \propto V^{-2} $. However, this does not help with the tuning problem discussed below.
[^23]: One could test this assumption by repeating our analysis e.g. in the UV model proposed in [@Burgess:2015nka].
[^24]: Note that the (bulk) flux transforms as $ \Phi \mapsto \Phi / a $, and so $ {\mathcal{N}}$ has to be readjusted accordingly. This, however, does not affect the relation between $ \hat{R} $ and $ V $.
[^25]: The case $ \gamma = 0 $ is still SI and identical to the discussion above after renaming $ \lambda + \tau \to \lambda $.
[^26]: They only considered the case without BLF, so we will do the same here.
[^27]: This indeed agrees with the finding in [@Burgess:2015kda] up to an irrelevant factor $-2$, which we think got somehow lost in [@Burgess:2015kda].
[^28]: Note that we checked this not only for the exponential tension coupling as discussed in the main text, but also for the analogous exponential BLF coupling.
[^29]: Recall that, since $ \epsilon \equiv \ell^2 / V $, one way of realizing the delta limit is to take $ V \to \infty $.
|
---
abstract: 'We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal $C^{*}$-algebras. As an example, we recover the dynamical ${\mathrm{SU}_{q}(2)}$ studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.'
address: |
University of Muenster\
Einsteinstr. 62, 48149 Muenster, Germany\
timmermt@math.uni-muenster.de
author:
- Thomas Timmermann
title: |
Free dynamical quantum groups\
and\
the dynamical quantum group ${\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)}$
---
[^1]
Introduction {#section:introuction}
============
Dynamical quantum groups were introduced by Etingof and Varchenko as an algebraic tool to study the quantum dynamical Yang-Baxter equation appearing in statistical mechanics [@etingof:book; @etingof:qdybe; @etingof:exchange]. Roughly, one can associate to every dynamical quantum group a monoidal category of dynamical representations, and to every solution $R$ of the dynamical Yang-Baxter equation a dynamical quantum group $A_{R}$ with a specific dynamical representation $\pi$ such that $R$ corresponds to a braiding on the monoidal category generated by $\pi$.
In this article, we introduce two families of dynamical quantum groups ${A^{B}_{\mathrm{o}}}(\nabla,F)$ and ${A^{B}_{\mathrm{u}}}(\nabla,F)$ which are natural generalizations of the free orthogonal and the free unitary quantum groups introduced by Wang and van Daele [@wang:universal; @wang:thesis]. Roughly, these dynamical quantum groups are universal with respect to the property that they possess a corepresentation $v$ such that $F$ becomes a morphism of corepresentations from the inverse of the transpose $v^{-{\mathsf{T}}}$ or from $(v^{-{\mathsf{T}}})^{-{\mathsf{T}}}$, respectively, to $v$.
For a specific choice of $B,\nabla,F$, the free orthogonal dynamical quantum group turns out to coincide with the dynamical analogue of ${\mathrm{SU}_{q}(2)}$ that arises from a trigonometric dynamical $R$-matrix and was studied by Koelink and Rosengren [@koelink:su2]. We refine the definition of this variant of ${\mathrm{SU}_{q}(2)}$ so that the resulting global dynamical quantum group includes the classical ${\mathrm{SU}(2)}$, the non-dynamical ${\mathrm{SU}_{q}(2)}$ of Woronowicz [@woron:0], the dynamical ${\mathrm{SU}_{q}(2)}$ and further interesting limit cases which can be recovered from the global object by suitable base changes.
In the non-dynamical case, free orthogonal and free unitary quantum groups are most conveniently constructed on the level of universal $C^{*}$-algebras, where Woronowicz’s theory of compact matrix quantum groups applies [@woron:1]. We shall, however, start on the purely algebraic level and then pass to the level of universal $C^{*}$-algebras, where the main problem is to identify a good definition of a dynamical quantum group.
These new classes of dynamical quantum groups give rise to several interesting questions, for example, whether it is possible to obtain a classification similar as in [@wang:classify], to determine their categories of representations as in [@banica:orthogonal] and [@banica:unitary], or to relate their representation theory to special functions as it was done in [@koelink:su2] in the special case of ${\mathrm{SU}_{q}(2)}$.
Let us now describe the organization and contents of this article in some more detail.
The first part of this article (§\[section:algebra\]) is devoted to the purely algebraic setting.
We start with a summary on dynamical quantum groups (§\[section:bg\]). Roughly, these objects can be regarded as Hopf algebras, that is, as algebras $A$ equipped with a comultiplication $\Delta$, counit $\epsilon$ and antipode $S$, where the field of scalars has been replaced by a commutative algebra $B$ equipped with an action of a group $\Gamma$. The comultiplication $\Delta$ does not take values in the ordinary tensor product $A\otimes A$, but in a product $A{\tilde{\otimes}}A$ that takes $B$ and $\Gamma$ into account, and the counit takes values in the crossed product algebra $B\rtimes \Gamma$ which is the unit for the product $-{\tilde{\otimes}}-$. If $B$ is trivial, however, these dynamical quantum groups are just $\Gamma$-graded Hopf algebras (§\[section:trivial\]). In general, we shall use the term $(B,\Gamma)$-Hopf algebroid instead of dynamical quantum group to be more precise.
The free orthogonal and unitary dynamical quantum groups are defined as follows. Let $B$ be a unital, commutative algebra with a left action of a group $\Gamma$, let $\nabla=(\gamma_{1},\ldots,\gamma_{n})$ be an $n$-tuple in $\Gamma$ and let $F \in {\mathrm{GL}_{n}}(B)$ such that $F_{ij} = 0$ whenever $\gamma_{i} \neq \gamma_{j}^{-1}$.
\[definition:intro-ao\] The *free orthogonal dynamical quantum* ${A^{B}_{\mathrm{o}}}(\nabla,F)$ is the universal algebra with a homomorphism $r
\times s\colon B\otimes B\to {A^{B}_{\mathrm{o}}}(\nabla,F)$ and a $v\in
{\mathrm{GL}_{n}}({A^{B}_{\mathrm{o}}}(\nabla,F))$ satisfying
- $v_{ij}r(b)s(b') = r(\gamma_{i}(b))
s(\gamma_{j}(b'))v_{ij}$ for all $b,b'\in B$ and $i,j
\in \{1,\ldots,n\}$,
- $r_{n}(\hat F)v^{-{\mathsf{T}}} =vs_{n}(F)$, where $v^{-{\mathsf{T}}}$ denotes the transpose of $v^{-1}$ and $
\hat F = (\gamma_{i}(F_{ij}))_{i,j}$.
\[theorem:intro-ao-hopf\] ${A^{B}_{\mathrm{o}}}(\nabla,F)$ can be equipped with the structure of a $(B,\Gamma)$-Hopf algebroid such that $\Delta(v_{ij}) =
\sum_{k} v_{ik} {\tilde{\otimes}}v_{kj}$, $ \epsilon(v_{ij}) =
\delta_{i,j}\gamma_{i}$, and $ S(v_{ij}) = (v^{-1})_{ij}$ for all $i,j$.
Assume now that $B$ is equipped with an involution and let $F\in {\mathrm{GL}_{n}}(B)$ such that $F^{*}=F$ and $F_{ij}=0$ whenever $\gamma_{i} \neq \gamma_{j}$.
The *free unitary dynamical quantum* ${A^{B}_{\mathrm{u}}}(\nabla,F)$ is the universal $*$-algebra with a homomorphism $r \times s\colon B\otimes B\to
{A^{B}_{\mathrm{u}}}(\nabla,F)$ and a unitary $v\in {\mathrm{GL}_{n}}({A^{B}_{\mathrm{u}}}(\nabla,F))$ satisfying the condition (a) above and (c) $\bar v$ is invertible and $r_{n}(\hat F)\bar v^{-{\mathsf{T}}} =vs_{n}(F)$.
${A^{B}_{\mathrm{u}}}(\nabla,F)$ can be equipped with the structure of a $(B,\Gamma)$-Hopf $*$-algebroid such that $\Delta(v_{ij}) =
\sum_{k} v_{ik} {\tilde{\otimes}}v_{kj}$, $ \epsilon(v_{ij}) =
\delta_{i,j}\gamma_{i}$, $ S(v_{ij}) = (v^{-1})_{ij}$ for all $i,j$.
The formulas for $\Delta(v_{ij})$ and $\epsilon(v_{ij})$ above imply that the matrices $v$ above are corepresentations of ${A^{B}_{\mathrm{o}}}(\nabla,F)$ and ${A^{B}_{\mathrm{u}}}(\nabla,F)$, respectively, and the conditions (b) and (c) assert that $F$ is an intertwiner from $v^{-{\mathsf{T}}}$ or $\bar v^{-{\mathsf{T}}}$, respectively, to $v$. Such intertwiner relations admit plenty functorial transformations which are studied systematically in §\[section:rn\], and yield short proofs of the results above in §\[section:ao\]. There, we also consider involutions on certain quotients ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ of ${A^{B}_{\mathrm{o}}}(\nabla,F)$ which are parameterized by an additional matrix $G\in {\mathrm{GL}_{n}}(B)$.
Interestingly, the square of the antipode on the dynamical quantum groups ${A^{B}_{\mathrm{o}}}(\nabla,F)$ and ${A^{B}_{\mathrm{u}}}(\nabla,F)$ can be described in terms of a natural family of characters $(\theta^{(k)})_{k}$ which, like the counit $\epsilon$, take values in $B\rtimes \Gamma$. This family is an analogue of Woronowicz’s fundamental family of characters on a compact quantum group.
As a main example of the constructions above, we recover the dynamical quantum group $\mathcal{F}_{R}(\mathrm{SU}(2))$ of Koelink and Rosengren [@koelink:su2] associated to a deformation parameter $q \neq 1$ as the free orthogonal dynamical quantum group ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$, where $B$ is the meromorphic functions on the plane, $\Gamma={\mathbb{Z}}$ acting by shifts, $\nabla=(1,-1)$ and $F=
\begin{pmatrix}
0 & 1 \\ \tilde f & 0
\end{pmatrix}
$, where $\tilde f$ is the meromorphic function $\lambda
\mapsto q^{-1}(q^{2\lambda}-q^{-2})/(q^{2\lambda}-1)$, and $G=
\begin{pmatrix}
0 & -1 \\ q^{-1} & 0
\end{pmatrix}
$. In §\[section:sud\], we show how this example can be refined such that the resulting dynamical quantum group ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ includes $\mathcal{F}_{R}(\mathrm{SU}(2))$ and, simultaneously, a number of interesting limit cases which can be recovered from the global object by suitable base changes.
The second part of this article (§\[section:universal\]) extends the definition of dynamical quantum groups to the level of universal $C^{*}$-algebras. Here, $B$ is assumed to be a unital, commutative $C^{*}$-algebra and $\Gamma$ acts via automorphisms. The main tasks is to find a $C^{*}$-algebraic analogue of the product $-{\tilde{\otimes}}-$ that describes the target of the comultiplication. As in the algebraic setting, we construct this product in two steps, by first forming a cotensor product with respect to the Hopf $C^{*}$-algebra $C^{*}(\Gamma)$ naturally associated to the group $\Gamma$ (§\[section:c\]), and then taking a quotient with respect to $B$ (§\[section:cb\]). Given the monoidal product, all definitions carry over from the algebraic setting to the setting of universal $C^{*}$-algebras easily (§\[section:free-c\]).
The purely algebraic level {#section:algebra}
==========================
Throughout this section, we assume all algebras and homomorphisms to be unital over a fixed common ground field, and $B$ to be a commutative algebra equipped with a left action of a group $\Gamma$.
Preliminaries on dynamical quantum groups {#section:bg}
-----------------------------------------
This subsection summarizes the basics of dynamical quantum groups used in this article. We introduce the monoidal category of $(B,\Gamma)$-algebras, then define $(B,\Gamma)$-Hopf algebroids, and finally consider base changes and the setting of $*$-algebras. Except for the base change, most of this material is contained in [@etingof:qdybe] and [@koelink:su2] in slightly different guise. We omit all proofs because they are straightforward.
Let $B^{{\mathrm{ev}}}=B\otimes B$. A *$B^{{\mathrm{ev}}}$-algebra* is an algebra with a homomorphism $r\times s\colon B^{{\mathrm{ev}}}\to A$, or equivalently, with homomorphisms $r_{A}=r,s_{A}=s\colon
B\to A$ whose images commute. A morphism of $B^{{\mathrm{ev}}}$-algebras is a $B^{{\mathrm{ev}}}$-linear homomorphism. Write $\Gamma^{{\mathrm{ev}}}=\Gamma\times \Gamma$ and let $e\in\Gamma$ be the unit. Given a $\Gamma^{{\mathrm{ev}}}$-graded algebra $A$, we write $\partial_{a} =
(\partial^{r}_{a},\partial^{s}_{a}) = (\gamma,\gamma')$ whenever $a\in A_{\gamma,\gamma'}$.
\[definition:bg-algebra\] A *$(B,\Gamma)$-algebra* is a $\Gamma^{{\mathrm{ev}}}$-graded $B^{{\mathrm{ev}}}$-algebra such that $(r\times s)(B^{{\mathrm{ev}}})
\subseteq A_{e,e}$ and $ar(b)= r(\partial^{r}_{a}(b))a$, $as(b)=s(\partial^{s}_{a}(b)) a$ for all $b \in B$, $a\in A$. A *morphism* of $(B,\Gamma)$-algebras is a morphism of $\Gamma^{{\mathrm{ev}}}$-graded $B^{{\mathrm{ev}}}$-algebras. We denote by ${\ensuremath
\mathbf{Alg}_{(B,\Gamma)}}$ the category of all $(B,\Gamma)$-algebras.
\[example:bg-unit\] Denote by $B\rtimes
\Gamma$ the crossed product, that is, the universal algebra containing $B$ and $\Gamma$ such that $e=1_{B}$ and $b\gamma \cdot b'\gamma' = b\gamma(b') \gamma\gamma'$ for all $b,b'\in B$, $\gamma,\gamma' \in \Gamma$. This is a $(B,\Gamma)$-algebra, where $\partial_{b\gamma}=(\gamma,\gamma)$ and $r(b)=s(b)=b$ for all $b\in B$, $\gamma\in \Gamma$.
The category of all $(B,\Gamma)$-algebras can be equipped with a monoidal structure [@maclane] as follows. Let $A$ and $C$ be $(B,\Gamma)$-algebras. Then the subalgebra $$\begin{aligned}
A {\stackrel{\Gamma}{\otimes}}C &:= \sum_{\gamma,\gamma',\gamma'' \in
\Gamma} A_{\gamma,\gamma'} \otimes
C_{\gamma',\gamma''} \subset A \otimes C\end{aligned}$$ is a $(B,\Gamma)$-algebra, where $\partial_{a\otimes c} = (\partial^{r}_{a},\partial^{s}_{c})$ for all $a\in A$, $c\in C$ and $(r\times s)(b\otimes b') =
r_{A}(b) \otimes s_{C}(b')$ for all $b,b'\in B$. Let $I\subseteq A{\stackrel{\Gamma}{\otimes}}C$ be the ideal generated by $\{s_{A}(b)\otimes 1 - 1\otimes r_{C}(b) : b\in
B\}$. Then $A {\tilde{\otimes}}C:=A{\stackrel{\Gamma}{\otimes}}C/I$ is a $(B,\Gamma)$-algebra again, called the *fiber product* of $A$ and $C$. Write $a{\tilde{\otimes}}c$ for the image of an element $a\otimes c$ in $A{\tilde{\otimes}}C$.
The product $(A,C) \mapsto A{\tilde{\otimes}}C$ is functorial, associative and unital in the following sense.
\[lemma:bg-monoidal\]
1. For all morphisms of $(B,\Gamma)$-algebras $\pi^{1} \colon A^{1}\to C^{1}$, $\pi^{2}\colon
A^{2}\to C^{2}$, there exists a morphism $\pi^{1}
\tilde\otimes \pi^{2} \colon A^{1} \tilde\otimes A^{2}
\to C^{1} \tilde\otimes C^{2}$, $a_{1} {\tilde{\otimes}}a_{2}
\mapsto \pi^{1}(a_{1}) {\tilde{\otimes}}\pi^{2}(a_{2})$.
2. For all $(B,\Gamma)$-algebras $A,C,D$, there is an isomorphism $(A \tilde\otimes C) \tilde\otimes D \to A
\tilde\otimes (C \tilde\otimes D)$, $(a {\tilde{\otimes}}c) {\tilde{\otimes}}d \mapsto a {\tilde{\otimes}}(c {\tilde{\otimes}}d)$.
3. For each $(B,\Gamma)$-algebra $A$, there exist isomorphisms $(B\rtimes \Gamma) \tilde\otimes A \to A$ and $A \tilde\otimes (B\rtimes \Gamma) \to A$, given by $b\gamma {\tilde{\otimes}}a \mapsto r(b)a$ and $a {\tilde{\otimes}}b\gamma
\mapsto s(b)a$, respectively.
Of course, the isomorphisms above are compatible in a natural sense.
\[remark:takeuchi\] The product $-{\tilde{\otimes}}-$ is related to the left and right Takeuchi products $-{_{B}\times} -$ and $-\times_{B}-$ as follows. Given a $B^{{\mathrm{ev}}}$-algebra $A$, we write ${_{\bullet}A}$ or $A_{\bullet}$ when we regard $A$ as a $B$-bimodule via $b \cdot a \cdot b':=r(b)s(b')a$ or $b
\cdot a \cdot b':=ar(b)s(b')$, respectively. Then the left and right Takeuchi products of $B^{{\mathrm{ev}}}$-algebras $A$ and $C$ are the $B^{{\mathrm{ev}}}$-algebras $$\begin{aligned}
A {} {_{B}\times} C &:= \left\{\sum_{i} a_{i} {\underset{B}{\otimes}}c_{i} \in {_{\bullet} A {\underset{B}{\otimes}}{_{\bullet}C}} \,\middle|\, \forall b\in B:
\sum_{i} a_{i}s_{A}(b) {\underset{B}{\otimes}}c_{i} = \sum_{i} a_{i} {\underset{B}{\otimes}}c_{i}r_{C}(b)
\right\}, \\
A \times_{B} C &:= \left\{\sum_{i} a_{i} {\underset{B}{\otimes}}c_{i} \in {A_{\bullet} {\underset{B}{\otimes}}C_{\bullet}}\,\middle|\, \forall b\in B:
\sum_{i} s_{A}(b)a_{i} {\underset{B}{\otimes}}c_{i} = \sum_{i} a_{i} {\underset{B}{\otimes}}r_{C}(b) c_{i}
\right\},
\end{aligned}$$ where the multiplication is defined factorwise and the embedding of $B^{{\mathrm{ev}}}$ is given by $b\otimes b'\mapsto
r_{A}(b) {\underset{B}{\otimes}}s_{C}(b')$. The assignments $(A,C)
\mapsto A {}{_{B}\times} C$ and $(A,C) \mapsto A
{\times_{B}} C$ extend to bifunctors on the category of $B^{{\mathrm{ev}}}$-algebras and turn it into a lax monoidal category [@day:quantum-cat]. The obvious forgetful functor $U$ from $(B,\Gamma)$-algebras to $B^{{\mathrm{ev}}}$-algebras is compatible with these products in the sense that for every pair of $(B,\Gamma)$-algebras $A,C$, the inclusion $A {\stackrel{\Gamma}{\otimes}}C \hookrightarrow A\otimes
C$ factorizes to inclusions of $A{\tilde{\otimes}}C$ into $ A
{}{_{B}\times} C$ and $A \times_{B} C$, yielding natural transformations from $U(-{\tilde{\otimes}}-)$ to $U(-) {_{B}\times}
U(-)$ and $U(-)\times_{B} U(-)$, respectively.
Briefly, a $(B,\Gamma)$-Hopf algebroid is a coalgebra in ${\ensuremath
\mathbf{Alg}_{(B,\Gamma)}}$ equipped with an antipode. To make this definition precise, we need two involutions on ${\ensuremath
\mathbf{Alg}_{(B,\Gamma)}}$. Given an algebra $A$, we denote by $A^{{\mathsf{op}}}$ its opposite, that is, the same vector space with reversed multiplication.
There exist automorphisms $(-)^{{\mathsf{op}}}$ and $(-)^{{\mathsf{co}}}$ of ${\ensuremath
\mathbf{Alg}_{(B,\Gamma)}}$ such that for each $(B,\Gamma)$-algebra $A$ and each morphism $\phi \colon A\to C$, we have $A^{{\mathsf{co}}}=A$ as an algebra and $$\begin{aligned}
(A^{{\mathsf{op}}})_{\gamma,\gamma'} &=
A_{\gamma^{-1},\gamma'{}^{-1}} \text{ for all }
\gamma,\gamma'\in \Gamma, & r_{A^{{\mathsf{op}}}} &= r_{A}, &
s_{A^{{\mathsf{op}}}} &= s_{A}, & \phi^{{\mathsf{op}}} &= \phi,
\\
(A^{{\mathsf{co}}})_{\gamma,\gamma'} &= A_{\gamma',\gamma}
\text{ for all } \gamma,\gamma'\in \Gamma, &
r_{A^{{\mathsf{co}}}} &= s_{A}, & s_{A^{{\mathsf{co}}}} &= r_{A}, &
\phi^{{\mathsf{co}}} &= \phi.
\end{aligned}$$ Furthermore, $(-)^{{\mathsf{op}}} \circ (-)^{{\mathsf{op}}} = \operatorname{id}$, $(-)^{{\mathsf{co}}} \circ (-)^{{\mathsf{co}}} = \operatorname{id}$, $(-)^{{\mathsf{op}}} \circ
(-)^{{\mathsf{co}}} = (-)^{{\mathsf{co}}} \circ (-)^{{\mathsf{op}}}$.
The automorphisms above are compatible with the monoidal structure as follows. Given $(B,\Gamma)$-algebras $A,C$, there exist isomorphisms $(A {\tilde{\otimes}}C)^{{\mathsf{op}}} \to (A^{{\mathsf{op}}}
{\tilde{\otimes}}C)^{{\mathsf{op}}}$ and $ (A {\tilde{\otimes}}C)^{{\mathsf{co}}} \to C^{{\mathsf{co}}}
{\tilde{\otimes}}A^{{\mathsf{co}}}$ given by $a {\tilde{\otimes}}c \mapsto a {\tilde{\otimes}}c$ and $a {\tilde{\otimes}}c \mapsto c {\tilde{\otimes}}a$, respectively. Moreover, $(B\rtimes \Gamma)^{{\mathsf{co}}} = B\rtimes \Gamma$ and there exists an isomorphism $S^{B\rtimes \Gamma} \colon
B\rtimes \Gamma \to (B\rtimes \Gamma)^{{\mathsf{op}}}$, $b\gamma
\mapsto \gamma^{-1} b$, and all of these isomorphisms and the isomorphisms in Lemma \[lemma:bg-monoidal\] are compatible in a natural sense.
\[definition:bg-hopf\] A *$(B,\Gamma)$-Hopf algebroid* is a $(B,\Gamma)$-algebra $A$ equipped with morphisms $\Delta
\colon A\to A{\tilde{\otimes}}A$, $\epsilon \colon A \to B\rtimes
\Gamma$, and $S \colon A \to A^{{\mathsf{co}},{\mathsf{op}}}$ such that the diagrams below commute, $$\begin{gathered}
\xymatrix@R=15pt@C=30pt{ A \ar[r]^(0.4){\Delta}
\ar[d]_{\Delta} & A {\tilde{\otimes}}A
\ar[d]^{\Delta {\tilde{\otimes}}\operatorname{id}} \\
A{\tilde{\otimes}}A \ar[r]_(0.4){\operatorname{id}{\tilde{\otimes}}\Delta} & A {\tilde{\otimes}}A
{\tilde{\otimes}}A, } \qquad \xymatrix@R=15pt@C=25pt{A {\tilde{\otimes}}A
\ar[d]_{\epsilon {\tilde{\otimes}}\operatorname{id}} & A \ar[l]_(0.4){\Delta}
\ar[r]^(0.4){\Delta} \ar[d]^{\operatorname{id}} & A
{\tilde{\otimes}}A \ar[d]^{\operatorname{id}{\tilde{\otimes}}\epsilon} \\
(B\rtimes \Gamma) {\tilde{\otimes}}A \ar[r]^(0.6){\cong} & A
&\ar[l]_(0.6){\cong} A {\tilde{\otimes}}(B\rtimes \Gamma),}\end{gathered}$$ $$\begin{gathered}
\xymatrix@R=15pt@C=20pt{A {\tilde{\otimes}}A \ar[d]_{S {\tilde{\otimes}}\operatorname{id}}
&& A \ar[ll]_(0.4){\Delta} \ar[rr]^(0.4){\Delta}
\ar[d]^{\epsilon} && A
{\tilde{\otimes}}A \ar[d]^{\operatorname{id}{\tilde{\otimes}}S} \\
A^{{\mathsf{co}},{\mathsf{op}}} {\tilde{\otimes}}A \ar[r]^(0.6){\check m} & A &
B\rtimes \Gamma \ar[l]_(0.6){\check s}
\ar[r]^(0.6){\hat r} & A &\ar[l]_(0.6){\hat m} A
{\tilde{\otimes}}A^{{\mathsf{co}},{\mathsf{op}}}}
\end{gathered}$$ where the linear maps $\hat m, \check m, \hat r,
\check s$ are given by $$\begin{aligned}
\hat m(a {\tilde{\otimes}}a') &= aa' = \check m (a{\tilde{\otimes}}a'), & \hat
r(b\gamma) &= r(b), & \check s(\gamma b) &= s(b)
\end{aligned}$$ for all $a,a'\in A, b\in B,\gamma\in \Gamma$.
A morphism of $(B,\Gamma)$-Hopf algebroids $(A,\Delta_{A},\epsilon_{A},S_{A})$, $(C,\Delta_{C},\epsilon_{C},S_{C})$ is a morphism of $(B,\Gamma)$-algebras $\pi \colon A \to C$ such that $\Delta_{C} \circ \pi = (\pi {\tilde{\otimes}}\pi) \circ
\Delta_{A}$, $\epsilon_{C} \circ \pi = \epsilon_{A}$, $S_{C} \circ \pi = \pi^{{\mathsf{co}},{\mathsf{op}}}\circ S_{A}$. We denote the category of all $(B,\Gamma)$-Hopf algebroids by ${\mathbf{Hopf}}_{(B,\Gamma)}$.
A $(B,\Gamma)$-Hopf algebroid reduces to an $\mathfrak{h}$-Hopf algebroid in the sense of [@koelink:su2] when $\mathfrak{h}$ is a commutative Lie algebra, $B$ is the algebra of meromorphic functions on the dual $\mathfrak{h}^{*}$, and $\Gamma=\mathfrak{h}^{*}$ acts by shifting the argument. Let us note that the axioms above can be weakened, see [@koelink:su2 Proposition 2.2], but our examples shall automatically satisfy the apparently stronger conditions above.
The $(B,\Gamma)$-algebra $B \rtimes \Gamma$ is a $(B,\Gamma)$-Hopf algebroid, where $\Delta(b \gamma) =
b\gamma {\tilde{\otimes}}\gamma = \gamma {\tilde{\otimes}}b\gamma$, $\epsilon(b\gamma) = b\gamma$, and $S( b\gamma) =
\gamma^{-1}b$ for all $b\in B,\gamma\in \Gamma$.
Let us comment on some straightforward properties of $(B,\Gamma)$-Hopf algebroids:
\[remarks:bg-hopf\] Let $(A,\Delta,\epsilon,S)$ be a $(B,\Gamma)$-Hopf algebroid.
1. If $\gamma\neq \gamma'$, then $\epsilon(A_{\gamma,\gamma'}) =0$ because $(B\rtimes
\Gamma)_{\gamma,\gamma'} =0$.
2. We have $\Delta(A)(1 {\tilde{\otimes}}A_{e,*}) = A{\tilde{\otimes}}A =
(A_{*,e} {\tilde{\otimes}}1)\Delta(A)$, where $A_{e,*}=\sum_{\gamma} A_{e,\gamma}$ and $A_{*,e} =
\sum_{\gamma} A_{\gamma,e}$. Indeed, by [@timmermann:measured Proposition 1.3.7], $$\begin{aligned}
\sum (xS(y_{(1)}) {\tilde{\otimes}}1)\Delta(y_{(2)}) &= x {\tilde{\otimes}}y
= \sum \Delta(x_{(1)})(1 {\tilde{\otimes}}S(x_{(2)})y)
\end{aligned}$$ for all $x\in A_{\gamma,\gamma'}, y\in
A_{\gamma',\gamma''}$, $\gamma,\gamma',\gamma'' \in
\Gamma$, where $\sum x_{(1)} {\tilde{\otimes}}x_{(2)}=\Delta(x)$ and $\sum y_{(1)} {\tilde{\otimes}}y_{(2)} = \Delta(y)$.
$(B,\Gamma)$-Hopf algebroids fit into the general definition of Hopf algebroids [@boehm:algebroids] as follows.
Let $(A,\Delta,\epsilon,S)$ be a $(B,\Gamma)$-Hopf algebroid. Denote by ${_{\bullet}\epsilon}$ and $\epsilon_{\bullet}$ the compositions of $\epsilon\colon A\to B\rtimes \Gamma$ with the linear maps $B\rtimes \Gamma \to B$ given by $b\gamma \mapsto
b$ and $\gamma b \mapsto b$, respectively, and denote by $_{\bullet}\Delta$ and $\Delta_{\bullet}$ the compositions of $\Delta$ with the natural inclusions $A {\tilde{\otimes}}A \to {_{\bullet}} A {\underset{B}{\otimes}}{_{\bullet}
A}$ and $A {\tilde{\otimes}}A \to {A_{\bullet}} {\underset{B}{\otimes}}{A_{\bullet}}$, respectively (see Remark \[remark:takeuchi\]).
1. The maps $_{\bullet }\epsilon,
\epsilon_{\bullet}\colon A\to B$ will in general not be homomorphisms, but satisfy $$\begin{aligned}
{_{\bullet}\epsilon}(ar({_{\bullet}\epsilon}(a'))&=
{_{\bullet}\epsilon}(aa') =
{_{\bullet}\epsilon}(as({_{\bullet}\epsilon}(a')), &
\epsilon_{\bullet}(r(\epsilon_{\bullet}(a))a') &=
\epsilon_{\bullet}(aa') =
\epsilon_{\bullet}(s(\epsilon_{\bullet}(a))a')
\end{aligned}$$ for all $a,a' \in A$. Indeed, since $\epsilon(a) =
{_{\bullet}\epsilon}(a)\partial_{a}$ for all homogeneous $a'\in A$, $$\begin{aligned}
{_{\bullet}\epsilon}(aa') \partial_{aa'} &=
{_{\bullet}\epsilon}(a) \partial_{a} \cdot
{_{\bullet}\epsilon}(a') \partial_{a'}
=\partial_{a}( {_{\bullet}\epsilon}(a'))
{_{\bullet}\epsilon}(a) \partial_{a}\partial_{a'} =
{_{\bullet}\epsilon}(ar({_{\bullet}\epsilon}(a'))) \partial_{aa'}
\end{aligned}$$ for all homogeneous $a,a'\in A$, and the remaining equations follow similarly.
2. One easily verifies that $({_{\bullet}A},{_{\bullet
}\Delta},{_{\bullet}\epsilon})$, $(A_{\bullet},\Delta_{\bullet},\epsilon_{\bullet})$ are $B$-corings, ${_{\bullet}\mathcal{A}}
:=(A,{_{\bullet}\Delta},{_{\bullet}\epsilon})$ is a left $B$-bialgebroid, and $\mathcal{A}_{\bullet}:=(A^{{\mathsf{co}}},\Delta_{\bullet},\epsilon_{\bullet})$ is a right $B$-bialgebroid in the sense of [@boehm:algebroids]. Using the relations $\check
s \circ \epsilon = s \circ \epsilon_{\bullet}$ and $\hat r \circ \epsilon = r \circ
{_{\bullet}\epsilon}$, one furthermore finds that $(\mathcal{A}_{\bullet},{_{\bullet}\mathcal{A}},S)$ is a Hopf algebroid over $B$. To make the match with Definition 4.1 in [@boehm:algebroids], one has to take $H,s_{L},t_{L},\Delta_{L},\epsilon_{L},s_{R},t_{R},\Delta_{R},\epsilon_{R},S$ equal to $A,s,r,{_{\bullet}\Delta},{_{\bullet}\epsilon},r,s,\Delta_{\bullet},\epsilon_{\bullet},S$, respectively.
Let $B$ and $C$ be commutative algebras with a left action of $\Gamma$ and let $\phi\colon B\to C$ be a $\Gamma$-equivariant homomorphism. We then obtain base change functors $\phi_{*}\colon {\ensuremath
\mathbf{Alg}_{(B,\Gamma)}}\to {\ensuremath
\mathbf{Alg}_{(C,\Gamma)}}$ and $\phi_{*}\colon {\mathbf{Hopf}}_{(B,\Gamma)}
\to {\mathbf{Hopf}}_{(C,\Gamma)}$ as follows. Let $A$ be a $(B,\Gamma)$-algebra. Regard $C$ as a $B$-module via $\phi$, and $A$ as a $B$-bimodule, where $b \cdot a \cdot b' = r(b)as(b')$ for all $b,b'\in B$, $a\in A$. Then the vector space $\phi_{*}(A):=C {\underset{B}{\otimes}}A {\underset{B}{\otimes}}C$ carries the structure of a $(C,\Gamma)$-algebra such that $$\begin{gathered}
(c {\underset{B}{\otimes}}a {\underset{B}{\otimes}}d)
(c' {\underset{B}{\otimes}}a' {\underset{B}{\otimes}}d') =
c \partial^{r}_{a}(c') {\underset{B}{\otimes}}aa' {\underset{B}{\otimes}}(\partial^{s}_{a'})^{-1}(d)d', \\
\begin{aligned}
\partial_{c{\underset{B}{\otimes}}a {\underset{B}{\otimes}}d} &= \partial_{a}, &
(r\times s)(c\otimes c') &= c{\underset{B}{\otimes}}1 {\underset{B}{\otimes}}c'
&&\text{ for all } c,c',d,d' \in C, a,a' \in A.
\end{aligned}\end{gathered}$$ Every morphism of $(B,\Gamma)$-algebras $\pi\colon A \to A'$ evidently yields a morphism of $(C,\Gamma)$-algebras $\phi_{*}(\pi) \colon \phi_{*}(A) \to \phi_{*}(A')$, $c{\underset{B}{\otimes}}a {\underset{B}{\otimes}}c'\mapsto c {\underset{B}{\otimes}}\pi(a) {\underset{B}{\otimes}}c'$, and the assignments $A \mapsto \phi_{*}(A)$ and $\phi
\mapsto \phi_{*}(\pi)$ form a functor $\phi_{*}\colon
{\ensuremath
\mathbf{Alg}_{(B,\Gamma)}}\to {\ensuremath
\mathbf{Alg}_{(C,\Gamma)}}$.
\[lemma:bg-cb\]
1. There exists a morphism of $(C,\Gamma)$-algebras $\phi^{(0)} \colon \phi_{*}(B\rtimes \Gamma) \to C\rtimes
\Gamma$, $c {\underset{B}{\otimes}}b\gamma {\underset{B}{\otimes}}c'
\mapsto c\phi(b) \gamma c' = c\phi(b)\gamma(c') \gamma$.
2. For all $(B,\Gamma)$-algebras $A,D$, there exists a unique morphism $\phi^{(2)}_{A,D} \colon \phi_{*}(A
{\tilde{\otimes}}D) \to \phi_{*}(A) {\tilde{\otimes}}\phi_{*}(D)$, $c{\underset{B}{\otimes}}(a {\tilde{\otimes}}d) {\underset{B}{\otimes}}c' \mapsto (c{\underset{B}{\otimes}}a{\underset{B}{\otimes}}1) {\tilde{\otimes}}(1 {\underset{B}{\otimes}}d {\underset{B}{\otimes}}c')$.
\[proposition:bg-cb\] Let $(A,\Delta,\epsilon,S)$ be a $(B,\Gamma)$-Hopf algebroid. Then $\phi_{*}(A)$ is $(C,\Gamma)$-Hopf algebroid with respect to the morphisms
1. $\Delta'\colon \phi_{*}(A)
\xrightarrow{\phi_{*}(\Delta)} \phi_{*}(A{\tilde{\otimes}}A)
\xrightarrow{\phi^{(2)}_{A,A}} \phi_{*}(A) {\tilde{\otimes}}\phi_{*}(A)$, given by $c {\underset{B}{\otimes}}a {\underset{B}{\otimes}}c' \mapsto
\sum_{i} (c{\underset{B}{\otimes}}a'_{i} {\underset{B}{\otimes}}1) {\tilde{\otimes}}(1
{\underset{B}{\otimes}}a''_{i} {\underset{B}{\otimes}}c')$ whenever $\Delta(a)=\sum
a'_{i}{\tilde{\otimes}}a''_{i}$;
2. $\epsilon'\colon \phi_{*}(A)
\xrightarrow{\phi_{*}(\epsilon)} \phi_{*}(B\rtimes
\Gamma) \xrightarrow{\phi^{(0)}} C\rtimes \Gamma$, given by $c {\underset{B}{\otimes}}a {\underset{B}{\otimes}}c' \mapsto \sum_{i} c
\phi(b_{i})\gamma_{i} c'$ whenever $\epsilon(a)=\sum_{i}
b_{i}\gamma_{i}$;
3. $S' \colon \phi_{*}(A) \to
(\phi_{*}A)^{{\mathsf{co}},{\mathsf{op}}}$ given by $c{\underset{B}{\otimes}}a{\underset{B}{\otimes}}c'
\mapsto c' {\underset{B}{\otimes}}S(a) {\underset{B}{\otimes}}c$.
The assignments $(A,\Delta,\epsilon,S) \mapsto
(\phi_{*}(A),\Delta',\epsilon',S')$ as above and $\pi \mapsto
\phi_{*}(\pi)$ evidently form a functor $\phi_{*} \colon
{\mathbf{Hopf}}_{(B,\Gamma)} \to {\mathbf{Hopf}}_{(C,\Gamma)}$.
The preceding definitions and results extend to $*$-algebras as follows. Assume that $B$ is a $*$-algebra and that $\Gamma$ preserves its involution.
A $(B,\Gamma)$-$*$-algebra is a $(B,\Gamma)$-algebra with an involution that is compatible with the grading and the involution on $B$, and a morphism of $(B,\Gamma)$-$*$-algebras is a morphism of $(B,\Gamma)$-algebra that preserves the involution. We denote by ${\ensuremath
\mathbf{*\text{-}Alg}_{(B,\Gamma)}}$ the category of all $(B,\Gamma)$-$*$-algebras. This subcategory of ${\ensuremath
\mathbf{Alg}_{(B,\Gamma)}}$ is monoidal because the crossed product $B\rtimes \Gamma$ is a $(B,\Gamma)$-$*$-algebra with respect to the involution given by $(b\gamma)^{*}=\gamma^{-1} b^{*}$, and for all $(B,\Gamma)$-$*$-algebras $A,C$, the fiber product $A{\tilde{\otimes}}C$ is a $(B,\Gamma)$-$*$-algebra with respect to the involution given by $(a {\tilde{\otimes}}c)^{*} =a^{*} {\tilde{\otimes}}c^{*}$.
\[definition:bg-s-algebra\] A $(B,\Gamma)$-Hopf $*$-algebroid is a $(B,\Gamma)$-Hopf algebroid $(A,\Delta,\epsilon,S)$ where $A$ is a $(B,\Gamma)$-$*$-algebra and $\Delta$ and $\epsilon$ are morphisms of $(B,\Gamma)$-$*$-algebras. A morphism of $(B,\Gamma)$-Hopf $*$-algebroids is a morphism of the underlying $(B,\Gamma)$-Hopf algebroid and $(B,\Gamma)$-$*$-algebras. We denote by ${\mathbf{Hopf}}_{(B,\Gamma)}^{*}$ the category of all $(B,\Gamma)$-Hopf $*$-algebroids.
If $(A,\Delta,\epsilon,S)$ is a $(B,\Gamma)$-Hopf $*$-algebroid, then $* \circ S \circ * \circ S = \operatorname{id}$; see [@koelink:su2 Lemma 2.9].
We denote by $\overline{A}$ the conjugate algebra of a complex algebra $A$; this is the set $A$ with conjugated scalar multiplication and the same addition and multiplication. Thus, the involution of a $*$-algebra $A$ is an automorphism $A\to \overline{A}^{{\mathsf{op}}}$.
\[lemma:bg-bar\] The category ${\ensuremath
\mathbf{*\text{-}Alg}_{(B,\Gamma)}}$ has an automorphism $\overline{(-)}$ such that for every $(B,\Gamma)$-$*$-algebra $A$ and every morphism of $(B,\Gamma)$-$*$-algebras $\phi \colon A\to C$, $$\begin{aligned}
(\overline{A})_{\gamma,\gamma'} &=
\overline{A_{\gamma,\gamma'}} \text{ for all }
\gamma,\gamma'\in \Gamma, & r_{\bar A} &=
r_{A}\circ \ast, & s_{\bar A} &= s_{A} \circ \ast, &
\overline{\phi} &= \phi.
\end{aligned}$$ Furthermore, $\overline{(-)} \circ \overline{(-)} =
\operatorname{id}$, $\overline{(-)} \circ (-)^{{\mathsf{op}}} = (-)^{{\mathsf{op}}} \circ
\overline{(-)}$, $\overline{(-)} \circ (-)^{{\mathsf{co}}} =
(-)^{{\mathsf{co}}} \circ \overline{(-)}$.
There exists an isomorphism $B\rtimes \Gamma \to
\overline{B\rtimes \Gamma}$, $b\gamma \mapsto
b^{*}\gamma$, and for each pair of $(B,\Gamma)$-$*$-algebras $A,C$, there exists an isomorphism $\overline{A {\tilde{\otimes}}C} \to \overline{A}
{\tilde{\otimes}}\overline{C}$, $a{\tilde{\otimes}}c \mapsto a{\tilde{\otimes}}c$.
Let also $C$ be a commutative $*$-algebra with a left action of $\Gamma$ and let $\phi\colon B\to C$ be a $\Gamma$-equivariant $*$-homomorphism. Then for every $(B,\Gamma)$-$*$-algebra $A$, the $(C,\Gamma)$-algebra $\phi_{*}(A)$ is a $(C,\Gamma)$-$*$-algebra with respect to the involution given by $(c {\underset{B}{\otimes}}a {\underset{B}{\otimes}}c')^{*}
=(\partial^{r}_{a})^{-1}(c)^{*} {\underset{B}{\otimes}}a^{*} {\underset{B}{\otimes}}\partial^{s}_{a}(c')^{*}$, and we obtain a functor $\phi_{*} \colon {\ensuremath
\mathbf{*\text{-}Alg}_{(B,\Gamma)}}\to{\ensuremath
\mathbf{Alg}^{*}_{(C,\Gamma)}}$. Likewise, we obtain a functor $\phi_{*} \colon {\mathbf{Hopf}}_{(B,\Gamma)}^{*}\to
{\mathbf{Hopf}}_{(C,\Gamma)}^{*}$.
The case of a trivial base algebra {#section:trivial}
----------------------------------
Assume for this subsection that $B={\mathbb{C}}$ equipped with the trivial action of $\Gamma$. Then the category of all $({\mathbb{C}},\Gamma)$-Hopf algebroids is equivalent to the comma category of all Hopf algebras over ${\mathbb{C}}\Gamma$ as follows.
Recall that the group algebra ${\mathbb{C}}\Gamma$ is a Hopf $*$-algebra with involution, comultiplication, counit and antipode given by $\gamma^{*}=\gamma^{-1}$, $\Delta_{{\mathbb{C}}\Gamma}(\gamma)=\gamma\otimes \gamma$, $\epsilon_{{\mathbb{C}}\Gamma}(\gamma)=1$, $S_{{\mathbb{C}}\Gamma}(\gamma) = \gamma^{-1}$ for all $\gamma\in \Gamma
\subset {\mathbb{C}}\Gamma$. Objects of the comma category ${\mathbf{Hopf}}_{{\mathbb{C}}\Gamma}$ are pairs consisting of a Hopf algebra $A$ and a morphism of Hopf algebras $A \to {\mathbb{C}}\Gamma$, and morphisms from $(A,\pi_{A})$ to $(C,\pi_{C})$ are all morphisms $A \xrightarrow{\phi} C$ such that $\pi_{C} \circ \phi = \pi_{A}$. Likewise, we define the comma category ${\mathbf{Hopf}}^{*}_{{\mathbb{C}}\Gamma}$ of Hopf $*$-algebras over ${\mathbb{C}}\Gamma$.
Note that a $({\mathbb{C}},\Gamma)$-algebra is just a $\Gamma\times\Gamma$-graded algebra and $A{\tilde{\otimes}}C = A
\stackrel{\Gamma}{\otimes} C \subseteq A\otimes C$ for all $({\mathbb{C}},\Gamma)$-algebras $A,C$. Moreover, ${\mathbb{C}}\rtimes \Gamma = {\mathbb{C}}\Gamma$, and for every $({\mathbb{C}},\Gamma)$-algebra $A$, the isomorphisms $({\mathbb{C}}\rtimes \Gamma) {\tilde{\otimes}}A \to A$ and $A {\tilde{\otimes}}({\mathbb{C}}\rtimes \Gamma) \to A$ are equal to $\epsilon_{{\mathbb{C}}\Gamma} \otimes \operatorname{id}$ and $\operatorname{id}\otimes
\epsilon_{{\mathbb{C}}\Gamma}$.
\[lemma:b-to-hopf\] Let $(A,\Delta,\epsilon,S)$ be a $({\mathbb{C}},\Gamma)$-Hopf algebroid and let $\epsilon':=\epsilon_{{\mathbb{C}}\Gamma}
\circ \epsilon \colon A \to {\mathbb{C}}$. The $(A,\Delta,\epsilon',S)$ is a Hopf algebra and $\epsilon
\colon A \to {\mathbb{C}}\Gamma$ is morphism of Hopf algebras.
The preceding observations easily imply that $(A,\Delta,\epsilon',S)$ is a Hopf algebra. To see that $\epsilon$ is a morphism of Hopf algebras, use the fact that $\Delta,\epsilon,S$ are $\Gamma\times
\Gamma$-graded.
\[lemma:b-from-hopf\] Let $(A,\Delta,\epsilon,S)$ be a Hopf algebra with a morphism $\pi \colon A \to {\mathbb{C}}\Gamma$. Then $A$ is a $({\mathbb{C}},\Gamma)$-algebra with respect to the grading given by $A_{\gamma,\gamma'} = \{ a\in A : (\pi \otimes
\operatorname{id}\otimes \pi)(\Delta(a)) = \gamma \otimes a \otimes
\gamma'\}$ for all $\gamma,\gamma' \in \Gamma$, and $(A,\Delta,\pi,S)$ is a $({\mathbb{C}},\Gamma)$-Hopf algebroid.
The formula above evidently defines a $\Gamma\times\Gamma$-grading on $A$. Coassociativity of $\Delta$ implies that $\Delta(A) \subseteq A {\tilde{\otimes}}A$. The remark preceding Lemma \[lemma:b-to-hopf\] and the relation $\epsilon_{{\mathbb{C}}\Gamma} \circ \pi =
\epsilon$ imply $(\pi {\tilde{\otimes}}\operatorname{id})\circ \Delta =\operatorname{id}=
(\operatorname{id}{\tilde{\otimes}}\pi) \circ \Delta$. Finally, in the notation of Definition \[definition:bg-hopf\], $\check m \circ (S
{\tilde{\otimes}}\operatorname{id}) \circ \Delta =m \circ (S\otimes \operatorname{id}) \circ
\Delta = \epsilon = \check s \circ \pi$ and similarly $\hat m \circ (\operatorname{id}{\tilde{\otimes}}S) \circ \Delta = \hat r \circ
\pi$.
Putting everything together, one easily verifies:
There exists an equivalence of categories ${\mathbf{Hopf}}_{({\mathbb{C}},\Gamma)}
\stackrel{{\mathbf{F}}}{\underset{{\mathbf{G}}}{\rightleftarrows}}
{\mathbf{Hopf}}_{{\mathbb{C}}\Gamma}$, where ${\mathbf{F}}(A,\Delta,\epsilon,S)= ((A,\Delta,\epsilon_{{\mathbb{C}}\Gamma}\circ \epsilon,S),\epsilon)$, ${\mathbf{F}}\phi= \phi$ and ${\mathbf{G}}((A,\Delta,\epsilon,S),\pi)= (A,\Delta,\pi,S)$ with the grading on $A$ defined as in Lemma \[lemma:b-from-hopf\], and ${\mathbf{G}}\phi= \phi$. Likewise, there exists an equivalence ${\mathbf{Hopf}}_{({\mathbb{C}},\Gamma)}^{*} \rightleftarrows
{\mathbf{Hopf}}^{*}_{{\mathbb{C}}\Gamma}$.
Let us next consider the base change from ${\mathbb{C}}$ to a commutative algebra $C$ along the unital inclusion $\phi
\colon {\mathbb{C}}\to C$ for a $({\mathbb{C}},\Gamma)$-Hopf algebroid $(A,\Delta,\epsilon,S)$.
The action of $\Gamma$ on $C$ and the morphism $\epsilon \colon A\to{\mathbb{C}}\Gamma$ turn $C$ into a left module algebra over the Hopf algebra $(A,\Delta,\epsilon_{{\mathbb{C}}\Gamma} \circ
\epsilon,S)$, and $\phi_{*}(A,\Delta,\epsilon,S)$ coincides with the Hopf algebroids considered in [@boehm:algebroids § 3.4.6] and [@kadison:pseudo-hopf Theorem 3.1], and is closely related to the quantum transformation groupoid considered in [@vainer Example 2.6].
Assume that $C$ is an algebra of functions on $\Gamma$ on which $\Gamma$ acts by left translations.
\[proposition:bg-lie\] Define $m\colon A \to \operatorname{End}(A)$ and $m_{r},m_{s} \colon C
\to \operatorname{End}(A)$ by $m(a')a=a'a$, $m_{r}(c)a = c
(\partial^{r}_{a})a$, $m_{s}(c)a = c(\partial^{s}_{a})a$ for all $a,a'\in A$, $c\in C$. Then there exists a homomorphism $\lambda \colon \phi_{*}(A)
\to \operatorname{End}(A)$, $c \otimes a \otimes c' \mapsto
m_{r}(c)m(a)m_{s}(c')$, and $\lambda$ is injective if $aA_{\gamma,\gamma'} \neq 0$ for all non-zero $a \in A$ and all $\gamma,\gamma' \in \Gamma$.
First, note that $$\begin{aligned}
m(a')m_{r}(c)a = a' c( \partial^{r}_{a})a =
c((\partial^{r}_{a'})^{-1}\partial^{r}_{a'a})a'a =
m_{r}(\partial^{r}_{a'}(c)) m(a)a
\end{aligned}$$ and likewise $m(a')m_{s}(c)=m_{s}(\partial^{s}_{a'}(c))m(a')$ for all $a,a'\in A$, $c\in C$. The existence of $\lambda$ follows. Assume that $aA_{\gamma,\gamma'} \neq 0$ for all non-zero $a \in A$ and all $\gamma,\gamma' \in
\Gamma$. Let $d:=\sum_{i} c_{i} \otimes a_{i} \otimes
c'_{i} \in \phi_{*}(A)$ be non-zero, where all $a_{i}$ are homogeneous. Identifying $C\otimes A \otimes C$ with a space of $A$-valued functions on $\Gamma\times \Gamma$ and using the assumption, we first find $\gamma,\gamma' \in \Gamma$ such that $a:=\sum_{i}
c_{i}(\partial^{r}_{a_{i}}\gamma)a_{i}c'_{i}(\gamma')$ is non-zero, and then an $a' \in
A_{\gamma,\gamma'}$ such that $\lambda(d)a' = aa' \neq 0$.
\[remark:bg-lie\] Regard elements of $C$ as functionals on ${\mathbb{C}}\Gamma$ via $c(\sum_{i} b_{i}\gamma_{i})=
\sum_{i} b_{i}c(\gamma_{i})$. Then $m_{r}(c)a = (c \circ
\epsilon \otimes \operatorname{id})(\Delta(a))$, $m_{s}(c)a =(\operatorname{id}\otimes c\circ \epsilon)(\Delta(a))$ for all $c\in C$, $a\in A$.
\[example:bg-lie\] Let $G$ be a compact Lie group, $\mathcal{O}(G)$ its Hopf algebra of representative functions [@timmermann:buch §1.2] and $T
\subseteq G$ a torus of rank $d$. We now apply Proposition \[proposition:bg-lie\], where
- $A=\mathcal{O}(G)$, regarded as a Hopf $({\mathbb{C}},\hat T)$-algebroid as in Lemma \[lemma:b-from-hopf\] using the homomorphism $\pi
\colon \mathcal{O}(G) \to \mathcal{O}(T)$ induced from the inclusion $T\subseteq G$, and the isomorphism $\mathcal{O}(T) \cong {\mathbb{C}}\hat T$,
- $C=U\mathfrak{t}$ is the enveloping algebra of the Lie algebra $\mathfrak{t}$ of $T$, regarded as a polynomial algebra of functions on $\hat T$ such that $X(\chi) = \frac{d}{dt}\big|_{t=0} \chi(e(tX))$, where $e\colon \mathfrak{t} \to T$ denotes the exponential map.
If we regard $U\mathfrak{t}$ as functionals on the algebra ${\mathbb{C}}\hat T \cong \mathcal{O}(T)$ as in Remark \[remark:bg-lie\], then $X(f)=\frac{d}{dt}\big|_{t=0} f(e(tX))$ and hence $m_{r},m_{s} \colon U\mathfrak{t}
\to \operatorname{End}(\mathcal{O}(G))$ are given by $$\begin{aligned}
(m_{r}(X)a)(x) &= \frac{d}{dt}\Big|_{t=0}
a(e(tX)x), & (m_{s}(X)a)(x) &=
\frac{d}{dt}\Big|_{t=0} a(xe(tX))
\end{aligned}$$ for all $X \in \mathfrak{t}$, $a\in \mathcal{O}(G)$, $x\in
G$. Thus $\lambda(\mathcal{O}(G))
\subseteq \operatorname{End}(\mathcal{O}(G))$ is the algebra generated by multiplication operators for functions in $\mathcal{O}(G)$ and by left and right differentiation operators along $T \subseteq G$.
If $G$ is connected, then $\mathcal{O}(G)$ has no zero-divisors and hence $\lambda$ is injective as soon as for all $\chi,\chi' \in \hat T$, there exists some non-zero $a \in \mathcal{O}(G)$ such that $a(xyz)=\chi(x)a(y)\chi'(z)$ for all $x,z \in T$ and $y\in
G$.
Intertwiners for $(B,\Gamma)$-algebras {#section:rn}
--------------------------------------
In this subsection, we study relations of the form used to define the free orthogonal and free unitary dynamical quantum groups ${A^{B}_{\mathrm{o}}}(\nabla,F)$ and ${A^{B}_{\mathrm{u}}}(\nabla,F)$, and show that such relations admit a number of natural transformations. Conceptually, these relations express that certain matrices are intertwiners or morphisms of corepresentations, and the transformations correspond to certain functors of corepresentation categories. Although elementary, these observations provide short and systematic proofs for the main results in the following subsection.
Regard ${M_{n}}(B)$ as a subalgebra of ${M_{n}}(B\rtimes \Gamma)$, and let $A$ be a $(B,\Gamma)$-algebra. Given a linear map $\phi \colon A\to C$ between algebras, we denote by $\phi_{n} \colon {M_{n}}(A) \to {M_{n}}(C)$ its entry-wise extension to $n\times n$-matrices.
\[definition:rn-intertwiner\] A matrix $u \in {M_{n}}(A)$ is *homogeneous* if there are $\gamma_{1},\ldots,\gamma_{n} \in A$ such that $u_{ij} \in
A_{\gamma_{i},\gamma_{j}}$ for all $i,j$. In that case, let $\partial_{u,i}:=\gamma_{i}$ for all $i$ and $\partial_{u}:=\operatorname{diag}(\gamma_{1},\ldots,\gamma_{n}) \in
{M_{n}}(B\rtimes \Gamma)$. An *intertwiner* for homogeneous matrices $u,v\in {M_{n}}(A)$ is an $F\in {\mathrm{GL}_{n}}(B)$ satisfying $\partial_{v}F\partial_{u}^{-1} \in {M_{n}}(B)$ and $r_{n}(\partial_{v}F\partial_{u}^{-1}) u = vs_{n}(F)$. We write such an intertwiner as $u\xrightarrow{F} v$ and let $\hat F:=\partial_{v} F\partial_{u}^{-1}$ if $u,v$ are understood.
If $u\xrightarrow{F} v$ and $v\xrightarrow{G} w$ are intertwiners, then evidently so are $v\xrightarrow{F^{-1}} u$ and $u
\xrightarrow{GF} w$.
We denote by ${\mathcal{R}_{n}}(A)$ the category of all homogeneous matrices in ${M_{n}}(A)$ together with their intertwiners as morphisms, and by ${\mathcal{R}_{n}}^{\times}(A)$ and ${\mathcal{R}_{n}}^{\times{\mathsf{T}}}(A)$ the full subcategories formed by all homogeneous $v$ in ${\mathrm{GL}_{n}}(A)$ or ${\mathrm{GL}_{n}}(A)^{{\mathsf{T}}}$, respectively.
Evidently, ${\mathcal{R}_{n}}(A)$ is a groupoid, and the assignment $A\mapsto {\mathcal{R}_{n}}(A)$ extends to a functor from $(B,\Gamma)$-algebras to groupoids.
We shall make frequent use of the following straightforward relations.
\[lemma:rn-grading\] Let $u,v\in {M_{n}}(A)$ be homogeneous, $F\in {M_{n}}(B)$ and $\hat F=\partial_{v}F\partial_{u}^{-1}$. Then $$\begin{aligned}
\hat F \in {M_{n}}(B) \ \Leftrightarrow \ (F_{ij}=0 \text{
whenever } \partial_{v,i} \neq \partial_{u,j}).\end{aligned}$$ Assume that these condition holds. Then $ \hat F =(\partial_{v,i}(F_{ij}))_{i,j} =
(\partial_{u,j}(F_{ij}))_{i,j}$ and $$\begin{aligned}
\hat F^{{\mathsf{T}}} &=
\partial_{u}F^{{\mathsf{T}}}\partial_{v}^{-1}, &
(\partial_{v}F)^{-{\mathsf{T}}} &= F^{-{\mathsf{T}}} \partial_{u}^{-1}, &
(F\partial_{u}^{-1})^{-{\mathsf{T}}} &= \partial_{v} F^{-{\mathsf{T}}}.
\end{aligned}$$ If $B$ is a $*$-algebra and $\Gamma$ preserves the involution, then $\overline{\partial_{v}F} = \overline{F} \partial_{u}^{-1}$ and $\overline{F\partial_{u}} = \partial_{v}^{-1} \overline{F}$.
Given $u,v \in {M_{n}}(A)$ such that $\partial^{s}_{u_{ik}}
= \partial^{r}_{v_{kj}}$ for all $i,k,j$, let $u {\tilde\boxtimes}v:=
(\sum_{k} u_{ik} {\tilde{\otimes}}v_{kj})_{i,j} \in {M_{n}}(A{\tilde{\otimes}}A)$.
\[lemma:rn-ed\] There exist functors $$\begin{aligned}
\bm{\epsilon} \colon {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(B\rtimes\Gamma), &&
&u &\mapsto \partial_{u}, & (u\xrightarrow{F} v)
&\mapsto
(\partial_{u} \xrightarrow{F} \partial_{v}), \\
\bm{\Delta} \colon {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(A{\tilde{\otimes}}A), && & u
&\mapsto u{\tilde\boxtimes}u, & (u\xrightarrow{F} v) &\mapsto
(u{\tilde\boxtimes}u \xrightarrow{F} v{\tilde\boxtimes}v), \\
(-)^{{\mathsf{op}}}\colon {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(A^{{\mathsf{op}}}), & & & u
&\mapsto u^{{\mathsf{op}}}:=u, &
(u\xrightarrow{F} v) &\mapsto (u^{{\mathsf{op}}} \xrightarrow{\hat
F} v^{{\mathsf{op}}}),
\end{aligned}$$ and $\partial_{u{\tilde\boxtimes}u} = \partial_{u}$, $\partial_{u^{{\mathsf{op}}}}
=\partial^{-1}_{u}$ for all $u\in {\mathcal{R}_{n}}(A)$.
For each $u\in {\mathcal{R}_{n}}(A)$, the matrices $\partial_{u}, u{\tilde\boxtimes}u,u^{{\mathsf{op}}}$ evidently are homogeneous, and for every intertwiner $u\xrightarrow{F} v$, Lemma \[lemma:rn-grading\] implies $$\begin{aligned}
r_{n}(\hat F)u {\tilde\boxtimes}u &= vs_{n}(F) {\tilde\boxtimes}u = v {\tilde\boxtimes}r_{n}(\hat F)u =
v{\tilde\boxtimes}vs_{n}(F), \\
r_{n}(\partial_{v^{{\mathsf{op}}}}\hat F \partial_{u^{{\mathsf{op}}}}^{-1})
u^{{\mathsf{op}}} &= r_{n}(F)^{{\mathsf{op}}}u^{{\mathsf{op}}} = (r_{n}(\hat
F)u)^{{\mathsf{op}}} = (vs_{n}(F))^{{\mathsf{op}}} = v^{{\mathsf{op}}}s_{n}(\hat
F)^{{\mathsf{op}}}. \end{aligned}$$ Functoriality of the assignments is evident.
\[lemma:rn-intertwiner\] There exist contravariant functors $$\begin{aligned}
(-)^{{\mathsf{T}},{\mathsf{co}}} \colon {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(A^{{\mathsf{co}}}), & u
&\mapsto u^{{\mathsf{T}},{\mathsf{co}}} := u^{{\mathsf{T}}}, & (u \xrightarrow{F} v)
&\mapsto (v^{{\mathsf{T}},{\mathsf{co}}} \xrightarrow{F^{{\mathsf{T}}}}
u^{{\mathsf{T}},{\mathsf{co}}}), \\
(-)^{-{\mathsf{co}}} \colon {\mathcal{R}_{n}}^{\times}(A) &\to {\mathcal{R}_{n}}(A^{{\mathsf{co}}}), & u
&\mapsto u^{-{\mathsf{co}}}:= u^{-1}, & (u \xrightarrow{F} v) &\mapsto
(v^{-{\mathsf{co}}} \xrightarrow{\hat F^{-1}} u^{-{\mathsf{co}}}),
\end{aligned}$$ and $\partial_{u^{{\mathsf{T}},{\mathsf{co}}}} =\partial_{u}$ and $\partial_{u^{-{\mathsf{co}}}} = \partial_{u}^{-1}$ for all $u$.
If $u \in {\mathcal{R}_{n}}(A)$, then $u^{{\mathsf{T}},{\mathsf{co}}}$ evidently is homogeneous as claimed. Assume $u \in
{\mathcal{R}_{n}}^{\times}(A)$. We claim that $u^{-{\mathsf{co}}}$ is homogeneous and $\partial_{u^{-{\mathsf{co}}}} = \partial_{u}^{-1}$. For each $i,j$, let $w_{ij}$ be the homogeneous part of $(u^{-1})_{ij}$ of degree $(\partial_{u,j}^{-1},\partial_{u,i}^{-1})$. Then $\sum_{l} u_{il}w_{lj}$ is homogeneous of degree $(\partial_{u,i}\partial_{u,j}^{-1},e)$ and coincides with the homogeneous part of the sum $\sum_{l}
u_{il}(u^{-1})_{lj}$ of the same degree for each $i,j$. Hence, $uw=uu^{-1}$ and the claim follows.
Let $u\xrightarrow{F} v$ be an intertwiner. Using Lemma \[lemma:rn-grading\], one easily verifies that $$\begin{aligned}
s_{n}(\partial_{u^{{\mathsf{T}},{\mathsf{co}}}}F^{{\mathsf{T}}}\partial_{v^{{\mathsf{T}},{\mathsf{co}}}}^{-1})v^{{\mathsf{T}}}
&= s_{n}(\hat F)^{{\mathsf{T}}}v^{{\mathsf{T}}} = (vs_{n}(F))^{{\mathsf{T}}}
= (r_{n}(\hat F)u)^{{\mathsf{T}}} = u^{{\mathsf{T}}}r_{n}(F^{{\mathsf{T}}}), \\
s_{n}(\partial_{u^{-{\mathsf{co}}}} \hat
F^{-1} \partial_{v^{-{\mathsf{co}}}}^{-1})v^{-1} &=
s_{n}(F^{-1})v^{-1} =u^{-1}r_{n}(\hat F^{-1}).
\end{aligned}$$ Finally, functoriality of the assignments is easily checked.
Forming suitable compositions, we obtain further co- or contravariant functors $$\begin{aligned}
(-)^{-{\mathsf{T}}} &= (-)^{{\mathsf{T}},{\mathsf{co}}}\circ (-)^{-{\mathsf{co}}}\colon
{\mathcal{R}_{n}}^{\times}(A) \to {\mathcal{R}_{n}}^{\times{\mathsf{T}}}(A), &
&\begin{cases}
u \mapsto u^{-{\mathsf{T}}}:=(u^{-1})^{{\mathsf{T}}}, \\
(u\xrightarrow{F} v) \mapsto (u^{-{\mathsf{T}}}
\xrightarrow{\hat F^{-{\mathsf{T}}}} v^{-{\mathsf{T}}}),
\end{cases}\\
(-)^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}} &= (-)^{-{\mathsf{co}}} \circ (-)^{{\mathsf{T}},{\mathsf{co}}} \colon
{\mathcal{R}_{n}}^{\times{\mathsf{T}}}(A) \to {\mathcal{R}_{n}}^{\times}(A), &
&\begin{cases}
u \mapsto u^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}}:=(u^{{\mathsf{T}}})^{-1}, \\
(u\xrightarrow{F} v) \mapsto (u^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}}
\xrightarrow{\hat F^{-{\mathsf{T}}}} v^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}})
\end{cases} \end{aligned}$$ and $$\begin{aligned}
(-)^{-{\mathsf{co}},{\mathsf{op}}} = (-)^{{\mathsf{op}}}\circ (-)^{-{\mathsf{co}}} \colon
{\mathcal{R}_{n}}^{\times}(A) \to {\mathcal{R}_{n}}(A^{{\mathsf{co}},{\mathsf{op}}}), \
\begin{cases}
u \mapsto (u^{-{\mathsf{co}}})^{{\mathsf{op}}}, \\ (u \xrightarrow{\! F\!} v)
\mapsto (v^{-{\mathsf{co}},{\mathsf{op}}} \xrightarrow{\!\! F^{-1}\!\! }
u^{-{\mathsf{co}},{\mathsf{op}}}),
\end{cases}
\end{aligned}$$ where $\partial_{u^{-{\mathsf{co}},{\mathsf{op}}}} =\partial_{u}$ and $\partial_{u^{-{\mathsf{T}}}} = \partial_{u^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}}} = \partial_{u}^{-1}$ for all $u$.
\[lemma:rn-commute\] The following relations hold: $$\begin{aligned}
\mathrm{i)} & \ \ (-)^{{\mathsf{op}}} \circ (-)^{-{\mathsf{T}}} = (-)^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}} \circ
(-)^{{\mathsf{op}}}, &
\mathrm{ii) } &\ \ (-)^{-{\mathsf{T}}} \circ (-)^{{\mathsf{op}}} = (-)^{{\mathsf{op}}}
\circ
(-)^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}}, \\
\mathrm{iii) } & \ \ (-)^{-{\mathsf{T}}} \circ \bm{\Delta} = \bm{\Delta} \circ
(-)^{-{\mathsf{T}}}, &
\mathrm{iv) } & \ \ (-)^{-{\mathsf{T}}} \circ (-)^{-{\mathsf{co}},{\mathsf{op}}} =
(-)^{-{\mathsf{co}},{\mathsf{op}}} \circ (-)^{-{\mathsf{T}}}.
\end{aligned}$$
i\) We first check that the compositions agree on objects. Let us write $v^{{\mathsf{op}}}$ if we regard $v\in {M_{n}}(A)$ as an element of ${M_{n}}(A^{{\mathsf{op}}})$. Then map ${M_{n}}(A) \to
{M_{n}}(A^{{\mathsf{op}}})$ given by $v \mapsto (v^{{\mathsf{T}}})^{{\mathsf{op}}} =
(v^{{\mathsf{op}}})^{{\mathsf{T}}}$ is an antihomomorphism and hence $(v^{-{\mathsf{T}}})^{{\mathsf{op}}} =
(v^{{\mathsf{T}},{\mathsf{op}}})^{-1} = (v^{{\mathsf{op}}})^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}}$ for all $v \in
{\mathrm{GL}_{n}}(A)$. The compositions also agree on morphisms because for every intertwiner $u\xrightarrow{F} v$, we have $\partial_{v^{-{\mathsf{T}}}}(\partial_{v}F\partial_{u}^{-1})^{-{\mathsf{T}}}\partial_{u^{-{\mathsf{T}}}}^{-1}
=
\partial_{v}^{-{\mathsf{T}}} \partial_{v}F^{-{\mathsf{T}}}\partial_{u}^{-1}
\partial_{u} = F^{-{\mathsf{T}}}$.
ii\) This equation follows similarly like i).
iii\) Let $u \in
{\mathcal{R}_{n}}^{\times}(A)$. Then $(u {\tilde\boxtimes}u)^{-{\mathsf{T}}} =u^{-{\mathsf{T}}}
{\tilde\boxtimes}u^{-{\mathsf{T}}}$ because $$\begin{aligned}
\sum_{k} (u{\tilde\boxtimes}u)_{ik} (u^{-{\mathsf{T}}} {\tilde\boxtimes}u^{-{\mathsf{T}}})_{jk}
= \sum_{k,l,m} u_{il} (u^{-1})_{mj} {\tilde{\otimes}}u_{lk}(u^{-1})_{km} = \delta_{i,j} 1 {\tilde{\otimes}}1.
\end{aligned}$$ and similarly $\sum_{k} (u^{-{\mathsf{T}}} {\tilde\boxtimes}u^{-{\mathsf{T}}})_{ki}
(u{\tilde\boxtimes}u)_{kj} = \delta_{i,j} 1{\tilde{\otimes}}1$. For morphisms, we have nothing to check because $\partial_{u{\tilde\boxtimes}u}
= \partial_{u}$.
iv\) This equation follows from the relation $(-)^{-{\mathsf{T}}}
\circ (-)^{{\mathsf{op}}} \circ (-)^{-{\mathsf{co}}} = (-)^{{\mathsf{op}}} \circ
(-)^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}} \circ (-)^{-{\mathsf{co}}} = (-)^{{\mathsf{op}}} \circ (-)^{-{\mathsf{co}}}
\circ (-)^{{\mathsf{T}},{\mathsf{co}}} \circ (-)^{-{\mathsf{co}}}$.
Assume for a moment that $(A,\Delta,\epsilon,S)$ is a Hopf $(B,\Gamma)$-algebroid.
\[definition:rn-corep\] A *matrix corepresentation* of $(A,\Delta,\epsilon,S)$ is a $v\in {\mathcal{R}_{n}}(A)$ for some $n\in
{\mathbb{N}}$ satisfying $\Delta_{n}(v)=v{\tilde\boxtimes}v$, $\epsilon_{n}(v)=\partial_{v}$, $S_{n}(v)=v^{-1}$.
\[lemma:rn-corep\] If $v\xrightarrow{F} w$ is a morphism in ${\mathcal{R}_{n}}(A)$ and $v$ is a matrix corepresentation, then so is $w$.
Applying the morphisms $\Delta,\epsilon,S$ and the functors $\bm{\Delta},\bm{\epsilon},(-)^{-{\mathsf{co}},{\mathsf{op}}}$ to $v
\xrightarrow{F} w$ or its inverse, we get intertwiners $w{\tilde\boxtimes}w \xrightarrow{F^{-1}} v{\tilde\boxtimes}v = \Delta_{n}(v)
\xrightarrow{F} \Delta_{n}(w)$, $\partial_{w}
\xrightarrow{F^{-1}} \partial_{v} = \epsilon_{n}(v)
\xrightarrow{F} \epsilon_{n}(w)$ and $w^{-{\mathsf{co}},{\mathsf{op}}}
\xrightarrow{F^{-1}} v^{-{\mathsf{co}},{\mathsf{op}}} = S_{n}(v)
\xrightarrow{F} S_{n}(w)$.
Let us now discuss the involutive case.
Given a $*$-algebra $C$ and a matrix $v \in {M_{n}}(C)$, we write $\overline{v}:=(v^{*}_{ij})_{i,j} = (v^{*})^{{\mathsf{T}}}$.
Assume that $B$ is a $*$-algebra, that $\Gamma$ preserves the involution, and that $A$ is a $(B,\Gamma)$-algebra. Then there exists an obvious functor ${\mathcal{R}_{n}}(A) \to {\mathcal{R}_{n}}(\bar A)$, given by $u \mapsto u$ and $(u \xrightarrow{F} v) \mapsto (u
\xrightarrow{\overline{F}} v)$. Composition with $(-)^{{\mathsf{op}}}$ gives a functor $$\begin{aligned}
(-)^{{\overline{{\mathsf{op}}}}} \colon {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(\overline{A}^{{\mathsf{op}}}), &
u &\mapsto u^{{\overline{{\mathsf{op}}}}}:= u^{{\mathsf{op}}}, & (u \xrightarrow{F} v)
&\mapsto (u^{{\overline{{\mathsf{op}}}}} \xrightarrow{\overline{\hat F}}
v^{{\overline{{\mathsf{op}}}}}),
\end{aligned}$$ and $\partial_{u^{{\overline{{\mathsf{op}}}}}} = \partial_{u}^{-1}$ for all $u$. For later use, we note the following relation.
\[lemma:rn-barop\] Let $u^{-{\mathsf{T}}}
\xrightarrow{F} v$ be an intertwiner in ${\mathcal{R}_{n}}^{\times}(A)\cap {\mathcal{R}_{n}}^{\times,{\mathsf{T}}}(A)$. Then $(v^{{\overline{{\mathsf{op}}}}})^{-{\mathsf{T}}} \xrightarrow{F^{*}} u^{{\overline{{\mathsf{op}}}}}$ is an intertwiner in ${\mathcal{R}_{n}}^{\times}(\overline{A}^{{\mathsf{op}}})\cap
{\mathcal{R}_{n}}^{\times,{\mathsf{T}}}(\overline{A}^{{\mathsf{op}}})$.
Subsequent applications of the functors $(-)^{{\overline{{\mathsf{op}}}}}$, $(-)^{-{\mathsf{T}}}$ yield intertwiners $(v^{-{\mathsf{T}}})^{{\overline{{\mathsf{op}}}}} =
(v^{{\overline{{\mathsf{op}}}}})^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}} \xrightarrow{\overline{\hat F}^{-1}}
u^{{\overline{{\mathsf{op}}}}}$ and $(u^{{\overline{{\mathsf{op}}}}})^{-{\mathsf{T}}}
\xrightarrow{\overline{F}^{-{\mathsf{T}}}=F^{-*}} v^{{\overline{{\mathsf{op}}}}}$.
Finally, assume that $A$ is a $(B,\Gamma)$-$*$-algebra. Then there exists a functor $$\begin{aligned}
(-)^{*,{\mathsf{co}}} \colon {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(A^{{\mathsf{co}}}), &
u &\mapsto u^{*,{\mathsf{co}}}:=u^{*}, &
(u \xrightarrow{F} v) &\mapsto (v^{*,{\mathsf{co}}}
\xrightarrow{\hat F^{*}} u^{*,{\mathsf{co}}}), \end{aligned}$$ because $s_{n}(F^{*})v^{*} =u^{*}r_{n}(\hat F^{*})$ for every intertwiner $u\xrightarrow{F} v$, and $\partial_{u^{*,{\mathsf{co}}}}
= \partial_{u}^{-1}$. Composing with $(-)^{{\mathsf{T}},{\mathsf{co}}}$ for $A^{{\mathsf{co}}}$ and with $(-)^{-{\mathsf{T}}}$, respectively, we get functors $$\begin{aligned}
\label{eq:rn-overline}
\overline{(-)}\colon {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(A), & u\mapsto
\overline{u} &= (u_{ij}^{*})_{i,j}, &
(u\xrightarrow{F} v) &\mapsto (\overline{u}
\xrightarrow{\overline{\hat F}} \overline{v}), \\
\label{eq:rn-star} {\mathcal{R}_{n}}(A) &\to {\mathcal{R}_{n}}(A), &
u\mapsto \overline{u} &= \overline{u}^{-{\mathsf{T}}} = \overline{u^{-{{{\begin{sideways}{\begin{sideways}$\scriptstyle\mathsf{T}$\end{sideways}}\end{sideways}}}}}}, &
(u\xrightarrow{F} v) &\mapsto (\overline{u}^{-{\mathsf{T}}}
\xrightarrow{F^{-*}} \overline{v}^{-{\mathsf{T}}}).\end{aligned}$$
The free orthogonal and free unitary dynamical quantum groups {#section:ao}
-------------------------------------------------------------
Using the preparations of the last subsection, we now show that the algebras ${A^{B}_{\mathrm{o}}}(\nabla,F)$ and ${A^{B}_{\mathrm{u}}}(\nabla,F)$ are $(B,\Gamma)$-Hopf algebroids as claimed in the introduction.
Let $B$ be a commutative algebra with an action of a group $\Gamma$ as before, and let $\gamma_{1},\ldots,\gamma_{n}
\in \Gamma$ and $\nabla=\operatorname{diag}(\gamma_{1},\ldots,\gamma_{n})
\in {M_{n}}(B\rtimes \Gamma)$.
Let $F \in {\mathrm{GL}_{n}}(B)$ be *$\nabla$-odd* in the sense that $\nabla F \nabla \in {M_{n}}(B)$. The first definition and theorem in the introduction can be reformulated as follows.
The *free orthogonal dynamical quantum group over $B$ with parameters $(\nabla,F)$* is the universal $(B,\Gamma)$-algebra ${A^{B}_{\mathrm{o}}}(\nabla,F)$ with a $v\in
{\mathcal{R}_{n}}^{\times}({A^{B}_{\mathrm{o}}}(\nabla,F))$ such that $\partial_{v} =
\nabla$ and $v^{-{\mathsf{T}}} \xrightarrow{F} v$ is an intertwiner.
\[theorem:ao-hopf\] The $(B,\Gamma)$-algebra ${A^{B}_{\mathrm{o}}}(\nabla,F)$ can be equipped with a unique structure of a $(B,\Gamma)$-Hopf algebroid such that $v$ becomes a matrix corepresentation.
The existence of morphisms $\Delta \colon A\to A{\tilde{\otimes}}A$, $\epsilon\colon A\to B\rtimes \Gamma$, $S \colon A\to
A^{{\mathsf{co}},{\mathsf{op}}}$ satisfying $\Delta_{n}(v) =v{\tilde\boxtimes}v$, $\epsilon_{n}(v) =\nabla$, $S_{n}(v) =v^{-1}$ follows from the universal property of $A$ and the relations $$\begin{gathered}
\bm{\Delta}(v^{-{\mathsf{T}}}\xrightarrow{F} v) =
((v{\tilde\boxtimes}v)^{-{\mathsf{T}}} \xrightarrow{F} v{\tilde\boxtimes}v), \quad
\bm{\epsilon}(v^{-{\mathsf{T}}} \xrightarrow{F} v) =
(\nabla^{-{\mathsf{T}}} \xrightarrow{F} \nabla), \\
(v^{-{\mathsf{T}}} \xrightarrow{F} v)^{-{\mathsf{co}},{\mathsf{op}}} =
( v^{-{\mathsf{co}},{\mathsf{op}}} \xrightarrow{F^{-1}} (v^{-{\mathsf{co}},{\mathsf{op}}})^{-{\mathsf{T}}});\end{gathered}$$ see Lemma \[lemma:rn-intertwiner\] and \[lemma:rn-commute\]. Straightforward calculations show that $(A,\Delta,\epsilon,S)$ is a $(B,\Gamma)$-Hopf algebroid.
\[remarks:ao\]
1. In the definition of ${A^{B}_{\mathrm{o}}}(\nabla,F)$, we may evidently assume that $\Gamma$ is generated by the diagonal components $\gamma_{1},\ldots,\gamma_{n}$ of $\nabla$.
2. Denote by $B_{0} \subseteq B$ the smallest $\Gamma$-invariant subalgebra containing the entries of $F$ and $F^{-1}$, and by $\iota\colon B_{0}\to B$ the inclusion. Then there exists an obvious isomorphism ${A^{B}_{\mathrm{o}}}(\nabla,F) \cong\iota_{*}{A^{B_{0}}_{\mathrm{o}}}(\nabla,F)$.
3. Let $H \in {\mathrm{GL}_{n}}(B)$ be $\nabla$-even and $\hat H=\nabla
H \nabla^{-1}$. Then there exists an isomorphism $
{A^{B}_{\mathrm{o}}}(\nabla,HF\hat
H^{{\mathsf{T}}}) \to {A^{B}_{\mathrm{o}}}(\nabla,F)$ of $(B,\Gamma)$-Hopf algebroids whose extension to matrices sends $v \in {A^{B}_{\mathrm{o}}}(\nabla,HF\hat
H^{{\mathsf{T}}})$ to $w:=r_{n}(\hat
H) vs_{n}(H)^{-1}\in {A^{B}_{\mathrm{o}}}(\nabla,F)$. Indeed, there exists such a morphism of $(B,\Gamma)$-algebras because in ${A^{B}_{\mathrm{o}}}(\nabla,F)$, we have intertwiners $v\xrightarrow{H}
w$, $v^{-{\mathsf{T}}} \xrightarrow{\hat H^{-{\mathsf{T}}}} w^{-{\mathsf{T}}}$ and $v^{-{\mathsf{T}}}\xrightarrow{F} v$, whence $w^{-{\mathsf{T}}}
\xrightarrow{H F\hat H^{{\mathsf{T}}}} w$, and this morphism is compatible with $\Delta,\epsilon,S$ because $w$ is a matrix corepresentation by Lemma \[lemma:rn-corep\]. A similar argument yields the inverse of this morphism.
Assume that $B$ carries an involution which is preserved by $\Gamma$, and let $F\in {\mathrm{GL}_{n}}(B)$ be self-adjoint and *$\nabla$-even* in the sense that $\nabla F \nabla^{-1}
\in {M_{n}}(B)$. The second definition and theorem in the introduction can be reformulated as follows.
\[definition:intro-au-hopf\] The *free unitary dynamical quantum group over $B$ with parameters $(\nabla,F)$* is the universal $(B,\Gamma)$-$*$-algebra ${A^{B}_{\mathrm{u}}}(\nabla,F)$ with a unitary $u
\in {\mathcal{R}_{n}}^{\times}({A^{B}_{\mathrm{u}}}(\nabla,F))$ such that $\partial_{v} =
\nabla$ and $(v^{-{\mathsf{T}}})^{-{\mathsf{T}}} \xrightarrow{F} v$ is an intertwiner.
\[theorem:intro-au-hopf\] The $*$-algebra ${A^{B}_{\mathrm{u}}}(\nabla,F)$ can be equipped with a unique structure of a $(B,\Gamma)$-Hopf $*$-algebroid such that $v$ becomes a matrix corepresentation.
To prove this result, we introduce an auxiliary $(B,\Gamma)$-algebra which does not involve the involution on $B$.
\[definition:au-prime\] We denote by ${A^{B}_{\mathrm{u}'}}(\nabla,F)$ the universal $(B,\Gamma)$-algebra with $v,w \in {\mathcal{R}_{n}}^{\times}(A)$ such that $\partial_{v}=\nabla$, $\partial_{w}=\nabla^{-1}$ and $v^{-{\mathsf{T}}} \xrightarrow{1}
w$, $w^{-{\mathsf{T}}} \xrightarrow{F} v$ are intertwiners.
Using the same techniques as in the proof of Theorem \[theorem:ao-hopf\], one finds:
\[proposition:au-prime-hopf\] The $(B,\Gamma)$-algebra ${A^{B}_{\mathrm{u}'}}(\nabla,F)$ can be equipped with a unique structure of a $(B,\Gamma)$-Hopf algebroid such that $v$ and $w$ become matrix corepresentations.
\[proposition:au-prime-star\] The $(B,\Gamma)$-algebra ${A^{B}_{\mathrm{u}'}}(\nabla,F)$ can be equipped with an involution such that it becomes a $(B,\Gamma)$-Hopf $*$-algebroid and $w=\bar v$.
Let $A:={A^{B}_{\mathrm{u}'}}(\nabla,F)$. By Lemma \[lemma:rn-barop\], we have intertwiners $(w^{{\overline{{\mathsf{op}}}}})^{-{\mathsf{T}}} \xrightarrow{1}
v^{{\overline{{\mathsf{op}}}}}$ and $(v^{{\overline{{\mathsf{op}}}}})^{-{\mathsf{T}}}
\xrightarrow{F^{*}=F} w^{{\overline{{\mathsf{op}}}}}$. The universal property of $A$ yields a homomorphism $j\colon A\to
\overline{A}^{{\mathsf{op}}}$ satisfying $j_{n}(v) = w^{{\overline{{\mathsf{op}}}}}$ and $j_{n}(w) = v^{{\overline{{\mathsf{op}}}}}$. Composition of $j$ with the canonical map $\bar A^{{\mathsf{op}}}\to A$ yields the desired involution, which is easily seen to be compatible with the comultiplication and counit.
Theorem \[theorem:intro-au-hopf\] now is an immediate corollary to the following result:
\[theorem:au-prime-iso\] There exists a unique $*$-isomorphism ${A^{B}_{\mathrm{u}}}(\nabla,F) \to
{A^{B}_{\mathrm{u}'}}(\nabla,F)$ whose extension to matrices sends $u$ to $v$.
One easily verifies that the universal properties of $A:={A^{B}_{\mathrm{u}}}(\nabla,F)$ and $A':={A^{B}_{\mathrm{u}'}}(\nabla,F)$ yield homomorphisms $A \to A'$ and $A' \to A$ whose extensions to matrices satisfy $u \mapsto v$ and $v\mapsto u$, $w \mapsto \bar u$, respectively.
The following analogues of Remarks \[remarks:ao\] apply to ${A^{B}_{\mathrm{u}}}(\nabla,F)$:
\[remarks:au\]
1. We may assume that $\Gamma$ is generated by the diagonal components of $\nabla$, and if $\iota \colon B_{0}\hookrightarrow
B$ denotes the inclusion of the smallest $\Gamma$-invariant $*$-subalgebra containing the entries of $F$ and $F^{-1}$, then ${A^{B}_{\mathrm{u}}}(\nabla,F) \cong
\iota_{*}{A^{B_{0}}_{\mathrm{u}}}(\nabla,F)$.
2. Let $H \in {\mathrm{GL}_{n}}(B)$ be $\nabla$-even and unitary, and let $\hat H=\nabla H \nabla^{-1}$. Then there exists an isomorphism $
{A^{B}_{\mathrm{u}}}(\nabla,HFH^{*}) \to {A^{B}_{\mathrm{u}}}(\nabla,F)$ of $(B,\Gamma)$-Hopf algebroids whose extension to matrices sends $u \in
{A^{B}_{\mathrm{u}}}(\nabla,HFH^{*})$ to $z:=r_{n}(\hat H) us_{n}(H)^{-1}\in
{A^{B}_{\mathrm{u}}}(\nabla,F)$. Indeed, there exists such a morphism of $(B,\Gamma)$-algebras because $z$ is a product of unitaries and in ${A^{B}_{\mathrm{u}}}(\nabla,F)$, we have intertwiners $u\xrightarrow{H}
z$, $\bar u^{-{\mathsf{T}}} \xrightarrow{H^{-*}} \bar
z^{-{\mathsf{T}}}$ by , and $\bar
u^{-{\mathsf{T}}}\xrightarrow{F} u$, whence $\bar z^{-{\mathsf{T}}}
\xrightarrow{H FH^{*}} z$, and this morphism is compatible with $\Delta,\epsilon,S$ because $z$ is a matrix corepresentation by Lemma \[lemma:rn-corep\]. A similar argument yields the inverse of this morphism.
We finally consider involutions on certain quotients of ${A^{B}_{\mathrm{o}}}(\nabla,F)$.
Assume that $F,G \in {\mathrm{GL}_{n}}(B)$ are $\nabla$-odd and $GF^{*}
=FG^{*}$. Let $Q := G(\nabla \bar G \nabla)$.
\[definition:matrix-hopf-involution\] The *free orthogonal dynamical quantum group* over $B$ with parameters $(\nabla,F,G)$ is the universal $(B,\Gamma)$-algebra ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ with a $v\in {\mathcal{R}_{n}}^{\times}(A)$ such that $\partial_{v} =\nabla$ and $v^{-{\mathsf{T}}} \xrightarrow{F} v$ and $v \xrightarrow{Q} v$ are intertwiners.
The algebra ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ depends only on $Q$ and not on $G$, but shall soon be equipped with an involution that does depend on $G$.
Evidently, there exists a canonical quotient map ${A^{B}_{\mathrm{o}}}(\nabla,F) \to
{A^{B}_{\mathrm{o}}}(\nabla,F,G)$, and $$\begin{aligned}
{A^{B}_{\mathrm{o}}}(\nabla,F,G) &\cong {A^{B}_{\mathrm{o}}}(\nabla,F) / (r(q)-s(q)) \quad \text{if } Q=\operatorname{diag}(q,\ldots,q),\\
{A^{B}_{\mathrm{o}}}(\nabla,F,G) &\cong {A^{B}_{\mathrm{o}}}(\nabla,F) / (r(q_{i})-s(q_{j})|
i,j=1,\ldots,n) \quad \text{if } Q=\operatorname{diag}(q_{1},\ldots,q_{n}),\end{aligned}$$ because in the second case $(r_{n}(\hat
Q)v)_{ij}=r_{n}(\gamma_{i}(q_{i}))v_{ij} =
v_{ij}r_{n}(q_{i})$ and $(vs_{n}(Q))_{ij} =
v_{ij}s_{n}(q_{j})$ in ${A^{B}_{\mathrm{o}}}(\nabla,F)$ for all $i,j$.
\[theorem:ao-prime-hopf\] The $(B,\Gamma)$-algebra ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ can be equipped with a unique structure of a $(B,\Gamma)$-Hopf $*$-algebroid such that $\bar v \xrightarrow{G} v$ becomes an intertwiner and $v$ a matrix corepresentation.
The existence of $\Delta,\epsilon,S$ follows similarly as in the case of ${A^{B}_{\mathrm{o}}}(\nabla,F)$; one only needs to observe that additionally, application of the functors $\bm{\Delta},\bm{\epsilon}$ and $(-)^{{\mathsf{co}},{\mathsf{op}}}$ to the intertwiner $(v \xrightarrow{Q} v)$ yield intertwiners $(v{\tilde\boxtimes}v \xrightarrow{Q} v{\tilde\boxtimes}v)$, $(\nabla
\xrightarrow{Q} \nabla)$ and $((v^{-1})^{{\mathsf{co}},{\mathsf{op}}}
\xrightarrow{Q} (v^{-1})^{{\mathsf{co}},{\mathsf{op}}})$
Let us prove existence of the involution. Let $w:=
r_{n}(\nabla G \nabla)^{-1} v s_{n}(G)$. Then there exist intertwiners $$\begin{gathered}
w \xrightarrow{G} v, \qquad (v \xrightarrow{G^{-1}}
w)\circ (v \xrightarrow{Q} v)
= (v \xrightarrow{\nabla \bar G\nabla} w), \\
(v \xrightarrow{\nabla \bar G\nabla} w)^{{\overline{{\mathsf{op}}}}}\circ (w
\xrightarrow{G} v)^{{\overline{{\mathsf{op}}}}} = \ (v^{{\overline{{\mathsf{op}}}}}
\xrightarrow{G} w^{{\overline{{\mathsf{op}}}}}) \circ (w^{{\overline{{\mathsf{op}}}}}
\xrightarrow{\nabla \bar G \nabla} v^{{\overline{{\mathsf{op}}}}}) =
(w^{{\overline{{\mathsf{op}}}}} \xrightarrow{Q} w^{{\overline{{\mathsf{op}}}}}), \\
(v^{-{\mathsf{T}}} \xrightarrow{F} v) \circ (w
\xrightarrow{(\nabla \bar G \nabla)^{-1}} v)^{-{\mathsf{T}}} =
(v^{-{\mathsf{T}}} \xrightarrow{F} v)\circ (w^{-{\mathsf{T}}}
\xrightarrow{G^{*}} v^{-{\mathsf{T}}}) = (w^{-{\mathsf{T}}}
\xrightarrow{FG^{*}=GF^{*}} v), \\
((v\xrightarrow{G^{-1}} w) \circ (w^{-{\mathsf{T}}}
\xrightarrow{GF^{*}} v))^{{\overline{{\mathsf{op}}}}} = (w^{-{\mathsf{T}}}
\xrightarrow{F^{*}} w)^{{\overline{{\mathsf{op}}}}} =
((w^{{\overline{{\mathsf{op}}}}})^{-{\mathsf{T}}} \xrightarrow{F} w^{{\overline{{\mathsf{op}}}}});
\end{gathered}$$ where we used Lemma \[lemma:rn-intertwiner\] in the last line. The universal property of $A:={A^{B}_{\mathrm{o}}}(\nabla,F,G)$ therefore yields a homomorphism $j\colon A\to
\overline{A}^{{\mathsf{op}}}$ such that $j_{n}(v)=w^{{\overline{{\mathsf{op}}}}}$, and this $j$ corresponds to a conjugate-linear antihomomorphism $A \to A$, $a \mapsto a^{*}$. To see that the map $a\mapsto a^{*}$ is involutive, we only need to check $\overline{w}=v$. The functor $\overline{(-)}$ of applied to $w
\xrightarrow{G} v$ yields $\overline{w}
\xrightarrow{\overline{\hat G}=\nabla \bar G\nabla}
\overline{v} = w$, and composition with $w\xrightarrow{G} v$ gives $\overline{w} \xrightarrow{Q}
v$. Hence, $\overline{w}=v$.
Finally, the involution is compatible with the comultiplication and counit because $w$ is a matrix corepresentation by Lemma \[lemma:rn-corep\].
\[remarks:ao-fg\]
1. The canonical quotient map ${A^{B}_{\mathrm{o}}}(\nabla,F) \to
{A^{B}_{\mathrm{o}}}(\nabla,F,G)$ is a morphism of $(B,\Gamma)$-Hopf algebroids.
2. Analogues of Remarks \[remarks:ao\] and \[remarks:au\] apply to ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$.
3. Note that ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ is the universal $(B,\Gamma)$-$*$-algebra with a $v\in {\mathcal{R}_{n}}^{\times}(A)$ such that $\partial_{v} =\nabla$ and $v^{-{\mathsf{T}}}
\xrightarrow{F} v$ and $\bar v \xrightarrow{G} v$ are intertwiners. Indeed, the composition of $\bar v
\xrightarrow{G} v$ with its image under the functor $\overline{(-)}$ in yields $v\xrightarrow{Q} v$.
4. If $F=G$, then $\overline{v} = v^{-{\mathsf{T}}}$ and hence $v$ is unitary. In general, assume that $H \in {\mathrm{GL}_{n}}(B)$ satisfies $\nabla H \nabla^{-1} \in {M_{n}}(B)$ and $\overline{H}H^{{\mathsf{T}}}\in {\mathbb{C}}\cdot G^{-1}F$. Then $u:=r_{n}(H^{-1})vs_{n}(\nabla^{-1}H\nabla)$ is a unitary matrix corepresentation whose entries generate ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ as a $(B,\Gamma)$-algebra. Indeed, $u
\xrightarrow{\nabla^{-1}H\nabla} v$ is an intertwiner, and applying $\overline{(-)}$ and $(-)^{-{\mathsf{T}}}$, respectively, we get $\overline{u}
\xrightarrow{\overline{H}} \overline{v} \xrightarrow{G}
v \xrightarrow{F^{-1}} v^{-{\mathsf{T}}} \xrightarrow{H^{{\mathsf{T}}}}
u^{-{\mathsf{T}}}$ which is scalar by assumption so that $\overline{u}=u^{-{\mathsf{T}}}$.
We finally consider a simple example; a more complex one is considered in §\[section:sud\].
\[example:ao-su\] Equip ${\mathbb{C}}[X]$ with an involution such that $X^{*}=X$ and an action of ${\mathbb{Z}}$ such that $X \stackrel{k}{\mapsto} X-k$ for all $k \in {\mathbb{Z}}$, and let $\gamma_{1}=1$, $\gamma_{2}=-1$, $\nabla = \operatorname{diag}(\gamma_{1},\gamma_{2})$ and $
F=G= \begin{pmatrix} 0 & -1 \\ 1 & 0
\end{pmatrix}$. Then $A^{{\mathbb{C}}[X]}_{\mathrm{o}}(\nabla,F,G) \cong
\iota_{*}(A^{{\mathbb{C}}}_{\mathrm{o}}(\nabla,F,G))$, where $\iota \colon {\mathbb{C}}\to {\mathbb{C}}[X]$ is the canonical map.
The algebra $A^{{\mathbb{C}}}_{\mathrm{o}}(\nabla,F,G)$ equipped with $\Delta,\epsilon_{{\mathbb{C}}\Gamma} \circ
\epsilon,S$ is a Hopf $*$-algebra by Lemma \[lemma:b-to-hopf\]. It is generated by the entries of a unitary matrix $v$ which satisfies $\overline{v} =
G^{-1}vG$ and therefore has the form $v=
\begin{pmatrix}
\alpha & -\gamma^{*} \\ \gamma & \alpha^{*}
\end{pmatrix}$. The relations $vv^{*}=1=v^{*}v$ then imply that $\alpha,\alpha^{*},\gamma,\gamma^{*}$ commute and $\alpha\alpha^{*}+\gamma\gamma^{*}=1$. Therefore, $A^{{\mathbb{C}}}_{\mathrm{o}}(\nabla,F,G)$ is isomorphic to the Hopf $*$-algebra $\mathcal{O}({\mathrm{SU}(2)})$ of representative functions on ${\mathrm{SU}(2)}$.
The algebra $A^{{\mathbb{C}}[X]}_{\mathrm{o}}(\nabla,F,G)\cong
\iota_{*}(A^{{\mathbb{C}}}_{\mathrm{o}}(\nabla,F,G))$ can be identified with the subalgebra of $\operatorname{End}(\mathcal{O}({\mathrm{SU}(2)}))$ generated by multiplication operators associated to elements of $\mathcal{O}({\mathrm{SU}(2)})$ and left or right invariant differentiation operators along the diagonal torus in ${\mathrm{SU}(2)}$; see Example \[example:bg-lie\].
The square of the antipode and the scaling character groups {#section:s}
-----------------------------------------------------------
The square of the antipode on the free dynamical quantum groups ${A^{B}_{\mathrm{o}}}(\nabla,F)$, ${A^{B}_{\mathrm{u}}}(\nabla,F)$, ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ can be described in terms of certain character groups as follows.
Recall the isomorphisms of Lemma \[lemma:bg-monoidal\] iii) and the anti-automorphism $S^{B\rtimes \Gamma}$ of $B\rtimes \Gamma$ given by $b\gamma \mapsto \gamma^{-1} b$.
\[definition:s-group\] Let $(A,\Delta$, $\epsilon,S)$ be a $(B,\Gamma)$-Hopf algebroid. A *character group* on $A$ is a family of morphisms $\theta=(\theta^{(k)} \colon A\to B\rtimes
\Gamma)_{k\in {\mathbb{Z}}}$ satisfying $(\theta^{(k)} {\tilde{\otimes}}\theta^{(l)})\circ \Delta = \theta^{k+l}$, $ \theta^{(0)}=
\epsilon$ and $\theta^{(k)} \circ S = S^{B\rtimes \Gamma}
\circ \theta^{(-k)}$ for all $k,l\in {\mathbb{Z}}$. We call a character group $\theta$ *scaling* if $S^{2} =
(\theta^{(1)} {\tilde{\otimes}}\operatorname{id}{\tilde{\otimes}}\theta^{(-1)}) \circ
\Delta^{(2)}$, where $\Delta^{(2)} = (\Delta {\tilde{\otimes}}\operatorname{id})
\circ \Delta = (\operatorname{id}{\tilde{\otimes}}\Delta) \circ \Delta$.
We construct scaling character groups using intertwiners of the form $u \xrightarrow{H}S^{2}_{n}(u)$ for suitable matrix corepresentations $u$.
\[lemma:s-group\] Let $(A,\Delta,\epsilon,S)$ be a $(B,\Gamma)$-Hopf algebroid, let $\theta=(\theta^{(k)} \colon A\to B\rtimes
\Gamma)_{k\in {\mathbb{Z}}}$ be a family of morphisms satisfying $(\theta^{(k)} {\tilde{\otimes}}\theta^{(l)})\circ \Delta
= \theta^{k+l}$ for all $k,l \in {\mathbb{Z}}$, and let $u
\in {\mathcal{R}_{n}}^{\times}(A)$ be a matrix corepresentation.
1. $S^{2}_{n}(u) = (u^{-{\mathsf{T}}})^{-{\mathsf{T}}}$.
2. Let $H =\partial_{u}^{-1}
\theta^{(1)}_{n}(u) $. Then $H \in
{\mathrm{GL}_{n}}(B)$, $\partial_{u}H\partial_{u}^{-1} \in {M_{n}}(B)$ and $\theta^{(k)}_{n}(u) = \partial_{u} H^{k}$ for all $k \in {\mathbb{Z}}$.
3. $\theta_{n}^{(0)}=\epsilon_{n}(u)$ and $\theta_{n}^{(k)}(S_{n}(u)) = S_{n}^{B\rtimes
\Gamma}(\theta^{(-k)}_{n}(u))$ for all $k \in
{\mathbb{Z}}$.
4. $S_{n}^{2}(u) = ((\theta^{(1)} {\tilde{\otimes}}\operatorname{id}{\tilde{\otimes}}\theta^{(-1)})\circ\Delta^{(2)})_{n}(u)$ if and only if $u\xrightarrow{H} S^{2}_{n}(u)$ is an intertwiner.
i\) The map ${M_{n}}(A) \to {M_{n}}(A)$ given by $x \mapsto
S_{n}(x)^{{\mathsf{T}}}$ is an antihomomorphism and therefore preserves inverses. Hence, $ S_{n}^{2}(u) =
S_{n}(u^{-{\mathsf{T}}})^{{\mathsf{T}}} = (S_{n}(u)^{{\mathsf{T}}})^{-{\mathsf{T}}} =
(u^{-{\mathsf{T}}})^{-{\mathsf{T}}}$.
ii\) Since each $\theta^{(k)}$ preserves the grading, there exists a family $(H_{k})_{k\in {\mathbb{Z}}}$ of elements of ${\mathrm{GL}_{n}}(B)$ satisfying $\partial_{u}H_{k}\partial_{u}^{-1} \in {M_{n}}(B)$ and $\theta^{(k)}_{n}(u) = \partial_{u}H_{k}$ for all $k \in {\mathbb{Z}}$. The assumption on $\theta$ implies that $H_{k}H_{l} = H_{k+l}$ for all $k,l\in {\mathbb{Z}}$, and consequently, $H_{k}=H_{1}^{k}$ for all $k\in {\mathbb{Z}}$.
iii\) By ii), $\theta_{n}^{(0)}(u)=\partial_{u}
=\epsilon_{n}(u)$ and $$\begin{aligned}
\theta_{n}^{(k)}(S_{n}(u)) =
\theta_{n}^{(k)}(u^{-1}) = \theta_{n}^{(k)}(u)^{-1} =
H^{-k} \partial_{u}^{-1} = S_{n}^{B\rtimes
\Gamma}(\partial_{u} H^{-k}) = S_{n}^{B\rtimes
\Gamma}(\theta_{n}^{(-k)}(u)).
\end{aligned}$$
iv\) This follows from the relation $$\begin{aligned}
((\theta^{(1)} {\tilde{\otimes}}\operatorname{id}{\tilde{\otimes}}\theta^{(-1)})\circ\Delta^{(2)})_{n}(u) &=
\theta^{(1)}_{n}(u) {\tilde\boxtimes}u {\tilde\boxtimes}\theta^{(-1)}_{n}(u) \\ &=
\partial_{u}H {\tilde\boxtimes}u {\tilde\boxtimes}\partial_{u}H^{-1} =
r_{n}(\partial_{u}H \partial_{u}^{-1}) u
s_{n}(H^{-1}). \qedhere
\end{aligned}$$
We first apply the lemma above to ${A^{B}_{\mathrm{o}}}(\nabla,F)$.
\[proposition:s-ao\] Let $F \in
{\mathrm{GL}_{n}}(B)$ be $\nabla$-odd. Then ${A^{B}_{\mathrm{o}}}(\nabla,F)$ has an intertwiner $v\xrightarrow{H} S^{2}(v)$ and a scaling character group $\theta$ such that $\theta^{(k)}_{n}(v) =
\nabla H^{k}$ for all $k\in {\mathbb{Z}}$, where $H=(\nabla F
\nabla)^{{\mathsf{T}}}F^{-1}$.
By Lemma \[lemma:s-group\] i), $(v^{-{\mathsf{T}}}
\xrightarrow{F}v) \circ (v^{-{\mathsf{T}}} \xrightarrow{F}
v)^{-{\mathsf{T}}} = S^{2}(v)\xrightarrow{H^{-1}} v$. To construct $\theta$, let $k \in {\mathbb{Z}}$ and $x=\nabla
H^{k}$. By Lemma \[lemma:rn-grading\], $x^{-{\mathsf{T}}} = (H^{-{\mathsf{T}}})^{k}
\nabla^{-1}$ and hence $$\begin{aligned}
(\nabla F \nabla) x^{-{\mathsf{T}}} = \nabla F \nabla
(\nabla^{-1} F^{-1} \nabla^{-1}F^{{\mathsf{T}}})^{k} \nabla^{-1}
= \nabla (\nabla^{-1}F^{{\mathsf{T}}}\nabla^{-1}F^{-1})^{k} F =
x F.
\end{aligned}$$ The universal property of ${A^{B}_{\mathrm{o}}}(\nabla,F)$ yields a morphism $\theta^{(k)} \colon {A^{B}_{\mathrm{o}}}(\nabla,F) \to B\rtimes
\Gamma$ such that $\theta^{(k)}_{n}(v)=x$. Using Lemma \[lemma:s-group\], one easily verifies that the family $(\theta^{(k)})_{k}$ is a scaling character group.
Assume that $B$ carries an involution which is preserved by $\Gamma$. We call a character group $(\theta^{(k)})_{k}$ on a $(B,\Gamma)$-Hopf $*$-algebroid *imaginary* if $\theta^{(k)} \circ * = * \circ
\theta^{(-k)}$ for all $k \in {\mathbb{Z}}$.
Let $F \in {\mathrm{GL}_{n}}(B)$ be $\nabla$-even. Then ${A^{B}_{\mathrm{u}}}(\nabla,F)$ has intertwiners $u \xrightarrow{F^{-1}}
S^{2}_{n}(u)$ and $\bar u \xrightarrow{(\nabla F
\nabla^{-1})^{{\mathsf{T}}}}S^{2}_{n}(\bar u)$, and an imaginary scaling character group $\theta$ such that $\theta^{(k)}_{n}(u) = \nabla F^{-k}$ and $\theta^{(k)}_{n}(\bar u) = F^{k{\mathsf{T}}}\nabla^{-1}$ for all $k\in {\mathbb{Z}}$.
By Lemma \[lemma:s-group\] i), the first intertwiner is the inverse of $S_{n}^{2}(u)=(u^{-{\mathsf{T}}})^{-{\mathsf{T}}}=\bar
u^{-{\mathsf{T}}}\xrightarrow{F} u$, and the second intertwiner is the inverse of $(\bar u^{-{\mathsf{T}}} \xrightarrow{F}
u)^{-{\mathsf{T}}}$. To construct $\theta$, let $k\in {\mathbb{Z}}$ and $x= \nabla F^{-k}$, $y=F^{k{\mathsf{T}}}\nabla^{-1}$. Using Lemma \[lemma:rn-grading\], we find $$\begin{aligned}
y&=x^{-{\mathsf{T}}}, & y^{-{\mathsf{T}}}&=x, &(\nabla F \nabla^{-1})
y^{-{\mathsf{T}}} &= (\nabla F \nabla^{-1}) x = \nabla F^{1-k} = x
F.
\end{aligned}$$ The universal property of the algebra ${A^{B}_{\mathrm{u}'}}(\nabla,F)$ and Theorem \[theorem:au-prime-iso\] yield a morphism $\theta^{(k)}\colon {A^{B}_{\mathrm{u}}}(\nabla,F) \to B\rtimes \Gamma$ such that $\theta^{(k)}_{n}(u)=x$ and $\theta_{n}^{(k)}(\bar u)=y$. Using Lemma \[lemma:s-group\], one easily verifies that the family $(\theta^{(k)})_{k}$ is a scaling character group. It is imaginary because by Lemma \[lemma:rn-grading\], $$\begin{aligned}
\overline{\theta^{(-k)}_{n}(u)} &= \overline{\nabla F^{k}}
= \overline{F}^{k} \nabla^{-1} = F^{{\mathsf{T}}k} \nabla^{-1} =
\theta^{(k)}_{n}(\bar u) \quad \text{for all } k\in
{\mathbb{Z}}. \qedhere
\end{aligned}$$
The case ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ requires some preparation. Let $F,G \in {\mathrm{GL}_{n}}(B)$ be $\nabla$-odd and $$\begin{aligned}
H&=(\nabla F
\nabla)^{{\mathsf{T}}}F^{-1}=\nabla^{-1}F^{{\mathsf{T}}}\nabla^{-1}F^{-1},
& Q&=G\nabla \bar G \nabla\end{aligned}$$ as before. We say that a diagram with arrows labeled by matrices commutes if for all possible directed paths with the same starting and ending point in the diagram, the products of the labels along the arrows coincide.
In the diagram below, (A) commutes if and only if (D) commutes, and (B) commutes if and only (C) commutes: $$\begin{aligned}
\xymatrix@C=70pt@R=25pt@l{ \bullet \ar@{<-}[r]^{\nabla
F^{-{\mathsf{T}}} \nabla} \ar@{<-}[d]_{\nabla \overline{G}
\nabla} \ar@{}[rd]|{\text{(D)}} & \bullet
\ar@{<-}[r]^{F} \ar@{<-}[d]|{\nabla^{-1} G^{{\mathsf{T}}}
\nabla^{-1}} \ar@{}[rd]|{\text{(B)}} & \bullet
\ar@{<-}[d]^{\nabla \overline{G} \nabla}
\\
\bullet \ar@{<-}[r]|{\nabla^{-1} \bar F^{-1}
\nabla^{-1}} \ar@{<-}[d]_{G} \ar@{}[rd]|{\text{(C)}}
&
\bullet \ar@{<-}[r]|{\overline{F}^{{\mathsf{T}}}} \ar@{<-}[d]|{\overline{G}^{{\mathsf{T}}}} \ar@{}[rd]|{\text{(A)}} & \bullet \ar@{<-}[d]^{G}\\
\bullet \ar@{<-}[r]_{\nabla F^{-{\mathsf{T}}} \nabla} &\bullet
\ar@{<-}[r]_{F} & \bullet }
\end{aligned}$$ If all squares commute, then $HQ=QH$, $\overline{G} \nabla
H^{-1} = \overline{H} \overline{G} \nabla$, and $Q F = F \nabla Q^{{\mathsf{T}}}\nabla^{-1}$.
Applying the transformation $X\mapsto X^{-{\mathsf{T}}}$ and reversing invertible arrows, one can obtain (D) from (A) and (C) from (B). If all small squares commute, then the three asserted relations follow from the commutativity of the large square, of the lower two squares, and of the left two squares, respectively.
Let $F,G \in {\mathrm{GL}_{n}}(B)$ be $\nabla$-odd. Assume that $FG^{*}=GF^{*}$ and $F^{*}(\nabla \bar G \nabla)^{*} =
(\nabla \bar G \nabla)F$, and let $H=
\nabla^{-1}F^{{\mathsf{T}}}\nabla^{-1} F^{-1}$. Then ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ has an intertwiner $v \xrightarrow{H}
S^{2}(v)$ and an imaginary scaling character group $(\theta^{(k)})_{k}$ such that $\theta^{(k)}_{n}(v) =
\nabla H^{k}$.
We can re-use the arguments in the proof of Proposition \[proposition:s-ao\] and only have to show additionally that $\nabla H^{k} \xrightarrow{Q} \nabla H^{k}$ is an intertwiner and that $\theta^{(-1)}_{n}(\bar v) =
\overline{\theta^{(1)}_{n}(v)}$. But by the lemma above, $(\nabla Q \nabla^{-1}) \nabla H^{k} = \nabla Q H^{k} =
\nabla H^{k} Q$ and $$\begin{aligned}
\theta^{(-1)}_{n}(\bar v) &= \theta^{(1)}_{n}(\overline{G} v
(\nabla \overline{G} \nabla)^{-1}) = \overline{G} \nabla H^{-1}
\nabla^{-1} \overline{G}^{-1} \nabla^{-1} = \overline{H}
\nabla^{-1} = \overline{\theta^{(1)}_{n}(v)}. \qedhere
\end{aligned}$$
Applying the functor to $v^{-{\mathsf{T}}}
\xrightarrow{F} v$, $\bar v \xrightarrow{G} v$, $v
\xrightarrow{\nabla \bar G \nabla} \bar v$, we obtain intertwiners $\bar v^{-{\mathsf{T}}} \xrightarrow{F^{*}}
\overline{v}$, $\bar v^{-{\mathsf{T}}} \xrightarrow{G^{*}}
v^{-{\mathsf{T}}}$, $v^{-{\mathsf{T}}} \xrightarrow{(\nabla \bar G
\nabla)^{*}} \bar v^{-{\mathsf{T}}}$, and the conditions $FG^{*}=GF^{*}$ and $F^{*}(\nabla \bar G \nabla)^{*} =
(\nabla \bar G \nabla)F$ amount to commutativity of the squares $$\begin{aligned}
\xymatrix@R=15pt@C=20pt{\bar v^{-{\mathsf{T}}} \ar[r]^{F^{*}}
\ar[d]_{G^{*}} & \bar v \ar[d]^{G} \\
v^{-{\mathsf{T}}} \ar[r]_{F} & v} \qquad\text{ and }\qquad
\xymatrix@R=15pt@C=20pt{v^{-{\mathsf{T}}}
\ar[r]^{F} \ar[d]_{(\nabla \bar G \nabla)^{*}}
& v \ar[d]^{\nabla \bar G \nabla} \\ \bar v^{-{\mathsf{T}}}
\ar[r]_{F^{*}} & \bar v.}
\end{aligned}$$ If $Q=G \nabla \bar G \nabla$ is scalar, then both conditions evidently are equivalent.
The full dynamical quantum group ${\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)}$ {#section:sud}
----------------------------------------------------------------------
In [@koelink:su2], Koelink and Rosengren studied a dynamical quantum group ${\mathcal{F}_{R}}({\mathrm{SU}(2)})$ that arises from a dynamical $R$-matrix via the generalized FRT-construction of Etingof and Varchenko. We first recall its definition, then show that this dynamical quantum group coincides with ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ for specific choice of $B,\Gamma,\nabla,F,G$, and finally construct a refinement that includes several interesting limit cases.
We shall slightly reformulate the definition of ${\mathcal{F}_{R}}({\mathrm{SL(2)}})$ and ${\mathcal{F}_{R}}({\mathrm{SU}(2)})$ given in [@koelink:su2 §2.2] so that it fits better with our approach.
Fix $q\in (0,1)$. Let ${{{\mathfrak{M}}}({\mathbb{C}})}$ be the algebra of meromorphic functions on the plane and let ${\mathbb{Z}}$ act on $B$ such that $b \stackrel{k}{\mapsto}$ $b_{(k)}:=b({\,\cdot\,}- k)$ for all $b\in B$, $k \in {\mathbb{Z}}$. Define $f \in {{{\mathfrak{M}}}({\mathbb{C}})}$ by $$\begin{aligned}
\label{eq:k-f}
f(\lambda) &=
q^{-1}\frac{q^{2(\lambda+1)}-q^{-2}}{q^{2(\lambda+1)}-1} =
\frac{q^{\lambda+2} -
q^{-(\lambda+2)}}{q^{\lambda+1}-q^{-(\lambda-1)}} \quad
\text{for all } \lambda\in {\mathbb{C}}.\end{aligned}$$ Then the $({{{\mathfrak{M}}}({\mathbb{C}})},{\mathbb{Z}})$-Hopf algebroid ${\mathcal{F}_{R}}({\mathrm{SL(2)}})$ is the universal $({{{\mathfrak{M}}}({\mathbb{C}})},{\mathbb{Z}})$-algebra with generators $\alpha,\beta,\gamma,\delta$ satisfying $$\begin{gathered}
\label{eq:k-0}
\begin{aligned}
\partial_{\alpha} &= (1,1), & \partial_{\beta} &= (1,-1), &
\partial_{\gamma} &=(-1,1), &\partial_{\delta} &= (-1,-1),
\end{aligned} \\
\label{eq:k-1}
\begin{aligned}
\alpha\beta &= s(f_{(1)})\beta \alpha, & \alpha\gamma &=
r(f)\gamma\alpha, & \beta\delta &= r(f)\delta\beta, &
\gamma\delta &= s(f_{(1)})\delta\gamma,
\end{aligned} \\
\label{eq:k-2}
\frac{r(f)}{s(f)} \delta\alpha - \frac{1}{s(f)}\beta\gamma =
\alpha\delta - r(f)\gamma\beta = \frac{r(f_{(1)})}{s(f_{(1)})}
\alpha\delta - r(f_{(1)})\beta\gamma = \delta\alpha -
\frac{1}{s(f_{(1)})} \gamma\beta = 1,\end{gathered}$$ and with comultiplication, counit and antipode given by $$\begin{gathered}
\label{eq:k-d}
\begin{aligned}
\Delta(\alpha) &= \alpha {\tilde{\otimes}}\alpha + \beta{\tilde{\otimes}}\gamma, & \Delta(\beta) &= \alpha {\tilde{\otimes}}\beta + \beta
{\tilde{\otimes}}\delta, \\
\Delta(\gamma) &= \gamma {\tilde{\otimes}}\alpha + \delta {\tilde{\otimes}}\gamma, & \Delta(\delta) &= \gamma {\tilde{\otimes}}\beta + \delta
{\tilde{\otimes}}\delta,
\end{aligned} \\ \label{eq:k-e}
\begin{aligned}
\epsilon(\alpha) &= \partial^{r}_{\alpha} =
\partial^{s}_{\alpha}, & \epsilon(\beta) =
\epsilon(\gamma) &= 0, & \epsilon(\delta) &= \partial^{r}_{\delta} =
\partial^{s}_{\delta},
\end{aligned} \\ \label{eq:k-s}
\begin{aligned}
S(\alpha) &= \frac{r(f)}{s(f)} \delta, & S(\beta) &=
-\frac{1}{s(f)}\beta, & S(\gamma) &= -r(f)\gamma, &
S(\delta) &= \alpha.
\end{aligned}\end{gathered}$$ Equip ${{{\mathfrak{M}}}({\mathbb{C}})}$ with the involution given by $b^{*}(\lambda) = \overline{b(\overline{\lambda})}$ for all $b\in {{{\mathfrak{M}}}({\mathbb{C}})}$, $\lambda \in
{\mathbb{C}}$. Then ${\mathcal{F}_{R}}({\mathrm{SL(2)}})$ can be equipped with an involution such that $$\begin{aligned}
\label{eq:k-star}
\alpha^{*} &= \delta, & \beta^{*} &= -q\gamma, & \gamma^{*}
&= -q^{-1}\beta, & \delta^{*}&= \alpha,\end{aligned}$$ and one obtains a $({{{\mathfrak{M}}}({\mathbb{C}})},{\mathbb{Z}})$-Hopf $*$-algebroid which is denoted by ${\mathcal{F}_{R}}({\mathrm{SU}(2)})$ [@koelink:su2].
\[proposition:sud\] Let $\nabla =
\operatorname{diag}(1,-1)$, $F =
\begin{pmatrix}
0 & -1 \\ f_{(1)}^{-1} & 0
\end{pmatrix}$, $G=
\begin{pmatrix}
0 & -1 \\ q^{-1} & 0
\end{pmatrix}$. Then there exist isomorphisms of $({{{\mathfrak{M}}}({\mathbb{C}})},{\mathbb{Z}})$-Hopf (\*-)algebroids $A^{{{{\mathfrak{M}}}({\mathbb{C}})}}_{\mathrm{o}}(\nabla,F) \to {\mathcal{F}_{R}}({\mathrm{SL(2)}})$ and $A^{{{{\mathfrak{M}}}({\mathbb{C}})}}_{\mathrm{o}}(\nabla,F,G) \to {\mathcal{F}_{R}}({\mathrm{SU}(2)})$ whose extensions to matrices map $v$ to $
\begin{pmatrix}
\alpha & \beta \\ \gamma & \delta
\end{pmatrix}
$.
First, note that the function $\lambda \mapsto
q^{\lambda}$ and hence also $f$ is self-adjoint, and that $$\begin{aligned}
\hat F:= \nabla F \nabla &=
\begin{pmatrix}
0 & -1 \\
f^{-1} & 0
\end{pmatrix}, & \hat G:=\nabla G \nabla &= G,
& FG^{*} &=
\begin{pmatrix}
1 & 0 \\ 0 & (qf_{(1)})^{-1}
\end{pmatrix} = GF^{*}.
\end{aligned}$$ Therefore, $A:=A_{\mathrm{o}}^{{{{\mathfrak{M}}}({\mathbb{C}})}}(\nabla,F)$ and $A_{\mathrm{o}}^{{{{\mathfrak{M}}}({\mathbb{C}})}}(\nabla,F,G)$ are well-defined. Since $\nabla G \nabla \overline{G} = G^{2}
=q^{-1} \in M_{2}({{{\mathfrak{M}}}({\mathbb{C}})})$, the latter algebra coincides with the former.
Write $v\in M_{2}(A)$ as $v=
\begin{pmatrix}
\alpha' & \beta' \\ \gamma' & \delta'
\end{pmatrix}
$ and write $'$–$'$ for the relations – with $\alpha',\beta',\gamma',\delta'$ instead of $\alpha,\beta,\gamma,\delta$. Then the relation $\partial_{v} = \nabla$ is equivalent to $'$. The relation $v^{-{\mathsf{T}}} = r_{2}(\hat
F^{-1})vs_{2}(F)$ is equivalent to $$\begin{aligned}
v^{-1}&=
\left(\begin{pmatrix}
0 & r(f) \\ -1 & 0
\end{pmatrix}
\begin{pmatrix}
\alpha' & \beta' \\ \gamma' & \delta'
\end{pmatrix}
\begin{pmatrix}
0 & -1 \\ s(f_{(1)}^{-1}) & 0
\end{pmatrix} \right)^{{\mathsf{T}}}
= \begin{pmatrix}
\frac{r(f)}{s(f)} \delta' & - \frac{1}{s(f)}\beta' \\
-r(f)\gamma' & \alpha'
\end{pmatrix},\end{aligned}$$ and multiplying out $v^{-1}v=1=vv^{-1}$ and using $'$, we find that this relation is equivalent to $'$ and $'$. Hence, there exists an isomorphism of $({{{\mathfrak{M}}}({\mathbb{C}})},{\mathbb{Z}})$-algebras $A \to {\mathcal{F}_{R}}({\mathrm{SL(2)}})$ sending $\alpha',\beta',\gamma',\delta'$ to $\alpha,\beta,\gamma,\delta$. This isomorphism is compatible with the involution, comultiplication, counit and antipode because $'$–$'$ are equivalent to $\Delta_{2}(v)=v{\tilde\boxtimes}v$, $\epsilon_{2}(v)=\partial_{v}$, $S_{2}(v)=v^{-1}$ and $$\begin{aligned}
\bar v&=r_{2}(\hat G^{-1})vs_{2}(G) =
\begin{pmatrix}
0 & q \\ -1 & 0
\end{pmatrix}
\begin{pmatrix}
\alpha' & \beta' \\ \gamma' & \delta'
\end{pmatrix}
\begin{pmatrix}
0 & -1 \\ q^{-1} & 0
\end{pmatrix}
=
\begin{pmatrix}
\delta' & -q\gamma' \\ -q^{-1}\beta' & \alpha'
\end{pmatrix}. \qedhere\end{aligned}$$
We now refine the definition above as follows. The first idea is to replace the base ${{{\mathfrak{M}}}({\mathbb{C}})}$ by the ${\mathbb{Z}}$-invariant subalgebra containing $f$ and $f^{-1}$. This subalgebra can be described in terms of the functions $x(\lambda)=q^{\lambda}$, $y(\lambda)=q^{-\lambda}$ and $z=x-y$ as follows. Since $f=z_{(-2)}/z_{(-1)}$, this subalgebra is generated by all fractions $z_{(k)}/z_{(l)}$, where $k,l \in {\mathbb{Z}}$, and since $z_{(-1)}-qz=(q^{-1}-q)q^{-\lambda}$, also by all fractions $x/z_{(k)}$ and $y/z_{(k)}$, where $k\in
{\mathbb{Z}}$. The second idea is to drop the relation $xy=1$ to allow the limit cases $\lambda \to \pm \infty$, and regard $x,y$ as canonical coordinates on ${\mathbb{C}}P^{1}$. Finally, we also regard $q$ as a variable.
Let us now turn to the details. Denote by $R \subset {\mathbb{C}}(Q)$ the localization of ${\mathbb{C}}[Q]$ with respect to $Q$ and the polynomials $$\begin{aligned}
S_{k} = (1-Q^{2k})/(1-Q^{2}) = 1 + Q^{2} + \cdots +
Q^{2(k-1)}, \quad \text{where } k \in
{\mathbb{N}}.\end{aligned}$$ Let ${\mathbb{Z}}$ act on the algebra ${\mathbb{C}}(Q,X,Y)$ of rational functions in $Q,X,Y$ by $$\begin{aligned}
Q_{(k)} &= Q, & X_{(k)} &= Q^{-k}X, & Y_{(k)}&= Q^{k}Y
&&\text{for all } k\in {\mathbb{Z}},\end{aligned}$$ where the lower index $(k)$ denotes the action of $k$. Denote by $B \subset {\mathbb{C}}(Q,X,Y)$ the subalgebra generated by $R$ and all elements $$\begin{aligned}
Z_{k,l} := (X-Y)_{(k)} / (X-Y)_{(l)}, \quad \text{where }
k,l \in {\mathbb{Z}}.\end{aligned}$$ We equip $B$ with the induced action of ${\mathbb{Z}}$ and the involution given by $Q=Q^{*}$ and $Z_{k,l}^{*} = Z_{k,l}$ for all $k,l\in {\mathbb{Z}}$. Note that this involution is the one inherited from ${\mathbb{C}}(Q,X,Y)$ when $Q=Q^{*}$ and either $X^{*}=X$, $Y^{*}=Y$ or $X^{*}=-X$, $Y^{*}=-Y$. Finally, let $$\begin{aligned}
\nabla&= (1,-1), &
F&=
\begin{pmatrix}
0 & -1 \\ Z_{0,-1} & 0
\end{pmatrix}, &
G&=
\begin{pmatrix}
0 & -Q \\ 1 & 0
\end{pmatrix}.\end{aligned}$$ Then $FG^{*}=G^{*}F$ and $G\nabla \bar G \nabla = G^{2} =
\operatorname{diag}(-Q,-Q)$.
\[definition:sud\] We let $\mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)}):={A^{B}_{\mathrm{o}}}(\nabla,F,G)$.
Thus, $\mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)})$ is generated by the entries $\alpha,\beta,\gamma,\delta$ of a $2\times 2$-matrix $v$ which satisfy the relations – with $Z_{-2,-1}$ and $Q$ instead of $f$ and $q$. This $(B,{\mathbb{Z}})$-Hopf $*$-algebroid aggregates several other interesting quantum groups and quantum groupoids which can be obtained by suitable base changes as follows.
Denote by $z\in {{{\mathfrak{M}}}({\mathbb{C}})}$ the function $\lambda \mapsto
q^{\lambda}-q^{-\lambda}$. Equip ${\mathbb{C}}(\lambda)$ with an involution such that $\lambda^{*}=\lambda$, and a ${\mathbb{Z}}$-action such that $\lambda_{(k)}=\lambda-k$. Let $\Omega=(0,1]\times [-\infty,\infty]$ and let ${\mathbb{Z}}$ act on $C(\Omega)$ by $g_{(k)}(q,t) = g(q,t-k)$ for all $g\in C(\Omega)$, $(q,t)\in \Omega$, $k\in {\mathbb{Z}}$.
There exist ${\mathbb{Z}}$-equivariant $*$-homomorphisms $$\begin{gathered}
\begin{aligned}
\mathrm{i)} & \!\! & \pi^{q}_{{{{\mathfrak{M}}}({\mathbb{C}})}} &\colon B \to {{{\mathfrak{M}}}({\mathbb{C}})},
&\!\! Q&\mapsto q, &\!\! Z_{k,l} &\mapsto
\frac{z_{(k)}}{z_{(l)}}, \qquad
\text{for } q\in (0,1) \cup (1,\infty),\\
\mathrm{ii)} & \!\! & \pi^{1}_{{{{\mathfrak{M}}}({\mathbb{C}})}} &\colon B \to
{\mathbb{C}}(\lambda), &\!\! Q &\mapsto 1, &\!\! Z_{k,l} &\mapsto
\frac{\lambda
-k}{\lambda-l}, \\
\mathrm{iii)} &\!\! & \pi_{\pm \infty} &\colon B \to R, &\!\! Q
&\mapsto Q, &\!\! Z_{k,l} &\mapsto \frac{Q^{\pm k}}{Q^{\pm
l}} = Q^{\pm k \mp l}, \\
\mathrm{iv)} & \!\! & \pi^{q}_{\pm \infty} &\colon B\to
{\mathbb{C}}, &\!\! Q &\mapsto q, &\!\! Z_{k,l} &\mapsto q^{\pm k
\mp l}, \qquad
\text{for } q \in (0,\infty), \\
\mathrm{v)} & \!\! & \pi_{\Omega} &\colon B\to C(\Omega), &\!\!
Q &\mapsto\left((q,t) \mapsto q\right), &\!\! Z_{k,l}
&\mapsto \left((q,t) \mapsto
\begin{cases}
\tfrac{q^{t-k}+q^{k-t}}{q^{t-l}+q^{l-t}}, & t\in {\mathbb{R}}, \\ q^{\pm
k \mp l}, & t =\pm \infty
\end{cases}\right).
\end{aligned}
\end{gathered}$$
i\) Restrict the homomorphism $\pi\colon
{\mathbb{C}}(Q,X,Y)\to{{{\mathfrak{M}}}({\mathbb{C}})}$ given by $Q\mapsto q$, $X\mapsto (\lambda \mapsto q^{\lambda})$, $Y \mapsto
(\lambda \mapsto q^{-\lambda})$ to $B$.
ii\) Use i) and the fact that for all $k,l\in {\mathbb{Z}}$ and $\lambda\in {\mathbb{C}}\setminus \{l\}$, $$\begin{aligned}
\lim_{q\to 1}
\pi^{q}_{{{{\mathfrak{M}}}({\mathbb{C}})}}(Z_{k,l})(\lambda)
= \lim_{q\to
1}\frac{q^{\lambda-k}-q^{k-\lambda}}{q^{\lambda-l}-q^{l-\lambda}}
= \frac{\lambda-k}{\lambda-l}.
\end{aligned}$$
iii\) Define $\pi\colon {\mathbb{C}}[Q,X,Y] \to R$ by $Q\mapsto
Q$, $X\mapsto 1$, $Y \mapsto 0$. Then $\pi$ extends to the localization $B$ of ${\mathbb{C}}[Q,X,Y]$, giving $\pi_{-\infty}$, because $\pi((X-Y)_{(k)})= Q^{-k}$ is invertible for all $k\in {\mathbb{Z}}$. This homomorphism $\pi_{-\infty}$ evidently is involutive, and ${\mathbb{Z}}$-equivariant because $\pi_{-\infty}(Z_{k+j,l+j})
= \pi_{-\infty}(Z_{k,l})$ for all $j\in {\mathbb{Z}}$. Similarly, one obtains $\pi_{+\infty}$.
iv\) Immediate from iii).
v\) Define $\pi\colon {\mathbb{C}}[Q,X,Y] \to C((0,1]\times
{\mathbb{R}})$ by $Q\mapsto ((q,t) \mapsto q)$, $X \mapsto
((q,t) \mapsto iq^{t})$, $Y \mapsto ((q,t) \mapsto
-iq^{t})$. Since $\pi((X-Y)_{(k)}) = i(q^{t-k}+q^{k-t})$ is invertible for all $t\in {\mathbb{R}}$, $k\in {\mathbb{Z}}$, this $\pi$ extends to $B$. Moreover, each $\pi(Z_{k,l})$ extends to a continuous function on $C(\Omega)$ as desired, giving $\pi_{\Omega}$.
Note that $\pi_{+\infty}^{1}=\pi_{-\infty}^{1}$. Using this map, we obtain for each algebra $C$ with an action by ${\mathbb{Z}}$ an ${\mathbb{Z}}$-equivariant homomorphism $\pi^{1}_{C}\colon B\to C$ sending $Q$ and each $Z_{k,l}$ to $1_{C}$.
\[proposition:sud-base-change\] There exist isomorphisms of Hopf $*$-algebroids as follows:
1. $(\pi^{q}_{{{{\mathfrak{M}}}({\mathbb{C}})}})_{*} \mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)}) \cong
\mathcal{F}_{R}({\mathrm{SU}(2)})$ for each $q\in (0,1) \cup
(1,\infty)$;
2. $(\pi^{q}_{-\infty})_{*} \mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)}) \cong
\mathcal{O}({\mathrm{SU}_{q}(2)})$ for each $q\in (0,\infty)$;
3. $(\pi^{q}_{\infty})_{*} \mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)}) \cong
\mathcal{O}({\mathrm{SU}_{q}(2)})^{{\mathsf{op}}}$ for each $q\in (0,\infty)$;
4. $(\pi^{1}_{{\mathbb{C}}[X]})_{*}\mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)})$ is isomorphic to the $({\mathbb{C}}[X],{\mathbb{Z}})$-Hopf $*$-algebroid in Example \[example:ao-su\].
i\) This is immediate from the definitions and Proposition \[proposition:sud\].
ii), iii) Let $\pi^{\pm}=\pi^{q}_{\pm\infty}$. Then $(\pi^{\pm})_{*}\mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)})$ is generated by the entries $\alpha',\beta',\gamma',\delta$ of a matrix $v'$ such that $\beta' = -q\gamma'{}^{*}$ and $\delta'=\alpha'{}^{*}$. Moreover, $v'{}^{-{\mathsf{T}}}
=\pi^{\pm}_{2}(F)^{-1}v'\pi^{\pm}_{2}(F)$ and $\bar v' =
\pi^{\pm}_{2}(G)^{-1}v'\pi^{\pm}_{2}(G)$, where $$\begin{aligned}
\pi^{-}_{2}(F) &=
\begin{pmatrix}
0 & -1 \\ \pi(Z_{0,-1}) & 0
\end{pmatrix}
= \begin{pmatrix}
0 & -1 \\ q^{-1} & 0
\end{pmatrix} =\pi^{\pm}_{2}(G), &
\pi^{+}_{2}(F) &=
\begin{pmatrix}
0 & -1 \\ q & 0
\end{pmatrix}.
\end{aligned}$$ In the case of $\pi_{-}$, we find that $v'$ is unitary, and obtain the usual presentation of $\mathcal{O}({\mathrm{SU}_{q}(2)})$. Multiplying out the relation $v'{}^{-{\mathsf{T}}} =\pi^{+}_{2}(F)^{-1}v'\pi^{+}_{2}(F)$, one easily verifies the assertion on $\pi^{+}$.
iv\) Immediate from the relations $(\pi^{1}_{{\mathbb{C}}[X]})_{2}(F)=(\pi^{1}_{{\mathbb{C}}[X]})_{2}(G)
=
\begin{pmatrix}
0 & -1 \\ 1 & 0
\end{pmatrix}$.
We expect most of the results of [@koelink:su2] to carry over from ${\mathcal{F}_{R}}({\mathrm{SU}(2)})$ to $\mathcal{O}({\mathrm{SU}^{\mathrm{dyn}}_{Q}(2)})$.
The level of universal $C^{*}$-algebras {#section:universal}
=======================================
Throughout this section, we shall only work with unital $C^{*}$-algebras. We assume all $*$-homomorphisms to be unital, and $B$ to be a commutative, unital $C^{*}$-algebra equipped with a left action of a discrete group $\Gamma$. Given a subset $X$ of a normed space $V$, we denote by $\overline{X} \subseteq v$ its closure and by $[X]\subseteq V$ the closed linear span of $X$.
The maximal cotensor product of $C^{*}$-algebras with respect to $C^{*}(\Gamma)$ {#section:c}
--------------------------------------------------------------------------------
This subsection reviews the cotensor product of $C^{*}$-algebras with respect to the Hopf $C^{*}$-algebra $C^{*}(\Gamma)$ and develops the main properties that will be needed in §\[section:cb\]. The material presented here is certainly well known to the experts, but we didn’t find a suitable reference.
We first recall a few preliminaries.
Let $A$ be a $*$-algebra. A *representation* of $A$ is a $*$-homomorphism into a $C^{*}$-algebra. Such a representation $\pi$ is *universal* if every other representation of $A$ factorizes uniquely through $\pi$. A universal representation exists if and only if for each $a\in A$, $$\begin{aligned}
|a|:= \sup\{ \|\pi(a)\| : \pi\text{ is a $*$-homomorphism
of $A$ into some $C^{*}$-algebra}\} < \infty.\end{aligned}$$ Indeed, if $|a|$ is finite for all $a \in A$, then the separated completion of $A$ with respect to $|\!-\!|$ carries a natural structure of a $C^{*}$-algebra, which is denoted by $C^{*}(A)$ and called the enveloping $C^{*}$-algebra of $A$, and the natural representation $A\to C^{*}(A)$ is universal.
The maximal tensor product of $C^{*}$-algebras $A$ and $C$ is the enveloping $C^{*}$-algebra of the algebraic tensor product $A\otimes C$, and will be denoted by $A{\hat{\otimes}}C$.
The full group $C^{*}$-algebra $C^{*}(\Gamma)$ of $\Gamma$ is the enveloping $C^{*}$-algebra of the group algebra ${\mathbb{C}}\Gamma$. We denote by $\Delta_{\Gamma} \colon C^{*}(\Gamma) \to
C^{*}(\Gamma) {\hat{\otimes}}C^{*}(\Gamma)$ the *comultiplication*, given by $\gamma\mapsto
\gamma\otimes \gamma$ for all $\gamma\in \Gamma$, and by $\epsilon_{\Gamma} \colon C^{*}(\Gamma) \to {\mathbb{C}}$ the *counit*, given by $\gamma \mapsto 1$ for all $\gamma\in \Gamma$. Clearly, $(\epsilon_{\Gamma}{\hat{\otimes}}\operatorname{id}) \Delta_{\Gamma} =\operatorname{id}= (\operatorname{id}{\hat{\otimes}}\epsilon_{\Gamma})\Delta_{\Gamma}$.
A completely positive (contractive) map, or brielfy c.p.(c.)-map, from a $C^{*}$-algebra $A$ to a $C^{*}$-algebra $C$ is a linear map $\phi \colon A\to C$ such that $\phi_{n}\colon {M_{n}}(A) \to {M_{n}}(C)$ is positive (and $\|\phi_{n}\|\leq 1$) for all $n \in {\mathbb{N}}$.
\[definition:c-cg\] A *$({\mathbb{C}},\Gamma)$-$C^{*}$-algebra* is a unital $C^{*}$-algebra $A$ with injective unital $*$-homomorphisms $\delta_{A}\colon A \to {C^{*}(\Gamma)}{\hat{\otimes}}A$ and $\bar \delta_{A}
\colon A \to A{\hat{\otimes}}{C^{*}(\Gamma)}$ such that $(\operatorname{id}{\hat{\otimes}}\delta_{A}) \circ \delta_{A} = (\Delta_{\Gamma}
{\hat{\otimes}}\operatorname{id}) \circ \delta_{A}$, $(\bar \delta_{A} {\hat{\otimes}}\operatorname{id}) \circ \bar \delta_{A} = (\operatorname{id}{\hat{\otimes}}\Delta_{\Gamma}) \circ \bar \delta_{A}$ and $(\delta_{A} {\hat{\otimes}}\operatorname{id}) \circ \bar \delta_{A} = (\operatorname{id}{\hat{\otimes}}\bar \delta_{A}) \circ \delta_{A}$. A *morphism* of $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras $A$ and $C$ is a unital $*$-homomorphism $\pi \colon A\to
C$ satisfying $\delta_{C} \circ \pi = (\operatorname{id}{\hat{\otimes}}\pi)
\circ \delta_{A}$ and $\bar \delta_{C} \circ \pi = (\pi
{\hat{\otimes}}\operatorname{id}) \circ \bar \delta_{A}$. We denote by ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$ the category of all $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras. Replacing $*$-homomorphisms by c.p.-maps, we define c.p.-maps of $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras and the category ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}$.
Let $A$ be a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra. Then $(\epsilon_{\Gamma} {\hat{\otimes}}\operatorname{id}) \circ \delta_{A} = \operatorname{id}_{A}$ because $$\begin{aligned}
\delta_{A} (\epsilon_{\Gamma} {\hat{\otimes}}\operatorname{id})\circ
\delta_{A} = (\epsilon_{\Gamma} {\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id}) \circ (\operatorname{id}{\hat{\otimes}}\delta_{A})\circ\delta_{A} =
((\epsilon_{\Gamma} {\hat{\otimes}}\operatorname{id}) \circ \Delta_{\Gamma}
{\hat{\otimes}}\operatorname{id}) \circ \delta_{A} = \delta_{A},\end{aligned}$$ and likewise $(\operatorname{id}{\hat{\otimes}}\epsilon_{\Gamma})\circ
\bar\delta_{A}=\operatorname{id}_{A}$.
Let $A$ and $C$ be $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras. Then the maximal tensor product $A{\hat{\otimes}}C$ is a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra with respect to $\delta_{A}
{\hat{\otimes}}\operatorname{id}$ and $\operatorname{id}{\hat{\otimes}}\bar \delta_{C}$, and the assignments $(A,C) \mapsto A{\hat{\otimes}}C$ and $(\phi,\psi)
\mapsto \phi{\hat{\otimes}}\psi$ define a product $-{\hat{\otimes}}-$ on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{({\mathrm{c.p.}})}$ that is associative in the obvious sense. Unless $\Gamma$ is trivial, this product can not be unital because it forgets $\bar\delta_{A}$ and $\delta_{C}$.
With respect to the restrictions of $\delta_{A}{\hat{\otimes}}\operatorname{id}$ and $\operatorname{id}{\hat{\otimes}}\bar \delta_{C}$, the subspace $$\begin{aligned}
A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C := \{ x \in A {\hat{\otimes}}C : (\bar
\delta_{A} {\hat{\otimes}}\operatorname{id})(x) = (\operatorname{id}{\hat{\otimes}}\delta_{C})(x) \} \subseteq A {\hat{\otimes}}C\end{aligned}$$ evidently is a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra again. Moreover, given morphisms of $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras $\phi\colon A\to C$ and $\psi \colon D \to E$, the product $\phi{\hat{\otimes}}\psi$ restricts to a morphism $\phi {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\psi \colon
A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D \to C{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}E$. We thus obtain a second product $-{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}-$ on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{({\mathrm{c.p.}})}$ that is associative in the natural sense, and unital in the following sense.
Regard ${C^{*}(\Gamma)}$ as a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra with respect to $\Delta_{\Gamma}$. Then for each $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra $A$, the maps $\delta_{A}$ and $\bar\delta_{A}$ are isomorphisms of $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras $$\begin{aligned}
\delta_{A} &\colon A \xrightarrow{\cong} {C^{*}(\Gamma)}{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A,&
\bar\delta_{A} & \colon A \xrightarrow{\cong} A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}{C^{*}(\Gamma)}.\end{aligned}$$ Indeed, they evidently are morphisms, and surjective because $$\begin{aligned}
x &=
(\epsilon_{\Gamma} {\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id})((\Delta_{\Gamma} {\hat{\otimes}}\operatorname{id})(x)) = (\epsilon_{\Gamma}
{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id}) ((\operatorname{id}{\hat{\otimes}}\delta_{A})(x)) =
\delta_{A}((\epsilon_{\Gamma} {\hat{\otimes}}\operatorname{id})(x))\end{aligned}$$ for each $x\in C^{*}(\Gamma) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A$ and likewise $y =
\bar \delta_{A}((\operatorname{id}{\hat{\otimes}}\epsilon_{\Gamma})(y))$ for each $y\in A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C^{*}(\Gamma)$.
We next construct a natural transformation $p \colon
(-{\hat{\otimes}}-) \to (-{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}-)$ which will be needed to prove associativity of the product of $(B,\Gamma)$-$C^{*}$-algebras in §\[section:cb\]. The construction is based on ideas taken from [@baaj:1 §7], and carries over from $C^{*}(\Gamma)$ to any Hopf $C^{*}$-algebra $H$ equipped with a Haar mean $H{\hat{\otimes}}H\to H$; see also [@maghfoul].
There exists a unique state $\nu$ on $C^{*}(\Gamma)
{\hat{\otimes}}C^{*}(\Gamma)$ such that $\nu(\gamma\otimes\gamma')=\delta_{\gamma,\gamma'}1$ for all $\gamma,\gamma' \in \Gamma$. Moreover, $\nu\circ
\Delta_{\Gamma} = \epsilon_{\Gamma}$ and $(\operatorname{id}{\hat{\otimes}}\nu) \circ (\Delta_{\Gamma}
{\hat{\otimes}}\operatorname{id})=(\nu{\hat{\otimes}}\operatorname{id}) \circ (\operatorname{id}{\hat{\otimes}}\Delta_{\Gamma})$.
This follows from [@maghfoul Theorem 0.1], but let us include the short direct proof. Uniqueness is clear. To construct $\nu$, denote by $(\epsilon_{\gamma})_{\gamma\in
\Gamma}$ the canonical orthonormal basis of $l^{2}(\Gamma)$, by $\lambda,\rho \colon C^{*}(\Gamma) \to
\mathcal{L}(l^{2}(\Gamma))$ the representations given by $\lambda(\gamma)\epsilon_{\gamma'}
=\epsilon_{\gamma\gamma'}$ and $\rho(\gamma)\epsilon_{\gamma'} =
\epsilon_{\gamma'\gamma^{-1}}$ for all $\gamma,\gamma'\in
\Gamma$, and by $\lambda \times \rho \colon C^{*}(\Gamma)
{\hat{\otimes}}C^{*}(\Gamma) \to \mathcal{L}(l^{2}(\Gamma))$ the representation given by $x \otimes y \mapsto
\lambda(x)\rho(y)$. Then $\nu:=\langle
\epsilon_{e}|(\lambda\times \rho)(-)\epsilon_{e}\rangle$ satisfies $\nu(\gamma\otimes
\gamma')=\delta_{\gamma,\gamma'}1$ for all $\gamma,\gamma'\in \Gamma$. The remaining equations follow easily.
1. For every $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra $A$, the maps $$\begin{aligned}
\bar p_{A}&:= (\operatorname{id}{\hat{\otimes}}\nu) ( \bar \delta_{A}
{\hat{\otimes}}\operatorname{id}) \colon A {\hat{\otimes}}C^{*}(\Gamma) \to A,
& p_{A} &:= (\nu {\hat{\otimes}}\operatorname{id}) ( \operatorname{id}{\hat{\otimes}}\delta_{A}) \colon C^{*}(\Gamma) {\hat{\otimes}}A \to A\end{aligned}$$ are morphisms in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}$ and satisfy $p_{A} \circ
\delta_{A}$ and $\bar p_{A}\circ \bar \delta_{A} = \operatorname{id}$.
2. The families $(p_{A})_{A}$ and $(\bar p_{A})_{A}$ are natural transformations from $-{\hat{\otimes}}{C^{*}(\Gamma)}$ and ${C^{*}(\Gamma)}{\hat{\otimes}}-$, respectively, to $\operatorname{id}$, regarded as functors on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}$.
i\) The map $p_{A}$ is a morphism in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}$ because $\bar \delta_{A} \circ p_{A} = (p_{A}
{\hat{\otimes}}\operatorname{id}) \circ \bar \delta_{A}$ and $$\begin{aligned}
\delta_{A} \circ p_{A} &= (\nu{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id})\circ(\operatorname{id}{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\delta_{A})(\operatorname{id}{\hat{\otimes}}\delta_{A}) \\
&=(\nu{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id}) \circ (\operatorname{id}{\hat{\otimes}}\Delta_{\Gamma} {\hat{\otimes}}\operatorname{id})\circ(\operatorname{id}{\hat{\otimes}}\delta_{A}) \\
&= (\operatorname{id}{\hat{\otimes}}\nu {\hat{\otimes}}\operatorname{id})\circ(\Delta_{\Gamma}
{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id})\circ(\operatorname{id}{\hat{\otimes}}\delta_{A})
\\
&=(\operatorname{id}{\hat{\otimes}}\nu {\hat{\otimes}}\operatorname{id})\circ(\operatorname{id}{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\delta_{A})\circ(\Delta_{\Gamma} {\hat{\otimes}}\operatorname{id}) =
(\operatorname{id}{\hat{\otimes}}p_{A})\circ(\Delta_{\Gamma} {\hat{\otimes}}\operatorname{id}).
\end{aligned}$$ Moreover, $p_{A} \circ \delta_{A} = (\nu {\hat{\otimes}}\operatorname{id})(\operatorname{id}{\hat{\otimes}}\delta_{A}) \delta_{A} = (\nu
\Delta_{\Gamma} {\hat{\otimes}}\operatorname{id}) \delta_{A} =
(\epsilon_{\Gamma} {\hat{\otimes}}\operatorname{id})\delta_{A} = \operatorname{id}$ and similarly $\bar p_{A} \circ \bar \delta_{A} = \operatorname{id}$.
ii\) This follows from the fact that $(\delta_{A})_{A}$ and $(\bar \delta_{A})_{A}$ are natural transformations.
1. Let $A,C$ be $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras. Then the map $$\begin{aligned}
p_{A,C} := (\operatorname{id}{\hat{\otimes}}\nu {\hat{\otimes}}\operatorname{id})\circ(\bar \delta_{A} {\hat{\otimes}}\delta_{C}) \colon
A{\hat{\otimes}}C\to A{\hat{\otimes}}C
\end{aligned}$$ is equal to $(\operatorname{id}{\hat{\otimes}}p_{C}) \circ (\bar \delta_{A}
{\hat{\otimes}}\operatorname{id})$ and $(\bar p_{A} {\hat{\otimes}}\operatorname{id})\circ(\operatorname{id}{\hat{\otimes}}\delta_{C})$, a morphism in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}$, and a conditional expectation onto $A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C\subseteq
A{\hat{\otimes}}C$ in the sense that $p_{A,C}(xyz) =
xp_{A,C}(y)z$ for all $x,z \in A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$ and $y \in
A{\hat{\otimes}}C$.
2. The family $(p_{A,C})_{A,C}$ is a natural transformation from $-{\hat{\otimes}}-$ to $-{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}-$, regarded as functors on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}\times
{\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}$.
i\) The equality follows immediately from the definitions and implies that $p_{A,C}$ is a morphism as claimed. Next, $p_{A,C}(A{\hat{\otimes}}C) \subseteq A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$ because $$\begin{aligned}
(\bar\delta_{A}{\hat{\otimes}}\operatorname{id})\circ p_{A,C} &= (\operatorname{id}{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\nu {\hat{\otimes}}\operatorname{id})\circ (\bar \delta_{A}
{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id})\circ(\bar \delta_{A}
{\hat{\otimes}}\delta_{C}) \\
&= (\operatorname{id}{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\nu {\hat{\otimes}}\operatorname{id})\circ (\operatorname{id}{\hat{\otimes}}\Delta_{\Gamma} {\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id})\circ(\bar \delta_{A} {\hat{\otimes}}\delta_{C}) \\
&= (\operatorname{id}{\hat{\otimes}}\nu {\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\operatorname{id})\circ (\operatorname{id}{\hat{\otimes}}\operatorname{id}{\hat{\otimes}}\Delta_{\Gamma} {\hat{\otimes}}\operatorname{id})\circ(\bar \delta_{A} {\hat{\otimes}}\delta_{C}) = (\operatorname{id}{\hat{\otimes}}\delta_{C}) \circ p_{A,C}.\end{aligned}$$ On the other hand, $ p_{A,C}(x) = (\bar p_{A}
{\hat{\otimes}}\operatorname{id}) ((\operatorname{id}{\hat{\otimes}}\delta_{C})(x)) = (\bar p_{A}
{\hat{\otimes}}\operatorname{id}) ((\delta_{A} {\hat{\otimes}}\operatorname{id})(x)) = x$ for all $x\in A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$. Thus, $p_{A,C}$ is a completely positive projection from $A{\hat{\otimes}}C$ onto $A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$ and hence a conditional expectation (see, e.g., [@brown-ozawa Proposition 1.5.7]).
ii\) Straightforward.
Denote by ${\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}\subseteq {\ensuremath
\mathbf{Alg}^{*}_{(C,\Gamma)}}$ the full subcategory formed by all $({\mathbb{C}},\Gamma)$-$*$-algebras that have an enveloping $C^{*}$-algebra. We shall need an adjoint pair of functors $$\begin{aligned}
\label{eq:c-functors}
\xymatrix@C=40pt{
{\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}\ar@<+3pt>[r]^{{C^{*}(-)}} & {\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}. \ar@<+3pt>[l]^{{(-)_{*,*}}}
}\end{aligned}$$
The functor ${C^{*}(-)}$ is defined as follows. Let $A \in
{\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}$. Using the universal property of $C^{*}(A)$, we obtain unique $*$-homomorphisms $\delta_{C^{*}(A)} \colon
C^{*}(A) \to {C^{*}(\Gamma)}{\hat{\otimes}}C^{*}(A)$ and $\bar
\delta_{C^{*}(A)} \colon C^{*}(A) \to C^{*}(A) {\hat{\otimes}}{C^{*}(\Gamma)}$ such that $\delta_{C^{*}(A)}(a) = \gamma \otimes a$ and $\bar \delta_{C^{*}(A)}(a) = a\otimes \gamma'$ for all $a\in
A_{\gamma,\gamma'},\gamma,\gamma' \in A$, and with respect to these $*$-homomorphisms, $C^{*}(A)$ becomes a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra. Moreover, every morphism $\pi\colon A\to C$ in ${\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}$ extends uniquely to a $*$-homomorphism $C^{*}(\pi) \colon C^{*}(A) \to
C^{*}(C)$ which is a morphism in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$.
The functor ${(-)_{*,*}}$ is defined as follows. Let $A$ be a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra and let $$\begin{aligned}
A_{\gamma,\gamma'}:= \{ a \in A : \delta(a) = \gamma
\otimes a, \bar\delta(a) = a\otimes \gamma'\} \subseteq A
\quad \text{for all } \gamma,\gamma' \in \Gamma.\end{aligned}$$ Then the sum $A_{*,*}:=\sum_{\gamma,\gamma'}
A_{\gamma,\gamma'} \subseteq A$ is a $({\mathbb{C}},\Gamma)$-$*$-algebra, and, every morphism $\pi\colon A \to {\mathbb{C}}$ of $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras restricts to a morphism $\pi_{*,*} \colon A_{*,*} \to C_{*,*}$ of $({\mathbb{C}},\Gamma)$-$*$-algebras. We thus obtain a functor ${(-)_{*,*}}\colon {\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}\to {\ensuremath
\mathbf{Alg}^{*}_{(C,\Gamma)}}$.
\[lemma:c-enveloping\] ${(-)_{*,*}}$ takes values in ${\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}$.
Let $A$ be a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra. Then for every $*$-representation $\pi$ of $A_{*,*}$, the restriction to the $C^{*}$-subalgebra $A_{e,e}$ is contractive and thus $\|\pi(a)\|^{2}=\|\pi(a^{*}a)\| \leq
\|a^{*}a\|=\|a\|^{2}$ for all $a\in A_{\gamma,\gamma'}$ $\gamma,\gamma'\in \Gamma$. Since such elements $a$ span $A_{*,*}$, we can conclude $|a'|<\infty$ for all $a'\in
A_{*,*}$.
For every $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra $A$, the morphisms $p_{A}$ and $\bar p_{A}$ yield a morphism $$\begin{aligned}
P_{A} &:= \bar p_{A} \circ (p_{A} {\hat{\otimes}}\operatorname{id}) = p_{A}
\circ (\operatorname{id}{\hat{\otimes}}\bar p_{A}) \colon {C^{*}(\Gamma)}{\hat{\otimes}}A
{\hat{\otimes}}{C^{*}(\Gamma)}\to A && \text{in} && {\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}^{{\mathrm{c.p.}}}.\end{aligned}$$
\[lemma:c-PA\]
1. Let $A \in {\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$. Then for all $\gamma,\gamma',\beta,\beta' \in \Gamma$, $$\begin{aligned}
P_{A}(\gamma \otimes A\otimes \gamma')
&=A_{\gamma,\gamma'}, & P_{A}(\beta \otimes
A_{\gamma,\gamma'} \otimes \beta') &=
\delta_{\beta,\gamma}\delta_{\beta',\gamma'}A_{\gamma,\gamma'},
& A&= \overline{A_{*,*}}.
\end{aligned}$$
2. Let $A \in {\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}$. Then $
C^{*}(A)_{\gamma,\gamma'} =
\overline{A_{\gamma,\gamma'}}$ for all $\gamma,\gamma'\in
\Gamma$.
We only prove i); assertion ii) follows similarly. First, $P_{A}(\gamma \otimes A \otimes \gamma') \subseteq
A_{\gamma,\gamma'}$ because $P_{A}$ is a morphism in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$ and $\Delta_{\Gamma}(\gamma'')=\gamma''\otimes
\gamma''$ for $\gamma''=\gamma,\gamma'$ . This inclusion, the relation $C^{*}(\Gamma) {\hat{\otimes}}A{\hat{\otimes}}C^{*}(\Gamma) = \overline{\sum_{\gamma,\gamma'}
\gamma\otimes A\otimes \gamma' }$ and continuity and surjectivity of $P_{A}$ imply $\overline{A_{*,*}}=A$. The equation $P_{A}(\beta \otimes A_{\gamma,\gamma'} \otimes
\beta') =
\delta_{\beta,\gamma}\delta_{\beta',\gamma'}A_{\gamma,\gamma'}$ follows from the definitions and implies that the inclusion $P_{A}(\gamma \otimes A \otimes \gamma')
\subseteq A_{\gamma,\gamma'}$ is an equality.
For every $A$ in ${\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}$ and $C$ in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$, we get canonical morphisms $\eta_{A} \colon A \to
C^{*}(A)_{*,*}$ in ${\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}$ and $\epsilon_{C} \colon
C^{*}(C_{*,*}) \to C$ in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$.
\[proposition:c-adjoints\] The functors ${C^{*}(-)}$ and ${(-)_{*,*}}$ are adjoint, where the unit and counit of the adjunction are the families $(\eta_{A})_{A}$ and $(\epsilon_{C})_{C}$, respectively. Furthermore, ${(-)_{*,*}}$ is faithful.
Let $A \in {\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}$ and $C\in {\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}$. Since the representation $A \to C^{*}(A)$ has dense image and is universal, the assignment $(C^{*}(A) \xrightarrow{\pi} C)
\mapsto (A \xrightarrow{\eta_{A}} C^{*}(A)_{*,*}
\xrightarrow{\pi_{*,*}} C_{*,*})$ yields a bijective correspondence between morphisms $C^{*}(A)\to C$ and morphisms $A \to C_{*,*}$. The functor ${(-)_{*,*}}$ is faithful because $A_{*,*} \subseteq A$ is dense.
\[remark:c-monoidal\] Similar arguments as in the proof of Lemma \[lemma:c-PA\] show that for all $A,C \in {\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$, $D,E \in {\ensuremath
\mathbf{*\text{-}Alg}^{0}_{({\mathbb{C}},\Gamma)}}$ and all $\gamma,\gamma'' \in \Gamma$, $$\begin{aligned}
(A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C)_{\gamma,\gamma''} &=
\overline{\sum_{\gamma'} A_{\gamma,\gamma'} \otimes
C_{\gamma',\gamma''}}, & (C^{*}(D) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C^{*}(E))_{\gamma,\gamma'} &= \overline{\sum_{\gamma'}
D_{\gamma,\gamma'} \otimes
E_{\gamma',\gamma''}}. \end{aligned}$$
A *short exact sequence* of $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra is a sequence of morphisms $J \xrightarrow{\iota} A \xrightarrow{\pi} C$ in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$ such that $\ker\iota=0$, $\iota(J)=\ker \pi$ and $\pi(A)=C$. A functor on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$ is *exact* if it maps short exact sequences to short exact sequences.
\[proposition:c-exact\] For every $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra $D$, the functors $-{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D$ and $D{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}-$ on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$ are exact.
If $J \xrightarrow{\iota} A \xrightarrow{\pi} C$ is a short exact sequence in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{({\mathbb{C}},\Gamma)}}$, then $J {\hat{\otimes}}D
\xrightarrow{\iota {\hat{\otimes}}\operatorname{id}} A{\hat{\otimes}}D
\xrightarrow{\pi{\hat{\otimes}}\operatorname{id}} C {\hat{\otimes}}D$ is exact (see, e.g., [@brown-ozawa Proposition 3.7]), whence $\ker (\iota {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id}) = 0$ and $$\begin{aligned}
\ker(\pi {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id}) = p_{A,D}(\ker (\pi {\hat{\otimes}}\operatorname{id})) &= p_{A,D}((\iota {\hat{\otimes}}\operatorname{id})(J{\hat{\otimes}}D)) \\ &=
(\iota {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id})(p_{J,D}(J{\hat{\otimes}}D)) = (\iota
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id})(J{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D), \\
(\pi {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id})(A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D) &= (\pi {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id})(p_{A, D}(A {\hat{\otimes}}D)) \\ &= p_{C,D} ((\pi {\hat{\otimes}}\operatorname{id})(A{\hat{\otimes}}D)) = p_{C,D}(C{\hat{\otimes}}D) = C{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D. \qedhere
\end{aligned}$$
The monoidal category of $(B,\Gamma)$-$C^{*}$-algebras {#section:cb}
------------------------------------------------------
We now define an analogue of $(B,\Gamma)$-$*$-algebras on the level of universal $C^{*}$-algebras, and construct a monoidal product which is unital and associative.
\[definition:c-bg-algebra\] A *$(B,\Gamma)$-$C^{*}$-algebra* is a $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra $A$ equipped with unital $*$-homomorphisms $r_{A}, s_{A}\colon B \to
A_{e,e}$ such that $A_{*,*}$ is a $(B,\Gamma)$-$*$-algebra with respect to the map $r_{A}\times s_{A} \colon B\otimes
B \to A_{e,e}$, $b\otimes b' \mapsto r_{A}(b)s_{A}(b')$. A *morphism* of $(B,\Gamma)$-$C^{*}$-algebras is a $B\otimes B$-linear morphism of $({\mathbb{C}},\Gamma)$-$C^{*}$-algebras. We denote by ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}$ the category of all $(B,\Gamma)$-$C^{*}$-algebras. Replacing $*$-homomorphisms by c.p.-maps, we define c.p.-maps of $(B,\Gamma)$-$C^{*}$-algebras and the category ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}^{{\mathrm{c.p.}}}$.
Denote by ${\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}\subseteq {\ensuremath
\mathbf{*\text{-}Alg}_{(B,\Gamma)}}$ the full subcategory formed by all $(B,\Gamma)$-$*$-algebras that have an enveloping $C^{*}$-algebra. This category is related to ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}$ as follows. If $C\in {\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}$, then $C_{*,*} \in
{\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}$ by Lemma \[lemma:c-enveloping\]. Conversely, if $A \in {\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}$, then $C^{*}(A)$ carries a natural structure of a $(B,\Gamma)$-$C^{*}$-algebra. The canonical maps $\eta_{A} \colon A \to C^{*}(A)_{*,*}$ and $\epsilon_{C} \colon C^{*}(C_{*,*}) \to C$ are morphisms in ${\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}$ and ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}$, respectively, and Proposition \[proposition:c-adjoints\] therefore implies:
\[corollary:c-bg-adjoints\] The assignments $A\mapsto C^{*}(A), \pi \mapsto
C^{*}(\pi)$ and $A \mapsto A_{*,*}, \pi \mapsto \pi_{*,*}$ form a pair of adjoint functors $$\begin{aligned}
\xymatrix@C=40pt{ {\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}\ar@<+3pt>[r]^{{C^{*}(-)}} &
{\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}\ar@<+3pt>[l]^{{(-)_{*,*}},} }
\end{aligned}$$ where the unit and counit of the adjunction are the families $(\eta_{A})_{A}$ and $(\epsilon_{C})_{C}$, respectively. Furthermore, ${(-)_{*,*}}$ is faithful.
Let $A$ and $C$ be $(B,\Gamma)$-$C^{*}$-algebras. Then the $({\mathbb{C}},\Gamma)$-$C^{*}$-algebra $A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$ is a $(B,\Gamma)$-$C^{*}$-algebra with respect to the $*$-homomorphisms $r \colon b\mapsto r_{A}(b) {\hat{\otimes}}1$ and $s \colon b' \mapsto 1
{\hat{\otimes}}s_{C}(b')$, and the assignments $(A,C) \mapsto
A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$ and $(\phi,\psi)\mapsto \phi{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\psi$ define a product $-{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}-$ on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}^{({\mathrm{c.p.}})}$ that is associative in the obvious sense. Using the map $$\begin{aligned}
t_{A,C} \colon B \to A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C, \quad b\mapsto
s_{A}(b){\hat{\otimes}}1 - 1{\hat{\otimes}}r_{C}(b),\end{aligned}$$ we define an ideal $(t_{A,C}(B)) \subseteq A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$. Since $t_{A,C}(B) \subseteq (A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C)_{e,e}$, the quotient $$\begin{aligned}
A {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C &:= (A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C)/(t_{A,C}(B)).\end{aligned}$$ inherits the $(B,\Gamma)$-$C^{*}$-algebra structure of $A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C$. For every pair of morphisms $\phi \colon
A\to C$ and $\psi \colon D\to E$ in ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}^{({\mathrm{c.p.}})}$, the morphism $\phi {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\psi$ maps $t_{A,D}(B)$ to $t_{C,E}(B)$ and thus factorizes to a morphism $\phi
{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}\psi \colon A{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}D \to C{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}E$. We thus obtain a product $-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-$ on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}^{({\mathrm{c.p.}})}$, and the canonical quotient map $q_{A,C} \colon A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C \to A{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C$ yields a natural transformation $q=(q_{A,C})_{A,C}$ from $-{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}-$ to $-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-$.
1. For all $(B,\Gamma)$-$C^{*}$-algebras $A,C$, we have $ [t_{A,C}(B) (A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C)] = (t_{A,C}(B)) =
[(A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C)t_{A,C}(B)]$. Indeed, a short calculation shows that for all $\gamma,\gamma',\gamma''
\in \Gamma$, $a \in A_{\gamma,\gamma'}$, $c\in
C_{\gamma',\gamma''}$, $b\in B$, $(a \otimes
c)t_{A,C}(b) = t_{A,C}(\gamma'(b))(a\otimes c)$, and now the assertion follows from Remark \[remark:c-monoidal\].
2. For every $(B,\Gamma)$-$C^{*}$-algebra $D$, the functors $-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}D$ and $D{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-$ on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}$ preserve surjections because the functors $-{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D$ and $D{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}-$ do so by Proposition \[proposition:c-exact\].
We show that the full crossed product $B\hat\rtimes
\Gamma:=C^{*}(B\rtimes \Gamma)$ is the unit for the product $-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-$. Denote by $\iota_{\Gamma} \colon
C^{*}(\Gamma) \to B\hat\rtimes \Gamma$ the natural inclusion.
1. For each $(B,\Gamma)$-$C^{*}$-algebra $A$, the $*$-homomorphisms $$\begin{aligned}
L_{A} &\colon A \xrightarrow{\delta_{A}} {C^{*}(\Gamma)}{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A \xrightarrow{\iota_{\Gamma} {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id}} (B\hat\rtimes \Gamma) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A
\xrightarrow{q_{B\hat\rtimes \Gamma,A}} (B\hat\rtimes
\Gamma) {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}A\end{aligned}$$ and $$\begin{aligned}
R_{A} &\colon A \xrightarrow{\bar \delta_{A}}
A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}{C^{*}(\Gamma)}\xrightarrow{\operatorname{id}{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\iota_{\Gamma}} A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}(B\hat\rtimes \Gamma)
\xrightarrow{q_{A,B\hat\rtimes \Gamma}} A {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}(B\hat\rtimes \Gamma),
\end{aligned}$$ are isomorphisms of $(B,\Gamma)$-$C^{*}$-algebras.
2. The families $R=(R_{A})_{A}$ and $L=(L_{A})_{A}$ form natural isomorphism from $\operatorname{id}$ to $((B\hat \rtimes
\Gamma){\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-)$ and $(-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}(B\hat \rtimes
\Gamma))$, respectively, regarded as functors on ${\ensuremath
\mathbf{C^{*}\text{-}Alg}_{(B,\Gamma)}}^{({\mathrm{c.p.}})}$.
One easily checks that each $L_{A}$ is a morphism of $(B,\Gamma)$-$C^{*}$-algebras and that $L=(L_{A})_{A}$ is a natural transformation. We show that $L_{A}$ is an isomorphism for every $(B,\Gamma)$-$C^{*}$-algebra $A$. The assertions concerning $R=(R_{A})_{A}$ then follow similarly.
To prove that $L_{A}$ is surjective, we only need to show that $(t_{B\hat\rtimes\Gamma,A}(B)) + C^{*}(\Gamma)
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A$ is dense in $(B\hat\rtimes \Gamma)
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A$. But by Remark \[remark:c-monoidal\], elements of the form $$\begin{aligned}
b\gamma \otimes a &= t_{B\hat\rtimes \Gamma,A}(b)
(\gamma \otimes a) + \gamma \otimes r_{A}(b)a, \quad
\text{where } b\in B,a\in
A_{\gamma,\gamma'},\gamma,\gamma' \in \Gamma,
\end{aligned}$$ are linearly dense in $(B\hat\rtimes \Gamma) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A$.
To prove that $L_{A}$ is injective, we only need to show that the intersection $$\begin{aligned}
J:=(\iota_{\Gamma}({C^{*}(\Gamma)}) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A) \cap
(t_{B\hat\rtimes \Gamma,A}(B)) \subseteq (B\hat\rtimes
\Gamma) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A
\end{aligned}$$ equals $0$. Since $J=\overline{J_{*,*}}$ by Lemma \[lemma:c-PA\], it suffices to show that $J_{\gamma,\gamma'} = 0$ for all $\gamma,\gamma'\in \Gamma$. Note that $J_{\gamma,\gamma'}
= [\gamma \otimes A_{\gamma,\gamma'}] \cap [(B\gamma
\otimes A_{\gamma,\gamma'})t_{B\hat\rtimes
\Gamma,A}(B)]$. For each $\gamma,\gamma' \in \Gamma$, define a linear map $R_{\gamma,\gamma'}\colon B\gamma
\otimes A_{\gamma,\gamma'} \to A_{\gamma,\gamma'}$ by $b\gamma \otimes a \mapsto r(b)a$. Then $R_{e,e}$ extends to a $*$-homomorphism on the $C^{*}$-subalgebra $B
{\hat{\otimes}}A_{e,e} \subseteq (B\hat\rtimes \Gamma)
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A$, and each $R_{\gamma,\gamma'}$ extends to a bounded linear map on $[B\gamma \otimes
A_{\gamma,\gamma'}] \subseteq (B\hat\rtimes \Gamma)
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}A$ because $$\begin{aligned}
\|R_{\gamma,\gamma'}(z)\|^{2} =
\|R_{\gamma,\gamma'}(z)R_{\gamma,\gamma'}(z)^{*}\| =
\|R_{e,e}(zz^{*})\| \leq \|zz^{*}\| = \|z\|^{2}
\end{aligned}$$ for all $z \in B\gamma \otimes A_{\gamma,\gamma'}$. Now, $R_{\gamma,\gamma'}(zt_{B\hat\rtimes\Gamma,A}(b)) = 0$ for all $z \in [B\gamma \otimes A_{\gamma,\gamma'}]$ and $b\in
B$, and $R_{\gamma,\gamma'}(\gamma \otimes a) = a$ for all $a \in A_{\gamma,\gamma'}$. Consequently, $J_{\gamma,\gamma'} =0$.
We now show that the product $-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-$ is associative. Let $A,C,D$ be $(B,\Gamma)$-$C^{*}$-algebras, denote by $a_{A,C,D} \colon (A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D
\to A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}(C{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D)$ the canonical isomorphism and let $$\begin{aligned}
\Phi_{A,C,D} := q_{\big(A{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C\big),D} \circ
(q_{A,C} {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id}) &\colon (A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C)
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D \to (A
{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C) {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}D, \\
\Psi_{A,C,D} := q_{A,\big(C{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}D\big)} \circ
(\operatorname{id}{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}q_{C,D}) &\colon A {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}(C{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D) \to A {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}(C{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}D).\end{aligned}$$
\[lemma:c-associativity\]
1. $\ker \Phi_{A,C,D}$ and $ \ker \Psi_{A,C,D}$ are generated as ideals by $t_{A,C}(B) \otimes 1_{D} + t_{\big(A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C,D\big)}(B)$ and $1_{A} \otimes
t_{C,D}(B) + t_{A,\big(C{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D\big)}(B)$, respectively.
2. There exists a unique isomorphism of $(B,\Gamma)$-$C^{*}$-algebras $\tilde a_{A,C,D} \colon
(A{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C) {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}D \to A {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}(C{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}D)$ such that $\tilde a_{A,C,D} \circ
\Phi_{A,C,D} =\Psi_{A,C,D} \circ a_{A,C,D}$.
i\) By Proposition \[proposition:c-exact\], $\ker (q_{A,C}
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id}_{D}) = (\ker q_{A,C}) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D =
(t_{A,C}(B)) {\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}D$, and $\ker q_{\big(A{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C\big),D}$ is generated as an ideal by $(q_{A,C}
{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}\operatorname{id}_{D})\big(t_{\big(A{\!\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}\!}C,D\big)}(B)\big)$. The assertion on $\Phi_{A,C,D}$ follows, and the assertion concerning $\Psi_{A,C,D}$ follows similarly.
ii\) Using i), one easily verifies that $a_{A,C,D}(\ker
\Phi_{A,C,D}) =\ker \Psi_{A,C,D}$. We thus get an isomorphism $\tilde a_{A,C,D}$ of $C^{*}$-algebras which is easily seen to be an isomorphism of $(B,\Gamma)$-$C^{*}$-algebras.
The family $(\tilde a_{A,C,D})_{A,C,D}$ is a natural isomorphism from $(-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-){\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-$ to $-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}(-{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}-)$.
By Lemma \[lemma:c-associativity\], we only need to check naturality which is straightforward.
Free dynamical quantum groups on the level of universal $C^{*}$-algebras {#section:free-c}
------------------------------------------------------------------------
Given the monoidal structure on the category of all $(B,\Gamma)$-$C^{*}$-algebras, the definitions in §\[section:bg\]–§\[section:ao\] carry over as follows:
A *compact $(B,\Gamma)$-Hopf $C^{*}$-algebroid* is a $(B,\Gamma)$-$C^{*}$-algebra $A$ with a morphism $\Delta
\colon A \to A {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}A$ satisfying
1. $(\Delta {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}\operatorname{id})\circ\Delta = (\operatorname{id}{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}\Delta)\circ\Delta$ (*coassociativity*),
2. $[\Delta(A)(1 \otimes A_{e,*})]=A {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}A =
[(A_{*,e} \otimes 1)\Delta(A)]$, where $A_{e,*} =
[\sum_{\gamma} A_{e,\gamma}] \subseteq A$ and $A_{*,e}=[\sum_{\gamma} A_{\gamma,e}] \subseteq A$ (*cancellation*).
A *counit* for a compact $(B,\Gamma)$-Hopf $C^{*}$-algebroid $(A,\Delta)$ is a morphism $\epsilon \colon
A \to B\hat\rtimes \Gamma$ of $(B,\Gamma)$-$C^{*}$-algebras satisfying $(\epsilon {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}\operatorname{id}) \circ \Delta =
\operatorname{id}_{A} = (\operatorname{id}{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}\epsilon) \circ \Delta$. A *morphism* of compact $(B,\Gamma)$-Hopf $C^{*}$-algebroids $(A,\Delta_{A})$ and $(C,\Delta_{C})$ is a morphism $\pi\colon A\to C$ satisfying $\Delta_{C}\circ
\pi = (\pi{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}\pi)\circ \Delta_{A}$. We denote the category of all compact $(B,\Gamma)$-Hopf $C^{*}$-algebroids by ${\mathbf{C^{*}}\text{-}\mathbf{Hopf}}_{(B,\Gamma)}$.
Denote by ${\mathbf{Hopf}}_{(B,\Gamma)}^{0}$ the full subcategory of ${\mathbf{Hopf}}_{(B,\Gamma)}^{*}$ formed by all $(B,\Gamma)$-Hopf $*$-algebroids $(A,\Delta,\epsilon,S)$ where $A \in
{\ensuremath
\mathbf{*\text{-}Alg}^{0}_{(B,\Gamma)}}$.
Let $(A,\Delta,\epsilon,S)
\in{\mathbf{Hopf}}_{(B,\Gamma)}^{*}$. Then $\Delta$ extends to a morphism of $(B,\Gamma)$-$C^{*}$-algebras $\Delta_{C^{*}(A)} \colon C^{*}(A) \to C^{*}(A)
{\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C^{*}(A)$ such that $(C^{*}(A),\Delta_{C^{*}(A)})$ is a compact $(B,\Gamma)$-Hopf $C^{*}$-algebroid with counit $C^{*}(\epsilon) \colon C^{*}(A) \to C^{*}(B\rtimes
\Gamma) = B\hat\rtimes \Gamma$.
The composition of $\Delta$ with the canonical map $A{\tilde{\otimes}}A \to C^{*}(A) {\!\underset{\scriptscriptstyle
B}{\stackrel{\scriptscriptstyle \Gamma}{{\hat{\otimes}}}}\!}C^{*}(A)$ extends to a morphism $\Delta_{C^{*}(A)}$ by the universal property of $C^{*}(A)$. Coassociativity of $\Delta$ and density of $A$ in $C^{*}(A)$ imply coassociativity of $\Delta_{C^{*}(A)}$, and cancellation follows from Remark \[remarks:bg-hopf\] ii).
The assignments $(A,\Delta,\epsilon,S) \mapsto
(C^{*}(A),\Delta_{C^{*}(A)})$ and $\pi \mapsto C^{*}(\pi)$ evidently form a functor ${\mathbf{Hopf}}_{(B,\Gamma)}^{0} \to
{\mathbf{C^{*}}\text{-}\mathbf{Hopf}}_{(B,\Gamma)}$.
We now apply this functor to the free unitary and free orthogonal dynamical quantum groups ${A^{B}_{\mathrm{u}}}(\nabla,F)$ and ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ introduced in Definition \[definition:intro-au-hopf\], Theorem \[theorem:intro-au-hopf\] and Definition \[definition:matrix-hopf-involution\], Theorem \[theorem:ao-prime-hopf\], respectively.
Let $\gamma_{1},\ldots,\gamma_{n} \in \Gamma$ and $\nabla =
\operatorname{diag}(\gamma_{1},\ldots,\gamma_{n}) \in {M_{n}}(B\rtimes
\Gamma)$.
Assume that $F \in {\mathrm{GL}_{n}}(B)$ be $\nabla$-even in the sense that $\nabla F\nabla^{-1} \in {M_{n}}(B)$. Then the $(B,\Gamma)$-Hopf $*$-algebroid ${A^{B}_{\mathrm{u}}}(\nabla,F)$ is generated by a copy of $B\otimes B$ and entries of a unitary matrix $v\in
{M_{n}}({A^{B}_{\mathrm{u}}}(\nabla,F))$ and therefore has an enveloping $C^{*}$-algebra. Applying the functor ${C^{*}(-)}$ and unraveling the definitions, we find:
$C^{*}({A^{B}_{\mathrm{u}}}(\nabla,F))$ is the universal $C^{*}$-algebra generated by a inclusion $r\times s$ of $B\otimes B$ and by the entries of a unitary $n\times n$-matrix $v$ subject to the relations
1. $v_{ij}r(b)=r(\gamma_{i}(b))v_{ij}$ and $v_{ij}s(b)=s(\gamma_{j}(b))v_{ij}$ for all $i,j$ and $b\in B$,
2. $v^{-{\mathsf{T}}} = \bar v$ is invertible and $r_{n}(\nabla F\nabla^{-1})\bar
v^{-{\mathsf{T}}}=vs_{n}(F)$.
It has the structure of a compact $(B,\Gamma)$-Hopf $C^{*}$-algebroid with counit, where for all $i,j$, $$\begin{aligned}
\label{eq:explicit}
\delta(v_{ij}) &= \gamma_{i} \otimes v_{ij}, &\bar
\delta(v_{ij}) &= v_{ij} \otimes \gamma_{j}, &
\Delta(v_{ij}) &= \sum_{k} v_{ik} \otimes v_{kj}, &
\epsilon(v_{ij}) &= \delta_{i,j}.
\end{aligned}$$
Let $F,G \in {\mathrm{GL}_{n}}(B)$ be $\nabla$-odd in the sense that $\nabla F \nabla, \nabla G\nabla \in {M_{n}}(B)$, and assume that $GF^{*}=FG^{*}$. If $G^{-1}F=\lambda \bar HH^{{\mathsf{T}}}$ for some $\lambda \in {\mathbb{C}}$ and some $\nabla$-even $H\in {\mathrm{GL}_{n}}(B)$, then ${A^{B}_{\mathrm{o}}}(\nabla,F,G)$ is generated by a copy of $B\otimes B$ and entries of a unitary matrix $u\in {M_{n}}({A^{B}_{\mathrm{o}}}(\nabla,F,G))$ by Remark \[remarks:ao-fg\] iii), and therefore has an enveloping $C^{*}$-algebra.
$C^{*}({A^{B}_{\mathrm{o}}}(\nabla,F,G))$ is is the universal $C^{*}$-algebra generated by a inclusion $r\times s$ of $B\otimes B$ and by the entries of an invertible $n\times
n$-matrix $v$ subject to the relations
1. $v_{ij}r(b)=r(\gamma_{i}(b))v_{ij}$ and $v_{ij}s(b)=s(\gamma_{j}(b))v_{ij}$ for all $i,j$ and $b\in B$,
2. $r_{n}(\nabla F\nabla) v^{-{\mathsf{T}}}=vs_{n}(F)$ and $r_{n}(\nabla G\nabla)\bar v = vs_{n}(G)$.
It carries the structure of a compact $(B,\Gamma)$-Hopf $C^{*}$-algebroid with counit such that holds.
*Acknowledgments.* I thank Erik Koelink for introducing me to dynamical quantum groups and for stimulating discussions, and the referee for helpful suggestions.
[10]{}
S. Baaj and G. Skandalis. -algèbres de [H]{}opf et théorie de [K]{}asparov équivariante. , 2(6):683–721, 1989.
T. Banica. Théorie des représentations du groupe quantique compact libre [${\rm O}(n)$]{}. , 322(3):241–244, 1996.
T. Banica. Le groupe quantique compact libre [${\rm U}(n)$]{}. , 190(1):143–172, 1997.
G. B[ö]{}hm. Hopf algebroids. In [*Handbook of algebra. [V]{}ol. 6*]{}, volume 6 of [*Handb. Algebr.*]{}, pages 173–235. Elsevier/North-Holland, Amsterdam, 2009.
N. P. Brown and N. Ozawa. , volume 88 of [*Graduate Studies in Mathematics*]{}. American Mathematical Society, Providence, RI, 2008.
B. Day and R. Street. Quantum categories, star autonomy, and quantum groupoids. In [*Galois theory, [H]{}opf algebras, and semiabelian categories*]{}, volume 43 of [*Fields Inst. Commun.*]{}, pages 187–225. Amer. Math. Soc., Providence, RI, 2004.
P. Etingof and F. Latour. , volume 29 of [*Oxford Lecture Series in Mathematics and its Applications*]{}. Oxford University Press, Oxford, 2005.
P. Etingof and A. Varchenko. Solutions of the quantum dynamical [Y]{}ang-[B]{}axter equation and dynamical quantum groups. , 196(3):591–640, 1998.
P. Etingof and A. Varchenko. Exchange dynamical quantum groups. , 205(1):19–52, 1999.
L. Kadison. Pseudo-[G]{}alois extensions and [H]{}opf algebroids. In [*Modules and comodules*]{}, Trends Math., pages 247–264. Birkhäuser Verlag, Basel, 2008.
E. Koelink and H. Rosengren. Harmonic analysis on the [${\rm SU}(2)$]{} dynamical quantum group. , 69(2):163–220, 2001.
S. Mac Lane. , volume 5 of [ *Graduate Texts in Mathematics*]{}. Springer-Verlag, New York, second edition, 1998.
M. Maghfoul. -algèbre de [W]{}oronowicz et moyenne de [H]{}aar. , 324(4):393–396, 1997.
D. Nikshych and L. Vainerman. Finite quantum groupoids and their applications. In [*New directions in Hopf algebras*]{}, volume 43 of [*Math. Sci. Res. Inst. Publ.*]{}, pages 211–262. Cambridge University Press, Cambridge, 2002.
T. Timmermann. Measured quantum groupoids associated to proper dynamical quantum groups. arXiv:1206.6744.
T. Timmermann. . EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008.
A. van Daele and S. Wang. Universal quantum groups. , 7(2):255–263, 1996.
S. [Wang]{}. . PhD thesis, University of California, Berkeley, 1993.
S. Wang. Structure and isomorphism classification of compact quantum groups [$A\sb u(Q)$]{} and [$B\sb u(Q)$]{}. , 48(3, suppl.):573–583, 2002.
S. L. Woronowicz. Compact matrix pseudogroups. , 111(4):613–665, 1987.
S. L. Woronowicz. Twisted [${\rm SU}(2)$]{} group. [A]{}n example of a noncommutative differential calculus. , 23(1):117–181, 1987.
[^1]: Supported by the SFB 878 “Groups, geometry and actions”
|
---
author:
- Magnus Fontes
title: 'Initial-Boundary Value Problems for Parabolic Equations.'
---
Introduction.
=============
In this paper we prove new existence and uniqueness results for weak solutions to non-homogeneous initial-boundary value problems for parabolic equations of the form
\[introeq\] $$\begin{aligned}
\frac{\partial u}{\partial t} - \nabla_x \cdot A(x,t,\nabla_x u)
&=f \quad
\mbox{in $\mathcal{D}'(Q_+)$}\\
u&=g \quad \mbox{on $ (\Omega \times \{0\}) \cup (\partial \Omega \times {\bf R}_+)$}.\end{aligned}$$
Here $\Omega$ is an open and bounded set in ${\bf R}^n$ and $Q_+ = \Omega \times {\bf R_+}$. Precise structural conditions for $A(\cdot,\cdot,\cdot)$ are given in Section 4, but the model is the following $p$-parabolic equation
\[introeqex\] $$\begin{aligned}
\frac{\partial u}{\partial t} - \nabla_x \cdot (|\nabla_x u|^{p-2}\nabla_x u)
&=f \quad
\mbox{in $\mathcal{D}'(Q_+)$}\\
u&=g \quad \mbox{on $ (\Omega \times \{0\}) \cup (\partial \Omega \times {\bf R}_+)$},\end{aligned}$$
with $1<p<\infty$.
The boundary data is prescribed on the whole parabolic boundary, $ (\Omega \times \{0\}) \cup (\partial \Omega \times {\bf R}_+)$, and we study the problem of finding the “largest possible” classes of boundary and source data such that (\[introeq\]) has a good meaning and is uniquely solvable.
In the case of the elliptic $p$-laplacian:
\[introeqexellip\] $$\begin{aligned}
- \nabla \cdot (|\nabla u|^{p-2}\nabla u)
&=f \quad
\mbox{in $\mathcal{D}'(\Omega)$}\\
u&=g \quad \mbox{on $ \partial \Omega $},\end{aligned}$$
it is well known that $W^{1,p}(\Omega)$ is a kind of golden mean. It has the useful property that:
Given $g \in W^{1,p}(\Omega)$, there exists a unique solution $u \in W^{1,p}(\Omega)$ to the $p$-laplace equation (\[introeqexellip\]) such that $u-g$ belongs to the closure of $\mathcal D (\Omega)$ in the $W^{1,p}(\Omega)$-norm topology. Furthermore the source data ($f$ in (\[introeqexellip\])) can then be taken as sums of first order derivatives of $L^{p/(p-1)}(\Omega)$-functions.
In this paper we construct an analogous optimal solution-space for equations of the type (\[introeq\]).
We point out that our results are new even in the linear case. In the linear case, where $p=2$ and we denote $W^{s,2}$ by $H^s$, it is well known (see e.g. [@LN-MG] Vol. II) that the parabolic solution and lateral boundary value spaces, replacing the “elliptic spaces” $H^{s}(\Omega)$ and $H^{s-1/2}(\partial \Omega)$, are $H^{s,s/2}(\Omega \times {\bf R}_+)$ and $H^{s-1/2,s/2-1/4}(\partial \Omega \times {\bf R}_+)$. The initial data on $\Omega \times \{0\}$ should then belong to $H^{s-1}(\Omega)$ and the natural source data space is $H^{s-2,s/2-1}(\Omega \times {\bf R}_+)$. With additional compatibility conditions for the coupling of the data in the “corners” of the space-time cylinder we then have unique solvability for the linear case when $s>1$ (see [@LN-MG], Vol. II). When $s=1$, the golden mean in the elliptic case, several difficulties arise in the parabolic case. One obvious difficulty is of course that we are in the borderline Sobolev imbedding case in the time direction (half-a-time derivative in $L^2({\bf R}_+,L^2(\Omega))$), and are thus for instance unable to define traces on $\Omega \times \{0\}$.
In Theorem \[th:mainlinear\] we give optimal results in the linear limiting case ($s=1$), and a complete description of the space of solutions (compare with the non-optimal results in e.g. [@LN-MG],[@LD-SL-UR] and [@KP]).
We use a similar construction of the solution space (with new technical complications) in the non-linear case when $p \neq 2$.
Our solution space for a general $p$, $1<p<\infty$, (see Definition \[def:xspace\]) is the sum of a Banach space carrying initial data and another Banach space carrying lateral boundary data. It is a dense subspace of the space of $L^p(Q_+)$-functions, having half order time derivatives in $L^2(Q_+)$ and first order space derivatives in $L^p(Q_+)$.
This statement requires some explanation and the appropriate distribution theory, allowing fractional differentiation in the time direction of general $L^p$-functions in a space-time half cylinder, is developed. This analytic framework makes it possible to give a precise meaning to the fractional integration by parts for the time derivatives that is one of the key tools in our method. We point out that we use two different half-a-time derivatives (adjoint to each other) and that demanding these different derivatives to belong to $L^2(Q_+)$ gives rise to different function spaces. In Section 4 we investigate the relations between these different function spaces and discuss some of their basic properties. It is for instance non-trivial to show that our function spaces are well behaved when we cut off (in a smooth way) in time. This is, apart from the fact that we are in the borderline Sobolev imbedding case in the time direction, due to the fact that they have non-homogeneous summability and regularity conditions, and that they are defined as spaces of distributions.
Most of these technical problems arise already for functions defined on the real line and half-line, and for clarity we have moved most of these arguments to an auxiliary section (Section 3) dealing with this case.
The main result of this paper is Theorem \[th:mainnonlinear\] which implies, among other things, that our solution space $X^{1,1/2}(Q_+)$ really is a true analog of the space $W^{1,p}(\Omega)$ for the elliptic $p$-laplacian, in the sense that:
Given $g \in X^{1,1/2}(Q_+)$ there exists a unique solution $u \in X^{1,1/2}(Q_+)$ to the $p$-parabolic equation (\[introeq\]) such that $u-g$ belongs to the closure of $\mathcal D (Q_+)$ in the $X^{1,1/2}(Q_+)$-norm topology. Furthermore the source data ($f$ in (\[introeq\])) can be taken as sums of first order space derivatives of $L^{p/(p-1)}(Q_+)$-functions and half-a-time derivatives of $L^2(Q_+)$-functions.
For simplicity we shall assume throughout the paper that the boundary of $\Omega $ is smooth, but this assumption is only used to prove that we can regularize functions near the lateral boundary so that the different spaces of test functions we use are dense in the corresponding function spaces (see Theorem \[th:densetestfunc\]).
Some analytical background.
===========================
We will use the fractional calculus presented in [@F]. Here we first give a brief review of the notation and some results. We then extend the calculus to space-time half-cylinders in order to be able to discuss initial-boundary value problems.
The Fourier transform on the Schwartz class ${\mathcal S}({\bf R}^n, {\bf C})$ is defined by $$\hat u (\xi) = \int_{{\bf R}^n} u(x)e^{-i2\pi x \cdot \xi }\, dx, \quad u \in
{\mathcal S}({\bf R}^n, {\bf C}).$$ The inverse will be denoted $$\check u (\xi) = \int_{{\bf R}^n} u(x)e^{i2\pi x \cdot \xi} \, dx, \quad u \in
{\mathcal S}({\bf R}^n, {\bf C}).$$ The isotropic fractional Sobolev spaces are defined as follows.
For $s \in {\bf R}$ and $1<p<\infty$ let $$H_p^s({\bf R}^n, {\bf C})= \{ u \in {\mathcal S'}({\bf R}^n, {\bf C});\;
((1+ |2\pi \xi |^2)^{s/2}
\hat u (\xi ))^{\vee} \in L^p({\bf R}^n,{\bf C}) \}.$$
They are separable and reflexive Banach spaces with the obvious norms. We will use the following multi-index notation. Let $\alpha =(\alpha_1, \dots ,\alpha_n) \in {\bf R}^n$ be an $n$-tuple. We write $\alpha > 0$ if $\alpha_j > 0,\; j=1, \dots , n$; $x^{\alpha}=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$ when $x \in {\bf R}^n$; $x^{\alpha}_+= {x_1^{\alpha_1}}_+\cdots {x_n^{\alpha_n}}_+$, (where $t_+= \max (0,t)$ for $t \in {\bf R}$, with a similar definition for $x^{\alpha}_-$) and $\Gamma (\alpha)= \Gamma(\alpha_1) \cdots \Gamma(\alpha_n)$, where $\Gamma$ denotes the gamma function. Furthermore we will sometimes write $k$ for the multi-index $(k,\dots,k)$, the interpretation should be clear from the context. We now define the classical Riemann-Liouville convolution operators.
For a multi-index $\alpha >0$, set $$D_{\pm}^{-\alpha} u = \chi_{\pm}^{\alpha -1}* u,\quad u \in {\mathcal S}({\bf R}^n,{\bf C}),$$ where the kernels $\chi_{\pm}^{\alpha -1}$, are given by $$\chi_{\pm}^{\alpha -1} = \Gamma (\alpha)^{-1} (\cdot)_{\pm}^{\alpha-1}.$$
We extend the definition of $D_{\pm}^{\alpha}$ to general multi-indices $\alpha \in {\bf R}^n$ in the usual way.
For $\alpha \in {\bf R}^n$ set $$D_{\pm}^{\alpha}u= D^k D_{\pm}^{\alpha-k}u, \quad u
\in {\mathcal S}({\bf R}^n,{\bf C}),$$ where we choose the multi-index $k \in \{0,1,2,\dots\}^n$ so that $k-\alpha >0$.
The definition is independent of the choice of $k$.
Although it is clear in this setting how the support of a function is affected under these mappings and also for instance that the operators map real valued functions to real valued functions, other features become transparent on the Fourier transform side.
Computing in ${\mathcal S'}({\bf R}^n,{\bf C})$, we have for all $\alpha \in {\bf R}^n $: $$D_{\pm}^{\alpha}u = ((0{\pm}i 2 \pi \xi )^{\alpha} \hat u (\xi))^{\vee},
\quad u \in {\mathcal S}({\bf R}^n,{\bf C}).$$
We will use the following space of test functions.
\[def:testfunctions\] Let $$\begin{gathered}
\mathcal F ({\bf R}^n,{\bf C}) \\
=\left\{ u \in C^{\infty}({\bf R}^n,{\bf C});\quad
\|u\|_{H^s_p({\bf R}^n,{\bf C})} < \infty,\;s \in {\bf R},\; 1< p<\infty
\right\}.\end{gathered}$$
$\mathcal F ({\bf R}^n,{\bf C})$ becomes a Fréchet space with the topology generated by, for instance, the following family of semi-norms $\| \cdot \|_{H^s_p({\bf R}^n,{\bf C})},$ $ s \in \{0,1,2,\dots\},$ $
p=1+2^k,$ $ k\in {\bf Z} $.
We have the following dense continuous imbeddings, $$\mathcal D ({\bf R}^n,{\bf C}) \hookrightarrow \mathcal S ({\bf R}^n,{\bf C})
\hookrightarrow \mathcal F ({\bf R}^n,{\bf C}) \hookrightarrow
\mathcal E ({\bf R}^n,{\bf C}).$$
An example of a function that belongs to $\mathcal F ({\bf R},{\bf C})$ but does not belong to $\mathcal S ({\bf R},{\bf C})$ is $x \mapsto 1/(1+x^2)$.
For $\alpha \geq 0$ we now define the fractional derivatives $$D_{\pm}^{\alpha} u = ((0 {\pm} i 2 \pi \xi )^{\alpha} \hat u )^{\vee}, \qquad u \in
\mathcal F ({\bf R}^n,{\bf C}).$$ The operators $D_+^{\alpha}$ and $D_-^{\alpha}$ are adjoint to each other and they are connected through the operator $$H^{\alpha} = \prod_{k=1}^n (\cos(\pi \alpha_k) \mbox{Id} +
\sin(\pi \alpha_k) H_k),$$ where Id is the identity operator and $H_k$ is the Hilbert transform with respect to the $k$th variable, i.e. $$H_k u(t)= \pi^{-1}\lim_{\epsilon \rightarrow +0} \int_{|s| \geq \epsilon}
\frac{u(t-se_k)}{s}\, ds, \quad u \in \mathcal F ({\bf R}^n,{\bf C}),$$ where $e_k$ is the usual canonical $k$th basis vector in ${\bf R}^n$. We have the following lemma.
\[lemma:compositionrulesfunctions\] For $\alpha \geq 0$, $D_{\pm}^{\alpha}$ are continuous linear operators on $\mathcal F ({\bf R}^n,{\bf C})$. For $\alpha \in {\bf R}^n$, $H^{\alpha}$ is an isomorphism on $\mathcal F ({\bf R}^n,{\bf C})$. For $\alpha, \beta \geq 0$ we have $$\begin{aligned}
D_{\pm}^{\alpha}D_{\pm}^{\beta} = D_{\pm}^{\alpha+\beta},\\
D_+^{\alpha}H^{\alpha} =D_-^{\alpha}.\end{aligned}$$ Furthermore all these operators commute on $\mathcal F ({\bf R}^n,{\bf C})$.
We note that for $\alpha \geq 0$ $$\label{eq:adjointnessDD}
\int_{{\bf R}^n} D_+^{\alpha} u \Phi \, dx =\int_{{\bf R}^n} u D_-^{\alpha} \Phi \, dx,
\quad u,\Phi \in \mathcal F ({\bf R}^n,{\bf C}),$$ and for $\alpha \in {\bf R}^n$ $$\label{eq:adjointnessHH}
\int_{{\bf R}^n} H^{\alpha} u \Phi \, dx =\int_{{\bf R}^n} u H^{-\alpha} \Phi \, dx,
\quad u,\Phi \in \mathcal F ({\bf R}^n,{\bf C}).$$
Now let $\mathcal F' ({\bf R}^n,{\bf C})$ denote the space of continuous linear functionals on $\mathcal F ({\bf R}^n,{\bf C})$, endowed with the weak$^*$ topology.
Inspired by (\[eq:adjointnessDD\]) and (\[eq:adjointnessHH\]), we extend the definition of $D_{\pm}^{\alpha}$ and $H^{\alpha}$ to $\mathcal F' ({\bf R}^n,{\bf C})$ by duality in the obvious way.
For $u \in \mathcal F' ({\bf R}^n,{\bf C})$ and $\alpha \geq 0$ let $$\langle D_{\pm}^{\alpha} u, \Phi \rangle
:= \langle u , D_{\mp}^{\alpha} \Phi \rangle ,
\quad \Phi \in \mathcal F ({\bf R}^n,{\bf C}),$$ and for $\alpha \in {\bf R}^n$ let $$\langle H^{\alpha} u, \Phi \rangle := \langle u, H^{-\alpha} \Phi \rangle ,
\quad \Phi \in \mathcal F ({\bf R}^n,{\bf C}).$$
The counterpart of Lemma \[lemma:compositionrulesfunctions\] is valid for $\mathcal F' ({\bf R}^n,{\bf C})$.
\[lemma:compositionrulesdist\] For $\alpha \geq 0$, $D_{\pm}^{\alpha}$ are continuous linear operators on $\mathcal F' ({\bf R}^n,{\bf C})$. For $\alpha \in {\bf R}^n$, $H^{\alpha}$ is an isomorphism on $\mathcal F' ({\bf R}^n,{\bf C})$. For $\alpha, \beta \geq 0$ we have $$\begin{aligned}
D_{\pm}^{\alpha}D_{\pm}^{\beta} = D_{\pm}^{\alpha+\beta},\\
D_+^{\alpha}H^{\alpha} =D_-^{\alpha}.\end{aligned}$$ Furthermore all these operators commute on $\mathcal F' ({\bf R}^n,{\bf C})$.
We recall that $D_{\pm}^{\alpha}$ and $H^{\alpha}$ all take real-valued functions (distributions) to real-valued functions (distributions), and from now on all functions and distributions will be real valued. We will denote the subspaces of real-valued functions and distributions simply by $\mathcal F ({\bf R}^n)$ and $\mathcal F' ({\bf R}^n)$.
In [@F] we studied parabolic operators on a space-time cylinder $Q=\Omega \times {\bf R}$, where $\Omega$ was a connected and open set in ${\bf R}^n$. We then introduced the following space of test functions.
Let $\mathcal F_{0,\cdot} (Q)$ denote the subspace of $\mathcal F ({\bf R}^n \times {\bf R})$ functions with support in $K \times {\bf R}$ for some compact subset $K \subset \Omega$.
We put a pseudo-topology on $\mathcal F_{0,\cdot} (Q)$ by specifying what sequential convergence means. We say that $\Phi_i \longrightarrow 0$ in $\mathcal F_{0,\cdot} (Q)$ if and only if the supports of all $\Phi_i$’s are contained in a fixed set $K \times {\bf R}$, where $K \subset \Omega$ is a compact subset, and $\| D^{\alpha} \Phi_i \|_{L^P(Q)} \longrightarrow 0$ as $i \longrightarrow \infty$ for all multi-indices $\alpha \in {\bf Z}_+^{n+1}$ and $1<p<\infty$.
The corresponding space of distributions is then defined as follows.
If $u$ is a linear functional on $\mathcal F_{0,\cdot} (Q)$, then $u$ is in $\mathcal{F'}_{\cdot,\cdot} (Q)$ if and only if for every compact set $K \subset \Omega$, there exist constants $C,p_1,\dots, p_N $ with $1<p_i<\infty,\quad i=1,\dots,N$ and multi-indices $\alpha_1,\dots,\alpha_N$ with $\alpha_i \in {\bf Z}_+^{n+1}, \quad
i=1,\dots,N$ such that $$|\langle u, \Phi \rangle| \leq C \sum_{i=1}^N \|D^{\alpha_i} \Phi \|_{L^{p_i}(Q)}$$ for all $\Phi \in \mathcal F_{0,\cdot} (Q)$ with support in $K \times {\bf R}$.
The motivation for these spaces is that they are invariant under fractional differentiation and Hilbert-transformation in the time variable, and ordinary differentiation in the space variables. In the given topologies, these operations are continuous.
For initial-boundary value problems, the parabolic operators will by defined on a space-time half-cylinder $Q_+=\Omega \times {\bf R}_+$, and we shall then need the following natural spaces of test functions defined on $Q_+$.
[**Remark.**]{} We shall use the same constructions on the real line and half-line, which can be thought of as the case $\Omega =\{0\}$ if we identify $\{0\} \times {\bf R}$ with ${\bf R}$ and $\{0\} \times {\bf R}_+$ with ${\bf R}_+$.
Let $\mathcal F_{0,\cdot} (Q_+)$ denote the space of those functions defined on $Q_+$ that can be extended to all of $Q$ as elements in $\mathcal F_{0,\cdot} (Q)$.
Furthermore let $\mathcal F_{0,0} (Q_+)$ denote the space of those functions defined on $Q_+$ that can be extended by zero to all of $Q$ as elements in $\mathcal F_{0,\cdot} (Q)$.
(A zero in the first position of course corresponds to zero boundary data on the lateral boundary and a zero in the second position corresponds to zero initial data.)
By using the construction in [@SE] of a (total) extension operator, we see that $\mathcal F_{0,\cdot} (Q_+)$ can be identified with the space of all smooth functions $\Phi$, defined on $Q_+$, with support in $K \times R_+$ for some compact subset $K \subset \Omega$ (i.e. they are zero on the complement, with respect to $Q_+$, of $K \times {\bf R}_+$), with $\| D^{\alpha} \Phi \|_{L^P(Q_+)} < \infty$ for all multi-indices $\alpha \in {\bf Z}_+^{n+1}$ and $1<p<\infty$.
Thus, we can put an intrinsic pseudo-topology on $\mathcal F_{0,\cdot} (Q_+)$ by defining that $\Phi_i \longrightarrow 0$ in $\mathcal F_{0,\cdot} (Q_+)$ if and only if the supports of all $\Phi_i$ are contained in a fixed set $K \times {\bf R}_+$, where $K \subset \Omega$ is a compact subset, and $\| D^{\alpha} \Phi_i \|_{L^P(Q_+)} \longrightarrow 0$ as $i \longrightarrow \infty$ for all multi-indices $\alpha \in {\bf Z}_+^{n+1}$ and $1<p<\infty$. Then $\mathcal F_{0,0} (Q_+)$ is a closed subspace of $\mathcal F_{0,\cdot} (Q_+)$ with the induced topology.
We also note that $\mathcal D (Q_+)$ is densely continuously imbedded in $\mathcal F_{0,0} (Q_+)$.
Connected with these spaces of test functions are the following spaces of distributions.
If $u$ is a linear functional on $\mathcal F_{0,\cdot} (Q_+)$, then $u$ is in $\mathcal{F'}_{\cdot,0} (Q_+)$ if and only if for every compact set $K \subset \Omega$, there exist constants $C,p_1,\dots, p_N $ with $1<p_i<\infty,\quad i=1,\dots,N$ and multi-indices $\alpha_1,\dots,\alpha_N$ with $\alpha_i \in {\bf Z}_+^{n+1}, \quad
i=1,\dots,N$ such that $$|\langle u, \Phi \rangle| \leq C \sum_{i=1}^N \|D^{\alpha_i} \Phi \|_{L^{p_i}}$$ for all $\Phi \in \mathcal F_{0,\cdot} (Q_+)$ with support in $K \times {\bf R}_+$.
Furthermore if $u$ is a linear functional on $\mathcal F_{0,0} (Q_+)$, then $u$ is in $\mathcal{F'}_{\cdot,\cdot} (Q_+)$ if and only if for every compact set $K \subset \Omega$, there exist constants $C,p_1,\dots, p_N $ with $1<p_i<\infty,\quad i=1,\dots,N$ and multi-indices $\alpha_1,\dots,\alpha_N$ with $\alpha_i \in {\bf Z}_+^{n+1}, \quad
i=1,\dots,N$ such that $$|\langle u, \Phi \rangle| \leq C \sum_{i=1}^N \|D^{\alpha_i} \Phi \|_{L^{p_i}(Q_+)}$$ for all $\Phi \in \mathcal F_{0,0} (Q_+)$ with support in $K \times {\bf R}_+$.
The importance of these spaces comes from the fact that, for a real-valued $\alpha \geq 0$, the operations $$\begin{aligned}
\frac{\partial_+^{\alpha}}{\partial t^{\alpha}}:=D_+^{(0,\dots,0,\alpha)}:
\mathcal{F}_{0,0}(Q_+) \longrightarrow
\mathcal{F}_{0,0}(Q_+)\\
\frac{\partial_-^{\alpha}}{\partial t^{\alpha}}:=D_-^{(0,\dots,0,\alpha)}:
\mathcal{F}_{0,\cdot}(Q_+) \longrightarrow
\mathcal{F}_{0,\cdot}(Q_+)\end{aligned}$$ are continuous. Ordinary differentiations with respect to the space variables are clearly also continuous operations on these spaces. We shall also use that the Hilbert-transform in the time variable $$h:= H^{(0,\dots,0,1/2)}:
\mathcal{F}_{0,0}(Q_+) \longrightarrow
\mathcal{F}_{0,\cdot}(Q_+),$$ is a continuous operator.
Extending these operators by duality in the obvious way we get that $$\begin{aligned}
\frac{\partial_+^{\alpha}}{\partial t^{\alpha}}: \mathcal{F'}_{\cdot,0}(Q_+) \longrightarrow
\mathcal{F'}_{\cdot,0}(Q_+),\\
\frac{\partial_-^{\alpha}}{\partial t^{\alpha}}: \mathcal{F'}_{\cdot,\cdot}(Q_+) \longrightarrow
\mathcal{F'}_{\cdot,\cdot}(Q_+),\\
h:
\mathcal{F'}_{\cdot,0}(Q_+) \longrightarrow
\mathcal{F'}_{\cdot,\cdot}(Q_+),\end{aligned}$$ and taking ordinary derivatives in the space variables, are continuous operations.
Using the total extension operator from [@SE], one can show that we can identify $\mathcal{F'}_{\cdot,0}(Q_+)$ with the space of $\mathcal{F'}_{\cdot,\cdot}(Q)$-distributions that are zero on $\Omega \times (-\infty,0)$.
Since $\mathcal D (Q_+)$ is densely continuously imbedded in $\mathcal{F}_{0,0}(Q_+)$, we get that $\mathcal{F'}_{\cdot,\cdot}(Q_+)$ is a continuously imbedded subspace of $\mathcal {D'}(Q_+)$.
We remark that the space $\mathcal{F'}_{\cdot,0}(Q_+)$ contains elements supported on $\Omega \times \{ 0 \}$. In fact $$\mathcal{F'}_{\cdot,\cdot}(Q_+) \simeq \mathcal{F'}_{\cdot,0}(Q_+)/
\mathcal{F^{\circ}}_{0,0}(Q_+),$$ where $\mathcal{F^{\circ}}_{0,0}(Q_+)=
\left\{ \xi \in \mathcal{F'}_{\cdot,0}(Q_+);\quad
\langle \xi,\Phi \rangle =0,\; \Phi \in \mathcal{F}_{0,0}(Q_+) \right\} $.
Finally, since $\mathcal{F}_{0,0}(Q_+)$ is densely continuously imbedded in $L^p(Q_+)$ when $1<p<\infty$, clearly $L^p(Q_+)$ is continuously imbedded in both $\mathcal{F'}_{\cdot,\cdot}(Q_+)$ and $\mathcal{F'}_{\cdot,0}(Q_+)$ when $1<p<\infty$. Thus $$\begin{aligned}
\frac{\partial_+^{\alpha}}{\partial t^{\alpha}}: L^p(Q_+) \longrightarrow
\mathcal{F'}_{\cdot,0}(Q_+)\\
\frac{\partial_-^{\alpha}}{\partial t^{\alpha}}: L^p(Q_+)\longrightarrow
\mathcal{F'}_{\cdot,\cdot}(Q_+),\end{aligned}$$ are well-defined continuous operations when $1<p<\infty$.
Auxiliary spaces on the real line and half-line.
================================================
We shall use the following auxiliary spaces defined on ${\bf R}$ and in the definition $\frac{\partial^{1/2}_- }{\partial t^{1/2}}$ should be understood in the $\mathcal F' ({\bf R})$ distribution sense.
For $1<p<\infty$, set $$B^{1/2}({\bf R})=\left\{ u \in L^p({\bf R});\;
\frac{\partial^{1/2}_- u}{\partial t^{1/2}}
\in L^2({\bf R})\right\}.$$
We equip these spaces with the following norms.
$$\|u\|_{B^{1,1/2}({\bf R})}:=
\|\frac{\partial^{1/2}_- u}{\partial t^{1/2}} \|_{L^2({\bf R})} +
\|u\|_{L^p({\bf R})}.$$
Computing in $\mathcal {F'}({\bf R})$ we see that we can represent these spaces as closed subspaces of the direct sums $L^2({\bf R})\oplus L^p({\bf R})$, and thus they are reflexive and separable Banach spaces in the topologies arising from the given norms.
If $\{\psi_{\epsilon}\}$ is a regularizing sequence it is clear that $$\|\psi_{\epsilon} \ast u \|_{B^{1/2}({\bf R})} \leq \|u\|_{B^{1/2}({\bf R})} \;,$$ and thus smooth functions are dense in $B^{1/2}({\bf R})$.
Due to the definition using distributions and to the inhomogeniety of our summability conditions, it is unfortunately not so easy to cut off in time and in this way show that $\mathcal{F}({\bf R})$ (or $\mathcal{D}({\bf R})$) is dense in $B^{1/2}({\bf R})$. Nevertheless this is true.
The space of testfunctions $\mathcal{F}({\bf R})$ is dense in $B^{1/2}({\bf R})$.
[**Proof.**]{} The proof is based on a non-linear version of the Riesz representation theorem.
We (temporarily) denote the closure of $\mathcal{F}({\bf R})$ in $B^{1/2}({\bf R})$ by $B^{1/2}_0({\bf R})$, and we shall show that $B^{1/2}_0({\bf R}) = B^{1/2}({\bf R})$.
Set $$T(u)= \frac{\partial u}{\partial t} +|u|^{p-2}u.$$
By fractional integration by parts $$\langle T(u), \Phi \rangle = \int_{{\bf R}}
\frac{\partial_+^{1/2}u}{\partial t^{1/2}}
\frac{\partial_-^{1/2}\Phi}{\partial t^{1/2}} +|u|^{p-2}u \Phi \, dt \; ; \quad
\Phi \in \mathcal F({\bf R}),$$ and Hölder’s inequality, it is clear that $$T : B^{1/2}({\bf R}) \longrightarrow B^{1/2}_0({\bf R})^*.$$ is continuous.
We notice that $$T : B^{1/2}_0({\bf R}) \longrightarrow B^{1/2}_0({\bf R})^*,$$ is weakly continuous and monotone (for definitions see \[KS\] or [@F]).
By M. Riesz’ conjugate function theorem, which says that the Hilbert transform $h$ is bounded from $L^p({\bf R})$ to $L^p({\bf R})$ (recall that $1<p<\infty$), we see that the operators $H^{\alpha}$ introduced above are isomorphisms on $B^{1/2}_0({\bf R})$.
Now for any $\alpha \in (0,1/2)$ we have $$\begin{aligned}
\langle T(u), H^{-\alpha}(u) \rangle \geq
\int_{{\bf R}}
\sin (\pi \alpha) \frac{\partial_+^{1/2}u}{\partial t^{1/2}}
\frac{\partial_+^{1/2}u}{\partial t^{1/2}}\\
+(\cos (\pi \alpha)
-\sin (\pi \alpha) C)|u|^p \, dt \quad ; u \in \mathcal F ({\bf R}),\end{aligned}$$ where $C <\infty$ is a constant such that $$\| h(u)\|_{L^p({\bf R})} \leq C \| u\|_{L^p({\bf R})}.$$ Choosing $\alpha \in (0,1/2)$ small enough we see that $H^{\alpha} \circ T$ is coercive. It follows that $T$ is a bijection (see [@F] for this functional-analytic result and similar arguments).
Thus given $u \in B^{1/2}({\bf R})$ there exists a unique $v \in B^{1/2}_0({\bf R})$ such that $T(u)=T(v)$ in $\mathcal F'({\bf R})$, i.e. $$\label{eq:modelopeq}
\frac{\partial (u-v)}{\partial t} +(|u|^{p-2}u -|v|^{p-2}v) =0.$$
This shows that the difference of elements with the same image has more regularity in time, namely $\frac{\partial (u-v)}{\partial t} \in L^{p/(p-1)}({\bf R})$.
The class of $L^p({\bf R})$ functions with derivatives in $L^{p/(p-1)}({\bf R})$ is stable under regularization and thus by a continuity argument we see that we can test with $\chi(u-v)$, where $\chi$ is a cut off function in time, in equation (\[eq:modelopeq\]). We get that (for a canonical continuous representative) $t \mapsto |u-v|(t)$ is decreasing. Since $u-v$ belongs to $L^p({\bf R})$, we conclude that $u=v$. The lemma follows. $\Box$
We are now in position to prove the following lemma.
\[lem:eqnorm\] If $u \in B^{1/2}({\bf R})$ then $$\iint_{{\bf R}\times{\bf R}} \left|\frac{u(s)-u(t)}{s-t}\right|^2 \, ds\,dt
=2 \pi \int_{{\bf R}} \left|\frac{\partial_-^{1/2} u}{\partial t^{1/2}} \right|^2 \, dt.$$
[**Proof.**]{} Since $\mathcal{F}({\bf R})$ is dense in $B^{1/2}({\bf R})$ we can compute using the Fourier transform. $$\begin{aligned}
\int_{{\bf R}} \left|\frac{\partial_-^{1/2} u}{\partial t^{1/2}} \right|^2 \, dt
=\int_{{\bf R}} 2\pi |\tau||\hat{u}|^2 \,d\tau \\
=\frac{1}{2 \pi} \iint_{{\bf R}\times {\bf R}}\frac{|1- e^{i2\pi \tau s}|^2}{s^2}
|\hat{u}(\tau)|^2 \,d\tau\,ds.\end{aligned}$$ Using Parseval’s formula the lemma follows. $\Box$
We note the following scaling and translation invariance $$\begin{aligned}
\label{eq:scaletransinv}\nonumber
\iint_{{\bf R}\times{\bf R}}
\left|\frac{u(a(s-b))-u(a(t-b))}{s-t}\right|^2 \, ds\,dt\\
=\iint_{{\bf R}\times{\bf R}} \left|\frac{u(s)-u(t)}{s-t}\right|^2
\, ds\,dt \; ; a,b \in {\bf R}.\end{aligned}$$
We also note the following fact.
The space $B^{1/2}({\bf R})$ is continuously imbedded in the space of functions with vanishing mean oscillation, $VMO({\bf R})$.
[**Proof.**]{} Let $I \subset {\bf R}$ denote a bounded interval and let $u_I$ denote the mean value of $u \in B^{1/2}({\bf R})$ over $I$. Then by Jensen’s inequality $$\label{eq:vmoh1/2}
\frac{1}{|I|} \int_I |u-u_I|^2\,dt
\leq \iint_{I \times I} \left|\frac{u(s)-u(t)}{s-t}\right|^2\,ds\,dt.$$ $\Box$
Using the form of the norm in Lemma \[lem:eqnorm\], we can now show that we have good estimates in the $B^{1/2}({\bf R})$-norm for the following cut-off operation.
\[lem:cutoff\] Let $\chi_n$ be the piecewise affine function that is one on $(-n,n)$, zero on $(-\infty,-2n) \cup (2n,\infty)$ and affine in between. Let $I_n=(-2n,2n)$ and for $u \in B^{1/2}({\bf R})$, denote the mean value of $u$ over $I_n$ by $u_{I_n}$. Then there exists a constant $C$ such that $$\begin{aligned}
\label{eq:cutoff}\nonumber
\iint_{{\bf R}\times{\bf R}} \left|\frac{\chi_n(u-u_{I_n})(s)-\chi_n(u-u_{I})(t)}{s-t}\right|^2
\, ds\,dt \\
\leq C \iint_{{\bf R}\times{\bf R}} \left|\frac{u(s)-u(t)}{s-t}\right|^2
\, ds\,dt \; ,\\
\| \chi_n(u-u_{I_n})\|^p_{L^p({\bf R})} \leq C \| u \|^p_{L^p({\bf R})}
\quad ; u \in B^{1/2}({\bf R}).\end{aligned}$$ Furthermore $\chi_n(u-u_{I_n}) \rightarrow u$ in $B^{1/2}({\bf R})$ as $n \longrightarrow \infty$.
[**Proof.**]{} The boundedness of the cut-off operation in the $L^p$-norm follows from Jensen’s inequality. For the $L^2$-part of the norm an elementary computation gives us $$\begin{aligned}
\nonumber
\iint_{{\bf R}\times{\bf R}} \left|\frac{\chi_n(u-u_{I_n})(s)-\chi_n(u-u_{I})(t)}{s-t}\right|^2
\, ds\,dt \\
\leq C \left\{ \frac{1}{|I_n|} \int_{I_n} |u-u_{I_n}|^2\,dt +
\iint_{{\bf R}\times{\bf R}} \left|\frac{u(s)-u(t)}{s-t}\right|^2
\, ds\,dt \right\},\end{aligned}$$ and thus (\[eq:cutoff\]) follows using (\[eq:vmoh1/2\]). That $\chi_n u \rightarrow u$ in $L^p({\bf R})$ is clear. If $u$ has compact support, since $p>1$, using Jensen’s inequality, we see that $\chi_n u_{I_n} \rightarrow 0$ in $L^p({\bf R})$. Since by Jensen’s inequality $\chi_n u_{I_n}$ is uniformly bounded in $L^p({\bf R})$, a density argument proves that $\chi_n(u-u_{I_n}) \rightarrow u$ in $L^p({\bf R})$. That $\chi_n(u-u_{I_n}) \rightarrow u$ for the $L^2$-part of the norm follows since by an elementary computation $$\begin{aligned}
\iint_{{\bf R}\times{\bf R}} \left|\frac{(1-\chi_n)(u-u_{I_n})(s)-(1-\chi_n)(u-u_{I})(t)}{s-t}\right|^2
\, ds\,dt \\
\leq C \left\{ \frac{1}{|I_n|} \int_{I_n} |u-u_{I_n}|^2\,dt +
\iint_{|t|>n} \left|\frac{u(s)-u(t)}{s-t}\right|^2
\, ds\,dt \right\}.\end{aligned}$$ The last term clearly tends to zero as $n$ tends to infinity. We only have to prove that also $$\frac{1}{|I_n|} \int_{I_n} |u-u_{I_n}|^2\,dt \longrightarrow 0$$ as $n \rightarrow \infty$. This is true since $$\begin{aligned}
\nonumber
\frac{1}{|I_n|} \int_{I_n} |u-u_{I_n}|^2\,dt
\leq \frac{1}{4n^2}\iint_{I_n \times I_n} |u(s)-u(t)|^2\,ds\,dt\\\nonumber
\leq C \left\{ \frac{\log^2 n}{n^2} \iint_{|s|,|t| \leq \log n}
\left|\frac{u(s)-u(t)}{s-t}\right|^2 \, ds \, dt \right.\\
\left.+ \iint_{|t| \geq \log n}
\left|\frac{u(s)-u(t)}{s-t}\right|^2 \, ds \, dt \right\},\end{aligned}$$ which clearly tends to zero as $n$ tends to infinity. $\Box$
[**Remark.**]{} We subtracted the mean value in the argument above in order not to have to rely on the fact that $u \in L^p({\bf R})$ when proving boundedness for the half-derivatives. This is crucial when we later use the same argument on functions defined in a space-time cylinder. In preparation for this we also note that, by regularizing, the lemma gives us an explicit sequence of $\mathcal{D} ({\bf R})$-functions tending to a given element in $B^{1/2}({\bf R})$.
We now introduce two sets of spaces defined on the real half-line.
Let $ B^{1/2}_0({\bf R}_+)$ be the space of functions defined on ${\bf R}_+$ that can be extended by zero as elements in $B^{1/2}({\bf R})$.
Furthermore let $ B^{1/2}({\bf R}_+)$ be the space of functions defined on ${\bf R}_+$ that can be extended as elements in $B^{1/2}({\bf R})$.
[**Remark.**]{} The space $ B^{1/2}_0({\bf R}_+)$ can of course be identified with the closed subspace of $ B^{1/2}({\bf R})$ of functions with support in ${\bf R}_+$.
We now give two simple lemmas, giving intrinsic descriptions of $
B^{1/2}_0({\bf R}_+)$ and $ B^{1/2}({\bf R}_+)$. We omit the proofs, which are straightforward elementary computations using the form of the norm in Lemma \[lem:eqnorm\].
\[lem:bhalf0norm\] The function space $B^{1/2}_0({\bf R}_+)$ is precisely the set of $L^p({\bf R}_+)$-functions such that the following norm is bounded: $$\begin{aligned}
\nonumber\label{eq:bhalf0norm}
\| u\|_{B^{1/2}_0({\bf R}_+)}:=\|u\|_{L^p({\bf R}_+)} +
\left\{ \int_{{\bf R}_+} \frac{u^2(t)}{t} \,dt \right.\\
\left. +
\iint_{{\bf R}_+ \times {\bf R}_+}
\left(\frac{u(s)-u(t)}{s-t}\right)^2 \, ds\,dt \right\}^{1/2}.\end{aligned}$$
The function space $B^{1/2}({\bf R}_+)$ is precisely the set of $L^p({\bf R}_+)$-functions such that the following norm is bounded: $$\label{eq:bhalfnorm}
\| u\|_{B^{1/2}({\bf R}_+)}:=\|u\|_{L^p({\bf R}_+)} +\left\{
\iint_{{\bf R}_+ \times {\bf R}_+}
\left(\frac{u(s)-u(t)}{s-t}\right)^2\, ds\,dt \right\}^{1/2}.$$ Furthermore, a continuous symmetric extension operator from $B^{1/2}({\bf R}_+)$ to $B^{1/2}({\bf R})$ is given by $E_S(u)(t)=u(|t|)$.
We have the following density results:
\[lem:densereal\] The space $\mathcal F ({\bf R}_+)$ is dense in $B^{1/2}({\bf R}_+)$ and $\mathcal F_0 ({\bf R}_+)$ is dense in $B^{1/2}_0({\bf R}_+)$.
[ **Proof.**]{} That $\mathcal F ({\bf R}_+)$ is dense in $B^{1/2}({\bf R}_+)$ follows immidiately from the fact that $\mathcal F ({\bf R})$ is dense in $B^{1/2}({\bf R})$. The argument to prove that $\mathcal F_0 ({\bf R}_+)$ is dense in $B^{1/2}_0({\bf R}_+)$ is a little more delicate. Given $u \in B^{1/2}_0({\bf R}_+)$, apriori we only know that there exists a sequence of testfunctions in $\mathcal F ({\bf R})$ approaching $u$ in the $B^{1/2}({\bf R})$-norm.
Given $u \in B^{1/2}_0({\bf R}_+)$ we will show that we can cut-off. Let $\chi_n$ be the piecewise affine function that is one on $(0,n)$, zero on $(2n,\infty)$ and affine in between. We will show that $\chi_n u \longrightarrow u$ in $B^{1/2}_0({\bf R}_+)$. Taking this for granted we can regularize with a regularizing sequence having support in ${\bf R}_+$ which gives us the lemma.
That $\chi_n u \longrightarrow u$ in $L^p({\bf R}_+)$ is clear. We now estimate the $L^2$-part of the norm. An elementary computation gives us $$\begin{aligned}
\nonumber
\int_{{\bf R}_+} \frac{((1-\chi_n)u)^2(t)}{t} \,dt
+ \iint_{{\bf R}_+ \times {\bf R}_+}
\frac{((1-\chi_n)u(s)-(1-\chi_n)u(t))^2}{(s-t)^2} \, ds\,dt \\ \nonumber
\leq C \left\{ \frac{1}{2n} \int_0^{2n} u^2(t) \,dt +
\iint_{(n,\infty) \times {\bf R}_+}
\left(\frac{u(s)-u(t)}{s-t}\right)^2\, ds\,dt \right.\\
\left. +\int_n^{\infty} \frac{u^2(t)}{t} \,dt \right\}.\quad\end{aligned}$$ The last two terms above clearly tend to zero as $n \rightarrow \infty$. To estimate the first term, we integrate by parts (we may assume that $u$ is smooth, it is the decay at infinity that is the issue). $$\begin{aligned}
\nonumber
\frac{1}{2n} \int_0^{2n} u^2(t) \,dt
= \frac{1}{2n} \int_0^{2n} \left( \int_0^{2n} \frac{u^2(s)}{s}\, ds -
\int_0^{t} \frac{u^2(s)}{s}\, ds \right) \, dt\\
\leq \frac{1}{2n} \int_{\log n}^{2n} \int_t^{2n}
\frac{u^2(s)}{s} \,ds \, dt
+ \frac{\log n}{2n} \int_0^{2n} \frac{u^2(s)}{s} \, ds,\end{aligned}$$ which clearly tends to zero as $n$ tends to infinity. The lemma follows. $\Box$
We now give the following equivalent characterization of $B^{1/2}_0({\bf R}_+)$.
A function $u \in L^p({\bf R}_+)$ belongs to $B^{1/2}_0({\bf R}_+)$ if and only if the $\mathcal F'_0 ({\bf R}_+)$-distribution derivative $\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}
\in L^2({\bf R}_+)$. Furthermore an equivalent norm on $B^{1/2}_0({\bf R}_+)$ is given by $$\|u\| = \| u\|_{L^p({\bf R}_+)} +
\|\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}\|_{L^2({\bf R}_+)}.$$
[**Remark.**]{} We recall that the $\mathcal F'_0 ({\bf R}_+)$-distribution derivative, apart from what happens inside ${\bf R}_+$, also controls what happens on the boundary $\{0\}$. The fact that $\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}
\in L^2({\bf R}_+)$ thus actually contains a lot of information about $u$’s behaviour at $0$.
[**Proof.**]{}
It is clear that a function in $B^{1/2}_0({\bf R}_+)$ has the $\mathcal F'_0 ({\bf R}_+)$-distribution derivative $\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}$ in $L^2({\bf R}_+)$.
On the other hand, let $E_0$ be the extension by zero operator. Then if $u \in L^p({\bf
R}_+)$ and the $\mathcal F'_0 ({\bf R}_+)$-distribution derivative $\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}
\in L^2({\bf R}_+)$ we have $$\int_{\bf R} E_0(u) \frac{\partial^{1/2}_-\Phi}{\partial t^{1/2}}\,dt
=\int_{\bf R} E_0(\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}) \Phi \,dt\;
;\; \Phi \in \mathcal F ({\bf R}).$$ This shows that the $\mathcal F' ({\bf R})$-distribution derivative $\frac{\partial^{1/2}_+ E_0(u)}{\partial t^{1/2}} $ belongs to $L^2({\bf R})$. An easy computation shows that $$\begin{aligned}
\nonumber
\int_{{\bf R}_+}
\left| \frac{\partial_+^{1/2} u}{\partial t^{1/2}} \right|^2 \, dt =
\int_{{\bf R}}
\left| \frac{\partial_+^{1/2} E_0(u)(t)}{\partial t^{1/2}} \right|^2 \,
dt \\
\sim
\iint_{ {\bf R}_+ \times {\bf R}_+ }
\left| \frac{E_0(u)(s)-E_0(u)(t)}{s-t}\right|^2 \, ds\,dt
+\int_{\bf R_+} \frac{E_0(u)^2}{t}\, dt.
\end{aligned}$$ Since $E_0(u)=u$ on $(0,\infty)$ the lemma follows. $\Box$
We now give a corresponding equivalent norm on $B^{1/2}({\bf R}_+)$.
If $u \in B^{1/2}({\bf R}_+)$, then the $\mathcal F' ({\bf R}_+)$-distribution derivative $\frac{\partial^{1/2}_- u}{\partial t^{1/2}}$ belongs to $L^2({\bf R}_+)$.
Furthermore an equivalent norm on $B^{1/2}({\bf R}_+)$ is given by $$\|u\| = \| u\|_{L^p({\bf R})} +
\|\frac{\partial^{1/2}_- u}{\partial t^{1/2}}\|_{L^2({\bf R}_+)}.$$
[**Remark.**]{} In contrast to the $\mathcal F'_0 ({\bf R}_+)$-distribution derivative, the $\mathcal F' ({\bf R}_+)$-distribution derivative that we use in this definition “does not see” what happens on the boundary, $\{0\}$.
[**Proof.**]{} Since $\mathcal F ({\bf R}_+)$ is dense in $B^{1/2}({\bf R}_+)$, it is enough to show that $$\int_{{\bf R}_+}
\left| \frac{\partial_-^{1/2} u}{\partial t^{1/2}} \right|^2 \, dt \sim
\iint_{ {\bf R}_+ \times {\bf R}_+ }
\left| \frac{u(s)-u(t)}{s-t}\right|^2 \, ds\,dt ,$$ for functions in $\mathcal F ({\bf R}_+)$, where $\sim$ means that the seminorms are equivalent.
For $p=2$ we (temporarily) denote the closure of $\mathcal F ({\bf R}_+)$ in the norm $$\|u \| =\|\frac{\partial_-^{1/2} u}{\partial t^{1/2}} \|_{L^2({\bf R}_+)}
+\|u\|_{L^2({\bf R}_+)},$$ by $H$.
It follows directely from the definitions, and the fact that $\mathcal F ({\bf R}_+)$ is dense in $B^{1/2}({\bf R}_+)$, that $B^{1/2}({\bf R}_+)$ is continuously imbedded in $H$.
We shall now show that in fact $H=B^{1/2}({\bf R}_+)$.
Let $T$ denote the operator $T:u \mapsto \frac{\partial u}{\partial t} +u$. Then $
T: B^{1/2}_0({\bf R}_+) \longrightarrow H^*
$ is continuous. This follows from fractional integration by parts, $$\begin{aligned}
\nonumber
\langle Tu, \Phi \rangle = \left( \frac{\partial_+^{1/2} u}{\partial t^{1/2}},
\frac{\partial_-^{1/2} \Phi}{\partial t^{1/2}} \right)_{L^2}
+\left(u,\Phi\right)_{L^2} \\
;\quad \Phi \in \mathcal F ({\bf R}_+), \; u \in \mathcal F_0 ({\bf R}_+),\end{aligned}$$ and the fact that $\mathcal F ({\bf R}_+)$ is dense in $H$ and that $\mathcal F_0 ({\bf R}_+)$ is dense in $B^{1/2}_0({\bf R}_+)$.
Now by the Hahn-Banach theorem, given $\xi \in H^*$ there exist elements $u, v \in L^2({\bf R}_+)$ such that $$\langle \xi, \Phi \rangle =
\left(u,\frac{\partial_-^{1/2} \Phi}{\partial t^{1/2}} \right)_{L^2} +
\left(v,\Phi\right)_{L^2}
;\quad \Phi \in \mathcal F ({\bf R}_+).$$
We can thus extend $\xi$ by zero to an element $E_0(\xi)$ of $B^{1/2}({\bf R})^*$. Since $ T:B^{1/2}({\bf R}) \rightarrow B^{1/2}({\bf R})^*$ is an isomorphism, we can find a unique element $ u \in B^{1/2}({\bf R})$ such that $Tu = E_0(\xi)$ in $\mathcal {F'} ({\bf R})$. But this holds if and only if $ u \in B^{1/2}_0({\bf R}_+)$ and $Tu = \xi$ in $\mathcal F'_0 ({\bf R}_+)$.
Thus $T: B^{1/2}_0({\bf R}_+) \longrightarrow H^*$ is an isomorphism.
Furthermore, by direct computation (or by interpolation (recall that $p=2$)), we know that $$T: B^{1/2}_0({\bf R}_+) \longrightarrow B^{1/2}({\bf R}_+)^*$$ is an isomorphism.
Since $\mathcal {F} ({\bf R}_+)$ is densely continuously imbedded in both $H$ and $B^{1/2}({\bf R}_+)$ and thus $H^*$ and $B^{1/2}({\bf R}_+)^*$ both are well defined subspaces in $\mathcal F'_0 ({\bf R}_+)$, we see that $H^*$ and $B^{1/2}({\bf R}_+)^*$ are identical as subspaces of $\mathcal F'_0 ({\bf R}_+)$ and equivalent as Hilbert spaces.
Since $B^{1/2}({\bf R}_+) \hookrightarrow H$, by Riesz representation theorem, this implies that $H$ and $B^{1/2}({\bf R}_+)$ have equivalent norms.
From a scaling argument it now follows that $$\int_{{\bf R}_+}
\left| \frac{\partial_-^{1/2} u}{\partial t^{1/2}} \right|^2 \, dt \sim
\iint_{ {\bf R}_+ \times {\bf R}_+ }
\left| \frac{u(s)-u(t)}{s-t}\right|^2 \, ds\,dt ,$$ for functions in $B^{1/2}({\bf R}_+)$. The lemma follows. $\Box$
Parabolic Equations.
====================
We shall consider operators of the form $$\label{eq:T}
Tu = \frac{\partial u}{\partial t} - \nabla_x\cdot A(x,t,\nabla_x u),$$ on a space-time cylinder $Q_+=\Omega \times {\bf R_+}$, where $\Omega$ is an open and bounded set in ${\bf R}^n$ with smooth boundary.
We shall assume the following structural conditions for the function $A: \Omega \times {\bf R}_+ \times {\bf R}^n \longrightarrow
{\bf R}^n$.
1. $Q_+ \ni (x,t) \mapsto A(x,t,\xi)$ is Lebesgue measurable for every fixed $\xi \in {\bf R}^n$.
2. ${\bf R}^n \ni \xi \mapsto A(x,t,\xi)$ is continuous for almost every $(x,t) \in Q_+$.
3. For every $\xi,\eta \in {\bf R}^n, \xi \neq \eta$ and almost every $(x,t) \in Q_+$, we have $$\label{eq:strictmonotonicity}
(A(x,t,\xi)-A(x,t,\eta),\xi -\eta) > 0.$$
4. There exists $p \in (1,\infty)$, a constant $\lambda >0$ and a function $ h \in L^1(Q_+)$ such that for every $\xi \in {\bf R}^n$ and almost every$(x,t) \in Q_+$: $$(A(x,t,\xi), \xi) \geq \lambda |\xi|^p - h(x,t).$$
5. There exists a constant $\Lambda \geq \lambda >0$ and a function $H \in L^{p/(p-1)}(Q_+)$ such that for every $\xi \in {\bf R}^n$ and almost every $(x,t)\in Q_+$: $$\label{eq:boundednesscond}
|A(x,t,\xi)| \leq \Lambda |\xi|^{p-1} + H(x,t).$$
The Carathéodory conditions 1 and 2 above guarantee that the function $Q \ni (x,t) \mapsto A(x,t,\Phi(x,t))$ is measurable for every function $\Phi \in L^p(Q_+,{\bf R}^n)$. Condition 3 is a strict monotonicity condition that gives us uniqueness results. Conditions 4 (coercivity) and 5 (boundedness) give us apriori estimates that imply existence results (see [@F]).
We now introduce some function spaces, and in their definitions $\partial_-^{1/2} /\partial t^{1/2}$ should be understood in the $\mathcal{F'}_{\cdot,\cdot}(Q)$ distribution-sense.
\[def:functionspaces\] For $1<p<\infty$, set $$\begin{aligned}
\nonumber
B^{1,1/2}_{\cdot,\cdot}(Q)=\left\{ u \in L^p(Q);\;
\frac{\partial^{1/2}_- u}{\partial t^{1/2}}
\in L^2(Q)\right. \\
, \left. \frac{\partial u}{\partial x_i} \in L^p(Q),
\, i=1,\dots ,n. \right\}.\end{aligned}$$
We equip these spaces with the following norms.
$$\|u\|_{B^{1,1/2}_{\cdot,\cdot}(Q)}=
\|\frac{\partial^{1/2}_- u}{\partial t^{1/2}} \|_{L^2(Q)} + \|u\|_{L^p(Q)}+
\sum_{i=1}^n \|\frac{\partial u}{\partial x_i} \|_{L^p(Q)}.$$
Computing in $\mathcal{F'}_{\cdot,\cdot}(Q)$ we see that we can represent these spaces as closed subspaces of the direct sum $L^2(Q)\oplus L^p(Q_)\oplus \cdots \oplus L^p(Q)$, and thus they are reflexive and separable Banach spaces in the topologies arising from the given norms.
Since the lateral boundary is smooth (in fact Lipschitz continuous suffices), we can extend an element in $B^{1,1/2}_{\cdot,\cdot}(Q)$ to all of ${\bf R}^n \times {\bf R}$ and then cut off in the space variables. By regularizing it is clear that functions smooth up to the boundary are dense in $B^{1,1/2}_{\cdot,\cdot}(Q)$. To show that $\mathcal F_{\cdot,\cdot}(Q)$ is dense in $B^{1,1/2}_{\cdot,\cdot}(Q)$ we only have to prove that we can “cut off” in time. This will follow as in Lemma \[lem:cutoff\] once we have the following result.
\[lem:equivnorms\] If $u \in B^{1,1/2}_{\cdot,\cdot}(Q)$, then $$\label{eq:equivnorms}
\iiint_{\Omega \times {\bf R}\times{\bf R}}
\left|\frac{u(x,s)-u(x,t)}{s-t}\right|^2 \, dx\,ds\,dt
=2\pi \iint_{Q} \left|\frac{\partial_-^{1/2} u}{\partial t^{1/2}} \right|^2 \,dx\, dt.$$
[**Proof.**]{}
That $\frac{\partial_-^{1/2}u}{\partial t^{1/2}}=v$ means that $$\begin{aligned}
\nonumber
\iint_Q u(x,t)\frac{\partial_+^{1/2}\Phi(x,t)}{\partial t^{1/2}}\,dx\,dt =
\iint_Q v(x,t)\Phi(x,t)\,dx\,dt \\
;\Phi \in \mathcal{F}_{0,\cdot}(Q).\end{aligned}$$ Now for almost every $x \in \Omega$, $\Omega \ni x \mapsto u(x,\cdot) \in L^p({\bf R})$ and $\Omega \ni x \mapsto v(x,\cdot) \in L^2({\bf R})$ are well defined. Let $S$ denote the set of common Lebesgue points. Since the Lebesgue points of a function can only increase by multiplication with a smooth function, by taking limits of mean values, we get that $$\begin{aligned}
\nonumber
\int_{\bf R} u(x,t)\frac{\partial_+^{1/2}\Phi(x,t)}{\partial t^{1/2}}\,dt =
\int_{\bf R} v(x,t)\Phi(x,t)\,dt \\
;\Phi \in \mathcal{F}_{0,\cdot}(Q),\end{aligned}$$ for all $x \in S$. This implies that for almost every $x\in \Omega$ the $L^p({\bf R})$ function $t \mapsto u(x,t)$ has half a derivative equal to $v(x,t) \in L^2({\bf R})$. So from the one-dimensional result it follows that $$\iint_{{\bf R}\times{\bf R}}
\left|\frac{u(x,s)-u(x,t)}{s-t}\right|^2 \,ds\,dt
=2\pi \int_{\bf R} \left|\frac{\partial_-^{1/2} u}{\partial t^{1/2}} \right|^2 \, dt,$$ for almost every $x\in \Omega$. Integrating with respect to $x$, the lemma follows. $\Box$
We conclude that:
The space of testfunctions $\mathcal{F}_{\cdot,\cdot}(Q)$ is dense in $B^{1,1/2}_{\cdot,\cdot}(Q)$.
We now introduce the following subspace that corresponds to zero boundary data on the lateral boundary $\partial \Omega \times {\bf R}$ and as $|t| \rightarrow \infty$.
Let $B^{1,1/2}_{0,\cdot}(Q)$ denote the closure of $\mathcal{F}_{0,\cdot}(Q)$ in the $B^{1,1/2}_{\cdot,\cdot}(Q)$-topology.
We shall work with the following two sets of function spaces on $Q_+$.
Let $B^{1,1/2}_{*,\cdot}(Q_+)$ denote the space of functions defined on $Q_+$ that can be extended to elements in $B^{1,1/2}_{*,\cdot}(Q)$.
Furthermore let $B^{1,1/2}_{*,0}(Q_+)$ denote the space of functions defined on $Q_+$ that can be extended by zero to elements in $B^{1,1/2}_{*,\cdot}(Q)$.
Here $*$ optionally stands for $\cdot$ or $0$. A zero in the first position corresponds to zero boundary data on the lateral boundary and a zero in the second position corresponds to zero initial data.
Clearly $B^{1,1/2}_{*,0}(Q_+)$ can be identified with a closed subspace of $B^{1,1/2}_{*,\cdot}(Q)$.
We give the following two simple lemmas concerning these spaces and, as in the case of the real line, we omit the easy proofs.
The function space $B^{1,1/2}_{*,0}(Q_+)$ becomes a Banach space with the norm $$\begin{aligned}
\nonumber\label{eq:bhalf0normRn}
\| u\|_{B^{1,1/2}_{*,0}(Q_+)}=\|u\|_{L^p(Q_+)} + \|\nabla_x u\|_{L^p(Q_+)}+
\left\{ \int_{Q_+} \frac{u^2(x,t)}{t} \,dt\, dx \right.\\
\left. +
\iiint_{\Omega \times {\bf R}_+ \times {\bf R}_+}
\left(\frac{u(x,s)-u(x,t)}{s-t} \right)^2 \,dx \,ds\,dt \right\}^{1/2}.\end{aligned}$$
The function space $B^{1,1/2}_{*,\cdot}(Q_+)$ becomes a Banach space with the norm $$\begin{aligned}
\nonumber\label{eq:bhalfnormRn}
\| u\|_{B^{1,1/2}_{\cdot,\cdot}(Q_+)}=\|u\|_{L^p(Q_+)} +\|\nabla_x u\|_{L^p(Q_+)} \\
+
\left\{
\iiint_{\Omega \times {\bf R}_+ \times {\bf R}_+}
\left(\frac{u(x,s)-u(x,t)}{s-t} \right)^2\,dx \, ds\,dt \right\}^{1/2}.\end{aligned}$$ Furthermore a continuous symmetric extension mapping from $B^{1,1/2}_{*,\cdot}(Q_+)$ to $B^{1,1/2}_{*,\cdot}(Q)$ is given by $E_S(u)(x,t)=u(x,|t|)$.
Computing in $\mathcal F'_{\cdot,0} (Q_+)$ we can give an equivalent characterization of $B^{1,1/2}_{\cdot,0}(Q_+)$.
A function $u \in L^p(Q_+)$ belongs to $B^{1,1/2}_{\cdot,0}(Q_+)$ if and only if the $\mathcal F'_{\cdot,0} (Q_+)$-distribution derivative $\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}$ belongs to $L^2(Q_+)$, and the $\mathcal F'_{\cdot,\cdot} (Q_+)$-distribution derivatives $\nabla_x u \in L^p(Q_+)$. Furthermore an equivalent norm on $B^{1,1/2}_{\cdot,0}({\bf R}_+)$ is then given by $$\|u\| = \| \nabla_x u\|_{L^p(Q_+)} +
\| u\|_{L^p(Q_+)}+
\|\frac{\partial^{1/2}_+ u}{\partial t^{1/2}}\|_{L^2(Q_+)}.$$
[**Proof.**]{} As on the real line. $\Box$
Using the corresponding result on the real half-line and the same type of argument as in the proof of Lemma \[lem:equivnorms\], we see that an equivalent norm on $B^{1,1/2}_{*,\cdot}(Q_+)$ is given by
$$\|u\| =
\|\frac{\partial^{1/2}_- u}{\partial t^{1/2}} \|_{L^2(Q_+)} + \|u\|_{L^p(Q_+)}+
\sum_{i=1}^n \|\frac{\partial u}{\partial x_i} \|_{L^p(Q_+)},$$
where $\frac{\partial^{1/2}_- }{\partial t^{1/2}}$ is understood in the $\mathcal{F'}_{\cdot,\cdot}(Q_+)$-distribution sense.
We have the following density results:
\[th:densetestfunc\] The space of testfunctions $\mathcal{F}_{\cdot,*}(Q_+)$ is dense in $B^{1,1/2}_{\cdot,*}(Q_+)$. Furthermore the space of testfunctions $\mathcal{F}_{0,*}(Q_+)$ is dense in $B^{1,1/2}_{0,*}(Q_+)$.
[**Proof.**]{} Since the boundary of $\Omega$ is smooth we have good extension operators in the space variables, and we can also translate the support of functions away from the lateral boundary without spreading the support in the time direction. The result thus follows exactly as in Lemma \[lem:densereal\]. $ \Box $
We point out the following result that follows immediately from the given norms.
The space $B^{1,1/2}_{*,0}(Q_+)$ is continuously imbedded in $B^{1,1/2}_{*,\cdot}(Q_+)$.
We also remark that the (semi)norms $\|\frac{\partial^{1/2}_- u}{\partial t} \|_{L^2(Q_+)}$ and $\|\frac{\partial^{1/2}_+ u}{\partial t} \|_{L^2(Q_+)}$ are not equivalent. In fact in Lemma \[lem:denseimbed\] below we show that $B^{1,1/2}_{0,0}(Q_+)$ is a dense subspace of $B^{1,1/2}_{0,\cdot}(Q_+)$. This is of course connected with the well known fact that if $u \in L^2(Q)$ and $\frac{\partial^{1/2}_- u}{\partial t^{1/2}} \in L^2(Q)$, it is in general impossible to define a trace on $\Omega \times \{ 0 \}$ (for instance the function $(x,t) \mapsto \log | \log |t||$ locally belongs to this space). Still a function in $ B^{1,1/2}_{\cdot,0}(Q_+)$ is of course zero on $\Omega \times \{ 0\}$ in the sense that
$$\label{eq:hardyineq}
\iint_{Q_+} \frac{u^2(x,t)}{t}\,dxdt < \infty.$$
We shall now discuss homogeneous data on the whole parabolic boundary.
Homogeneous data.
-----------------
We introduce the following space of $\mathcal{F'}_{\cdot,\cdot}(Q)$-distributions defined globally in time, but supported in $Q_+$.
Let $$B^{-1,-1/2}_{\cdot,0}(Q_+):= \left\{ \xi \in B^{1,1/2}_{0,\cdot}(Q)^*;\quad
\xi = 0 \; \mbox{in} \; \Omega \times (-\infty,0) \right\}$$
From Theorem 4.3 and Theorem 4.4 in [@F] follows
\[th:homglobal\] For $T$ as defined in (\[eq:T\]), satisfying the structural conditions (1)–(5), $$T: B^{1,1/2}_{0,0}(Q_+) \longrightarrow B^{-1,-1/2}_{\cdot,0}(Q_+)$$ is a bijection.
We shall now show that $B^{-1,-1/2}_{\cdot,0}(Q_+)$ can be identified with the dual space of $B^{1,1/2}_{0,\cdot}(Q_+)$.
\[lem:bdual\] We can identify $
B^{-1,-1/2}_{\cdot,0}(Q_+)$ with $B^{1,1/2}_{0,\cdot}(Q_+)^*$.
[**Remark.**]{} Note that we here identify a subspace of $\mathcal{F'}_{\cdot,\cdot}(Q)$ with a subspace of $\mathcal{F'}_{\cdot,0}(Q_+)$.
[**Proof.**]{} Given $\xi \in B^{1,1/2}_{0,\cdot}(Q_+)^*$ we have (by the Hahn-Banach theorem) $u_0 \in L^2(Q_+)$ and $u_i \in L^{p'}(Q_+)$, $i=1,\dots,n$ such that $$\langle \xi, \Phi \rangle =
\iint_{Q_+} u_0 \frac{\partial^{1/2}_- \Phi}{\partial t} +
\sum_{i=1}^n u_i \frac{\partial \Phi}{\partial x_i}\,dx dt ;\quad
\Phi \in \mathcal{F}_{0,\cdot}(Q_+).$$ It is thus clear that we can extend this $\xi$ to all of $\mathcal{F}_{0,\cdot}(Q)$ by zero. Set $$\langle \xi_0, \Phi \rangle =
\iint_{Q} E_0(u_0) \frac{\partial^{1/2}_- \Phi}{\partial t} +
\sum_{i=1}^n E_0(u_i) \frac{\partial \Phi}{\partial x_i}\,dx dt ;\quad
\Phi \in \mathcal{F}_{0,\cdot}(Q),$$ where $E_0$ denotes the operator that extends a function with $0$ to all of $Q$. The mapping $B^{1,1/2}_{0,\cdot}(Q_+)^* \ni \xi \mapsto
\xi_0 \in B^{-1,-1/2}_{\cdot,0}(Q_+)$ is clearly injective, but it is also surjective. This follows since given $\xi \in B^{-1,-1/2}_{\cdot,0}(Q_+)$, by Theorem \[th:homglobal\] above, there exists a (unique) $u_{\xi} \in B^{1,1/2}_{0,0}(Q_+)$ such that $$\frac{\partial u_{\xi}}{\partial t} -\nabla_x \cdot (|\nabla_x u_{\xi}|^{p-2}
\nabla_x u_{\xi}) = \xi,$$ i.e. $$\nonumber
\langle \xi, \Phi \rangle =
\iint_{Q} \frac{\partial^{1/2}_+ u_{\xi}}{\partial t}
\frac{\partial^{1/2}_- \Phi}{\partial t}$$ $$+(|\nabla_x u_{\xi}|^{p-2}
\nabla_x u_{\xi})\cdot \nabla_x \Phi \,dx dt ;\quad
\Phi \in \mathcal{F}_{0,\cdot}(Q),$$ and we see that $\xi$ has the required form. $\Box$
Thus we can reformulate Theorem \[th:homglobal\].
\[th:homognonlin\] For $T$ as defined in (\[eq:T\]), satisfying the structural conditions (1)–(5), $$T: B^{1,1/2}_{0,0}(Q_+) \longrightarrow B^{1,1/2}_{0,\cdot}(Q_+)^*$$ is a bijection.
[**Remark.**]{} This theorem of course means that given $\xi \in B^{1,1/2}_{0,\cdot}(Q_+)^*$ there exists a unique $u \in B^{1,1/2}_{0,0}(Q_+)$ such that $$\label{eq:tuxi}
\langle T(u), \Phi \rangle = \langle \xi , \Phi \rangle \; ; \quad
\Phi \in B^{1,1/2}_{0,\cdot}(Q_+).$$ Which means precisely that $$\begin{aligned}
\nonumber
\langle \xi, \Phi \rangle =
\iint_{Q_+} \frac{\partial^{1/2}_+ u_{\xi}}{\partial t}
\frac{\partial^{1/2}_- \Phi}{\partial t}
+ A(x,t,\nabla_x u) \cdot \nabla_x \Phi \,dx dt \\
;\quad \Phi \in \mathcal{F}_{0,\cdot}(Q_+),\end{aligned}$$ since $\mathcal{F}_{0,\cdot}(Q_+)$ is dense in $B^{1,1/2}_{0,\cdot}(Q_+)$.
The following structure theorem for our source data space is an immediate consequence of the Hahn-Banach theorem.
Given $\xi \in B^{1,1/2}_{0,\cdot}(Q_+)^*$ there exist functions $u_0 \in L^2(Q_+)$ and $u_1, \dots, u_n \in L^{p/(p-1)}(Q_+)$ such that $$\xi = \frac{\partial^{1/2}_+ u_0}{\partial t}+ \sum_{i=1}^n \frac{\partial u_i}{\partial x_i}$$ in $\mathcal F'_{\cdot,0}(Q_+)$.
Our next result implies that in general it is actually enough to test our equations with $\mathcal{F}_{0,0}(Q_+)$ instead of $\mathcal{F}_{0,\cdot}(Q_+)$.
\[lem:denseimbed\] The continuous imbedding $$B^{1,1/2}_{0,0}(Q_+) \hookrightarrow B^{1,1/2}_{0,\cdot}(Q_+)$$ is dense.
[**Proof.**]{} It is enough to show that if $\xi \in B^{1,1/2}_{0,\cdot}(Q_+)^*$ and $\langle \xi,\Phi \rangle =0$ for all $ \Phi \in B^{1,1/2}_{0,0}(Q_+)$, then $\xi =0$.
Now given $\xi \in B^{1,1/2}_{0,\cdot}(Q_+)^*$, by Theorem \[th:homognonlin\], there exists a unique $u_{\xi} \in B^{1,1/2}_{0,0}(Q_+)$ such that $$\frac{\partial u_{\xi}}{\partial t} - \nabla_x \cdot(|\nabla_x u_{\xi}|^{p-2}
\nabla_x u_{\xi}) = \xi.$$ Now if $\langle \xi, \Phi \rangle =0$ for all $\Phi \in B^{1,1/2}_{0,0}(Q_+)$, then with $\Phi=u_{\xi}$ we get $$\iint_{Q_+}|\nabla_x u_{\xi}|^p \,dx\,dt =0.$$ By the Poincaré inequality $u_{\xi}=0$, and so $\xi=0$. $\Box$
Non-homogeneous initial data.
-----------------------------
We will first introduce the space that will carry the initial data. In the definition, all derivatives should be understood in the $\mathcal{F'}_{\cdot,\cdot}(Q_+)$-distribution sense.
Let $$\begin{aligned}
\nonumber
B_I(Q_+)= \left\{ u \in B^{1,1/2}_{0,\cdot}(Q_+)\cap
C_b([0,\infty),L^2(\Omega)) \right.\\
\left. ;\frac{\partial u}{\partial t}
\in L^{p'}({\bf R}_+, W^{-1,p'}(\Omega)) \right\} .\end{aligned}$$
Here $C_b([0,\infty),L^2(\Omega))$ denotes the space of bounded continuous functions from $[0,\infty)$ into $L^2(\Omega)$, and $\frac{\partial u}{\partial t}
\in L^{p'}({\bf R}_+, W^{-1,p'}(\Omega))$ means exactly that $$|\langle u, \frac{\partial \Phi}{\partial t} \rangle |
\leq C \|\nabla_x \Phi \|_{L^p(Q_+)}\quad; \quad \Phi \in \mathcal{F}_{0,0}(Q_+),$$ for some constant $C>0$. The smallest possible constant is by definition $\|\frac{\partial u}{\partial t}\|_{L^{p'}({\bf R}_+, W^{-1,p'}(\Omega))}$.
We equip $B_I(Q_+)$ with the following norm $$\begin{aligned}
\nonumber
\|u\|_{B_I(Q_+)}:= \|u\|_{B^{1,1/2}_{0,\cdot}(Q_+)}+
\sup_{t \in {\bf R}_+} \| u(\cdot,t) \|_{L^2(\Omega)} \\
+ \|\frac{\partial u}{\partial t}\|_{L^{p'}({\bf R}_+, W^{-1,p'}(\Omega))}.\end{aligned}$$
Using Theorem \[th:homognonlin\] and the monotonicity of $A(x,t,\cdot)$ we shall now prove that we always have a unique solution in $B_I(Q_+)$ to the following initial value problem.
Given $u_0 \in L^2(\Omega)$, there exists a unique element $u \in B_I(Q_+)$ such that
\[eq:initialvalues\] $$\begin{aligned}
\frac{\partial u}{\partial t} - \nabla_x\cdot A(x,t,\nabla_x u)
&=0 \quad
\mbox{in $\mathcal{F'}_{\cdot,\cdot}(Q_+)$}\\
u&=u_0 \quad \mbox{on $ \Omega \times \{0\} $}.\end{aligned}$$
[**Proof.**]{}
Uniqueness follows immediately from the monotonicity of $A(x,t,\cdot)$ by pairing with a cut off function in time multiplied with the difference of two solutions. To prove existence we first note that if $u_0 \in \mathcal D (\Omega)$, we can extend it for instance to a smooth testfunction $U_0 \in \mathcal D (\Omega \times (-2,2))$ such that $U_0(x,t)=u_0(x)$ when $-1<t<1$.
Since $\frac{\partial U_0}{\partial t} \in B^{1,1/2}_{0,\cdot}(Q_+)^*$, by Theorem \[th:homognonlin\], we know that there exists a unique $w \in B^{1,1/2}_{0,0}(Q_+)$ such that $$\frac{\partial w}{\partial t} - \nabla_x\cdot A(x,t,\nabla_x w + \nabla_x U_0)
=-\frac{\partial U_0}{\partial t} \quad
\mbox{in $ B^{1,1/2}_{0,\cdot}(Q_+)^*$}.$$ Then clearly $u= (w+U_0) \in B^{1,1/2}_{0,\cdot}(Q_+)$ solves (\[eq:initialvalues\]), and the initial value is taken in the sense that $$\iint_{\Omega \times (0,1)} \frac{(u(x,t)-u_0(x))^2}{t}\,dx\,dt < \infty.$$ By standard arguments it follows from (\[eq:initialvalues\]) that $u \in B_I(Q_+)$ and so the initial data is actually taken in $C_b([0,\infty),L^2(\Omega))$-sense.
Given $u_0 \in L^2(\Omega)$ we now choose a sequence $\mathcal D(\Omega) \ni u^n_0 \longrightarrow u_0$ in $L^2(\Omega)$.
Let $u^n$ denote the solution of (\[eq:initialvalues\]) with initial data $u^n_0$. By testing with $u^n \chi$, where $\chi$ is a standard cut off function in time, in (\[eq:initialvalues\]), we get that $$\sup_{t \in {\bf R}_+} \int_{\Omega} (u^n -u^m)^2(x,t)\,dx \leq
\int_{\Omega} (u^n_0 -u^m_0)^2(x)\,dx.$$ It is also clear that $\|\nabla_x u^n \|_{L^p(Q_+)}$ is bounded by a constant independent of $n$.
Finally we note that we can extend $u^n$ symmetrically to $Q$ and the extended function $E_S(u^n) \in B^{1,1/2}_{0,\cdot}(Q)$ will satisfy $\frac{\partial E_S(u^n)}{\partial t}
\in L^{p'}({\bf R}, W^{-1,p'}(\Omega))$.
We then have $$\iint_Q \frac{\partial_-^{1/2} E_S(u^n)}{\partial t^{1/2}}
\frac{\partial_-^{1/2} \Phi_k}{\partial t^{1/2}} \,dx\,dt
= \int_{\bf R} \langle
\frac{\partial E_S(u^n)}{\partial t}, h(\Phi_k) \rangle \,dt,$$ for a sequence $\mathcal {F}_{0,\cdot}(Q) \ni \Phi_k \rightarrow E_S(u^n)$ in $B^{1,1/2}_{0,\cdot}(Q)$.
This implies that $\|\frac{\partial_-^{1/2} E_S(u^n)}{\partial t^{1/2}}\|_{L^2(Q_+)}$ is bounded by a constant independent of $n$.
We conclude that $\| u^n \|_{B_I(Q_+)} \leq C$, where $C <\infty $ is a constant independent of $n$.
We can now extract a weakly convergent subsequence and in fact, as we have seen, we actually have strong convergence in $C_b([0,\infty),L^2(\Omega))$ and thus the limit function satisfies the initial conditions.
Finally a Minty argument using the monotonicity of $A(x,t,\cdot)$ shows that the limit function solves (\[eq:initialvalues\]). The theorem follows. $\Box$
Fully non-homogeneous initial-boundary values.
----------------------------------------------
We shall now introduce the function space that will carry both initial and lateral boundary data.
Since we have continuous imbeddings $B^{1,1/2}_{0,\cdot}(Q_+)\hookrightarrow B^{1,1/2}_{\cdot,\cdot}(Q_+)$ and\
$B_I(Q_+) \hookrightarrow B^{1,1/2}_{\cdot,\cdot}(Q_+)$, the following definition makes sense.
\[def:xspace\] Let $$X^{1,1/2}(Q_+)= B^{1,1/2}_{\cdot,0}(Q_+) + B_I(Q_+),$$ be equipped with the norm $$\|u \|_{X^{1,1/2}(Q_+)} = \inf_{(u_1,u_2) \in K_u} \left(
\|u_1\|_{B^{1,1/2}_{\cdot,0}(Q_+)} + \| u_2\|_{B_I(Q_+)}\right),$$ where the infimum is taken over the set $$K_u=\left\{ (u_1,u_2);\; u_1+u_2 =u,\; u_1 \in B^{1,1/2}_{\cdot,0}(Q_+),\;
u_2 \in B_I(Q_+) \right\}.$$
The following imbeddings are immediate $$\begin{aligned}
\| u\|_{X^{1,1/2}(Q_+)} \leq \|u\|_{B_{\cdot,0}^{1,1/2}(Q_+)};\quad
u \in B_{\cdot,0}^{1,1/2}(Q_+),\\
\| u\|_{X^{1,1/2}(Q_+)} \leq \|u\|_{B_I(Q_+)};\quad
u \in B_I(Q_+),\\
\| u\|_{B^{1,1/2}_{\cdot,\cdot}(Q_+)} \leq C \| u\|_{X^{1,1/2}(Q_+)};\quad
u \in X^{1,1/2}(Q_+).\end{aligned}$$
For an element in $X^{1,1/2}(Q_+)$ we can always define the trace on $\Omega \times \{0\}$.
\[th:tracex12\] There exists a continuous linear and surjective trace operator $$Tr_0:X^{1,1/2}(Q_+) \longrightarrow L^2(\Omega).$$ There also exists a bounded extension operator $$E : L^2(\Omega) \longrightarrow X^{1,1/2}(Q_+)$$ such that $Tr_0 \circ E = Id_{L^2(\Omega)}$.
[**Proof.**]{} Given $u \in X^{1,1/2}(Q_+)$, there exist $u_1 \in B_{\cdot,0}^{1,1/2}(Q_+)$ and $u_2 \in B_I(Q_+)$ such that $u=u_1+u_2$. Since $ u_2 \in B_I(Q_+) \Longrightarrow u_2 \in
C_b([0,+\infty),L^2(\Omega))$, $u_2|_{\Omega \times \{0\}}$ is a well defined element of $L^2(\Omega)$. We now define $u|_{\Omega \times \{0\}}=
u_2|_{\Omega \times \{0\}}$. We have to show that this is independent of the decomposition of $u$, but if we have two different decompositions $u_1+u_2 = v_1+v_2$ as above, then $(u_2-v_2) \in B_I(Q_+) \cap B_{\cdot,0}^{1,1/2}(Q_+)$, which implies that $$\iint_{\Omega \times (0,+\infty)} \frac{(u_2-v_2)^2(x,t)}{t}\,dxdt < +\infty,$$ and so $u_2(\cdot,0)=v_2(\cdot,0)$ since they both belong to $C_b([0,+\infty),L^2(\Omega))$.
Now $$\|u(\cdot,0)\|_{L^2(\Omega)} = \| u_2(\cdot,0)\|_{L^2(\Omega)}
\leq C \|u_2 \|_{B_I(Q_+)},$$ for any decomposition $u=u_1+u_2$ as above. Taking the infimum over all such decompositions gives: $$\| u(\cdot,0)\|_{L^2(\Omega)} \leq C \|u\|_{X^{1,1/2}(Q_+)},\;
u \in X^{1,1/2}(Q_+).$$
Now given $u_0 \in L^2(\Omega)$, let $E(u_0)$ be the (unique) solution in $B_I(Q_+)$ of the initial value problem:
$$\begin{aligned}
\frac{\partial u}{\partial t} - \nabla_x \cdot (|\nabla_x u|^{p-2}\nabla_x u)
&=0 \quad
\mbox{in $Q_+=\Omega \times {\bf R_+}$}\\
u&=u_0 \quad \mbox{on $ \Omega \times \{0\} $}.\end{aligned}$$
Clearly this extension map satisfies $Tr_0 \circ E = Id_{L^2(\Omega)}$ and furthermore $$\|E (u_0)\|_{B_I(Q_+)} \leq C \|u_0\|_{L^2(\Omega)},$$ and thus $$\|E (u_0)\|_{X^{1,1/2}(Q_+)} \leq C \|u_0\|_{L^2(\Omega)}.$$ $\Box$
[**Remark.**]{} Note that if $p=2$ the extension map is linear.
We have the following imbedding: $$\| u\|_{B^{1,1/2}_{\cdot,0}(Q_+)} \leq C \| u\|_{X^{1,1/2}(Q_+)}; \; u
\in B^{1,1/2}_{\cdot,0}(Q_+).$$
[**Proof.**]{} If $u \in B^{1,1/2}_{\cdot,0}(Q_+)$, and $u=u_1+u_2$ with $u_1 \in B_{\cdot,0}^{1,1/2}(Q_+)$ and $u_2 \in B_I(Q_+)$, then $u_2(\cdot,0)=0$ since $u_2 \in B_{\cdot,0}^{1,1/2}(Q_+) \cap
B_I(Q_+)$. Thus $u_2$ can be extended by zero to all of $Q$. Since, by a continuity argument, $$\| \frac{\partial_+^{1/2} u_2}{\partial t} \|_{L^2(Q_+)}^2
= -\int_{{\bf R}_+} \langle \frac{\partial u_2}{\partial t}, h(u_2)
\rangle \,dt, \quad
u_2 \in B^{1,1/2}_{\cdot,0}(Q_+) \cap B_I(Q_+).$$ We get $$\| u_1\|_{ B_{\cdot,0}^{1,1/2}(Q_+)} + \|u_2\|_{ B_I(Q_+)}$$ $$\geq C\left( \| u_1\|_{ B_{\cdot,0}^{1,1/2}(Q_+)} +
\|u_2\|_{ B_{\cdot,0}^{1,1/2}(Q_+)} \right)$$ $$\geq C\|u_1+u_2\|_{ B_{\cdot,0}^{1,1/2}(Q_+)}=
C\|u \|_{ B_{\cdot,0}^{1,1/2}(Q_+)},$$ where $C>0$. Taking the infimum concludes the proof. $\Box$ We immediately get the following
There exist constants $C_1,C_2>0$ such that $$C_1 \| u \|_{ B_{0,0}^{1,1/2}(Q_+)} \leq \| u \|_{ X^{1,1/2}(Q_+)}
\leq C_2 \| u \|_{ B_{0,0}^{1,1/2}(Q_+)};\; u \in
\mathcal F_{0,0}(Q_+).$$ Thus $B_{0,0}^{1,1/2}(Q_+)$ is the closure of $\mathcal F_{0,0}(Q_+)$ in the $X^{1,1/2}(Q_+)$-norm topology.
We are now ready to state our main theorem.
\[th:mainnonlinear\] Given $f \in B_{0,\cdot}^{1,1/2}(Q_+)^*$ and $g \in X^{1,1/2}(Q_+)$, there exists a unique element $u \in X^{1,1/2}(Q_+)$ such that
\[eq:proofnonlinear\] $$\begin{aligned}
\frac{\partial u}{\partial t} - \nabla_x \cdot (A(x,t,\nabla_x u))
&=f \quad
\mbox{in}\; \mathcal F'_{\cdot,\cdot}(Q_+) \\
u-g \in & \, B_{0,0}^{1,1/2}(Q_+).\end{aligned}$$
[**Proof.**]{}
Let $w=u-g$. Then (\[eq:proofnonlinear\]) is equivalent to
\[eq:proofhomnonlinear\] $$\begin{aligned}
\frac{\partial w}{\partial t} - \nabla_x \cdot (A(x,t,\nabla_x (w+g)))
&=f-\frac{\partial g}{\partial t} \; \mbox{in}\; \mathcal F'_{\cdot,\cdot}(Q_+)\\
w \in & \, B_{0,0}^{1,1/2}(Q_+).\end{aligned}$$
Here $\frac{\partial g}{\partial t} \in \mathcal{F'}_{\cdot,\cdot}(Q_+)$ has a unique extension to an element in $B_{0,\cdot}^{1,1/2}(Q_+)^*$. In fact, if $g \in X^{1,1/2}(Q_+)$, we can write $g=g_1+g_2$, where $g_1 \in B_{\cdot,0}^{1,1/2}(Q_+)$ and $g_2 \in B_I(Q_+)$. Thus $$\nonumber
|\langle g, \frac{\partial \Phi}{\partial t} \rangle | =
|\langle g_1, \frac{\partial \Phi}{\partial t} \rangle +
\langle g_2, \frac{\partial \Phi}{\partial t} \rangle |$$ $$\leq C \left( \|g_1\|_{ B_{\cdot,0}^{1,1/2}(Q_+)} +
\| g_2\|_{B_I(Q_+)} \right) \| \Phi \|_{B_{0,\cdot}^{1,1/2}(Q_+)};\;
\Phi \in \mathcal{F}_{0,0}(Q_+).$$ Since, by Lemma \[lem:denseimbed\] and Theorem \[th:densetestfunc\], $\mathcal F_{0,0}(Q_+)$ is dense in $B_{0,\cdot}^{1,1/2}(Q_+)$, it is clear that we have a unique extension. If the function $A(\cdot,\cdot,\cdot)$ satisfies the structural conditions 1–5 given above, then also $A(\cdot,\cdot,\cdot + g)$, with $g \in X^{1,1/2}(Q_+)$, satisfies the same structural conditions (with new constants $\lambda , \Lambda$ and functions $H,h$ depending on $g$). Thus Theorem \[th:homognonlin\], and the remark following Theorem \[th:homognonlin\], tell us that (\[eq:proofhomnonlinear\]) has a unique solution. This implies that $u=w+g$ is the unique solution to (\[eq:proofnonlinear\]). $\Box$
[**Remark.**]{} Note that since $\mathcal D(Q_+)$ is densely continuously imbedded in $\mathcal F_{0,0}(Q_+)$ it is equivalent to demand that (\[eq:proofnonlinear\]) should hold in $\mathcal D'(Q_+)$.
We shall conclude with a comment on the linear case.
The function spaces we have introduced so far coincides with well known function spaces existing in the literature when $p=2$. When $p=2$ we shall follow existing notation and replace $B$ with $H$ for all spaces (for instance if $p=2$ we shall write $H^{1,1/2}_{0,\cdot}(Q_+)$ instead of $B^{1,1/2}_{0,\cdot}(Q_+)$ and so on).
The Sobolev space $H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times {\bf R}_+)$ below is defined by pull-backs in local charts on $\partial \Omega$.
\[th:lateraltrace\] If $p=2$ there exists a linear, continuous and surjective trace operator $$Tr: X^{1,1/2}(Q_+) \longrightarrow
H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times
{\bf R}_+).$$ There also exists a continuous and linear extension operator $$E: H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times {\bf R}_+) \longrightarrow
X^{1,1/2}(Q_+),$$ such that $Tr \circ E =
Id|_{H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times {\bf R})} $.
[**Proof.**]{} Using a partition of unity argument and the Fourier multiplier operators $$m_s(D)u = ((1+i2\pi\tau +4\pi^2|\xi|^2)^{-s}\hat{u})^{\vee};\quad s\in {\bf R},$$ which preserves forward support in time, and have the property that $$m_s(D)\left(L^2({\bf R}^n \times {\bf R})\right)=
H_{\cdot,\cdot}^{2s,s}({\bf R}^n \times {\bf R}),$$ we can construct continuous linear operators: $$Tr: H^{1,1/2}_{\cdot,0}(Q_+) \longrightarrow
H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times
{\bf R}_+)$$ and $$E: H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times {\bf R}_+) \longrightarrow
H^{1,1/2}_{\cdot,0}(Q_+),$$ such that $Tr \circ E =Id|_{H_{\cdot,\cdot}^{1/2,1/4}
(\partial \Omega \times {\bf R})} $. Now given $u \in X^{1,1/2}(Q_+)$, let $u=u_1+u_2$ where $u_1 \in
H^{1,1/2}_{\cdot,0}(Q_+)$ and $u_2 \in H_I(Q_+)$. We define $u|_{\partial \Omega \times {\bf R}_+} =
u_1|_{\partial \Omega \times {\bf R}_+}$. This definition is independent of the decomposition of $u$. In fact, if $u_1+u_2 =v_1+v_2$ are two decompositions as above, then $u_1-v_1 \in L^2({\bf R}_+, H_0^1(\Omega))$, and so $(u_1-v_1)|_{\partial \Omega \times {\bf R}}=0$.
Now $$\|Tr(u)\|_{H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times {\bf R}_+)} \leq
C\| u_1\|_{ H^{1,1/2}_{\cdot,0}(Q_+)},$$ for any decomposition. Taking the infimum proves the continuity of $Tr$. The continuity of the extension operator $E$ follows from the imbedding $H_{\cdot,0}^{1,1/2}(Q_+) \hookrightarrow X^{1,1/2}(Q_+)$. $\Box$
Combining our trace theorems with Theorem \[th:mainnonlinear\] gives us in the linear case:
\[th:mainlinear\] If $$Tu = \frac{\partial u}{\partial t} -\nabla_x \cdot (A(x,t,\nabla_x u)),$$ is a linear operator, satisfying the structural conditions 1–5 above, then $$\nonumber
X^{1,1/2}(Q_+) \ni u \mapsto (Tu, u|_{\partial \Omega \times {\bf R}_+},
u|_{\Omega \times \{0\}})$$ $$\in H^{1,1/2}_{0,\cdot}(Q_+)^* \times
H_{\cdot,\cdot}^{1/2,1/4}(\partial \Omega \times {\bf R}_+)\times L^2(\Omega),$$ is a linear isomorphism.
[**ACKNOWLEDGEMENTS.**]{}
I thank Johan R[å]{}de for useful remarks and stimulating discussions in connection with this work, and Anders Holst, Per-Anders Ivert and Stefan Jakobsson for reading and commenting on this paper.
[WWWWWW99]{} M. Fontes [*A Monotone Operator Method for Elliptic-Parabolic Equations*]{}, Comm. in PDE, Vol. 25, 3&4, 2000, pp. 681–702. M. Fontes: [*Initial-Boundary Value Problems for Parabolic Equations*]{} Institut Mittag-Leffler, Report no 22 (2000). S. Kaplan. [*Abstract boundary value problems for linear parabolic equations*]{}, Ann. Scoula Norm. Sup. Pisa Cl. Sci. (4) , 1966, pp. 395–419. O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva. [*Linear and quasi-linear equations of parabolic type*]{}, Transl. Math. Monogr. , Amer. Math. Soc. Providence, Rhode Island, 1968. J.L. Lions, E. Magenes. [*Problémes aux limites non homogénes et applications I–II*]{}, Dunod, Paris, 1968. R.T. Seeley [*Extension of $C^{\infty}$-functions defined in a half plane*]{}, Proc. Amer. Math. Soc. 15, 1964, pp. 625–626.
|
---
abstract: 'Results from the first lattice QCD analysis of vacuum correlators of local hadronic currents using dispersion relations are presented. We have explored the vector, pseudoscalar, axial, and scalar meson channels, and the proton-like and delta-like baryon channels. The lattice results are shown to agree qualitatively with experimental results in channels where experimental data exist, and shed insight into interacting instanton approximations and sum rule calculations in the other channels.'
address:
- |
Kellogg Laboratory, California Institute of Technology, 106-38,\
Pasadena, California 91125 U. S. A.
- |
T-8 Group, MS B-285, Los Alamos National Laboratory,\
Los Alamos, New Mexico 87545 U. S. A.
- |
FM-15, Department of Physics, University of Washington,\
Seattle, Washington 98195 U. S. A.
- |
Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics,\
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 U. S. A.
author:
- 'M.-C. Chu, J. M. Grandy, S. Huang, and J. W. Negele [^1]'
title: 'Lattice Analysis of Two-Point Hadronic Correlators in the QCD Vacuum'
---
INTRODUCTION
============
Two point correlators of local interpolating hadronic currents are a useful means of studying the QCD vacuum. These correlators have been measured experimentally for the vector and axial channels, and have been computed phenomenologically using interacting instanton approximations[@Shuryak89] and QCD sum rules[@Ioffe; @FZ81] in the remaining channels. The current experimental and phenomenological predictions of the correlators have been reviewed extensively by Shuryak[@Shuryak92]. Lattice calculations provide a means of investigating the correlators from the first principles of QCD, although at present the lattice work is in an exploratory stage. We therefore emphasize the qualitative features of the comparison between our lattice results and the predictions and demonstrate the ability of lattice calculations to refine sum rule calculations.
‘?=?
-------------- ------------------------------------------------------------ --------------------------------------------------------------------------- ----------------------------
Vector $J^V_\mu = \bar{u}\gamma_\mu d $ $\left\langle 0|T\left[ J^V_\mu(x) \bar{J}^V_\mu(0) $ {{1}\over{12\pi^2}}$
\right] |0\right\rangle $
Axial $J_\mu^A = \bar{u}\gamma_\mu \gamma_5 d $ $\left\langle 0|T\left[ J_\mu^A(x) \bar{J}_\mu^A(0) $ {{1}\over{12\pi^2}}$
\right]|0\right\rangle $
Pseudoscalar $J^P = \bar{u}\gamma_5 d $ $\left\langle 0|T\left[ J^P(x) \bar{J}^P(0) $ {{3s}\over{8\pi^2}}$
\right]|0\right\rangle $
Scalar $J^S = \bar{u} d $ $\left\langle 0|T\left[ J^S(x) \bar{J}^S(0) $ {{3s}\over{8\pi^2}}$
\right]|0\right\rangle $
Nucleon $J^N = \epsilon_{abc}(u^a C\gamma_\mu u^b) \gamma_\mu ${{1}\over{4}}\,{\rm Tr}\, \left(\left\langle 0| $ {{s^2}\over{64\pi^4}}$
\gamma_5 d^c $ T\left[ J^N(x) \bar{J}^N(0)\right]|0\right\rangle
x_\nu \gamma^\nu \right)$
Delta $J^\Delta_\mu = \epsilon_{abc} (u^a C\gamma_\mu u^b) u^c $ ${{1}\over{4}}\,{\rm Tr}\, \left(\left\langle 0| $ {{3s^2}\over{256\pi^4}}$
T\left[ J^\Delta_\mu(x) \bar{J}^\Delta_\mu(0) \right]|0
\right\rangle x_\nu \gamma^\nu \right)$
-------------- ------------------------------------------------------------ --------------------------------------------------------------------------- ----------------------------
The hadronic currents we use and their correlators are listed in Table 1. The extra $(x_\nu \gamma^\nu)$ factor for the baryons selects the part which is stable in the chiral limit. In the next section we briefly review the existing predictions of the correlators. Section 3 describes the method of our lattice calculations. Finally, we present our lattice results and describe the comparisons with predictions.
CORRELATOR PREDICTIONS
======================
The correlator in the vector channel is determined directly from hadron production in $e^+e^-$ annihilation experiments. The axial correlator is also known experimentally from the isospin-violating $\tau \rightarrow 3\pi$ reaction, although the direct experimental determination is limited to $Q^2 < 1.5\,{\rm GeV}^2$ due to the mass of the $\tau$. In the pseudoscalar channel the correlator is fit phenomenologically using the dispersion form described below. For the scalar, the experimental data is sketchy, so the best means of predicting the correlator is by using the interacting instanton approximation (AII)[@Shuryak89]. For the baryons, the threshold and current coupling parameters obtained from sum rule calculations[@Ioffe; @FZ81] are entered into the dispersion relations to obtain predicted correlations.
The form of the correlator in momentum space, $\hat R(q) = \int d^4x\,
e^{iqx} R(x)$, is given by the standard dispersion relation[@ours] $$\hat R(q)
= \left\{\matrix{ 1\cr 3q^2\cr-iq^\mu\gamma_\mu\cr}\right\}
\left( \int ds {f(s)\over s-q^2} + \ldots \right)$$ where contact terms and terms with other Dirac structures have been omitted. The top term in the brackets is used for the pseudoscalar and scalar channels, the middle term for the vector and axial, and the bottom term for the baryons. The physical spectral density $f(s)$ is phenomenologically approximated using a resonance contribution from the ground state $f_r(s) =
\lambda^2\delta(s-M^2)$ where $M$ is the resonance mass and $\lambda$ is the current coupling to the resonance state, and a contribution from a continuum of excited states above the threshold energy $s_0$, $f_c(s) = f_p(s) \theta(s-s_0)$. We use asymptotic freedom to approximate the perturbative correlators with the corresponding free correlators in Table 1. In position space the correlators are given by $R(x) = R_r(x) + R_c(x)$ where $$R_r(x)
= \lambda^2 M g_r(x) K_p(Mx)$$ $$R_c(x) = \int^{\infty}_{s_0} ds\ f_p(s)\, g_c(s,x) K_p(\sqrt{s}\,x) \, .$$ The functions $g_r$ and $g_c$ are listed in Table 2. The Bessel function orders are $p=1$ for mesons and $p=2$ for baryons. This form for $R(x)$ is the basis for phenomenological fits by Shuryak[@Shuryak92] and also our fits to lattice data.
‘?=?
------------ -------------- -------------------
V,A $3M^2x^{-1}$ $3s^{3/2} x^{-1}$
P,S $x^{-1}$ $s^{1/2} x^{-1}$
N,$\Delta$ $M$ $s$
------------ -------------- -------------------
: Correlator Fitting Functions[]{data-label="tab:fitfns"}
LATTICE CORRELATORS
===================
Background
----------
We compute lattice correlators on $16$ independent $16^3\times 24$ configurations at $\beta=5.7$, with a lattice spacing normalized by the proton mass of $a=.168\, {\rm fm}$. Hard wall boundary conditions are imposed at the end time slices with periodic boundary conditions in the three spatial directions. Propagators, generated using a localized source in the central time slice, have been previously computed by Soni [*et al.*]{}[@Soni]. Free particle correlators, $R_0(x)$, are calculated analytically on a very large lattice, $(48)^4$, to eliminate boundary effects at separations below $2\,$ fm. In practice, we compare $R(x)/R_{0}(x)$ on the lattice with predictions of the same, in order to more closely observe the onset of asymptotic freedom and the effects of quark interactions.
Lattice Artifacts
-----------------
=3.0in =2.9in
It is necessary to account for lattice artifacts when extracting physical results from the computed quantities. First, periodic boundary conditions introduce leakage between Brillouin zones surrounding the point sources. This leakage is corrected by subtracting contributions from image sources from the lattice data, and self-consistently fitting a curve through the corrected points. The Cartesian lattice introduces artificial directional anisotropy. To counteract this we select only points near the body diagonal, $\hat
d= {{1}\over{\sqrt{3}}} (1,1,1)$, so that $\hat x \cdot \hat d \ge
0.9$. In this direction, the free propagator agrees closely with the continuum result, and we thus believe that the interacting results are most reliable in this direction. In our current phase we use separations $x$ within the central time slice to avoid contamination from the hard wall, and we plan in the future to include a few slices from the center. For the purposes of fitting, we consider $R(x)$ as a function of the magnitude of the separation and combine lattice separations in one-lattice-unit bins. It is desirable to normalize the ratio $R(x)/R_{0}(x)$ to unity at an infinitesimal separation but we actually normalize at our smallest separation, $\sqrt{3}a =
0.29\,{\rm fm}$.
=3.8in =2.9in
We also must contend with the inability to compute propagators at the physical pion mass. We compute the correlators at quark masses of $m_q =\,$ 351, 199, 110, 67, and 25 MeV and extrapolate the correlators at the lightest four quark masses using a best quadratic fit to the quark mass $m_q=8\,{\rm MeV}$ which corresponds to the physical pion mass of $140 \,{\rm MeV}$. In the pseudoscalar channel which becomes infinite in the chiral limit the logarithm of the correlator is extrapolated. As an example, the extrapolation of the pseudoscalar is explicitly plotted (Fig. 1). In the other channels only the extrapolated results are shown.
=3.8in =2.9in
RESULTS
=======
In the vector channel, it is remarkable[@Shuryak92] that the ratio $R(x) \over R_{0}(x)$ remains close to 1 over the range of separations plotted, although $R(x)$ falls by several orders of magnitude. The lattice calculation (Fig. 2a) is consistent with this result, and has similar features to the experimental result. In the axial channel (Fig. 2b) the lattice data are fitted with the continuum distribution $R_c(x)$ only since we cannot resolve a resonance contribution from the continuum with our data. Again, the lattice result is qualitatively similar to the phenomenology but the rising tail, due to pion mixing, is difficult to reproduce. In the pseudoscalar channel the extrapolation is shown explicitly, and the extrapolated result at the physical pion mass is close to Shuryak’s fit. The scalar channel, not plotted here, is difficult to extrapolate and subject to large statistical errors. As in the axial channel we see no clear resonance contribution for the scalar.
There are no direct experimental determinations of the baryon correlators, so we compare in figure 3 the lattice results with the dispersion fits based on sum rule calculations by Farrar [*et al.*]{} [@FZ81] (dotdash) and Ioffe[@Ioffe] (dashes). The good dispersion fits to the baryon data and the wide disparity between the sum rule calculations suggest that future lattice results can be used to refine sum rule calculations.
This exploratory calculation of two-point hadronic correlators produces qualitative agreement with available experimental results and phenomenological fits in most channels. This agreement motivates further, improved lattice calculations which can be used in tandem with previously established methods to understand the behavior of hadronic correlators in the QCD vacuum.
[9]{} E. Shuryak, [*Nucl. Phys.*]{} [**B328**]{}, 102 (1989).
B. L. Ioffe, [*Nucl. Phys.*]{} [**B188**]{}, 317 (1981); V. M. Belyaev and B. L. Ioffe, [*Sov. Phys. JETP*]{} [**83**]{}, 976 (1982). G. Farrar, H. Zhoang, A. A. Ogloblin and I. R. Zhitnitsky, [*Nucl. Phys.*]{} [**B311**]{}, 585 (1981). E. Shuryak, Stony Brook preprint SUNY-NTG-91/45, to appear in [*Rev. Mod. Phys.*]{} (1992). M.-C. Chu, J. M. Grandy, S. Huang, and J. W. Negele, MIT Preprint CTP\#2113 (1992) (hep-lat 9208030). A. Soni, [*National Energy Research Supercomputer Center Buffer*]{} [**14**]{}, 23 (1990). Relevant details are presented in C. Bernard, T. Draper, G. Hockney and A. Soni, [*Phys. Rev.*]{} [**D38**]{}, 3540 (1988).
[^1]: This work is supported in part by funds provided by the U. S. Department of Energy (D. O. E.) under contracts \# DE-AC02-76ER03069 and \# DE-FG06-88ER40427, and the National Science Foundation under grant \# PHY 88-17296.
|
---
abstract: 'We are conducting a multi-wavelength (radio, optical, and X-ray) observational campaign to classify, morphologically and physically, a sample of 55 flat-spectrum radio sources dominated by structure on kpc-scales. This sample contains 22 compact-/medium-sized symmetric object candidates, a class of objects thought to be the early stages of the evolution of radio galaxies. The vast majority of the remaining objects have core-plus-one-sided-jet structures, half of which present sharply bent jets, probably due to strong interactions with the interstellar medium of the host galaxies. Once the observational campaign is completed, we will constrain evolutionary theories of radio galaxies at their intermediate stages and possibly understand the physics of the hypothesized narrow line region in active galactic nuclei, given our advantageous statistical position.'
address:
- |
Universidade da Madeira, Centro de Ciências Matemáticas,\
Caminho da Penteada, 9000 Funchal, Portugal
- |
Instituto de Física de Cantabria (CSIC-Universidad de Cantabria),\
Facultad de Ciencias, 39005 Santander, Spain
- 'University of Durham, Dep. of Physics, South Road, Durham DH1 3CE, UK'
- |
University of Ioannina, Section of Astro-Geophysics, Dep. of Physics,\
45110 Ioannina, Greece
- 'University of Manchester, Nuffield Radio Astronomy Laboratories, Jodrell Bank, Macclesfield, Cheshire SK11 9DL, UK'
- 'Instituto de Astrofísica de Canarias, c/ Via Láctea s/n, 38200 La Laguna, Tenerife, Spain'
author:
- 'P. Augusto'
- 'J.I. Gonzalez-Serrano'
- 'A.C. Edge'
- 'N.A.B. Gizani'
- 'P.N. Wilkinson'
- 'I. Perez-Fournon'
title: 'The kpc-scale radio source population'
---
radio continuum: general ,galaxies: active, evolution, ISM, jets ,quasars: general 98.54.Aj ,98.54.Gr ,98.62.Lv ,98.62.Nx
Introduction {#intro}
============
@Augetal98 conducted the first systematic search for flat-spectrum radio sources with dominant structure on 90–300 mas angular scales (0.2–2 kpc linear scales at $z>0.2$): gravitational lenses and compact-/medium-sized symmetric objects (CSO/MSOs), in particular. The selected sample of 55 such radio sources is described in Section \[sample\]. In Section \[RGqua\] we discuss results, from this sample, pertaining to CSO/MSOs. These are symmetric double or triple sources, with sizes smaller than 15 kpc (e.g., @Reaetal96a [@Reaetal96b]). CSOs ($<1$ kpc; aged $10^3$–$10^4$ years) and MSOs (1–15 kpc; aged $10^5$–$10^6$ years) present compact lobes ($<20$ mas) having, overall, $\alpha < 0.75$ ($S_{\nu} \propto \nu^{-\alpha}$), and are probably the precursors of the large radio galaxies which they resemble. VLBI surveys have unveiled a significant population of eighteen 0.01–0.1 kpc CSOs, which constitute $\sim6\%$ of a complete flux-limited sample of 293 flat-spectrum radio sources (CJF; @Tayetal96a [@Tayetal96b]). In the last section, we mention the potential of our sample in terms of understanding the hypothesized kpc-sized narrow-line region (NLR) in active galactic nuclei (AGN; e.g., @Rob96).
The Sample {#sample}
==========
Starting from the total of $\sim4800$ sources in the Jodrell-VLA Astrometric Survey (e.g., @Patetal92) and the first part of the Cosmic Lens All Sky Survey (e.g., @Broetal98), @Augetal98 have first established a parent sample containing 1665 strong ($S_{\rm 8.4\: GHz}>100$ mJy), flat-spectrum ($\alpha_{1.4}^{4.85}<0.5$) radio sources. From this sample, 55 sources were selected in accordance with an extra resolution criterion as described in @Augetal98. The completeness of this latter sample depends on both the separation and the flux density ratio of the components of each radio source in the parent sample. Unresolved single-component sources would be rejected.
With regard to the spectral properties of the 55-source sample [@Augetal98], 45 sources have power-law radio spectra down to the lowest measured frequency (which is 365 MHz for 31 of the objects and 151 MHz for 14 of them), 3 sources present complex spectra, and 7 have spectra peaked at $\sim300$ MHz. It is relevant that only two of the fourteen 0.2–1 kpc CSO candidates in Table \[CSOMSO\] can be classified as GHz-Peaked Spectrum Sources (GPSs), peaking at $\sim0.5$–10 GHz. From the same table, only three CSO candidates have a peak at $\sim300$ MHz. Hence, the statement “every CSO is a GPS source” [@Bicetal97] seems incorrect.
There are two main populations of radio sources uncovered on kpc scales by @Augetal98. These consist of 22 CSO/MSOs and 30 core-plus-one-sided-jet (CJ) sources.
It is unfortunate that, for the vast majority of the 55 sources, information on any optical counterparts comes only from the Palomar Observatory Sky Survey (POSS) plates. Using POSS identifications, we have compared [@Augetal98] the abundance of blue stellar objects, red stellar objects, galaxies, and empty fields for the 55-source and the parent 1665-source samples. We have found that the fraction of blue stellar objects in the 55-source sample is half of the fraction of such objects in the 1665-source sample. Furthermore, the fraction of galaxies is three times larger in the 55-source sample. For the other two identification types examined, the results are comparable in both samples (one-third are empty fields and one-eighth are red stellar objects).
Thus, it seems that selecting kpc structure in the radio leads to selecting structure in the optical. There seems to exist a global bias against radio/optical unresolved sources. As regards redshift information, we have $<\!\!z\!\!> \: \sim 0.7$ for the 19 out of the 55 sources that have spectroscopic data. The faintest sources (namely, the 18 sources that correspond to POSS empty fields) still need redshift determinations, suggesting that the average redshift of the sample will increase. Unfortunately for the discussion on this paper, very few of the CSO/MSOs have measured redshifts (Table \[CSOMSO\]).
Compact/Medium Symmetric Objects {#RGqua}
================================
The sample of 22 CSO/MSO candidates found by @Augetal98 — Table \[CSOMSO\] — contains 9 certain CSOs and two certain MSOs. Most likely, six of the remaining sources are MSOs, leaving five sources that could be either. The fact that for sources at $z>0.2$ we have selected the ones dominated by structures on 0.2–2 kpc scales suggests a bias against the population of MSOs within our sample. Since most flat spectral-index sources will consist of a pair of compact lobes (plus, possibly, a core), they will be included in our sample only if their sizes are $\leq0\rlap{.}''3$. Much larger sources (like B0824+355 in Table \[CSOMSO\]), consisting of very weak jets and low surface brightness extended lobes, are probably the exception. We believe that most MSOs in our sample will have sizes of $\sim$1–2 kpc, much like the confirmed MSO B0205+722 (Table \[CSOMSO\]). Note that even if we allow for sources with $z<0.2$, this will only favour the increase in number of small MSOs, since for the same angular dimensions seen, a lower redshift will translate into a smaller linear size.
@Augetal98 have shown that the 55-source sample includes every CSO from the 1665-source parent sample having a 160–300 mas separation (0.4–1 kpc for sources at $z>0.2$) between compact components with a flux-density ratio of 7:1 or smaller. CSOs containing compact lobes with similar flux-density ratios are included in the sample down to a separation of 90 mas (0.2–0.6 kpc at $z>0.2$). The key issue now is to review evidence for why virtually all CSOs present in the 1665-source parent sample are at most the 14 found by @Augetal98 among their 55-source sample. Typically, CSOs have weak cores (weaker than any of the lobes) and, hence, it is the lobes that are the ‘components’ that will go through the selection criterion of @Augetal98. It is very rare to find a CSO with lobes presenting flux density ratios greater than 7. In fact, there are not any of these cases among the 0.2–1 kpc CSOs in Table \[CSOMSO\] or the eighteen 0.01–0.1 kpc CJF CSOs discussed here. Therefore, we believe that @Augetal98 have selected virtually all of the CSOs present in the 1665-source parent sample; these are shown in Table \[CSOMSO\]. In any case, for the discussion of this paper, we performed simulations (see below), which estimate the effects of the ‘resolution’ criterion on the CJF sample, before making any comparison between our CSO-fraction and that of the CJF. The simulations give results that are consistent with the ‘typical’ morphology of CSOs just presented.
Conservatively, taking a maximal number of 0.2–1 kpc CSOs as 14 (Table \[CSOMSO\], including the sources classified as ‘question marks’) out of a parent sample containing 1665 sources, only $\sim0.8\%$ of flux-limited samples seem to be such CSOs. It seems, then, that these are six times less common than 0.01–0.1 kpc CSOs (which constitute $\sim6\%$ of CJF). Both the 0.2–1 kpc and the 0.01–0.1 kpc CSOs are dominated by components $<20$ mas in size. Is the number difference due to luminosity evolution alone? Strong luminosity evolution takes place during the time that the 0.01-kpc scale CSOs grow to be 100-kpc scale radio sources (e.g., @Reaetal96a [@Reaetal96b]). @KaiAle97 have proposed a model in which the luminosity of double sources decreases proportionally to the square root of their size. If this relation applies continuously as the source evolves from the 0.01-kpc to the 100-kpc scale, then in the evolution from a 0.01–0.1 kpc to a 0.2–1 kpc CSO, size increases by a factor of $\sim10$ and hence the luminosity decreases by $\sim3$. Given that all CJF sources have $S_{\rm 5 \: GHz} > 350$ mJy and, like the 55-source sample, have $\alpha_{1.4}^{4.85}<0.5$, our flux-density criterion $S_{\rm 8.4 \: GHz} > 100$ mJy allows a sampling $\sim3$ times fainter, cancelling out the predicted luminosity evolution. Before rushing to other evolutionary explanations, we note that our selection process included a resolution criterion not present in CJF. Hence, we need to find out how many of the 18 CJF CSOs would remain in the CJF if it had an equivalent resolution criterion. The simplest way to do this is to use models fitted to the 18 CJF CSOs, expand the separation of the components by a factor of 10, and check whether they meet the criteria for inclusion in our sample. This will only give indicative results, of course. The models and maps are found in the literature from the VLBI surveys, except for three models that we crudely produced from the available maps.
Using the program [FAKE]{} in the Caltech VLBI package [@Pea91], we have performed a test for the reliability of selection (details in @Augetal98). Eleven out of the ‘order-of-magnitude-expanded’ 18 CJF 0.01–0.1 kpc CSOs would be in our sample. To contemplate the possibility that some of our 0.2–1 kpc CSOs might have been selected by a lucky combination of observational conditions, we also ran [FAKE]{} on the 14 such CSOs in Table \[CSOMSO\]. All of them are reliably in our sample.
The revised frequencies of CSOs are then $\sim0.8\%$ (14/1665) in our sample and $\sim4\%$ (11/293) in CJF. Since five of the fourteen 0.2–1 kpc CSO candidates in Table \[CSOMSO\] could be $>$1 kpc MSOs, a conservative factor of $\sim5$ still remains between the abundance of CSOs in both samples. To explain this difference, we suggest evolution of the lobes in CSOs as they grow — self-similar growth of radio galaxies: the lobes start off as compact hot spots when 0.01–0.1 kpc apart and expand until they grow $\sim100$ kpc apart, as in normal radio galaxies. The number of 0.2–1 kpc young radio galaxies seems to be less than the number with sizes 0.01–0.1 kpc due to the resolution criterion used to select the 55-source sample in @Augetal98: only double (or triple) sources with compact ($<20$ mas) components are in the sample. The extended lobes of Compact Steep Spectrum ($\alpha>0.5$) radio sources, the dominant radio sources on 0.2–1 kpc scales, cannot be selected by the resolution criterion of @Augetal98.
---------- ------ ------------- ---------------- --------------- --------
Linear size Classification POSS id. z
(kpc)
0046+316 300 0.09 CSO G; 15$^m$ 0.015
0112+518 650 (MSO) EF
0116+319 75 0.08 CSO G; 16$^m$ 0.0592
0205+722 600 3 MSO G; 18$^m$ 0.895
0225+187 225 ? EF
0233+434 120 CSO EF
0352+825 44 CSO G; 15.5$^m$
0638+357 400 (MSO) EF
0732+237 175 CSO EF
0817+710 225 ? EF
0819+082 275 ? RSO; 19.3$^m$
0824+355 2000 11 MSO RSO; 19.6$^m$ 2.249
1010+287 75 CSO EF
1058+245 900 (MSO) EF
1212+177 100 CSO RSO; 20$^m$
1233+539 240 ? BSO; 19$^m$
1504+105 110 CSO ?; 16$^m$
1628+216 800 (MSO) ?
1801+036 1200 (MSO) G; 17$^m$
1928+681 120 CSO BSO; 20.5$^m$
1947+677 500 (MSO) EF
2345+113 275 ? G; 19$^m$
---------- ------ ------------- ---------------- --------------- --------
: The 22 compact/medium symmetric object (CSO/MSO) candidates found in the 55-source sample of @Augetal98. The linear size is calculated using $H_{0}$=75 km s$^{-1}$ Mpc$^{-1}$ and $q_0=0.5$. For the vast majority of sources, without redshift information, formal classification is not possible. Nevertheless, independent of redshift, any object smaller than 175 mas is a CSO. Furthermore, assuming $z>0.2$ for the remaining objects, angular sizes larger than 350 mas identify MSOs. Palomar Observatory Sky Survey (POSS) identifications are with galaxies (G), empty fields (EF), and blue or red stellar objects (BSO,RSO).[]{data-label="CSOMSO"}
Future {#RGqua.NLR}
======
Once redshifts are determined for the remaining 36 of the 55 sources, we will not only classify CSO/MSOs correctly, according to their sizes, but also determine the linear (projected) sizes of the CJs. Most of these CJs might also show evidence for strong shocks in the NLR. Half of the CJs in the 55-source sample contain sharply bent jets that bend by more than $90^{\circ}$, in some cases more than once. This hints at strong interactions with the interstellar medium of the host galaxies. Altogether, the CSOs, MSOs, and CJs in our sample will give us clues about the composition and density of the NLR in galaxies because of their interactions with the NLRs of their hosts. Due to our good statistics, this might be a useful step forward towards understanding the standard model of AGN as a whole, and the NLR in particular.
[999]{} Augusto, P., Wilkinson, P.N., & Browne, I.W.A., 1998, MNRAS, 299, 1159.
Bicknell, G.V., Dopita, M.A., & O’Dea, P.O, 1997, ApJ, 485, 112.
Browne, I.W.A. et al., 1998, in: Bremer, M., Jackson, N. & Perez-Fournon, I. (eds), Observational Cosmology with the New Radio Surveys, Kluwer Academic, Dordrecht, p. 305.
Kaiser, C.R. & Alexander, P., 1997, MNRAS, 286, 215.
Patnaik, A.R., Browne, I.W.A., Wilkinson, P.N., & Wrobel, J.M., 1992, MNRAS, 254, 655.
Pearson, T.J., 1991, BAAS, 23, 991.
Readhead, A.C.S., Taylor, G.B., Xu, W., Pearson, T.J., Wilkinson, P.N., & Polatidis, A.G., 1996a, ApJ, 460, 612.
Readhead, A.C.S., Taylor, G.B., Pearson, T.J., & Wilkinson, P.N., 1996b, ApJ, 460, 634.
Robson, I., 1996, Active Galactic Nuclei, John Wiley and Sons, Chichester, England.
Taylor, G.B., Vermeulen, R.C., Readhead, A.C.S., Pearson, T.J., Henstock, D.R., & Wilkinson, P.N., 1996a, ApJS, 107, 37.
Taylor, G.B., Vermeulen, R.C., Readhead, A.C.S., Pearson, T.J., Henstock, D.R., & Wilkinson, P.N., 1996b, in: Snellen, I., Schilizzi, R.T., Rottgering, H.J.A., & Bremer, M.N. (eds), 2$^{\rm nd}$ Workshop on Gigahertz Peaked Spectrum and Compact Steep Spectrum Radio Sources, Univ. Leiden.
|
---
abstract: 'A brief review is given of some recent works where baryogenesis and dark matter have a common origin within the $U(1)$ extensions of the standard model and of the minimal supersymmetric standard model. The models considered generate the desired baryon asymmetry and the dark matter to baryon ratio. In one model all of the fundamental interactions do not violate lepton number, and the total $B-L$ in the Universe vanishes. In addition, one may also generate a normal hierarchy of neutrino masses and mixings in conformity with the current data. Specifically one can accommodate $\theta_{13}\sim 9^{\circ}$ consistent with the data from Daya Bay reactor neutrino experiment.'
address: |
Department of Physics, Northeastern University, Boston, MA 02115-5000, USA\
w.feng@northeastern.edu, p.nath@neu.edu
author:
- 'Wan-Zhe Feng, Pran Nath'
title: 'Baryogenesis and Dark Matter in $U(1)$ Extensions'
---
Introduction
============
Three of the important puzzles in cosmology relate to the origin of baryon asymmetry in the Universe, the nature of dark matter and the cosmic coincidence that the amount of dark matter and visible matter are comparable. The fact that dark matter and visible matter are comparable in size points to the possibility of a common origin of the two. Here we discuss classes of models where baryon asymmetry and dark matter have a common origin within the framework of $U(1)$ extensions of the standard model (SM) and of the minimal supersymmetric standard model (MSSM) [@Feng:2013wn; @Feng:2013zda; @Feng:2012jn].
The basic tenets of generating matter over anti-matter are well-known and consist of three conditions [@Sakharov:1967dj]: the existence of baryon (or lepton) number violation, the presence of C and CP violating interactions, and out of equilibrium processes. One suggestion for explaining the comparable size of dark matter and visible matter is the so-called asymmetric dark matter hypothesis [@Kaplan:2009ag] where the dark particles are in thermal equilibrium with the SM (MSSM) particles in the early universe, and thus their chemical potentials are of the same order. The satisfaction of dark matter and visible matter ratio ${\Omega_{\rm DM} }/{ \Omega_{\rm B}}\approx 5.5$ [@Ade:2013sjv] can then be achieved via a constraint on the dark matter mass (for reviews see [@review]). More specifically, the asymmetry can transfer from the visible sector to the dark sector via the asymmetry transfer interaction $\mathcal{L}_{\rm asy} = \frac{1}{M^n_{\rm asy}}\mathcal{O}_{\rm DM} \mathcal{O}_{\rm asy}$ [@Kaplan:2009ag], where $M_{\rm asy}$ is the scale of the interaction, $\mathcal{O}_{{\rm asy}}$ is an operator constructed from SM (MSSM) fields which carries a non-vanishing $B-L$ quantum number while $\mathcal{O}_{{\rm DM}}$ carries the opposite $B-L$ quantum number. This interaction would decouple at some temperature greater than the dark matter mass. As the Universe cools down, the dark matter asymmetry freezes at the order of the baryon asymmetry, which explains the observed relation between the amount of baryon and dark matter. In [@Feng:2012jn] we discussed asymmetric dark matter in the $U(1)_{L_\mu - L_\tau}$ and $U(1)_{B-L}$ Stueckelberg extensions of the SM and of MSSM [@Stueckelberg]. In what follows we discuss two model classes where baryon asymmetry and dark matter have a common origin (for related works see [@DtoV; @CoG]). For the first model class, dark matter is generated via the decay of some primordial fields and the asymmetry created by the CP violating decays is then transferred to the visible sector via the asymmetry transfer interaction [@Feng:2013wn]. In the second model class, leptogenesis takes place with all the fundamental interactions conserving lepton number and leptogenesis consists in generating equal and opposite lepton numbers in the visible and dark sectors [@Feng:2013zda]. Subsequently the sphaleron processes transmute a part of the lepton asymmetry into baryon asymmetry. In this model class the total $B-L$ number in the Universe is exactly conserved. In the model classes referred to above the stability of dark matter is protected by the $U(1)$ gauge symmetry. A kinetic mixing between the $U(1)$ and $U(1)_Y$ gauge bosons allows for dissipation of the symmetric component of dark matter through the exchange of the $U(1)$ gauge boson. An alternative way of depleting the symmetric component of dark matter is assuming that the $U(1)$ gauge boson is massless (dark photon). Majorana mass terms for dark particles are forbidden. Consequently, the dark matter asymmetry generated in the early universe would not be washed out by oscillations.
Baryogenesis from Dark Sector
=============================
We first discuss the model class where primordial fields decay into dark matter and create an asymmetry. The dark matter asymmetry then transmutes into lepton and baryon asymmetries.
The model
---------
Here we work in a supersymmetric framework.[^1] We assume that in the early universe there exist several $\hat{N}_i$ fields ($i \geq 2$) with masses $M_i$, where $\hat N=(N, \tilde{N})$ and $N$ is the Majorana field and $\tilde N$ is the super-partner field. The scalar field of the lightest $\hat{N}_i$ superfields could play the role of the inflaton, and $\hat{N}_i$ can also be right-handed neutrinos as suggested in earlier works. The dark sector is comprised of $(\hat X, \hat{X}^c, \hat X', \hat{X}'^c)$ which are charged under the gauge group $U(1)_x$ with charges $(+1,-1,-1,+1)$ while the MSSM fields are not charged under $U(1)_x$. We assume the $\hat{N}_i$ carry a non-vanishing lepton number $+2$, $\hat X, \hat X'$ carry lepton number $-1$ and $\hat{X}^c, \hat{X}'^c$ carry lepton number $+1$. The superpotential of the model is given by $$W=
\lambda_i \hat{N}_i \hat{X} \hat{X}' + \frac{1}{M_{\rm asy}^2} \hat{X}\hat{X}' (L H_u)^2
+ m \hat X \hat{X}^c + m' \hat X' \hat{X}'^c\,,
\label{Wfull}$$ where the couplings $\lambda_i$ are assumed to be complex. $W$ is invariant under both $U(1)_x$ and lepton number, and the first term is responsible for generating an asymmetry in the dark sector whereas the second term is responsible for transferring the asymmetry generated in the dark sector to the visible sector. Finally we add mass terms for $\hat{N}_i$ to the superpotential, i.e., a term $W \sim \frac{1}{2} M_i \hat{N}_i \hat{N}_i$, which violates lepton number.
We assume the mass hierarchy $M_i \gg m+ m'$ so that in the early universe, and the out-of-equilibrium decays of $\hat{N}_i$ generates dark matter through $N_i \to X \tilde X', \tilde X X', \bar{X} \tilde X'^*, \tilde X^* \bar{X}'$ and $\tilde{N}_i \to X X', \bar X \bar X'$. Further, the CP violation due to the complex couplings $\lambda_i$ generates an excess of $X,X'$ over their anti-particles $\bar X, \bar X'$ carrying the opposite lepton numbers. Thus the decays of $N_i$ produce a lepton number asymmetry in the dark sector. The lepton asymmetry generated in this fashion in the dark sector is then transferred to the visible sector through the asymmetry transfer interaction, and thus leptogenesis occurs. Finally, a part of lepton number asymmetry of the visible sector then transmutes to baryon number asymmetry via the sphaleron interactions. In the simplest model we have $i=2$, and we assume $\hat{N}_2$ mass $M_2$ is much larger than $\hat{N}_1$ mass $M_1$.
![Loop diagrams responsible for the genesis of dark matter asymmetry from the decay of $N_1$ to final states $X\tilde X'$ and there are similar diagrams for the decay of the $N_1$ to the final states $\tilde X X'$, and for the decay of $\tilde N_1$ to $X X'$ and to $\tilde{X} \tilde{X}'$.[]{data-label="DAsy"}](FDNXXP.pdf)
The dark matter asymmetry arises from the interference of the one-loop diagrams shown in Fig. \[DAsy\] with the tree-level diagrams, similar to the conventional leptogenesis diagrams [@Early]. The asymmetries, i.e., the excess of $\hat{X},\hat{X}'$ over their anti-particles $\overline{\hat{X}},\overline{\hat{X}'}$ are measured by $\epsilon_{X\tilde{X}'}, \epsilon_{\tilde{X}X'}, \epsilon_{XX'}, \epsilon_{\tilde{X} \tilde{X}'}$ [@Feng:2013wn] where the lower indices of $\epsilon$ denote the final state particles. There are two types of loops involved: vertex contribution and wave contribution as shown in Fig \[DAsy\]. It’s straight forward to compute the above asymmetry parameters $\epsilon_{X\tilde{X}'}$ etc. It turns out that the contributions of the vertex diagrams and the wave diagrams satisfy the following relations $$\begin{gathered}
\epsilon_{X\tilde{X}'}^{vertex}=\epsilon_{\tilde{X}X'}^{vertex}=\epsilon_{XX'}^{vertex}=\epsilon_{\tilde{X}\tilde{X}'}^{vertex}\equiv\epsilon^{vertex}\,,\\
\epsilon_{X\tilde{X}'}^{wave}=\epsilon_{\tilde{X}X'}^{wave}=\epsilon_{XX'}^{wave}=\epsilon_{\tilde{X}\tilde{X}'}^{wave}\equiv\epsilon^{wave}\,.\end{gathered}$$ Specifically, we have $$\begin{aligned}
\label{vw1}
\epsilon^{vertex} & =-\frac{1}{8\pi}\frac{{\rm Im}(\lambda_{1}^{2}\lambda_{2}^{*2})}{|\lambda_{1}|^{2}}\frac{M_{2}}{M_{1}}\ln\frac{M_{1}^{2}+M_{2}^{2}}{M_{2}^{2}}\,,\\
\epsilon^{wave} & =-\frac{1}{8\pi}\frac{{\rm Im}(\lambda_{1}^{2}\lambda_{2}^{*2})}{|\lambda_{1}|^{2}}\frac{M_{1}(M_{1}+M_{2})}{M_{2}^{2}-M_{1}^{2}}\,.
\label{vw2}\end{aligned}$$ Thus the total asymmetry parameter is the sum of the vertex and the wave contributions and in the limit $M_{2}\gg M_{1}$, we obtain $$\epsilon=\epsilon^{vertex}+\epsilon^{wave}\approx-\frac{1}{4\pi}\frac{{\rm Im}(\lambda_{1}^{2}\lambda_{2}^{*2})}{|\lambda_{1}|^{2}}\frac{M_{1}}{M_{2}}\,.
\label{epsi}$$
The total excess of $X,\tilde{X},X',\tilde{X'}$ over $\bar{X},\tilde{X}^{*},\bar{X}',\tilde{X'}^{*}$ generated by the decay of $\hat{N}_1$ is given by $\Delta n_X \approx 2\kappa s \epsilon \big/ g_*$, where $s$ is the entropy, $g_* \approx 228.75$ is the entropy degrees of freedom for MSSM, and $\kappa$ is a washout factor due to inverse processes $X+\tilde X', \tilde X+ X' \to N$ and $X+ X', \tilde X+ \tilde X' \to \tilde N$ and in our analysis we set $\kappa =0.1$. The excess of $\hat{X}, \hat{X}'$ then give rise to a non-vanishing $(B-L)$-number in the early universe: $(B-L)_{\rm t} = (+1)\times \Delta n_X \approx 2\kappa s \epsilon \big/ g_*$, where $(B-L)_{\rm t}$ is the total $B-L$ in the Universe and $+1$ indicates each of $X, X'$ carries a $B-L$ number $+1$.
The $B-L$ asymmetry generated in the visible sector through the asymmetry transfer interaction can be obtained by using the standard thermal equilibrium method introduced in [@Harvey:1990qw]. For very high temperatures the MSSM fields are ultra-relativistic, hence MSSM fields and dark particles are in thermal equilibrium, which gives rise to relations among their chemical potentials [@Harvey:1990qw; @Feng:2012jn]. These relations allow us to express the chemical potentials of all the MSSM fields in terms of the chemical potential of one single field, e.g., $\mu_{L}$, the chemical potential of the left-handed lepton doublet. Similarly other quantities of interest, i.e., the total lepton number $L$, the total baryon number $B$, and the net $B-L$ in the visible sector can all be expressed in terms of $\mu_{L}$. Specifically we have $(B-L)_{\rm v} =-\tfrac{237}{7}\mu_L$, where $(B-L)_{\rm v}$ is the $B-L$ in the visible sector. Here we assume the asymmetry transfer interaction would decouple above the supersymmetry breaking scale, thus the asymmetry would transfer from the dark sector to the visible sector when all of the MSSM particles are active in the thermal bath. Hence dark particles are in thermal equilibrium with all of the MSSM particles, which gives $\mu_{\hat{X}}+\mu_{\hat{X}'}= -\mu_{\hat{X}^c} -\mu_{\hat{X}'^c} = -\frac{22}{7}\mu_{L}$. Thus the total dark particle number is given by $X = \frac{44}{79} (B-L)_{\rm v}$.
The dark matter mass is determined using the constraint $\Omega_{\rm DM} / \Omega_{\rm B} = (X \,m_{\rm DM}) / (B \,m_{\rm B}) \approx 5.5$, where $m_{\rm DM}$ is the mass of the dark matter particle and $m_B$ is the baryon mass which is taken to be $m_{\rm B}\sim 1$ GeV. An important subtlety here is that although the total dark particle number is fixed after the asymmetry transfer interaction decouples, the total baryon number changes after this decoupling because of the sphaleron processes. As explained in detail in [@Feng:2012jn], the total baryon number to be used in the computation of $\Omega_{\rm DM} / \Omega_{\rm B}$ is $B_{\rm final}$ after the sphaleron processes decouple. Thus one has $m_{\rm DM} = (B_{\rm final} / X) \cdot 5.5~{\rm GeV}$ where $B_{\rm final} = \frac{30}{97} (B-L)_{\rm v} \approx 0.31 (B-L)_{\rm v}$ [@Feng:2012jn]. This leads to $m_{\rm DM} \approx 3.01~{\rm GeV}$. The astrophysical constraint $B_{\rm final} / s \sim 6 \times 10^{-10}$ [@Beringer:1900zz], can be satisfied with $\epsilon \sim 4 \times 10^{-6}$, which sets bounds for the complex couplings $\lambda_i$ and the ratio $M_1/M_2$.
Physics of the dark sector
--------------------------
In order to achieve a viable model one needs to dissipate the symmetric component of dark matter. This can be achieved by gauge kinetic energy mixing of $U(1)_x$ and $U(1)_Y$ [@Holdom:1985ag]. The thermally produced dark matter and its anti-matter can annihilate efficiently into SM particles through the $Z'$ boson exchange with a Breit-Wigner enhancement [@DMRD; @NathWig; @Celis:2016ayl]. The kinetic mixing does not generate couplings between the photon and dark sector particles and thus dark matter carries no milli-charge. Consequently there are no experimental constraints from the limits on milli-charges on the parameter $\delta$ which enters in the gauge kinetic energy mixing of $U(1)_x$ and $U(1)_Y$. Thus the strongest experimental constraints on the $Z'$ boson mass and its coupling to the visible sector come from corrections to $g_{\mu}-2$ as well as LEP II constraints. These lead to the limit $\delta \lesssim 0.001$. With such constraints, one can deplete the symmetric component of dark matter in sufficient amounts, i.e., less than $10\%$ of the total dark matter relic abundance.
An alternative way of depleting the symmetric component of dark matter is assuming that the $U(1)_x$ gauge boson is massless (dark photon). Then the symmetric component of the dark matter could annihilate into the $U(1)_x$ dark photons and become radiation in the early universe. As shown in [@Blennow:2012de], the constraints on the number of extra effective neutrino species $\Delta N_{\rm eff}$, can be satisfied for a large class of asymmetric dark matter models.
Such dark matter can scatter from quarks within a nucleon through the t-channel exchange of the $Z'$ boson. The spin-independent dark matter-nucleon cross section can be approximately written as $\sigma_{{\rm SI}}\sim 4\delta^{2} g_{x}^{2} g_{Y}^{2} \cos^{4}\theta_{W}\mu_{n}^{2} \big/ \pi m_{Z'}^{4}$, where $\mu_{n}$ is the dark matter-nucleon reduced mass. For our model we find $\sigma_{{\rm SI}}\sim10^{-37}~{\rm cm}^{2}$, which is just on the edge of sensitivity of the CRESST I experiment [@Angloher:2002in]. Thus improved experiment in the future in the low dark matter mass region with better sensitivities should be able to test the model.
As in the supersymmetric case, the $U(1)_x$ gaugino $\chi$ is given a soft mass $\mathcal{L}_{\chi} = m_{\chi} \bar \chi \chi$. It can then decay into $X\tilde X$ or $X' \tilde X'$ via the supersymmetric interaction $\mathcal{L} \sim \chi X \tilde X+ \chi X' \tilde X' + h.c.$, where we assume $m_{\chi} > m_X + m_{\tilde X}$. Thus the gaugino $\chi$ decays into dark particles and is removed from the low energy spectrum. One important aspect of the supersymmetric case is that it presents a multi-component picture of dark matter. The total dark matter relic abundance consists of dark sector particles $(\hat X, \hat{X}^c, \hat X', \hat{X}'^c)$ as well as the conventional lightest supersymmetric particle with R-parity, i.e., the (lightest) neutralino. There exists a significant part of the parameter space of MSSM where the relic density of neutralinos can be 10% or less of the current relic density [@Feng:2012jn]. The analysis of [@Feng:2012jn] shows that even with 10% of the relic density, the neutralino dark matter would be still accessible in dark matter searches. Thus this feature also offers a direct test of the model in neutralino dark matter searches. However, the leptonic dark matter would be difficult to see in direct searches for dark matter as well as in collider experiments because of its small couplings to the visible sector via the $Z'$ boson exchange. Future colliders with higher sensitivity and accuracy may have the possibility to explore the $Z'$ boson with tiny couplings to the SM particles.
Cogenesis of Baryon Asymmetry and Dark Matter
=============================================
We discuss now a model class [@Feng:2013zda] where leptogenesis takes place with all fundamental interactions not violating lepton number, and the total $B-L$ number in the Universe vanishes. Such leptogenesis leads to equal and opposite lepton numbers in the visible sector and the dark sector. Part of the lepton number generated in the visible sector subsequently transfers to the baryonic sector via sphaleron interactions.
The model
---------
We begin by considering the set of fields $N_i, \psi, \phi, X, X'$ with lepton number assignments $(0, +1, -1, +1/2, +1/2)$. Here $N_i$ ($i \geq 2$) are Majorana fermions, $\psi, X,X'$ are Dirac fields and $\phi$ is a complex scalar field. The fields $N_i, \psi, \phi$ are heavy and will decay into lighter fields and eventually disappear. The dark sector is constituted of two fermionic fields $X,X'$, which as indicated above each carry a lepton number $+1/2$ and are oppositely charged under the dark sector gauge group $U(1)_x$ with gauge charges $(+1,-1)$. All other fields are neutral under $U(1)_x$. We assume their interactions to have the following form which conserve both the lepton number and the $U(1)_x$ gauge symmetry $$\mathcal{L} =\lambda_i \bar N_i \psi \phi + \beta\, \bar \psi L H
+ \gamma\, \phi \bar X^c X' + h.c.\,,
\label{1.1}$$ where the couplings $\lambda_i$ are assumed to be complex and the couplings $\beta,\gamma$ are assumed to be real. In addition we add mass terms so that $$- \mathcal{L}_m= M_i \bar N_i N_i + m_1 \bar \psi \psi + m_2^2 \phi^* \phi
+m_X \bar X X + m_{X'} \bar X' X' \,.
\label{1.2}$$ Here $N_i$ have Majorana masses, while $\psi, X, X'$ have Dirac masses. We assume the mass hierarchy $M_i \gg m_1 + m_2$, $m_1\sim m_2 \gg m_X+ m_{X'}$. Consistent with the above constraint, $m_1, m_2$, the masses of $\psi$ and $\phi$, could span a wide range from TeV scale to scales much higher.
In the early universe, the out-of-equilibrium decays of the heavy Majorana fields $N_i$ produce a heavy Dirac field $\psi$ and a heavy complex scalar field $\phi$. The CP violation due to the complex couplings $\lambda_i$ generates an excess of $\psi,\phi$ over their anti-particles $\bar{\psi},\phi^*$ which carry the opposite lepton numbers. Since the lepton number carried by $\psi$ and $\phi$ always sums up to zero, the out-of-equilibrium decays of $N_i$ do not generate an excess of lepton number in the Universe. Further, $\psi$ and $\phi$ (as well as their anti-particles) produced in the decay of the Majorana fields $N_i$ will sequentially decay, with $\psi$ (and its anti-particle) decaying into the visible sector fields and $\phi$ (and its anti-particle) decaying into the dark sector fields. Their decays thus produce a net lepton asymmetry in the visible sector and a lepton asymmetry of opposite sign in the dark sector. We note that the absence of the decays $\psi \to \bar{X}+ X'$ and $\phi^* \to L+H$ guarantees that leptonic asymmetries of equal and opposite sign are generated in the visible and in the dark sectors. Indeed, right after the heavy Majorana fermions $N_i$ have decayed completely, and created the excess of $\psi,\phi$ over $\bar{\psi},\phi^*$, equal and opposite lepton numbers are already assigned to the visible sector and the dark sector. It is clear from the above analysis that there is no violation of lepton number in the entire process of generating the leptonic asymmetries. We further note that while sphaleron interactions are active during the period when the leptogenesis and the genesis of (asymmetric) dark matter occur, they are not responsible for creating a net $B-L$ number in the visible sector, though they do play a role in transmuting a part of the lepton number into baryon number in the visible sector.
One can estimate on general grounds the mass of the dark particles in this model for the cosmic coincidence to occur. Since the total $B-L$ in the Universe vanishes, the $B-L$ number in the visible sector is equal in magnitude and opposite in sign to the lepton number created in the visible sector right after $N_i$ have completely decayed (the decay of $N_i$ does not generate any baryon asymmetry), and thus is equal to the lepton number in the dark sector, i.e., $ (B-L)_{\rm v} = L_{\rm d}$ where the indices ${\rm v,d}$ denote the visible sector and the dark sector respectively. We are interested in the relative density of particle species at the time when the sphaleron interactions go out of the thermal equilibrium. After the decoupling of the sphaleron interactions $B$ and $L$ are separately conserved and correspond to the $B$ and $L$ seen today. Recall the final (currently observed) value of the baryon number density $B_{\rm final}\approx 0.31 (B-L)_{\rm v}$ [@Feng:2012jn], assuming that $X$ and $X'$ have the same mass, we obtain $m_X = m_{X'} \approx 0.85~{\rm GeV}$.
![Generation of asymmetry in $\psi,\phi$ over their antiparticles $\bar\psi,\phi^*$ from the decay of the Majorana field $N_1$. The lepton number is conserved in these processes.[]{data-label="Cog"}](FDG1.pdf)
We turn now to the detail of the generation of the asymmetry between $\psi,\phi$ and $\bar{\psi}, \phi^*$. We assume there are two Majorana fields $N_1$ and $N_2$ with $N_2$ mass $M_2$ being much larger than the $N_1$ mass $M_1$, i.e., $M_2 \gg M_1$. The diagrams that contribute to it are shown in \[Cog\] where the Majorana particles $N_i$ decay into the Dirac fermion $\psi$ and the complex scalar $\phi$ with $\psi$ and $\phi$ carrying opposite lepton numbers while the Majorana fields $N_i$ carry no lepton number. In this case the asymmetry arising from the excess of $\psi,\phi$ over $\bar{\psi}, \phi^*$ is given by $$\epsilon
=\frac{\Gamma(N_{1}\to \psi\phi)-\Gamma(N_{1}\to\bar\psi\phi^*)}
{\Gamma(N_{1}\to \psi\phi)+\Gamma(N_{1}\to\bar\psi\phi^*)} \simeq
-\frac{1}{8\pi}\frac{{\rm Im}(\lambda_{1}^{2}\lambda_{2}^{*2})}{|\lambda_{1}|^{2}}\frac{M_{1}}{M_{2}}
\,,
\label{ASM}$$ where we have included both the vertex contribution and the wave contribution. Since the dark sector does not communicate with the visible sector, $(B-L)_v$ is equal in magnitude and opposite in sign to the lepton number generated in the visible sector: $(B-L)_{\rm v} = -L_{\rm v} \approx -0.4\kappa \epsilon s / g_*$. where $s$ is the entropy, $\kappa$ is the washout factor and we take $\kappa = 0.1$ and $g_* = 106.75$. Using again $B_{\rm final} \approx 0.31 (B-L)_{\rm v}$, one estimates $| \epsilon | \sim 5 \times 10^{-6}$. The supersymmetric extension of this model is straightforward, as discussed in [@Feng:2013zda].
Phenomenology of the model
--------------------------
In a manner similar to what was discussed earlier, the symmetric component of dark matter would be sufficiently depleted by annihilating via the $Z'$ gauge boson into SM particles (or annihilating into $U(1)_x$ dark photons), which ensures the asymmetric dark matter to be the dominant component of the current dark matter relic abundance.
An interesting implication of this model class arises in the neutrino sector. Here we add three families of right-handed neutrinos. We assume the coupling $\beta$ is family-dependent, i.e., $\beta \to \beta_i$ where $i=1,2,3$ correspond to $e,\mu,\tau$, c.f., \[1.1\] so the Lagrangian reads $$\mathcal{L}' =
\beta_i \bar \psi_R L_i H
+ \beta_{ij}'' \bar \nu_{iR} L_j H
+ \mu_i' \bar \nu_{iR} \psi_L
+ h.c.\,.$$ After spontaneous breaking of the electroweak symmetry, the mass terms take the form $\mathcal{L}_m = \vec{\nu}_R^T\,\mathcal{M} \,\vec{\nu}_L + h.c.$, where $\vec{\nu}_R^T = \left(\bar{\nu}_{R}^{e},\bar{\nu}_{R}^{\mu},\bar{\nu}_{R}^{\tau},\bar{\psi}_{R}\right)$, and $\vec{\nu}_L^T = \left(\nu_{L}^{e},\nu_{L}^{\mu},\nu_{L}^{\tau},\psi_{L}\right)$. For simplicity we assume a symmetrical form for the neutrino mass terms so that $$\mathcal{L}_{m}^\nu=
\vec{\nu}_R^T
\left(\begin{array}{cccc}
m_{\nu_{e}} & 0 & 0 & \mu_{1}\\
0 & m_{\nu_{\mu}} & 0 & \mu_{2}\\
0 & 0 & m_{\nu_{\tau}} & \mu_{3}\\
\mu_1 & \mu_2 & \mu_3 & m_{1}
\end{array}\right)
\vec{\nu}_L
+h.c.\,,
\label{neutrino-matrix2}$$ \[neutrino-matrix2\] contains no direct mixings among the neutrino flavor states. However, their mixings with the field $\psi$ automatically leads to neutrino flavor mixings for the mass diagonal states. To exhibit this mixing we diagonalize the matrix of \[neutrino-matrix2\] by an orthogonal transformation. By setting $m_{\nu_{e}}=10^{-11},m_{\nu_{\mu}}=1.7\times10^{-10},m_{\nu_{\tau}}=2\times10^{-9},m_1=2000,
\mu_1=3.6\times10^{-5},\mu_2=8.9\times10^{-5},\mu_3=5.9\times10^{-4}$ (all masses in GeV) the three neutrino masses in the mass diagonal basis are $m_{3} \approx 4.8 \times10^{-2}~{\rm eV},
m_{2} \approx 1.2 \times10^{-2}~{\rm eV},
m_{1} \approx 4.2 \times10^{-3}~{\rm eV}$, which is the normal hierarchy of neutrino masses [@Beringer:1900zz] while the mass of the heavy field $\psi$ is still $\sim m_{1}$. For the neutrino mixings we obtain $\sin^2\theta_{12} \approx 0.30\,, \ \sin^2\theta_{23} \approx 0.36\,,\ \sin^2\theta_{13} \approx 0.024$, which is in good accord with the experimental determination of the mixing angles. Specifically the model is consistent with the result from the Daya Bay reactor neutrino experiment [@An:2013zwz] of $\theta_{13} \sim 9^\circ$. It is interesting that the model provides an explanation of the neutrino mixings at a fundamental level. The neutrino mixings arise as a consequence of the interaction of the neutrinos with the primordial Dirac field $\psi$ which enters in leptogenesis which points to the cosmological origin of neutrino mixings.
![Flavor changing processes $\ell_i \to \ell_j \gamma$ via the charged Higgs and $Y$ loop.[]{data-label="muega"}](muega.pdf)
Other implications of the model involve flavor changing processes. For the supersymmetric version of the model, after spontaneous breaking one has interactions of the charged Higgs $H^+$ with charged leptons and $Y$: $${\cal L}_{H\ell\psi}= \beta_i \bar Y \ell_i H^+ + h.c.\,,$$ where $\ell_i$ denotes the charged leptons and $Y$ is a chiral field with lepton number $-1$ [@Feng:2013zda]. Such interactions will give rise to $\ell_i \to \ell_j \gamma$ processes, where a charged lepton $\ell_i$ converts into a charged lepton $\ell_j$ via exchange of $Y$ while a photon is emitted by the charged Higgs inside the loop, see \[muega\]. Assuming $m_Y^2 \gg m_{H^+}^2$, we obtain the decay rate of the flavor changing process $\ell_i \to \ell_j \gamma$ to be $$\d \Gamma_{\ell_i \to \ell_j \gamma} = \frac{\alpha_{\rm em} (\beta_i \beta_j)^2}{(16 \pi^2)^2} \frac{m_i^3}{M_Y^2}\,,
\label{DW}$$ where $m_i$ is the mass of the decaying charged lepton and we have used $m_i \gg m_j$. The current experimental bounds constrain the couplings to be $\beta_1 \sim \beta_2 \lesssim 3\times 10^{-3}$ and $\beta_3 \lesssim 2 \times 10^{-4} / \beta_1$ for $M_Y \sim 1~{\rm TeV}$. One can expect observable effects in these flavor changing processes in future experiments with improved sensitivities.
Conclusion
==========
The comparable size of dark matter and visible matter in the Universe points to a possible common origin of the two. Here we discussed two classes of models. In the first model class, the dark matter is generated from the decay of some primordial fields. The asymmetry is generated in the dark sector by the CP violating decays, and then transfer to the visible sector via the asymmetry transfer interaction. In the second model class all of the fundamental interactions conserve lepton number, and leptogenesis occurs when equal and opposite lepton numbers are generated in the visible sector and dark sector. Subsequently the sphaleron processes transmute a part of lepton asymmetry to baryon asymmetry. In this model class the total $B-L$ number in the Universe is exactly conserved. Phenomenological aspects of these models were also discussed.
Acknowledgments {#acknowledgments .unnumbered}
===============
Research was supported in part by NSF Grants PHY-1314774 and PHY-1620575.
[0]{}
W. Z. Feng, A. Mazumdar and P. Nath, Phys. Rev. D [**88**]{}, no. 3, 036014 (2013). W. Z. Feng and P. Nath, Phys. Lett. B [**731**]{}, 43 (2014). W. Z. Feng, P. Nath and G. Peim, Phys. Rev. D [**85**]{}, 115016 (2012). A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. [**5**]{}, 32 (1967) \[JETP Lett. [**5**]{}, 24 (1967)\]. D. E. Kaplan, M. A. Luty and K. M. Zurek, Phys. Rev. D [**79**]{}, 115016 (2009). P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], Astron. Astrophys. [**571**]{}, A1 (2014).
H. Davoudiasl and R. N. Mohapatra, New J. Phys. [**14**]{}, 095011 (2012); K. Petraki and R. R. Volkas, Int. J. Mod. Phys. A [**28**]{}, 1330028 (2013); K. M. Zurek, Phys. Rept. [**537**]{}, 91 (2014). For recent work on Stueckelberg $U(1)$ see W. Z. Feng, Z. Liu and P. Nath, JHEP [**1604**]{}, 090 (2016). For earlier works see B. Kors and P. Nath, Phys. Lett. B [**586**]{}, 366 (2004); JHEP [**0412**]{}, 005 (2004); JHEP [**0507**]{}, 069 (2005); W. Z. Feng, G. Shiu, P. Soler and F. Ye, JHEP [**1405**]{}, 065 (2014); W. Z. Feng, G. Shiu, P. Soler and F. Ye, Phys. Rev. Lett. [**113**]{}, 061802 (2014). J. Shelton and K. M. Zurek, Phys. Rev. D [**82**]{}, 123512 (2010); N. Haba and S. Matsumoto, Prog. Theor. Phys. [**125**]{}, 1311 (2011); M. R. Buckley and L. Randall, JHEP [**1109**]{}, 009 (2011). H. Davoudiasl, D. E. Morrissey, K. Sigurdson and S. Tulin, Phys. Rev. Lett. [**105**]{}, 211304 (2010); P. H. Gu, M. Lindner, U. Sarkar and X. Zhang, Phys. Rev. D [**83**]{}, 055008 (2011); P. Fileviez P¨¦rez and H. H. Patel, Phys. Lett. B [**731**]{}, 232 (2014). M. Fukugita and T. Yanagida, Phys. Lett. B [**174**]{}, 45 (1986); H. Murayama, H. Suzuki, T. Yanagida and J. Yokoyama, Phys. Rev. Lett. [**70**]{}, 1912 (1993); L. Covi, E. Roulet and F. Vissani, Phys. Lett. B [**384**]{}, 169 (1996); W. Buchmuller and M. Plumacher, Int. J. Mod. Phys. A [**15**]{}, 5047 (2000)
J. A. Harvey and M. S. Turner, Phys. Rev. D [**42**]{}, 3344 (1990).
J. Beringer [*et al.*]{} \[Particle Data Group Collaboration\], Phys. Rev. D [**86**]{}, 010001 (2012).
B. Holdom, Phys. Lett. B [**166**]{}, 196 (1986); Phys. Lett. B [**259**]{}, 329 (1991). K. Griest and D. Seckel, Phys. Rev. D [**43**]{}, 3191 (1991); P. Gondolo and G. Gelmini, Nucl. Phys. B [**360**]{}, 145 (1991). P. Nath and R. L. Arnowitt, Phys. Rev. Lett. [**70**]{}, 3696 (1993); Phys. Lett. B [**299**]{}, 58 (1993); D. Feldman, Z. Liu and P. Nath, Phys. Rev. D [**79**]{}, 063509 (2009). A. Celis, W. Z. Feng and M. Vollmann, arXiv:1608.03894 \[hep-ph\]. M. Blennow, E. Fernandez-Martinez, O. Mena, J. Redondo and P. Serra, JCAP [**1207**]{}, 022 (2012) G. Angloher [*et al.*]{}, Astropart. Phys. [**18**]{}, 43 (2002). F. P. An [*et al.*]{} \[Daya Bay Collaboration\], Phys. Rev. Lett. [**112**]{}, 061801 (2014)
[^1]: In the non-supersymmetric case, the simplest model with interaction $\mathcal{L} \sim \lambda_i N_i \bar X^c X' + h.c.$, where $\lambda_i$ are complex coupling constants, $N_i$ are complex scalars and $i\geq2$, $X,X'$ are Dirac fermions, does not work. Namely, the decay of $N_i$ as well as $N_i^*$ does not generate an asymmetry between $X,X'$ and $\bar{X},\bar{X}'$.
|
---
abstract: 'We investigate hyperfine induced electron spin and entanglement dynamics in a system of two quantum dot spin qubits. We focus on the situation of zero external magnetic field and concentrate on approximation-free theoretical methods. We give an exact solution of the model for homogeneous hyperfine coupling constants (with all coupling coefficients being equal) and varying exchange coupling, and we derive the dynamics therefrom. After describing and explaining the basic dynamical properties, the decoherence time is calculated from the results of a detailed investigation of the short time electron spin dynamics. The result turns out to be in good agreement with experimental data.'
author:
- 'B. Erbe and J. Schliemann'
title: 'Hyperfine induced spin and entanglement dynamics in Double Quantum Dots: A homogeneous coupling approach'
---
Introduction
============
Quantum dot spin qubits are among the most promising and most intensively investigated building blocks of possible future solid state quantum computation systems [@LossDi98; @Hanson07]. One of the major limitations of the decoherence time of the confined electron spin is its interaction with surrounding nuclear spins by means of hyperfine interaction [@KhaLossGla02; @KhaLossGla03; @expMarcus; @Koppens05; @Petta05; @Koppens06; @Koppens08; @Braun05]. For reviews the reader is referred to Refs. [@SKhaLoss03; @Zhang07; @Klauser07; @Coish09; @Taylor07]. Apart from this adverse aspect, hyperfine interaction can act as a resource of quantum information processing [@Taylor03; @SchCiGi08; @SchCiGi09; @ChriCiGi09; @ChriCiGi07; @ChriCiGi08]. For the above reasons it is of key interest to understand the hyperfine induced spin dynamics.
Most of the work into this direction, for single as well as double quantum dots, has been carried out under the assumption of a strong magnetic field coupled to the central spin system. This allows for a perturbative treatment or a complete neglect of the electron-nuclear “flip-flop” part of the Hamiltonian, yielding great simplification [@KhaLossGla02; @KhaLossGla03; @Coish04; @Coish05; @Coish06; @Coish08]. In the present paper we consider the case of zero magnetic field where such approximations fail, and we therefore concentrate on exact methods.
In the case of a single quantum dot spin qubit the usual Hamiltonian describing hyperfine interaction with surrounding nuclei is integrable by means of Bethe ansatz as devised by Gaudin several decades ago[@Gaudin; @John09; @BorSt071; @BorSt09]. In the following we shall refer to that sytem also as the Gaudin model. Nevertheless exact results are rare also here because the Bethe ansatz equations are very hard to handle. Hence there are mainly three different routes in order to gain some exact results: (i) Restriction of the initial state to the one magnon sector [@KhaLossGla02; @KhaLossGla03], (ii) restriction to small system sizes enabling progress via exact numerical diagonalizations [@SKhaLoss02; @SKhaLoss03], and (iii) restrictions to the hyperfine coupling constants [@BorSt07; @ErbS09]. In the present paper we will follow the third route and study in detail the electron spin as well as the entanglement dynamics in a double quantum dot model with partially homogeneous couplings: The hyperfine coupling constants are chosen to be equal to each other, whereas the exchange coupling is arbitrary. Although the assumption of homogeneous hyperfine constants (being the same for each spin in the nuclear bath) is certainly a great simplification of the true physical situation, models of this type offer the opportunity to obtain exact, approximation-free results which are scarce otherwise. Moreover, such models have been the basis of several recent theoretical studies leading to concrete predictions [@SchCiGi08; @SchCiGi09; @ChriCiGi09; @ChriCiGi08].
The paper is organized as follows: In Sec. \[model\] we introduce the Hamiltonian of the hyperfine interaction and derive the spin and entanglement dynamics for homogeneous hyperfine coupling constants. In Sec. \[dynamics\] we study the spin and entanglement dynamics for different exchange couplings and bath polarizations. For the completely homogeneous case of the exchange coupling being the same as the hyperfine couplings we find an empirical rule describing the transition from low polarization dynamics to high polarization dynamics. The latter shows a jump in the amplitude when varying the exchange coupling away from complete homogenity. This effect as well as features like the periodicity of the dynamics are explained by analyzing the level spacings and their contributions to the dynamics. In Sec. \[decoherence\] we extract the decoherence time from the dynamics by investigating the scaling behaviour of the short time electron spin dynamics. The result turns out to be in good agreement with experimental findings.
Model and formalism {#model}
===================
The hyperfine interaction in a system of two quantum dot spin qubits is described by the Hamiltonian $$\label{1}
H= \vec{S}_1 \cdot \sum_{i=1}^N A_i^1 \vec{I}_i
+ \vec{S}_2 \cdot \sum_{i=1}^N A_i^2 \vec{I}_i + {J_{ex}}\vec{S}_1 \cdot \vec{S}_2 ,$$ where ${J_{ex}}$ denotes the exchange coupling between the two electron spins $\vec S_1$, $\vec S_2$, and $A_i^1 $, $A_i^2 $ are the coupling parameters for their hyperfine interaction with the surrounding nuclear spins $\vec I_i$.
In a realistic quantum dot these quantities are proportional to the square modulus of the electronic wave function at the sites of the nuclei and therefore clearly spatially dependent $$\label{cpl}
A_i^{j}=A_i v \left|\psi^{j}(\vec{r}_i)\right|^2,$$ where $v$ is the volume of the unit cell containing one nuclear spin and $\psi^{j}(\vec{r}_i)$ is the electronic wave function of electron $j=1,2$ at the site of $i$-th nucleus. The quantity $A_i$ denotes the hyperfine coupling strength which depends on the respective nuclear species through the nuclear gyromagnetic ratio [@Coish09]. It should be stressed that these can have different lengths. In a GaAs quantum dot for example all Ga and As isotopes carry the same nuclear spin $I_i=3/2$, whereas in an InAs quantum dot the In isotopes carry a nuclear spin of $I_i=9/2$ [@SKhaLoss03]. In any case the Hamiltonian obviously conserves the total spin $\vec{J}=\vec{S} + \vec{I}$, where $\vec{S}=\vec{S}_1 + \vec{S}_2$ and $\vec{I}=\sum_{i=1}^N \vec{I}_i$.
The model to be studied in this paper now results by neglecting the spatial variation of the hyperfine coupling constants and choosing them to be equal to each other $A^1_i=A^2_i=A/N$. Variation of the exchange coupling between the two central spins ${J_{ex}}$ then gives rise to an inhomogeneity in the system. Hence the two electron spins are interacting with a common nuclear spin bath. Moreover, if small variations of the coupling constants would be included, degenerate energy levels would slightly split and give rise to a modified [*long-time*]{} behavior of the system. In our quantitative studies to be reported on below, however, we focus on the [*short-time*]{} properties where decoherence phenomena take place. Indeed, in section \[decoherence\] we obtain realistic $T_{2}$ decoherence time scales in an almost analytical fashion.
In consistency with the homogenous couplings we choose the length of the bath spins to be equal to each other. For simplicity we restrict the nuclear spins to $I_i=1/2$. We expect our results to be of quite general nature not strongly depeding on this choice [@John09]. Note that both, the square $\vec S^{2}$ of the total central spin as well as the square $\vec I^{2}$ of the total bath spin are separately conserved quantities.
Considering the two electrons to interact with a common nuclear spin bath as in our model corresponds to a physical situation where the electrons are comparatively near to each other. This leads to the question whether our model is also adapted to the case of two electrons in one quantum dot, rather than in two nearby quantum dots. Assuming perfect confinement, in the former case one of the two electrons would be forced into the first excited state, which typically has a zero around the dot center. Thus, the coupling constants near the very center of the dot would clearly be different for the two electrons. Therefore our model is more suitable for the description of two electrons in two nearby quantum dots than for the case of two electrons in one dot.
Let us now turn to the exact solution of our homogeneous coupling model and calculate the spin and entanglement dynamics from the eigensystem. In what follows we shall work in subspaces of a fixed eigenvalue of $J^z$. Thus, the expectation values of the $x$- and $y$-components of the central and nuclear spins vanish, and we only have to consider their $z$-components.
If all hyperfine couplings are equal to each other $A^1_i=A^2_i=A/N$, the Hamiltonian (\[1\]) can be rewritten in the following way $$\label{5}
H=H_{\operatorname{hom}}+\left({J_{ex}}-\frac{A}{N} \right)\vec{S}_1 \cdot \vec{S}_2$$ with $$\label{2}
H_{\operatorname{hom}}=\frac{A}{2N}\left( \vec{J}^2 - \vec{S}^2_1
- \vec{S}^2_2 - \vec{I}^2\right).$$ Omitting the quantum numbers corresponding to a certain Clebsch-Gordan decomposition of the bath, the eigenstates are labelled by $J,m,S$ associated with the operators $\vec{J}^2, J^z, \vec{S}^2$. The two central spins couple to $S=0,1$. Hence the eigenstates of $H$ are given by triplet states ${\lvert J,m,1\rangle}$, corresponding to the coupling of a spin of length one to an arbitrary spin, and a singlet state ${\lvert J,m,0\rangle}$. The explicit expressions are given by (\[eig1\], \[eig2\], \[eig3\]) in appendix A.
The corresponding eigenvalues read as follows:
\[4\] $$\begin{aligned}
H {\lvert I+1,m,1\rangle}&=&\left( \frac{A}{N}I+\frac{{J_{ex}}}{4} \right)
{\lvert I+1,m,1\rangle}\\ \label{4b}
H {\lvert I,m,1\rangle}&=&\left( \frac{{J_{ex}}}{4}-\frac{A}{N}\right)
{\lvert I,m,1\rangle}\\ \label{4c}
H {\lvert I-1,m,1\rangle}&=&\left(-\frac{A}{N}I+\frac{{J_{ex}}}{4}-\frac{A}{N}
\right) {\lvert I-1,m,1\rangle}\\
H {\lvert I,m,0\rangle}&=& -\frac{3}{4}{J_{ex}}{\lvert I,m,0\rangle}\end{aligned}$$
Now we are ready to evaluate the time evolution of the central spins and their entanglement from the eigensystem of the Hamiltonian. We consider initial states ${\lvert \alpha\rangle}$ of the form ${\lvert \alpha\rangle}={\lvert \alpha_1\rangle}{\lvert \alpha_2\rangle}$, where ${\lvert \alpha_1\rangle}$ is an arbitrary central spin state and ${\lvert \alpha_2\rangle}$ is a product of $N$ states ${\lvert \uparrow\rangle},{\lvert \downarrow\rangle}$.
The physical significance of this choice becomes clear by rewriting the electron-nuclear coupling parts of the Hamiltonian in terms of creation and annihilation operators: $$\label{flipflop}
\vec{S}_i\vec{I}_j=\frac{1}{2}\left(S_i^+I_j^- + S_i^-I_j^+\right)+S_i^z I_j^z$$ Obviously the second term does not contribute to the dynamics for initial states which are simple product states. Hence by considering initial states of the above form, we mainly study the influence of the flip-flop part on the dynamics of the system. This is exactly the part which is eliminated by considering a strong magnetic field like in Refs. [@KhaLossGla02; @KhaLossGla03; @Coish04; @Coish05; @Coish06; @Coish08].
As the $2^N$ dimensional bath Hilbert space is spannend by the $\vec{I}^2$ eigenstates, every product state can be written in terms of these eigenstates. If $N_D \leq N/2$ is the number of down spins in the bath, it follows
$$\label{8}
{\lvert \underbrace{\downarrow \ldots \downarrow}_{N_D} \uparrow \ldots \uparrow\rangle}
= \sum_{k=0}^{N_D} \sum_{\left\{S_i\right\}} c_k^{\left\{S_i\right\}}
{\lvert \underbrace{\frac{N}{2}-k}_{I},\frac{N}{2}-N_D,\left\{S_i\right\}\rangle},$$
where the quantum numbers $\lbrace S_i \rbrace$ are due to a certain Clebsch-Gordan decomposition of the bath. In (\[8\]) we assumed the first $N_D$ spins to be flipped, which is no loss of generality due to the homogeneity of the couplings. For the following discussions it is convenient to introduce the bath polarization $p_b=\left(N-2N_D \right)/N $.
Using (\[8\]) and inverting (\[eig1\], \[eig2\], \[eig3\]), the time evolution can be calculated by writing ${\lvert \alpha\rangle}$ in terms of the above eigenstates and applying the time evolution operator. Using (\[eig1\], \[eig2\], \[eig3\]) again and tracing out the bath degrees of freedom we arrive at the reduced density matrix $\rho(t)$, which enables to evaluate the expectation value $\langle S^z_{1/2} (t) \rangle$ and the dynamics of the entanglement between the two central spins.
As a measure for the entanglement we use the concurrence [@Wootters97] $$C(t)=\operatorname{max}\lbrace0,\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4}\rbrace,$$ where $ \lambda_i$ are the eigenvalues of the non-hermitian matrix $\rho(t) \tilde{\rho}(t)$ in decreasing order. Here $\tilde{\rho}(t)$ is given by $\left(\sigma_y \otimes \sigma_y \right)\rho^*(t) \left( \sigma_y \otimes \sigma_y \right) $, where $\rho^*(t)$ denotes the complex conjugate of $\rho(t)$. The coefficients $c_k^{\left\{S_i\right\}}$ are of course products of Clebsch-Gordan coefficients, which enter the time evolution through the quantity $$d_k=\sum_{\lbrace S_i \rbrace}\left( c_k^{\lbrace S_i \rbrace}\right)^2$$ and usually have to be calculated numerically. The main advantage in considering $I_i=1/2$ is now that in this case a closed expression for $d_k$ can be derived [@BorSt07]: $$\label{10}
d_k =\frac{N_D!(N-N_D)!(N-2k+1)}{(N-k+1)!k!}$$
For further details on the calculation of the time dependent reduced density matrix and the dynamical quantities derived therefrom we refer the reader to appendix B.
Finally, it is a simple but remarkable difference between our one bath system with two central spins and the homogeneous Gaudin model of a single central spin [@SKhaLoss03; @BorSt07], that even if we choose ${\lvert \alpha_2\rangle}$ as an $\vec{I}^2$ eigenstate and hence fix $k$ in (\[8\]) to a single value, due to the higher number of eigenvalues the resulting dynamics can not be described by a single frequency.
Basic dynamical properties {#dynamics}
==========================
We now give an overview over basic dynamical features of the system considered. Due to the homogeneous couplings, the dynamics of the two central spins can be read off from each other.
Hence the following discussion of the dynamics will be restricted to $\langle S^z_1(t) \rangle$.
Electron spin dynamics
----------------------
In Figs. \[Fig:evenodd1\], \[Fig:evenodd2\] we consider the completely homogeneous case ${J_{ex}}=A/N$ and plot the dynamics for ${\lvert \alpha\rangle}={\lvert \Uparrow \Downarrow\rangle}, {\lvert T_+\rangle},{\lvert T_0\rangle}$ and varying polarization $p_b\approx 2\% - 30\%$. A polarization of $30 \%$ does not seem to be particularly high, but the behavior typical for high polarizations occurs indeed already at such a value. We omit the singlet case because it is an eigenstate of the system. In Fig. \[Fig:evenodd1\] the number of spins is even, whereas in Fig. \[Fig:evenodd2\] an odd number is chosen. Note that we measure the time $t$ in rescaled units $\hbar/(A/2N)$ depending on the number of bath spins [@Note1]. Similarly to the homogeneous Gaudin system [@SKhaLoss03; @BorSt07], from Figs. \[Fig:evenodd1\], \[Fig:evenodd2\] we see that the dynamics for an even number of spins is periodic with a periodicity of $\pi$ (in rescaled time units), whereas an odd number of spins leads to a periodicity of $2 \pi$. This is the case for ${J_{ex}}$ being any integer multiple of $A/N$. These characteristics can of course be explained by analyzing the level spacings in the different situations. For example, for an even number of bath spins, all level spacings are even multiples of $A/2N$ [@Note1], resulting in dynamics periodic with $\pi$. However, if the number of spins is odd, we get even and odd level spacings (in units of $A/2N$), giving a period of $2 \pi$. For the given case of completely homogeneous couplings the dynamics can be nicely characterized: The number of local extrema for an even number of bath spins within a complete period, as well as for an odd number of bath spins within half a period, is in both cases given by $N-2N_D+1$.
This – so far empirical – rule holds for all initial central spin states and is illustrated in Figs. \[Fig:evenodd1\] and \[Fig:evenodd2\].
Let us now investigate the spin dynamics for varying exchange coupling, i.e. the case ${J_{ex}}\neq A/N$. Note that for the initial central spin state ${\lvert \alpha_1\rangle}={\lvert T_0\rangle}$ this inhomogeneity has no influence on the spin dynamics since ${\lvert T_0\rangle}$ is an eigenstate of $\vec{S}_1 \cdot \vec{S}_2$ and $$\left[ H_{\operatorname{hom}},\vec{S}_1 \cdot \vec{S}_2 \right]=0.$$ In Fig. \[Fig:evenoddJ1\] the dynamics for ${\lvert \alpha_1\rangle}={\lvert \Uparrow \Downarrow\rangle}$ and varying exchange coupling is plotted. In the upper two panels we consider the case of low polarization $p_b \approx 10\%$ for an even and an odd number of spins. The remaining two panels show the dynamics for high polarization $p_b \approx 30 \%$. In Fig. \[Fig:evenoddJ2\] the plots are ordered likewise for a more general linear combination of ${\lvert \Uparrow \Downarrow\rangle}$ and ${\lvert T_0\rangle}$ , ${\lvert \alpha_1\rangle}=(1/\sqrt{13})\left( 2 {\lvert \Uparrow \Downarrow\rangle} + 3 {\lvert \Downarrow \Uparrow\rangle} \right) $.
From Figs. \[Fig:evenoddJ1\], \[Fig:evenoddJ2\] we see that if the exchange coupling is an odd multiple of $A/2N$, the even-odd effect described above does not occur and we have periodicity of $2 \pi$. In both of the aforementioned situations the time evolutions are symmetric with respect to the middle of the period, which is a consequence of the invariance of the underlying Hamiltonian under time reversal. For a more general exchange coupling, the periodicity, along with the mirror symmetry, of the dynamics is broken on the above time scales.
Considering the case of low polarization, neither the dynamics of initial states with a product nor the one of states with an entangled central spin state dramatically changes if ${J_{ex}}$ is varied. However, if the polarization is high, the spin is oscillating with mainly one frequency proportional to ${J_{ex}}$.
Furthermore the amplitude of the oscillation is larger for the case ${J_{ex}}\neq A/N$ than for the completely homogeneous case. This behaviour can be understood as follows: If the polarization is high $d_{N_D} \approx 1$, whereas $d_k \approx 0$ for $k \neq N_D$. This means that calculating the spin and entanglement dynamics, we only have to consider the term $k=N_D$. An evaluation of the coeffcients for the different frequencies now shows that the main contribution results from $E_{T_0}-E_S = (A/N)-{J_{ex}}$ in obvious notation. Hence if the polarization is more and more increased, this is the only frequency left. If ${J_{ex}}=(A/N)$, the two associated eigenstates are degenerate so that in this case the main contribution to the dynamics is constant. This explains why the amplitude of the high polarization dynamics in Figs. \[Fig:evenoddJ1\], \[Fig:evenoddJ2\] is big compared to the one in Figs. \[Fig:evenodd1\], \[Fig:evenodd2\]. For further details the reader is referred to appendix B.
Entanglement dynamics
---------------------
In Figs. \[Fig:con\_hom1\], \[Fig:con\_hom\] the concurrence dynamics $C(t)$ for ${\lvert \alpha_1\rangle}={\lvert \Uparrow \Downarrow\rangle}, {\lvert T_+\rangle}$ is plotted for the same polarizations as in Figs. \[Fig:evenoddJ1\], \[Fig:evenoddJ2\] and varying exchange coupling.
It is interesting that in the second case the concurrence drops to zero for certain periods of time. This is very similar for the case ${\lvert \alpha_1\rangle}={\lvert T_0\rangle}$ not shown above. As already explained concerning the spin dynamics, the exchange coupling ${J_{ex}}$ of course has no influence because ${\lvert T_+\rangle}$ is an eigenstate of $\vec{S}_1 \cdot \vec{S}_2$.
It is an interesting fact now that for ${\lvert \alpha_1\rangle}={\lvert \Uparrow \Downarrow\rangle}$ and a small polarization changing from $\vert {J_{ex}}\vert > 1$ to $\vert {J_{ex}}\vert <1$ increases the maximum value of the function $C(t)$. Furthermore we see from Fig. \[Fig:con\_hom1\] that surprisingly the entanglement is much smaller for the completely homogeneous case ${J_{ex}}= A/N$ than for ${J_{ex}}\neq A/N$ even for low polarization.
Decoherence and its quantification {#decoherence}
==================================
Depending on the choice of the exchange coupling, the dynamics of the one bath model can either be symmetric and periodic or without any regularities. It is now not entirely obvious to determine in how far these dynamics constitute a process of decoherence. Considering for example the spin dynamics for an integer ${J_{ex}}$ and an even number of bath spins shown in Fig. \[Fig:evenodd1\], one can either regard the decay of the spin as decoherence or, especially due to the symmetry of the function, as part of a simple periodic motion. In Ref. [@BorSt07] the first zero of $\langle S^z_1(t) \rangle$ has been considered as a measure for the decoherence time. In Fig. \[Fig:Jsubzero\] we illustrate examples of the spin dynamics on short time scales for ${J_{ex}}\geq 0$, ${J_{ex}}<0$ and a varying number of bath spins. For ${J_{ex}}\geq 0$ this procedure is straightforward meaning that $\langle S^z_1(t) \rangle$ crosses the horizontal line $\langle S^z_1 \rangle=0$ before reaching its first minimum with $\langle S^z_1(t) \rangle<0$. However, for ${J_{ex}}<0$ and a sufficiently small number of bath spins, as seen from the lower panel of Fig. \[Fig:Jsubzero\], such a first minimum is attained before the first actual zero $\langle S^z_1(t) \rangle=0$. This first zero occurs indeed at much large times $t$ whose scaling behavior as a function of system size $N$ is clearly different from the zero positions found for ${J_{ex}}\geq 0$, as we have checked in a detailed analysis. Thus, our evaluation scheme needs to be modified for ${J_{ex}}<0$. An obvious way out of this problem is to either consider large enough spin baths where such an effect does not occur, or to evaluate the intersection with alternative “threshold level” $\langle S^z_1 \rangle>0$. In Fig. \[Fig:Jsubzero\] we have chosen $\langle S^z_1 \rangle=0.2$, which will be the basis of our following investigation. As a further alternative, one could also consider the position of the first minimum of $\langle S^z_1(t) \rangle$. Hence, strictly speaking, it is not per se the first zero of $\langle S^z_1(t) \rangle<0$ which is a measure for the decoherence time, but the scaling behavior of the dynamics on short time scales. Following the route described above, in Fig. \[Fig:scale\] we plot the positions (measured in units of $\hbar/(A/2N)$) of the first zeroes of $\langle S^z_1(t) \rangle$ for ${J_{ex}}\geq 0$, and of the first intersections with the threshold level shown in Fig. \[Fig:Jsubzero\] for ${J_{ex}}<0$, on a double logarithmic scale. We choose a weakly polarized bath $N=2N_D+2\Rightarrow p_b=2/N$, approaching the completely unpolarized case for $N\to\infty$. The absolute values of the positions for ${J_{ex}}\geq 0$ and ${J_{ex}}<0$ differ slightly from each other, which results from the fact that the intersection with the threshold level at $0.2$ happens closer to zero than with the usual threshold level $\langle S^z_1 \rangle=0$. Nevertheless, the scaling behavior is very similar in all cases, and each curve can nicely be fitted by a power law $\propto (N+2)^\nu$ with $\nu\approx -0.5$, a result similar to the one found for the homogeneous Gaudin system with only one central spin [@BorSt07].
In a GaAs quantum dot the electron spins usually interact with approximately $N=10^6$ nuclei. Assuming the hyperfine coupling strength to be of the order of $A=10^{-5}$eV, as realistic for GaAs quantum dots [@SKhaLoss03], this results in a time scale of $Nh/(\pi A)= 1.31 \cdot 10^{-4} $s. If we now use the above scaling behaviour $1/\sqrt{N+2}$, we get a decoherence time of $131$ns, which fits quite well with the experimental data [@expAwschalom; @Koppens05; @Petta05; @Koppens08]. This is an interesting result not only with respect to the validity of our model: As explained following equation (\[flipflop\]), generally decoherence results “directly” from the electron-nuclear flip-flop terms and through the superposition of product states from the z terms. Above we calculate the decoherence time for ${\lvert \alpha_1\rangle} ={\lvert \Uparrow \Downarrow\rangle}$, where the influence of the z terms is eliminated. The fact that we are able to reproduce the decoherence times suggests that the decoherence time caused by the flip-flop terms is equal or smaller than the one resulting from the z parts of the Hamiltonian. It should be stressed that we calculate the decoherence time of an individual electron $T_2$ here. In Ref. [@Merkulov02] the decoherence time of an ensemble of dots $T_2^*$ has been calculated yielding $1$ns for a GaAs quantum dot with $10^5$ nuclear spins.
It is now a well-known fact for the Gaudin system that the decaying part of the dynamics decreases with increasing polarization [@SKhaLoss03]. A numerical evaluation shows that this is also the case for two central spins. As explained in the context of Figs. \[Fig:evenodd1\], \[Fig:evenodd2\], \[Fig:evenoddJ1\], \[Fig:evenoddJ2\] the oscillations of our one bath model become more and more coherent with increasing polarization. Together with the above results for the decoherence this means that, although the homogeneous couplings are a strong simplification of the physical reality, our homogeneous coupling model shows rather realistic dynamical characteristics on the relevent time scales. This is plausible because artifacts of the homogeneous couplings, like the periodic revivals, set in on longer time scales.
Conclusion
==========
In conclusion we have studied in detail the hyperfine induced spin and entanglement dynamics of a model with homogeneous hyperfine coupling constants and varying exchange coupling, based on an exact analytical calculation.
We found the dynamics to be periodic and symmetric for ${J_{ex}}$ being an integer multiple of $A/N$ or an odd multiple of $A/2N$, where the period depents on the number of bath spins. We explained this periodicity by analyzing the level spectrum. For ${J_{ex}}=A/N$ we found an empirical rule which charaterizes the dynamics for varying polarization. We have seen that for low polarizations the exchange coupling has no significant influence, whereas in the high polarization case the dynamics mainly consists of one single frequency proportional to ${J_{ex}}$. It is not possible to entangle the central spins completely in the setup considered in this article.
Following Ref. [@BorSt07] we extracted the decoherence time by analyzing the scaling behaviour of the first zero. In the case of negative exchange coupling the dynamics strongly changes on short time scales and instead of the first zero we considered the intersection of the dynamics with another threshold level parallel to the time axis. Both cases yield the same result which is in good agreement with experimental data. Hence the scaling behaviour of the short time dynamics can be regarded as a good indicator for the decoherence time.
This work was supported by DFG program SFB631. J. S. acknowledges the hospitality of the Kavli Institute for Theoretical Physics at the University of California at Santa Barbara, where this work was reaching completion and was therefore supported in part by the National Science Foundation under Grant No. PHY05-51164.
Diagonalization of the homogeneous coupling model
=================================================
The eigenstates of $H_{\operatorname{hom}}$ can be found directly by iterating the well known expressions[@Schwabl] for coupling an arbitrary spin to a spin $S=1/2$. Two of these states lie in the triplet sector:
$$\begin{gathered}
\label{eig1}
\nonumber {\lvert I+1,m,1\rangle} = \sqrt{\frac{I+m+1}{2I+2}\cdot \frac{I+m}{2I+1}}{\lvert I,m-1\rangle}{\lvert T_+\rangle} \\
\nonumber + \sqrt{\frac{I+m+1}{I+1}\cdot \frac{I-m+1}{2I+1}}{\lvert I,m\rangle}{\lvert T_0\rangle}\\
+ \sqrt{\frac{I-m+1}{2I+2}\cdot \frac{I-m}{2I+1}}{\lvert I,m+1\rangle}{\lvert T_-\rangle}\\
\nonumber {\lvert I-1,m,1\rangle}= \sqrt{\frac{I-m}{2I}\cdot \frac{I-m+1}{2I+1}}{\lvert I,m-1\rangle}{\lvert T_+\rangle} \\
\nonumber - \sqrt{\frac{I-m}{I}\cdot \frac{I+m}{2I+1}}{\lvert I,m\rangle}{\lvert T_0\rangle}\\
+ \sqrt{\frac{I+m}{2I}\cdot \frac{I+m+1}{2I+1}}{\lvert I,m+1\rangle}{\lvert T_-\rangle}\end{gathered}$$
As already mentioned in the text, the states are labelled by the quantum numbers $J,m,S$ corresponding to the operators $\vec{J}^2,J^z,\vec{S}^2$. The rest of the quantum numbers due to a certain Clebsch-Gordan decomposition of the bath is omitted. For the eigenstates of the central spin term $\vec{S}_1 \cdot \vec{S}_2$ we used the standard notation:
\[Trip\] $$\begin{aligned}
{\lvert T_+\rangle}&=&{\lvert \Uparrow \Uparrow\rangle} \\
{\lvert T_0\rangle}&=&\frac{1}{\sqrt{2}}\left({\lvert \Uparrow \Downarrow\rangle}+{\lvert \Downarrow \Uparrow\rangle} \right) \\
{\lvert T_-\rangle}&=&{\lvert \Downarrow \Downarrow\rangle} \\
{\lvert S\rangle}&=&\frac{1}{\sqrt{2}}\left({\lvert \Uparrow \Downarrow\rangle}-{\lvert \Downarrow \Uparrow\rangle} \right) \end{aligned}$$
The remaining two eigenstates are superpositions of singlet and triplet states. As the expressions are rather cumbersome, it is convenient to introduce the following notation in order to abbreviate the Clebsch-Gordan coefficients:
$$\begin{aligned}
\left\lbrace \mu^1_1,\mu^1_2,\mu^1_3,\mu^1_4 \right\rbrace &=& \left\lbrace \sqrt{\frac{I+m}{2I}\cdot\frac{I-m+1}{2I+1}},\sqrt{\frac{I+m}{2I}\cdot\frac{I+m}{2I+1}},\sqrt{\frac{I-m}{2I}\cdot \frac{I-m}{2I+1}},\sqrt{\frac{I-m}{2I}\cdot\frac{I+m+1}{2I+1}} \right\rbrace \\
\left\lbrace \mu^2_1,\mu^2_2,\mu^2_3,\mu^2_4 \right\rbrace &=& \left\lbrace \sqrt{\frac{I-m+1}{2I+2}\cdot \frac{I+m}{2I+1}},\sqrt{\frac{I-m+1}{2I+2}\cdot\frac{I-m+1}{2I+1}},\sqrt{\frac{I+m+1}{2I+2}\cdot \frac{I+m+1}{2I+1}},\sqrt{\frac{I+m+1}{2I+2}\cdot\frac{I-m}{2I+1}} \right\rbrace\end{aligned}$$
With this definitions the superposition states can be written as:$$\begin{aligned}
{\lvert 1\rangle}&=& \mu^1_1 {\lvert I,m-1\rangle}{\lvert T_+\rangle}+\frac{\mu^1_3-\mu^1_2}{\sqrt{2}}{\lvert I,m\rangle}{\lvert T_0\rangle}\\
&-&\mu^1_4{\lvert I,m+1\rangle}{\lvert T_-\rangle}+\frac{\mu^1_3+\mu^1_2}{\sqrt{2}}{\lvert I,m\rangle}{\lvert S\rangle}\end{aligned}$$ $$\begin{aligned}
{\lvert 2\rangle}&=& \mu^2_1 {\lvert I,m-1\rangle}{\lvert T_+\rangle}+\frac{\mu^2_2-\mu^2_3}{\sqrt{2}}{\lvert I,m\rangle}{\lvert T_0\rangle}\\
&-&\mu^2_4{\lvert I,m+1\rangle}{\lvert T_-\rangle}-\frac{\mu^2_3+\mu^2_2}{\sqrt{2}}{\lvert I,m\rangle}{\lvert S\rangle}\end{aligned}$$ These states are degenerate with respect to $H_{\operatorname{hom}}$, hence we are left with the simple task to find a superposition of ${\lvert 1\rangle}$ and ${\lvert 2\rangle}$, which eliminates ${\lvert I,m\rangle}{\lvert S\rangle}$. Obviously this is given by $$\begin{aligned}
{\lvert I,m,1\rangle} = \frac{1}{N_T} \left( \frac{\sqrt{2}}{\mu^1_2+\mu^1_3}{\lvert 1\rangle}+ \frac{\sqrt{2}}{\mu^2_2+\mu^2_3}{\lvert 2\rangle}\right),\end{aligned}$$ where $N_T= \sqrt{-(I+1)^{-1}+I^{-1}+4}$ is the normalization constant. Inserting ${\lvert 1\rangle}$ and ${\lvert 2\rangle}$ this reads: $$\begin{aligned}
\label{eig2}
\nonumber {\lvert I,m,1\rangle} &=& \frac{1}{N_T} \sum_{i=1}^2 \left( \frac{\sqrt{2}\mu^i_1}{\mu^i_2+\mu^i_3}{\lvert I,m-1\rangle}{\lvert T_+\rangle} \right. \\
\nonumber &+& (-1)^{i+1}\frac{\mu^i_3-\mu^i_2}{\mu^i_2+\mu^i_3}{\lvert I,m\rangle}{\lvert T_0\rangle}\\
&-& \left. \frac{\sqrt{2} \mu^i_4}{\mu^i_2+\mu^i_3}{\lvert I,m+1\rangle}{\lvert T_-\rangle} \right) \end{aligned}$$ Together with the singlet state $$\label{eig3}
{\lvert I,m,0\rangle}={\lvert I,m\rangle}{\lvert S\rangle}$$ this solves our problem of diagonalizing the one bath homogeneous coupling Hamiltonian. Furthermore (\[eig1\]) and (\[eig2\]) give a solution to the very general problem of coupling an arbitrary spin to a spin $S=1$.
Calculation of the time-dependent reduced density matrix
========================================================
Let $H$ be a time-independent Hamiltonian acting on a product Hilbert space ${\mathcal{H}}=\otimes_{i=1}^N {\mathcal{H}}_i$. We denote its eigenvectors by ${\lvert \psi_i\rangle}$ and the corresponding eigenvalues by $E_i$. In the following we calculate the time-dependent reduced density matrix for an initial state which is a pure state and derive the time evolution $\left\langle O_i(t)\right\rangle$ associated with an operator $O_i$ acting on ${\mathcal{H}}_i$. Then we consider the Hamiltonian (\[5\]) and give some more details on the corresponding calculations for our model.
As the eigenstates of $H$ span the whole Hilbertspace ${\mathcal{H}}$, the initial state ${\lvert \alpha\rangle}$ of the system described by $H$ can be written as $$\label{eigwr}
{\lvert \alpha\rangle}=\sum_{i} \alpha_i {\lvert \psi_i\rangle}.$$ The time evolution of the initial state results from the application of the time evolution operator $U=e^{-\frac{i}{\hbar}Ht}$. It follows: $$\begin{aligned}
\label{time}
\nonumber {\lvert \alpha(t)\rangle}{\langle \alpha(t) \lvert}&=&{\lvert U \alpha\rangle} {\langle U \alpha \lvert} \\
\nonumber &=& \sum_{ij} \alpha_i \alpha^*_j {\lvert U \psi_i\rangle} {\langle U \psi_j \lvert} \\
&=& \sum_{ij} \alpha_i \alpha^*_j e^{-\frac{i}{\hbar} \left(E_i-E_j \right)t } {\lvert \psi_i\rangle}{\langle \psi_j \lvert}\end{aligned}$$ As $O_i$ acts on ${\mathcal{H}}_i$, the other degrees of freedom have to be traced out $$\rho_i(t)={\operatorname{Tr}}_{{\mathcal{H}}\setminus {\mathcal{H}}_i}\left({\lvert \alpha(t)\rangle}{\langle \alpha(t) \lvert} \right),$$ finally giving the time evolution of the operator: $$\label{ttime}
\left\langle O_i(t) \right\rangle = {\operatorname{Tr}}_{{\mathcal{H}}_i}\left( \rho_i(t) O_i \right)$$ Usually such calculations are done numerically, but for our homogeneous coupling model it is possible to derive exact analytical expressions for the dynamics of the two central spins.
Following the general scheme, we have to write the initial state in terms of energy eigenstates first. As explained in the text, we consider ${\lvert \alpha\rangle}={\lvert \alpha_1\rangle}{\lvert \alpha_2\rangle}$, where ${\lvert \alpha_1\rangle}$ is an arbitrary central spin state and ${\lvert \alpha_2\rangle}$ is a product state in the bath Hilbertspace ${\mathcal{H}}_N$. Using (\[8\]) it follows:
$$\label{ap21}
{\lvert \alpha_1\rangle}{\lvert \alpha_2\rangle}= \sum_{k=0}^{N_D} \sum_{\left\{S_i\right\}} c_k^{\left\{S_i\right\}} {\lvert \alpha_1\rangle}{\lvert \frac{N}{2}-k,\frac{N}{2}-N_D,\left\{S_i\right\}\rangle}$$
The eigenstates (\[eig1\], \[eig2\], \[eig3\]) are given in terms of product states between a basis element from (\[Trip\]) and an $\vec{I}^2$ eigenstate. Hence we can find the coefficients of (\[eigwr\]) by solving (\[eig1\], \[eig2\], \[eig3\]) for these states and inserting them into (\[ap21\]). If we arrange the coefficients from (\[eig1\], \[eig2\], \[eig3\]) into a $4 \times 4$ matrix $V$ according to $$\label{V}
V=\left( \begin{array}{c|cccc}
&{\lvert T'_+\rangle}&{\lvert T'_0\rangle}&{\lvert T'_-\rangle}&{\lvert S'\rangle}\\\hline
{\lvert I+1,m,1\rangle}&\ddots&&& \\
{\lvert I,m,1\rangle}&&\ddots&&\\
{\lvert I-1,m,1\rangle}&&&\ddots&\\
{\lvert I,m,0\rangle}&&&&\ddots
\end{array}\right),$$ this is simply done by transposing $V$. Here ${\lvert T'_+\rangle}={\lvert I-1,m\rangle}{\lvert T_+\rangle}$ and analogously for the other states. In order to abbreviate the following expressions we denote the energy eigenstates by ${\lvert \psi_i\rangle}$ as in the general considerations above and number with respect to (\[V\]). Analogously we introduce the shorthand notation ${\lvert i\rangle}$ for the basis states (\[Trip\]).
In order to avoid further coefficients we choose ${\lvert \alpha_1\rangle}$ to be the $j$-th element of (\[Trip\]) and find the following expression for the decomposition of the initial state into energy eigenstates $$\label{dec}
{\lvert j\rangle}{\lvert \alpha_2\rangle}= \sum_{l=1}^4 \sum_{k=0}^{N_D} \sum_{\left\{S_i\right\}} c_k^{\left\{S_i\right\}} V^T_{jl} {\lvert \psi_l\rangle},$$ where it is has to be noted that the elements $V^T_{jl}$ and the eigenstates ${\lvert \psi_l\rangle}$ depent on the quantum numbers the sums run over. Hence in our case the coefficients $\alpha_i$ and the eigenstates ${\lvert \psi_i\rangle}$ in fact have more than one index. Inserting (\[dec\]) and (\[eig1\], \[eig2\], \[eig3\]) in (\[time\]) and tracing out the bath degrees of freedom, we finally arrive at the reduced density matrix of the two central spins $$\begin{gathered}
\label{rho}
\rho(t):= {\operatorname{Tr}}_{{\mathcal{H}}_N}\left( {\lvert \alpha(t)\rangle}{\langle \alpha(t) \lvert} \right)= \\
\nonumber \sum_{k=0}^{N_D} \underbrace{\sum_{\left\lbrace S_i\right\rbrace }\left( c_k^{\left\lbrace S_i\right\rbrace }\right)^2 }_{d_k} \sum_{l,m,n,o=1}^4 V_{jl}^T V_{jm}^T V_{ln} V_{mo} e^{-\frac{i}{\hbar}\left(E_l -E_m \right)t }{\lvert n\rangle}{\langle o \lvert}.\end{gathered}$$ If we now choose $O_1 = S^z_1$, we have to trace out the second central spin. Inserting the result into (\[ttime\]) then gives rise to the time evolution $\left\langle S^z_1(t) \right\rangle $. This is given by (\[rho\]) with $n=o$, multiplied by coefficients resulting from the eigenvalues of $S^z_1$. As mentioned in the text, for high polarizations $d_k \approx 0$ if $k \neq N_D$. Fixing $l,m$ we can calculate the contribution of the respective frequency by evaluating the remaining sum over $n$. If the polarization is strongly increased, all frequencies are suppressed except for $E_2-E_4=E_{T_0}-E_S$.
[99]{}
D. Loss and D. P. DiVincenzo, Phys. Rev. A **57**, 120 (1998).
R. Hanson, L .P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Rev. Mod. Phys. [**79**]{}, 1217 (2007).
A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. **88**, 186802 (2002).
A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. B **67**, 195329 (2003).
A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Nature **435**, 925 (2005).
F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H. Willems van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, Science [**309**]{}, 1346 (2005).
J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,and A. C. Gossard, Science [**309**]{}, 2180 (2005).
F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, Nature [**442**]{}, 766 (2006).
F. H. L. Koppens, K. C. Nowack, and L. M. K. Vandersypen, Phys. Rev. Lett. [**100**]{}, 236802 (2008).
P. F. Braun, X. Marie, L. Lombez, B. Urbaszek, T. Amand, P. Renucci, V. K. Kalevick, K. V. Kavokin, O. Krebs, P. Voisin, and Y. Masumoto, Phys. Rev. Lett. [**94**]{}, 116601 (2005).
J. Schliemann, A.V. Khaetskii, and D. Loss , J. Phys.: Condens. Mat. **15**, R1809 (2003).
W. Zhang, N. Konstantinidis, K. A. Al-Hassanieh, and V. V. Dobrovitski, J. Phys.: Condens. Mat. [**19**]{}, 083202 (2007).
D. Klauser, D. V. Bulaev, W. A. Coish, and D. Loss, in *Semiconductor Quantum Bits*, edited by O. Benson and F. Henneberger (Pan Stanford Publishing, 2008)
W. A. Coish and J. Baugh, phys. stat. sol. B [**246**]{}, 2203 (2009).
J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, and M. D. Lukin, Phys. Rev. B **76**, 035315 (2007).
J. M. Taylor, A. Imamoglu, and M. D. Lukin, Phys. Rev. Lett. **91**, 246802 (2003).
H. Schwager, J. I. Cirac, and G. Giedke, Phys. Rev. B **81**, 045309 (2010)
H. Schwager, J. I. Cirac, and G. Giedke, arXiv:0903.1727 (2009).
H. Christ, J. I. Cirac, and G. Giedke, Solid State Sciences **11**, 965-969 (2009).
H. Christ, J. I. Cirac, and G. Giedke, Phys. Rev. B **75**, 155324 (2007).
H. Christ, J. I. Cirac, and G. Giedke, Phys. Rev. B **78**, 125314 (2008).
W. A. Coish and D. Loss, Phys. Rev. B **70**, 195340 (2004).
W. A. Coish and D. Loss, Phys. Rev. B **72**, 125337 (2005).
D. Klauser, W.A. Coish and D. Loss, Phys. Rev. B **73**, 205302 (2006).
D. Klauser, W.A. Coish and D. Loss, Phys. Rev. B **78**, 205301 (2006).
M. Gaudin, J. Phys. (Paris) **37**, 1087 (1976).
M. Bortz and J. Stolze, Phys. Rev. B **76**, 014304 (2007).
J. Schliemann, Phys. Rev. B **81**, 081301(R) (2010).
M. Bortz, S. Eggert, and J. Stolze, Phys. Rev. B **81**, 035315 (2010)
J. Schliemann, A. V. Khaetskii, and D. Loss, Phys. Rev. B **66**, 245303 (2002).
M. Bortz and J. Stolze, J. Stat. Mech. P06018 (2007).
B. Erbe and H.-J. Schmidt, J. Phys. A: Math. Theor. **43**, 085215 (2010).
W. K. Wootters, Phys. Rev. Lett. **80**, 2245 (1998).
Note that, compared to the “natural” time unit $[\hbar/(A/N)]$, we introduce a factor $2$ on the time scale. This is convenient in order to compare our results with those for the homogeneous Gaudin system given in Ref. [@BorSt07].
J.M. Kikkawa and D. Awschalom, Phys. Rev. Lett. **80**, 4313 (1998)
I.A. Merkulov, Al.L. Efros and M. Rosen, Phys. Rev. B **65**, 205309 (2002)
F. Schwabl, *Quantum Mechanics*, (Springer, Berlin 2002) chapter 10.3.
|
---
abstract: |
*Random $s$-intersection graphs* have recently received much interest in a wide range of application areas. Broadly speaking, a random $s$-intersection graph is constructed by first assigning each vertex a set of items in some *random* manner, and then putting an undirected edge between all pairs of vertices that share at least $s$ items (the graph is called a *random intersection graph* when $s=1$). A special case of particular interest is a *uniform random $s$-intersection graph*, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a *binomial random $s$-intersection graph*, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random $s$-intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random $s$-intersection graphs).
In this paper, for binomial/uniform random $s$-intersection graphs, we present results related to $k$-connectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zero–one laws for the following three properties: (i) $k$-vertex-connectivity, (ii) $k$-edge-connectivity and (iii) the property of minimum vertex degree being at least $k$.
author:
- |
Jun Zhao Osman Yağan Virgil Gligor\
Carnegie Mellon University[^1]
title: |
On $k$-Connectivity and Minimum Vertex Degree\
in Random $s$-Intersection Graphs
---
**Keywords—**Random intersection graph, random key graph, connectivity, secure sensor network.
Introduction
============
*Random $s$-intersection graphs* have received considerable attention recently [@Rybarczyk; @bloznelis2013; @Assortativity; @mil10; @ZhaoYaganGligor; @Perfectmatchings; @JZISIT14; @Bloznelis201494; @ball2014; @fullver; @r4; @Y; @X]. In such a graph, each vertex is equipped with a set of items in some *random* manner, and two vertices establish an undirected edge in between if and only if they have at least $s$ items in common. A large amount of work [@r1; @herdingRKG; @PES:6114960; @YaganThesis; @Shang; @ZhaoAllerton; @virgil; @GodehardtJaworski; @ryb3; @zz; @2013arXiv1301.0466R; @CohenThesis; @ZhaoCDC; @DBLP:journals/corr/abs-1301-7320; @Jaworski20062152; @ISIT; @BloznelisD13; @Nikoletseas:2008:LIS:1414105.1414429; @RSA:RSA20005; @Karonski99] study the case of $s$ being $1$, under which the graphs are simply referred to as *random intersection graphs*.
Random ($s$-)intersection graphs have been used to model secure wireless sensor networks [@Rybarczyk; @JZISIT14; @adrian; @ISIT; @virgil; @YaganThesis; @qcomp_kcon; @ZhaoAllerton; @ANewell; @zz; @2013arXiv1301.0466R], wireless frequency hopping [@ZhaoAllerton], epidemics in human populations [@ball2014; @mil10], small-world networks [@5383986], trust networks [@virgillncs; @Ysb], social networks [@Assortativity; @bloznelis2013; @mil10; @ZhaoYaganGligor; @r4] such as collaboration networks [@Assortativity; @bloznelis2013; @mil10] and common-interest networks [@ZhaoYaganGligor; @r4]. Random intersection graphs also motivated Beer *et al.* [@beer2011vertex-journal; @beer2011vertex-conf] to introduce a general concept of [*vertex random graphs*]{} that subsumes any graph model where [*random*]{} features are assigned to vertices, and edges are drawn based on deterministic relations between the features of the vertices.
Among different models of random $s$-intersection graphs, two widely studied models are the so-called *uniform random $s$-intersection graph* and *binomial random $s$-intersection graph* defined in detail below.
Graph models
------------
#### Uniform random $s$-intersection graph.
A *uniform random $s$-intersection graph*, denoted by $G_s(n,K_n,P_n)$, is defined on $n$ vertices as follows. Each vertex *independently* selects $K_n$ different items *uniformly at random* from a pool of $P_n$ distinct items. Two vertices have an edge in between if and only if they share at least $s$ items. The notion “uniform” means that all vertices have the same number of items (but likely different sets of items). Here $K_n$ and $P_n$ are both functions of $n$, while $s$ does not scale with $n$. It holds that $1\leq s\leq K_n \leq
P_n$. Under $s=1$, the graph is also known as a *random key graph* [@mobihocQ1; @yagan; @5383986].
#### Binomial random $s$-intersection graph.
A *binomial random $s$-intersection graph*, denoted by $H_s(n,t_n,P_n)$, is defined on $n$ vertices as follows. Each item from a pool of $P_n$ distinct items is assigned to each vertex *independently* with probability $t_n$. Two vertices establish an edge in between if and only if they have at least $s$ items in common. The term “binomial” is used since the number of items assigned to each vertex follows a binomial distribution with parameters $P_n$ (the number of trials) and $t_n$ (the success probability in each trial). Here $t_n$ and $P_n$ are both functions of $n$, while $s$ does not scale with $n$. Also it holds that $1\leq
s \leq P_n$.
Problem Statement.
------------------
Our goal in this paper is to investigate properties related to *$k$-connectivity* and *minimum vertex degree* of random $s$-intersection graphs (*$k$-vertex-connectivity* and *$k$-edge-connectivity* are called together as $k$-connectivity for convenience). In particular, we wish to answer the following question:
For a uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ (resp., a binomial random $s$-intersection graph $H_s(n,t_n,P_n)$), with parameters $K_n$ (resp., $t_n$) and $P_n$ scaling with the number of vertices $n$, what is the asymptotic behavior of the probabilities for $G_s(n,K_n,P_n)$ (resp., $H_s(n,t_n,P_n)$) (i) being $k$-vertex-connected, (ii) being $k$-edge-connected, and (iii) having a minimum vertex degree at least $k$, respectively, as $n$ grows large? A graph is said to be $k$-vertex-connected if the remaining graph is still connected despite the deletion of at most $(k-1)$ arbitrary vertices, and $k$-edge-connectivity is defined similarly for the deletion of edges [@Bollobas]; with $k=1$, these definitions reduce to the standard notion of [*graph connectivity*]{} [@citeulike:4012374; @Z]. The degree of a vertex is defined as the number of edges incident on it. The three graph properties considered here are related to each other in that $k$-vertex-connectivity implies $k$-edge-connectivity, which in turn implies that the minimum vertex degree is at least $k$ [@Bollobas].
Summary of Results.
-------------------
We summarize our results below, first for a uniform random $s$-intersection graph and then for a binomial random $s$-intersection graph. Throughout the paper, both $s$ and $k$ are positive integers and do not scale with $n$. Also, naturally we consider $1 \leq s \leq K_n \leq P_n$ for graph $G_s(n,K_n,P_n)$ and $1 \leq s \leq P_n$ for graph $H_s(n,t_n,P_n)$. We use the standard Landau asymptotic notation $\Omega(\cdot), \omega(\cdot), O(\cdot),
o(\cdot),\Theta(\cdot)$. $\mathbb{P}[\mathcal {E}]$ denotes the probability that event $\mathcal {E}$ happens.
**$k$-Connectivity & minimum vertex degree in uniform random $s$-intersection graphs:**\
For a uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $P_n = \Omega(n)$, with sequence $\alpha_n$ defined by $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n +
{(k-1)} \ln \ln n + {\alpha_n}}{n}, \label{crit}\end{aligned}$$ then as $n \to \infty$, if $ \alpha_n \to \alpha^{\star} \in
[-\infty, \infty]$, the following convergence results hold:\
$
\mathbb{P} \left[\hspace{1.5pt}G_s(n,K_n,P_n)\textrm{ is
$k$-vertex-connected}.\hspace{1pt}\right] \to e^{-
\frac{e^{-\alpha^{\star}}}{(k-1)!}}$,\
$ \mathbb{P} \left[\hspace{1pt}G_s(n,K_n,P_n) \textrm{ is
$k$-edge-connected}.\hspace{1.5pt}\right] \to e^{-
\frac{e^{-\alpha^{\star}}}{(k-1)!}}$,\
and $$\begin{aligned}
\mathbb{P}\left[
\begin{array}{l} \hspace{-3pt} G_s(n,K_n,P_n) \textrm{ has a minimum\hspace{-3pt}}
\\ \hspace{-3pt}\textrm{vertex degree at least }k.
\end{array} \right] & \to e^{- \frac{e^{-\alpha^{\star}}}{(k-1)!}} . ~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
**$k$-Connectivity & minimum vertex degree in binomial random $s$-intersection graphs:**\
For a binomial random $s$-intersection graph $H_s(n,t_n,P_n)$ under $P_n = \Omega(n)$ for $s\geq 2$ or $P_n = \Omega(n^c)$ for $s=1$ with some constant $c>1$, with sequence $\beta_n$ defined by $$\begin{aligned}
\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} & = \frac{\ln n + {(k-1)}
\ln \ln n + {\beta_n}}{n}, \nonumber\end{aligned}$$ then as $n \to \infty$, if $ \beta_n \to \beta^{\star} \in [-\infty,
\infty]$, the following convergence results hold:\
$ \mathbb{P}
\left[\hspace{1.5pt}H_s(n,t_n,P_n)\textrm{ is
$k$-vertex-connected}.\hspace{1.5pt}\right] \to
e^{- \frac{e^{-\beta^{\star}}}{(k-1)!}}$,\
$ \mathbb{P} \left[\hspace{1.5pt}H_s(n,t_n,P_n) \textrm{ is
$k$-edge-connected}.\hspace{1.5pt}\right] \to e^{-
\frac{e^{-\beta^{\star}}}{(k-1)!}}$,\
and $$\begin{aligned}
\mathbb{P}\left[
\begin{array}{l} \hspace{-3pt} H_s(n,t_n,P_n) \textrm{ has a minimum\hspace{-3pt}}
\\ \hspace{-3pt}\textrm{vertex degree at least }k.
\end{array} \right] & \to e^{- \frac{e^{-\beta^{\star}}}{(k-1)!}} . ~~~~~~~~~~~~~~~\nonumber\end{aligned}$$
Since the probability $e^{- \frac{e^{-\alpha^{\star}}}{(k-1)!}}$ (resp., $e^{- \frac{e^{-\beta^{\star}}}{(k-1)!}}$) equals $1$ if $\alpha^{\star} = \infty$ (resp., $\beta^{\star} = \infty$) and $0$ if $ \alpha^{\star} = -\infty$ (resp., $ \beta^{\star} = -\infty$), the above results of asymptotically exact probabilities also imply the corresponding zero–one laws, where a zero–one law [@yagan] means that the probability that the graph has certain property asymptotically converges to $0$ under some conditions and converges to $1$ under some other conditions.
Comparison with related work.
-----------------------------
Table \[table:related-work\] summarizes relevant work in the literature on uniform/binomial random $s$-intersection graphs in terms of $k$-vertex-connectivity, $k$-edge connectivity, and the property of minimum vertex degree being at least $k$.
Among the related work, Bloznelis and Rybarczyk [@Bloznelis201494] recently also derived the asymptotically exact probabilities of *uniform* random $s$-intersection graphs (but not of *binomial* random $s$-intersection graphs) for the three properties above (the easily implied results on $k$-edge-connectivity were not explicitly mentioned). Yet, when $s$ is a constant or $O(1)$ as in many applications, their results require $K_n = O\big((\ln
n)^{\frac{1}{5s}}\big)$ for $k$-connectivity ([$k$-vertex-connectivity]{} and [$k$-edge-connectivity]{}), under a scaling the same as in Equation (\[crit\]). In other words, the one-law part of $k$-connectivity is as follows: under certain conditions including $K_n = O\big((\ln
n)^{\frac{1}{5s}}\big)$, if $\frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}} = \frac{\ln n + {(k-1)} \ln \ln n +
\omega(1)}{n}$, then a uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ is $k$-connected with a probability converging to $1$ as $n\to\infty$. From $K_n = O\big((\ln n)^{\frac{1}{5s}}\big)$ and $\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} = \frac{\ln n +
{(k-1)} \ln \ln n + \omega(1)}{n}$, it is straightforward to derive $P_n = O\big(n^{\frac{1}{s}} (\ln n)^{-\frac{3}{5s}}\big)$; i.e., $P_n =\widetilde{O}\big(n^{\frac{1}{s}}\big)$ ignoring the $\ln n$ terms. However, in secure wireless sensor network applications where uniform random $s$-intersection graphs are widely investigated, conditions $K_n = O\big((\ln
n)^{\frac{1}{5s}}\big)$ and $P_n
=\widetilde{O}\big(n^{\frac{1}{s}}\big)$ are both likely impractical because $K_n$ and $P_n$ are often at least on the order of $\ln n$ and $n$, respectively, to ensure that the network has reasonable resiliency against sensor capture attacks [@adrian; @virgil; @YaganThesis]. The results reported in this paper cover the practical range where $P_n$ is at least on the order of $n$.
[![width 1.15pt]{}l|l|l|l|l![width 1.15pt]{}]{} & Property & Results & Work\
& & & & **this paper**\
& & & &
--------------------------------------------
[@Bloznelis201494] (only for
$K_n = O\big((\ln n)^{\frac{1}{5s}}\big)$)
--------------------------------------------
\
& &
-----------------------------
connectivity &
min. vertex degree $\geq 1$
-----------------------------
& exact probabilities &
--------------------------------------------
[@Perfectmatchings] (only for
$K_n = O\big((\ln n)^{\frac{1}{5s}}\big)$)
--------------------------------------------
\
& & & exact probabilities & [@ZhaoCDC]\
& & & zero–one laws & [@zz; @ISIT]\
& & & exact probabilities & [@ryb3]\
& & & zero–one laws & [@r1; @yagan]\
& & & exact probabilities &\
& & & zero–one laws &\
& & & exact probabilities &\
& & & zero–one laws &\
& & & exact probabilities & [@ZhaoCDC]\
& & & zero–one laws & [@zz]\
& & & exact probabilities & [@2013arXiv1301.0466R]\
& & & zero–one laws & [@CohenThesis; @Shang]\
Roadmap.
--------
We organize the rest of the paper as follows. We detail the main results in Section \[sec:res\]. Sections \[sec:basic:ideas\] and \[sec:prf:thm:bin\] detail the steps of establishing the theorems. We conclude the paper in Section \[sec:Conclusion\]. The Appendix provides additional arguments used in proving the theorems.
\[table:related-work\]
[![width 1.15pt]{}l|l|l|l![width 1.15pt]{}]{} Graph & Property & Results & Work\
& & exact probabilities & **this paper**\
& & zero–one law & [[@ZhaoYaganGligor; @ISIT] ]{}\
& & exact probabilities & **this paper**\
& & zero–one law & [[@yagan_onoff]]{}\
& &exact probabilities & **this paper**\
& & zero–one law & [[@zz](implicitly)]{}\
& & exact probabilities & [@ryb3]\
& & zero–one law &[@r1; @yagan]\
$\begin{array}{l} \textrm{Random ($1$-)intersection graph}
\\[-2pt] \textrm{$G_1(n,K_n,P_n)$}
\end{array}$
Main Results {#sec:res}
============
Below we explain the main results of uniform random $s$-intersection graphs and binomial random $s$-intersection graphs, respectively.
Results of uniform random $s$-intersection graphs.
--------------------------------------------------
The following theorem presents results on $k$-connectivity and minimum vertex degree in a uniform random $s$-intersection graph $G_s(n,K_n,P_n)$.
\[thm:uni\] For a uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $$\begin{aligned}
P_n = \Omega(n),\vspace{-3pt} \label{eqPnOmegan}\end{aligned}$$ with sequence $\alpha_n$ defined by $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n +
{(k-1)} \ln \ln n + {\alpha_n}}{n},
\vspace{-5pt}\label{thm:uni:eq:edge}\end{aligned}$$ it holds that $$\begin{aligned}
& \lim\limits_{n \to \infty} \mathbb{P}[\hspace{2pt} G_s(n,K_n,P_n)
\textrm{ is
$k$-vertex-connected.} \hspace{2pt} ]
\nonumber \\ & = \lim\limits_{n \to \infty}
\mathbb{P}[\hspace{2pt} G_s(n,K_n,P_n) \textrm{ is
$k$-edge-connected.} \hspace{2pt} ]
\nonumber \\ & = \lim\limits_{n \to \infty} \mathbb{P}\left[
\begin{array}{l} \hspace{-4pt} G_s(n,K_n,P_n) \textrm{ has a minimum\hspace{-4pt}}
\\ \hspace{-4pt}\textrm{vertex degree at least }k.
\end{array} \right] \nonumber \\ & = \begin{cases} e^{- \frac{e^{-\alpha ^*}}{(k-1)!}}, \hspace{0.1cm} \textrm{if
}\lim\limits_{n \to \infty}{\alpha_n} = \alpha^* \in
(-\infty,\infty),
\\0, \hspace{1.1cm} \textrm{if }\lim\limits_{n \to
\infty}{\alpha_n} = -\infty, \\ 1, \hspace{1.1cm} \textrm{if
}\lim\limits_{n \to \infty}{\alpha_n} = \infty.
\end{cases} \nonumber\end{aligned}$$
For graph $G_s(n,K_n,P_n)$, Theorem \[thm:uni\] presents the asymptotically exact probabilities and zero–one laws for the following three properties: (i) $k$-vertex connectivity, (ii) $k$-edge-connectivity and (iii) the property of minimum vertex degree being at least $k$.
By Lemma \[qn-dist\] on Page , under (\[eqPnOmegan\]) and (\[thm:uni:eq:edge\]) with constrained $|\alpha_n |= O(\ln \ln n )$, we can show that the left hand side of (\[thm:uni:eq:edge\]), i.e., $\frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}} $, is asymptotically equivalent to the edge probability of graph $G_s(n,K_n,P_n)$. As given in Lemma 13 of the full version [@fullver], with $q_n$ denoting the edge probability of graph $G_s(n,K_n,P_n)$, if condition (\[thm:uni:eq:edge\]) is replaced by $q_n = \frac{\ln n + {(k-1)}
\ln \ln n + {\alpha_n}}{n}$, and condition (\[eqPnOmegan\]) is kept unchanged, then all results in Theorem \[thm:uni\] still follow. Hence, the uniform random $s$-intersection graph model under condition (\[eqPnOmegan\]) exhibits the same behavior as the well-known Erdős-Rényi graph model [@citeulike:4012374], in the sense that for each of (i) $k$-vertex-connectivity, (ii) $k$-edge-connectivity and (iii) the property of minimum vertex degree being at least $k$, a common point for the phase transition from a zero-law to a one-law occurs when the edge probability equals $\frac{\ln n + {(k-1)} \ln
\ln n}{n}$.
[^2]
For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under ..., the probability that the vertex connectivity, the edge connectivity, and the minimum vertex degree all equal converges to 1 as $n\to \infty$.
Results for binomial random $s$-intersection graphs.
----------------------------------------------------
The following theorem presents results on $k$-connectivity and minimum vertex degree in a binomial random $s$-intersection graph $H_s(n,t_n,P_n)$.
\[thm:bin\] For a binomial random $s$-intersection graph $H_s(n,t_n,P_n)$ under $$\begin{aligned}
\begin{cases}
P_n = \Omega(n), &\textrm{ for } s \geq 2, \\
P_n = \Omega(n^c)\textrm{ for some constant }c>1, &\textrm{ for } s
= 1,
\end{cases}
\label{thm:bin:eq:P}\end{aligned}$$ with sequence $\beta_n$ defined by $$\begin{aligned}
\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} & = \frac{\ln n + {(k-1)}
\ln \ln n + {\beta_n}}{n}, \label{thm:bin:eq:edge}\end{aligned}$$ it holds that $$\begin{aligned}
& \lim\limits_{n \to \infty} \mathbb{P}[\hspace{2pt} H_s(n,t_n,P_n)
\textrm{ is
$k$-vertex-connected.} \hspace{2pt} ]
\nonumber \\ & = \lim\limits_{n \to \infty}
\mathbb{P}[\hspace{2pt} H_s(n,t_n,P_n) \textrm{ is
$k$-edge-connected.} \hspace{2pt} ]
\nonumber \\ & = \lim\limits_{n \to \infty} \mathbb{P}\left[
\begin{array}{l} \hspace{-4pt} H_s(n,t_n,P_n) \textrm{ has a minimum\hspace{-4pt}}
\\ \hspace{-4pt}\textrm{vertex degree at least }k.
\end{array} \right] \nonumber \\ & = \begin{cases} e^{- \frac{e^{-\beta ^*}}{(k-1)!}}, \hspace{0.1cm} \textrm{if
}\lim\limits_{n \to \infty}{\beta_n} = \beta^* \in (-\infty,\infty),
\\0, \hspace{1.1cm} \textrm{if }\lim\limits_{n \to
\infty}{\beta_n} = -\infty, \\ 1, \hspace{1.1cm} \textrm{if
}\lim\limits_{n \to \infty}{\beta_n} = \infty.
\end{cases} \nonumber\end{aligned}$$
For graph $H_s(n,t_n,P_n)$, Theorem \[thm:bin\] presents the asymptotically exact probabilities and zero–one laws for the following three properties: (i) $k$-vertex connectivity, (ii) $k$-edge-connectivity and (iii) the property of minimum vertex degree being at least $k$.
By Lemma 12 of the full version [@fullver], under (\[thm:bin:eq:P\]) and (\[thm:bin:eq:edge\]) with constrained $|\beta_n| = O(\ln \ln n )$, we can show that the left hand side of (\[thm:bin:eq:edge\]), i.e., $\frac{1}{s!} \cdot
{t_n}^{2s}{P_n}^{s} $, is asymptotically equivalent to the edge probability of graph $H_s(n,t_n,P_n)$. As given in Lemma 14 of the full version [@fullver], with $\rho_n$ denoting the edge probability of graph $H_s(n,t_n,P_n)$, if condition (\[thm:bin:eq:edge\]) is replaced by $\rho_n = \frac{\ln n +
{(k-1)} \ln \ln n + {\beta_n}}{n}$, and condition (\[thm:bin:eq:P\]) is kept unchanged, then all results in Theorem \[thm:bin\] still follow. Therefore, the binomial random $s$-intersection graph model under condition (\[thm:bin:eq:P\]) exhibits the same behavior with Erdős-Rényi graph model, in the sense that for each of (i) $k$-vertex-connectivity, (ii) $k$-edge-connectivity, and (iii) the property of minimum vertex degree being at least $k$, a common point for the phase transition from a zero-law to a one-law occurs when the edge probability equals $\frac{\ln n + {(k-1)} \ln
\ln n}{n}$.
The condition (\[thm:bin:eq:P\]) has $P_n = \Omega(n)$ for $s \geq
2$, and requires a stronger one for $s=1$: $P_n = \Omega(n^c)$ for some constant $c>1$. The range $P_n = \Theta(n)$ is covered by $P_n
= \Omega(n)$, but not by $P_n = \Omega(n^c)$ with $c>1$. For $s=1$ and $P_n = \Theta(n)$, results for $k$-vertex-connectivity, $k$-edge connectivity, and the property of minimum vertex degree being at least $k$ use a scaling different from (\[thm:bin:eq:edge\]), as given by [@zz Theorem 4].
For binomial random $s$-intersection graph $G_s(n,K_n,P_n)$ under ..., the probability that the vertex connectivity, the edge connectivity, and the minimum vertex degree all equal converges to 1 as $n\to \infty$.
Lemmas
======
For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $$\begin{aligned}
P_n = \omega\big(n^{\frac{1}{s}} (\ln n)^{-\frac{1}{s}}\big)\end{aligned}$$ with sequence $a_n$ defined by $$\begin{aligned}
\frac{1}{s!} \cdot
\frac{{[K_n(K_n-1)\ldots(K_n-(s-1))]}^2}{{P_n}^{s}} &
= \frac{\ln n + {(k-1)} \ln \ln n + {a_n}}{n}.\end{aligned}$$ then $$\begin{aligned}
& \lim\limits_{n \to \infty}\mathbb{P}[\hspace{2pt} G_s(n,K_n,P_n)
\textrm{ has
a minimum vertex degree at least $k$.} \hspace{2pt} ]
\nonumber \\ & =
\begin{cases}0, &\textrm{if
}\lim\limits_{n \to \infty}{a_n} = -\infty, \\ 1, &\textrm{if
}\lim\limits_{n \to \infty}{a_n} = \infty, \\ e^{- \frac{e^{-a
^*}}{(k-1)!}}, &\textrm{if }\lim\limits_{n \to \infty}{a_n} = a^*
\in (-\infty,\infty). \end{cases} \nonumber\end{aligned}$$
Establishing Theorem \[thm:uni\] {#sec:basic:ideas}
================================
Theorem \[thm:uni\] in the special case of $s = 1$ is proved by us [@ZhaoCDC]. Below we explain the steps of establishing Theorem \[thm:uni\] for $s \geq 2$. In Section \[sec:confine:alpha\], we show that $|\alpha_n|$ can be confined as $O(\ln \ln n) $ in proving Theorem \[thm:uni\]. In Section \[sec:del:vem\], we consider the relationships between vertex connectivity, edge connectivity, and minimum vertex degree.
[![width 1.15pt]{}l|l|l|l|l![width 1.15pt]{}]{} Symbol & Meaning & & Symbol & Meaning\
$\kappa_v$ & vertex connectivity & & $\mathcal{V}_n $ & the set of vertices: $\{v_1, v_2, \ldots, v_n\}$\
$\kappa_e$ & edge connectivity & & $S_i$ & the number of items on vertex $v_i$\
$\delta$ & minimum vertex degree & & $E_{ij}$ &\
$q_n$ & edge probability & & $r_n$ & $\min \big( {\big \lfloor \frac{P_n}{K_n} \big
\rfloor}, \big \lfloor \frac{n}{2} \big \rfloor \big)$\
Confining $|\alpha_n|$. {#sec:confine:alpha}
-----------------------
To confine $|\alpha_n|$ as $O(\ln \ln n) $ in proving Theorem \[thm:uni\], we will demonstrate $$\begin{aligned}
\textrm{Theorem \ref{thm:uni} under }|\alpha_n | = O(\ln \ln n)
\Rightarrow \textrm{Theorem \ref{thm:uni}}. \label{cp_alph}\end{aligned}$$ Note that $k$-vertex-connectivity, $k$-edge-connectivity, and the property of minimum vertex degree being at least $k$ are all monotone increasing[^3]. For any monotone increasing property $\mathcal {I}$, the probability that a spanning subgraph (resp., supergraph) of graph $G$ has $\mathcal {I}$ is at most (resp., at least) the probability of $G$ having $\mathcal {I}$. Therefore, to show (\[cp\_alph\]), it suffices to prove the following lemma.
\[graph\_Gs\_cpl\]
**(a)** For graph $G_s(n,K_n,P_n)$ under $P_n = \Omega(n)$ and $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n +
{(k-1)} \ln \ln n + {\alpha_n}}{n} \label{al1-parta}\end{aligned}$$ with $\lim_{n \to \infty}\alpha_n = -\infty$, there exists graph $G_s(n,\widetilde{K_n},\widetilde{P_n})$ under $\widetilde{P_n} =
\Omega(n)$ and $$\begin{aligned}
\frac{1}{s!} \cdot
\frac{{\widetilde{K_n}}^{2s}}{{\widetilde{P_n}}^{s}} & = \frac{\ln
n + {(k-1)} \ln \ln n + {\widetilde{\alpha_n}}}{n} \label{al0-parta}\end{aligned}$$ with $\lim_{n \to \infty}\widetilde{\alpha_n} = -\infty$ and $\widetilde{\alpha_n} = -O(\ln \ln n)$, such that there exists a graph coupling[^4] under which $G_s(n,K_n,P_n)$ is a spanning subgraph of $G_s(n,\widetilde{K_n},\widetilde{P_n})$.
**(b)** For graph $G_s(n,K_n,P_n)$ under $P_n = \Omega(n)$ and $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n +
{(k-1)} \ln \ln n + {\alpha_n}}{n} \label{al1}\end{aligned}$$ with $\lim_{n \to \infty}\alpha_n = \infty$, there exists graph $G_s(n,\widehat{K_n},\widehat{P_n})$ under $\widehat{P_n} =
\Omega(n)$ and $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{\widehat{K_n}}^{2s}}{{\widehat{P_n}}^{s}}
& = \frac{\ln n + {(k-1)} \ln \ln n + {\widehat{\alpha_n}}}{n}
\label{al0}\end{aligned}$$ with $\lim_{n \to \infty}\widehat{\alpha_n} = \infty$ and $\widehat{\alpha_n} = O(\ln \ln n)$, such that there exists a graph coupling under which $G_s(n,K_n,P_n)$ is a spanning supergraph of $G_s(n,\widehat{K_n},\widehat{P_n})$.
The proof of Lemma \[graph\_Gs\_cpl\] is provided in Section \[sec\_graph\_Gs\_cpl\] in the Appendix.
Relationships between vertex connectivity, edge connectivity, and minimum vertex degree. {#sec:del:vem}
----------------------------------------------------------------------------------------
Recall that the vertex connectivity of a graph is defined as the minimum number of vertices needing to be deleted to have the remaining graph disconnected, and the edge connectivity is defined similarly for the deletion of edges [@Bollobas]. For graph $G_s(n,K_n,P_n)$, we use $\kappa_v, \kappa_e$ and $\delta$ to denote the vertex connectivity, the edge connectivity, and the minimum vertex degree, respectively. Then $k$-vertex-connectivity, $k$-edge-connectivity, and the property of minimum vertex degree being at least $k$, are given by events $\kappa_v \geq k$, $\kappa_e \geq k$, and $\delta \geq k$, respectively. For any graph, the vertex connectivity is at most the edge connectivity, and the edge connectivity is at most the minimum vertex degree [@Bollobas; @ZhaoISIT2014; @FJYGISIT2014]. Therefore, $\kappa_v \leq \kappa_e \leq \delta$ holds. Then $$\begin{aligned}
\mathbb{P}[\hspace{2pt} \kappa_v \geq k\hspace{2pt}] & \leq \mathbb{P}[ \hspace{2pt}\kappa_e \geq k\hspace{2pt}] \leq \mathbb{P}[\hspace{2pt}\delta \geq k\hspace{2pt}], \label{eq_kpv1}
\end{aligned}$$ and $$\begin{aligned}
\mathbb{P}[ \kappa_v \geq k ] &=
\mathbb{P}[\delta \geq k ] -
\mathbb{P}[\hspace{2pt}(\kappa_v < k) \cap (\delta \geq k)
\hspace{2pt}] \nonumber \\ & \geq \mathbb{P}[\hspace{2pt}\delta
\geq k \hspace{2pt}] - \sum_{\ell = 0}^{k-1}
\mathbb{P}[\hspace{2pt}(\kappa_v = \ell) \cap (\delta > \ell
)\hspace{2pt}]. \label{eq_kpv2}\end{aligned}$$
We will prove Lemmas \[lem:mvd\] and \[lem:mvd:con\] below. We explain that based with (\[eq\_kpv1\]) and (\[eq\_kpv2\]), Lemmas \[lem:mvd\] and \[lem:mvd:con\] imply Theorem \[thm:uni\] under $|\alpha_n | = O(\ln \ln n)$. We discuss the two cases respectively: (a) $\lim_{n \to \infty} \alpha_n = -\infty$, and (b) $\lim_{n \to \infty} \alpha_n = \alpha^{\star} \in (-\infty,
\infty]$.
\(a) For $\lim_{n \to \infty} \alpha_n = = -\infty$, under conditions (\[eqPnOmegan\]) and (\[thm:uni:eq:edge\]) with $|\alpha_n| = O(\ln \ln n)$ in Theorem \[thm:uni\], we use Lemma \[lem:mvd\] to derive $$\begin{aligned}
\lim_{n \to \infty} \mathbb{P}[\delta \geq k] & = 0.
\label{mnd-del-k}\end{aligned}$$ From (\[eq\_kpv1\]) and (\[mnd-del-k\]), it follows that $$\begin{aligned}
\lim_{n \to \infty} \mathbb{P}[ \kappa_v \geq k] & = \lim_{n \to \infty} \mathbb{P}[ \kappa_e \geq k] = 0. \label{eq_kpv-del}
\end{aligned}$$ Hence, for $\lim_{n \to \infty} \alpha_n = = -\infty$, the result in Theorem \[thm:uni\] follow in view of (\[mnd-del-k\]) and (\[eq\_kpv-del\]).
\(b) For $\lim_{n \to \infty} \alpha_n = \alpha^{\star} \in (-\infty, \infty]$,
Therefore, the proof is completed once we show Lemmas \[lem:mvd\] and \[lem:mvd:con\] below. Note that since $k$ is a constant, condition (\[thm:uni:eq:edge\]) with $|\alpha_n| = O(\ln \ln n)$ in Theorem \[thm:uni\] implies condition (\[pe-lnn-lnlnn-pm\]) in Lemma \[lem:mvd:con\].
\[lem:mvd\]
For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $P_n = \Omega(n)$, if there exists sequence $\alpha_n$ satisfying $|\alpha_n| = O(\ln \ln n)$ such that $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n +
{(k-1)} \ln \ln n + {\alpha_n}}{n},\nonumber\end{aligned}$$ then with $\delta$ denoting the minimum vertex degree, it holds that $$\begin{aligned}
\lim\limits_{n \to \infty} \mathbb{P}[\delta \geq k] & =
e^{- \frac{e^{-\alpha^{\star}}}{(k-1)!}}, \textrm{ if }\lim_{n \to \infty} \alpha_n = \alpha^{\star} \in [-\infty, \infty].
\label{mnd}\end{aligned}$$
\[lem:mvd:con\] For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $P_n = \Omega(n)$ and $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n
\pm O(\ln \ln n)}{n} ,\label{pe-lnn-lnlnn-pm}\end{aligned}$$ then with $\kappa_v$ denoting the vertex connectivity and $\delta$ denoting the minimum vertex degree, it holds for constant integer $\ell$ that $$\begin{aligned}
\mathbb{P}[\hspace{2pt}(\kappa_v = \ell) \cap (\delta > \ell
)\hspace{2pt}] &= o(1). \label{mnd_kcon}\end{aligned}$$
We detail the proof of Lemma \[lem:mvd:con\] below.
The proof of Lemma \[lem:mvd:con\].
-----------------------------------
For graph $G_s(n,K_n,P_n)$, let the set of vertices be $
\mathcal{V}_n = \{ v_1, v_2, \ldots , v_n \}$. Also, for $i=1,2,\ldots, n$, we let $S_i$ denote the set of items on vertex $v_i$. We introduce event ${\mathcal{E} (\boldsymbol{J})}$ in the following manner: $$\begin{aligned}
{\mathcal{E} (\boldsymbol{J})} &= \bigcup_{\begin{subarray}{c} T
\subseteq \mathcal{V}_n, \\ |T| \geq 2.\end{subarray}} ~ \left[
\hspace{2pt} |\cup_{v_i \in T} S_i|~\leq~{J}_{ |T|}
\hspace{2pt}\right], \label{eq:E_n_defnex}\end{aligned}$$ where $\boldsymbol{J} =[{J}_{2} , {J}_{3}, \ldots, {J}_{n } ]$ is an $(n-1)$-dimensional integer valued array, with $J_{i}$ defined through $$\begin{aligned}
J_{i} &=
\begin{cases}
\max\{ \left \lfloor (1+\varepsilon_1) K_n \right \rfloor ,
\left \lfloor \lambda_1 K_n i \right \rfloor \},~~i=2,\ldots, r_n,\\
\left \lfloor\mu_1 P_n \right \rfloor,~~~~~~~~~~~~~~~~~~~~~~~~~i=r_n+1, \ldots, n,
\end{cases} \label{olp_xjdef}\end{aligned}$$ for an arbitrary constant $0<\varepsilon_1<1$ and some positive constants $\lambda_1, \mu_1$ in Lemma \[prp:EJ\] below, where $r_n
:= \min \big( {\big \lfloor \frac{P_n}{K_n} \big \rfloor}, \big
\lfloor \frac{n}{2} \big \rfloor \big)$.
By a crude bounding argument, we get $$\begin{aligned}
{{\mathbb{P}}\left[{ \hspace{2pt}(\kappa_v = {\ell}) \cap (\delta > {\ell}
)\hspace{2pt}}\right]}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
\nonumber \leq {{\mathbb{P}}\left[{\hspace{2pt}{\mathcal{E} (\boldsymbol{J})}}\right]} +
{{\mathbb{P}}\left[{ (\kappa_v = {\ell}) \cap (\delta > {\ell} )
\cap\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}}\right]}.\end{aligned}$$ Hence, a proof of Lemma \[lem:mvd:con\] consists of proving two lemmas below. Under (\[thm:uni:eq:edge\]) with $|\alpha_n| = O(\ln
\ln n)$, we have $\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} =
\frac{\ln n \pm O(\ln \ln n)}{n} = o(1)$ and $\frac{K_n}{P_n} =
o(1)$, enabling us to use Lemmas \[prp:EJ\] and \[prp-kvl-del-EJ\].
\[prp:EJ\] If $ P_n = \Omega(n)$ and $\frac{K_n}{P_n} = o(1)$, then for an arbitrary constant $0<\varepsilon_1<1$ and some selected positive constants $\lambda_1,\mu_1$, it holds that $$\begin{aligned}
{{\mathbb{P}}\left[{\mathcal{E} (\boldsymbol{J})}\right]} = o(1).
\label{eq:OneLawAfterReductionPart1}\end{aligned}$$
By [@ZhaoYaganGligor Proposition 3], positive constants $\lambda_1$ and $ \mu_1$ are selected to ensure $$\begin{gathered}
\lambda_1, \mu_1 < \frac{1}{2} \nonumber \\
\max \left ( 2 \lambda_1 \sigma , \lambda_1 \left(
\frac{e^2}{\sigma} \right) ^{\frac{ \lambda_1 }{ 1 - 2 \lambda_1 } }
\right ) < 1, \nonumber\end{gathered}$$ and $$\begin{aligned}
\max \left ( 2 \left ( \sqrt{\mu_1} \left ( \frac{e}{ \mu_1 } \right
)^{\mu_1} \right )^\sigma, \sqrt{\mu_1} \left ( \frac{e}{ \mu_1 }
\right)^{\mu_1} \right ) < 1 . \nonumber\end{aligned}$$
\[prp-kvl-del-EJ\] For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $P_n = \Omega(n)$ and $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n
\pm O(\ln \ln n)}{n} , \label{cn-qn-2lnn-n}\end{aligned}$$ then $$\begin{aligned}
{{\mathbb{P}}\left[{ (\kappa_v = {\ell}) \cap (\delta >
{\ell} ) \cap\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}}\right]} =
o(1). \label{eq:OneLawAfterReductionPart2}\end{aligned}$$
The proof of Lemma \[prp-kvl-del-EJ\] is given in Section \[sec:prf:prop:OneLawAfterReductionPart2\] in the Appendix.
In proving, we have conditions $P_n = \Omega(n)$ and (\[thm:uni:eq:edge\]) with $\alpha_n = o(\ln n )$. As shown below, these conditions imply (\[cn-qn-2lnn-n\]).
Proposition \[prp-kvl-del-EJ\] is proved in Section \[sec:prf:prop:OneLawAfterReductionPart2\].
Under conditions $P_n = \Omega(n)$ and $\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} = \frac{\ln n \pm O(\ln \ln n) }{n}$, we use Lemma \[qn-dist\] to derive $$\begin{aligned}
q_n & = \frac{\ln n \pm O(\ln \ln n) }{n}. \nonumber\end{aligned}$$ Therefore, it holds that $$\begin{aligned}
q_n & \leq \frac{2 \ln n}{n}, \textrm{ for all $n$ sufficiently large}, \label{qnev-lnn-n}\end{aligned}$$ and there exists a positive constant $c_0$ such that $$\begin{aligned}
q_n & \geq \frac{\ln n - c_0 \ln \ln n }{n}, \textrm{ for all $n$ sufficiently large}. \label{qnev-lnn-c0lnln-n}\end{aligned}$$
For graph $G_q(n,K_n,P_n)$ under $\lim\limits_{n \to
\infty}\frac{K_n}{\ln n} = \infty$, with sequence $\beta_n$ defined by $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n +
{(k-1)} \ln \ln n + {\beta_n}}{n}, \nonumber\end{aligned}$$ where $k$ is a positive integer that does not scale with $n$, then as $n \to \infty$, $$\begin{aligned}
\mathbb{P}\bigg[\hspace{-3pt}
\begin{array}{c} G_q(n,K_n,P_n)\\ \textrm{is
$k$-connected.} \end{array}\hspace{-3pt}
\bigg] \hspace{-2pt}\to\hspace{-2pt}
\begin{cases}0, \hspace{28pt}\textrm{if
}\lim\limits_{n \to \infty}{\beta_n} \hspace{-2pt}=\hspace{-2pt}
-\infty, \\ 1, \hspace{28pt}\textrm{if }\lim\limits_{n \to
\infty}{\beta_n} \hspace{-2pt}=\hspace{-2pt} \infty, \\ e^{-
\frac{e^{-\beta ^*}}{(k-1)!}}, \textrm{if
}\hspace{-2pt}\lim\limits_{n \to \infty}\hspace{-2pt}{\beta_n} =
\hspace{-2pt}\beta^*\hspace{-2pt} \in\hspace{-2pt} (-\infty,\infty).
\end{cases} \nonumber\end{aligned}$$
\[thm:exact\_qcomposite2\] For graph $G_q(n,K_n,P_n)$ under $K_n
\hspace{-2pt}=\hspace{-2pt} \omega\big((\ln n)^3\big)$, $\frac{{K_n}^2}{P_n} = o(1)$ and $\frac{K_n}{P_n} =
o\big(\frac{1}{n}\big)$, with sequence $\alpha_n$ defined by $$\begin{aligned}
\frac{1}{q!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{q} & = \frac{\ln n
+ {(k-1)} \ln \ln n + {\alpha_n}}{n}, \nonumber\end{aligned}$$ where $k$ is a positive integer that does not scale with $n$, then as $n \to \infty$, $$\begin{aligned}
\mathbb{P}\bigg[
\begin{array}{c} G_q(n,K_n,P_n)\\ \textrm{is
$k$-connected.} \end{array}
\bigg] \to
\begin{cases} 1, &\textrm{if }\lim_{n \to \infty}{\alpha_n} =
\infty, \\ e^{- \frac{e^{-\alpha ^*}}{(k-1)!}}, &\textrm{if }\lim_{n
\to \infty}{\alpha_n} = \alpha^*, \\ 0, &\textrm{if }\lim_{n \to
\infty}{\alpha_n} = -\infty,\end{cases} \nonumber\end{aligned}$$ and $$\begin{aligned}
\mathbb{P}\bigg[
\begin{array}{c} G_q(n,K_n,P_n)\\ \textrm{is
$k$-robust.} \end{array}
\bigg] \to
\begin{cases} 1, &\textrm{if }\lim_{n \to \infty}{\alpha_n} =
\infty, \\ e^{- \frac{e^{-\alpha ^*}}{(k-1)!}}, &\textrm{if }\lim_{n
\to \infty}{\alpha_n} = \alpha^*, \\ 0, &\textrm{if }\lim_{n \to
\infty}{\alpha_n} = -\infty.\end{cases} \nonumber\end{aligned}$$
Establishing Theorem \[thm:bin\] {#sec:prf:thm:bin}
================================
Similar to the idea of confining $|\alpha_n |$ in Theorem \[thm:uni\], here we confine $|\beta_n |$ as $O(\ln \ln n)$ in Theorem \[thm:bin\]. Specifically, we will demonstrate $$\begin{aligned}
\textrm{Theorem \ref{thm:bin} under }|\beta_n | = O(\ln \ln n)
\Rightarrow \textrm{Theorem \ref{thm:bin}}. \label{cp_alph_bin}\end{aligned}$$ Since $k$-vertex-connectivity, $k$-edge-connectivity, and the property of minimum vertex degree being at least $k$, are all monotone increasing, then to show (\[cp\_alph\_bin\]), it suffices to prove the following lemma.
\[graph\_Hs\_cpln\]
**(a)** For graph $H_s(n,t_n,P_n)$ under $$\begin{aligned}
\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} & = \frac{\ln n + {(k-1)}
\ln \ln n + {\beta_n}}{n} \label{al0-parta-Hs-od}\end{aligned}$$ with $\lim_{n \to \infty}\beta_n = -\infty$, there exists graph $H_s(n,\widetilde{t_n}, \widetilde{P_n})$ under $$\begin{aligned}
\frac{1}{s!} \cdot {\widetilde{t_n}}^{2s}{\widetilde{P_n}}^{s} & =
\frac{\ln n + {(k-1)} \ln \ln n + {\widetilde{\beta_n}}}{n}
\label{al0-parta-Hs}\end{aligned}$$ with $\lim_{n \to \infty}\widetilde{\beta_n} = -\infty$ and $\widetilde{\beta_n} = -O(\ln \ln n)$ such that there exists a graph coupling under which $H_s(n,t_n,P_n)$ is a spanning subgraph of $H_s(n,\widetilde{t_n},\widetilde{P_n}) $.
**(b)** For graph $H_s(n,t_n,P_n)$ under $$\begin{aligned}
\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} & = \frac{\ln n + {(k-1)}
\ln \ln n + {\beta_n}}{n} \label{al0-parta-Hs-pb-od}\end{aligned}$$ with $\lim_{n \to \infty}\beta_n = \infty$, there exists graph $H_s(n,\widehat{t_n}, \widehat{P_n})$ under $$\begin{aligned}
\frac{1}{s!} \cdot {\widehat{t_n}}^{2s}{\widehat{P_n}}^{s} & =
\frac{\ln n + {(k-1)} \ln \ln n + {\widehat{\beta_n}}}{n}
\label{al0-parta-Hs-pb}\end{aligned}$$ with $\lim_{n \to \infty}\widehat{\beta_n} = \infty$ and $\widehat{\beta_n} = O(\ln \ln n)$ such that there exists a graph coupling under which $H_s(n,t_n,P_n)$ is a spanning supergraph of $H_s(n,\widehat{t_n},\widehat{P_n})$.
The proof of Lemma \[graph\_Hs\_cpln\] is detailed in Section \[sec:pro:graph\_Hs\_cpln\] in the Appendix.
\[lem:bin:mvd\] For binomial random $s$-intersection graph $H_s(n,t_n,P_n)$ under $P_n = \omega(\ln n)$, with sequence $\delta_n$ defined by $$\begin{aligned}
\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} & = \frac{\ln n + {(k-1)}
\ln \ln n + {\delta_n}}{n}, \label{thm:bin:eq:edge:del}\end{aligned}$$ then $$\begin{aligned}
\lim\limits_{n \to \infty} \mathbb{P}[\hspace{2pt} H_s(n,t_n,P_n)
\textrm{ has
a minimum vertex degree at least $k$.} \hspace{2pt} ]
& = 0, &\textrm{if
}\lim\limits_{n \to \infty}{\delta_n} = -\infty. \nonumber\end{aligned}$$
We use $\rho_n$ to denote the edge probability in binomial random $s$-intersection graph $H_s(n,t_n,P_n)$.
By [@Rybarczyk Proposition 2], it holds that $$\begin{aligned}
\rho_n & = \frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} \cdot \big[1\pm
O({t_n}^{2} P_n) - O({P_n}^{-1})\big].\label{thm:bin:eq:edge:rho}\end{aligned}$$
Using $\lim\limits_{n \to \infty}{\delta_n} = -\infty$ in (\[thm:bin:eq:edge:del\]), we have $$\begin{aligned}
\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} & = O\big(n ^{-1} \ln
n\big), \label{thm:bin:eq:edge:rho4}\end{aligned}$$ resulting in $$\begin{aligned}
{t_n}^{2} P_n & = o\bigg(\frac{1}{\ln n}\bigg).
\label{thm:bin:eq:edge:rho2}\end{aligned}$$
Applying (\[thm:bin:eq:edge:rho2\]) and $P_n = \omega(\ln n)$ to (\[thm:bin:eq:edge:rho\]), it follows that $$\begin{aligned}
\rho_n & = \frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} \cdot \bigg[1\pm
o\bigg(\frac{1}{\ln n}\bigg)\bigg]. \label{thm:bin:eq:edge:rho3}\end{aligned}$$ Then we use (\[thm:bin:eq:edge:del\]) and (\[thm:bin:eq:edge:rho4\]) in (\[thm:bin:eq:edge:rho3\]) to further obtain $$\begin{aligned}
\rho_n & = \frac{\ln n + {(k-1)} \ln \ln n + {\delta_n} \pm
o(1)}{n}. \label{thm:bin:eq:edge:rho5}\end{aligned}$$
In graph $H_s(n,t_n,P_n)$, let $N_{h}$ be the number of vertices with degree $h$ for non-negative integer $h$. Then $\mathbb{E}[N_h]$ denoting the expectation of $N_h$, is given by $$\begin{aligned}
\mathbb{E}[N_h] & = \binom{n-1}{h} {\rho_n}^h .
\label{thm:bin:eq:edge:rho6}\end{aligned}$$
$t_n P_n = \omega\big((\ln n)^3\big)$
Now we use Theorem \[thm:uni\] to prove Theorem \[thm:bin\] with confined $|\beta_n | = O(\ln \ln n)$. Here, the main idea is to exploit a [*coupling*]{} result between the uniform $s$-intersection graph and a binomial $s$-intersection graph. Let $\mathcal {I}^{*}$ denote either one of the following graph properties: $k$-vertex-connectivity, $k$-edge-connectivity, and the property of minimum vertex degree being at least $k$. With $K_n^{-}$ and $K_n^{+}$ defined by $$\begin{aligned}
K_n^{\pm} & =
t_n P_n \pm \sqrt{3\ln n (\ln n + t_n P_n)}
\label{KnminandKnplus},\vspace{-5pt}\end{aligned}$$ we have from Lemma \[lem:cp\] that if $t_n P_n = \omega(\ln n )$, then $$\begin{aligned}
\label{propIud} & \mathbb{P} \big[\hspace{2pt}\textrm{Graph
}G_s(n,K_n^{-},P_n)\textrm{
has $\mathcal {I}^{*}$}.\hspace{2pt}\big] - o(1) \\
& \leq \mathbb{P} \big[\hspace{2pt}\textrm{Graph
}H_s(n,t_n,P_n)\textrm{ has $\mathcal {I}^{*}$}.
\hspace{2pt}\big] \nonumber \\
& \leq \mathbb{P} \big[\hspace{2pt}\textrm{Graph
}G_s(n,K_n^{+},P_n)\textrm{ has $\mathcal
{I}^{*}$}.\hspace{2pt}\big] \vspace{-5pt} + o(1).\nonumber
\end{aligned}$$
Under conditions (\[thm:bin:eq:P\]), (\[thm:bin:eq:edge\]), and $|\beta_n | = O(\ln \ln n)$, we now show that $t_n P_n = \omega(\ln n )$. From (\[thm:bin:eq:edge\]) and $|\beta_n | = O(\ln \ln n)$, we first get $$\begin{aligned}
\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} & = \frac{\ln n \pm O(\ln
\ln n) }{n} = \frac{\ln n}{n} \cdot [1\pm o(1)] \vspace{-5pt} .
\label{tn2sPnslnneq}\end{aligned}$$ From (\[thm:bin:eq:P\]) and (\[tn2sPnslnneq\]), it follows that $$\begin{aligned}
&\label{eq_tnPn_c} t_n P_n = \sqrt{ {t_n}^{2}{P_n}} \cdot \sqrt{P_n
} \\ & =
\begin{cases}
\big\{ {s! n^{-1} \ln n} \cdot [1\pm o(1)] \big\}^{\frac{1}{2s}} \cdot \sqrt{\Omega(n)} , &\textrm{for } s \geq 2, \\
\big\{ {s! n^{-1} \ln n} \cdot [1\pm o(1)] \big\}^{\frac{1}{2s}} \cdot \sqrt{\Omega(n^c)} , &\textrm{for } s = 1,
\end{cases} \nonumber \\ &
=
\begin{cases}
\Omega\big(n^{\frac{1}{2}-\frac{1}{2s}} (\ln n)^{\frac{1}{2s}}\big) , &\textrm{for } s \geq 2, \\
\Omega\big(n^{\frac{c-1}{2}} (\ln n)^{\frac{1}{2}} \big), &\textrm{for } s = 1,\vspace{-5pt}
\end{cases}\nonumber\end{aligned}$$ yielding $t_n P_n = \omega(\ln n )$ in view of $c>1$, so we can use (\[propIud\]). Using (\[KnminandKnplus\]) and (\[eq\_tnPn\_c\]), we further obtain $$\begin{aligned}
\label{KnsPntn2s} \frac{{(K_n^{\pm})}^{2s}}{{P_n}^{s}} & =
\frac{{\big[ t_n P_n \pm \sqrt{3\ln n (\ln n + t_n P_n)}\hspace{1pt}\big]}^{2s}}{{P_n}^{s}}
\\ & = \frac{(t_n P_n )^{2s}}{{P_n}^{s}} \cdot \Bigg[1 \pm \sqrt{\frac{3\ln n}{t_n P_n} \bigg(\frac{\ln n}{t_n P_n} + 1\bigg)} \hspace{2pt} \Bigg]^{2s}
\nonumber \\ & = {t_n}^{2s} {P_n}^{s} \cdot \bigg[1 \pm o\bigg(\frac{1}{\ln n}\bigg)\bigg], \nonumber\end{aligned}$$ where in the last step we use $t_n P_n = \omega\big((\ln n
)^3\big)$, which follows from (\[eq\_tnPn\_c\]) due to constant $c>1$.
Applying (\[thm:bin:eq:edge\]) and (\[tn2sPnslnneq\]) to (\[KnsPntn2s\]), we have $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{(K_n^{\pm })}^{2s}}{{P_n}^{s}} & =
\frac{\ln n + {(k-1)} \ln \ln n + {\beta_n \pm o(1)}}{n}. \label{KnsPntn2sbtn}\end{aligned}$$ In view of (\[KnsPntn2sbtn\]) and $P_n = \Omega(n )$, we use Theorem \[thm:uni\] to obtain $$\begin{aligned}
\label{KnsPntn2sbtnud} &\lim_{n \to \infty} \mathbb{P}
\big[\hspace{2pt}\textrm{Graph }G_s(n,K_n^{\pm },P_n)\textrm{ has
$\mathcal {I}^{*}$}.\hspace{2pt}\big] \\ & = e^{- \frac{e^{-\lim_{n
\to \infty} [\beta_n \pm o(1)]}}{(k-1)!}} = e^{- \frac{e^{-\lim_{n
\to \infty}\beta_n}}{(k-1)!}}.\nonumber
\end{aligned}$$
The proof of Theorem \[thm:bin\] is completed by (\[propIud\]) and (\[KnsPntn2sbtnud\]).
Conclusion {#sec:Conclusion}
==========
Random $s$-intersection graphs have been used in a wide range of applications. Two extensively studied models are a uniform random $s$-intersection graph and a binomial random $s$-intersection graph. In this paper, for a uniform/binomial random $s$-intersection graph, we derive exact asymptotic expressions for the probabilities of the following three properties: (i) $k$-vertex-connectivity, (ii) $k$-edge-connectivity and (iii) the property that each vertex has degree at least $k$.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported in part by CMU CyLab under grant CNS-0831440 from the National Science Foundation. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.
[10]{}
F. G. Ball, D. J. Sirl, and P. Trapman, “Epidemics on random intersection graphs,” [*The Annals of Applied Probability*]{}, vol. 24, pp. 1081–1128, June 2014.
M. Bradonjić, A. Hagberg, N. Hengartner, and A. Percus, “Component evolution in general random intersection graphs,” in [*Workshop on Algorithms and Models for the Web Graph (WAW)*]{}, 2010.
M. Bloznelis, “Degree and clustering coefficient in sparse random intersection graphs,” [*The Annals of Applied Probability*]{}, vol. 23, no. 3, pp. 1254–1289, 2013.
M. Bloznelis, J. Jaworski, and V. Kurauskas, “Assortativity and clustering of sparse random intersection graphs,” [*The Electronic Journal of Probability*]{}, vol. 18, no. 38, pp. 1–24, 2013.
J. Zhao, O. Yağan, and V. Gligor, “$k$-[C]{}onnectivity in secure wireless sensor networks with physical link constraints — the on/off channel model,” *arXiv e-prints*, arXiv:1206.1531 \[cs.IT\], 2012.
P. Marbach, “A lower-bound on the number of rankings required in recommender systems using collaborativ filtering,” in [*IEEE Conference on Information Sciences and Systems (CISS)*]{}, 2008.
M. Bloznelis and K. Rybarczyk, “$k$-Connectivity of uniform $s$-intersection graphs,” [*Discrete Mathematics*]{}, vol. 333, no. 0, pp. 94–100, 2014.
M. Bloznelis and T. [Ł]{}uczak, “Perfect matchings in random intersection graphs,” [*Acta Mathematica Hungarica*]{}, vol. 138, no. 1-2, pp. 15–33, 2013.
J. Zhao, O. Yağan, and V. Gligor, “On $k$-connectivity and minimum vertex degree in random $s$-intersection graphs,” *arXiv e-prints*, arXiv:1409.6021v1 \[math.CO\], 2014 (full version of this paper: http://arxiv.org/pdf/1409.6021v1.pdf).
M. Farrell, T. Goodrich, N. Lemons, F. Reidl, F. Villaamil, and B. Sullivan, “[Hyperbolicity, degeneracy, and expansion of random intersection graphs]{},” *arXiv e-prints*, arXiv:1409.8196 \[cs.SI\], 2014.
A. Barbour and G. Reinert , “The shortest distance in random multi-type intersection graphs,” *arXiv e-prints*, arXiv:1001.5357 \[math.PR\], 2010.
M. Bloznelis, J. Jaworski, and K. Rybarczyk, “Component evolution in a secure wireless sensor network,” [*Networks*]{}, vol. 53, pp. 19–26, January 2009.
J. Zhao, O. Yağan, and V. Gligor, “On topological properties of wireless sensor networks under the $q$-composite key predistribution scheme with on/off channels,” in [*IEEE International Symposium on Information Theory (ISIT)*]{}, 2014.
A. Newell, H. Yao, A. Ryker, T. Ho, and C. Nita-Rotaru,“Node-Capture Resilient Key Establishment in Sensor Networks: Design Space and New Protocols,” *ACM Computing Surveys (CSUR)*, vol. 47, no. 2, pp. 24:1–24:34, 2014.
J. Zhao, O. Yağan, and V. Gligor, “Results on vertex degree and $k$-connectivity in uniform s-intersection graphs,” Technical Report CMU-CyLab-14-004, CyLab, Carnegie Mellon University, 2014.
H. Chan, A. Perrig, and D. Song, “Random key predistribution schemes for sensor networks,” in [*IEEE Symposium on Security and Privacy*]{}, May 2003.
L. Eschenauer and V. Gligor, “A key-management scheme for distributed sensor networks,” in [*ACM Conference on Computer and Communications Security (CCS)*]{}, 2002.
O. Yağan, [*Random Graph Modeling of Key Distribution Schemes in Wireless Sensor Networks*]{}. PhD thesis, Department of Electrical and Computer Engineering, University of Maryland, 2011.
J. Zhao, O. Yağan, and V. Gligor, “Secure $k$-connectivity in wireless sensor networks under an on/off channel model,” in [*IEEE International Symposium on Information Theory (ISIT)*]{}, 2013.
K. Rybarczyk, “Sharp threshold functions for the random intersection graph via a coupling method,” [*The Electronic Journal of Combinatorics*]{}, vol. 18, pp. 36–47, 2011.
K. [Rybarczyk]{}, “[The coupling method for inhomogeneous random intersection graphs]{},” *arXiv e-prints*, arXiv:1301.0466 \[math.CO\], 2013.
J. Zhao, O. Yağan, and V. Gligor, “Connectivity in secure wireless sensor networks under transmission constraints,” in [*Allerton Conference on Communication, Control, and Computing*]{}, 2014.
M. Bradonjić, A. Hagberg, N. W. Hengartner, N. Lemons, and A. G. Percus, “The phase transition in inhomogeneous random intersection graphs,” *arXiv e-prints*, arXiv:1301.7320 \[cs.DM\], 2013.
S. Blackburn, D. Stinson, and J. Upadhyay, “On the complexity of the herding attack and some related attacks on hash functions,” [*Designs, Codes and Cryptography*]{}, vol. 64, no. 1-2, pp. 171–193, 2012.
J. Zhao, O. Yağan, and V. Gligor, “On the strengths of connectivity and robustness in general random intersection graphs,” in [*IEEE Conference on Decision and Control (CDC)*]{}, 2014.
M. Deijfen and W. Kets, “Random intersection graphs with tunable degree distribution and clustering,” [*Probability in the Engineering and Informational Sciences*]{}, vol. 23, pp. 661–674, 2009.
E. Godehardt and J. Jaworski, “Two models of random intersection graphs for classification,” [*Exploratory data analysis in empirical research*]{}, pp. 67–81, 2003.
K. Rybarczyk, “Diameter, connectivity and phase transition of the uniform random intersection graph,” [*Discrete Mathematics*]{}, vol. 311, 2011.
K. B. Singer-Cohen, [*Random intersection graphs*]{}. PhD thesis, Department of Mathematical Sciences, The Johns Hopkins University, 1995.
Y. Shang, “[On the isolated vertices and connectivity in random intersection graphs]{},” *International Journal of Combinatorics*, vol. 2011, ID 872703, 2011.
J. Jaworski, M. Karoński, and D. Stark, “The degree of a typical vertex in generalized random intersection graph models,” [*Discrete Mathematics*]{}, vol. 306, no. 18, pp. 2152 – 2165, 2006.
M. Bloznelis and J. Damarackas, “Degree distribution of an inhomogeneous random intersection graph.,” [*The Electronic Journal of Combinatorics*]{}, vol. 20, no. 3, p. P3, 2013.
K. Rybarczyk, “Constructions of independent sets in random intersection graphs,” [*Theoretical Computer Science*]{}, vol. 524, no. 0, pp. 103 – 125, 2014.
S. Nikoletseas, C. Raptopoulos, and P. Spirakis, “Large independent sets in general random intersection graphs,” [*Theororetical Computer Science*]{}, vol. 406, pp. 215–224, Oct. 2008.
D. Stark, “The vertex degree distribution of random intersection graphs,” [*Random Structures & Algorithms*]{}, vol. 24, no. 3, pp. 249–258, 2004.
M. Karoński, E. R. Scheinerman, and K. B. Singer-Cohen, “On random intersection graphs: The subgraph problem,” [*Combinatorics, Probability and Computing*]{}, vol. 8, pp. 131–159, Jan. 1999.
S. Blackburn and S. Gerke, “Connectivity of the uniform random intersection graph,” [*Discrete Mathematics*]{}, vol. 309, no. 16, August 2009.
O. Yağan and A. M. Makowski, “Zero–one laws for connectivity in random key graphs,” [*IEEE Transactions on Information Theory*]{}, vol. 58, pp. 2983–2999, May 2012.
J. Zhao, O. Yağan, and V. Gligor, “On asymptotically exact probability of $k$-connectivity in random key graphs intersecting [Erdős–Rényi]{} graphs,” *arXiv e-prints*, arXiv:1409.6022 \[math.CO\], 2014.
O. Yğan and A. M. Makowski, “Random key graphs — Can they be small worlds?,” in [*International Conference on Networks and Communications (NETCOM)*]{}, pp. 313 -318, 2009.
V. Gligor, A. Perrig, and J. Zhao, “Brief encounters with a random key graph,” [*Lecture Notes in Computer Science*]{}, vol. 7028, pp. 157–161, 2013.
A. Goel, S. Khanna, S. Raghvendra, and H. Zhang, “Connectivity in random forests and credit networks,” in [*ACM-SIAM Symposium on Discrete Algorithms (SODA)*]{}, 2015.
E. Beer, J. A. Fill, S. Janson, and E. R. Scheinerman, “On vertex, edge, and vertex-edge random graphs,” [*The Electronic Journal of Combinatorics*]{}, vol. 18, no. 1, p. P110, 2011.
E. Beer, J. A. Fill, S. Janson, and E. R. Scheinerman, “On vertex, edge, and vertex-edge random graphs,” in [*ACM-SIAM Analytic Algorithmics and Combinatorics (ANALCO)*]{}, 2011.
K. Censor-Hillel, M. Ghaffari, G. Giakkoupis, B. Haeupler, and F. Kuhn, “Tight bounds on vertex connectivity under vertex sampling,” in [*ACM-SIAM Symposium on Discrete Algorithms (SODA)*]{}, 2015.
P. Erdős and A. Rényi, “On random graphs, [I]{},” [*Publicationes Mathematicae (Debrecen)*]{}, vol. 6, pp. 290–297, 1959.
J. Zhao, “Minimum node degree and $k$-[c]{}onnectivity in wireless networks with unreliable links,” in [*IEEE International Symposium on Information Theory (ISIT)*]{}, 2014.
F. Yavuz, J. Zhao, O. Yağan, and V. Gligor, “On secure and reliable communications in wireless sensor networks: [T]{}owards $k$-connectivity under a random pairwise key predistribution scheme,” in [*IEEE International Symposium on Information Theory (ISIT)*]{}, 2014.
B. Bollob[á]{}s, [*Random graphs*]{}. Cambridge Studies in Advanced Mathematics, 2001.
S. Janson, T. [Ł]{}uczak, and A. Ruci[ń]{}ski, [*Random graphs*]{}. Wiley-Interscience Series on Discrete Mathematics and Optimization, 2000.
Appendix
========
We first present in Section \[sec:add:lemma\] additional lemmas used in proving the theorems. Afterwards, we detail the proofs of the lemmas.
Additional lemmas. {#sec:add:lemma}
------------------
\[fact:logn1\] For constant $h$, if $|\alpha_n| = O(\ln \ln n)$, then $$\begin{aligned}
\frac{\ln n + h \ln \ln n + {\alpha_n}}{n} & \sim \frac{\ln
n}{n}. \nonumber\end{aligned}$$
\[lem:logn2\] For constants $s$ and $k$, if $$\begin{aligned}
\frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}} & = \frac{\ln n + {(k-1)} \ln \ln n
+ {\alpha_n}}{n}\nonumber\end{aligned}$$ with $|\alpha_n| = O(\ln \ln n)$, we have
- $K_n = \Theta\big({P_n}^{\frac{1}{2}} n^{-\frac{1}{2s}} (\ln n)^{\frac{1}{2s}} \big)$,
- under $P_n = \Omega(n)$, then $K_n = \Omega\big( (\ln n)^{\frac{1}{2s}} \big)$, and
- under $P_n = \Omega(n^c)$ for constant $c$, then $K_n = \Omega\big( n^{\frac{c}{2} - \frac{1}{2s}} (\ln n)^{\frac{1}{2s}} \big)$.
Some additional lemmas are given below. The relation “$\sim$” stands for an asymptotical equivalence; i.e., $f_n \sim g_n$ means $\lim_{n \to
\infty}({f_n }/{g_n })=1$.
\[lem:logn2\] If $\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} = \frac{\ln
n \pm O(\ln \ln n) }{n}$ and $P_n = \Omega(n^c)$ for constant $c$, then $K_n = \Omega\big( n^{\frac{c}{2} - \frac{1}{2s}} (\ln
n)^{\frac{1}{2s}} \big)$.
\[lem:eval:qn\] If $\frac{{K_n}^2}{P_n} = o(1)$ and $ K_n = \omega(1)$, then $$\begin{aligned}
q_n & = \frac{1}{s!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{s} \cdot
\bigg[1\pm O\bigg(\frac{{K_n}^2}{P_n}\bigg) \pm
O\bigg(\frac{1}{K_n}\bigg)\bigg]. \nonumber\end{aligned}$$
\[qn-dist\]
The following properties (a) and (b) hold, where $q_n$ is the edge probability in uniform random $s$-intersection graph $G_s(n,K_n,P_n)$.
- If $P_n = \Omega(n)$ and $\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} = \frac{\ln n \pm O(\ln \ln n) }{n}$, then $q_n \sim \frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}}$ and $\big| \hspace{2pt} q_n - \frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \big| = o\big(\frac{1}{n}\big)$.
- If $P_n = \Omega(n)$ and $q_n = \frac{\ln n \pm O(\ln \ln n) }{n}$, then $q_n \sim \frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}}$ and $\big| \hspace{2pt} q_n - \frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \big| = o\big(\frac{1}{n}\big)$.
$$\begin{aligned}
\bigg| \hspace{2pt} q_n - \frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \bigg| & =
o\bigg(\frac{1}{n}\bigg). \nonumber\end{aligned}$$
$$\begin{aligned}
\bigg| \hspace{2pt} q_n - \frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \bigg| & =
o\bigg(\frac{1}{n}\bigg). \nonumber\end{aligned}$$
If $\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} = \frac{\ln n
\pm O(\ln \ln n) }{n}$ and $P_n = \Omega(n)$, then $q_n \sim
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}}$ and $\big|
\hspace{2pt} q_n - \frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}}
\hspace{2pt} \big| = o\big(\frac{1}{n}\big)$, where $q_n$ is the edge probability in uniform random $s$-intersection graph $G_s(n,K_n,P_n)$.
$$\begin{aligned}
q_n \sim \frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}}, \nonumber\end{aligned}$$ and for $s \geq 2$, it holds that $$\begin{aligned}
\bigg| \hspace{2pt} q_n - \frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \bigg| =
o\bigg(\frac{1}{n}\bigg). \nonumber\end{aligned}$$
$$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & \sim \frac{\ln
n}{n}, \nonumber\end{aligned}$$
$$\begin{aligned}
K_n & \geq \frac{1}{2} \cdot \bigg(\frac{s! \ln
n}{n}\bigg)^{\frac{1}{2s}} \cdot n^{\frac{1}{2s} +
\frac{\epsilon}{2}}\nonumber = \frac{1}{2} (s!)^{\frac{1}{2s}}
n^{\frac{\epsilon}{2}} (\ln n)^{\frac{1}{2s}}\end{aligned}$$
$$\begin{aligned}
q_n & = \frac{1}{s!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{s} \cdot
\bigg[1\pm o\bigg(\frac{1}{\ln n}\bigg) \bigg].\end{aligned}$$
\[lem-unig1\]
For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under (\[eqPnOmegan\]) and (\[thm:uni:eq:edge\]) with constrained $\alpha_n = o(\ln n )$, the edge probability $q_n$ satisfies $$\begin{aligned}
q_n & \sim \frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} .\end{aligned}$$
\[lem\_prob\_Eij\_S1r\]
For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $K_n = \omega(1)$, the following properties (a) (b) and (c) hold for $i = r+1, r+2, \ldots, n$ (i.e., vertex $v_i \notin \{v_1, v_2,
\ldots, v_r\}$), where $E_{ij}$ denotes the event that an edge exists between vertices $v_i$ and $v_j$, $S_i$ is the number of items on vertex $v_i$, and $q_n$ is the edge probability.
- If $|\bigcup_{j=1}^{r} S_j| \geq \lfloor (1+{\varepsilon_1}) K_n \rfloor$ for some positive constant $\varepsilon_1$, then for any positive constant $\varepsilon_2 < (1+{\varepsilon_1})^s -
1$, it holds for all $n$ sufficiently large that $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{- q_n (1+\varepsilon_2)}.\end{aligned}$$
- If $|\bigcup_{j=1}^{r} S_j| \geq \lfloor \lambda_1 r K_n \rfloor$ for some positive constant $\lambda_1$, then for any positive constant $\lambda_2 < {\lambda_1}^s$, it holds for all $n$ sufficiently large that $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{- \lambda_2 r q_n}.\end{aligned}$$
- If $|\bigcup_{j=1}^{r} S_j| \geq \lfloor \mu_1 P_n \rfloor$ for some positive constant $\mu_1$, then for any positive constant $\mu_2 < (s!)^{-1}{\mu_1}^s$, it holds for all $n$ sufficiently large that $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{- \mu_2 K_n}.\end{aligned}$$
\[olp\_lem1\]
For uniform random $s$-intersection graph $G_s(n,K_n,P_n)$ under $P_n = \Omega(n)$, $K_n = \omega(1)$ and $r_n := \min \big( {\big
\lfloor \frac{P_n}{K_n} \big \rfloor}, \big \lfloor \frac{n}{2} \big
\rfloor \big) = \omega(1)$, the following properties (a) (b) and (c) hold for any constant integer $R \geq 2$, where $\varepsilon_1$, $\lambda_1$ and $\mu_1$ are specified in Lemma \[prp:EJ\], and events $\mathcal{A}_{{\ell},r}$ and $\mathcal{E} (\boldsymbol{J})$ are defined in Sections 6.3 and 3.3, respectively.
- Let $ \varepsilon_3$ be any positive constant with $ \varepsilon_3 < (1+\varepsilon_1)^s-1$. For all $n$ sufficiently large, it holds for $r = 2, 3, \ldots, R$ that $$\begin{aligned}
{{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} \hspace{1.5pt}\cap\hspace{1.5pt} \overline{\mathcal{E} (\boldsymbol{J})} }\right]} &\leq
r^{r-2} {q_n}^{r-1} ( r {q_n})^{{\ell} } e^{- q_n n (1+\varepsilon_3)}
. \nonumber\end{aligned}$$
- Let $\lambda_2$ be any positive constant with $\lambda_2 < {\lambda_1}^s$. For all $n$ sufficiently large, it holds for $r = R + 1, R + 2, \ldots, r_n $ that $$\begin{aligned}
{{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} \hspace{2pt}\cap\hspace{2pt} \overline{\mathcal{E} (\boldsymbol{J})} }\right]} &\leq
r^{r-2} {q_n}^{r-1} e^{- \lambda_2 r q_n n /3}
. \nonumber\end{aligned}$$
- Let $\mu_2$ be any positive constant with $\mu_2 < (s!)^{-1}{\mu_1}^s$. For all $n$ sufficiently large, it holds for $r = r_n + 1, r_n + 2, \ldots, \lfloor
\frac{n-{\ell}}{2} \rfloor $ that $$\begin{aligned}
{{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} \hspace{2pt}\cap\hspace{2pt} \overline{\mathcal{E} (\boldsymbol{J})} }\right]} &\leq
e^{- \mu_2 K_n n /3}
. \nonumber\end{aligned}$$
\[lem:cp\]
Let $K_n^{-}$ and $K_n^{+}$ denote $t_n P_n - \sqrt{3\ln n (\ln n +
t_n P_n)}$ and $t_n P_n + \sqrt{3\ln n (\ln n + t_n P_n)}$, respectively. If $t_n P_n = \omega(\ln n )$, then for any monotone increasing graph property $\mathcal {I}$, it holds that $$\begin{aligned}
& \mathbb{P} \big[\hspace{2pt}\textrm{Graph }G_s(n,K_n^{-},P_n)\textrm{
has $\mathcal {I}$}.\hspace{2pt}\big] - o(1) \nonumber \\
& \leq \mathbb{P} \big[\hspace{2pt}\textrm{Graph
}H_s(n,t_n,P_n)\textrm{ has $\mathcal {I}$}.
\hspace{2pt}\big] \nonumber \\
& \leq \mathbb{P} \big[\hspace{2pt}\textrm{Graph
}G_s(n,K_n^{+},P_n)\textrm{ has $\mathcal {I}$}.\hspace{2pt}\big] +
o(1). \nonumber
\end{aligned}$$
\[lem-bing1\]
The following properties (a) and (b) hold, where $\rho_n$ is the edge probability in binomial random $s$-intersection graph $H_s(n,t_n,P_n)$.
- Under (\[thm:bin:eq:P\]) and $ \frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} = \frac{\ln n \pm O(\ln \ln n) }{n}$, then $\rho_n \sim \frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} $ and $\big| \hspace{2pt} \rho_n - \frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} \hspace{2pt} \big| = o\big(\frac{1}{n}\big)$.
- Under (\[thm:bin:eq:P\]) and $\rho_n = \frac{\ln n \pm O(\ln \ln n) }{n}$, then $\rho_n \sim \frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} $ and $\big| \hspace{2pt} \rho_n -\frac{1}{s!} \cdot {t_n}^{2s}{P_n}^{s} \big| = o\big(\frac{1}{n}\big)$.
Proof of Lemma \[graph\_Gs\_cpl\] {#sec_graph_Gs_cpl}
---------------------------------
#### Proving property (a).
We define $\widetilde{\alpha_n}^*$ by $$\begin{aligned}
\widetilde{\alpha_n}^* & = \max\{\alpha_n, -\ln \ln n\},
\label{al2-parta}\end{aligned}$$ and define $\widetilde{K_n}^*$ such that $$\begin{aligned}
\frac{1}{s!} \cdot \frac{({\widetilde{K_n}^{*}})^{2s}}{{{P_n}}^{s}}
& = \frac{\ln n + {(k-1)} \ln \ln n + \widetilde{\alpha_n}^*}{n}.
\label{al3-parta}\end{aligned}$$ We set $$\begin{aligned}
\widetilde{K_n} & : = \big\lfloor \widetilde{K_n}^* \big\rfloor,
\label{al4-parta}\end{aligned}$$ and $$\begin{aligned}
\widetilde{P_n} & : = P_n. \label{al5-parta}\end{aligned}$$
From (\[al1\]) (\[al2-parta\]) and (\[al3-parta\]), it holds that $$\begin{aligned}
K_n \leq \widetilde{K_n}^*. \label{Kn1-parta}\end{aligned}$$ Then by (\[al4-parta\]) (\[Kn1-parta\]) and the fact that $K_n$ and $\widetilde{K_n}$ are both integers, it follows that $$\begin{aligned}
K_n \leq \widetilde{K_n}. \label{al6-parta}\end{aligned}$$ From (\[al5-parta\]) and (\[al6-parta\]), by [@Rybarczyk Lemma 3], there exists a graph coupling under which $G_s(n,K_n,P_n)$ is a spanning subgraph of $G_s(n,\widetilde{K_n},\widetilde{P_n})$. Therefore, the proof of property (a) is completed once we show $\widetilde{\alpha_n}$ defined in $(\ref{al0-parta})$ satisfies $$\begin{aligned}
\lim_{n \to \infty}\widetilde{\alpha_n} & = - \infty, \label{al8-parta} \\
\widetilde{\alpha_n} & = - O(\ln \ln n). \label{al7-parta}\end{aligned}$$
We first prove (\[al8-parta\]). From (\[al0-parta\]) (\[al3-parta\]) and (\[al4-parta\]), it holds that $$\begin{aligned}
\widetilde{\alpha_n} \leq \widetilde{\alpha_n}^*, \label{haa-parta}\end{aligned}$$ which together with (\[al2-parta\]) and $\lim_{n \to
\infty}\alpha_n = -\infty$ yields (\[al8-parta\]).
Now we establish (\[al7-parta\]). From (\[al4-parta\]), we have $\widetilde{K_n} > \widetilde{K_n}^* - 1$. Then from (\[al0-parta\]) and (\[al5-parta\]), it holds that $$\begin{aligned}
\label{aph1-parta}\widetilde{\alpha_n} = n \cdot \frac{1}{s!} \cdot \frac{{\widetilde{K_n}}^{2s}}{{{P_n}}^{s}} - [\ln n + {(k-1)} \ln \ln n]~~~ \\
> n \cdot \frac{1}{s!} \cdot \frac{{(\widetilde{K_n}^* - 1)}^{2s}}{{{P_n}}^{s}} - [\ln n + {(k-1)} \ln \ln n] .\nonumber\end{aligned}$$ By $\lim_{n \to \infty}\alpha_n =- \infty$, it holds that $\alpha_n
\leq 0$ for all $n$ sufficiently large. Then from (\[al2-parta\]), it follows that $$\begin{aligned}
\widetilde{\alpha_n}^* = - O(\ln \ln n), \label{widetilde-al2-parta}\end{aligned}$$ which along with Lemma \[lem:logn2\], equation (\[al3-parta\]) and condition $P_n = \Omega(n)$ induces $$\begin{aligned}
\widetilde{K_n}^* & = \Omega\big( (\ln n)^{\frac{1}{2s}} \big). \label{aph5-parta}\end{aligned}$$ Hence, we have $\lim_{n \to \infty} \widetilde{K_n}^* = \infty$ and it further holds for all $n$ sufficient large that $$\begin{aligned}
{(\widetilde{K_n}^* - 1)}^{2s} > ({\widetilde{K_n}^{*}})^{2s} - 3s
({\widetilde{K_n}^{*}})^{2s-1}. \label{aph2-parta}\end{aligned}$$ Applying (\[aph2-parta\]) to (\[aph1-parta\]) and then using (\[al3-parta\]), Lemma \[lem:logn2\] and $P_n = \Omega(n)$, it follows that $$\begin{aligned}
\label{widetilde-al-parta} & \widetilde{\alpha_n} \\
& > \frac{n}{s!}\hspace{-1pt} \cdot\hspace{-1pt}
\frac{ ({\widetilde{K_n}^{*}})^{2s}
\hspace{-1pt}-\hspace{-1pt} 3s ({\widetilde{K_n}^{*}})^{2s-1}}{{{P_n}}^{s}}
\hspace{-1pt}-\hspace{-1pt} [\ln n \hspace{-1pt}+\hspace{-1pt} {(k\hspace{-1pt}-\hspace{-1pt}1)} \ln \ln n] \nonumber \\
& = \widetilde{\alpha_n}^* - \frac{3s}{s!} \cdot n \cdot \Theta\big({P_n}^{-\frac{1}{2}} n^{-\frac{2s-1}{2s}} (\ln n)^{\frac{2s-1}{2s}} \big) \nonumber \\
& = \widetilde{\alpha_n}^* - O\big(n^{-\frac{1}{2}+\frac{1}{2s}} (\ln n)^{1-\frac{1}{2s}}\big).\nonumber\end{aligned}$$
As noted at the beginning of Section \[sec:basic:ideas\], our proof is for $s \geq 2$ since the case of $s = 1$ already is proved by us [@ZhaoCDC]. Using $s \geq 2$ in (\[widetilde-al-parta\]), it holds that $ \widetilde{\alpha_n} >
\widetilde{\alpha_n}^* + o(1)$, which along with (\[haa-parta\]) and (\[widetilde-al2-parta\]) yields (\[al7-parta\]).
#### Proving property (b).
We define $\widehat{\alpha_n}^*$ by $$\begin{aligned}
\widehat{\alpha_n}^* & = \min\{\alpha_n, \ln \ln n\}, \label{al2}\end{aligned}$$ and define $\widehat{K_n}^*$ such that $$\begin{aligned}
\frac{1}{s!} \cdot \frac{({\widehat{K_n}^{*}})^{2s}}{{{P_n}}^{s}} &
= \frac{\ln n + {(k-1)} \ln \ln n + \widehat{\alpha_n}^*}{n}.
\label{al3}\end{aligned}$$ We set $$\begin{aligned}
\widehat{K_n} & : = \big\lceil \widehat{K_n}^* \big\rceil,
\label{al4}\end{aligned}$$ and $$\begin{aligned}
\widehat{P_n} & : = P_n. \label{al5}\end{aligned}$$
From (\[al1\]) (\[al2\]) and (\[al3\]), it holds that $$\begin{aligned}
K_n \geq \widehat{K_n}^*. \label{Kn1}\end{aligned}$$ Then by (\[al4\]) (\[Kn1\]) and the fact that $K_n$ and $\widehat{K_n}$ are both integers, it follows that $$\begin{aligned}
K_n \geq \widehat{K_n}. \label{al6}\end{aligned}$$ From (\[al5\]) and (\[al6\]), by [@Rybarczyk Lemma 3], there exists a graph coupling under which $G_s(n,K_n,P_n)$ is a spanning supergraph of $G_s(n,\widehat{K_n},\widehat{P_n})$. Therefore, the proof of property (b) is completed once we show $\widehat{\alpha_n}$ defined in $(\ref{al0})$ satisfies $$\begin{aligned}
\lim_{n \to \infty}\widehat{\alpha_n} & = \infty, \label{al8} \\
\widehat{\alpha_n} & = O(\ln \ln n). \label{al7}\end{aligned}$$
We first prove (\[al8\]). From (\[al0\]) (\[al3\]) and (\[al4\]), it holds that $$\begin{aligned}
\widehat{\alpha_n} \geq \widehat{\alpha_n}^*, \label{haa}\end{aligned}$$ which together with (\[al2\]) and $\lim_{n \to \infty}\alpha_n =
\infty$ yields (\[al8\]).
Now we establish (\[al7\]). From (\[al4\]), we have $\widehat{K_n} < \widehat{K_n}^* + 1$. Then from (\[al0\]) and (\[al5\]), it holds that $$\begin{aligned}
\label{aph1} \widehat{\alpha_n} = n \cdot \frac{1}{s!} \cdot
\frac{{\widehat{K_n}}^{2s}}
{{{P_n}}^{s}} - [\ln n + {(k-1)} \ln \ln n]~~~ \\
< n \cdot \frac{1}{s!}
\cdot \frac{{(\widehat{K_n}^* + 1)}^{2s}}{{{P_n}}^{s}}
- [\ln n + {(k-1)} \ln \ln n] .\nonumber\end{aligned}$$ By $\lim_{n \to \infty}\alpha_n = \infty$, it holds that $\alpha_n
\geq 0$ for all $n$ sufficiently large. Then from (\[al2\]), it follows that $$\begin{aligned}
\widehat{\alpha_n}^* = O(\ln \ln n), \label{widehat-al2}\end{aligned}$$ which along with Lemma \[lem:logn2\], equation (\[al3\]) and condition $P_n = \Omega(n)$ induces $$\begin{aligned}
\widehat{K_n}^* & = \Omega\big( (\ln n)^{\frac{1}{2s}} \big). \label{aph5}\end{aligned}$$ Hence, we have $\lim_{n \to \infty} \widehat{K_n}^* = \infty$ and it further holds for all $n$ sufficient large that $$\begin{aligned}
{(\widehat{K_n}^* + 1)}^{2s}< ({\widehat{K_n}^{*}})^{2s} + 3s
({\widehat{K_n}^{*}})^{2s-1}. \label{aph2}\end{aligned}$$ Applying (\[aph2\]) to (\[aph1\]) and then using (\[al3\]), Lemma \[lem:logn2\] and $P_n = \Omega(n)$, it follows that $$\begin{aligned}
\label{widehat-al} & \widehat{\alpha_n} \\
\nonumber & < \frac{n}{s!} \hspace{-1pt}\cdot\hspace{-1pt} \frac{ ({\widehat{K_n}^{*}})^{2s}\hspace{-1pt} +\hspace{-1pt} 3s ({\widehat{K_n}^{*}})^{2s-1}}{{{P_n}}^{s}} \hspace{-1pt}-\hspace{-1pt} [\ln n \hspace{-1pt}+\hspace{-1pt} {(k\hspace{-1pt}-\hspace{-1pt}1)} \ln \ln n] \nonumber \\
& = \widehat{\alpha_n}^* + \frac{3s}{s!} \cdot n \cdot \Theta\big({P_n}^{-\frac{1}{2}} n^{-\frac{2s-1}{2s}} (\ln n)^{\frac{2s-1}{2s}} \big) \nonumber \\
& = \widehat{\alpha_n}^* + O\big(n^{-\frac{1}{2}+\frac{1}{2s}} (\ln
n)^{1-\frac{1}{2s}}\big). \nonumber\end{aligned}$$
As noted at the beginning of Section \[sec:basic:ideas\], our proof is for $s \geq 2$ since the case of $s = 1$ already is proved by Rybarczyk [@ryb3]. Using $s \geq 2$ in (\[widehat-al\]), it holds that $ \widehat{\alpha_n} < \widehat{\alpha_n}^* + o(1)$, which along with (\[haa\]) and (\[widehat-al2\]) yields (\[al7\]).
The proof of Lemma \[prp-kvl-del-EJ\]. {#sec:prf:prop:OneLawAfterReductionPart2}
--------------------------------------
By the analysis in [@ZhaoYaganGligor Section IV], we obtain [@ZhaoYaganGligor Equation (148)]. Namely, with some events defined as follows:
- $\mathcal{C}_{r}$: event that the induced subgraph of $G_s(n, K_n, P_n)$ defined on vertex set $\{v_1, v_2, \ldots, v_r\}$ is connected,
- $\mathcal{B}_{\ell,r}$: event that any vertex in $\{v_{r+1}, v_{r+2}, \ldots, v_{r+\ell} \}$ has an edge with at least one vertex in $\{v_{1}, v_{2}, \ldots, v_{r}\}$,
- $\mathcal{D}_{\ell,r}$: event that any vertex in $\{v_{r+\ell+1}, v_{r+\ell+2}, \ldots, v_{n} \}$ and any vertex in $\{v_{1}, v_{2}, \ldots, v_{r}\}$ has no edge in between, and
- $\mathcal{A}_{\ell, r}$: event that events $\mathcal{C}_{r}$, $\mathcal{B}_{\ell,r}$ and $\mathcal{D}_{\ell,r}$ all happen,
it holds that $$\begin{aligned}
\label{eq:BasicIdea+UnionBound2} {{\mathbb{P}}\left[{(\kappa ={\ell}) ~\cap~ (\delta > {\ell}) ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})} }\right]}~~~~~~~~~~~~~ \\
\leq \sum_{r=2}^{ \lfloor \frac{n-{\ell}}{2} \rfloor } {n \choose
{\ell} }{ {n-{\ell}} \choose r} ~ {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})}}\right]} . \nonumber\end{aligned}$$
The proof of Lemma \[prp-kvl-del-EJ\] is completed once we show the following three results: $$\begin{aligned}
\sum_{r=2}^{ R} {n \choose {\ell} }{ {n-{\ell}} \choose r} ~ {{\mathbb{P}}\left[{
\mathcal{A}_{{\ell},r} ~\cap~ \overline{\mathcal{E}
(\boldsymbol{J})}}\right]} = o(1) , \label{prf-e1}
\\ \sum_{r=R+1}^{ r_n} {n \choose {\ell} }{ {n-{\ell}} \choose
r} ~ {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~ \overline{\mathcal{E}
(\boldsymbol{J})}}\right]} = o(1) , \label{prf-e2}\end{aligned}$$ and $$\begin{aligned}
\sum_{r=r_n+1}^{ \lfloor \frac{n-{\ell}}{2}\rfloor } {n \choose
{\ell} }{ {n-{\ell}} \choose r} ~ {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})}}\right]} = o(1) , \label{prf-e3}\end{aligned}$$ where $r_n = \min \big( {\big \lfloor \frac{P_n}{K_n} \big \rfloor},
\big \lfloor \frac{n}{2} \big \rfloor \big)$.
From condition (\[cn-qn-2lnn-n\]), it follows that $
\frac{K_n}{P_n} = o(1)$, yielding $r_n = \omega(1)$. From conditions (\[cn-qn-2lnn-n\]) and $P_n = \Omega(n)$, we use Lemma \[lem:logn2\] to derive $K_n = \omega(1)$. Therefore, we have $P_n
= \Omega(n)$, $K_n = \omega(1)$ and $r_n = \omega(1)$, enabling us to use Lemma \[olp\_lem1\].
In addition, given conditions (\[cn-qn-2lnn-n\]) and $P_n =
\Omega(n)$, we use Lemma \[qn-dist\] to obtain $$\begin{aligned}
q_n & = \frac{\ln n \pm O(\ln \ln n)}{n}
.\label{cn-qn-2lnn-n2-newqn}\end{aligned}$$ Hence, it holds that $$\begin{aligned}
q_n & \leq \frac{2\ln n}{n}, \textrm{ for all $n$ sufficiently
large},\label{cn-qn-2lnn-n2-newqn-1}\end{aligned}$$ and there exists constant $c_0$ such that $$\begin{aligned}
q_n & \geq \frac{\ln n - c_0 \ln \ln n}{n}, \textrm{ for all $n$
sufficiently large}.\label{cn-qn-2lnn-n2-newqn-2}\end{aligned}$$
### Establishing (\[prf-e1\]).
From ${n \choose {\ell}} \leq n^{{\ell} }$, ${ n-{\ell} \choose r}
\leq n^{ r}$ and property (a) of Lemma \[olp\_lem1\], it follows that $$\begin{aligned}
\label{olp_zja} & {n \choose {\ell} }{ {n-{\ell}} \choose r} {{\mathbb{P}}\left[{
\mathcal{A}_{{\ell},r} ~\cap~ \overline{\mathcal{E}
(\boldsymbol{J})}}\right]} \\ &\leq n^{{\ell} } \cdot n^{ r} \cdot r^{r-2}
{q_n}^{r-1} ( r q_n)^{\ell} \cdot e^{-q_n n (1+\varepsilon_3) }
\nonumber \\ & = r ^{\ell} r^{r-2} \cdot n^{{\ell}+r} {q_n}^{
{\ell}+r -1} \cdot e^{-q_n n (1+\varepsilon_3) } . \nonumber\end{aligned}$$
Applying (\[cn-qn-2lnn-n2-newqn\]) and (\[cn-qn-2lnn-n2-newqn-2\]) to (\[olp\_zja\]), we get $$\begin{aligned}
& {n \choose {\ell} }{ {n-{\ell}} \choose
r} {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~ \overline{\mathcal{E} (\boldsymbol{J})}}\right]} \nonumber \\
&\leq r ^{\ell} r^{r-2} n^{{\ell}+r} \bigg(\hspace{-2pt} \frac{2 \ln n }{n}\hspace{-1pt}\bigg)^{ {\ell}+r -1} \hspace{-1pt} e^{-(1+\varepsilon_3)(\ln n - c_0 \ln \ln n)} \nonumber \\
& \leq 2^{ {\ell}+r -1} r ^{\ell+r-2} n^{-\varepsilon_3} (\ln n)^{\ell + r -1 +c_0 (1+\varepsilon_3)} \nonumber \\
& = o(1). \nonumber\end{aligned}$$ Since $R$ is a constant, (\[prf-e1\]) clearly follows.
$$\begin{aligned}
q_n & = \sum_{r=s}^{K_n} \mathbb{P}[|S_{i} \cap S_{j}| = r] =
\sum_{r=s}^{K_n}
\frac{\binom{K_n}{r}\binom{P_n-K_n}{K_n-r}}{\binom{P_n}{K_n}}.
\nonumber\end{aligned}$$
$q_n \sim \Omega \big(\frac{\ln n }{n}\big)$
$|\alpha_n| = O(\ln \ln n)$
$$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} & \sim \frac{\ln
n}{n}, \nonumber\end{aligned}$$
$$\begin{aligned}
\frac{{K_n}^2}{P_n}\ & \sim \bigg(\frac{s!\ln
n}{n}\bigg)^{\frac{1}{s}} , \nonumber\end{aligned}$$
$$\begin{aligned}
K_n & \sim \bigg(\frac{s! \ln n}{n}\bigg)^{\frac{1}{2s}} \cdot
(P_n)^{\frac{1}{2}}, \nonumber\end{aligned}$$
$$\begin{aligned}
K_n & \geq \frac{1}{2} \cdot \bigg(\frac{s! \ln
n}{n}\bigg)^{\frac{1}{2s}} \cdot n^{\frac{1}{2s} +
\frac{\epsilon}{2}}\nonumber = \frac{1}{2} (s!)^{\frac{1}{2s}}
n^{\frac{\epsilon}{2}} (\ln n)^{\frac{1}{2s}}\end{aligned}$$
$$\begin{aligned}
q_n & = \frac{1}{s!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{s} \cdot
\bigg[1\pm o\bigg(\frac{1}{\ln n}\bigg) \bigg].\end{aligned}$$
### Establishing (\[prf-e2\]).
From ${n \choose {\ell}} \leq n^{{\ell} }$, ${ n-{\ell} \choose r}
\leq \left( \frac{e (n-{\ell}) }{r}\right)^r \leq \left( \frac{e n
}{r}\right)^r$ and property (b) of Lemma \[olp\_lem1\], we have $$\begin{aligned}
&\label{olpewrwfr2tae} {n \choose {\ell} }{ {n-{\ell}} \choose
r} {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~ \overline{\mathcal{E}
(\boldsymbol{J})}}\right]} \\ &\leq n^{\ell} \cdot \left( \frac{e
(n-{\ell}) }{r}\right)^r \cdot r^{r-2}{q_n}^{r-1} e^{- \lambda_2 r
{q_n} n / 3} \nonumber \\ & \leq n^{{\ell}+r} e^{r} {q_n}^{r-1}
e^{- \lambda_2 r {q_n} n / 3}.\nonumber\end{aligned}$$
Applying (\[cn-qn-2lnn-n2-newqn\]) and (\[cn-qn-2lnn-n2-newqn-2\]) to (\[olpewrwfr2tae\]), we get $$\begin{aligned}
\label{olp_an3} & {n \choose {\ell} }{ {n-{\ell}} \choose
r} {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~ \overline{\mathcal{E}
(\boldsymbol{J})}}\right]} \\ &\leq n^{{\ell}+r} e^{r}
\cdot \left(\frac{2\ln n}{n}\right)^{r-1} \cdot
e^{- \lambda_2 r (\ln n - c_0 \ln \ln n) / 3} \nonumber \\
& \leq
n^{{\ell}+1} \cdot \big(2en^{-\lambda_2 /3} (\ln n)^{c_0\lambda_2/3
+ 1}\big)^r. \nonumber\end{aligned}$$ Given $2en^{-\lambda_2 /3} (\ln n)^{c_0\lambda_2/3 + 1} = o(1)$ and (\[olp\_an3\]), we obtain $$\begin{aligned}
\nonumber & \sum_{r=R+1}^{ r_n } {n \choose {\ell} }{
{n-{\ell}} \choose r} {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})}}\right]} \\ & \leq
\sum_{r=R+1}^{ \infty } n^{{\ell}+1} \cdot \big(2en^{-\lambda_2 /3} (\ln n)^{c_0\lambda_2/3 + 1}\big)^r
\nonumber \\ & = n^{{\ell}+1} \cdot
\frac{\big(2en^{-\lambda_2 /3} (\ln n)^{c_0\lambda_2/3 +
1}\big)^{R+1}}{1- 2en^{-\lambda_2 /3} (\ln n)^{c_0\lambda_2/3 + 1}}
\nonumber \\
& \sim n^{{\ell}+1 -\lambda_2 (R+1) /3 }
\big(2e (\ln n)^{c_0\lambda_2/3 + 1}\big)^{R+1} .\label{olp_an3ar}\end{aligned}$$ We pick constant $R \geq \frac{3({\ell}+1)}{\lambda_2}$ so that ${\ell}+1 -\lambda_2
(R+1) /3 \leq -\frac{\lambda_2}{3}$. As a result, we obtain $$\begin{aligned}
\textrm{R.H.S. of
(\ref{olp_an3ar})} & = o(1) \nonumber\end{aligned}$$ and thus establish (\[prf-e2\]).
### Establishing (\[prf-e3\]).
From ${n \choose
{\ell}} \leq n^{{\ell} }$ and property (c) of Lemma \[olp\_lem1\], it holds that $$\begin{aligned}
\label{sumrrrn1}&\sum_{r=r_n+1}^{ \lfloor \frac{n-{\ell}}{2}\rfloor
} {n \choose {\ell} }{ {n-{\ell}} \choose r} ~ {{\mathbb{P}}\left[{
\mathcal{A}_{{\ell},r} ~\cap~ \overline{\mathcal{E}
(\boldsymbol{J})}}\right]} \\ & \leq n^{\ell} \cdot e^{- \mu_2 K_n n
/3} \cdot \sum_{r=r_n+1}^{ \lfloor \frac{n-{\ell}}{2}\rfloor } {
{n-{\ell}} \choose r} . \nonumber\end{aligned}$$ Given conditions $P_n = \Omega(n)$ and (\[cn-qn-2lnn-n\]), we use Lemma \[lem:logn2\] to derive $$\begin{aligned}
K_n = \Omega\big( n^{\frac{1}{2} - \frac{1}{2s}} (\ln
n)^{\frac{1}{2s}} \big) = \omega(1), \nonumber\end{aligned}$$ which yields $$\begin{aligned}
\mu _2 K_n / 3 \geq 2 \ln 2, \textrm{ for all $n$ sufficiently
large}. \label{sumrrrn1c}\end{aligned}$$ We have $$\begin{aligned}
\sum_{r=r_n+1}^{ \lfloor \frac{n-{\ell}}{2}\rfloor } { {n-{\ell}}
\choose r} \leq \sum_{r=r_n+1}^{ \lfloor \frac{n-{\ell}}{2}\rfloor
} { n \choose r } \leq \sum_{r=0}^{n} { n \choose r } = 2^n.
\label{sumrrrn1b}\end{aligned}$$
Applying (\[sumrrrn1c\]) and (\[sumrrrn1b\]) to (\[sumrrrn1\]), we finally obtain $$\begin{aligned}
&\sum_{r=r_n+1}^{ \lfloor \frac{n-{\ell}}{2}\rfloor } {n \choose
{\ell} }{ {n-{\ell}} \choose r} ~ {{\mathbb{P}}\left[{ \mathcal{A}_{{\ell},r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})}}\right]}\nonumber \\ & \leq
n^{\ell} \cdot 2^n \cdot e^{- \mu_2 K_n n /3} \nonumber \\ & =
e^{\ell \ln n + n \ln 2 - \mu_2 K_n n /3}
\nonumber \\
& \leq e^{\ell \ln n - n \ln 2 }, \textrm{ for all $n$ sufficiently large}.\nonumber\end{aligned}$$ The result (\[prf-e3\]) clearly follows with $n \to \infty$.
Proof of Lemma \[graph\_Hs\_cpln\] {#sec:pro:graph_Hs_cpln}
----------------------------------
**(a)** $$\begin{aligned}
\widetilde{P_n} = P_n, \label{wPntdn}\end{aligned}$$ and $$\begin{aligned}
\widetilde{\beta_n} = \max\{\beta_n, -\ln \ln n\}. \label{wPntdn2}\end{aligned}$$
Given (\[wPntdn2\]) and $\lim_{n \to \infty}\beta_n = -\infty$, we clearly obtain $\lim_{n \to \infty}\widetilde{\beta_n} = -\infty$ and $\widetilde{\beta_n} = -O(\ln \ln n)$.
It holds from (\[wPntdn2\]) that $\widetilde{\beta_n} \geq
\beta_n$, which along with (\[al0-parta-Hs-od\]) (\[al0-parta-Hs\]) and (\[wPntdn\]) yields $t_n \leq \widetilde{t_n}$. Under $t_n \leq \widetilde{t_n}$ and $ \widetilde{P_n} = P_n$, by [@zz Section 3], there exists a graph coupling under which $H_s(n,t_n,P_n)$ is a spanning subgraph of $H_s(n,\widetilde{t_n},\widetilde{P_n}) $.
**(b)** We set $$\begin{aligned}
\widehat{P_n} = P_n, \label{wPntdn-pb}\end{aligned}$$ and $$\begin{aligned}
\widehat{\beta_n} = \min\{\beta_n, \ln \ln n\}. \label{wPntdn2-pb}\end{aligned}$$
Given (\[wPntdn2-pb\]) and $\lim_{n \to \infty}\beta_n = \infty$, we clearly obtain $\lim_{n \to \infty}\widehat{\beta_n} = \infty$ and $\widehat{\beta_n} = O(\ln \ln n)$.
It holds from (\[wPntdn2-pb\]) that $\widehat{\beta_n} \leq
\beta_n$, which along with (\[al0-parta-Hs-pb-od\]) (\[al0-parta-Hs-pb\]) and (\[wPntdn-pb\]) yields $t_n \geq \widehat{t_n}$. Under $t_n \geq \widehat{t_n}$ and $ \widehat{P_n} = P_n$, by [@zz Section 3], there exists a graph coupling under which $H_s(n,t_n,P_n)$ is a spanning supergraph of $H_s(n,\widehat{t_n},\widehat{P_n}) $.
Proof of Lemma \[lem:logn2\].
-----------------------------
From condition $$\begin{aligned}
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} = \frac{\ln n \pm
O(\ln \ln n) }{n} \sim \frac{\ln n}{n}, \label{eqslnn-Kpsn}\end{aligned}$$ it holds that $$\begin{aligned}
\frac{{K_n}^2}{P_n}\ & = \Theta\big( n^{-\frac{1}{s}} (\ln
n)^{\frac{1}{s}} \big) , \label{tp-KnPn}\end{aligned}$$ which along with condition $P_n = \Omega(n^c)$ yields $$\begin{aligned}
K_n & = \sqrt{P_n \cdot \Theta\big( n^{-\frac{1}{s}} (\ln
n)^{\frac{1}{s}} \big)} = \Omega\Big( n^{\frac{c}{2} - \frac{1}{2s}}
(\ln n)^{\frac{1}{2s}} \Big) . \label{neq-om-Kn2s}\end{aligned}$$
Proof of Lemma \[qn-dist\].
---------------------------
**(a)** We still have (\[eqslnn-Kpsn\]) and (\[tp-KnPn\]) here. Then setting $c$ as $1$ in (\[neq-om-Kn2s\]), it holds that $$\begin{aligned}
K_n & = \Omega\Big( n^{\frac{1}{2} - \frac{1}{2s}} (\ln
n)^{\frac{1}{2s}} \Big) . \label{tp-Knsim}\end{aligned}$$ Given (\[tp-KnPn\]) and (\[tp-Knsim\]), we use [@qcomp_kcon Lemma 1] and [@ZhaoYaganGligor Lemma 8] to have $$\begin{aligned}
q_n & = \begin{cases} \frac{1}{s!} \big( \frac{{K_n}^2}{P_n}
\big)^{s} \big[1\hspace{-1pt}\pm\hspace{-1pt}
O\big(\frac{{K_n}^2}{P_n}\big) \hspace{-1pt}\pm\hspace{-1pt}
O\big(\frac{1}{K_n}\big)\big], &\textrm{\hspace{-2pt}for }s\geq 2,
\vspace{3pt}
\\ \frac{{K_n}^2}{P_n} \big[1 \hspace{-1pt}\pm\hspace{-1pt}
O\big(\frac{{K_n}^2}{P_n}\big)\big], &\textrm{\hspace{-2pt}for }s=1.
\end{cases} \label{tp-Knsim-qnn}\end{aligned}$$
Now we use (\[tp-KnPn\]) (\[tp-Knsim\]) and (\[tp-Knsim-qnn\]) to derive $q_n \sim \frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}}$ and $\big| \hspace{2pt} q_n -
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \big|
= o\big(\frac{1}{n}\big)$.
First, (\[tp-KnPn\]) and (\[tp-Knsim\]) imply $\frac{{K_n}^2}{P_n} = o(1)$ and $K_n = \omega(1)$, respectively, which are used in (\[tp-Knsim-qnn\]) to derive $q_n \sim
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}}$. Second, applying (\[tp-KnPn\]) and (\[tp-Knsim\]) directly to (\[tp-Knsim-qnn\]), we obtain the following two cases:
\(i) For $s\geq 2$, it holds that $$\begin{aligned}
&\bigg|q_n -\frac{1}{s!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{s}\bigg|
\nonumber \\ & = \Theta\bigg(\hspace{-1pt}\frac{\ln
n}{n}\hspace{-1pt} \bigg) \Big[\hspace{-2pt}\pm\hspace{-1pt}
O\big(n^{-\frac{1}{s}}(\ln n)^{\frac{1}{s}}\big)
\hspace{-1pt}\pm\hspace{-1pt} O\Big( n^{\frac{1}{2s} - \frac{1}{2}}
(\ln n)^{-\frac{1}{2s}} \hspace{-1pt}\Big)\hspace{-1pt}\Big]
\nonumber \\ & = \pm o\bigg(\frac{1}{n}\bigg) . \nonumber\end{aligned}$$
\(ii) For $s=1$, it holds that $$\begin{aligned}
&\bigg|q_n-\frac{{K_n}^2}{P_n} \bigg| = \pm
O\Bigg(\bigg(\frac{\ln n}{n} \bigg)^2 \Bigg) = \pm
o\bigg(\frac{1}{n}\bigg) . \nonumber\end{aligned}$$
Summarizing cases (i) and (ii) above, we have proved property (a) of Lemma \[qn-dist\].
**(b)** By [@bloznelis2013 Lemma 6], the edge probabilty $q_n$ satisfies $$\begin{aligned}
q_n \leq \frac{\big[\binom{K_n}{s}\big]^2}{\binom{P_n}{s}}.
\label{qnKnPn}\end{aligned}$$ From (\[qnKnPn\]) and condition $q_n = \frac{\ln n \pm O(\ln \ln
n) }{n}$, it holds that $$\begin{aligned}
\frac{\big[\binom{K_n}{s}\big]^2}{\binom{P_n}{s}} \geq \frac{\ln
n}{n} \cdot [1-o(1)] , \nonumber\end{aligned}$$ which along with $\frac{\big[\binom{K_n}{s}\big]^2}{\binom{P_n}{s}}
\leq \frac{1}{s!} \cdot \frac{{K_n}^{2s} }{(P_n - s +1)^s}$ and $P_n = \Omega(n)$ leads to $$\begin{aligned}
{K_n}^{2s} & \geq s! (P_n \hspace{-1pt}-\hspace{-1pt} s \hspace{-1pt}+\hspace{-1pt}1)^s \cdot \frac{\ln n}{n} [1\hspace{-1pt}-\hspace{-1pt}o(1)] = \Omega(n^{s-1} \ln n) . \nonumber\end{aligned}$$ Therefore, it follows that $$\begin{aligned}
K_n & = \Omega\Big( n^{\frac{1}{2} - \frac{1}{2s}} (\ln
n)^{\frac{1}{2s}} \Big) . \label{tp-Knsim-cb}\end{aligned}$$
Note that for some $n$, if $P_n < 2K_n -s $, then two vertices share at least $s$ items with probability $1$, resulting in $q_n = 1$. Therefore, given condition $q_n = \frac{\ln n \pm O(\ln \ln n)
}{n}$, we know that for all $n$ sufficiently large, $P_n \geq 2K_n
-s $ holds, so the probability that two vertices share exactly $s$ items is expressed by ${\binom{K_n}{s} \binom{P_n - K_n}{K_n -
s}}\big/{\binom{P_n}{K_n}}$. Then $$\begin{aligned}
\label{qnnewptv} q_n & \geq \mathbb{P}[\hspace{2pt}\textrm{Two vertices share exactly $s$ items.}\hspace{2pt}] \\
& = {\binom{K_n}{s} \binom{P_n - K_n}{K_n - s}}\bigg/{\binom{P_n}{K_n}} \nonumber \\
& = \frac{1}{s!} \cdot \bigg[\prod_{i=0}^{s-1}(K_n-i)\bigg]^2 \cdot
\frac{\prod_{i=0}^{K_n- s-1}(P_n-K_n)}{\prod_{i=0}^{K_n-1}(P_n-i) }
\nonumber \\
& \geq \frac{1}{s!} \cdot \frac{(K_n-s+1)^{2s}}{{P_n}^s} ,
\nonumber\end{aligned}$$ which together with condition $q_n = \frac{\ln n \pm O(\ln \ln n)
}{n}$ implies $$\begin{aligned}
\frac{(K_n-s+1)^2}{P_n} = O\big( n^{-\frac{1}{s}} (\ln n)^{\frac{1}{s}} \big). \label{tp-Knsim-cb2}\end{aligned}$$ From (\[tp-Knsim-cb\]) and the fact that $s$ is a constant, it holds that $K_n-s+1 \sim K_n$, which with (\[tp-Knsim-cb2\]) yields $$\begin{aligned}
\frac{{K_n}^2}{P_n} = O\big( n^{-\frac{1}{s}} (\ln n)^{\frac{1}{s}} \big). \label{tp-Knsim-cb3}\end{aligned}$$
Now we use (\[tp-Knsim-cb3\]) (\[tp-Knsim-cb\]) and (\[tp-Knsim-qnn\]) to derive $q_n \sim \frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}}$ and $\big| \hspace{2pt} q_n -
\frac{1}{s!} \cdot \frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \big|
= o\big(\frac{1}{n}\big)$, in a way similar to proving property (a) above.
First, (\[tp-Knsim-cb3\]) and (\[tp-Knsim-cb\]) imply $K_n =
\omega(1)$ and $\frac{{K_n}^2}{P_n} = o(1)$, respectively, which are used in (\[tp-Knsim-qnn\]) to derive $q_n \sim \frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}}$. Second, applying (\[tp-Knsim-cb3\]) and (\[tp-Knsim-cb\]) directly to (\[tp-Knsim-qnn\]), we still have cases (i) and (ii) in the proof of property (a) above. Hence, finally we also obtain $\big| \hspace{2pt} q_n - \frac{1}{s!} \cdot
\frac{{K_n}^{2s}}{{P_n}^{s}} \hspace{2pt} \big| =
o\big(\frac{1}{n}\big)$. Then property (b) is proved.
Proof of Lemma \[lem\_prob\_Eij\_S1r\].
---------------------------------------
Recall that $E_{ij}$ denotes the event that an edge exists between vertices $v_i$ and $v_j$, and $S_i$ is the number of items on vertex $v_i$. Event $E_{ij}$ occurs if and only if $|S_i \cap S_j| \geq s$. Therefore, event $\bigcap_{j=1}^{r}\overline{E_{ij}} $ is equivalent to $\bigcap_{j=1}^{r} \big(|S_i \cap S_j| < s\big)$, which clearly is implied by event $ \big|S_i \cap \big(\bigcup_{j=1}^{r} S_j\big)\big| < s $. Then $$\begin{aligned}
\label{Pj1r} &
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}}
\hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
\\ & \leq \mathbb{P}\bigg[ \hspace{2pt} \bigg|S_i \cap \bigg(\bigcup_{j=1}^{r} S_j\bigg)\bigg| < s \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r \hspace{2pt}\bigg]
\nonumber \\
& \leq 1 - \mathbb{P}\bigg[ \hspace{2pt}\bigg|S_i \cap
\bigg(\bigcup_{j=1}^{r} S_j\bigg)\bigg| = s \bigg| \hspace{2pt} S_1,
S_2, \ldots, S_r\hspace{2pt}\bigg]
\nonumber \\
& = 1 - \frac{\binom{|\bigcup_{j=1}^{r} S_j|}{s}\binom{P_n-s}{K_n-s}}{\binom{P_n}{K_n}} \nonumber \\
& = 1 - \frac{\binom{|\bigcup_{j=1}^{r} S_j|}{s}\binom{K_n}{s}}{\binom{P_n}{s}} \nonumber \\
& \leq e^{- \frac{\binom{|\bigcup_{j=1}^{r}
S_j|}{s}\binom{K_n}{s}}{\binom{P_n}{s}}} . \nonumber\end{aligned}$$
First, we have (\[qnKnPn\]) by [@bloznelis2013 Lemma 6]. Applying (\[qnKnPn\]) to (\[Pj1r\]), we obtain $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{-\frac{\binom{|\bigcup_{j=1}^{r}
S_j|}{s}}{\binom{K_n}{s}} q_n} . \label{Pj1r2}\end{aligned}$$
Now we prove properties (a), (b), and (c) of Lemma \[lem\_prob\_Eij\_S1r\], respectively.
\(a) Given condition $K_n = \omega(1)$, it follows that $\lfloor
(1+{\varepsilon_1}) K_n \rfloor > s $ for all $n$ sufficiently large. For property (a), we have condition $|\bigcup_{j=1}^{r} S_j|
\geq \lfloor (1+{\varepsilon_1}) K_n \rfloor$, which is used in (\[Pj1r2\]) to derive $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq
e^{-\frac{\binom{\lfloor(1+\varepsilon_1)K_n\rfloor}{s}}{\binom{K_n}{s}}
q_n}. \label{Pj1r3}\end{aligned}$$ We have $$\begin{aligned}
\label{epsKns} \frac{\binom{\lfloor(1+\varepsilon_1)
K_n\rfloor}{s}}{\binom{K_n}{s}} & =
\frac{\prod_{i=0}^{s-1}\big\{\lfloor(1+\varepsilon_1) K_n\rfloor -
i\big\}}{\prod_{i=0}^{s-1}(K_n-i)} \\
& \geq \bigg[\frac{(1+\varepsilon_1) K
_n-1-s}{K_n}\bigg]^s.\nonumber\end{aligned}$$
Given conditions $ \varepsilon_2 < (1+\varepsilon_1)^s-1$ and $K_n = \omega(1)$, it follows that $K_n \geq \frac{s+1}{1+\varepsilon_1 - \sqrt[s]{1+\varepsilon_2}} $ for all $n$ sufficiently large, yielding $$\begin{aligned}
\label{epsKns2} \frac{(1+\varepsilon_1) K _n-1-s}{K_n}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
\geq (1+\varepsilon_1)- (s+1) \cdot \frac{1+\varepsilon_1 -
\sqrt[s]{1+\varepsilon_2}}{s+1} = \sqrt[s]{1+\varepsilon_2}.
\nonumber\end{aligned}$$ Applying (\[epsKns2\]) to (\[epsKns\]), we obtain $$\begin{aligned}
\frac{\binom{\lfloor(1+\varepsilon_1)K_n\rfloor}{s}}{\binom{K_n}{s}}
& \geq \big( \sqrt[s]{(1+\varepsilon_2)} \hspace{2pt} \big)^s =
1+\varepsilon_2, \nonumber\end{aligned}$$ which is substituted into (\[Pj1r3\]) to induce $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{- q_n ( 1+\varepsilon_2)}. \nonumber\end{aligned}$$
\(b) Given condition $K_n = \omega(1)$, it follows that $\lfloor
\lambda_1 r K_n \rfloor > s $ for all $n$ sufficiently large. For property (b), we have condition $|\bigcup_{j=1}^{r} S_j| \geq
\lfloor \lambda_1 r K_n \rfloor$, which is used in (\[Pj1r2\]) to derive $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{-\frac{\binom{\lfloor \lambda_1 r K_n
\rfloor}{s}}{\binom{K_n}{s}} q_n}. \label{Pj1r3b}\end{aligned}$$ We have $$\begin{aligned}
\frac{\binom{\lfloor\lambda_1 r K_n\rfloor}{s}}{\binom{K_n}{s}}
&\hspace{-2pt} =\hspace{-2pt}
\frac{\prod_{i=0}^{s-1}(\lfloor\lambda_1 r K_n \rfloor \hspace{-1pt}
- \hspace{-1pt} i)}{\prod_{i=0}^{s-1}(K_n-i)} \hspace{-2pt} \geq
\hspace{-2pt} \bigg(\frac{\lambda_1 r K
_n\hspace{-1pt}-\hspace{-1pt}1\hspace{-1pt}-\hspace{-1pt}s}{K_n}\bigg)^s\hspace{-2pt}.
\label{epsKnsb}\end{aligned}$$
Given conditions $ \lambda_2 < {\lambda_1}^s$ and $K_n = \omega(1)$, it follows that $K_n \geq \frac{s+1}{\lambda_1 - \sqrt[s]{\lambda_2}} \geq \frac{s+1}{r(\lambda_1 - \sqrt[s]{\lambda_2})}$ for all $n$ sufficiently large, inducing $$\begin{aligned}
\frac{\lambda_1 r K
_n\hspace{-1.5pt}-\hspace{-1.5pt}1\hspace{-1.5pt}-\hspace{-1.5pt}s}{K_n}
\hspace{-1.5pt}\geq\hspace{-1.5pt} \lambda_1 r\hspace{-1.5pt}
-\hspace{-1.5pt} (s\hspace{-1.5pt}+\hspace{-1.5pt}1)
\frac{r(\lambda_1 \hspace{-1.5pt}-\hspace{-1.5pt}
\sqrt[s]{\lambda_2}\hspace{2pt})}{s+1}
\hspace{-1.5pt}=\hspace{-1.5pt} \sqrt[s]{\lambda_2} r.
\label{epsKns2b}\end{aligned}$$ Applying (\[epsKns2b\]) to (\[epsKnsb\]), we obtain $ {\binom{\lfloor\lambda_1 r K_n \rfloor}{s}}\Big/{\binom{K_n}{s}}
\geq \big( \sqrt[s]{\lambda_2} r \hspace{2pt} \big)^s = \lambda_2
r^s \geq \lambda_2 r,$ which is substituted into (\[Pj1r3b\]) to induce $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{-\lambda_2 r q_n}. \nonumber\end{aligned}$$
\(c) From $P_n \geq K_n = \omega(1)$, it follows that $P_n =
\omega(1)$. Then $\lfloor\mu_1 P_n \rfloor > s $ for all $n$ sufficiently large. For property (c), we have condition $|\bigcup_{j=1}^{r} S_j| \geq \lfloor\mu_1 P_n\rfloor$, which is used in (\[Pj1r\]) to derive $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{-\frac{\binom{\lfloor\mu_1 P_n
\rfloor}{s}\binom{K_n}{s}}{\binom{P_n}{s}} }. \label{Pj1r3c}\end{aligned}$$ We have $$\begin{aligned}
\label{epsKnsc} & \frac{\binom{\lfloor\mu_1
P_n\rfloor}{s}\binom{K_n}{s}}{\binom{P_n}{s}} \\ & \geq
\frac{(s!)^{-1}(\lfloor\mu_1 P_n\rfloor -s )^s \cdot
(s!)^{-1}(K_n-s)^s}{(s!)^{-1}(P_n)^s} \nonumber \\ & \geq
\frac{1}{s!} \cdot \bigg(\frac{\mu_1 P_n -1 -s}{P_n}\bigg)^s \cdot
(K_n-s)^s.\nonumber\end{aligned}$$ Given $0 < \mu_2 < (s!)^{-1}{\mu_1}^s$ and $P_n \geq K_n = \omega(1)$, it follows that $P_n \geq \frac{s+1}{\mu_1 - \sqrt{\mu_1}\sqrt[2s]{s! \mu_2}}$ and $K_n \geq \frac{s+1}{1 - \sqrt{\frac{s! \mu_2}{{\mu_1}^s}} }$ for all $n$ sufficiently large, inducing $$\begin{aligned}
\label{epsKns2c} \frac{\mu_1 P_n - 1 -s}{P_n}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\ \geq \mu_1 - (s+1)
\cdot \frac{\mu_1 - \sqrt{\mu_1}\sqrt[2s]{s! \mu_2}}{s+1} =
\sqrt{\mu_1}\sqrt[2s]{s! \mu_2},\nonumber\end{aligned}$$ and $$\begin{aligned}
\label{epsKns2c2} (K_n-s)^s~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\ \geq K_n-s \geq K_n-K_n\Bigg(1 - \sqrt{\frac{s!
\mu_2}{{\mu_1}^s}}\hspace{2pt}\Bigg) = \sqrt{\frac{s!
\mu_2}{{\mu_1}^s}} K_n.\nonumber\end{aligned}$$ Applying (\[epsKns2c\]) and (\[epsKns2c2\]) to (\[epsKnsc\]), we obtain $$\begin{aligned}
\frac{\binom{\lfloor\mu_1
P_n\rfloor}{s}\binom{K_n}{s}}{\binom{P_n}{s}} \hspace{-1pt} &
\hspace{-1pt} \geq \hspace{-1pt} \frac{1}{s!} \hspace{-.5pt}
(\sqrt{\mu_1}\sqrt[2s]{s! \mu_2})^s \hspace{-1pt} \sqrt{\frac{s!
\mu_2}{{\mu_1}^s}} K_n \hspace{-1pt} = \hspace{-1pt} \mu_2 K_n,
\nonumber\end{aligned}$$ which is substituted into (\[Pj1r3c\]) to induce $$\begin{aligned}
\mathbb{P}\bigg[\hspace{2pt}\bigcap_{j=1}^{r}\overline{E_{ij}} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
& \leq e^{-\mu_2 K_n}. \nonumber\end{aligned}$$
Proof of Lemma \[olp\_lem1\].
-----------------------------
We consider events $\mathcal{B}_{\ell,r} $, $ \mathcal{D}_{\ell,r}$ and $\mathcal{A}_{{\ell},r}$ defined in Section 6.3. By definitions, we have $$\begin{aligned}
\mathcal{B}_{\ell,r}&:= \bigcap_{i=r+1}^{r+\ell} \bigcup_{j =
1}^{r}E_{ij}, \nonumber
\\ \mathcal{D}_{\ell,r} &:= \bigcap_{i =
r+\ell+1}^{n}\bigcap_{j = 1}^{r} \overline{E_{ij}}, \nonumber\end{aligned}$$ and $$\begin{aligned}
\mathcal{A}_{\ell,r} &:= \mathcal{B}_{\ell,r} \cap \mathcal{C}_{r}
\cap \mathcal{D}_{\ell,r} . \nonumber\end{aligned}$$ Then considering that given $S_1,S_2, \ldots, S_r$, events $\mathcal{B}_{\ell,r}$ and $\mathcal{D}_{\ell,r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})}$ are conditionally independent, we obtain $$\begin{aligned}
& \label{allbounds} {{\mathbb{P}}\left[{ \mathcal{A}_{\ell, r} \cap \overline{\mathcal{E} (\boldsymbol{J})}}\right]}
\\
& = {{\mathbb{P}}\left[{ \mathcal{C}_{r} \cap \mathcal{B}_{\ell,r} \cap \mathcal{D}_{\ell,r} \cap
\overline{\mathcal{E} (\boldsymbol{J})} }\right]} \nonumber \\
& = \sum_{\begin{subarray} ~S_1,S_2, \ldots, S_r: \\ ~\mathcal{C}_{r} \textnormal{ happens.} \end{subarray}}\bigg\{ \mathbb{P}[ S_1,\hspace{-1pt} S_2,\hspace{-1pt}
\ldots,\hspace{-1pt} S_r] \mathbb{P}[ \mathcal{B}_{\ell,r} \hspace{-1pt} \boldsymbol{\mid}\hspace{-1pt} S_1,\hspace{-1pt} S_2,\hspace{-1pt}
\ldots,\hspace{-1pt} S_r] \nonumber \\
& ~~~~~~~~~~~~~\mathbb{P}[\hspace{2pt} \mathcal{D}_{\ell,r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}
\boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r] \bigg\}. \nonumber\end{aligned}$$
We have $$\begin{aligned}
\mathbb{P}[ \mathcal{B}_{\ell,r} \hspace{-1pt} \boldsymbol{\mid}\hspace{-1pt} S_1,\hspace{-1pt} S_2,\hspace{-1pt}
\ldots,\hspace{-1pt} S_r]
& \hspace{-1pt} = \hspace{-1pt} \Bigg\{ \hspace{-1pt}
\mathbb{P}\bigg[ \hspace{-1pt}\bigcup_{j=1}^{r} E_{ij} \hspace{1pt}
\bigg| \hspace{1pt} S_1,\hspace{-1pt} S_2,\hspace{-1pt}
\ldots,\hspace{-1pt} S_r\hspace{-1pt}\bigg]\hspace{-1pt}
\Bigg\}^{\ell}. \nonumber\end{aligned}$$ By the union bound, $$\begin{aligned}
&\mathbb{P}\bigg[\hspace{2pt}\bigcup_{j=1}^{r} E_{ij} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2, \ldots, S_r\bigg]
\nonumber \\ &\leq \sum_{j=1}^{r} \mathbb{P}[ E_{ij} \hspace{2pt}
\boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r] =
\sum_{j=1}^{r} \mathbb{P}[ E_{ij} ] = r q_n. \nonumber\end{aligned}$$ Then $$\begin{aligned}
\mathbb{P}[\hspace{2pt} \mathcal{B}_{\ell,r} \hspace{2pt} \boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r]
& \leq \min\{(r q_n)^{\ell}, 1\} . \label{boundBrleq}\end{aligned}$$ We have $$\begin{aligned}
& \mathbb{P}[\hspace{2pt} \mathcal{D}_{\ell,r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}
\boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r] \nonumber\\ &
= \Bigg\{ \mathbb{P}\bigg[\hspace{2pt} \bigg(\bigcap_{j=1}^{r}
\overline{E_{ij}} \bigg)~\cap~ \overline{\mathcal{E}
(\boldsymbol{J})} \hspace{2pt} \bigg| \hspace{2pt} S_1, S_2,
\ldots, S_r\bigg] \Bigg\}^{n-\ell-r}.\nonumber\end{aligned}$$ By Lemma \[lem\_prob\_Eij\_S1r\], for all $n$ sufficiently large,
- for $r = 2, 3, \ldots, R$, it holds that $$\begin{aligned}
& \mathbb{P}[\hspace{2pt} \mathcal{D}_{\ell,r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}
\boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r] \nonumber \\ &
\quad \leq e^{- q_n (1+\varepsilon_2)(n-\ell-r)} \leq e^{- q_n n
(1+\varepsilon_3)}.\nonumber\end{aligned}$$ To see this, pick any $\varepsilon_3 < (1+ \varepsilon_1)^s-1$, and use Lemma \[lem\_prob\_Eij\_S1r\] with $\varepsilon_3 < \varepsilon_2 < (1+ \varepsilon_1)^s-1$.
- for $r = 2, 3, \ldots, r_n $, it holds that $$\begin{aligned}
& \mathbb{P}[\hspace{2pt} \mathcal{D}_{\ell,r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}
\boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r] \nonumber \\
& \quad \leq e^{- \lambda_2 r q_n (n-\ell-r)} \leq e^{- \lambda_2
r q_n n /3}.\nonumber\end{aligned}$$
- for $r = r_n + 1, r_n + 2, \ldots, \lfloor
\frac{n-{\ell}}{2} \rfloor $, it holds that $$\begin{aligned}
& \mathbb{P}[\hspace{2pt} \mathcal{D}_{\ell,r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}
\boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r] \nonumber \\
& \quad \leq e^{- \mu_2 K_n (n-\ell-r)} \leq e^{- \mu_2 K_n n
/3}.\nonumber\end{aligned}$$
For simplicity, we use $\Lambda$ to summarize the upper bounds on $\mathbb{P}[\hspace{2pt} \mathcal{D}_{\ell,r} ~\cap~
\overline{\mathcal{E} (\boldsymbol{J})} \hspace{2pt}
\boldsymbol{\mid} \hspace{2pt} S_1, S_2, \ldots, S_r]$ in cases (a) (b) and (c) above; i.e., $\Lambda = e^{- q_n n
(1+\varepsilon_3)}$ for $r = 2, 3, \ldots, R$, and $e^{- \lambda_2
r q_n n /3}$ for $r = R+1, R+2, \ldots, r_n $, and $e^{- \mu_2 K_n n
/3}$ for $r = r_n + 1, r_n + 2, \ldots, \lfloor
\frac{n-{\ell}}{2} \rfloor $. In view of $\Lambda$, (\[boundBrleq\]) and ${{\mathbb{P}}\left[{ \mathcal{C}_{r}}\right]} \leq \min\{ r^{r-2} {q_n}^{r-1}, 1\}$ by [@ZhaoYaganGligor Lemma 11], we obtain from (\[allbounds\]) that $$\begin{aligned}
& \nonumber {{\mathbb{P}}\left[{ \mathcal{A}_{\ell, r} \cap \overline{\mathcal{E} (\boldsymbol{J})}}\right]}
\\
& \leq \sum_{\begin{subarray} ~S_1,S_2, \ldots, S_r: \\ ~\mathcal{C}_{r} \textnormal{ happens.} \end{subarray}}\bigg\{ \mathbb{P}[ S_1,\hspace{-1pt} S_2,\hspace{-1pt}
\ldots,\hspace{-1pt} S_r]\cdot \min\{(r q_n)^{\ell}, 1\} \cdot \Lambda \bigg\} \nonumber \\
& = {{\mathbb{P}}\left[{\mathcal{C}_{r}}\right]}\cdot \min\{(r q_n)^{\ell}, 1\} \cdot \Lambda \\
& \leq \min\{ r^{r-2} {q_n}^{r-1}, 1\} \cdot \min\{(r q_n)^{\ell}, 1\} \cdot \Lambda , \nonumber\end{aligned}$$ which clearly completes the proof of Lemma \[olp\_lem1\].
Proof of Lemma \[lem-bing1\]
----------------------------
Since $s$ is a constant, from condition $$\begin{aligned}
\frac{1}{s!} \cdot {{t_n}^{2s}}{{P_n}^{s}} = \frac{\ln n \pm
O(\ln \ln n) }{n} = \Theta\bigg(\frac{\ln n}{n} \bigg),
\label{eqslnn-Kpsn-brig}\end{aligned}$$ it holds that $$\begin{aligned}
{{t_n}^2}{P_n} & = \Theta\Bigg( \bigg(\frac{\ln n}{n}\bigg)^{\frac{1}{s}} \Bigg) = o(1) .\label{tp-KnPn-brig}\end{aligned}$$ Note that under (\[thm:bin:eq:P\]), we have $P_n = \Omega(n)$ for $s\geq 2$ or $P_n = \Omega(n^c)$ for $s=1$ with some constant $c>1$. Hence, under (\[thm:bin:eq:P\]) and $(\ref{tp-KnPn-brig})$, we use [@Rybarczyk Proposition 2] and obtain $$\begin{aligned}
\label{tp-KnPn-brig2} \rho_n = \binom{P_n}{s} \cdot {t_n}^{2s} \cdot \big[1 \pm O({{t_n}^2}{P_n} )\big]~~~~~~~~~~~~ \\
= \begin{cases} \frac{1}{s!} {{t_n}^{2s}}{{P_n}^{s}} \big[1 \pm O({{t_n}^2}{P_n} ) \pm O({P_n}^{-1})\big] , &\textrm{for } s \geq 2, \\
{t_n}^2 P_n \big[1 \pm O({{t_n}^2}{P_n} )\big] , &\textrm{for } s = 1.
\end{cases}\nonumber\end{aligned}$$ Applying $P_n = \Omega(n)$ and $(\ref{tp-KnPn-brig})$ to $(\ref{tp-KnPn-brig2})$, we derive $\rho_n \sim \frac{1}{s!}
\cdot {{t_n}^{2s}}{{P_n}^{s}}$ and $\rho_n = \frac{1}{s!} \cdot
{{t_n}^{2s}}{{P_n}^{s}} + o\big(\frac{1}{n}\big)$.
Evaluating the Edge Probability $p_{q}$ {#eval}
=======================================
We present Lemma \[lem\_eval\_psq\] to evaluate the edge probability $p_{q}$.
\[lem\_eval\_psq\]
The properties (a) and (b) below hold.
\(a) If $\lim\limits_{n \to \infty}\frac{{K_n}^2}{P_n} = 0$ and $K_n
\geq q$ for all $n$ sufficiently large, then $$\begin{aligned}
p_{q} & = \frac{{[K_n(K_n-1)\ldots(K_n-(q-1))]}^2}{q!{P_n}^{q}}
\cdot \bigg[1\pm O\bigg(\frac{{K_n}^2}{P_n}\bigg)\bigg]. \label{pu1}\end{aligned}$$
\(b) If $\lim\limits_{n \to \infty}\frac{{K_n}^2}{P_n} = 0$ and $\lim\limits_{n \to \infty} K_n = \infty$, then $$\begin{aligned}
p_{q} & = \frac{1}{q!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{q}
\cdot \bigg[1\pm O\bigg(\frac{{K_n}^2}{P_n}\bigg) \pm
O\bigg(\frac{1}{K_n}\bigg)\bigg]. \label{pu2}\end{aligned}$$
The Proof of Lemma \[lem\_eval\_psq\] {#sec:lem_eval_psq}
-------------------------------------
We demonstrate properties (a) and (b) of Lemma \[lem\_eval\_psq\] below.
***Establishing Property (a):***
By definition, the edge probability is expressed by $$\begin{aligned}
p_u & = \sum_{r=q}^{K_n} \mathbb{P}[|S_{i} \cap S_{j}| = r].
\nonumber\end{aligned}$$
From $\lim\limits_{n \to \infty}\frac{{K_n}^2}{P_n} = 0$, it holds that $P_n \geq 2K_n$ for all $n$ sufficiently large. Then $$\begin{aligned}
\mathbb{P}[|S_{i} {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}S_{j}| =
r]& =
\frac{\binom{K_n}{r}\binom{P_n-K_n}{K_n-r}}{\binom{P_n}{K_n}}.
\label{psiju}\end{aligned}$$
We will demonstrate the following (\[eq\_psijq\]) and (\[eq\_psiju\]). $$\begin{aligned}
\mathbb{P}[|S_{i} {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}S_{j}| = q] & =
\frac{{[K_n(K_n-1)\ldots(K_n-(q-1))]}^2}{q!{P_n}^{q}} \nonumber \\ &
\quad \times \bigg[1\pm O\bigg(\frac{{K_n}^2}{P_n}\bigg)\bigg],
\label{eq_psijq}\end{aligned}$$ and $$\begin{aligned}
\frac{\sum_{r=q+1}^{K_n} \mathbb{P}[|S_{i} \cap S_{j}| =
r]}{\mathbb{P}[|S_{i} \cap S_{j}| = q]} & =
O\bigg(\frac{{K_n}^2}{P_n}\bigg). \label{eq_psiju}\end{aligned}$$ Once (\[eq\_psijq\]) and (\[eq\_psiju\]) is proved, with $\lim\limits_{n \to \infty}\frac{{K_n}^2}{P_n} = 0$, we have $$\begin{aligned}
p_u & = \sum_{r=q}^{K_n} \mathbb{P}[|S_{i} \cap S_{j}| = r]
\nonumber
\\ & = \mathbb{P}[|S_{i} {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}S_{j}| = q] \cdot \bigg\{1+\frac{\sum_{r=q+1}^{K_n} \mathbb{P}[|S_{i} \cap S_{j}| =
r]}{\mathbb{P}[|S_{i} \cap S_{j}| = q]}\bigg\} \nonumber
\\ & = \frac{{[K_n(K_n-1)\ldots(K_n-(q-1))]}^2}{q!{P_n}^{q}} \cdot
\bigg[1\pm O\bigg(\frac{{K_n}^2}{P_n}\bigg)\bigg]; \nonumber\end{aligned}$$ i.e., the property (a) of Lemma \[lem\_eval\_psq\] holds.
We start with showing (\[eq\_psijq\]). From (\[psiju\]), it follows that $$\begin{aligned}
& \mathbb{P}[|S_{i} {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}S_{j}| = q] \nonumber
\\ & = \frac{\binom{K_n}{q}\binom{P_n-K_n}{K_n-q}}{\binom{P_n}{K_n}}
\nonumber
\\ &
=
\frac{1}{q!} \bigg[\frac{K_n!}{(K_n-q)!}\bigg]^2 \cdot
\frac{(P_n-K_n)!}{(P_n-2K_n+q)!} \cdot
\frac{(P_n-K_n)!}{P_n!}. \label{psijq}\end{aligned}$$ Then $$\begin{aligned}
& \mathbb{P}[|S_{i} {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}S_{j}| = q] \bigg/ \bigg\{
\frac{{[K_n(K_n-1)\ldots(K_n-(q-1))]}^2}{q!{P_n}^{q}} \bigg\}
\label{psb}
\\ \quad & = {P_n}^{q} \cdot
\frac{(P_n-K_n)!}{(P_n-2K_n+q)!} \cdot
\frac{(P_n-K_n)!}{P_n!} \nonumber
\\ \quad & = {P_n}^{q} \cdot
\bigg[ \prod_{\ell = 0}^{K_n-q-1} (P_n -K_n - \ell)\bigg] \cdot
\prod_{\ell = 0}^{K_n-1} \frac{1}{P_n - \ell} \nonumber
\\ \quad & = \prod_{\ell = 0}^{K_n-1}\frac{P_n}{P_n - \ell}
\prod_{\ell = 0}^{K_n-q-1}\frac{P_n -K_n - \ell}{P_n} \nonumber
\\ \quad & = \Bigg[{\prod_{\ell = 0}^{K_n-q-1}\bigg(1-\frac{K_n +
\ell}{P_n}\bigg)}\Bigg]\Bigg/\Bigg[{\prod_{\ell =
0}^{K_n-1}\bigg(1-\frac{\ell}{P_n}\bigg)}\Bigg]. \label{p0}\end{aligned}$$ By [@ZhaoYaganGligor Fact 2(b)], we have $$\begin{aligned}
\prod_{\ell = 0}^{K_n-q-1}\bigg(1-\frac{K_n + \ell}{P_n}\bigg) &
\geq \bigg(1-\frac{2K_n}{P_n}\bigg)^{K_n} \geq 1 -
\frac{2{K_n}^2}{P_n} \label{p1}\end{aligned}$$ and $$\begin{aligned}
{\prod_{\ell =
0}^{K_n-1}\bigg(1-\frac{\ell}{P_n}\bigg)} & \geq
\bigg(1-\frac{K_n}{P_n}\bigg)^{K_n} \geq 1 - \frac{{K_n}^2}{P_n}.
\label{p2}\end{aligned}$$ In addition, $$\begin{aligned}
\prod_{\ell = 0}^{K_n-q-1}\bigg(1-\frac{K_n + \ell}{P_n}\bigg) &
\leq 1 \vspace{-2pt} \label{p3}\end{aligned}$$ and $$\begin{aligned}
{\prod_{\ell =
0}^{K_n-1}\bigg(1-\frac{\ell}{P_n}\bigg)}
\bigg(1-\frac{K_n}{P_n}\bigg)^{K_n} & \leq 1 \vspace{-2pt}
\label{p4}.\end{aligned}$$
Applying (\[p1\]) and (\[p4\]) to (\[p0\]), we obtain $$\begin{aligned}
\textrm{(\ref{psb})} & \geq 1 - \frac{2{K_n}^2}{P_n} \vspace{-2pt}
\label{p5}.\end{aligned}$$ From $\lim\limits_{n \to \infty}\frac{{K_n}^2}{P_n} = 0$, it holds that for all $n$ sufficiently large $$\begin{aligned}
\bigg(1 + \frac{2{K_n}^2}{P_n}\bigg) \bigg(1 -
\frac{{K_n}^2}{P_n}\bigg) = 1 + \frac{{K_n}^2}{P_n} - 2
\bigg(\frac{{K_n}^2}{P_n}\bigg)^2 \geq 1. \vspace{-2pt} \nonumber\end{aligned}$$ Then applying (\[p2\]) and (\[p3\]) to (\[p0\]), we obtain $$\begin{aligned}
\textrm{(\ref{psb})} & \leq \frac{1}{1 - \frac{{K_n}^2}{P_n}}\leq
1 + \frac{2{K_n}^2}{P_n} \vspace{-2pt} \label{p6}.\end{aligned}$$ Clearly, (\[eq\_psijq\]) is proved in view of (\[p5\]) and (\[p6\]).
We now demonstrate (\[eq\_psiju\]). From (\[psiju\]), $$\begin{aligned}
& \frac{\sum_{r=q+1}^{K_n}\mathbb{P}[|S_{ij}| = r]}{\mathbb{P}[|S_{ij}| = q]}\nonumber \\
& = \sum_{r=q+1}^{K_n} \bigg[{\binom{K_n}{r}
\binom{P_n-K_n}{K_n-r}}\bigg/{\binom{K_n}{q}\binom{P_n-K_n}{K_n-q}}
\bigg]
\nonumber \\
& = \sum_{r=q+1}^{K_n} \bigg\{ \frac{q!}{r!}
\bigg[\frac{(K_n-q)!}{(K_n-r)!}\bigg]^2
\frac{(P_n-2K_n+q)!}{(P_n-2K_n+r)!}\bigg\} \nonumber \\
& \leq \sum_{r=q+1}^{K_n} \bigg[ \frac{1}{(r-q)!} \cdot
{K_n}^{2(r-q)}
\cdot (P_n-2K_n)^{q-r} \bigg] \nonumber \\
& \leq e^{\frac{{K_n}^2}{P_n-2K_n}} - 1 \quad \textrm{(by Taylor
series)} \nonumber \\ & = O\bigg(\frac{{K_n}^2}{P_n}\bigg).
\nonumber \vspace{-2pt}\end{aligned}$$ Property (a) of Lemma \[lem\_eval\_psq\] is proved with (\[eq\_psijq\]) and (\[eq\_psiju\]).
***Establishing Property (b):***
From condition $\lim\limits_{n \to \infty} K_n = \infty$, we have $K_n
> q$ for all $n$ sufficiently large, leading to $$\begin{aligned}
K_n(K_n-1)\ldots(K_n-(q-1)) & \geq {K_n}^q \cdot
\bigg(1-\frac{q}{K_n}\bigg)^q \nonumber.\end{aligned}$$ By [@ZhaoYaganGligor Fact 2(b)], we further obtain $$\begin{aligned}
K_n(K_n-1)\ldots(K_n-(q-1)) & \geq {K_n}^q \cdot
\bigg(1-\frac{q^2}{K_n}\bigg),\nonumber\end{aligned}$$ which together with $K_n(K_n-1)\ldots(K_n-(q-1)) \leq {K_n}^q$ yields $$\begin{aligned}
K_n(K_n-1)\ldots(K_n-(q-1)) & = {K_n}^q \cdot \bigg[1-
O\bigg(\frac{1}{K_n}\bigg)\bigg]. \label{Xns}\end{aligned}$$
Given $\lim\limits_{n \to \infty}\frac{{K_n}^2}{P_n} = 0$ and $K_n
> q$ for all $n$ sufficiently large, we use property (a) of Lemma \[lem\_eval\_psq\] to obtain (\[pu1\]). Substituting (\[Xns\]) into (\[pu1\]), we get $$\begin{aligned}
p_{q} & = {K_n}^q \cdot \bigg[1- O\bigg(\frac{1}{K_n}\bigg)\bigg]
\cdot \bigg[1\pm O\bigg(\frac{{K_n}^2}{P_n}\bigg)\bigg],\nonumber\end{aligned}$$ which along with $\frac{{K_n}^2}{P_n} = o(1)$ leads to (\[pu2\]).
With “$\Leftrightarrow$” meaning the equivalence relations between events, it is clear that $$\begin{aligned}
(d = h) \Leftrightarrow \bigg[ (L_h \neq 0) {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}\bigg(\bigcap_{i=0}^{h-1}L_i = 0\bigg) \bigg]. \label{dh}\end{aligned}$$
First, from (\[XnYn3\]),
$$\begin{aligned}
\mathbb{P}[d = h] & \leq \mathbb{P}[L_{h-1} = 0] . \nonumber\end{aligned}$$
Given ??, we use $\mathbb{P}[A {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}B] \geq \mathbb{P}[A] -
\mathbb{P}[\overline{B}]$ and the union bound to derive $$\begin{aligned}
\mathbb{P}[d = h] & = \mathbb{P}\bigg[ (L_h \neq 0) {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}(L_{h-1} =
0) {\hspace{2pt} \mathlarger{\cap}
\hspace{2pt}}\bigg(\bigcap_{i=0}^{h-2}L_i = 0\bigg) \bigg] \nonumber \\
& \geq \mathbb{P}[L_{h-1} = 0] - \mathbb{P}[L_{h} = 0] -
\mathbb{P}\bigg[\bigg(\bigcup_{i=0}^{h-1}L_i \neq 0\bigg)\bigg]
\nonumber \\
& \geq \mathbb{P}[L_{h-1} = 0] - \mathbb{P}[L_{h} = 0] -
\sum_{i=0}^{h-1} \mathbb{P}[L_i \neq 0] . \nonumber\end{aligned}$$
Relaxing the Condition on $K_n$ in Theorem \[thm:exact\_qcomposite2\]
=====================================================================
Note that we have $\lim\limits_{n \to \infty} K_n = \infty$ in Theorem \[thm:exact\_qcomposite2\]. A future research is to relax such condition. Without such condition, property (a) of Lemma \[lem\_eval\_psq\] presents an evaluation of the edge probability $p_u$ by (\[pu1\]) under $\lim\limits_{n \to
\infty}\frac{{K_n}^2}{P_n} = 0$.
Given (\[pu1\]), we may replace condition (\[XnYn\]) in Theorem \[thm:exact\_qcomposite2\] by $$\begin{aligned}
\frac{{[K_n(K_n-1)\ldots(K_n-(q-1))]}^2}{q!{P_n}^{q}} & =
\frac{\ln n + {(h-1)} \ln \ln n + {\gamma_n}-\ln[(h-1)!]}{n} \label{XnYn2}\end{aligned}$$ for positive integer $h$ and sequence $\gamma_n$ satisfying (\[gamn\]) and having $\lim\limits_{n \to \infty}\gamma_n$ ($\lim\limits_{n \to \infty}\gamma_n \in [-\infty,\infty]$). Then a future work is to prove that under (\[XnYn2\]) and some condition on $K_n$ weaker than $\lim\limits_{n \to \infty} K_n = \infty$, properties (a) and (b) in Theorem \[thm:exact\_qcomposite2\] still follow. Note that a required condition on $K_n$ is that $K_n \geq q$ for all $n$ sufficiently large, to ensure $\textrm{L.H.S. of
(\ref{XnYn2})}$ is positive for all $n$ sufficiently large.
We discuss the required condition on $K_n$ weaker than $\lim\limits_{n \to \infty} K_n = \infty$. First, to ensure $\textrm{L.H.S. of (\ref{XnYn2})}$ is positive for all $n$ sufficiently large, we have $$\begin{aligned}
\textrm{$K_n \geq q$ for all $n$ sufficiently large}. \label{Xs1}\end{aligned}$$ Second, we explain below $$\begin{aligned}
\textrm{$K_n \geq 2$ for all $n$ sufficiently large}. \label{Xs2}\end{aligned}$$ By contradiction, if such condition does not hold, there exists a subsequence $m$ of $n$ ($n=1,2,\ldots$) such that $X_m = 1$ for all $m$ sufficiently large (note that $m \to \infty$ if and only if $n
\to \infty$). $X_m = 1$ and (\[Xs1\]) lead to $q=1$. Then from (\[XnYn2\]), we obtain $$\begin{aligned}
\frac{{K_n}^2}{P_n} & = \frac{\ln n + {(h-1)} \ln \ln n +
{\gamma_n}-\ln[(h-1)!]}{n} ,\nonumber\end{aligned}$$ and $$\begin{aligned}
\frac{{X_m}^2}{Y_m} & = \frac{\ln m + {(h-1)} \ln \ln m +
{\gamma_m}-\ln[(h-1)!]}{m} . \label{Xs3}\end{aligned}$$ From $X_m = 1$ and (\[Xs3\]), we derive $$\begin{aligned}
Y_m & = \frac{m}{\ln m + {(h-1)} \ln \ln m +
{\gamma_m}-\ln[(h-1)!]} . \label{Xs4}\end{aligned}$$ From $X_m = 1$ and $q=1$, graph $G_q(m,X_m,Y_m)$ has a positive minimum vertex degree (i.e., no vertex is isolated) if and only if
Under (\[XnYn2\]), we can prove that (\[limXnYn\]) still holds, and there exists some sequence $\gamma_n^* = \gamma_n \pm o(1)$ such that (\[XnYn3\]) still holds. For $i=0,1,\ldots,q-1$, for all $n$ sufficiently large we have $K_n
- i
\geq \frac{2K_n}{q+1} $ from $K_n \geq q$, so $\textrm{L.H.S. of (\ref{XnYn2})} \geq \frac{1}{q!}
\big( \frac{{K_n}^2}{P_n} \big)^{q}
\big(\frac{2}{q+1}\big)^{2q} $ follows. Similar to (\[leq\]), with $\gamma_n $ satisfying (\[gamn\]), we derive from (\[XnYn2\]) that for all $n$ sufficient large, $\textrm{L.H.S. of (\ref{XnYn2})}
\leq \frac{\ln n + h \ln \ln n}{n} = O\big(\frac{\ln n}{n}\big)$. Clearly, we obtain $\frac{{K_n}^2}{P_n} = \big[O\big(\frac{\ln
n}{n}\big) \cdot q! \cdot \big(\frac{q+1}{2}\big)^{2q}\big] =
O\big(\big(\frac{\ln n}{n}\big)^{{1}/{q}}\big)$; i.e., (\[limXnYn\]) holds. Then using (\[limXnYn\]) (\[XnYn2\]) and $\textrm{L.H.S. of (\ref{XnYn2})} = O\big(\frac{\ln n}{n}\big)$ in property (a) of Lemma \[lem\_eval\_psq\], we obtain
$$\begin{aligned}
K_n - i & \geq K_n - (q-1) \geq K_n - \frac{q-1}{q+1} \cdot K_n =
\frac{2K_n}{q+1} .\end{aligned}$$
$$\begin{aligned}
\textrm{L.H.S. of (\ref{XnYn2})} \geq
\frac{1}{q!{P_n}^{q}}\bigg(\frac{2K_n}{q+1}\bigg)^{2q} =
\frac{1}{q!} \bigg( \frac{{K_n}^2}{P_n} \bigg)^{q} \cdot
\bigg(\frac{2}{q+1}\bigg)^{2q} .\end{aligned}$$
As just explained, we still have (\[limXnYn\]). Then it follows that $\lim\limits_{n \to \infty}\frac{{K_n}^2}{P_n} = 0$, which allows us to use property (a) of Lemma \[lem\_eval\_psq\]. Hence, (\[pu1\]) holds.
Substituting (\[limXnYn\]) (\[XnYn2\]) and (\[lnnn\]) to (\[pu1\]), we obtain $$\begin{aligned}
p_{q} & = \frac{\ln n + {(h-1)} \ln \ln n + {\gamma_n}}{n} \pm
O\bigg(\frac{\ln n}{n}\bigg) \cdot
O\Bigg(\bigg(\frac{\ln n}{n}\bigg)^{\frac{1}{q}} \Bigg). \nonumber \\
& = \frac{\ln n + {(h-1)} \ln \ln n + {\gamma_n} \pm
O\big(n^{-\frac{1}{q}}(\ln n)^{\frac{q+1}{q}}\big)}{n}.
\label{thm_eq_psXnPn3b}\end{aligned}$$ With $\gamma_n^*$ defined by (\[XnYn3\]), from (\[thm\_eq\_psXnPn3b\]), it holds that $\gamma_n^* = \gamma_n \pm
o(1)$.
Numerical Experiments {#sec:expe}
=====================
We present numerical experiments below to support the theoretical results. For graph $G_q(n,K,P)$, where $n$ is the number of vertices, $K$ is the number of items for each vertex, and $P$ is the item pool size, we set $n=1,000$, $P=20,000$ and $q=2$. Figure \[fig\] depicts both the analytical and experimental curves for the probability that graph $G_q(n,K,P)$ has a minimum vertex degree at least $\ell$ with $\ell = 2$ or $\ell = 7$. For the analytical curves, we use the results in Theorem \[thm:exact\_qcomposite2\]; namely, we first compute $\gamma$ satisfying $$\begin{aligned}
\frac{1}{q!} \bigg( \frac{{K}^2}{P} \bigg)^{q} & = \frac{\ln n +
{(k-1)} \ln \ln n + {\alpha}}{n}. \nonumber\end{aligned}$$ Then we use $e^{- \frac{e^{-\alpha}}{(k-1)!}}$ as the analytical value for the probability that graph $G_q(n,K,P)$ has a minimum vertex degree at least $\ell$. For the experimental curves, we generate $1,000$ samples of $G_q(n,K,P)$ and count the times where the minimum vertex degree is no less than $\ell$. We divide the counts by $1,000$ to derive the empirical probabilities. As illustrated in Figure \[fig\], the simulation results confirm our theoretical findings.
![A plot of the probability that the minimum vertex degree of graph $G_q(n,K,P)$ is $\ell$-connected for $\ell = 2$ or $\ell = 7$ with $n=1,000$, $P=20,000$ and $q=2$. []{data-label="fig"}](fig.eps){height="30.00000%"}
Results for Graph $\mathbb{G}_1$ {#sec:g1}
================================
For graph $\mathbb{G}_1$ (i.e., $\mathbb{G}_q$ in the special case of $q=1$), in our work [@mobihocQ1], we have derived asymptotically exact probabilities for $k$-connectivity and the property that the minimum vertex degree is at least $k$ with an arbitrary $k$ in graph $\mathbb{G}_1$. Compared with Theorem \[thm:exact\_qcomposite2\] and Corollary \[cor:exact\_qcomposite\] for $\mathbb{G}_q$ in this paper, our results for $\mathbb{G}_1$ in [@mobihocQ1] does not need the condition $\frac{{K_n}^2}{P_n} =
o(1)$, and only requires a weaker condition: $P_n \geq 3K_n $ for all $n$ sufficiently large. For completeness, we present Theorem 1 in [@mobihocQ1] as follows, which can be viewed as an analog of Corollary \[cor:exact\_qcomposite\] in this paper:
\[thm:mobihocQ1\] Consider a positive integer $k$ and scalings $K_n : \mathbb{N}_0
\rightarrow \mathbb{N}_0,P_n : \mathbb{N}_0 \rightarrow
\mathbb{N}_0$ and $p_n : \mathbb{N}_0 \rightarrow (0,1]$, with $P_n
\geq 3K_n $ for all $n$ sufficiently large. Let the sequence $\alpha: \mathbb{N}_0
\rightarrow \mathbb{R}$ be defined through $$\begin{aligned}
p_{e,1} & = \frac{\ln n + {(k-1)} \ln \ln n + {\alpha_n}}{n}.
\nonumber\end{aligned}$$ For $\lim_{n \to \infty} \alpha_n = \alpha ^* \in [-\infty,
\infty]$, the properties (a) and (b) below hold.
[(a)]{} If $ K_n = \omega(1)$, then as $n \to \infty$, $$\begin{aligned}
\mathbb{P}\left[ \begin{array}{c} \textrm{The
minimum~vertex~degree} \\ \mbox{of graph
$\mathbb{G}\iffalse_{on}\fi$ is at least }k.
\end{array} \right] &
\to e^{- \frac{e^{-\alpha ^*}}{(k-1)!}} . \nonumber \end{aligned}$$ If $P_n = \Omega (n)$, then as $n \to \infty$, $$\begin{aligned}
\mathbb{P} \left[\textrm{Graph }\mathbb{G}\iffalse_{on}\fi \textrm{
is $k$-connected}.\hspace{2pt}\right] & \to
e^{- \frac{e^{-\alpha ^*}}{(k-1)!}} . \nonumber \end{aligned}$$
Note that $p_{e,1}$ is $p_{e,q}$ with $q=1$, and is the probability that two vertices have a link in between in graph $\mathbb{G}_1$. In the statement of Theorem 1 in [@mobihocQ1], $\alpha ^* $ belongs to $(-\infty, \infty)$. However, an easy monotonicity argument can lead to results for $\alpha ^*= -\infty$ and $\alpha ^*= \infty$. Hence, in Theorem \[thm:mobihocQ1\] above, $\alpha ^* $ can fall in $[-\infty, \infty]$. In establishing Theorem 1 in [@mobihocQ1] (i.e., Theorem \[thm:mobihocQ1\] above), we have shown that the number of vertices with an arbitrary degree in graph $\mathbb{G}_1$ asymptotically converges to a Poisson distribution. Using the idea similar to that of proving property (b) of Theorem \[thm:exact\_qcomposite2\] in this paper, we also establish the asymptotic probability distribution for the minimum vertex degree and for the connectivity of graph $\mathbb{G}_1$. Therefore, we present the following theorem on graph $\mathbb{G}_1$, which is an analog of Theorem \[thm:exact\_qcomposite2\] in this paper:
Consider scalings $K: \mathbb{N}_0 \rightarrow \mathbb{N}_0,P:
\mathbb{N}_0 \rightarrow \mathbb{N}_0$ and $p: \mathbb{N}_0
\rightarrow (0,1]$ with $P_n \geq 3K_n $ for all $n$ sufficiently large. For $$\begin{aligned}
p_{e,1} & = \frac{\ln n \pm O(\ln \ln n)}{n},\nonumber\end{aligned}$$ (i.e., $\frac{n p_{e,1} - \ln n}{\ln \ln n}$ is bounded for all $n$), the following properties (a) and (b) for graph $\mathbb{G}_1\iffalse_{on}^{(q)}\fi$ hold.
**(a)** If $ K_n = \omega(1)$, the number of vertices in $\mathbb{G}_1\iffalse_{on}\fi$ with an arbitrary degree converges to a Poisson distribution as $n \to \infty$.
**(b)** Defining $\ell$ and $\beta_n$ by $$\begin{aligned}
\ell : = \bigg\lfloor \frac{np_{e, 1} - \ln n + (\ln \ln n) / 2}{\ln
\ln n} \bigg\rfloor + 1, \nonumber\end{aligned}$$ and $$\begin{aligned}
\beta_n : = np_{e, 1} - \ln n - (\ell-1)\ln\ln n, \nonumber\end{aligned}$$ we obtain that if $ K_n = \omega(1)$, with $\mu$ denoting the minimum vertex degree of graph $\mathbb{G}_1$,
- $(\mu \neq \ell)\cap (\mu \neq \ell-1)$ 0 as $n \to \infty$;
- if $\lim_{n \to \infty} \beta_n = \beta ^* \in (-\infty, \infty)$, then as $n \to
\infty$, $$\begin{aligned}
\begin{cases} \mu = \ell \textrm{ with a probability converging to }
e^{- \frac{e^{-\beta ^*}}{(k-1)!}}, \\
\mu = \ell - 1\textrm{ with a probability tending to }\Big(
1 - e^{- \frac{e^{-\beta
^*}}{(k-1)!}} \Big);\nonumber
\end{cases}\end{aligned}$$
- if $ \lim_{n \to \infty} \beta_n = \infty$, then as $n \to
\infty$, $$\begin{aligned}
\begin{cases} \mu = \ell\textrm{ with a probability approaching to }1, \\
\mu \neq \ell\textrm{ with a probability going to }0;
\end{cases}\nonumber \hspace{20pt}\textrm{and}\end{aligned}$$
- if $ \lim_{n \to \infty} \beta_n = - \infty$, then as $n \to
\infty$, $$\begin{aligned}
\hspace{-27pt}\begin{cases} \mu = \ell - 1\textrm{ with a probability tending to }1, \\
\mu \neq \ell - 1\textrm{ with a probability converging to }0;
\end{cases}\nonumber\end{aligned}$$
and that if $P_n = \Omega (n)$, with $\nu$ denoting the connectivity of graph $\mathbb{G}_1$,
- $(\nu \neq \ell)\cap (\nu \neq \ell-1)$ 0 as $n \to \infty$;
- if $\lim_{n \to \infty} \beta_n = \beta ^* \in (-\infty, \infty)$, then as $n \to
\infty$, $$\begin{aligned}
\begin{cases} \nu = \ell \textrm{ with a probability converging to }
e^{- \frac{e^{-\beta ^*}}{(k-1)!}}, \\
\nu = \ell - 1\textrm{ with a probability tending to }\Big(
1 - e^{- \frac{e^{-\beta
^*}}{(k-1)!}} \Big);\nonumber
\end{cases}\end{aligned}$$
- if $ \lim_{n \to \infty} \beta_n = \infty$, then as $n \to
\infty$, $$\begin{aligned}
\begin{cases} \nu = \ell\textrm{ with a probability approaching to }1, \\
\nu \neq \ell\textrm{ with a probability going to }0;
\end{cases}\nonumber \hspace{20pt}\textrm{and}\end{aligned}$$
- if $ \lim_{n \to \infty} \beta_n = - \infty$, then as $n \to
\infty$, $$\begin{aligned}
\hspace{-27pt}\begin{cases} \nu = \ell - 1\textrm{ with a probability tending to }1, \\
\nu \neq \ell - 1\textrm{ with a probability converging to }0.
\end{cases}\nonumber\end{aligned}$$
Related Work {#related}
============
For graph $G_q(n, K_n, P_n)$, Bloznelis and [Ł]{}uczak [@Perfectmatchings] recently consider the following three properties: (i) the graph has a minimum vertex degree at least $1$; (ii) the graph has a perfect matching; (iii) the graph is connected. They present that when $n$ is even, for each of these three properties, its probability converges to $\exp\Big\{-
\exp\big\{-\lim\limits_{n \to \infty}\alpha_n\big\}\Big\}$ as $n\to\infty$, under conditions $p_u =\frac{\ln n + \alpha_n}{n} $ ($p_u$ is the edge probability), $p_u = O\big(\frac{\ln n}{n}\big)$ and $\Big\{(q+2)\big[\binom{K_n}{q}\big]^5(\ln \ln
n)^2\Big\}^{\frac{3K_n-q}{3(K_n-q)}} \leq (\ln n)^{\beta}$ for some $\beta \in (0,1)$.
Bloznelis [@bloznelis2013] investigates the clustering coefficient of graph $G_q(n, K_n, P_n)$ and show that the chance of two neighbors of a given vertex $v$ to be adjacent decays as $c d^{-1}$, where $c$ is a positive constant and $d$ is the degree of vertex $v$. Bloznelis *et al.* [@Assortativity] study the correlation coefficient of degrees of adjacent vertices in $G_q(n, K_n, P_n)$. Bloznelis *et al.* [@Rybarczyk] demonstrate that a connected component in $G_q(n,
K_n, P_n)$ with at at least a constant fraction of $n$ emerges as $n
\to \infty$ when the edge probability $p_{u,q}$ exceeds $1/n$.
Conclusion {#sec:Conclusion}
==========
In this paper, we derive
[^1]: The authors are with ECE Department and CyLab, Carnegie Mellon University, USA. Emails: {junzhao, oyagan, virgil}@andrew.cmu.edu
[^2]: In an Erdős-Rényi graph [@citeulike:4012374; @Gilbert], an edge between each pair of vertices exists independently with the same probability.
[^3]: A graph property is called monotone increasing if it holds under the addition of edges [@Bollobas; @JansonLuczakRucinski7].
[^4]: As used by Rybarczyk [@zz; @2013arXiv1301.0466R], a coupling of two random graphs $G_1$ and $G_2$ means a probability space on which random graphs $G_1'$ and $G_2'$ are defined such that $G_1'$ and $G_2'$ have the same distributions as $G_1$ and $G_2$, respectively. If $G_1'$ is a spanning subgraph (resp., supergraph) of $G_2'$, we say that under the coupling, $G_1$ is a spanning subgraph (resp., supergraph) of $G_2$, which yields that for any monotone increasing property $\mathcal {I}$, the probability of $G_1$ having $\mathcal {I}$ is at most (resp., at least) the probability of $G_2$ having $\mathcal
{I}$.
|
---
abstract: |
The answer to the question in the title is: in search of new physics beyond the Standard Model, for which there are many motivations, including the likely instability of the electroweak vacuum, dark matter, the origin of matter, the masses of neutrinos, the naturalness of the hierarchy of mass scales, cosmological inflation and the search for quantum gravity. So far, however, there are no clear indications about the theoretical solutions to these problems, nor the experimental strategies to resolve them. It makes sense now to prepare various projects for possible future accelerators, so as to be ready for decisions when the physics outlook becomes clearer. Paraphrasing George Harrison, “ If you don’t [*yet*]{} know where you’re going, any road [*may*]{} take you there."\
\
[*Contribution to the 2017 Hong Kong UST IAS Programme and Conference on High-Energy Physics.*]{}\
\
KCL-PH-TH-2017-18, CERN-TH-2017-080
address: |
Theoretical Particle Physics and Cosmology Group, Physics Department,\
KingÕs College London, London WC2R 2LS, UK;\
Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland\
John.Ellis@cern.ch
author:
- JOHN ELLIS
title: '**WHERE IS PARTICLE PHYSICS GOING?**'
---
Introduction
============
The bedrock upon which our search for new physics beyond the Standard Model (SM) is founded is our ability to make precise predictions within the Standard Model, notably for the LHC experiments. The predictions of many hard higher-order perturbative QCD calculations have been confirmed, as seen in Fig. \[fig:heaven\], providing confidence in predictions for the production of the Higgs boson [@Mistlberger], and for the backgrounds to many searches for new physics.
![\[fig:heaven\] *Many SM processes have been measured at the LHC, and have cross sections that are generally in excellent agreement with QCD calculations [@ATLASSM].* ](heaven){height="6cm"}
The Flavour Sector
==================
Many measurements in the flavour sector are also consistent with the predictions of the Cabibbo-Kobayashi-Maskawa (CKM) model [@CKMfitter; @UTfit], e.g., there are many consistent measurements of the unitarity triangle, as seen in the left panel of Fig. \[fig:CKM\]. Historically, the angle $\gamma$ has been the least constrained experimentally, but the LHCb Collaboration has recently published a combined measurement [@LHCbgamma] that dominates the world average and is consistent with the other unitarity triangle measurements.
![\[fig:CKM\] *Left panel: Compilation of experimental constraints on the CKM unitarity triangle [@CKMfitter]. Compilation of constraints on possible new physics contributions to operator coefficients [@Wolfgang].*](CKMfitter "fig:"){height="5.5cm"} ![\[fig:CKM\] *Left panel: Compilation of experimental constraints on the CKM unitarity triangle [@CKMfitter]. Compilation of constraints on possible new physics contributions to operator coefficients [@Wolfgang].*](flavouranomaly "fig:"){height="5.1cm"}\
That said, there are several anomalies in the flavour sector of varying significance. For example, there are strengthening indications of violations of $e/\mu$ lepton universality in $B \to K e^+ e^-$ and $B \to K \mu^+ \mu^-$ decays, [@Kll] and of $\tau/(\ell = e$ or $\mu)$ universality in $B \to D^{(*)} \tau \nu$ decays [@Dtaunu] - to which my attitude is ‘wait and see’, as lepton non-universality has held up very well so far. Much attention has been attracted to the $P_5^\prime$ angular distribution in $B \to K^* \mu^+ \mu^-$ decay [@P5prime], which may be accompanied by an anomaly in the $q^2$ distribution in $B \to \phi \mu^+ \mu^-$ decay, leading to the constraints on possible new physics contributions to operator coefficients shown in the right panel of Fig. \[fig:CKM\] [@Wolfgang]. These both appear at $q^2 \lesssim 5$ GeV$^2$, and I do not know how seriously to take them, in view of my lack of understanding of the non-perturbative QCD corrections in this region. My ignorance also makes it difficult for me to judge the significance of the apparent discrepancy between theory [@CTS; @Buras] and experiment for $\epsilon^\prime/\epsilon$. Finally, a new kid on the flavour block has been the interesting search for $H \to \mu \tau$ decay [@Htaumu] discussed below, though this may be reverting towards the SM with the latest Run 2 results [@CMSHtaumu2].
Higgs Physics
=============
The Higgs Mass
--------------
The most fundamental Higgs measurement is that of its mass. The combined LHC Run 1 results of ATLAS and CMS based on $H$ decays into $\gamma \gamma$ and $Z Z^* \to 2 \ell^+ 2 \ell^-$ yielded [@ATLAS+CMS] $$m_H \; = \; 125.09 \pm 0.21 ({\rm stat.}) \pm 0.11 ({\rm syst.}) \, ,
\label{mHRun1}$$ and the preliminary CMS result from Run 2 is consistent with this, with slightly smaller errors [@CMSRun2]: $$m_H \; = \; 125.26 \pm 0.20 ({\rm stat.}) \pm 0.08 ({\rm syst.}) \, ,
\label{mHRun2}$$ It is noteworthy that statistical uncertainties dominate, and we can look forward to substantial reductions in the future, determining $m_H$ at the [*per mille*]{}. Accurate knowledge of the Higgs mass is important for precision tests of Standard Model (and other) predictions and, as discussed later, is crucial for understanding the (in/meta)stability of the electroweak vacuum.
Higgs Couplings
---------------
The couplings of the Higgs boson to Standard Model particles are completely specified and, consequently, there are definite predictions for its production processes and decay branching ratios [@LHCHXSWG]. Concretely, one expects gluon-gluon fusion to dominate over vector-boson fusion, production in association with a vector boson and in association with a $t {\bar t}$ pair. The dominant $H$ decay mode is predicted to be into $b {\bar b}$, with much smaller branching ratios for $\gamma \gamma$ and $Z Z^* \to 2 \ell^+ 2 \ell^-$.
Much progress was made in Run 1 probing these predictions [@ATLAS+CMS2], but much remains to be done. Higgs decays to $\gamma \gamma, ZZ^*, WW^*$ and $\tau^+ \tau^-$ have been measured in gluon-gluon fusion, and there is solid evidence for vector-boson fusion, but the associated production mechanisms have yet to be confirmed. Moreover, there is no confirmation yet of the expected dominant $H \to b {\bar b}$ decay mode: LHC evidence is at the level of 2.6 $\sigma$ [@Tevatronbbbar], and the Tevatron experiments have reported evidence at the 2.8-$\sigma$ level. There is indirect evidence for the expected $H t {\bar t}$ vertex via the measurements of gluon-gluon fusion and $H \to \gamma \gamma$ decay, but no significant evidence via associated $H t {\bar t}$ or single $H t ({\bar t})$ production. Also on the agenda is the search for $H \to \mu^+ \mu^-$, which is predicted in the SM to appear at a level close to the current experimental sensitivity.
Fig. \[fig:Mepsilon\] is one way of displaying the available information on Higgs couplings [@EY; @ATLAS+CMS2]. It is a characteristic prediction of the SM that the couplings to other particles should be related to their masses, $\propto m_f$ for fermions and $\propto m_V^2$ for massive vector bosons. The black solid line is a fit where $m \to m^{(1+ \epsilon)}$ in the couplings: we see that the combined ATLAS and CMS data are highly consistent with the SM expectation that $\epsilon = 0$, shown as the blue dashed line.
![*A fit by the ATLAS and CMS Collaborations to a parametrization of the mass-dependence of the Higgs couplings: $m \to m^{(1+ \epsilon)}$ [@ATLAS+CMS2]. The Standard Model predictions are connected by a dotted line, the red line is the best fit, and the green and yellow bands represent the 68 and 95% CL fit ranges.*[]{data-label="fig:Mepsilon"}](Mepsilon){width="7cm"}
The couplings in Fig. \[fig:Mepsilon\] are all flavour-diagonal. The SM predicts that flavour-violating Higgs couplings should be very small, but measurements of flavour-violating processes at low energies would allow [*either*]{} $H \to \mu \tau$ [*or*]{} $H \to e \mu$ with branching ratio $\lesssim 10$%, whereas the branching ratio for $H \to e \mu$ must be $\lesssim 10^{-5}$ [@BEI]. The was some excitement after Run 1 when the combined CMS and ATLAS data indicated a possible 2-$\sigma$ excess [@Htaumu]. This has not reappeared in early Run 2 data [@CMSHtaumu2], but remains an open question.
Elementary Higgs Boson, or Composite?
=====================================
There has been a long-running theoretical debate whether the Higgs boson could be as elementary as the other particles in the SM, or whether it might be composite. The elementary option encounters quadratically-divergent loop corrections to the mass of the Higgs boson, which are frequently (usually?) postulated to be cancelled by supersymmetric particles [@susy] appearing at the TeV scale [@hierarchy] - which have not yet been seen.
On the other hand, the composite option has been favoured by many with memories of the (composite) Cooper pairs underlying superconductivity, and the (composite) pions associated with quark-antiquark condensation in QCD [@techni]. A composite Higgs would require a novel set of strong interactions, and early models tended to have a scalar particle much heavier than the Higgs that has been discovered, and to be in tension with the precision electroweak data. These difficulties can be circumvented by postulating that the Higgs is a pion-like pseudo-Nambu-Goldstone boson of a partially-broken larger symmetry that is restored at some higher energy scale [@PNGBH].
A phenomenological framework that is convenient for characterizing the experimental constraints on such as possibility is provided by the following form of effective Lagrangian that preserves a custodial SU(2)$_V$ symmetry that guarantees $\rho \equiv m_W/m_Z \cos \theta_W = 1$ up to quantum corrections [@NLEL]: $$\begin{aligned}
{\cal L} & = & \frac{v^2}{4} {\rm Tr} D_\mu \Sigma D^\mu \Sigma \left(1 + 2 \kappa_V \frac{H}{v} + b \frac{H^2}{v^2} + \dots \right)
- m_i \bar{\psi}^i_L \Sigma \left( \kappa_F \frac{H}{v} + \dots \right) + {\rm h.c.} \nonumber \\
&& + \frac{1}{2} \partial_\mu H \partial^\mu H + \frac{1}{2} H^2 + d_3 \frac{1}{6} \left( \frac{3 m_H^2}{v} \right) H^3
+ d_4 \frac{1}{24} \left( \frac{3 m_H^2}{v} \right) H^4 + ... \, ,
\label{nonlinearH}\end{aligned}$$ where $H$ is the field of the physical Higgs boson and the massive vector bosons are parametrized by the $2 \times 2$ matrix $\Sigma = \exp( i \frac{\sigma_a \pi_a}{v} )$. The terms in (\[nonlinearH\]) are normalized so that the coefficients $\kappa_V, b, \kappa_F, d_i = 1$ in the SM. The question for experiment is whether any of these coefficients exhibit a deviation that might be a signature of some composite Higgs model.
As seen in the left panel of Fig. \[fig:compo\], measurements of Higgs properties (yellow and orange ellipses) and precision electroweak data (blue ellipses) play complementary roles in constraining the $H$ couplings to vector bosons $\kappa_V$ and fermions $\kappa_F$ in (\[nonlinearH\]) [@Gfitter]. These constraints can be translated into lower limits on the possible compositeness scale in various models, as seen in the right panel of Fig. \[fig:compo\] [@SS].
![*Left panel: A fit of the LHC $H$ couplings to vector bosons and fermions $(\kappa_V, \kappa_F)$ using $H$ measurements (orange and yellow ellipses), and in combination with precision electroweak data (blue ellipses) [@Gfitter]. Right panel: Constraints from LHC Run 1 and early Run 2 data on the compositeness scale in various models [@SS].*[]{data-label="fig:compo"}](Gfitterkappas "fig:"){width="5.5cm"} ![*Left panel: A fit of the LHC $H$ couplings to vector bosons and fermions $(\kappa_V, \kappa_F)$ using $H$ measurements (orange and yellow ellipses), and in combination with precision electroweak data (blue ellipses) [@Gfitter]. Right panel: Constraints from LHC Run 1 and early Run 2 data on the compositeness scale in various models [@SS].*[]{data-label="fig:compo"}](VS "fig:"){width="5.75cm"}
Stability of the Electroweak Vacuum
===================================
If the Higgs is indeed elementary, the measurements (\[mHRun1\], \[mHRun2\]) of $m_H$, combined with those of $m_t$, raise important questions about the stability and history of the electroweak vacuum, suggesting the necessity of new physics beyond the SM [@Medellin]. The issue is that the Higgs quartic self-coupling $\lambda$ is renormalized not only by itself, which tends to increase it as the energy/mass scale increases, but also by the Higgs coupling to the top quark, which tends to drive it to smaller (even negative) values at higher scales $Q$, as seen in the left panel of Fig. \[fig:down\]GeV [@DDEEGIS]. At leading order: $$\lambda (Q) \; \simeq \; \lambda (v) - \frac{3 m_t^4}{2 \pi v^4} \log \left( \frac{Q}{v} \right) \, ,
\label{down}$$ The right panel of Fig. \[fig:down\] displays the results of one calculation of the regions of the $(m_H, m_t)$ plane where the electroweak vacuum is stable, metastable or unstable, and yields the following estimate of the ‘tipping point’ $\Lambda_I$ where $\lambda$ goes negative [@BDGGSSS]: $$\begin{aligned}
\log_{10} \left( \frac{\Lambda_I}{\rm GeV}\right) & = & 9.4 + 0.7 \left(\frac{m_H}{\rm GeV} -125.15 \right) \nonumber \\
& - & 1.0 \left( \frac{m_t}{\rm GeV} - 173.34 \right) + 0.3 \left( \frac{\alpha_s(m_Z) - 0.1184}{0.0007} \right) \, .
\label{LambdaI}\end{aligned}$$ The dominant uncertainty in the calculation of $\Lambda_I$ is due to that in $m_t$, followed by that in $\alpha_s(m_Z)$ (which enters in higher order in the calculation), the uncertainty due to the measurement of $m_H$ being relatively small. The final result is an estimate $$\log_{10} \left( \frac{\Lambda_I}{\rm GeV} \right) \; = 9.4 \pm 1.1 \, ,
\label{LambdaIvalue}$$ indicating that we are (probably) doomed, unless some new physics intervenes.
![*Left panel: Top quark loops renormalize the Higgs self-coupling $\lambda$ negatively, suggesting that it takes negative values at field values $\gtrsim 10^{9}$ GeV[@DDEEGIS], leading to instability of the Higgs potential in the SM. Right panel: Measurements of $m_t$ and $m_H$ indicate that the SM vacuum is probably metastable, although there are important uncertainties in $m_t$ and $\alpha_s$ [@BDGGSSS].*[]{data-label="fig:down"}](GoingNegative.png "fig:"){width="6.1cm"} ![*Left panel: Top quark loops renormalize the Higgs self-coupling $\lambda$ negatively, suggesting that it takes negative values at field values $\gtrsim 10^{9}$ GeV[@DDEEGIS], leading to instability of the Higgs potential in the SM. Right panel: Measurements of $m_t$ and $m_H$ indicate that the SM vacuum is probably metastable, although there are important uncertainties in $m_t$ and $\alpha_s$ [@BDGGSSS].*[]{data-label="fig:down"}](Buttazzo.png "fig:"){width="6cm"}
Some people discount this ‘problem’ on the grounds that the prospective lifetime of the vacuum is much longer than its age. However, there is another issue, namely that fluctuations in the Higgs field in the very early Universe would have been much larger than now, and would probably have driven almost everywhere in the Universe into an anti-De Sitter phase from which there would have been no escape [@aDS]. One could postulate that our piece of the Universe happened to be extraordinarily lucky and avoid this fate, but it seems more plausible that some new physics intervenes before the instability scale $\Lambda_I$. Possible such remedies include higher-dimensional operators in the SM effective field theory (see the next Section), a non-minimal Higgs coupling to gravity, or a threshold for new physics such as supersymmetry [@ER] (see later).
The SM Effective Field Theory
=============================
An alternative way of analyzing the Higgs and other data is to assume that all the known particles (including the Higgs boson) are SM-like, and look for the effects of physics beyond the SM via an effective field theory (the SMEFT) containing higher-dimensional SU(2)$\times$U(1)-invariant operators constructed out of SM fields, e.g., of dimension 6 [@SMEFT]: $${\cal L}_{eff} \; = \; \sum_n \frac{c_n}{\Lambda^2} {\cal O}_n \, ,
\label{dim6}$$ where the characteristic scale of new physics is described by $\Lambda$, with the $c_n$ being unknown dimensionless coefficients. Data on Higgs properties, precision electroweak data, triple-gauge couplings (TGCs), etc., can all be combined to constrain the SMEFT operator coefficients in a unified and consistent way. Table \[tab:dim6\] shows which observables currently provide the greatest sensitivities to some of these operators [@ESY].
\[tab:dim6\]
The left panel of Fig. \[fig:SMEFT\] shows how the coefficients of the SMEFT operators in Table \[tab:dim6\] were constrained by Run 1 Higgs data including kinematical variables (blue bar) and by Run 1 measurements of TGCs (red bar) [@ESY]. The green bar gives the resulting ranges when each operator is switched on individually, and the black bar is for a global fit marginalizing over all the listed operators. The right panel of Fig. \[fig:SMEFT\] manifests the complementarity between the Higgs and LEP-2 TGC data for constraining the anomalous couplings $\delta g_{1Z}$ and $\delta g_\gamma$ [@Falkowski]. Because of its power to constrain new physics appearing in many observables in a consistent way, the SMEFT is the preferred framework for assessing the sensitivities of future analyses of precision LHC measurements to physics beyond the SM, whose motivations are discussed in the next Section.
![*Left panel: The 95% CL ranges for fits to individual SMEFT operator coefficients (green bars), and the marginalised 95% CL ranges for global fits combining data on the LHC $H$ signal strength data with the kinematic distributions for associated $H+V$ production (blue bars), or with the LHC TGC data (red bars), and combining all the data (black bars) [@ESY]. Right panel: The 68 and 95% CL ranges allowed by a fit to the anomalous TGCs $(\delta g_{1,z}, \delta \kappa_\gamma)$ using LEP-2 TGC data (orange and yellow), LHC Higgs data (green) and their combination (blue) [@Falkowski].*[]{data-label="fig:SMEFT"}](SMEFTL "fig:"){width="6.5cm"} ![*Left panel: The 95% CL ranges for fits to individual SMEFT operator coefficients (green bars), and the marginalised 95% CL ranges for global fits combining data on the LHC $H$ signal strength data with the kinematic distributions for associated $H+V$ production (blue bars), or with the LHC TGC data (red bars), and combining all the data (black bars) [@ESY]. Right panel: The 68 and 95% CL ranges allowed by a fit to the anomalous TGCs $(\delta g_{1,z}, \delta \kappa_\gamma)$ using LEP-2 TGC data (orange and yellow), LHC Higgs data (green) and their combination (blue) [@Falkowski].*[]{data-label="fig:SMEFT"}](Falkowskietal "fig:"){width="5.5cm"}
The Standard Model is not Enough [@Bond]
========================================
There are many reasons to anticipate the existence of physics beyond the SM, of which I list just 7 here. 1) The prospective instability of the electroweak vacuum discussed earlier. 2) The astrophysical and cosmological necessity for dark matter. 3) The origin of matter itself, i.e., the cosmological baryon asymmetry. 4) The masses of neutrinos. 5) The naturalness of the hierarchy of mass scales in physics. 6) A mechanism (or replacement) for cosmological inflation to explain the great size and age of the Universe. 7) A quantum theory of gravity.
The good news is that LHC experiments are tackling most of these issues during Run 2. The bad news is that there is no consensus among theorists how to resolve them. Until recently, supersymmetry found the most theoretical favour, but the negative results from early Run 2 supersymmetry searches have caused some to waver. Not me, however - I still think that it is the most comprehensive and promising framework for new physics beyond the SM. In the words of the famous World War 1 cartoon [@cartoon] “If you knows of a better ’ole, go to it.” I do not, so I will stay in the supersymmetric ’ole.
Supersymmetry
=============
Indeed, I would even argue that Run 2 of the LHC has provided us with 3 new motivations for supersymmetry. i) It stabilizes the electroweak vacuum [@ER]. ii) It made a successful prediction for the Higgs mass, namely that it should weigh $\lesssim 130$ GeV in simple models [@susymH]. iii) It predicted correctly that the Higgs couplings measured at the LHC should be within a few % of their SM values [@EHOW]. These new motivations are additional to the classic ones from the naturalness of the mass hierarchy [@hierarchy], the availability of a natural dark matter candidate [@EHNOS], the welcome help of supersymmetry in making grand unification possible [@susyGUTs], and its apparent necessity in string theory, which I regard as the only serious candidate for a quantum theory of gravity.
At this point, I must ’fess up to two pieces of bad news. One is that theorists have also not reached any consensus on the most promising supersymmetric model, largely because there is no favoured scenario for supersymmetry breaking. Alternatives range from models in which this is assumed to be universal at some GUT scale (such as the CMSSM) to models in which all the soft supersymmetry-breaking parameters are treated entirely phenomenologically as unknown parameters at the electroweak scale (the pMSSM). The other piece of bad news is that the LHC experiments have found not even a hint of supersymmetry, despite many searches making different assumptions about the supersymmetric spectrum [^1].
In the following, the negative results of the searches are combined with other measurements to constrain the parameter spaces of a couple of representative supersymmetric models.
Probing a Supersymmetric SU(5) GUT
----------------------------------
The first model we study here is a supersymmetric SU(5) GUT in which the soft supersymmetry-breaking gaugino masses are assumed to be universal at the GUT scale, whereas the soft supersymmetry-breaking scalar masses are generation-independent but allowed to be different for the spartners of fermions in the $\mathbf{\bar{5}}$ and $\mathbf{10}$ representations [@MCSU5].
Fig. \[fig:glsq\] displays the regions of the $(m_{\tilde g}, m_{\tilde \chi^0_1})$ plane (left panel) and the $(m_{\tilde u_R}, m_{\tilde \chi^0_1})$ plane (right panel) that are allowed in a global fit in this supersymmetric SU(5) GUT at the 95% CL (blue contours) and favoured at the 68% CL (red contours), as well as the best-fit point (green stars). The black lines are the nominal 95% CL limits set by LHC searches, assuming simplified decay patterns with 100% branching ratios, and the coloured shadings represent the actual dominant decays found in different regions of parameter space. We see in the left panel that gluino masses $\gtrsim 1900$ GeV are indicated, with a best-fit value of $\simeq 2400$ GeV, whereas the ${\tilde u_R}$ mass may be $\sim 400$ GeV lighter. One curiosity is a small strip in the right panel where $m_{\tilde u_R} - m_{\tilde \chi^0_1}$ is small and $m_{\tilde u_R} \lesssim 650$ GeV. In this strip the dark matter (DM) density is brought into the range allowed by astrophysics and cosmology by squark-neutralino coannihilation, and this compressed-spectrum region is on the verge of exclusion by LHC searches.
\
\
The best-fit spectrum in this SU(5) GUT model is shown in Fig. \[fig:bestfit\]. We see that all the squarks have masses below $\sim 2200$ GeV at the best-fit point, where they would be within the range of future LHC runs. This analysis included the results from the first $\sim 13$/fb of LHC data at 13 TeV, and Fig. \[fig:compare\] compares the profiled $\chi^2$ likelihood functions for $m_{\tilde g}$ (left panel) and $m_{\tilde u_R}$ (right panel) found in this analysis (solid blue lines) with those found in an analysis restricted to 8 TeV data (dashed blue lines) [^2]. We see that, whilst the 13 TeV have had a significant impact, they have not yet been a game-changer. There is still plenty of room for discovering supersymmetry in future LHC runs in this model, though there are no guarantees!
One of the interesting experimental possibilities in this and related models is that the next-to-lightest supersymmetric particle (NLSP) might be the lighter stau slepton, with a mass that could be so close to that of the ${\tilde \chi^0_1}$ that it might have a long enough lifetime to decay at a separated vertex, or even escape from the detector as a massive charged non-relativistic particle, as illustrated in Fig. \[fig:staudecay\] [@MCSU5].
Probing the Minimal Anomaly-Mediated Supersymmetry-Breaking Model
-----------------------------------------------------------------
Another model we have studied recently is the minimal anomaly-mediated supersymmetry-breaking (mAMSB) model [@MCmAMSB]. In this case, the supersymmetric spectrum is relatively heavy. If one assumes that the lightest supersymmetric particle (LSP) is a wino that provides all the cosmological DM, it must weigh about 3 TeV, leading to a relatively heavy spectrum as seen in the left panel of Fig. \[fig:mAMSB\], though the spectrum could be lighter if the LSP is a Higgsino, or if it provides only a fraction of the dark matter, as seen in the right panel of Fig. \[fig:mAMSB\]. We also see that the soft supersymmetry-breaking scalar mass $m_0$ in the mAMSB model must be quite large if the LSP provides all the dark matter, $m_0 \gtrsim 4$ TeV, though it could be smaller if there is some other contribution to the dark matter.
Fig. \[fig:reaches\] displays the reaches of the LHC and a 100-TeV $pp$ collider (FCC-hh) in the $(m_{\tilde g}, m_{\tilde \chi^0_1})$ plane (left panel) and the $(m_{\tilde q_R}, m_{\tilde \chi^0_1})$ plane (right panel) in the mAMSB [@MCmAMSB]. We see that most of the allowed region of the mAMSB parameter space lies beyond the reach of the LHC, though it may be within reach of FCC-hh [@FCC-hh-BSM].
Direct Dark Matter Searches
===========================
Besides missing-energy searches at the LHC, the best prospects for exploring supersymmetry may be in the direct search for dark matter via scattering on nuclei in deep-underground laboratories [@DDMAspen]. Possible ranges of the LSP mass and the spin-independent cross section for LSP scattering on a proton target, $\sigma_p^{\rm SI}$, in the supersymmetric SU(5) and mAMSB models discussed above are shown in the left [@MCSU5] and right [@MCmAMSB] panels of Fig. \[fig:directDM\], respectively. In both panels the range of $\sigma_p^{\rm SI}$ excluded by the latest results from the PandaX [@PandaX] and LUX [@LUX] experiments is shaded green. The estimated sensitivities of the planned LZ [@LZ] and XENON1/nT [@XENON] experiments are also shown, as is the neutrino ‘floor’ below which neutrino-induced backgrounds dominate. As in previous plots, the ranges allowed at the 95% CL (favoured at the 68% CL) are surrounded by blue and red contours, respectively, while the coloured shadings within them correspond to different mechanisms for bringing the LSP density into the cosmological range (discussed in [@MCSU5; @MCmAMSB], and the best-fit points are marked by green stars.
We see that values of $\sigma_p^{\rm SI}$ anywhere from the present experimental limit down to below the neutrino ‘floor’ are possible in both the SU(5) and mAMSB cases. There are decent prospects for discovering direct DM scattering in the LZ and XENON1/nT experiments, but again no guarantees.
It is interesting to compare the sensitivities of LHC searches for mono-jet and other searches with those of direct searches for DM scattering, which can be done in the frameworks of simplified models for DM [@SDMM]. The results of the comparison depend, in particular, on the form of the coupling of the intermediate particle mediating the interactions between the DM and SM particles. Fig. \[fig:DMcomparison\] compares the sensitivities of LHC mono-jet and $\sigma_p^{\rm SI}$ constraints in the case of a vector-like mediator (left panel) and LHC mono-jet searches and constraints on the spin-dependent scattering cross section, $\sigma_p^{\rm SD}$, in the case of an axial-vector mediator (right panel) [@CMSSDMM]. We see that in the vector-like case the direct DM searches currently have more sensitivity except for small DM masses, whereas in the axial-vector case the LHC has greater sensitivity over a wide range of DM masses. These examples illustrate the complementarity of the LHC and direct searches in the quest for dark matter.
A Plea for Patience
===================
The LHC will continue to operate for another 15 to 20 years, with the objective of gathering two orders of magnitude more data than those analyzed so far. Thus it has many opportunities to discover new physics beyond the Standard Model, e.g., in Higgs studies and in searches for new particles beyond the Standard Model such as supersymmetry and/or dark matter. Some lovers of superymmetry may be tempted to lose faith. However, it is worth remembering that the discovery of the Higgs boson came 48 years after it was postulated, whereas the first interesting supersymmetric models in four dimensions were written down at the end of 1973 [@susy], only just over 43 years ago! Moreover, the discovery of gravitational waves came just 100 years after they were predicted. Sometimes one must be patient.
In the mean time, what are the prospects for new accelerators to follow the LHC?
Electron-Positron Colliders
===========================
Fig. \[fig:electronpositron\] shows the estimated luminosities as functions of the centre-of-mass energy for various projected $e^+ e^-$ colliders. We see that linear colliders (ILC [@ILC], CLIC [@CLIC]) could reach higher energies, but circular colliders (CEPC [@CEPC], FCC-ee [@FCC-ee]) could provide higher luminosities at low energies. This means that CLIC, in particular, might be the accelerator of choice if future LHC runs reveal some new particles with masses $\lesssim 1$ TeV, or if the emphasis will be on probing decoupled new physics via SMEFT effects that grow with the centre-of-mass energy [@ERSY-CLIC], whereas FCC-ee would be advantageous [@EY-FCC-ee] if high-precision Higgs and $Z$ measurements are to be prioritized.
The left panel of Fig. \[fig:FCC-eeCLIC\] compares the estimated sensitivities of FCC-ee and ILC measurements of Higgs and electroweak precision measurements to the coefficients of some dimension-6 operators in the SMEFT [@EY-FCC-ee]. The green bars are for fits to individual operator coefficients, and the red bars are after marginalization in global fits. We see that both FCC-ee (darker bars) and ILC (lighter bars) could reach far into the multi-TeV region. The right panel of Fig. \[fig:FCC-eeCLIC\] shows the estimated sensitivities of CLIC measurements to other combinations of dimension-6 SMEFT operators [@ERSY-CLIC], highlighting the advantages conferred by high-energy running at CLIC.
Higher-Energy Proton-Proton Colliders
=====================================
Circular colliders with circumferences approaching 100 km are being considered in China (CEPC/SppC [@CEPC]) and as a possible future CERN project (FCC-ee/hh [@FCC]). One could imagine filling the tunnel with two successive accelerators, as was done with LEP and then the LHC in CERN’s present 27-km tunnel.
Fig. \[fig:FCC-pp\] provides two illustrations of the possible physics reach of the FCC-hh project for a $pp$ collider. In the left panel we see the ways in which various Higgs production cross sections grow by almost two orders of magnitude with the centre-of-mass energy [@FCC-hh-H], offering many possibilities for high-precision measurements of Higgs production mechanisms and decay modes in collisions at 100 TeV. In particular, these might offer the opportunity to make the first accurate direct measurements of the triple-Higgs coupling. In the right panel we see the discovery reaches for squark and gluino discovery at FCC-pp [@FCC-hh-BSM]. The reaches for both these sparticles extend beyond 10 TeV and offer, e.g., the prospects for detecting the heavy spectrum of the mAMSB model shown in Fig. \[fig:reaches\].
In my opinion, the combination of high precision and large kinematic reach offered by large circular colliders is unbeatable as a vision for the future of high-energy physics, offer the twin possibilities of exploring the 10 TeV scale directly in $pp$ collisions at centre-of-mass energies up to 100 TeV and indirectly via the high-precision $e^+ e^-$ measurements mentioned in the previous Section.
Summary
=======
Despite the impressive progress already made, many things are still to be learnt about the Higgs boson, including its expected dominant $b \bar{b}$ decay modes, rare decays into lighter particles and the triple-Higgs coupling. The best tool for interpreting Higgs and other electroweak measurements is the SMEFT, and possible future $e^+ e^-$ colliders offer good prospects for higher-precision measurements beyond the sensitivities of the LHC.
Like that of Mark Twain, rumours of the death of supersymmetry are exaggerated. I still think that it is the best framework for TeV-scale physics beyond the SM at the TeV scale. Simple supersymmetric models have been coming under increasing pressure from LHC searches, but other models with heavier spectra are still quite healthy. There are good prospects for discovering supersymmetry in future LHC runs and in direct dark matter detection experiments, but no guarantees. Maybe we will have to wait for a future higher-energy $pp$ collider before discovering or abandoning supersymmetry?
In the mean time, we look forward to whatever indications the full LHC Run 2 date may provide before choosing what collider we would like to build next, but the answer to the question in the title may well be “round in circles".
Acknowledgments {#acknowledgments .unnumbered}
===============
The author’s work was supported partly by the STFC Grant ST/L000326/1. He thanks his collaborators on topics discussed here, and thanks the Institute of Advanced Study of the Hong Kong University of Science and Technology for its kind hospitality.
[0]{} For recent high points in this effort, see C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Phys. Rev. Lett. [**114**]{} (2015) 212001 doi:10.1103/ PhysRevLett.114.212001 \[arXiv:1503.06056 \[hep-ph\]\]; J. Currie, E. W. N. Glover and J. Pires, Phys. Rev. Lett. [**118**]{} (2017) no.7, 072002 doi:10.1103/PhysRevLett.118.072002 \[arXiv:1611.01460 \[hep-ph\]\]. ATLAS Collaboration, [https://twiki.cern.ch/twiki/bin/view/AtlasPublic/]{} [StandardModelPublicResults]{}; see also CMS Collaboration, [https://twiki.cern.ch/]{} [twiki/bin/view/CMSPublic/PhysicsResultsCombined]{}.
J. Charles [*et al.*]{} \[CKMfitter Collaboration\], Phys. Rev. D [**91**]{} (2015) no.7, 073007\
doi:10.1103/PhysRevD.91.073007 \[arXiv:1501.05013 \[hep-ph\]\],\
and [http://ckmfitter.in2p3.fr]{}.
UTfit Collaboration, [http://www.utfit.org/UTfit/]{}.
R. Aaij [*et al.*]{} \[LHCb Collaboration\], JHEP [**1612**]{} (2016) 087\
doi:10.1007/JHEP12(2016)087 \[arXiv:1611.03076 \[hep-ex\]\]. For a recent review, see S. Bifani, [https://indico.in2p3.fr/event/13763/]{} [session/9/contribution/104/material/slides/0.pdf]{}.
For a recent review, see G. Wormser, [https://indico.in2p3.fr/event/13763/]{} [session/9/contribution/105/material/slides/0.pdf]{}.
S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, JHEP [**1606**]{} (2016) 092 doi:10.1007/JHEP06(2016)092 \[arXiv:1510.04239 \[hep-ph\]\]. W. Altmannshofer, C. Niehoff, P. Stangl and D. M. Straub, arXiv:1703.09189 \[hep-ph\]. Z. Bai [*et al.*]{} \[RBC and UKQCD Collaborations\], Phys. Rev. Lett. [**115**]{} (2015) no.21, 212001 doi:10.1103/PhysRevLett.115.212001 \[arXiv:1505.07863 \[hep-lat\]\]. A. J. Buras, M. Gorbahn, S. JŠger and M. Jamin, JHEP [**1511**]{} (2015) 202 doi:10.1007/JHEP11(2015)202 \[arXiv:1507.06345 \[hep-ph\]\]. V. Khachatryan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**749**]{} (2015) 337 doi:10.1016/j.physletb.2015.07.053 \[arXiv:1502.07400 \[hep-ex\]\]; G. Aad [*et al.*]{} \[ATLAS Collaboration\], JHEP [**1511**]{} (2015) 211 doi:10.1007/JHEP11(2015)211 \[arXiv:1508.03372 \[hep-ex\]\]. CMS Collaboration, [https://cds.cern.ch/record/2159682/files/]{}\
[HIG-16-005-pas.pdf]{}.
G. Aad [*et al.*]{} \[ATLAS and CMS Collaborations\], Phys. Rev. Lett. [**114**]{} (2015) 191803 doi:10.1103/PhysRevLett.114.191803 \[arXiv:1503.07589 \[hep-ex\]\]. S. Oda, [https://indico.in2p3.fr/event/13763/session/0/contribution/39/]{}\
[material/slides/0.pdf]{}.
D. de Florian [*et al.*]{} \[LHC Higgs Cross Section Working Group\], arXiv:1610.07922 \[hep-ph\]. G. Aad [*et al.*]{} \[ATLAS and CMS Collaborations\], JHEP [**1608**]{} (2016) 045 doi:10.1007/JHEP08(2016)045 \[arXiv:1606.02266 \[hep-ex\]\]. T. Aaltonen [*et al.*]{} \[CDF and D0 Collaborations\], Phys. Rev. D [**88**]{} (2013) no.5, 052014 doi:10.1103/PhysRevD.88.052014 \[arXiv:1303.6346 \[hep-ex\]\]. J. Ellis and T. You, JHEP [**1306**]{} (2013) 103 doi:10.1007/JHEP06(2013)103 \[arXiv:1303.3879 \[hep-ph\]\]. G. Blankenburg, J. Ellis and G. Isidori, Phys. Lett. B [**712**]{} (2012) 386 doi:10.1016/j.physletb.2012.05.007 \[arXiv:1202.5704 \[hep-ph\]\]. J. Wess and B. Zumino, Phys. Lett. [**49B**]{} (1974) 52 doi:10.1016/0370-2693(74)90578-4; Nucl. Phys. B [**70**]{} (1974) 39 doi:10.1016/0550-3213(74)90355-1; Nucl. Phys. B [**78**]{} (1974) 1 doi:10.1016/0550-3213(74)90112-6. L. Maiani, [*Proc. Summer School on Particle Physics*]{}, Gif-sur-Yvette, 1979 (IN2P3, Paris, 1980) p. 3; G. ’t Hooft, in: G. ’t Hooft et al., eds., [*Recent Developments in Field Theories*]{} (Plenum Press, New York, 1980); E. Witten, Nucl. Phys. [**B188**]{} (1981) 513; R.K. Kaul, Phys. Lett. [**109B**]{} (1982) 19.
See, for example: S. Weinberg, Phys. Rev. D [**13**]{} (1976) 974. See, for example: N. Arkani-Hamed, A. G. Cohen, E. Katz and A. E. Nelson, JHEP [**0207**]{} (2002) 034 \[arXiv:hep-ph/0206021\]. G. F. Giudice, C. Grojean, A. Pomarol and R. Rattazzi, JHEP [**0706**]{} (2007) 045 doi:10.1088/1126-6708/2007/06/045 \[hep-ph/0703164\]; R. Contino, C. Grojean, M. Moretti, F. Piccinini and R. Rattazzi, JHEP [**1005**]{} (2010) 089 doi:10.1007/JHEP05(2010)089 \[arXiv:1002.1011 \[hep-ph\]\]. M. Baak [*et al.*]{} \[Gfitter Group\], Eur. Phys. J. C [**74**]{} (2014) 3046\
doi:10.1140/epjc/s10052-014-3046-5 \[arXiv:1407.3792 \[hep-ph\]\]. V. Sanz and J. Setford, arXiv:1703.10190 \[hep-ph\]. J. Ellis, J. R. Espinosa, G. F. Giudice, A. Hoecker and A. Riotto, Phys. Lett. B [**679**]{} (2009) 369 doi:10.1016/j.physletb.2009.07.054 \[arXiv:0906.0954 \[hep-ph\]\]. G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori and A. Strumia, JHEP [**1208**]{} (2012) 098 \[arXiv:1205.6497 \[hep-ph\]\]. D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio and A. Strumia, JHEP [**1312**]{} (2013) 089 doi:10.1007/JHEP12(2013)089 \[arXiv:1307.3536 \[hep-ph\]\]. A. Hook, J. Kearney, B. Shakya and K. M. Zurek, JHEP [**1501**]{} (2015) 061 doi:10.1007/JHEP01(2015)061 \[arXiv:1404.5953 \[hep-ph\]\]. J. R. Ellis and D. Ross, Phys. Lett. B [**506**]{} (2001) 331 \[arXiv:hep-ph/0012067\]. W. Buchmuller and D. Wyler, Nucl. Phys. B [**268**]{} (1986) 621; A. Pomarol and F. Riva, JHEP [**1401**]{} (2014) 151 \[arXiv:1308.2803 \[hep-ph\]\]. J. Ellis, V. Sanz and T. You, JHEP [**1503**]{} (2015) 157 doi:10.1007/JHEP03(2015)157 \[arXiv:1410.7703 \[hep-ph\]\]. A. Falkowski, M. Gonzalez-Alonso, A. Greljo and D. Marzocca, Phys. Rev. Lett. [**116**]{} (2016), 011801 doi:10.1103/PhysRevLett.116.011801 \[arXiv:1508.00581 \[hep-ph\]\]. Paraphrasing J. Bond, [http://www.imdb.com/title/tt0143145/fullcredits/]{}.
B. Bairnsfather, [http://www.independent.co.uk/news/uk/home-news/the-]{}\
[captain-who-gave-britain-its-ultimate-weapon-during-world-war-one-]{}\
[laughter-9833596.html]{}.
J. R. Ellis, G. Ridolfi and F. Zwirner, Phys. Lett. B [**257**]{} (1991) 83; H. E. Haber and R. Hempfling, Phys. Rev. Lett. [**66**]{} (1991) 1815; Y. Okada, M. Yamaguchi and T. Yanagida, Prog. Theor. Phys. [**85**]{} (1991) 1. J. R. Ellis, S. Heinemeyer, K. A. Olive and G. Weiglein, JHEP [**0301**]{} (2003) 006 doi:10.1088/1126-6708/2003/01/006 \[hep-ph/0211206\]. J. R. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. A. Olive and M. Srednicki, Nucl. Phys. B [**238**]{} (1984) 453. doi:10.1016/0550-3213(84)90461-9 J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. [**B260**]{} (1991) 131; U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. [**B260**]{} (1991) 447; P. Langacker and M. Luo, Phys. Review [**D44**]{} (1991) 817; C. Giunti, C. W. Kim and U. W. Lee, Mod. Phys. Lett. A [**6**]{} (1991) 1745. E. Bagnaschi [*et al.*]{}, Eur. Phys. J. C [**77**]{} (2017) no.2, 104 doi:10.1140/epjc/s10052-017-4639-6 \[arXiv:1610.10084 \[hep-ph\]\]. E. Bagnaschi [*et al.*]{}, arXiv:1612.05210 \[hep-ph\]. T. Golling [*et al.*]{}, \[arXiv:1606.00947 \[hep-ph\]\]. P. Cushman [*et al.*]{} \[WIMP Dark Matter Direct Detection Working Group\], arXiv:1310.8327 \[hep-ex\]. A. Tan [*et al.*]{} \[PandaX-II Collaboration\], Phys. Rev. Lett. [**117**]{} (2016) no.12, 121303 doi:10.1103/PhysRevLett.117.121303 \[arXiv:1607.07400 \[hep-ex\]\]. D. S. Akerib [*et al.*]{} \[LUX Collaboration\], Phys. Rev. Lett. [**118**]{} (2017) no.2, 021303 doi:10.1103/PhysRevLett.118.021303 \[arXiv:1608.07648 \[astro-ph.CO\]\]. D. S. Akerib [*et al.*]{} \[LZ Collaboration\], arXiv:1509.02910 \[physics.ins-det\]. E. Aprile [*et al.*]{} \[XENON Collaboration\], JCAP [**1604**]{} (2016) no.04, 027 doi:10.1088/1475-7516/2016/04/027 \[arXiv:1512.07501 \[physics.ins-det\]\]. O. Buchmueller, M. J. Dolan and C. McCabe, JHEP [**1401**]{} (2014) 025 doi:10.1007/JHEP01(2014)025 \[arXiv:1308.6799 \[hep-ph\]\]; O. Buchmueller, S. A. Malik, C. McCabe and B. Penning, Phys. Rev. Lett. [**115**]{} (2015) no.18, 181802 doi:10.1103/PhysRevLett.115.181802 \[arXiv:1505.07826 \[hep-ph\]\]; S. A. Malik [*et al.*]{}, [*White Paper from 2014 Brainstorming Workshop held at Imperial College London*]{}, Phys. Dark Univ. [**9-10**]{} (2015) 51 doi:10.1016/j.dark.2015.03.003 \[arXiv:1409.4075 \[hep-ex\]\]; J. Abdallah [*et al.*]{}, Phys. Dark Univ. [**9-10**]{} (2015) 8 doi:10.1016/j.dark.2015.08.001 \[arXiv:1506.03116 \[hep-ph\]\]; M. Bauer [*et al.*]{}, [*White Paper from 2016 Brainstorming Workshop held at Imperial College London*]{}, arXiv:1607.06680 \[hep-ex\]; D. Abercrombie [*et al.*]{} \[ATLAS/CMS Dark Matter Forum\], arXiv:1507.00966 \[hep-ex\]. A. Boveia [*et al.*]{} \[LHC Dark Matter Working Group\], arXiv:1603.04156 \[hep-ex\]. CMS Collaboration, [https://cds.cern.ch/record/2208044/files/DP2016[\_]{}057.pdf]{}.
G. Aarons [*et al.*]{} \[ILC Collaboration\], arXiv:0709.1893 \[hep-ph\]; ILC TDR, H. Baer, T. Barklow, K. Fujii, Y. Gao, A. Hoang, S. Kanemura, J. List and H. E. Logan [*et al.*]{}, arXiv:1306.6352 \[hep-ph\]; D. M. Asner [*et al.*]{}, arXiv:1310.0763 \[hep-ph\]. CLIC CDR, eds. M. Aicheler, P. Burrows, M. Draper, T. Garvey, P. Lebrun, K. Peach, N. Phinney, H. Schmickler, D. Schulte and N. Toge, CERN-2012-007,\
[http://project-clic-cdr.web.cern.ch/project-CLIC-CDR/]{}; M. J. Boland [*et al.*]{} \[CLIC and CLICdp Collaborations\], doi:10.5170/CERN-2016-004 arXiv:1608.07537 \[physics.acc-ph\]; H. Abramowicz [*et al.*]{}, arXiv:1608.07538 \[hep-ex\]. CEPC-SPPC Study Group, IHEP-CEPC-DR-2015-01, IHEP-TH-2015-01, IHEP-EP-2015-01. M. Bicer [*et al.*]{} \[TLEP Design Study Working Group Collaboration\], JHEP [**1401**]{} (2014) 164\[arXiv:1308.6176 \[hep-ex\]\]; see also [http://tlep.web.cern.ch/content/]{}\
[machine-parameters]{}.
J. Ellis, P. Roloff, V. Sanz and T. You, arXiv:1701.04804 \[hep-ph\]. J. Ellis and T. You, JHEP [**1603**]{} (2016) 089 doi:10.1007/JHEP03(2016)089 \[arXiv:1510.04561 \[hep-ph\]\]. FCC Collaboration, [https://fcc.web.cern.ch/Pages/default.aspx]{}.
R. Contino [*et al.*]{}, arXiv:1606.09408 \[hep-ph\].
[^1]: On the other hand, they have found no hint of any other physics beyond the SM, despite a similar myriad of searches.
[^2]: The grey lines are for the NUHM2 model - see [@MCSU5] for details.
|
---
abstract: 'A fully quantum treatment of Einstein’s Brownian motion is given, showing in particular the role played by the two original requirements of translational invariance and connection between dynamics of the Brownian particle and atomic nature of the medium. The former leads to a clearcut relationship with Holevo’s result on translation-covariant quantum-dynamical semigroups, the latter to a formulation of the fluctuation-dissipation theorem in terms of the dynamic structure factor, a two-point correlation function introduced in seminal work by van Hove, directly related to density fluctuations in the medium and therefore to its atomistic, discrete nature. A microphysical expression for the generally temperature dependent friction coefficient is given in terms of the dynamic structure factor and of the interaction potential describing the single collisions. A comparison with the Caldeira Leggett model is drawn, especially in view of the requirement of translational invariance, further characterizing general structures of reduced dynamics arising in the presence of symmetry under translations.'
author:
- Francesco
- Bassano
title: 'On the quantum description of Einstein’s Brownian motion'
---
Introduction {#sec:introduction}
============
Right a century has passed since Albert Einstein published the first of a series of papers on the theory of Brownian movement [@Einstein; @EinsteinDOVER], a pioneering work attempting to provide a suitable theoretical framework for the description of a long-standing experimental puzzle [@Brown]. Einstein’s investigation has gone much beyond the explanation of an interesting experiment, proving a milestone in the understanding of statistical mechanics of non-equilibrium processes, motivating and inspiring physical and mathematical research on stochastic processes. By now the term Brownian motion is ubiquitously found in the physical literature, both at quantum and classical level, used as a kind of keyword in a wealth of situations relying on a description in terms of mathematical structures or physical concepts akin to those first appeared in the explanation of Einstein’s Brownian motion. In this paper we address the question of a proper quantum description of Brownian motion in the sense of Einstein, i.e., the motion of a massive test particle in a homogeneous fluid made up of much lighter particles. In doing so we actually go back to Einstein’s real motivation in facing Brownian motion, i.e., to demonstrate the molecular, discrete nature of matter. His aim was in fact to give a decisive argument probing the correctness of the molecular-kinetic conception of heat, a question he considered most important, as stressed in the very last sentence of the paper, actually quite emphatic in the original German version: *Möge es bald einem Forscher gelingen, die hier aufgeworfene, für die Theorie der Wärme wichtige Frage zu entscheiden!* [@citation]
In contrast with previous approaches and results, based either on a modelling of the environment aiming at exact solubility given a certain suitable phenomenological Ansatz [@CLPhysicaA83], or on an axiomatic approach relying on mathematical input [@LindbladQBM; @Sandulescu], or on the exploitation of semiclassical correspondence [@DiosiEL95], we will base our microscopic analysis on the two key features of Einstein’s Brownian motion: homogeneity of the background medium, reflected into the property of translational invariance, and the atomic nature of matter responsible for density fluctuations, showing up in a suitable formulation of the fluctuation-dissipation relationship. Translational invariance comes about because of the homogeneity of the fluid and the translational invariance of the interaction potential between test particle and elementary constituents of the fluid. This fundamental symmetry property leads to important restrictions both on the expression for possible interactions and on the structure of the completely positive generator of a quantum-dynamical semigroup describing the Markovian reduced dynamics. The first key point is therefore to consider the proper type of translational invariance interaction leading to Einstein’s Brownian motion, thus fixing the relevant correlation function appearing in the structure of the generator of the quantum-dynamical semigroup, which turns out to be the so-called dynamic structure factor and provides the natural formulation of the fluctuation-dissipation relationship for the case of interest, first put forward by van Hove in an epochal paper [@vanHove]. Given that the dynamics can be fairly assumed to be Markovian, the second key point is the characterization of the structure of generators of quantum-dynamical semigroups covariant under a suitable symmetry group, in this case $\mathbb{R}$, i.e., translations, which has been recently given in most relevant work by Holevo [@HolevoJMP].
The present paper partially builds on previous work [@art3; @art4; @art5], putting it in a wider conceptual and theoretical framework, providing the previously unexplored connection to the fluctuation-dissipation theorem and further comparing this kinetic approach to quantum dissipation with the one by Caldeira Leggett, also in view of recent criticism on the realm of validity of the last approach [@SolsPhysicaA94; @AlickiOSID04]. In this way a new, different approach to the quantum description of decoherence and dissipation is put forward, which, though obviously not universally valid, could provide a direct connection between a precise microphysical model and reduced dynamics for a wide class of open quantum systems, characterized by suitable symmetries. While universality might often be a fancy, loose word in such a complex framework, this precise microphysical modelling makes a close, quantitative comparison between present [@ZeilingerQBM-exp; @ZeilingerQBM-th1; @garda03; @ZeilingerQBM-th2] and next generation experiments on decoherence and dissipation in principle feasible.
The paper is organized as follows: in Sect. \[sec:transl-invar\] we introduce the basic possible translationally invariant interactions, putting into evidence their effect on the structure of the reduced dynamics, also in comparison with previous models in the literature; in Sect. \[sec:fluct-diss-theor\] we point out the relevant interaction for the description of Einstein’s quantum Brownian motion, showing the related expression of the fluctuation-dissipation theorem; in Sect. \[sec:quant-descr-einst\] we come to the formulation of Einstein’s quantum Brownian motion putting into evidence the general microphysical expression for the friction coefficient in terms of a suitable autocorrelation function; in Sect. \[sec:conclusions-outlook\] we finally comment on our results and discuss possible future developments.
Translational Invariance {#sec:transl-invar}
========================
As a first step we characterize the general structure of microscopic Hamiltonians leading to a translationally invariant reduced dynamics for the test particle. Due to translational invariance the test particle has to be free apart form the interaction with the fluid, subject at most to a potential linearly depending on position, e.g. a constant gravitational field, so that in particular it has a continuous spectrum. The fluid is supposed to be stationary and homogeneous, and for simplicity, without loss of generality, possessing inversion symmetry, so that energy, momentum and parity are constants of motion.
Characterization of translationally invariant interactions {#sec:char-transl-invar}
----------------------------------------------------------
The microscopic Hamiltonian may be written in the form $$\label{eq:1}
H_{{\rm \scriptscriptstyle PM}}=H_{{\rm \scriptscriptstyle
P}}+H_{{\rm \scriptscriptstyle
M}}+V_{{\rm \scriptscriptstyle PM}},$$ where the subscripts P and M stand for particle and matter respectively, while $H_{{\rm \scriptscriptstyle P}}$ and $H_{{\rm
\scriptscriptstyle M}}$ satisfy the aforementioned constraints. The key point is the characterization of a suitable translationally invariant interaction potential, which we put forward in the formalism of second quantization. This non-relativistic field theoretical approach is the natural one in order to account for statistics and more generally many-particle features of the background macroscopic system, also proving useful in microphysical calculations [@art6] and allowing to deal not only with the one-particle sector of the Fock-space in which the fields referring to the test particle are described. The interaction potential between test particle and matter will have the general form $$\label{eq:2}
V_{{\rm \scriptscriptstyle PM}}=
\int d^3 \! \bm{x} \int d^3 \! \bm{y} \,
A_{{\rm \scriptscriptstyle P}} (\bm{x})
t (\bm{x} - \bm{y})
A_{{\rm \scriptscriptstyle M}} (\bm{y}) ,$$ where $t (\bm{x})$ is a $\mathbb{C}$-number, in the following applications short range, interaction potential; $A_{{\rm
\scriptscriptstyle P}} (\bm{x})$ is a self-adjoint operator built in terms of the field $$\label{eq:3}
\varphi (\bm{x})= \int \frac{d^3 \! \bm{p}}{(2\pi\hbar)^{3/2}} \,
e^{\frac{i}{\hbar}\bm{p}\cdot\bm{x}} a_{\bm{p}}$$ satisfying canonical commutation or anticommutation relations, according to the spin of the test particle; similarly $A_{{\rm
\scriptscriptstyle M}} (\bm{y})$ a self-adjoint operator given by a function of the field $$\label{eq:4}
\psi (\bm{y})= \int \frac{d^3 \! \bm{\eta}}{(2\pi\hbar)^{3/2}} \,
e^{\frac{i}{\hbar}\bm{\eta}\cdot\bm{y}} b_{\bm{\eta}}$$ pertaining to the macroscopic system and obeying suitable commutation or anticommutation relations. Eq. can be most meaningfully rewritten in terms of the Fourier transform of the interaction potential $$\label{eq:5}
\tilde{t} (\bm{q})=\int \frac{d^3 \! \bm{x}}{(2\pi\hbar)^{3}} \,
e^{{i\over\hbar}\bm{q}\cdot \bm{x}}
t (\bm{x})
,$$ where the continuous parameter $\bm{q}$, to be seen as a momentum transfer, has a natural group theoretical meaning as label of the irreducible unitary representations of the group of translations, as to be stressed later on, thus coming to the equivalent expression $$\label{eq:6}
V_{{\rm \scriptscriptstyle PM}}=
\int d^3 \! \bm{q} \,
\tilde{t} (\bm{q})
A_{{\rm \scriptscriptstyle P}} (\bm{q})
A_{{\rm \scriptscriptstyle M}}^{\scriptscriptstyle \dagger} (\bm{q}),$$ where the operators $A_{{\rm \scriptscriptstyle P}} (\bm{q})$ and $A_{{\rm \scriptscriptstyle M}} (\bm{q})$ are defined according to $$\label{eq:7}
A_{{\rm \scriptscriptstyle P/M}} (\bm{q})=\int d^3 \! \bm{x} \,
e^{-\frac{i}{\hbar}\bm{q}\cdot\bm{x}} A_{{\rm \scriptscriptstyle P/M}} (\bm{x}),$$ so that in particular because of the self-adjointness of $A_{{\rm
\scriptscriptstyle P/M}} (\bm{x})$ one has the identity $$\label{eq:8}
A_{{\rm \scriptscriptstyle P/M}}^{\scriptscriptstyle \dagger} (\bm{q})=A_{{\rm \scriptscriptstyle P/M}} (-\bm{q})$$ and similarly $$\label{eq:9}
\tilde{t}^{*} (\bm{q})=\tilde{t} (-\bm{q}),$$ because of the reality of the interaction potential. Translational invariance of the interaction, leading to the invariance of $V_{{\rm
\scriptscriptstyle PM}}$ under a global translation, is obvious in because the coupling through the potential only depends on the relative positions of the two local operator densities, and comes about in because the operators in simply transform under a phase $\exp({\frac{i}{\hbar}\bm{q}\cdot\bm{a}})$ under a translation of step $\bm{a}$. The relationship between and can be most easily seen by analogy with the following identity exploiting the fact that the Fourier transform is a unitary transformation $$\label{eq:10}
\langle f| v*g \rangle=
\langle \tilde{f}| \widetilde{v*g} \rangle=
\langle \tilde{f}|\tilde{v}\tilde{g} \rangle=\int d^3 \! \bm{q} \,
\tilde{v}(\bm{q})\tilde{f}(-\bm{q})\tilde{g}(\bm{q}) ,$$ where $f$, $v$ and $g$ are real functions, $\tilde{f}$ denotes the Fourier transform and $*$ the convolution product.
We will now consider two general types of physically meaningful translationally invariant couplings, corresponding to quite distinct situations. The first is a density-density coupling, given by the identifications $$\label{eq:11}
A_{{\rm \scriptscriptstyle P}} (\bm{x})=\varphi^{\scriptscriptstyle \dagger}
(\bm{x})\varphi (\bm{x})\equiv N_{{\rm \scriptscriptstyle P}} (\bm{x}),$$ $N_{{\rm \scriptscriptstyle P}} (\bm{x})$ being the number-density operator for the test particles, and $$\label{eq:12}
A_{{\rm \scriptscriptstyle M}} (\bm{x})=\psi^{\scriptscriptstyle \dagger}
(\bm{x})\psi (\bm{x})\equiv N_{{\rm \scriptscriptstyle M}} (\bm{x})$$ respectively. One therefore has $$\label{eq:13}
V_{{\rm \scriptscriptstyle PM}}=
\int d^3 \! \bm{x} \int d^3 \! \bm{y} \,
N_{{\rm \scriptscriptstyle P}} (\bm{x})
t (\bm{x} - \bm{y})
N_{{\rm \scriptscriptstyle M}} (\bm{y}) ,$$ or equivalently setting $$\label{eq:14}
A_{{\rm \scriptscriptstyle P}} (\bm{q})=\int d^3 \! \bm{x} \,
e^{-\frac{i}{\hbar}\bm{q}\cdot\bm{x}} N_{{\rm \scriptscriptstyle P}}
(\bm{x})=
\int \frac{d^3 \! \bm{k}}{(2\pi\hbar)^{3}} \, a_{\bm{k}}^{\scriptscriptstyle \dagger}a_{\bm{k}+\bm{q}}$$ and introducing the $\bm{q}$-component of the number-density operator $\rho_{\bm{q}}$ [@Lovesey; @Stringari] $$\label{eq:15}
A_{{\rm \scriptscriptstyle M}} (\bm{q})=\int d^3 \! \bm{x} \,
e^{-\frac{i}{\hbar}\bm{q}\cdot\bm{x}} N_{{\rm \scriptscriptstyle M}}
(\bm{x})=
\int \frac{d^3 \! \bm{\eta}}{(2\pi\hbar)^{3}} \,
b_{\bm{\eta}}^{\scriptscriptstyle \dagger}b_{\bm{\eta}+\bm{q}}\equiv \rho_{\bm{q}}$$ the alternative expression $$\label{eq:16}
V_{{\rm \scriptscriptstyle PM}}=
\int d^3 \! \bm{q} \,
\tilde{t} (\bm{q})
A_{{\rm \scriptscriptstyle P}} (\bm{q})
\rho_{\bm{q}}^{\scriptscriptstyle \dagger} (\bm{q}).$$ Note that an interaction of the form or equivalently , besides being translationally invariant, commutes with the number operators $N_{{\rm \scriptscriptstyle P}}$ and $N_{{\rm
\scriptscriptstyle M}}$, so that the elementary interaction events do bring in exchanges of momentum between the test particle and the environment, but the number of particles or quanta in both systems are independently conserved, thus typically describing an interaction in terms of collisions.
The other type of interaction we shall consider is a density-displacement coupling, corresponding to the expressions $$\label{eq:17}
A_{{\rm \scriptscriptstyle P}} (\bm{x})=\varphi^{\scriptscriptstyle \dagger}
(\bm{x})\varphi (\bm{x})\equiv N_{{\rm \scriptscriptstyle P}} (\bm{x}),$$ as above for the particle, and $$\label{eq:18}
A_{{\rm \scriptscriptstyle M}} (\bm{x})=
\int \frac{d^3 \! \bm{\eta}}{(2\pi\hbar)^{3}} \,
(b_{\bm{\eta}}+b_{-\bm{\eta}}^{\scriptscriptstyle \dagger})
e^{\frac{i}{\hbar}\bm{\eta}\cdot\bm{x}}
\equiv u(\bm{x})$$ for the macroscopic system, where $u(\bm{x})$ is often called displacement operator [@Ashcroft; @SchwablQMII], thus leading to $$\label{eq:19}
V_{{\rm \scriptscriptstyle PM}}=
\int d^3 \! \bm{x} \int d^3 \! \bm{y} \,
N_{{\rm \scriptscriptstyle P}} (\bm{x})
t (\bm{x} - \bm{y})
u(\bm{y}) ,$$ or in terms of the Fourier transformed quantities and $$\label{eq:20}
A_{{\rm \scriptscriptstyle M}} (\bm{q})=b_{\bm{q}}+b_{-\bm{q}}^{\scriptscriptstyle \dagger}=u(\bm{q}),$$ to the equivalent expression $$\label{eq:21}
V_{{\rm \scriptscriptstyle PM}}=
\int d^3 \! \bm{q} \,
\tilde{t} (\bm{q})
A_{{\rm \scriptscriptstyle P}} (\bm{q})
u^{\scriptscriptstyle \dagger} (\bm{q})=
\int d^3 \! \bm{q} \,
\tilde{t} (\bm{q})
A_{{\rm \scriptscriptstyle P}} (\bm{q})
(b_{\bm{q}}+b_{-\bm{q}}^{\scriptscriptstyle \dagger}).$$ Contrary to or , the interaction considered in or does not preserve the number of quanta of the macroscopic system and rather than a collisional interaction describes, e.g., a Fröhlich-type interaction between electron and phonon [@Mahan].
Before showing the relationship between the above introduced translationally invariant interactions and corresponding structures of master-equation in the Markovian, weak-coupling limit, we briefly discuss the connection with the most famous Caldeira Leggett model for the quantum description of dissipation and decoherence. Despite, or equivalently because of, its widespread use and relevance in applications, it is well worth trying to elucidate the basic physics behind the model, at least restricted to specific situations. In the standard formulation of the Caldeira Leggett model (see for example [@Petruccione; @IngoldLNPH02; @Weiss99]) the Hamiltonian for the environment is given in first quantization by the expression $$\label{eq:22}
H_{{\rm \scriptscriptstyle M}}=\sum_{i=1}^{N}
\left(
\frac{p_i^2}{2 m_i} + \frac{1}{2} m_i \omega_i^2 x_i^2
\right),$$ which should describe a set of independent harmonic oscillators, while the interaction term is given by (here and in the following we denote one-particle operators referring to the test particle with a hat) $$\label{eq:23}
V_{{\rm \scriptscriptstyle PM}}=-{\hat{\mathsf{x}}}\sum_{i=1}^{N}c_i x_i +
{\hat{\mathsf{x}}}^2\sum_{i=1}^{N}\frac{c_i^2}{2 m_i\omega_i^2},$$ typically focusing on a one-dimensional system, where the first term is a position-position coupling and the second one is justified as a counter-term necessary in order to restore the physical frequencies of the dynamics of the microsystem, given e.g. by a Brownian particle. In the absence of an external potential for the test particle it is also observed that translational invariance, explicitly broken by and , can be recovered by suitably fixing the otherwise arbitrary coupling constants $c_i$ to be given by [@IngoldLNPH02] $$\label{eq:24}
c_i=m_i\omega_i^2,$$ which should not affect the relevant results which actually only depend on the so-called spectral density $$\label{eq:25}
J (\omega)=\sum_{i=1}^{N}\frac{c_i^2}{2 m_i\omega_i} \delta (\omega-\omega_i),$$ which as a matter of fact is phenomenologically fixed. Since the original idea behind the model is to give an effective description of quantum dissipation in which the phenomenological quantities are to be fixed by comparison with the classical model, thus working in a semiclassical spirit, recovery of quantum Brownian motion in the sense of Einstein in the case of a test particle in a homogeneous medium is a natural requirement, and in fact the master-equation obtained from the Caldeira Leggett model with the Ohmic prescription for is considered as the standard quantum description of Brownian motion. Nonetheless, as stressed in [@SolsPhysicaA94], despite the aforementioned ad hoc adjustments the Caldeira Leggett model does not comply with one of the basic features of Brownian motion, i.e., translational invariance, and in fact also previous work has focused on how to recover translational invariance in the quantum description of dissipation [@Gallis93]. In their analysis the authors of [@SolsPhysicaA94] try to recover a modified, translationally invariant version of the Caldeira Leggett model by exploiting a suitable limit of an interaction of the density-displacement type considered above in or equivalently . While this model might be the correct one for other physical systems, we claim the Einstein’s quantum Brownian motion corresponds to a density-density coupling and we now see how the Caldeira Leggett model is related to the long-wavelength limit of a density-density coupling. Let us in fact consider and restricting the expressions to the one-particle sector for the test particle and to the $N$-particle sector for the macroscopic system, thus obtaining, using a first quantization formalism as in the Caldeira Leggett model, $$\label{eq:26}
V_{{\rm \scriptscriptstyle PM}}=\sum_{i=1}^{N} t
({\hat{\mathsf{x}}}-\bm{x}_i)=
\int d^3 \! \bm{q} \,
\tilde{t} (\bm{q})
\sum_{i=1}^{N} e^{-\frac{i}{\hbar}\bm{q}\cdot({\hat{\mathsf{x}}}-\bm{x}_i)}.$$ Considering only small momentum transfers and thus taking the long-wavelength limit of the expression, corresponding to a collective response of the macroscopic medium, one obtains up to second order $$\label{eq:27}
V_{{\rm \scriptscriptstyle PM}}
\, {\buildrel {\rm \scriptscriptstyle LWL} \over {\approx}} \,
N \int d^3 \! \bm{q} \, \tilde{t} (\bm{q})
%\\
- \frac{1}{2\hbar^2} \int d^3 \! \bm{q} \, \tilde{t} (\bm{q})
\sum_{i=1}^{N} [\bm{q}\cdot({\hat{\mathsf{x}}}-\bm{x}_i)]^2
+ O (q^4),$$ where the term linear in $\bm{q}$ has dropped because of inversion symmetry. Further exploiting isotropy, so that $\tilde{t}
(\bm{q})=\tilde{t} (q)$ and recalling the relationships $$\label{eq:28}
\int d^3 \! \bm{q} \, \tilde{f} (\bm{q})=
\left.
f (\bm{x})
\right|_{\bm{x}=0}
\quad \mathrm{and} \quad
\int d^3 \! \bm{q} \, q_i^2 \tilde{f} (\bm{q})=-\hbar^2
\left.
\frac{\partial^2 f}{\partial x_i^2}(\bm{x})
\right|_{\bm{x}=0},$$ one has $$\label{eq:29}
V_{{\rm \scriptscriptstyle PM}}
\, {\buildrel {\rm \scriptscriptstyle LWL} \over {\approx}} \,
N t (0)-\frac{1}{3}\Delta_2t (0)\, {\hat{\mathsf{x}}}\cdot\sum_{i=1}^{N}
\bm{x}_i
%\\
+\frac{1}{6}\Delta_2t (0)\,\sum_{i=1}^{N}\bm{x}_i^2
+\frac{N}{6}\Delta_2t (0)\,{\hat{\mathsf{x}}}^2+ O (q^4).$$ Here one easily recognizes the Caldeira Leggett model, though with some constraints and modifications. First of all, as evident from and also stressed in [@SolsPhysicaA94], translational invariance is preserved in the long-wavelength limit only provided that all terms up to a given order in $\bm{q}$ are consistently kept, and this also applies to any calculation put forward by means of . This explains the appearance of the so-called counter-term in , as well as the relationship required in order to apparently restore translational invariance. The symmetry requirement thus strictly fixes the relationship between coefficients. However a position-position coupling such as the one appearing in is the common feature of the long-wavelength limit of a density-density coupling with a generic, not necessarily harmonic, potential. In the case in which the potential is harmonic, i.e. $$\label{eq:30}
t (\bm{x})=\frac{1}{2}m\omega^2 \bm{x}^2,$$ one obtains from $$\label{eq:31}
V_{{\rm \scriptscriptstyle PM}}
\, {\buildrel {\rm \scriptscriptstyle LWL} \over {\approx}} \,
\frac{1}{2}m\omega^2 \sum_{i=1}^{N} (\bm{x}_i^2 + {\hat{\mathsf{x}}}^2)-
m\omega^2 {\hat{\mathsf{x}}}\cdot\sum_{i=1}^{N} \bm{x}_i$$ as in [@IngoldLNPH02]. Let us note how in the test particle couples to the collective coordinate $$\label{eq:32}
\bm{X}=\sum_{i=1}^{N} \bm{x}_i$$ of the macroscopic system, proportional to its center of mass. In a truly quantum picture of Einstein’s Brownian motion, the gas has to be described by identical particles (or mixtures thereof), so that one cannot introduce different masses and different coupling constants. According to or in a density-density interaction the test particle is differently coupled to the various $\bm{q}$-components of the number-density operator for the macroscopic system $\rho_{\bm{q}}$, depending on the specific expression of the interaction potential $t (\bm{x})$. Of course this is no more relevant when interpreting the harmonic oscillators as representatives of possible modes of the macroscopic system. Here and in the following we are not aiming at a general critique of the Caldeira Leggett model, which obviously has big merits, let alone its historical meaning as a pioneering work in research on quantum dissipation. Rather, focusing on the particular and at the same time paradigmatic example of the quantum description of Einstein’s Brownian motion, we want to put into evidence the possible detailed microscopic physics behind the model, especially in view of natural symmetry requirements, thus also opening the way for alternative ways to look at and cope with dissipation and decoherence in quantum mechanics, especially overcoming the limitation to Gaussian statistics inherent in the Caldeira Leggett model. The relevance that the microphysical coupling actually has in determining which physical phenomena can be correctly described by a given model has also been stressed in [@AlickiOSID04], where an analysis is made of pure decoherence without dissipation, indicating that a full density-density coupling rather than a position-position coupling as in the Caldeira Leggett model (in the paper correctly formalized in terms of a Bose field) should provide the proper way to describe pure, recoilless decoherence.
Structure of translation-covariant quantum-dynamical semigroups {#sec:struct-transl-covar}
---------------------------------------------------------------
We now come back to the translationally invariant interactions given by and , or and , showing the master-equations they lead to in the Markovian, weak-coupling limit. To do this we first observe that because of homogeneity of the underlying medium and translational invariance of the interaction potential, the reduced dynamics of the test particle must also be invariant under translations, so that the generator of the quantum-dynamical semigroup driving the dynamics of the test particle, i.e., giving the master-equation, must comply with the general characterization of translation-covariant generators of quantum-dynamical semigroups given in recent, seminal work by Holevo [@HolevoRMP32; @HolevoRMP33; @HolevoRAN; @HolevoJMP]. Given the unitary representation ${\hat{\mathsf{U}}}
(\bm{a})=\exp({-\frac{i}{\hbar}\bm{a}\cdot{\hat{\mathsf{p}}}})$, $\bm{a}\in \mathbb{R}^3$ of the group of translations $\mathbb{R}^3$ in the test particle Hilbert space, a mapping $\mathcal{L}$ acting on the statistical operators in this space is said to be translation-covariant if it commutes with the action of the unitary representation, i.e. $$\label{eq:33}
\mathcal{L}[{\hat{\mathsf{U}}}
(\bm{a}){\hat \varrho}{\hat{\mathsf{U}}}^{\scriptscriptstyle \dagger}
(\bm{a})]={\hat{\mathsf{U}}}
(\bm{a})\mathcal{L}[{\hat \varrho}]{\hat{\mathsf{U}}}^{\scriptscriptstyle \dagger}
(\bm{a}),$$ for any statistical operator ${\hat \varrho}$ and any translation $\bm{a}$. Needless to say the notion of covariance under a given symmetry group has proved very powerful not only in characterizing mappings such as quantum-dynamical semigroups and operations, but also observables, especially in the generalized sense of POVM [@Grabowski; @HolevoNEW]. In the specific case of generators of translation-covariant quantum-dynamical semigroups the result of Holevo, while obviously fitting in the general framework set by the famous Lindblad result [@GoriniJMP76; @Lindblad], goes beyond it giving much more detailed information on the possible structure of operators appearing in the Lindblad form, information conveyed by the symmetry requirements and relying on a quantum generalization of the Levy-Kintchine formula. Referring to the papers by Holevo for the related mathematical details (see also [@vienna] for a brief résumé), the physically relevant structure of the generator is given by $$\label{eq:34}
\mathcal{L}[{\hat \varrho}]=-\frac{i}{\hbar}[H
({\hat{\mathsf{p}}}),{\hat \varrho}]
+\mathcal{L}_{G}[{\hat \varrho}]+\mathcal{L}_{P}[{\hat \varrho}],$$ with $H ({\hat{\mathsf{p}}})$ a self-adjoint operator which is only a function of the momentum operator of the test particle; the so-called Gaussian part $\mathcal{L}_{G}$ is given by $$\begin{aligned}
\label{eq:35}
\mathcal{L}_{G}[{\hat \varrho}]
=&-{i \over \hbar}
\left[{\hat{\mathsf{y}}}_0+
H_{\mathrm{\scriptscriptstyle eff}} ({\hat{\mathsf{x}}},{\hat{\mathsf{p}}})
,{\hat \varrho}
\right]
\\ \nonumber
&+\sum_{k=1}^{r}
\left[K_k{\hat \varrho}K_k^{\dagger} -\frac{1}{2}\left\{K_k^{\dagger}K_k,{\hat \varrho}\right\} \right],\end{aligned}$$ where $$\begin{aligned}
K_k &={\hat{\mathsf{y}}}_k+L_k ({\hat{\mathsf{p}}}) ,
\\
{\hat{\mathsf{y}}}_k &=\sum_{i=1}^{3}a_{ki}{\hat{\mathsf{x}}}_i \quad
k=0,\ldots, r\leq 3 \quad a_{ki}\in \mathbb{R},
\\
H_{\mathrm{\scriptscriptstyle eff}}
({\hat{\mathsf{x}}},{\hat{\mathsf{p}}})&=\frac{\hbar}{2i}\sum_{k=1}^{r}
({\hat{\mathsf{y}}}_k L_k ({\hat{\mathsf{p}}}) -L_k^{\dagger} ({\hat{\mathsf{p}}}){\hat{\mathsf{y}}}_k)\end{aligned}$$ and the remaining Poisson part takes the form $$\label{eq:36}
\mathcal{L}_{P}[{\hat \varrho}]=
\int d\mu (\bm{q})\sum_{j=1}^{\infty}
\left[e^{{i\over\hbar}\bm{q}\cdot{\hat{\mathsf{x}}}}
L_j(\bm{q},{\hat{\mathsf{p}}})
{\hat \varrho}
L^{\dagger}_j(\bm{q},{\hat{\mathsf{p}}})e^{-{i\over\hbar}\bm{q}\cdot{\hat{\mathsf{x}}}}
%\right.
%\\
%\left.
- \frac 12
\left \{
L^{\dagger}_j(\bm{q},{\hat{\mathsf{p}}})L_j(\bm{q},{\hat{\mathsf{p}}}),{\hat \varrho}
\right \}
\right],$$ with $d\mu (\bm{q})$ a positive measure, ${\hat{\mathsf{x}}}$ and ${\hat{\mathsf{p}}}$ position and momentum operators for the test particle respectively. As it can be seen the characterization is quite powerful, so that the only freedom left is in the choice of a few coefficients and functions of the momentum operator of the test particle ${\hat{\mathsf{p}}}$. These can be fixed either referring to microphysical calculations, or relying on a suitably guessed phenomenological Ansatz. In this kind of reduced dynamics the information on the macroscopic system the test particle is interacting with is essentially encoded in a suitable, possibly operator-valued, two-point correlation function of the macroscopic system appearing in the formal Lindblad structure. The key physical point is then the identification of the relevant two-point correlation function, depending both on the coupling between test particle and reservoir, and on a characterization of the equilibrium state of the reservoir.
Physical examples {#sec:physical-examples}
-----------------
The case of density-density coupling given by and , when the reservoir is given by a free quantum gas, has been dealt with in [@art3; @art4; @art5], and the relevant test particle correlation function turns out to be the so-called dynamic structure factor [@Lovesey; @Stringari] $$\label{eq:37}
S (\bm{q},E)=\frac{1}{2\pi\hbar}\frac{1}{N}
\int dt \,
e^ {{i\over\hbar}Et}
\langle \rho_{\bm{q}}^{\scriptscriptstyle \dagger}\rho_{\bm{q}} (t) \rangle,$$ which can be written in an equivalent way as $$\label{eq:38}
S (\bm{q},E)=\frac{1}{N}\sum_{mn}\frac{e^{-\beta
E_n}}{\mathcal{Z}}
|\langle m |\rho_{\bm{q}}| n\rangle|^2 \delta (E+E_m-E_n),$$ where contrary to the usual conventions, momentum and energy are considered to be positive when transferred to the test particle, on which we are now focusing our attention, rather than on the macroscopic system. The master-equation then takes the form $$\begin{aligned}
\label{eq:39}
\frac{d{\hat \varrho}}{dt}=
&{}-
{i \over \hbar}[
{\hat{\mathsf{H}}}_0
,
{\hat \varrho}
]
\\ \nonumber
&{}+
{2\pi \over\hbar}
(2\pi\hbar)^3
n
\int d^3\!
\bm{q}
\,
{
| \tilde{t} (q) |^2
}
\Biggl[
e^{{i\over\hbar}\bm{q}\cdot{\hat{\mathsf{x}}}}
\sqrt{
S(\bm{q},E (\bm{q},{\hat{\mathsf{p}}}))
}
{\hat \varrho}
\sqrt{
S(\bm{q},E (\bm{q},{\hat{\mathsf{p}}}))
}
e^{-{i\over\hbar}\bm{q}\cdot{\hat{\mathsf{x}}}}
-
\frac 12
\left \{
S(\bm{q},E (\bm{q},{\hat{\mathsf{p}}})),
{\hat \varrho}
\right \}
\Biggr],\end{aligned}$$ where ${\hat{\mathsf{H}}}_0$ is the free particle Hamiltonian, $n$ the density of the homogeneous gas, and the dynamic structure factor appears operator-valued: in fact the energy transfer in each collision, which is given by $$\label{eq:40}
E (\bm{q},\bm{p})=\frac{(\bm{p}+\bm{q})^2}{2M}-\frac{\bm{p}^2}{2M},$$ with $M$ the mass of the test particle, is turned into an operator by replacing $\bm{p}$ with ${\hat{\mathsf{p}}}$. For the case of a free gas of particles obeying Maxwell-Boltzmann statistics the dynamic structure factor takes the explicit form $$\label{eq:41}
S_{\rm \scriptscriptstyle MB}(\bm{q},E)
=
\sqrt{\frac{\beta m}{2\pi}}
{
1
\over
q
}
e^{-{
\beta
\over
8m
}
{
(2mE + q^2)^2
\over
q^2
}}$$ with $\beta$ the inverse temperature and $m$ the mass of the gas particles. A density-displacement type of coupling as in and has been dealt with in [@SpohnJSP77; @SpohnRMP], considering an environment essentially given by a phonon bath. The relevant test particle correlation function in these kind of models is given by the following spectral function [@Lovesey; @SchwablQMII] $$\label{eq:42}
S (\bm{q},E)=\frac{1}{2\pi\hbar}
\int dt \,
e^ {{i\over\hbar}Et}
\langle u^{\scriptscriptstyle \dagger}(\bm{q})u(\bm{q},t) \rangle,$$ given by a linear combination of correlation functions of the form $$\label{eq:43}
A(\bm{q},E)=\frac{1}{2\pi\hbar}
\int dt \,
e^ {{i\over\hbar}Et}
\langle b_{\bm{q}}^{\scriptscriptstyle \dagger}b_{\bm{q}} (t) \rangle,$$ which can also be written [@Griffin] $$\label{eq:44}
A(\bm{q},E)=\sum_{mn}\frac{e^{-\beta
E_n}}{\mathcal{Z}}
|\langle m |b_{\bm{q}}| n\rangle|^2 \delta (E+E_m-E_n).$$ Contrary to the smooth expression of the dynamic structure factor for a free quantum gas given in , the spectral function has the highly singular structure $$\label{eq:45}
S (\bm{q},E)=[1+N_{\beta} (\hbar\omega_{\bm{q}})]\delta
(E+\hbar\omega_{\bm{q}})+N_{\beta} (\hbar\omega_{\bm{q}})\delta
(E-\hbar\omega_{\bm{q}}),$$ with $$\label{eq:46}
N_{\beta} (\hbar\omega_{\bm{q}})=\frac{1}{e^{\beta\hbar\omega_{\bm{q}}}-1},$$ where the exact frequencies $\hbar\omega_{\bm{q}}$ of the phonon appear. The smooth energy dependence of the test particle correlation function used in the derivation of , allowing an exact treatment in the case of a free gas of Maxwell-Boltzmann particles, here no longer applies, and in fact the master-equation has only been worked out for the diagonal matrix elements of the statistical operator in the momentum representation. Setting $\varrho
(\bm{p})\equiv \langle \bm{p}|{\hat \varrho}|\bm{p}\rangle$ one has $$\label{eq:47}
\frac{d\varrho}{dt}(\bm{p})=\int d^3 \! \bm{q} \,
| \tilde{t} (q) |^2
\Biggl[
S(\bm{q},E (\bm{q},\bm{p}-\bm{q}))
\varrho (\bm{p}-\bm{q})
-
S(\bm{q},E (\bm{q},\bm{p}))\varrho (\bm{p})
\Biggr],$$ using the notation introduced in . One immediately sees that both and fit in the general expression for the Poisson part of the generator of a translation-covariant quantum-dynamical semigroup given by Holevo, with the $|L_j(\bm{q},{\hat{\mathsf{p}}})|^2$ operators replaced by the spectral functions and respectively, the integration measure $d\mu (\bm{q})$ corresponding to the Lebesgue measure with a weight given by the square modulus of the Fourier transform of the interaction potential. It is here already apparent that the presented results and , pertaining to the Poisson part of the general structure of generator of a translation-covariant quantum-dynamical semigroup , go beyond the limitation to Gaussian statistics typical of the Caldeira Leggett model.
The relevant correlation function for these translation-covariant master-equations thus appears to be given by the Fourier transform with respect to energy of the time-dependent autocorrelation function of the operator of the macroscopic system appearing in the interaction potential $V_{{\rm \scriptscriptstyle PM}}$ when written in the form , i.e. $$\label{eq:48}
S (\bm{q},E)=\frac{1}{2\pi\hbar}
\int dt \,
e^ {{i\over\hbar}Et}
\langle A_{{\rm \scriptscriptstyle M}}^{\scriptscriptstyle
\dagger}(\bm{q}) A_{{\rm \scriptscriptstyle M}}(\bm{q},t)
\rangle.$$ The parameter $\bm{q}$ one integrates over in , with a weight given by the square modulus of the Fourier transform of the interaction potential appearing in , is to be seen as an element of the translation group, physically corresponding to the possible momentum transfers in the single collisions. The key difference between the two models lies in the physical meaning of the different correlation functions. The dynamic structure factor is linked to the so-called density fluctuations spectrum, accounting for particle number conservation of the macroscopic system. This connection to density fluctuations brings into play the other key feature of Einstein’s Brownian motion, i.e., the molecular, discrete nature of matter. As we shall see shortly, the smooth correlation function arising in connection with this density-density coupling allows us to take a diffusive limit of the reduced dynamics, thus obtaining the quantum description of Einstein’s Brownian motion. On the contrary in the typically quantized spectrum of a harmonic oscillator appears, thus leading to the singular function , so that as stressed in [@SpohnRMP] rather than a diffusion equation one necessarily has a jump process.
Fluctuation-dissipation theorem {#sec:fluct-diss-theor}
===============================
In the previous paragraph we have tried to point out and analyze the typical structures for the quantum description of dissipation and decoherence in the Markovian case that come into play when the first of the two key features of Einstein’s Brownian motion mentioned in Sect. \[sec:introduction\] is taken into account, i.e., translational invariance. We now focus on the second key feature, i.e., the connection with the discrete nature of matter, which Einstein actually wanted to demonstrate. As already hinted at the end of Sect. \[sec:transl-invar\], in the present paper we substantiate the claim that the correct description of Einstein’s Brownian motion is obtained considering a density-density coupling. As we shall see in detail in Sect. \[sec:quant-descr-einst\] this happens thanks to the fact that the two-point correlation function appearing in the master-equation in this case is the dynamic structure factor , where the Fourier transform of the number-density operator $\rho_{\bm{q}}$, as given in , appears. This function is in fact directly related to the density fluctuations in the medium, as it can be seen writing it, rather than in the form , relevant for the comparison between the different types of translational invariance interactions and related master-equations, in the following way [@Lovesey]: $$\label{eq:49}
S (\bm{q},E)=\frac{1}{2\pi\hbar}
\int dt \int d^3 \! \bm{x} \,
e^ {\frac{i}{\hbar}(E t -
\bm{q}\cdot\bm{x})}
G (\bm{x},t),$$ i.e., as Fourier transform with respect to energy and momentum transfer of the time dependent density correlation function $$\label{eq:50}
G (\bm{x},t)=\frac{1}{N}\int d^3 \! \bm{y} \,
\left \langle
N_{{\rm \scriptscriptstyle M}}(\bm{y})
N_{{\rm \scriptscriptstyle M}}(\bm{x}+\bm{y},t)
\right \rangle.$$ Here the connection with density fluctuations and therefore discrete nature of matter is manifest. Introducing the real correlation functions $$\label{eq:51}
\begin{split}
\phi^{-} (\bm{q},t)&=\frac{i}{\hbar
N}\langle[\rho_{\bm{q}}(t),\rho_{\bm{q}}^{\scriptscriptstyle
\dagger}]\rangle
\\
\phi^{+} (\bm{q},t)&=\frac{1}{\hbar
N}\langle\{\rho_{\bm{q}}(t),\rho_{\bm{q}}^{\scriptscriptstyle
\dagger}\}\rangle,
\end{split}$$ where $\{,\}$ denotes the anticommutator, the fluctuation-dissipation theorem can be formulated in terms of the dynamic structure factor as follows $$\label{eq:52}
\begin{split}
\phi^{-} (\bm{q},t)&=-\frac{2}{\hbar}\int^{0}_{-\infty} dE\,
\sin \left( \frac{E}{\hbar}t\right)\left( 1-e^{\beta E}\right)S
(\bm{q},E)
\\
\phi^{+} (\bm{q},t)&=-\frac{2}{\hbar}\int^{0}_{-\infty} dE\,
\cos \left( \frac{E}{\hbar}t\right)\coth\left(
\frac{\beta}{2}E\right) \left( 1-e^{\beta E}\right)S
(\bm{q},E).
\end{split}$$ We stress once again that contrary to the usual perspective in linear response theory, we are here concerned with the reduced dynamics of the test particle, so that we take as positive momentum and energy transferred to the particle. The dynamic structure factor can also be directly related to the dynamic response function $\chi'' (\bm{q},E)$ [@Stringari], according to $$\label{eq:53}
\begin{split}
S (\bm{q},E)&=\frac{1}{2\pi}\left[1- \coth\left(
\frac{\beta}{2}E\right)\right]\chi'' (\bm{q},E)
\\
&=\frac{1}{\pi}\frac{1}{1-e^{\beta E}}\chi'' (\bm{q},E),
\end{split}$$ the relationship leading to the important fact that while the dynamic response function is an odd function of energy, the dynamic structure factor obeys the so-called detailed balance condition $$\label{eq:54}
S (\bm{q},E)=e^{-\beta E}S (-\bm{q},-E),$$ a property granting the existence of a stationary state for the master-equation , as shown in [@art5]. In terms of the dynamic response function the fluctuation-dissipation theorem can also be written $$\label{eq:55}
\begin{split}
\phi^{-} (\bm{q},t)&=-\frac{2}{\pi\hbar}\int^{0}_{-\infty} dE\,
\sin \left( \frac{E}{\hbar}t\right)\chi'' (\bm{q},E)
\\
\phi^{+} (\bm{q},t)&=-\frac{2\pi}{\hbar}\int^{0}_{-\infty} dE\,
\cos \left( \frac{E}{\hbar}t\right)\coth\left(
\frac{\beta}{2}E\right)\chi'' (\bm{q},E),
\end{split}$$ a formulation that will prove useful for later comparison with the Caldeira Leggett model. The most significant formulation of the so-called fluctuation-dissipation theorem for the physics we are considering is however neither nor , but is to be traced back to a seminal paper by van Hove [@vanHove; @SchwablQMII]. In fact he showed that the scattering cross-section of a microscopic probe off a macroscopic sample can be written in Born approximation in the following way $$\label{eq:56}
\frac{d^2 \sigma}{d\Omega_{p'} dE_{p'}} (\bm{p}) =
\left({2\pi\hbar}\right)^6
\left(\frac{M}{2\pi\hbar^2}\right)^2
\frac{p'}{p}
{
| \tilde{t} (q) |^2
}
S (\bm{q},E)
,$$ where a particle of mass $M$ changes its momentum from $\bm{p}$ to $\bm{p}' = \bm{p}+\bm{q}$ scattering off a medium with dynamic structure factor $S (\bm{q},E)$. This is the most pregnant formulation of the fluctuation-dissipation relationship for the case of a test particle interacting through collisions with a macroscopic fluid. The energy and momentum transfer to the particle, characterized by the expression of the scattering cross-section at l.h.s. of are related to the density fluctuations of the macroscopic fluid appearing through the dynamic structure factor at r.h.s. of . One of the basic ideas of Einstein’s Brownian motion, i.e., the discrete nature of matter, once again appears in the formulation of the fluctuation-dissipation relationship. From the comparison between and one sees that the reduced dynamics is actually driven by the collisional scattering cross-section, in particular the last term of can also be written $$\label{eq:57}
-\frac{n}{2M}\{|{\hat{\mathsf{p}}}|\sigma ({\hat{\mathsf{p}}}),{\hat \varrho}\},$$ where $\sigma (\bm{p})$ is the total macroscopic scattering cross-section obtained from the differential expression for a test particle with incoming momentum $\bm{p}$. The term can be seen quite naturally as a loss term in a kinetic equation, and in fact is actually to be seen as a quantum version of the linear Boltzmann equation [@art7]. Besides this, from the direct relation between scattering cross-section and dynamic structure factor one reads on physical grounds the positivity of the correlation function, a property exploited in in order to take the square root.
We now compare the above formulations of the fluctuation-dissipation theorem with the ones encountered in the long-wavelength limit of the density-density coupling type of translationally invariant interaction, which as shown in Sect. \[sec:transl-invar\] is strongly related to the Caldeira Leggett model. In the long-wavelength limit the $\bm{q}$-component of the number-density operator becomes $$\label{eq:58}
\rho_{\bm{q}}
\, {\buildrel {\rm \scriptscriptstyle LWL} \over {\approx}} \,
N -\frac{i}{\hbar} \bm{q}\cdot\sum_{i=1}^{N}
\bm{x}_i+ O (q^2),$$ and once again the collective coordinate $\bm{X}=\sum_{i=1}^{N}
\bm{x}_i$ introduced in is put into evidence. The relevant correlation functions then become $$\label{eq:59}
\begin{split}
\phi^{-}_{ij} (\bm{q},t)&=\frac{i}{\hbar
N}\langle[\bm{X}_{i}(t),\bm{X}_{j}]\rangle
\\
\phi^{+}_{ij} (\bm{q},t)&=\frac{1}{\hbar
N}\langle\{\bm{X}_{i}(t),\bm{X}_{j}\}\rangle,
\end{split}$$ the indexes $i$ and $j$ here denoting Cartesian components of the collective coordinate . Introducing accordingly the spectral function $$\label{eq:60}
S_{ij} (E)=\frac{1}{2\pi\hbar}\frac{1}{N}
\int dt \,
e^ {{i\over\hbar}Et} \langle\bm{X}_{j} \bm{X}_{i}(t)\rangle,$$ the fluctuation-dissipation theorem reads $$\label{eq:61}
\begin{split}
\phi^{-}_{ij} (t)&=-\frac{2}{\hbar}\int^{0}_{-\infty} dE\, \sin
\left( \frac{E}{\hbar}t\right)\left( 1-e^{\beta E}\right) S_{ij}
(E)
\\
\phi^{+}_{ij} (t) &=-\frac{2}{\hbar}\int^{0}_{-\infty} dE\, \cos
\left( \frac{E}{\hbar}t\right)\coth\left(
\frac{\beta}{2}E\right) \left( 1-e^{\beta E}\right) S_{ij}
(E).
\end{split}$$ With the help of the response function $\chi_{ij}'' (E)$ $$\label{eq:62}
S_{ij} (E)=\frac{1}{\pi}\frac{1}{1-e^{\beta E}}\chi_{ij}'' (E),$$ the relations can be also written as $$\label{eq:63}
\begin{split}
\phi^{-}_{ij} (t)&=-\frac{2}{\pi\hbar}\int^{0}_{-\infty} dE\,
\sin \left( \frac{E}{\hbar}t\right)\chi_{ij}'' (E)
\\
\phi^{+}_{ij} (t) &=-\frac{2}{\pi\hbar}\int^{0}_{-\infty} dE\,
\cos \left( \frac{E}{\hbar}t\right)\coth\left(
\frac{\beta}{2}E\right) \chi_{ij}'' (E).
\end{split}$$ While a formulation of the fluctuation-dissipation theorem like the van Hove relation is missing in this long-wavelength limit, the relations , involving expectation values of commutator and anticommutator of the components of the collective coordinates, are the ones to be compared with the typical relations used in order to introduce the so-called spectral density in the Caldeira Leggett model. In fact if all coupling constants $c_i$ are put equal to $c$, as should be forced upon in the case of Einstein’s quantum Brownian motion, in which the particle interacts through collisions with a collection of identical, indistinguishable particles, the spectral density, when expressed in terms of energy $E$ rather than frequency $\omega$, would be related to the response function $\chi'' (E)$ for a one dimensional system according to $$\label{eq:64}
J (E)=\frac{c^2}{\pi}\chi'' (E).$$ The relation , first intuitively guessed in [@CLAP83], actually shows how in the friction coefficient, usually phenomenologically introduced through the spectral density, features of both the single interaction events and the reservoir do appear. In Sect. \[sec:quant-descr-einst\] we will give a microscopic expression for the friction coefficient in the case of Einstein’s quantum Brownian motion, in which both features do appear: the coupling through the Fourier components of the interaction potential, and the reservoir through certain values of the dynamic structure factor.
Quantum description of Einstein’s Brownian motion {#sec:quant-descr-einst}
=================================================
Relying on the premises of Sect. \[sec:transl-invar\] and \[sec:fluct-diss-theor\] we now come to the master-equation for the quantum description of Einstein’s Brownian motion. The requirement of translational invariance has been settled in Sect. \[sec:transl-invar\], while the connection between reduced dynamics of the test particle and density fluctuations in the medium, coming about because of its discrete nature, has been taken into account in Sect. \[sec:fluct-diss-theor\], considering a density-density coupling and thus coming to . The last step to be taken is to consider the test particle much more massive than the particles making up the gas, i.e., the Brownian limit $m/M \ll
1$, which in turn implies considering both small energy and momentum transfers, similarly to the classical case [@Uhlenbeck48]. We therefore start from and consider a free gas of Maxwell-Boltzmann particles, so that taking the limiting expression of when the ratio between the masses is much smaller than one, or equivalently considering small energy transfers, i.e. $$\label{eq:65}
S^{\scriptscriptstyle\infty}_{\rm \scriptscriptstyle
MB}(\bm{q},E)
=
\sqrt{\frac{\beta m}{2\pi}}
{
1
\over
q
}
e^{
-{
\beta
\over
8m
}
q^2
}
e^{
-\frac{\beta}{2}
E
},$$ one obtains the master-equation [@art3; @art4; @art5]
$$\begin{aligned}
\label{eq:66}
\frac{d{\hat \varrho}}{dt}=
&{}-
{i \over \hbar}[
{\hat{\mathsf{H}}}_0
,
{\hat \varrho}
]
\\ \nonumber
&{}+
{2\pi \over\hbar}
(2\pi\hbar)^3
n
\sqrt{\frac{\beta m}{2\pi}}
\int d^3\!
\bm{q}
\,
\frac{| \tilde{t} (q) |^2}{q}
e^{-\frac{\beta}{8m}\left( 1+2\frac{m}{M}\right) q^2}
\Biggl[
e^{{i\over\hbar}\bm{q}\cdot{\hat{\mathsf{x}}}}
e^{-{\beta\over 4M}\bm{q}\cdot{\hat{\mathsf{p}}}}
{\hat \varrho}
e^{-{\beta\over 4M}\bm{q}\cdot{\hat{\mathsf{p}}}}
e^{-{i\over\hbar}\bm{q}\cdot{\hat{\mathsf{x}}}}
- {1\over 2}
\left \{
e^{-{\beta\over 2M}\bm{q}\cdot{\hat{\mathsf{p}}}}
,
{\hat \varrho}
\right \}
\Biggr],\end{aligned}$$
which in the limit of small momentum transfer leads, of necessity as can be seen from the Gaussian contribution in Holevo’s result but also from previous work [@LindbladQBM; @Sandulescu], to a Caldeira Leggett type master-equation, however without shortcomings related to the lack of preservation of positivity of the statistical operator. The master-equation takes the form $$\label{eq:67}
{
d {\hat \varrho}
\over
dt
}
=
-
{i\over\hbar}
[
{{\hat{\mathsf{H}}}_0}
,{\hat \varrho}
]
-
{i\over\hbar}
\frac{\eta}{2}
\sum_{i=1}^3
\left[
{\hat{\mathsf{x}}}_i ,
\left \{
{\hat{\mathsf{p}}}_i,{\hat \varrho}
\right \}
\right] -
{
D_{pp}
\over
\hbar^2
}
\sum_{i=1}^3
\left[
{\hat{\mathsf{x}}}_i,
\left[
{\hat{\mathsf{x}}}_i,{\hat \varrho}
\right]
\right]
-
{
D_{xx}
\over
\hbar^2
}
\sum_{i=1}^3
\left[
{\hat{\mathsf{p}}}_i,
\left[
{\hat{\mathsf{p}}}_i,{\hat \varrho}
\right]
\right]
,$$
with $$\label{eq:68}
D_{pp}=\frac{M}{\beta}\eta\quad \text{and}\quad
D_{xx}=\frac{\beta\hbar^2}{16 M}\eta .$$ The friction coefficient $\eta$ is uniquely determined on the basis of the microscopic information on interaction potential and correlation function of the macroscopic system, according to $$\label{eq:69}
\eta=
\frac{\beta}{2M}
{2\pi \over\hbar}
(2\pi\hbar)^3
n
\int d^3\!
\bm{q}
\,
| \tilde{t} (q) |^2 \,
\frac{q^2}{3} S (\bm{q},E=0),$$ the factor 3 being related to the space dimensions, or equivalently $$\label{eq:70}
\eta=
\frac{\beta}{2M}
{2\pi \over\hbar}
(2\pi\hbar)^2
n
\int d^3\!
\bm{q}
\,
| \tilde{t} (q) |^2 \,
\frac{q^2}{3}
\frac{1}{N}
\int dt \,
\langle \rho_{\bm{q}}^{\scriptscriptstyle \dagger}\rho_{\bm{q}} (t) \rangle,$$ thus proving in a specific physical case of interest the so-called standard wisdom expecting the decoherence and dissipation rate to be connected with the value at zero energy of some suitable spectral function [@AlickiOSID04]. Introducing the Fourier transform of the gradient of the number-density operator, which we indicate by $\nabla \rho_{\bm{q}}$ $$\label{eq:71}
\nabla \rho_{\bm{q}}\equiv \bm{q}\rho_{\bm{q}} = -i\hbar
\int d^3 \! \bm{x}\,
e^{-\frac{i}{\hbar}\bm{q}\cdot\bm{x}} \nabla N_{{\rm \scriptscriptstyle M}}(\bm{x}),$$ the friction coefficient can also be written in terms of the time dependent autocorrelation function of $\nabla \rho_{\bm{q}}$ according to $$\label{eq:72}
\eta=
\frac{\beta}{6M}
{2\pi \over\hbar}
(2\pi\hbar)^2
n
\int d^3\!
\bm{q}
\,
| \tilde{t} (q) |^2 \,
\frac{1}{N}
\int dt \,
\langle \nabla\rho_{\bm{q}}^{\scriptscriptstyle \dagger}\cdot\nabla\rho_{\bm{q}}
(t) \rangle.$$ It is worth noticing how, contrary to the usual Caldeira Leggett model, the friction coefficient will generally exhibit an explicit temperature dependence, being related both to the expectation value of the operators $\rho_{\bm{q}}$ and to the interaction potential. No energy cutoff needs to be introduced, since all quantities appearing in the calculations remain finite, being directly linked to the relevant physical properties of the macroscopic system the test particle is interacting with. Note that introducing the thermal momentum spread $$\label{a}
{\Delta p}^2_{\rm
\scriptscriptstyle th}=\frac{M}{\beta}$$ and the square thermal wavelength $$\label{b}
{\Delta x}^2_{\rm
\scriptscriptstyle th}=\frac{\beta\hbar^2}{4M}$$ satisfying the minimum uncertainty relation $$\label{c}
{\Delta p}_{\rm \scriptscriptstyle
th}{\Delta x}_{\rm \scriptscriptstyle th}=\frac{\hbar}{2}$$ the coefficients given in can also be expressed in the form $$\label{d}
D_{pp}=\eta {\Delta p}^2_{\rm
\scriptscriptstyle th}\quad \text{and}\quad
D_{xx}=\frac{\eta}{4} {\Delta x}^2_{\rm
\scriptscriptstyle th}.$$
The main difference between and the master-equation introduced by Caldeira and Leggett for the description of quantum Brownian motion, apart from the microphysical expression for the appearing coefficients, lies in the appearance of the last contribution, given by a double commutator with the momentum operator of the Brownian particle, and corresponding to position diffusion. This term, which here appears in the expansion for small energy and momentum transfer of the dynamic structure factor, is directly linked to preservation of positivity of the statistical operator, and in fact in the past many different amendments of the Caldeira Leggett master-equation have been proposed in the literature introducing a term of this kind [@art3; @debate; @reply], even though it is not obvious how to actually experimentally check the relevance of this term, essentially quantum in origin, as can also be seen from and . In recent work [@art7] it has been shown how this contribution might lead in the strong friction limit to a typically quantum correction to Einstein’s diffusion coefficient, only relevant at low temperatures, thus opening the way to the conception of future experiments in which to possibly check the correction, as considered in [@art11].
Conclusions and outlook {#sec:conclusions-outlook}
=======================
In the present paper a fully quantum approach to the description of Brownian motion in the sense of Einstein, i.e., considering a massive test particle interacting through collisions with a background of much lighter ones, has been presented. The two cardinal requirements determining the quantum description of the reduced dynamics are translational invariance and the connection with the discrete, atomistic nature of the medium, along the lines of Einstein’s original confrontation with the problem. The former implies the choice of a translationally invariant interaction potential and leads to the requirement of translation covariance for the quantum-dynamical semigroup giving the time evolution, according to Holevo’s results [@HolevoJMP] as seen in Sect. \[sec:transl-invar\]; the latter relates the dynamics to the density fluctuations in the fluid, expressed in terms of the dynamic structure factor, first introduced by van Hove [@vanHove], and ensuring the physically most telling formulation of the fluctuation-dissipation theorem for the considered case, as seen in Sect. \[sec:fluct-diss-theor\]. A comparison has been drawn whenever possible between the present approach and the famous Caldeira Leggett model for the treatment of decoherence and dissipation in quantum mechanics, showing how the Caldeira Leggett model may arise as long-wavelength limit of a density-density coupling preserving translational invariance. This accounts in particular for the limitation to Gaussian statistics inherent in the Caldeira Leggett model or variants thereof. At variance with the Caldeira Leggett model a new microphysical expression for the friction coefficient has been given, relating it to the Fourier transform of the interaction potential and a suitable autocorrelation function as seen in Sect. \[sec:quant-descr-einst\]. No need of renormalizations or energy cutoffs appears in the treatment. Furthermore physical realizations of the Poisson component of the general structure of generator of a translation-covariant quantum-dynamical semigroup has been presented, going beyond the typical restriction to Gaussian statistics.
Even though focusing on the specific issue of Einstein’s quantum Brownian motion, the general results presented in Sect. \[sec:transl-invar\] and \[sec:fluct-diss-theor\], providing a clearcut connection between expression of the translationally invariant interaction and precise structure of the associated reduced Markovian dynamics, fulfilling the natural and physically compelling requirement of translation covariance, further clarifying the relevant correlation function of the environment and its connection to the fluctuation-dissipation theorem, should provide a general framework for a precise description of dissipation and decoherence in quantum mechanics, also allowing for a direct connection with microscopic quantities. Now that experimental quantitative tests of decoherence begin to be within reach (see for example [@HarochePRL96-HarocheRMP02; @WinelandNature00; @PritchardPRL01; @ZeilingerQBM-exp] or [@KieferNEW; @Petruccione] for more general references), the next challenge for the theoretical analysis is in fact no more an effective, phenomenological description of the phenomenon, but rather a full-fledged microphysical analysis, in which both phenomena of dissipation and decoherence can be correctly described.
Francesco Petruccione thanks Università di Milano and Bassano Vacchini thanks University of KwaZulu-Natal for kind hospitality. Bassano Vacchini is grateful to Prof. L. Lanz for his support during the whole work and to Prof. A. Barchielli for very useful discussions on the subject of this paper. This work was financially supported by INFN, and by MURST under Cofinanziamento and FIRB.
[99]{} , [Ann. Phys. (Leipzig)]{} [**17**]{}, 549 (1905). , *Investigations on the Theory of the Brownian Movement* (Dover, New York, 1956). , Philosophical Magazine N. S. [**4**]{}, 161 (1828). *It is to be hoped that some enquirer may succeed shortly in solving the problem suggested here, which is so important in connection with the theory of Heat.* (Ref. [@EinsteinDOVER], p. 18). A. O. Caldeira and A. J. Leggett, [Physica A]{} [**121**]{}, 587 (1983). G. Lindblad, [Rep. Math. Phys.]{} [**10**]{}, 393 (1976). , [Ann. Phys. (N. Y.)]{} [**173**]{}, 277 (1987). , [Europhys. Lett.]{} [**30**]{}, [63]{} (1995). , [J. Math. Phys.]{} [**37**]{}, [1812]{} (1996). , [ Phys. Rev.]{} [**95**]{}, [249]{} (1954). , [Phys. Rev. Lett.]{} [**84**]{}, 1374 (2000). , [Phys. Rev. E]{} [**63**]{}, 066115 (2001). , J. Math. Phys. [**42**]{}, 4291 (2001). , [Physica A]{} [**212**]{}, 181 (1994). R. Alicki, [Open Sys. & Information Dyn.]{} [**11**]{}, 53 (2004). K. Hornberger, S. Uttenthaler, B. Brezger, L. Hackermüller, M. Arndt and A. Zeilinger, [Phys. Rev. Lett.]{} [**90**]{}, 160401 (2003). L. Hackermüller, K. Hornberger, B. Brezger, A. Zeilinger, and M. Arndt, [Appl. Phys. B]{} [**77**]{}, 781 (2003). B. Vacchini, J. Mod. Opt. [**51**]{}, 1025 (2004). K. Hornberger, J. E. Sipe and M. Arndt, Physical Review A, in press. , [Int. J. Mod. Phys. A]{} [**17**]{}, 435 (2002). , [*Theory of Neutron Scattering from Condensed Matter*]{} (Clarendon Press, Oxford, 1984). , *Bose-Einstein Condensation* (Clarendon Press, Oxford, 2003). , *Solid State Physics* (Saunders College Publishing, Philadelphia, 1976). , *Advanced Quantum Mechanics* 2nd ed. (Springer, Berlin, 2003). *Many-particle physics* 3rd ed. (Kluwer Academic/Plenum Publisher, New York, 2000). H. P. Breuer and F. Petruccione, [*The Theory of Open Quantum Systems*]{} (Oxford University Press, Oxford, 2002). , n *Coherent Evolution in Noisy Environments*, A. Buchleitner and K. Hornberger Eds., Lecture Notes in Physics, Vol. 611, (Springer, Berlin, 2002), pag. 1. U. Weiss, *Quantum Dissipative Systems* 2nd ed. (World Scientific, Singapore, 1999). M. R. Gallis, [Phys. Rev. A ]{} [**48**]{}, 1028 (1993). , [Rep. Math. Phys.]{}, [**32**]{}, 211 (1993). , [Rep. Math. Phys.]{}, [**33**]{}, 95 (1993). , Izvestija RAN, Ser. Mat. [**59**]{}, 205 (1995). P. Busch, M. Grabowski and P. J. Lahti, *Operational Quantum Physics*, Lecture Notes in Physics, Vol. m31, 2ed. (Springer, Berlin, 1997). , [*Statistical Structure of Quantum Theory*]{}, Lecture Notes in Physics, Vol. m67, (Springer, Berlin, 2001). , [J. Math. Phys.]{} [**17**]{}, [821]{} (1976). , [Commun. Math. Phys.]{} [**48**]{}, [119]{} (1976). B. Vacchini, [Int. J. Theor. Phys.]{}, in press. , J. Stat. Phys. [**17**]{}, 385 (1977). , [Rev. Mod. Phys.]{} [**53**]{}, 569 (1980). , *Excitations in a Bose-condensed Liquid* (Cambridge University Press, Cambridge, 1993). , [Phys. Rev. E]{} [**66**]{}, 027107 (2002). A. O. Caldeira and A. J. Leggett, [Ann. Phys. (N. Y.)]{} [**149**]{}, 374 (1983). , in *Studies in statistical mechanics*, Vol. V, edited by J. De Boer, (North-holland, Amsterdam, 1970). V. Ambegaokar, [Ber. Bunsenges Phys. Chem.]{} [**95**]{}, 400 (1991); [L. Diósi]{}, [Europhys. Lett.]{} [**22**]{}, [1]{} (1993); [L. Diósi]{}, [Europhys. Lett.]{} [**30**]{}, [63]{} ([1995]{}); [A. Tameshtit and J. E. Sipe]{}, [Phys. Rev. Lett.]{} [**77**]{}, 2600 (1996); [W. J. Munro and C. W. Gardiner]{}, [Phys. Rev. A]{} [**53**]{}, 2633 (1996); [S. Gao]{}, [Phys. Rev. Lett.]{} [**79**]{}, 3101 (1997); [H. M. Wiseman and W. J. Munro]{}, [*ibid*]{}. [**80**]{}, 5702 (1998); [S. Gao]{}, [*ibid*]{}. [**80**]{}, 5703 (1998); [G. W. Ford and R. F. O’Connell]{}, [*ibid*]{}. [**82**]{}, 3376 (1999); [S. Gao]{}, [*ibid*]{}. [**82**]{}, 3377 (1999). , [Phys. Rev. Lett.]{} [**87**]{}, 028901 (2001); [B. Vacchini]{}, [*ibid*]{}. [**87**]{}, 028902 (2001). A. Ferraro, S. Olivares, N. Piovella, B. Vacchini and R. Bonifacio, in preparation. , [Phys. Rev. Lett.]{} [**77**]{}, 4887 (1996); [J. M. Raimond, M. Brune and S. Haroche]{}, [Rev. Mod. Phys.]{} [**73**]{}, 565 (2001). , [Nature]{} [**403**]{}, 269 (2000). , [Phys. Rev. Lett.]{} [**86**]{}, 2191 (2001). , [*Decoherence and the Appearance of a Classical World in Quantum Theory*]{} 2nd ed. (Springer, Berlin, 2003).
|
---
abstract: 'Let $p$ be an odd prime, and fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)$. We give a $p$-local homotopy decomposition for the loop space of the complex Stiefel manifold $W_{n,m}$. Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the $p$-exponent of $W_{n,m}$. Upper bounds for $p$-exponents in the stable range $2m<n$ and $0<m\leq (p-1)(p-2)$ are computed as well.'
author:
- Piotr Beben
bibliography:
- 'stiefel-exp.bib'
title: 'Homotopy Decompositions of Looped Stiefel manifolds, and their Exponents'
---
Introduction
============
Fix $p$ to be an odd prime. Throughout this paper we assume that all spaces have been localized at $p$, and we set $q=2(p-1)$. When a reference to the cell structure of a space is made, we will be referring to a given $p$-local cell structure. For $p$-localizations of $CW$-complexes in particular, it will be implicit that the $p$-local cell structure being used is the one induced by localizing.
We shall use the term *fibration* to refer to both homotopy fibrations and fibrations in the strict sense. When stating the homology (or cohomology) of a space without specifying the coefficients, this will be taken to mean that the statement holds for both $\mathbb{Z}_{p}$-homology and ${\ensuremath{\mathbb{Z}_{(p)}}}$-homology.
Let $W_{n,m}$ be the *complex Stiefel manifold*, the group quotient $SU(n)/SU(n-m)$. Our first theorem provides a nontrivial homotopy decomposition for the loop space of low rank Stiefel manifolds.
\[MAIN1\] Fix integers $n$ and $m$ such that $0<m<n\leq (p-1)(p-2)$. Then there exists a product decomposition $$\Omega W_{n,m}\simeq \displaystyle\prod_{1\leq i\leq p-1} \Omega D_{i}$$ such that for each $1\leq i\leq (p-1)$, $D_{i}$ is an $H$-space whose homology is the exterior algebra $${\ensuremath{H_{*}(D_{i})}}\cong\Lambda(y_{2(n-m+i)-1},y_{2(n-m+i)-1+q},...,y_{2(n-m+i)-1+k_{i}q}),$$ where $k_{i}$ is the largest integer such that $2(n-m+i)-1+k_{i}q\leq 2n-1$.
We should also mention that the complex Stiefel manifolds are not $H$-spaces in general (even in the $p$-local sense), so one should not hope that the above decomposition will hold before looping. In particular, it is not clear whether the above decomposition is an $H$-space decomposition.
Theorem \[MAIN1\] is based on a decomposition of the unitary group $SU(n)$ for arbitrary $n$ as a product of indecomposable spaces as in [@MNT2] and [@T2]. Our approach expands on ideas of Theriault [@T2], and uses a fundamental construction of Cohen and Neisendorfer [@CN1] to give a decomposition that is functorial and enjoys good naturality properties (see Theorem \[T2\]). The real analog of this decomposition is provided in Theorem \[T3\]. In a similar vein to Theorem \[MAIN1\], this leads us to low rank homotopy decompositions for the loop spaces of *real Stiefel manifolds* $V_{n,m}=SO(n)/SO(n-m)$ and *symplectic Stiefel manifolds* $X_{n,m}=Sp(n)/Sp(n-m)$, stated as Theorems \[MAIN2\] and \[MAIN3\].
It is clear that these decompositions have an application towards computing $p$-exponents. Recall for an arbitrary space $X$ the $p$-exponent $exp_{p}(X)$ of $X$ is defined as the smallest power $p^t$ that annihilates the $p$-primary torsion of $\pi_{*}(X)$. Then we have the following.
\[MAIN4\] Fix $0<m\leq (p-1)(p-2)$ and assume either $2m<n$ or $0<m<n\leq (p-1)(p-2)$. Let $k$ be the number of cells in the suspended stunted complex projective space $\Sigma\mathbb{C}P^{n-1}_{m}$ that are in dimensions of the form $(2n-1-iq)$ for $0\leq i<p-1$. Then $$exp_{p}(W_{n,m})\leq p^{n-1+(k-1)}.$$
Furthermore, if $k>1$ and $0<m<n\leq (p-1)(p-2)$, and there exists a cell of dimension $(2n-1-iq)$ in $\Sigma\mathbb{C}P^{n-1}_{m}$ such that $i>0$ and $(2n-1-iq)$ is divisible by $p$, then $$exp_{p}(W_{n,m})\leq p^{n-1+(k-2)}.$$
We will see (Remark \[rProd\]) that the precise bound $\exp_{p}(W_{n,m})=p^{n-1}$ holds whenever $m\leq p-1$. By using Theorems \[MAIN2\] and \[MAIN3\] we can also compute $p$-exponent bounds for the real and symplectic Stiefel manifolds within certain dimensional ranges. These results are stated as Theorems \[MAIN5\] and \[MAIN6\] without proof.
Preliminary Facts About Finite $H$-spaces
=========================================
Let $\mathcal{C}$ be the sub-category of spaces and continuous maps defined as follows. The objects in $\mathcal{C}$ are $p$-localizations of path-connected $CW$-complexes $X$, where $X$ consists of no more than $p-2$ odd dimensional cells and no even cells, and the morphisms are continuous maps between these spaces. Let $\mathcal{D}$ be the category of $p$-local finite $H$-spaces spaces and $H$-maps. In this section we recall Cohen and Neisendorfer’s [@CN1] construction of a functor between these categories, which will be of fundamental use in our proof of Theorem \[MAIN1\].
\[T1\] Let $X$ be a space in $\mathcal{C}$. There exists a functor [$M\colon\mathcal{C}
{\longrightarrow}\mathcal{D}$]{} such that:
[$H_{*}(M(X))$]{} $\cong$ $\Lambda ({\ensuremath{\widetilde{H}_{*}(X)}})$, and there is a functorial map [$\iota\colonX
{\longrightarrow}M(X)$]{} that induces an inclusion of generating sets on homology;
there exist functorial maps [$M(X)\stackrel{s}
{\longrightarrow}\Omega\Sigma X\stackrel{r}{\longrightarrow}M(X)$]{} such that the composition $r\circ s$ is homotopic to the identity;
the composition [$X\stackrel{\iota}
{\longrightarrow}M(X)\stackrel{s}{\longrightarrow}\Omega\Sigma X$]{} induces the inclusion [${\ensuremath{H_{*}(X)}}\stackrel{}
{\longrightarrow}T({\ensuremath{\widetilde{H}_{*}(X)}})$]{} on homology. ${\hfill\square}$
The $H$-space structures for the spaces under the image of $M$ are induced by the retraction in part $(ii)$ of Theorem \[T1\]. The functor $M$ takes certain cofibrations to fibrations, as is stated in the following proposition from [@CN1].
\[T1B\]
Let $X$ and $Y$ be spaces in $\mathcal{C}$. Let $X'$ be a $p$-local subcomplex of $X$, and $X''$ be the cofibre of the inclusion [$X'
{\longrightarrow}X$]{}.
There exists a fibration ${\ensuremath{M(X')\stackrel{}
{\longrightarrow}M(X)\stackrel{}{\longrightarrow}M(X'')}}$;
if $X$ and $Y$ have $l$ and $m$ cells such that $l+m\leq p-2$, then $M(X\vee Y)$ is homotopy equivalent to $M(X)\times M(Y)$. ${\hfill\square}$
Observe that Theorem \[T1\] implies $M(S^{2n-1})=S^{2n-1}$. Hence Proposition \[T1B\] implies a cofibration sequence [$X'\stackrel{}
{\longrightarrow}X\stackrel{}{\longrightarrow}S^{2n-1}$]{} gives a fibration sequence [$M(X')\stackrel{}
{\longrightarrow}M(X)\stackrel{}{\longrightarrow}S^{2n-1}$]{}.
Decomposition of Looped Stiefel Manifolds
=========================================
Complex Stiefel Manifolds
-------------------------
The following decomposition of the suspended complex projective space is due to Mimura, Nishida, and Toda [@MNT1].
\[L1\] For each positive integer $n$, there exists a wedge decomposition $$\Sigma\mathbb{C}P^{n-1}\simeq\displaystyle\bigvee_{1\leq i\leq p-1}C_{i}$$ with $${\ensuremath{\widetilde{H}_{*}(C_{i})}}\cong\{x_{2i+1},x_{(2i+1)+q},...,x_{(2i+1)+k_{i}q}\},$$ where $k_{i}$ is the largest integer such that $(2i+1)+k_{i}q\leq 2n-1$. These decompositions are natural with respect to inclusions $j:$[$\Sigma\mathbb{C}P^{n-m-1}
{\hookrightarrow}\Sigma\mathbb{C}P^{n-1}$]{}. ${\hfill\square}$
The *stunted complex projective space* $\mathbb{C}P^{n}_{m}$ is the cofibre of the inclusion ${\ensuremath{\mathbb{C}P^{n-m}\stackrel{j}
{\longrightarrow}\mathbb{C}P^{n}}}$. By the naturality of the above decompositions, $j$ splits as a wedge of maps [$C^{\prime}_i\stackrel{j_i}
{\longrightarrow}C_i$]{} for $1\leq i\leq p-1$. Then the cofibre $\Sigma\mathbb{C}P^{n-1}_{m}$ of $j$ splits as a wedge of $p-1$ spaces that are the homotopy cofibres of the maps $j_i$. We record this as the following corollary.
\[C1\] For each pair of positive integers $m<n$, there exists a wedge decomposition $$\Sigma\mathbb{C}P^{n-1}_{m}\simeq\displaystyle\bigvee_{1\leq i\leq p-1}A_{i}$$ with $${\ensuremath{\widetilde{H}_{*}(A_{i})}}\cong\{x_{2(n-m+i)-1},x_{2(n-m+i)-1+q},...,x_{2(n-m+i)-1+k_{i}q}\},$$ where $k_{i}$ is the largest integer such that $2(n-m+i)-1+k_{i}q\leq 2n-1$. ${\hfill\square}$
A decomposition of the unitary group $SU(n)$ for arbitrary $n$ as a product of indecomposable spaces was given by Mimura, Nishida, and Toda in [@MNT2]. The decompositions were of the form $SU(n)\simeq\prod^{p-1}_{i=1}\bar{B}_{i}$ such that [$H_{*}(\bar{B}_{i})$]{}$\cong\Lambda(x_{2i+1},x_{(2i+1)+q},...,x_{(2i+1)+k_{i}q})$, and $k_{i}$ is the largest integer such that $(2i+1)+k_{i}q\leq 2n-1$. A similar decomposition of $SU(n)$ is given by Theriault [@T2] for $n\leq (p-1)(p-3)$, but this time each of the factors are generated by the functor $M$. We recover Theriault’s decomposition for the slightly larger dimensional range $n\leq (p-1)(p-2)$. Along with this, we have the additional property that our decomposition is natural with respect to the inclusion [$SU(n-m)\stackrel{\tilde{j}}
{\longrightarrow}SU(n)$]{} of $(n-m)$-frames into $n$-frames. This also presents an advantage over Mimura’s, Nishida’s, and Toda’s decomposition in the sense that their decompositions are not known to be natural. Another advantage is that the maps between corresponding factors in these decompositions fit into certain fibration sequences, as is stated in part $(iii)$ of Theorem \[T2\].
\[T2\] Fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)$. Then there exists a homotopy commutative diagram of product decompositions $$\diagram
\prod^{p-1}_{i=1}B^{\prime}_{i}\dto^{\simeq}\rto^{\prod g_{i}}
&\prod^{p-1}_{i=1}B_{i}\dto^{\simeq}\\
SU(n-m)\rto^{\tilde{j}}
&SU(n),
\enddiagram$$ such that the following properties hold:
[$H_{*}(B_{i})$]{}$\cong\Lambda(x_{2i+1},x_{(2i+1)+q},...,x_{(2i+1)+k_{i}q})$, where $k_{i}$ is the largest integer such that $(2i+1)+k_{i}q\leq 2n-1$;
[$H_{*}(B^{\prime}_{i})$]{}$\cong\Lambda(x_{2i+1},x_{(2i+1)+q},...,x_{(2i+1)+k^{\prime}_{i}q})$, where $k^{\prime}_{i}$ is the largest integer such that $(2i+1)+k^{\prime}_{i}q\leq 2(n-m)-1$;
There exist fibrations [$B^{'}_{i}\stackrel{g_i}
{\longrightarrow}B_{i}\stackrel{}{\longrightarrow}D_{i}$]{}, where $D_{i}$ is an $H$-space such that [$H_{*}(D_{i})$]{}$\cong\Lambda(x_{(2i+1)+(k^{\prime}_{i}+1)q},...,x_{(2i+1)+k_{i}q})$.
Fix an integer $i$ such that $1\leq i\leq p-1$. Let $C^{\prime}_{i}$ and $C_{i}$ be the corresponding summands in the wedge decompositions of $\Sigma\mathbb{C}P^{n-m-1}$ and $\Sigma\mathbb{C}P^{n-1}$ in Lemma \[L1\]. For each natural number $k$, there exists a map [$\Sigma\mathbb{C}P^{k-1}
{\longrightarrow}SU(k)$]{} that induces on homology an isomorphism onto the generating set of ${\ensuremath{H_{*}(SU(k))}}\cong\Lambda({\ensuremath{H_{*}(\Sigma\mathbb{C}P^{k-1})}})$. These maps are natural in the sense that we have the following commutative diagrams $$\label{D1less}
\diagram
\Sigma\mathbb{C}P^{n-m-1}\dto^{}\rto^{j}
&\Sigma\mathbb{C}P^{n-1}\dto^{}\\
SU(n-m)\rto^{\tilde{j}}
&SU(n).
\enddiagram$$ Take the compositions $h\colon$[$C_{i}\stackrel{}
{\longrightarrow}\Sigma\mathbb{C}P^{n-1}\stackrel{}{\longrightarrow}SU(n)$]{} and $h^{\prime}\colon$ [$C^{\prime}_{i}\stackrel{}
{\longrightarrow}\Sigma\mathbb{C}P^{n-m-1}\stackrel{}{\longrightarrow}SU(n-m)$]{}. Combining the diagram in (\[D1less\]) with the naturality of the decompositions in Lemma \[L1\], we have a map [$C^{\prime}_i\stackrel{j_{i}}
{\longrightarrow}C_i$]{} such that the following diagram homotopy commutes $$\label{D1}
\diagram
C^{\prime}_{i}\dto^{h^{\prime}}\rto^{j_{i}}
&C_{i}\dto^{h}\\
SU(n-m)\rto^{\tilde{j}}
&SU(n).
\enddiagram$$ Since $SU(n)$ and $SU(n-m)$ are homotopy associative $H$-spaces, and $\tilde{j}$ is an $H$-map, from the universal property of the James construction we obtain a homotopy commutative diagram $$\diagram
\Omega\Sigma C^{\prime}_{i}\dto^{\bar{h}^{\prime}}\rto^{\Omega\Sigma j_{i}}
&\Omega\Sigma C_{i}\dto^{\bar{h}}\\
SU(n-m)\rto^{\tilde{j}}
&SU(n),
\enddiagram$$ where $\bar{h}$ and $\bar{h}^{\prime}$ are $H$-maps extending the maps $h$ and $h^{\prime}$. Since $1\leq n\leq (p-1)(p-2)$, the space $C_{i}$ consists of less than $p-1$ odd dimensional cells. Thus we can apply Theorem \[T1\] to obtain an $H$-space $B_{i}=M(C_{i})$, a map [$C_{i}\stackrel{\iota}
{\longrightarrow}B_{i}$]{} that induces an inclusion of generating sets on homology, and a map [$B_{i}\stackrel{s}
{\longrightarrow}\Omega\Sigma C_{i}$]{} with a left homotopy inverse. Similarly we obtain an $H$-space $B^{\prime}_{i}=M(C^{\prime}_{i})$, and maps $\iota^{\prime}$ and $s^{\prime}$ with similar properties. The map [$C^{\prime}_{i}\stackrel{j_{i}}
{\longrightarrow}C_{i}$]{} induces an $H$-map [$B^{\prime}_{i}\stackrel{g_{i}}
{\longrightarrow}B_{i}$]{} via the functor $M$, and we have the following homotopy commutative diagram $$\label{D2}
\diagram
B^{\prime}_{i}\dto^{s^{\prime}}\rto^{g_{i}}
&B_{i}\dto^{s}\\
\Omega\Sigma C^{\prime}_{i}\dto^{\bar{h}^{\prime}}\rto^{\Omega\Sigma j_{i}}
&\Omega\Sigma C_{i}\dto^{\bar{h}}\\
SU(n-m)\rto^{\tilde{j}}
&SU(n),
\enddiagram$$ where the top square commutes because of the functorial property of the maps in Theorem \[T1\] $(ii)$. Using part $(iii)$ of Theorem \[T1\], $\bar{h}\circ s$ induces an inclusion of the generating set of [$H_{*}(B_{i})$]{} into the generating set of [$H_{*}(SU(n))$]{}. Similarly $\bar{h}^{\prime}\circ s^{\prime}$ induces an inclusion of the generating set of [$H_{*}(B^{\prime}_{i})$]{} into the generating set of [$H_{*}(SU(n-m))$]{}.
Taking the product of diagrams (\[D2\]) for every integer $i$ such that $1\leq i\leq (p-1)$, we obtain the following homotopy commutative diagram. $$\diagram
\prod^{p-1}_{i=1}B^{\prime}_{i}\dto^{\prod f^{\prime}_{i}}\rto^{\prod g_{i}}
&\prod^{p-1}_{i=1}B_{i}\dto^{\prod f_{i}}\\
\prod^{p-1}_{i=1}SU(n-m)\dto^{mult.}\rto^{\prod \tilde{j}}
&\prod^{p-1}_{i=1}SU(n)\dto^{mult.}\\
SU(n-m)\rto^{\tilde{j}}
&SU(n),
\enddiagram$$ where the left and right vertical compositions induce isomorphisms on the generating sets of the respective homology rings. Dualizing to mod-$p$ cohomology, both vertical compositions induce algebra maps that are isomorphisms on generating sets, so they both induce isomorphisms on mod-$p$ cohomology. Therefore both vertical compositions in the above diagram are homotopy equivalences.
Finally, for each $1\leq i\leq p-1$, let $\bar{A}_{i}$ be the cofibre of the inclusion [$C^{\prime}_{i}\stackrel{j_{i}}
{\longrightarrow}C_{i}$]{}. Then $\bar{A}_{i}$ consists of no more than $p-2$ odd dimensional cells. Applying Proposition \[T1B\] to the cofibration sequence [$C^{\prime}_{i}\stackrel{j}
{\longrightarrow}C_{i}\stackrel{}{\longrightarrow}\bar{A}_i$]{} for each integer $i$, we obtain fibration sequences $${\ensuremath{B^{\prime}_{i}\stackrel{g_{i}}
{\longrightarrow}B_{i}\stackrel{}{\longrightarrow}D_{i}}},$$ where $D_{i}=M(\bar{A}_{i})$.
\[R1\]
Notice that the spaces $\bar{A}_{i}$ such that $D_{i}=M(\bar{A}_{i})$ are (with indices rearranged) precisely the summands in the wedge decomposition of $\Sigma\mathbb{C}P^{n-1}_{m}$ in Corollary \[C1\].
We now prove one of our main theorems.
Applying Theorem \[T2\] we obtain a diagram of fibration sequences $$\label{D6}
\diagram
\prod^{p-1}_{i=1}\Omega D_{i}\dto^{\ell}\rto^{}
&\prod^{p-1}_{i=1}B^{\prime}_{i}\dto^{\simeq}\rto^{\prod g_{i}}
&\prod^{p-1}_{i=1}B_{i}\dto^{\simeq}\\
\Omega W_{n,m}\rto^{}
&SU(n-m)\rto^{\tilde{j}}
&SU(n),
\enddiagram$$ for some induced map of fibres $\ell$. This diagram implies that the map $\ell$ is a homotopy equivalence by the $5$-lemma.
Real Stiefel Manifolds {#sReal}
----------------------
Where localized at an odd prime $p$, there is a difference in the homology of $SO(n)$ when $n$ is even as opposed to odd. That is, we have homology isomorphisms $$\label{eHlgy1}
{\ensuremath{H_{*}(SO(2k+1))}}\cong\Lambda(x_{3},x_{7},...,x_{4k-1})$$ and $$\label{eHlgy2}
{\ensuremath{H_{*}(SO(2k))}}\cong\Lambda(x_{3},x_{7},...,x_{4k-5},\bar{x}_{2k-1}).$$ The inclusion of $(n-m)$-frames into $n$-frames [$SO(n-m)\stackrel{\tilde{j}}
{\longrightarrow}SO(n)$]{} induces on homology the algebra map that sends each generator $x_{i}\in{\ensuremath{H_{*}(SO(n-m))}}$ to the corresponding generator $x_{i}\in{\ensuremath{H_{*}(SO(n))}}$, and if $n-m$ is even, the generator $\bar{x}_{n-m-1}\in{\ensuremath{H_{*}(SO(n-m))}}$ is mapped trivially.
For $n=2k$ it is well known (Theorem $6.5$ in reference [@MT]) that there exists a decomposition $$\label{eDecomp}
SO(2k)\simeq S^{2k-1}\times SO(2k-1).$$
Harris [@Harris] showed there are decompositions $$\label{eHarris}
SU(2k)\simeq SO(2k+1)\times (SU(2k)/Sp(k))$$
that are natural with respect to the inclusions [$SO(2(k-k')+1)\stackrel{\tilde{j}}
{\longrightarrow}SO(2k+1)$]{} and [$SU(2(k-k'))\stackrel{\tilde{j}}
{\longrightarrow}SU(2k)$]{} for $k'\leq k$. With this we can prove the following homotopy decomposition as an application of Theorem \[T2\]. A general form of this decomposition was found by Mimura, Nishida, and Toda [@MNT2], but The same advantages hold in our decomposition as was the case for the special unitary groups $SU(n)$ in the previous section.
\[T3\] Fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)+1$, and let $r={\ensuremath{\left\lfloor \frac{p-1}{2} \right\rfloor}}$. Then there exists a homotopy commutative diagram of product decompositions $$\label{eSquare}
\diagram
X^{\prime}\times\prod^{r}_{i=1}\mathcal{B}^{\prime}_{i}\dto^{\simeq}\rto^{\bar{g}\times\prod g_{i}}
&X\times\prod^{r}_{i=1}\mathcal{B}_{i}\dto^{\simeq}\\
SO(n-m)\rto^{\tilde{j}}
&SO(n)
\enddiagram$$ such that the following properties hold.
[$H_{*}(\mathcal{B}_{i})$]{}$\cong\Lambda(x_{2i+1},x_{(2i+1)+2q},...,x_{(2i+1)+2k_{i}q})$, where $k_{i}$ is the largest integer such that $(2i+1)+2k_{i}q\leq 2n-3$;
[$H_{*}(\mathcal{B}^{\prime}_{i})$]{}$\cong\Lambda(x_{2i+1},x_{(2i+1)+2q},...,x_{(2i+1)+2k^{\prime}_{i}q})$, where $k^{\prime}_{i}$ is the largest integer such that $(2i+1)+2k^{\prime}_{i}q\leq 2(n-m)-3$;
There exist fibrations [$\mathcal{B}^{\prime}_{i}\stackrel{f_i}
{\longrightarrow}\mathcal{B}_{i}\stackrel{}{\longrightarrow}\mathcal{D}_{i}$]{}, ${\ensuremath{H_{*}(\mathcal{D}_{i})}}\cong\Lambda(x_{(2i+1)+2(k^{\prime}_{i}+1)q},...,x_{(2i+1)+2k_{i}q})$, and $\mathcal{D}_{i}$ is an $H$-space;
The map [$X^{\prime}\stackrel{\bar{g}}
{\longrightarrow}X$]{} is the trivial map;
If $n-m$ is even, $X^{\prime}=S^{n-m-1}$, and if $n-m$ is odd, then $X^{\prime}$ is a point;
If $n$ is even, $X=S^{n-1}$, and if $n$ is odd, then $X$ is a point.
Throughout this proof let us fix $n$ and $n-m$ both odd, and $0<m<n\leq (p-1)(p-2)$. Let $r={\ensuremath{\left\lfloor \frac{p-1}{2} \right\rfloor}}$. Recall from Theorem \[T2\] the decompositions of the special unitary groups $SU(n-1)$ and $SU(n-m-1)$ - as products of $p-1$ factors $B^{\prime}_{i}$ and $B_{i}$ respectively - and recall the homology of each of the factors in these decompositions. Restricting to the odd factors, we have a homotopy commutative square $$\diagram
\prod^{r}_{i=1}B^{\prime}_{2i-1}\dto^{}\rto^{\prod g_{2i-1}}
&\prod^{r}_{i=1}B_{2i-1}\dto^{}\\
SU(n-m-1)\rto^{\tilde{j}}
&SU(n-1).
\enddiagram$$ Since $n$ and $n-m$ are odd, $SO(n-m)$ and $SO(n)$ are retracts of $SU(n-m-1)$ and $SU(n-1)$. The naturality of this retraction implies we have the following homotopy commutative square $$\label{eD1}
\diagram
\prod^{r}_{i=1}B^{\prime}_{2i-1}\dto^{\simeq}_{\phi'}\rto^{\prod g_{2i-1}}
&\prod^{r}_{i=1}B_{2i-1}\dto^{\simeq}_{\phi}\\
SO(n-m)\rto^{\tilde{j}}
&SO(n),
\enddiagram$$ where we observe that the vertical maps induce isomorphisms on homology, so they are homotopy equivalences. We complete the proof for $n$ and $n-m$ both odd by setting $\mathcal{B}_{i}=B_{2i-1}$, $\mathcal{B}^{\prime}_{i}=B_{2i-1}$, $f_{i}=g_{2i-1}$, $\mathcal{D}_{i}=D_{2i-1}$, and applying Theorem \[T2\].
To complete the proof for the other cases, we keep $n$ and $n-m$ odd. For convenience set $f=\prod^{r}_{i=1} f_{i}$, $A=\prod^{r}_{i=1}\mathcal{B}_{i}$, and $A^{\prime}=\prod^{r}_{i=1}\mathcal{B}^{\prime}_{i}$. On homology [$SO(n-m+1)\stackrel{\tilde{j}}
{\longrightarrow}SO(n-m+2)$]{} sends the generator $\bar{x}_{n-m}$ trivially, so the homology Serre exact sequence for the fibration sequence ${\ensuremath{\Omega S^{n-m+1}\stackrel{\delta}
{\longrightarrow}SO(n-m+1)\stackrel{\tilde{j}}{\longrightarrow}SO(n-m+2)
\stackrel{\pi}{\longrightarrow}S^{n-m+1}}}$ implies $\delta_{*}$ sends the bottom generator of [$H_{*}(\Omega S^{n-m+1})$]{} to $c\cdot\bar{x}_{n-m}$ for some integer $c$ prime to $p$. Thus the Hurewicz image of the composition $\iota\colon{\ensuremath{S^{n-m}\stackrel{E}
{\longrightarrow}\Omega S^{n-m+1}\stackrel{\delta}{\longrightarrow}SO(n-m+1)}}$ is $c\cdot\bar{x}_{n-m}$. By exactness of the homotopy long exact sequence $\tilde{j}\circ\iota$ is null homotopic, implying the composition [$S^{n-m}\stackrel{\iota}
{\longrightarrow}SO(n-m+1)\stackrel{\tilde{j}}{\longrightarrow}SO(n)$]{} is also nullhomotopic. Since [$A^{\prime}\stackrel{\phi'}
{\longrightarrow}SO(n-m)$]{} is a homotopy equivalence, [$SO(n-m)\stackrel{\tilde{j}}
{\longrightarrow}SO(n-m+1)$]{} induces an inclusion of algebras on homology, and the Hurewicz image of $\iota_{*}$ is $c\cdot\bar{x}_{n-m}$, then the composition $\theta'=\iota\cdot(\tilde{j}\circ\phi')\colon{\ensuremath{S^{n-m}\times A^{\prime}\stackrel{}
{\longrightarrow}SO(n-m+1)}}$ induces an isomorphism on homology, and so it is a homotopy equivalence. Similarly we have a map [$S^{n}\stackrel{\iota}
{\longrightarrow}SO(n+1)$]{} whose Hurewicz image is $d\cdot\bar{x}_{n-m}$ for some $d$ prime to $p$. Thus $\theta=\iota\cdot(\tilde{j}\circ\phi)\colon{\ensuremath{S^{n}\times A\stackrel{}
{\longrightarrow}SO(n+1)}}$ is a homotopy equivalence.
Taking products we obtain the following homotopy commutative diagram $$\label{eD4}
\diagram
S^{n-m}\times A^{\prime}\dto^{\iota\times\phi'}\rto^{*\times f}
&*\times A\dto^{*\times\phi}\\
(SO(n-m+1))^{2}\dto^{mult.}\rto^(0.6){\tilde{j}\times\tilde{j}}
&(SO(n))^{2}\dto^{mult.}\\
SO(n-m+1)\rto^(0.6){\tilde{j}}
&SO(n),
\enddiagram$$ where the bottom square commutes since $\tilde{j}$ is an $H$-map. Consider the following diagram $$\label{eD5}
\diagram
A^{\prime}\dto^{\simeq}_{\phi'}\rto^{*\times\mathbbm{1}}
&S^{n-m}\times A^{\prime}\dto^{\simeq}_{\theta'}\rto^(0.6){*\times f}
&A\dto^{\simeq}_{\phi}\rto^(0.4){*\times\mathbbm{1}}
&S^{n}\times A\dto^{\simeq}_{\theta}\\
SO(n-m)\rto^{\tilde{j}}
&SO(n-m+1)\rto^(0.6){\tilde{j}}
&SO(n)\rto^(0.4){\tilde{j}}
&SO(n+1).
\enddiagram$$ The proof will be complete if this diagram homotopy commutes. Here the left and right squares homotopy commute by the construction of $\theta$ and $\theta'$, and the middle square is the outer part of the diagram in (\[eD4\]).
Theorem \[T3\] allows us to decompose the loop spaces of low rank real Stiefel manifolds $V_{n,m}=O(n)/O(n-m)$ as follows.
\[MAIN2\] Fix integers $n$ and $m$ such that $0<m<n\leq(p-1)(p-2)+1$. Let $r={\ensuremath{\left\lfloor \frac{p-1}{2} \right\rfloor}}$. Then there exists a product decomposition $$\Omega V_{n,m}\simeq X^{\prime}\times\Omega X\times\displaystyle\prod^{r}_{i=1} \Omega\mathcal{D}_{i}$$ where each $\mathcal{D}_{i}$ is the $H$-space $D_{2i-1}$ from Theorem \[T3\], and
If $n-m$ is even, $X^{\prime}=S^{n-m-1}$, and if $n-m$ is odd, then $X^{\prime}$ is a point;
If $n$ is even, $X=S^{n-1}$, and if $n$ is odd, then $X$ is a point.
The proof is similar to that of Theorem \[MAIN1\]. Applying the diagram in (\[eSquare\]) from Theorem \[T3\], and noting that $\bar{g}$ is the trivial map, we obtain a diagram of fibration sequences $$\diagram
X^{\prime}\times\Omega X\times\prod^{r}_{i=1}\Omega\mathcal{D}_{i}\dto^{\ell}\rto^{}
&X^{\prime}\times\prod^{r}_{i=1}\mathcal{B}^{\prime}_{i}\dto^{\simeq}\rto^{\bar{g}\times\prod g_{i}}
&X\times\prod^{r}_{i=1}\mathcal{B}_{i}\dto^{\simeq}\\
\Omega V_{n,m}\rto^{}
&SO(n-m)\rto^{\tilde{j}}
&SO(n)
\enddiagram$$ for some induced map of fibres $\ell$. Since the middle and right vertical maps are homotopy equivalences, the map $\ell$ is a homotopy equivalence by the $5$-lemma. Finally as we saw in the proof of Theorem \[T3\], $\mathcal{D}_{i}=D_{2i-1}$ where each $D_{2i-1}$ is one the $H$-spaces from Theorem \[MAIN1\].
Symplectic Stiefel Manifolds {#sSymp}
----------------------------
Harris [@Harris] showed that localized at odd primes $p$, there is a natural homotopy equivalence $$Sp(n)\simeq Spin(2n+1),$$ where the *spinor group* $Spin(2n+1)$ is the simply connected cover of $SO(2n+1)$. Since (integrally) we have $\pi_{1}(SO(2n+1))\cong {\ensuremath{\mathbb{Z}_{2}}}$, then $\pi_{1}(SO(2n+1))=0$ when localized at an odd prime $p$. Thus there is a natural $p$-local homotopy equivalence $$Spin(2n+1)\simeq SO(2n+1).$$
With this information we can use Theorem \[T3\] to decompose $Sp(n)$ when $n<\frac{1}{2}(p-1)(p-2)$. In a similar manner as before we decompose the loop spaces of low rank symplectic Stiefel manifolds $X_{n,m}=Sp(n)/Sp(n-m)$. This is stated as follows.
\[MAIN3\] Fix integers $k$ and $j$ such that $0<j<k\leq\frac{1}{2}(p-1)(p-2)$. Let $r={\ensuremath{\left\lfloor \frac{p-1}{2} \right\rfloor}}$. Then there exists a product decomposition $$\Omega X_{k,j}\simeq \displaystyle\prod^{r}_{i=1} \Omega\mathcal{D}_{i}$$ where each $\mathcal{D}_{i}$ is the $H$-space $D_{2i-1}$ from Theorem \[T3\], for $n=2k+1$ and $m=2j$. $~{\hfill\square}$
Exponents
=========
As an application of our decompositions of $\Omega W_{n,m}$ we compute upper bounds for the $p$-exponents of $W_{n,m}$ in the range $0<m<n\leq (p-1)(p-2)$. The $p$-exponents in the stable range $0<m\leq (p-1)(p-2)$ and $2m<n$ will also be considered, though using different methods.
Recall that the integral James number $U(n,m)$ of $W_{n,m}$ is defined as the degree of the map [$\mathbbm{Z}
{\longrightarrow}\mathbbm{Z}$]{} induced by the projection [$W_{n,m}
{\longrightarrow}W_{n,1}=S^{2n-1}$]{} on $\pi_{2n-1}$, and the $p$-local James number $U_{(p)}(n,m)$ is the $p$-component of $U(n,m)$. The proof of part $(1)$ of the following proposition can be found in Proposition $(7.2)$ of [@Beben], and Proposition $(6.3)$ of [@MNT2]. Part $(2)$ is an easy consequence of part $(1)$, and can be found in Theorem $(7.1)$ of [@Beben], or with the use of $K$-theory in [@Crabb].
\[tJN\] Let the space $A$ be a summand in the splitting of a suspended stunted complex projective space in Corollary \[C1\]. Suppose $A$ has $l<p-1$ cells, with the bottom cell in dimension $2r+1$, and hence the top cell in dimension $2r+1+(l-1)q$. Let $J$ be the unique integer in the range $0\leq J\leq p-1$ such that $r+J(p-1)$ is divisible by $p$, and take the map [$M(A)\stackrel{\tilde{\nu}}
{\longrightarrow}S^{2r+1+(l-1)q}$]{} induced by the quotient [$A\stackrel{\nu}
{\longrightarrow}S^{2r+1+(l-1)q}$]{}.
If $l-1\leq J$, then $\tilde{\nu}$ induces a degree $p^{l-1}$ from [${\ensuremath{\mathbb{Z}_{(p)}}}{\longrightarrow}{\ensuremath{\mathbb{Z}_{(p)}}}$]{} on $\pi_{2r+1+(l-1)q}$. Otherwise if $l-1=J+1$, then $\tilde{\nu}$ induces a degree $p^{t}$ for some integer $0\leq t\leq l-2$, and if $l-1>J+1$, then $\tilde{\nu}$ induces a degree $p^{t}$ for some integer $1\leq t\leq l-2$.
Fix $0<m\leq (p-1)(p-2)$ and assume either $2m<n$ or $0<m<n\leq (p-1)(p-2)$. Pick $A$ to be the summand of $\Sigma\mathbb{C}P^{n-1}_{m}$ that has its top cell in dimension $2n-1$. Then the degree of $\tilde{\nu}_{*}$ on $\pi_{2n-1}$ is equal to the $p$-local James number $U_{(p)}(n,m)$.
Cconsequently, whenever there exists a cell of dimension $(2n-1-iq)$ in $\Sigma\mathbb{C}P^{n-1}_{m}$ such that $i>0$ and $(2n-1-iq)$ is divisible by $p$, then $U_{(p)}(n,m)\leq p^{l-2}$. Otherwise $U_{(p)}(n,m)= p^{l-1}$.
${\hfill\square}$
We use the following proposition, proven in [@T4].
\[MV\] Take a fibration [$F\stackrel{i}
{\longrightarrow}E\stackrel{r}{\longrightarrow}B$]{} with $r$ an $H$-map between the $H$-spaces $E$ and $B$. Suppose there exists a map [$s\colonB
{\longrightarrow}E$]{} such that the composition [$r\circ s\colonB
{\longrightarrow}B$]{} is a $p^{t}$-power map for some integer $t$. Then there exists a fibration $${\ensuremath{B\{t\}\stackrel{}
{\longrightarrow}F\times B\stackrel{}{\longrightarrow}E}},$$ where $B\{t\}$ is the homotopy fibre of the $p^{t}$-power map [$r\circ s\colonB
{\longrightarrow}B$]{}.$~{\hfill\square}$
The following lemma will be used to prove part of Theorem \[MAIN4\].
\[E1\] Let $A$ be a summand in the wedge decomposition of $\Sigma\mathbb{C}P^{n-1}_{m}$ in Corollary \[C1\], and let $J$ be the unique integer in the range $1\leq J\leq p$ such that $r+J(p-1)$ is divisible by $p$. If $l-1\leq J$, then $exp_{p}(M(A))\leq p^{r+(l-1)p}$. Otherwise if $l-1>J$, then $exp_{p}(M(A))\leq p^{r+(l-1)p-1}$.
We shall use $exp_{p}(S^{2k+1})=p^{k}$ and $exp_{p}(S^{2k+1}\{p^t\})=p^{t}$ (Cohen, Moore, and Neisendorfer [@CMN; @N3]), which holds for odd primes $p$ and all integers $k\geq 0$.
Suppose $l-1\leq J$. Fix some $k\leq l-1$ and let $A^{k}$ denote the $(2r+1+kq)$-skeleton of $A=A^{l-1}$. We proceed by induction by assuming that $exp_{p}(M(A^{k-1}))\leq p^{r+(k-1)p}$. The base case $k=1$ holds since $M(A^{0})=M(S^{2r+1})=S^{2r+1}$. For the induction step, note that because $A$ is a summand in the wedge decomposition of a suspended stunted complex projective space, so is its skeleton $A^{k}$. Then by Proposition \[tJN\] we have a map [$S^{2r+1+kq}\stackrel{\alpha}
{\longrightarrow}M(A^{k})$]{} such that the composition [$S^{2r+1+kq}\stackrel{\alpha}
{\longrightarrow}M(A^{k})\stackrel{\tilde{\nu}}{\longrightarrow}S^{2r+1+kq}$]{} is a degree $p^{k}$ map, where $\tilde{\nu}$ is induced by the quotient [$A^{k}\stackrel{\nu}
{\longrightarrow}S^{2r+1+kq}$]{}. Since we are localizing at an odd prime $p$, then $S^{2r+1+kq}$ is an $H$-space, and so this composition is also a $p^{k}$-power map. Applying Proposition \[MV\] to the fibration [$M(A^{k-1})\stackrel{}
{\longrightarrow}M(A^{k})\stackrel{\tilde{\nu}}{\longrightarrow}S^{2r+1+kq}$]{}, there is the following fibration. $${\ensuremath{S^{2r+1+kq}\{p^{k}\}\stackrel{}
{\longrightarrow}M(A^{k-1})\times S^{2r+1+kq}\stackrel{}{\longrightarrow}M(A^{k})}}.$$ So by the homotopy long exact sequence for this fibration and our inductive assumption $$\begin{aligned}
exp_{p}(M(A^{k}))\leq &
exp_{p}(S^{2r+1+kq}\{p^{k}\})\cdot max(exp_{p}(M(A^{k-1})),exp_{p}(S^{2r+1+kq}))\\
\leq &p^{k}\cdot max(p^{r+(k-1)p},p^{r+k(p-1)})\\
= &p^{k}\cdot p^{r+k(p-1)} = p^{r+kp},\end{aligned}$$ where $max(p^{r+(k-1)p},p^{r+k(p-1)})=p^{r+k(p-1)}$ since we assume $k\leq l-1<p-1$. Hence $exp_{p}(M(A))\leq p^{r+(l-1)p}$.
For the case $l-1>J$, the induction starts at the base case $k=J$, where we have shown that $exp_{p}(M(A^{J}))\leq p^{r+Jp}$. If $J<k\leq l-1$, then by Theorem \[tJN\] we have a map $\alpha$ such that the composition [$S^{2r+1+kq}\stackrel{\alpha}
{\longrightarrow}M(A^{k})\stackrel{\tilde{\nu}}{\longrightarrow}S^{2r+1+kq}$]{} is a $p^{k-1}$-power map. The rest of the induction is the same as the previous case.
Even though we failed to obtain analogous decompositions of $\Omega W_{n,m}$ for most choices of $n$ and $m$ in the stable range $m\leq (p-1)(p-2)$ and $2m<n$, fortunately there is a work-around. Together with Lemma \[tJN\], the following lemma allows us to calculate $p$-exponent bounds in this stable range. The results are similar to what could be achieved if such decompositions in reality existed:
\[lE2\] Fix $p-1<m\leq (p-1)(p-2)$ and $2m<n$. Let [$W_{n,m}\stackrel{\pi}
{\longrightarrow}W_{n,p-1}$]{} be the projection map. Then there exists a space $B$, a map [$\Omega B\stackrel{\alpha}
{\longrightarrow}\Omega W_{n,m}$]{}, and a homotopy equivalence [$\Omega W_{n,p-1}\stackrel{h}
{\longrightarrow}\Omega B$]{} such that the composition [$\Omega B\stackrel{\alpha}
{\longrightarrow}\Omega W_{n,m}\stackrel{\Omega\pi}{\longrightarrow}\Omega W_{n,p-1}
\stackrel{h}{\longrightarrow}\Omega B$]{} is a $p^{t}$-power map, and $p^{t}$ is equal to the maximum of the set of James numbers ${\ensuremath{\left\{ U_{(p)}(n-i,m-i)\,|\,0\leq i<p-1 \right\}}}$.
We have the following homotopy commutative diagram $$\diagram
\bigvee_{i=0}^{p-2}A_{i}\dto^{\simeq}\rto^{\vee q_{i}}
&\bigvee_{i=0}^{p-2}S^{2n-1-2i}\dto^{\simeq}\\
\Sigma\mathbb{C}P^{n-1}_{m}\dto^{}\rto^{q}
&\Sigma\mathbb{C}P^{n-1}_{p-1}\dto^{}\\
W_{n,m}\rto^{\pi}
&W_{n,p-1},
\enddiagram$$ where the vertical homotopy equivalences are due to Corollary \[C1\] (and we index so that $A_{i}$ has the $(2n-1-2i)$-cell in its top dimension), and the top vertical maps [$A_{i}\stackrel{q_{i}}
{\longrightarrow}S^{2n-1-2i}$]{} in the wedge are the quotient maps. Using the Hilton-Milnor theorem, $\prod_{i}\Omega A_{i}$ and $\prod_{i}\Omega S^{2n-1-2i}$ are retracts of $\Omega(\bigvee_{i}\Omega A_{i})$ and $\Omega(\prod_{i}\Omega S^{2n-1-2i})$, and these retractions are natural with respect to the map $\Omega\vee q_{i}$ (restricting to $\prod \Omega q_{i}$). Thus looping the above diagram one obtains $$\diagram
\prod_{i=0}^{p-2}\Omega A_{i}\dto^{}\rto^{\prod \Omega q_{i}}
&\prod_{i=0}^{p-2}\Omega S^{2n-1-2i}\dto^{\simeq}\\
\Omega W_{n,m}\rto^{\Omega\pi}
&\Omega W_{n,p-1}.
\enddiagram$$
In the stable range $p-1<m\leq (p-1)(p-2)$ and $2m<n$, the second part of Proposition \[tJN\] implies the multiplication induced by each [$A_{i}\stackrel{q_{i}}
{\longrightarrow}S^{2n-1-2i}$]{} on $\pi_{2n-1-2i}$ is equal to the multiplication induced by the projection [$W_{n-i,m-i}\stackrel{\pi}
{\longrightarrow}S^{2n-1-2i}$]{}. Hence for each integer $0\leq i<p-1$ we have maps [$S^{2n-1-2i}\stackrel{\beta_{i}}
{\longrightarrow}A_{i}$]{} such that each composition [$S^{2n-1-2i}\stackrel{\beta_{i}}
{\longrightarrow}A_{i}\stackrel{q_{i}}{\longrightarrow}S^{2n-1-2i}$]{} is a degree $p^t$ map, where $p^{t}=\max{\ensuremath{\left\{ U_{(p)}(n-i,m-i)\,|\,0\leq i<p-1 \right\}}}$. Since odd spheres are $p$-local $H$-spaces, the loopings of these compositions are $p^t$-power maps. The lemma follows by setting $B=\prod_{0\leq i<p-1}S^{2n-1-2i}$.
\[rProd\] In the proof of Lemma \[lE2\] we showed $\Omega W_{n,p-1}\simeq \prod_{0\leq i<p-1}\Omega S^{2n-1-2i}$. With a similar argument one can show $\Omega W_{n,m}\simeq \prod_{0\leq i<m}\Omega S^{2n-1-2i}$ when $m\leq p-1$, which reproduces a specific case of a more general result due to Kumpel [@Kumpel]. Thus $\exp_{p}(W_{n,m})=p^{n-1}$ when $m\leq p-1$.
We now prove Theorem \[MAIN4\].
Let us first consider the case $0<m<n\leq (p-1)(p-2)$. By Theorem \[MAIN1\] and Remark \[R1\] we have the product decomposition $\Omega W_{n,m}\simeq\prod^{p-1}_{i=1}\Omega M(A_{i})$, where each $A_{i}$ is a summand in the wedge decomposition of $\Sigma\mathbb{C}P^{n-1}_{m}$ in Corollary \[C1\], and we index so that $A_{i}$ has the $(2(n-m+i)-1)$-cell in its bottom dimension when $i\leq m$, and is trivial if $i>m$. Therefore $$exp_{p}(W_{n,m})= max\{exp_{p}(M(A_{i}))|1\leq i\leq p-1\}.$$ Let $t_{i}$ be the number of cells in $A_{i}$. By Lemma \[E1\] we have the exponent bounds $$\label{Bound}
exp_{p}(M(A_{i}))\leq p^{n-m+i-1+(t_{i}-1)p}.$$ We see that this exponent bound is the greatest when $j$ is the integer such that $A_{j}$ has the $(2n-1)$-cell in its top dimension. Therefore $exp_{p}(W_{n,m})\leq p^{n-1+(t_{j}-1)}$. Note that $t_{j}=k$, where $k$ is the number of cells in $\Sigma\mathbb{C}P^{n-1}_{m}$ that are in dimensions of the form $(2n-1-iq)$ for $0\leq i<p-1$. Hence $exp_{p}(W_{n,m})\leq p^{n-1+(k-1)}$.
When $A_{j}$ has a cell in a dimension divisible by $p$, then Lemma \[E1\] implies the bound can be improved to $exp_{p}(M(A_{j}))\leq p^{n-1+(t_{j}-2)}$. Still this bound is at least as large as all the bounds in (\[Bound\]) for $i\neq j$, though possibly no longer strictly as large. Therefore $exp_{p}(W_{n,m})\leq p^{n-1+(t_{j}-2)}=p^{n-1+(k-2)}$ in this case.
For the last case take $2m<n$ and $0<m\leq (p-1)(p-2)$. If $m\leq p-1$, then by Remark \[rProd\]$\exp_p(W_{n,m})=p^{n-1}$ and we are done. So let us assume $m>p-1$. Note there exists a fibration $${\ensuremath{W_{n-(p-1),m-(p-1)}\stackrel{}
{\longrightarrow}W_{n,m}\stackrel{\pi}{\longrightarrow}W_{n,p-1}}}.$$ By Lemma \[lE2\] there is a space $B$ and a homotopy equivalence [$\Omega W_{n,p-1}\stackrel{h}
{\longrightarrow}\Omega B$]{} such that the composition $${\ensuremath{\Omega B\stackrel{\alpha}
{\longrightarrow}\Omega W_{n,m}\stackrel{\Omega\pi}{\longrightarrow}\Omega W_{n,p-1}
\stackrel{h}{\longrightarrow}\Omega B}}$$ is a $p^{t}$-power map, and $p^{t}$ is equal to the maximum of the set of James numbers $${\ensuremath{\left\{ U_{(p)}(n-j,m-j)\,|\,0\leq j<p-1 \right\}}}.$$ Since $2m<n$, then $2(m-j)<n-j$, and an upper bound for each of the James numbers in this set are known by Theorem \[tJN\]. That is, $$\label{setOfBounds}
U_{(p)}(n-j,m-j)\leq p^{t_{j}-1}$$ where $t_{j}$ is the number of cells in $\Sigma\mathbb{C}P^{n-1-j}_{m-j}$ for dimensions of the form $(2(n-j)-1-iq)$. Therefore the maximum of the bounds in (\[setOfBounds\]) happens when $j=0$, implying $p^{t}\leq p^{t_{0}-1}$.
Now take the following homotopy commutative diagram of homotopy fibrations $$\label{presidentsChoice}
\diagram
W_{n-(p-1),m-(p-1)}\rto^{}\dto^{\ell}
&W_{n,m}\ddouble\rto^{\pi}
&W_{n,p-1}\dto^{h}\\
F\rto^{}
&W_{n,m}\rto^{f}
&B
\enddiagram$$ where the map $f$ is the composition $h\circ\pi$, and $F$ is the homotopy fibre of $f$. Since the middle and right vertical maps are homotopy equivalences, the lift $\ell$ is also a homotopy equivalence by the $5$-lemma. Now applying Proposition \[MV\] to the bottom fibration, and using the homotopy equivalences in (\[presidentsChoice\]), we obtain the bound $$\label{expbound2}
\exp_{p}(W_{n,m})\leq p^{t_{0}-1}\cdot \max(\exp_{p}(W_{n-(p-1),m-(p-1)}), \exp_{p}(W_{n,p-1})).$$
Repeat the above argument to get bounds $$\label{expbound3}
\exp_{p}(W_{n-j(p-1),m-j(p-1)})\leq p^{t_{0,j}-1}\cdot \max(\exp_{p}(W_{n-(j+1)(p-1),m-(j+1)(p-1)}), \exp_{p}(W_{n-j(p-1),p-1}))$$ where $m-(j+1)(p-1)>0$ and $t_{0,j}$ is the number of cells in $\Sigma\mathbb{C}P^{n-1-j(p-1)}_{m-j(p-1)}$ in dimensions of the form $(2(n-j(p-1))-1-iq)$. Note $t_{0,0}=t_{0}$ and $t_{0,j+1}<t_{0,j}=t_{0,j+1}+1$. By Remark \[rProd\] we have $$\exp_{p}(W_{n-j(p-1),p-1})=p^{n-1-j(p-1)}.$$
We induct on the bound in (\[expbound3\]) starting with the base case $j=t_{0}-1$, where $0<m-(t_{0}-1)(p-1)\leq p-1$, and then apply Remark \[rProd\]. The inductive assumption is $$\exp_{p}(W_{n-(j+1)(p-1),m-(j+1)(p-1)})\leq p^{n-1-(j+1)(p-1)+(t_{0,j+1}-1)}.$$
Since $0<m\leq (p-1)(p-2)$, $t_{0,j+1}<t_{0}\leq p-2$, and so $$\exp_{p}(W_{n-(j+1)(p-1),m-(j+1)(p-1)})< \exp_{p}(W_{n-j(p-1),p-1}).$$
Then using the bound in (\[expbound3\]) $$\exp_{p}(W_{n-j(p-1),m-j(p-1)})\leq p^{t_{0,j}-1}\cdot \exp_{p}(W_{n-j(p-1),p-1})=p^{n-1-j(p-1)+(t_{0,j}-1)}.$$
Therefore by induction $$\exp_{p}(W_{n,m})\leq p^{n-1+(t_{0}-1)}.$$
We finish off by giving analogous exponent bounds for real and symplectic Stiefel manifolds. These follow from the decompositions in Theorems \[MAIN2\] and \[MAIN3\], and the same argument used to prove Theorem \[MAIN4\].
\[MAIN5\] Fix $0<m<n\leq (p-1)(p-2)+1$ and let $n$ be odd. Let $k$ be the number of cells in $\Sigma\mathbb{C}P^{n-2}_{m}$ that are in dimensions of the form $(2n-3-iq)$ for $0\leq i<p-1$. Then $$exp_{p}(V_{n,m})\leq p^{n-2+(k-1)}$$ and $$exp_{p}(V_{n+1,m})\leq p^{n-2+(k-1)}.$$
Furthermore, if $k>1$ and there exists a cell of dimension $(2n-3-iq)$ in $\Sigma\mathbb{C}P^{n-2}_{m}$ such that $i>0$ and $(2n-3-iq)$ is divisible by $p$, then $$exp_{p}(V_{n,m})\leq p^{n-2+(k-2)}$$ and $$exp_{p}(V_{n+1,m})\leq p^{n-2+(k-2)}.$$ $~{\hfill\square}$
\[MAIN6\] Fix $0<j<k\leq \frac{1}{2}(p-1)(p-2)$ and let $n=2k+1$ and $m=2j$. Let $k$ be the number of cells in $\Sigma\mathbb{C}P^{n-2}_{m}$ that are in dimensions of the form $(2n-3-iq)$ for $0\leq i<p-1$. Then $$exp_{p}(X_{k,j})\leq p^{n-2+(k-1)}.$$
Furthermore, if $k>1$ and there exists a cell of dimension $(2n-3-iq)$ in $\Sigma\mathbb{C}P^{n-2}_{m}$ such that $i>0$ and $(2n-3-iq)$ is divisible by $p$, then $$exp_{p}(X_{k,j})\leq p^{n-2+(k-2)}.$$ $~{\hfill\square}$
|
---
abstract: 'In this work we provide a non-perturbative description of the phenomenon of dynamical mass generation in the case of quantum electrodynamics in $2+1$ dimensions. We will use the Kugo-Ojima-Nakanishi formalism to conclude that the physical Hilbert space of the asymptotic photon field is the same as that of the Maxwell-Chern-Simons.'
author:
- |
G. B. de Gracia[^1], B. M. Pimentel [^2], L. Rabanal [^3]\
*[Instituto de Física Teórica (IFT), Universidade Estadual Paulista (UNESP)]{}*\
*[Rua Dr. Bento Teobaldo Ferraz, 271, Bloco II, Barra Funda]{}*\
*[CEP 01140-070-São Paulo, SP, Brazil]{}*\
title: |
**Dynamical mass generation in QED$_3$:\
A non-perturbative approach**
---
Introduction
============
It is widely known that quantum electrodynamics in $2+1$ dimensions (QED$_3$) has important applications in condensed matter physics. The predominant example is the quantum Hall effect (QHE), where a pure topological Chern-Simons (CS) term [@hall] is commonly added to model the response of the quantum Hall ground state to low energy perturbations as an effective theory [@hall1], but it can also be used to study the behavior of ultracold matter in optical lattices [@cond]. Nevertheless, this theory also has outstanding properties from the theoretical point of view.
What is special about $2+1$ spacetime dimensions? Let us consider first the theory in the absence of fermions. By naive dimensional analysis we note a big difference with respect to the $3+1$ case. The vector potential $A_{\mu}$ has dimension 1 (in units of mass) in any $d$-dimensional spacetime. As a consequence, if we write the Lagrangian in the form $$\mathcal{L}_{QED_d} = -\frac{1}{4e^2}F_{\mu\nu}F^{\mu\nu} + A_{\mu}J^{\mu},
\label{qedaction}$$ then we realize that the coupling constant $e^2$ is dimensionless in $3+1$ but dimensionful in other dimension $d \ne 3+1$. In particular, in $d = 2+1$, the effective dimensionless coupling would be $e'^2 = e^2/E$, where $E$ is the energy scale. In the ultraviolet (UV) regime, $E$ tends to infinity and the coupling $e'$ goes to zero implying that the theory is superrenormalizable and always asymptotically free, i.e., this theory describes *free photons in the UV*. In this sense, the UV does not matter at all. On the other hand, the theory is always *strongly coupled in the infrared* (IR) because $e' \rightarrow \infty$ as $E\rightarrow 0$. Consequently, the IR limit of the theory becomes a playground for developing ideas to tackle more realistic problems as confinement in quantum chromodynamics (QCD) [@herbut; @grignani1; @grignani2] or gapped boundary phases in topological insulators (TI) [@seibergwitten].
Another interesting property of the theory in this dimensionality is related to the existence of magnetic monopoles. Whenever we have a $U(1)$ gauge field we have a new current
$$\mathcal{J}^{\mu} \propto \epsilon^{\mu\nu\rho}F_{\nu\rho},
\label{TopCurr}$$
which is identically conserved without imposing the equations of motion, i.e., it is not a Noether current. Its conservation is equivalent to the Bianchi identity $dF = 0$, where $F$ is the two-form field strength. This follows simply by the symmetry of partial derivatives which contributes to zero when contracted with a Levi-Civita symbol if $A_\mu$ is globally well-defined. A natural question is: Who is charged under the charge $$\mathcal{Q} = \int d^2x \mathcal{J}^0?
\label{TopCharg}$$ If we replace (\[TopCurr\]) in (\[TopCharg\]) we obtain that $\mathcal{Q}$ is equal to a magnetic flux from which we conclude that *magnetic monopoles are charged under* $\mathcal{Q}$. This charge is known as the vortex charge because Abrikosov-Nielsen-Olesen (ANO) vortices carry it when the theory is put in the Higgs phase [@borokhov]. Moreover, *the vector potential* $A_{\mu}$ *can be dualized to a free scalar* $\sigma$ *in the UV*. It is known as the dual photon field. The construction of the dual theory is carried out analogously as the electric-magnetic duality of Maxwell theory in $3+1$ spacetime dimensions, namely, $$Z = \int \mathcal{D}A_{\mu} \exp\left(-\int_x\frac{F^2}{4e^2}\right) \rightarrow \int \mathcal{D}\sigma\mathcal{D}F_{\mu\nu} \exp\left[\int_x \left(-\frac{F^2}{4e^2} + \frac{i}{4\pi} \sigma \epsilon^{\mu\nu\rho}\partial_{\mu}F_{\nu\rho}\right)\right],$$ where the dual photon $\sigma$ has been introduced as a Lagrange constraint in order to be able to treat the field strength as the integration variable [@polchinski]. After integrating out the field strength through its equation of motion we obtain $$Z_{\text{dual}} = \int D\sigma \exp\left( -\int_x \frac{e^2}{8\pi^2}(\partial\sigma)^2 \right).
\label{DualPhotonPathIntegral}$$ It can be shown straightforwardly that the conserved Noether current of this dual theory under the shift symmetry $\sigma \rightarrow \sigma + \text{const}$, coincides with the current (\[TopCurr\]). This in turn implies that $F_{\alpha\beta} \propto \epsilon_{\alpha\beta\mu}\partial^{\mu}\sigma$. Consequently, $\partial^{\alpha}F_{\alpha\beta} = 0$ and the theory describes free photons in accordance with our previous discussion of the UV.
We can also add to the action (\[qedaction\]) Chern-Simons (CS) or topological terms. Although, these terms do not described any dynamics and have zero degrees of freedom, they can have effects on the degeneracy of the ground state of the theory with interesting consequences [@chen]. When added, the theory is known as Maxwell-Chern-Simons (MCS) theory and it is gapped, i.e., *the photon is massive*. After having understood that the theory is strongly coupled in the IR, we could, effectively, drop out the Maxwell term and conclude that *the theory is a topological quantum field theory* (TQFT) *in the IR limit* [@dunne].
Now, interesting things start happening when matter (either fermions, bosons or both) is taken into consideration. In the above-mentioned effective description of the theory in the IR, we can write $$\mathcal{L} = \mathcal{L}_{\text{CS}} + \mathcal{L}_{\text{Fermion}} \quad \text{or} \quad \mathcal{L} = \mathcal{L}_{\text{CS}} + \mathcal{L}_{\text{Scalar}},$$ because the Maxwell term disappears. Obviously, *all dynamics arise from matter*. However, they are no longer TQFT but *believed to be*[^4] non-trivial conformal field theories (CFT) when their masses are tune to zero in a IR fixed point. If this were true, there is a possibility of studying topological changing phase transitions by relevant deformations, e.g., mass deformations, between TQFT’s, $$TQFT_1 \xleftarrow{\text{Relev. Deform.}} CFT \xrightarrow{\text{Relev. Deform.}} TQFT_2.$$ Yet, there is no free lunch. There is a subtlety with massive fermions in $2+1$ dimensions. Their path integral description presents *parity anomaly*, that is, parity is a symmetry at the classical level but is not at quantum level [@witten]. Among the several ways this anomaly can arise, one can understand it through the $1$-loop term in the low energy approximation of the Euclidean path integration [@alvarez; @niemi; @redlich] $$-\frac{1}{2} \text{Tr} \left( \frac{1}{i\gamma_{\mu}\partial_{\mu} - m_e}\gamma_{\nu}A_{\nu} \frac{1}{i\gamma_{\mu}\partial_{\mu} - m_e}\gamma_{\delta}A_{\delta}\right) = \frac{1}{2}\int \frac{d^3p}{(2\pi)^3} A_{\mu}(p)\Gamma_{\mu\nu}(p,m_e)A_{\nu}(p),$$ with $$\Gamma_{\mu\nu}(p,m_e) = -\int \frac{d^3k}{(2\pi)^3} \frac{\text{Tr}\left[\gamma_{\mu}(\gamma_{\rho}(p_{\rho}+k_{\rho})+m_e)\gamma_{\nu}(-\gamma_{\delta}k_{\delta}-m_e)\right]}{\left[(p+k)^2+m_e^2\right]^2(k^2+m_e^2)^2}, \quad p \ll m_e.$$ At zero temperature, the contribution of the anomaly to the effective action (in Minkowski signature) is of the form $$S_{\text{eff}}[A,m_e]^{(T=0)} = \cdots + \frac{1}{2} \frac{1}{4\pi} \frac{m_e}{|m_e|} \int d^3x \ \epsilon^{\mu \nu \beta}A_\mu \partial_\nu A_\beta + \cdots,$$ whereas at finite temperature, after imposing the anti-periodic conditions of Dirac fermions $\psi(0,\boldsymbol{x}) = - \psi(\beta,\boldsymbol{x})$, we obtain $$S_{\text{eff}}[A,m_e]^{(T\ne0)} = \cdots + \frac{1}{2} \frac{1}{4\pi} \frac{m_e}{|m_e|} \tanh\left(\frac{|m_e|}{T}\right)\int_0^{1/T} dt\int d^2x \ \epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho} + \cdots.$$ Clearly, CS terms have arisen and the breaking of parity depends on the sign of $m$.
A detailed study of all the above-mentioned points and additional exact results in lattice models, e.g, weak duality [@kramers; @peskin], have led to the conjeture of the existence of a web of dualities in $2+1$ spacetime dimensions with possible connections to the realization of 3D bosonization [@webduality1; @webduality2]. Hence, this theory deserves to be investigated further.
This work investigates some properties of QED$_3$ within the covariant operator formalism of quantum field theory. We call it the Kugo-Ojima-Nakanishi (KON) formalism [@kugoojima; @Nakanishi; @Nak1; @Laut]. Firstly, we want to address the problem of a dynamical mass generation for the photon arising from the interaction with generic charged particles, that is, either bosons or fermions in a given specific representation. In fact, this phenomenon is expected to happen since in this dimension the appearance of a mass term is in accordance with the local symmetries of the theory if one considers a discrete symmetry breaking scenario, e.g., parity anomaly in the presence of massive fermions.
The standard way to gap the photon is considering the MCS theory from the outset. In other words, by “adding by hand” a bare topological mass term. However, we argue that this procedure is not necessary and that QED$_3$ per se provides us these terms *dynamically*. Under this perspective the conventional low energy quantum Hall effect field description [@mar; @tong] would arise naturally from the situation of bidimensional electrons interacting with initially massless photons. The interaction changes the dispersion relation of the photon and the electromagnetic correlations become to fall faster through the material medium.
This property of QED$_3$ was intensively studied within perturbation theory (PT). In this framework, however, it was uncertain whether a renormalized mass of the photon actually existed. If the Pauli-Villars regularization method was used, the photon could either acquire an effective mass or remain masless which, by themselves, are two contradictory results. In fact, this kind of problem appears in the conventional perturbation theory when the regularization techniques are wrongly applied. Nevertheless, in [@Pim2] it was shown that if Pauli-Villars regularization is correctly applied, no problem arises and the photon becomes massive. The controversy was finally completely solved (of course, only in PT) by using the causal perturbation theory [@Pim1] where by construction no regularization is needed.
This paper is organized as follows. In section 2 we consider, following the ideas of [@Nakanishi], the “pre-Maxwell-Chern-Simons model” to derive the non-perturbative two-point function of the gauge field, a “massive” combination of field operators, an asymtoptic constraint for the matter currents, and a general condition for the existence of a renormalized mass of the photon with arbitrary matter currents. In section 3 we compare our result with the one obtained from PT for the particular case of fermionic matter in bidimensional representation. Finally, in section 4, the asymtoptic structure is constructed revealing that our “massive” combination has indeed a dynamically generated massive character. The conclusions and the outlook are presented in section 5. The metric signature $+--$ is used throughout.
Effective mass of the photon in 2+1 dimensions
==============================================
Let us start with the following Lagrangian density within the KON formalism $$\mathcal{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu} + \frac{m}{4}\epsilon^{\mu \nu \rho}F_{\mu\nu}A_{\rho} + B\partial^\mu A_\mu + \frac{1}{2}\alpha B^2 + J^{\mu}A_{\mu} + \mathcal{L}_M.
\label{eq:LagrangianGeneralCurrentPMCS}$$ In the above expression, $\mathcal{L}_M$ is a generic matter Lagrangian density, $J^{\mu}$ is an *arbitrary* $2+1$ dimensional matter current that breaks parity and $B$ is an auxiliary field that keeps track of the gauge fixing condition via the gauge parameter $\alpha$. Needless to say, $\mathcal{L}$ is invariant under the gauge transformations $$\delta A_\mu(x) = \partial_\mu \Lambda(x), \qquad \Box \Lambda = 0, \qquad \delta B(x) = 0,$$ wherein $\Lambda$ is a c-number.
We are interested in the behavior of this theory in the limit $m\rightarrow 0$. In 3 + 1 spacetime dimensions without spontaneous symmetry breaking (SSB), the renormalized mass of the photon is constrained to vanish as the bare mass goes to zero. This follows by the Johnson’s theorem [@jon]. We want to follow this line of thought in 2 + 1 dimensions in order to show that in the limit $m\rightarrow 0$ a renormalized mass $m_r$ for the photon exists. It is the aim of this paper to derive a general mathematical expression for this statement (cf. (\[eq:MassRenormalization\])).
The Heisenberg equations of motion read $$\begin{aligned}
\partial^{\mu}A_{\mu} + \alpha B &= 0 \label{eq:GaugeCondition}\\
\partial_{\mu} F^{\mu\nu} + m\epsilon^{\nu\mu\beta}\partial_{\mu}A_{\beta} - \partial^{\nu} B &= -J^{\nu} \label{eq:EquationForCurrent}\\
\partial_{\mu}J^{\mu} = 0. \label{eq:CurrentConserv}\end{aligned}$$ Applying $\partial_{\nu}$ to (\[eq:EquationForCurrent\]) and using (\[eq:CurrentConserv\]) we determine the equation of motion for the $B$-field $$\Box B = 0.
\label{eq:B-Field}$$ Hence, as usual in the case of an Abelian theory, the subsidiary condition necessary to identify the physical space $\mathfrak{F}_{\text{phys}}$ is given by $$B^+(x) | \text{phys} \rangle = 0, \quad \forall | \text{phys} \rangle \in \mathfrak{F}_{\text{phys}}.$$
In order to give a non-perturbative description of the dynamical mass generation phenomenon, let us first determine the vacuum expectation values of the commutation relations of the Heisenberg fields $A_{\mu}$. Equal-time commutation relations, quantum equations of motion and symmetries is all what we need. Although, an exact answer for them in the presence of interactions is almost impossible, the spectral representation method helps us to extract valuable information. In particular, it guides the construction of the asymptotic fields of the theory which represent the in/out Fock spaces $\mathfrak{F}$[^5].
Since the matter current is gauge invariant it has vanishing projection with the auxiliary $B$-field, that is, $\left[ J^\mu(x), B(y) \right]=0$ or $J^{\mu}(x)|0\rangle \in \mathfrak{F}_{\text{phys}}$. From this, together with the sourced equations of motion and the zero norm character of $B(x)$, we find that (see Appendix) $$\bigg( \Box^x\eta^{\alpha\nu} + m\epsilon^{\alpha\mu\nu}\partial_{\mu}^x \bigg) \bigg( \Box^y\eta^{\beta\sigma} + m\epsilon^{\beta\mu\sigma}\partial_{\mu}^y \bigg) \big[ A_{\nu}(x), A_{\sigma}(y) \big] = \left[ J^{\alpha}(x), J^{\beta}(y) \right].
\label{RelationBetweenSpectralFunctions}$$ This result means that the spectral function for the full two-point function of the gauge field are related to the corresponding spectral function of the arbitrary matter current. In particular, the asymptotic structure of the latter imposes constraints on the former. A useful constraint can be derived by applying a trick based on reference [@des]. Considering a renormalized mass $m_r$, we can find an asymtoptic parity breaking condition for the current and a pure massive physical discrete pole excitation by means of the expression $$\Big(\Box+m^2_r \Big)\mathcal{U}^{\mu}=\Big( m_rJ^{\mu}+\epsilon^{\mu \alpha \nu}\partial_\alpha J_\nu \Big).
\label{Trick}$$ If the asymptotic field $\mathcal{U}_{\mu}$ is to describe a purely massive field then it must satisfy the Proca conditions $$\quad \partial_\mu \ \mathcal{U}^{\mu}=0 \quad \text{and} \quad \left(\Box+m^2_r\right)\mathcal{U}^{\mu}=0,$$ and must be physical in the following sense $$\left[ {\cal{U}}^\mu(x), B(y) \right] = 0.$$ Hence, an asymptotic condition for the matter current follows immediately $$\epsilon_{\mu \nu \alpha}\partial^\nu J^{\alpha}_{ \text{as}}=-m_r J_\mu^{\text{as}}.
\label{AsymptoticConditionForCurrents}$$ This constraint will help us to fix some constants below whereas the identification of the asymptotic field is devoted to section 4.
Going back to equation , we can find a general result for the vacuum expectation value of the gauge field commutator as follows $$\begin{aligned}
\langle 0 | \left[ A_{\mu}(x), A_{\nu}(y) \right] | 0 \rangle &= a\left(\eta_{\mu\nu} + \frac{1}{m^2}\partial_{\mu}\partial_{\nu} - \frac{1}{m}\epsilon_{\mu\nu\sigma}\partial^{\sigma} \right)\Delta(x-y;m^2) \nonumber \\
&\quad+ \left( b\partial_{\mu}\partial_{\nu} + c\epsilon_{\mu\nu\beta}\partial^{\beta} \right) \Delta(x-y;0) + f\partial_{\mu}\partial_{\nu}E(x-y;0) \nonumber \\
&\quad- i\int^{\infty}_{0}ds \left[ \rho(s) \left( \eta_{\mu\nu} + s^{-1}\partial_{\mu}\partial_{\nu} \right) + \widetilde{\rho}(s) \epsilon_{\mu\nu\beta}\partial^{\beta} \right] \Delta(x-y;s),
\label{eq:SpectralRepresentation1}\end{aligned}$$ where the Green’s functions are defined by the following Cauchy data $$\begin{aligned}
\Box \Delta(x-y; s) &= -s\Delta(x-y; s), \quad \Delta(x-y; s)|_0 = 0, \quad \partial_0^x\Delta(x-y; s)|_0 = -\delta^2(x-y) \\
\big(\Box+s\big) E(x-y; s) &= \Delta(x-y; s), \quad E(x-y; s)|_0=0, \quad (\partial_0^x)^3E(x-y;s)|_0 = -\delta^2(x-y),
\label{CauchyData}\end{aligned}$$ with the subscript $|_0$ meaning $|_{x_0=y_0}$. In fact, the first two lines in belong to the kernel of the differential operator in the left-hand side of , that is, it is the solution in the absence of matter currents. The last term is the non-homogeneous part of the solution which arises due to the presence of matter currents, its specific form is fixed by current conservation (\[eq:CurrentConserv\]).
By imposing the gauge fixing condition (\[eq:GaugeCondition\]), the relation $f=-i\alpha$ is obtained. Using the initial condition $[A_k(x),\partial_0 A_l(y)]|_{0}=-i\eta_{kl}\ \delta^2(x-y)$ we have $$-i = -a - i\int^\infty_{0+}ds \ \rho(s), \quad\quad \frac{a}{m^2} + b = i\int^\infty_{0}ds \ s^{-1}\rho(s),
\label{spectral1}$$ and using $[A_k(x),A_l(y)]|_{0} = 0$ we have $$c - \frac{a}{m} = i\int^\infty_{0}ds \ \tilde{\rho}(s).$$ These results have been completely general so far but we can study particular solutions of them motivated by physical facts. Henceforth, we shall fix $a=0$, as is done in the spontaneous symmetry breaking context [@Nak2], since in the MCS theory as well as in QED$_3$ there is just one asymptotic transverse physical excitation with a given mass. If $a\ne 0$, it would imply the existence of an additional asymptotic particle in the physical sector besides the radiatively generated one, namely, the one when parity breaking matter fields are considered in consistency with the Wilsonian perspective. However, this conclusion leads to a violation of the number of degrees of freedom in the theory and, thus, it is not allowed. Consequently, $$b = i\int^\infty_{0}ds \ s^{-1}\rho(s), \qquad c = i\int^\infty_{0}ds \ \tilde{\rho}(s), \qquad \int^{\infty}_{0^+} ds \ \rho(s) = 1.
\label{spectral2}$$
All in all, we obtain the following non-perturbative result $$\begin{aligned}
\langle 0 | \left[ A_{\mu}(x), A_{\nu}(y) \right] | 0 \rangle &= i \left( L\partial_{\mu}\partial_{\nu} + R\epsilon_{\mu\nu\beta}\partial^{\beta} \right) \Delta(x-y;0) -i\alpha\partial_{\mu}\partial_{\nu}E(x-y;0) \nonumber \\
&\quad- i\int^{\infty}_{0}ds \left[ \rho(s) \left( \eta_{\mu\nu} + s^{-1}\partial_{\mu}\partial_{\nu} \right) + \widetilde{\rho}(s) \epsilon_{\mu\nu\beta}\partial^{\beta} \right] \Delta(x-y;s)
\label{eq:CommutationAFinalResult}\end{aligned}$$ where we have defined the quantities $L \equiv -ib$ and $R \equiv -ic$.
Starting from we will soon derive a relation between the bare and renormalized masses below but, before procedding, it is important to establish a non-trivial connection between the spectral functions $\rho(s)$ and $\tilde{\rho}(s)$. As usual, we shall decompose the spectral functions in their discrete and continuum parts $$\rho(s) = Z \delta (s - m^2_r) + \sigma(s), \quad\quad \tilde{\rho}(s) = s^{-1/2}\tilde Z \delta (s - m^2_r) + s^{-1/2}\tilde{\sigma}(s).
\label{eq:DiscreteContinuousContributions}$$ From equation and its general solution , it is possible to compute the vacuum expectation value for the matter current. In fact, $$\begin{aligned}
\langle 0 | \left[ J_{\mu}(x), J_{\nu}(y) \right] | 0 \rangle = -i\int_0^{\infty} ds \ s\left(s-m^2\right) \rho_{\mu \nu}(x,y;s),
\label{MatterCurrent}\end{aligned}$$ where we have defined the spectral density, $\rho_{\mu \nu}(x,y;s)$, of the matter current as $$\rho_{\mu \nu}(x,y;s) = \left[ \rho(s) \left( \eta_{\mu\nu} + s^{-1}\partial_{\mu}\partial_{\nu} \right) + \widetilde{\rho}(s) \epsilon_{\mu\nu\beta}\partial^{\beta} \right] \Delta(x-y;s).$$ The form of was, of course, expected by construction. Imposing the constraint (\[AsymptoticConditionForCurrents\]), we obtain that the following relation holds asymptotically $$\epsilon^{\nu\alpha\mu} \partial_{\alpha}\langle 0 | \left[ J^{\text{as}}_{\mu}(x), J^{\text{as}}_{\nu}(y) \right] | 0 \rangle = - m_r \langle 0 | \left[ J_{\text{as}}^{\nu}(x), J^{\text{as}}_{\nu}(y) \right] | 0 \rangle.$$ Thus, choosing only the discrete parts in we have $$\int_0^{\infty} ds \ s\left(s+m^2\right) \left[ s^{-1/2}\tilde Z\epsilon^{\nu\alpha\mu}\epsilon_{\mu\nu\beta}\partial_{\alpha}\partial^{\beta} + 2m_r Z \right] \delta(s - m^2_r) \Delta(x-y;s) = 0,$$ from which it follows that $$\tilde Z = \text{sgn}(m_r)Z.$$ For completeness, after plugging this result back in equation (\[eq:DiscreteContinuousContributions\]), we get from (\[spectral2\]) that $$L = \frac{Z}{m_r^2} + \int^{\infty}_{0} ds \ \frac{\sigma(s)}{s}, \quad\quad R = \frac{Z}{m_r} + \int^{\infty}_{0} ds \ s^{-1/2}\tilde{\sigma}(s), \quad\quad 1 = Z + \int^{\infty}_{0} ds \ \sigma(s).$$
Now, acting with the differential operator $\Box\eta^{\mu\gamma} +m\epsilon^{\mu \beta \gamma}\partial_\beta$ on the two-point function (\[eq:CommutationAFinalResult\]) we obtain for the left-hand side, by using the equations of motion (\[eq:GaugeCondition\]) and (\[eq:EquationForCurrent\]), the following result[^6] $$\begin{aligned}
\langle 0 | \left[ (1-\alpha)\partial^{\gamma}B(x) - J^{\gamma}(x), A_{\nu}(y) \right] | 0 \rangle &= (1-\alpha) \partial^{\gamma}_x\langle 0 | \left[ B(x) , A_{\nu}(y) \right] | 0 \rangle - \langle 0 |\left[ J^{\gamma}(x) , A_{\nu}(y) \right] | 0 \rangle \nonumber \\
&= i(1-\alpha)\partial^{\gamma}\partial_{\nu}\Delta(x-y;0) + \cdots.\end{aligned}$$ Thus, together with similar manipulations for the right-hand side, we have $$\begin{aligned}
i(1-\alpha)\partial^{\gamma}\partial_{\nu}\Delta(x-y;0) + \cdots &= -imR\partial^{\gamma}\partial_{\nu}\Delta(x-y;0) -i\alpha\partial^{\gamma}\partial_{\nu}\Delta(x-y;0) \nonumber \\
&\quad-i\int_0^{\infty} ds\ \rho(s) \left( -s\delta^{\gamma}_{\nu} - \partial^{\gamma}\partial_{\nu} + m\epsilon_{\nu}^{~\beta\gamma}\partial_{\beta} \right)\Delta(x-y;s) \nonumber \\
&\quad-i\int_0^{\infty} ds\ \widetilde{\rho}(s) \left( -s\epsilon^{\gamma}_{~\nu\beta}\partial^{\beta} - m\partial^{\gamma}\partial_{\nu} - ms\delta^{\gamma}_{\nu} \right)\Delta(x-y;s) \nonumber.
\label{PreResult}\end{aligned}$$ After considering the spatial components $\gamma, \nu = i, j$ at equal times and using the Cauchy data (\[CauchyData\]), it follows that $$0 = im\epsilon_{j}^{~0i}\delta^2(\Vec{x}-\Vec{y})\int_0^{\infty}ds\ \rho(s) - i\epsilon^{i}_{~j0}\delta^2(\Vec{x}-\Vec{y})\int_0^{\infty} ds\ s \tilde{\rho}(s) ,$$ or $$m = \int_0^{\infty} ds\ s \tilde{\rho}(s).$$ Replacing in this result we get straightforwardly that $$m = Zm_r + \int^{\infty}_0 ds \ s^{1/2}\tilde{\sigma}(s).
\label{eq:MassRenormalization}$$
This is the most important result of this paper. We interpret (\[eq:MassRenormalization\]) as a *non-perturbative model-dependent* relation between the bare and renormalized mass of the photon. It shows a new property which is intimately related to the dimensionality of the model. In fact, in the limit of vanishing bare mass $m\rightarrow 0$, the renormalized photon mass $m_r$ *does not a priori vanish* and it depends on the continuous part of the spectral function $\tilde{\rho}(s)$ which arose only because we were working in 2 + 1 dimensions. It is worthwhile to mention that a similar equation relating the renormalized with the bare mass arises in 3 + 1 dimensions, the so-called Johnson’s theorem. However, in that case, we conclude that in the limit $m\rightarrow 0$, the renormalized mass must vanish unless the matter current has massless discrete spectrum. A well-known example for the latter statement occurs in the presence of spontaneous symmetry breaking where gauge bosons can be massive [@Nakanishi].
The next step is to identify what kind of matter current may produce a non-vanishing $\tilde \sigma(s)$. Certainly, it must break discrete symmetry even in the limit of vanishing bare mass since we are interested in dynamical mass generation. Although [@mcs] mentioned an explicit perturbative non-discrete symmetry breaking example in which scalar matter has nonvanishing $\tilde \sigma(s)$, it turns out that it is proportional to the bare mass, thus, the photon remains massless in the presence of scalars. Consequently, we are left with massive fermions in bidimensional representation. Since the source of the parity breaking comes from the mass term in the Dirac Lagrangian, it is expected that the topological mass generation depends strongly on the fermion mass. In the next section, our assumptions are verified perturbatively and in section 4 we show that we arrive at a massless discrete pole structure when considering $m_r \to 0$. In fact, the specific low energy prescription used to manipulate the equations $(6)$ and $(11)$ loses its sense in the limit $m_r \to 0$ since we cannot postulate an asymtoptic excitation such as $ \mathcal{U}^{\mu}(x)$ that explicitly violates parity without a discrete symmetry breaking Lagrangian. It can be perturbatively shown that without topological as well as fermion bare masses they are not radiatively generated [@jac]. On the other hand, in the presence of any of those terms, the other is dynamically obtained. Since they break discrete symmetries, the previous discussion is in agreement with the Wilsonian perspective.
Perturbation Theory
===================
Let us denote the following smooth limit $$\lim_{m \to 0}\tilde \rho(s) = \tilde \rho (s)_{\text{QED}_3},$$ where the right-hand side represents the desired QED$_3$ parity breaking contribution. We can extract from the computations made for the vacuum polarization tensor in QED$_3$ using causal perturbation theory [@Pim1] the following result $$\lim\limits_{m\to 0} \int ds \ s^{1/2}\tilde{\sigma}(s) = \text{Im} \left( \frac{e^2m_e}{4\pi ^2} \int_{4m_e^2}^{\infty} ds \ s^{-3/2} \log \left( \frac{1 - \sqrt{s/4m_e^2}}{1 + \sqrt{s/4m_e^2}} \right) \right).$$ As discussed in the previous section, the continuous part is non-vanishing in the limit of $m \to 0$ due to the presence of the electron mass $m_e$ which manifests as a symmetry breaking term. Using we obtain the one-loop result $$m_r = \frac{e^2}{4\pi}\text{sgn}(m_e).$$
Asymptotic Structure
====================
Having established the non-perturbative description of the phenomenon of dynamical mass generation of a gauge field through interactions with matter in 2 + 1 dimensions, we are ready to perform an analysis of the asymptotic structure of the theory.
First, we extract the discrete spectrum of (\[eq:CommutationAFinalResult\]) assuming asymptotic completeness [@mcs] $$\begin{aligned}
\langle 0 | \left[ A_{\mu}(x), A_{\nu}(y) \right] | 0 \rangle &\xrightarrow{\text{Disc. Spectr.}} i \left( L\partial_{\mu}\partial_{\nu} - R\epsilon_{\mu\nu\beta}\partial^{\beta} \right) \Delta(x-y;0) -i\alpha\partial_{\mu}\partial_{\nu}E(x-y;0) \nonumber \\
&\qquad\qquad\quad- i Z \left( \eta_{\mu\nu} + \frac{1}{m_r^2}\partial_{\mu}\partial_{\nu} - \frac{1}{m_r} \epsilon_{\mu\nu\beta}\partial^{\beta} \right) \Delta(x-y;m_r^2).\end{aligned}$$ We next define the asymptotic field of the Heisenberg operator $A_{\mu}$ as $A_{\mu}^{\text{as}} = Z^{-1/2}A_{\mu}$ and the renormalized gauge parameter as $\alpha_r = Z^{-1} \alpha$ in terms of which the commutator for $A_{\mu}^{\text{as}}$ reads $$\begin{gathered}
\big[ A_{\mu}^{\text{as}}(x), A_{\nu}^{\text{as}}(y) \big] = \\ i \left[ \left( \frac{1}{m_r^2} + Z^{-1}\int^{\infty}_{0} ds \ \frac{\sigma(s)}{s} \right) \partial_{\mu}\partial_{\nu} - \left( \frac{1}{m_r} + Z^{-1}\int^{\infty}_{0} ds \ s^{-1/2}\tilde{\sigma}(s) \right) \epsilon_{\mu\nu\beta}\partial^{\beta} \right] \Delta(x-y;0) \\
\quad-i\alpha_r\partial_{\mu}\partial_{\nu}E(x-y;0) - i \left( \eta_{\mu\nu} + \frac{1}{m_r^2}\partial_{\mu}\partial_{\nu} - \frac{1}{m_r} \epsilon_{\mu\nu\beta}\partial^{\beta} \right) \Delta(x-y;m_r^2).
\label{eq:CommutationAAsymptotic}\end{gathered}$$ In view of (\[eq:B-Field\]) we define the asymptotic field $B^{\text{as}} = B$ because it is just a free field.
Having determined (\[eq:CommutationAAsymptotic\]), we are in position to distinguish between massive and massless spectrum by decomposing $A_{\mu}^{\text{as}}$ in terms of the following fields $$\mathcal{U}^{\mu} = \frac{1}{m_r} \left( \epsilon^{\mu\nu\sigma}\partial_\nu A_\sigma^{\text{as}}- \frac{\partial^\mu B^{\text{as}}}{m_r} \right) , \qquad \cal{A}^\mu = A^\mu_{\text{as}}- \tilde{\cal{U}}^\mu.$$ The non-physical part $\cal{A}^\mu$ is purely massless while the transverse part is physical, massive and its commutator is given by $$\big[ \mathcal{U}_{\mu}(x),\mathcal{U}_{\nu}(y) \big] = -i \left( \eta_{\mu\nu} + \frac{1}{m_r^2}\partial_{\mu}\partial_{\nu} - \frac{1}{m_r} \epsilon_{\mu\nu\beta}\partial^{\beta} \right) \Delta(x-y;m_r^2).$$ Note that this expression recovers the physical Hilbert space of the MCS theory. Therefore, we conclude that the *Chern-Simons mass term has been induced by the interaction of the photon with matter*. This result is compatible with the discussion given after equation (\[Trick\]) since $\cal{U}^\mu$ represents our massive pole.
The important point of our result is that this phenomenon does not occur via an “eating" process. In fact, it is an intrinsic characteristic of the dimensionality and the topological properties of the model. The fermionic and gauge degrees of freedom must remain the same separately. It means that the latter can not have both massive and massless poles in order to preserve its degrees of freedom before and after the interaction. It is known that in 2 + 1 dimensions both MCS and Maxwell fields have one local excitation due to its Hamiltonian similarity. We have shown that the massive excitation is physical in the sense of $ \big[ {\cal{U}}^\mu(x), B(y) \big]=0$. So it must represent the unique observable degree of freedom.
It is also important to mention that the emergence of a Chern-Simons term can be understood as a topological Higgs mechanism [@top]. It is expected since every mass generation can be expressed as a kind of Higgs phenomenon [@Nakanishi].
Furthermore, we can show that in the massless limit the asymtoptic field recovers the well-known discrete massless pole structure. To see this, we use the definition of the renormalized mass and its Taylor expansion given by $$\Delta(x-y,m_r)=\Delta(x-y,0)-E(x-y,0)m_r^2+ \cdots.$$ After the redefinition of variables $$A_{\mu}^{\text{as}}(x)\to A_{\mu}^{\text{as}}(x)-\frac{1}{2}\left( Z^{-1}\int^{\infty}_{0} ds \ \frac{\sigma(s)}{s} \right)\partial_{\mu}B^{\text{as}}(x),$$ we get [@Nakanishi] $$\begin{aligned}
\left[ A_{\mu}^{\text{as}}(x), A_{\nu}^{\text{as}}(y) \right] =
-i\alpha_r\partial_{\mu}\partial_{\nu}E(x-y;0) - i \left( \eta_{\mu\nu}\Delta(x-y;0)-\partial_{\mu}\partial_{\nu}E(x-y;0) \right). \end{aligned}$$
Conclusion
==========
Throughout this work a dynamical mass generation for QED$_3$ was verified first by means of the Heisenberg equations of motion valid in all Hilbert space. Later, we obtained this same result by studying the asymptotic two-point structure of the renormalized photon fields whose physical part is the same as that of the Maxwell-Chern-Simons theory. This last observation allows us to talk about a dynamically generated topological mass term.
This result was previously obtained in the perturbative approach but here we had the opportunity to make some general observations which are characteristic of the non-perturbative treatment. The appearance of this massive excitation was expected because the Wilsonian perspective strongly indicates it since the addition of a Chern-Simons topological mass term is a natural generalization to QED in $D=2+1$ dimensions if we are in a parity breaking scenario. So, we also pointed out the importance of coupling with bidimensional massive fermions for the occurence of the mass generation phenomena.
The asymptotic structure was obtained and the massive excitation recovered is the one previously found by means of the operator equations of motion. We also show how to circumvent the Johnson’s theorem in order to have a dynamically generated renormalized mass to the photon field. The method employed is indeed consistent since the massless structure could be continuosly reached in the limit $m_r \to 0$.
Finally, we have pointed out throughout the introduction of this work that these models have interesting properties when studied in their dual language. It would be interesting to know how the notion of duality can be formulated within the KON formalism. This investigation is reserved to another paper [@future].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank the referee for the comments and suggestions to improve the manuscript significantly. G. B. de Gracia and L. Rabanal thank CAPES for support, and B. M. Pimentel thanks CNPq for partial support.
Remarks on covariant quantization of the interacting Abelian gauge theory
=========================================================================
In this appendix we shall derive equation . In section 2 we learned that in an abelian gauge theory with linear covariant gauge fixing and arbitrary matter current, $B$ satisfies a massless free-field equation . Consequently, we can obtain an integral representation for $B(y)$ $$B(y) = \int d^3z \left[ \partial_0^z \Delta(y-z;0)B(z) - \Delta(y-z;0)\partial_0 B(z) \right].
\label{BIntegral}$$ Owing to the $z^0$ independence of , we can compute four-dimensional commutation relations of the form $\left[ \Phi(x),B(y) \right]$ by using evaluated at $z^0 = x^0$ and the equal-time commutations relations. In particular, we have $\left[B(x), B(y)\right] = 0$ and $$\begin{aligned}
\left[ A_{\mu}(x),B(y) \right] &= \left[ A_{\mu}(x), \int d^3z \left[ \partial_0^z \Delta(y-z;0)B(z) - \Delta(y-z;0)\partial_0 B(z) \right]\right] \nonumber \\
&= -i\partial_{\mu}\Delta(x-y;0).
\label{A2}\end{aligned}$$ for the Abelian gauge field. This suggests a remarkable similarity between the field $B(x)$ and the generator of local gauge transformations. In fact, $\left[B(x), \phi(y)\right] = \phi(x)\Delta(x-y,0)$ and $\left[\psi(x), B(y) \right] = e\psi(x)\Delta(x-y,0)$ for scalar and fermion fields, respectively. See [@Nakanishi] for more details.
From it follows immediately, by the symmetry of the product of two derivatives, that $$\left[ F_{\mu\nu}(x),B(y) \right] = 0.
\label{A3}$$ Moreover, from , and we get $$\begin{aligned}
\left[ J^{\nu}(x), B(y)\right] &= -\left[ \partial_{\mu} F^{\mu\nu}(x) + m\epsilon^{\nu\mu\beta}\partial_{\mu}A_{\beta}(x) - \partial^{\nu} B(x), B(y)\right] \nonumber \\
&= -m\epsilon^{\nu\mu\beta}\partial_{\mu}^x \left[A_{\beta}(x), B(y)\right] \nonumber \\
&= im\epsilon^{\nu\mu\beta}\partial_{\mu}\partial_{\beta}\Delta(x-y;0) \nonumber \\
&= 0.\end{aligned}$$ We interpret this result as the statement of gauge invariance for the matter current. In fact, any field $\Psi(x)$ that satisfies $\left[\Psi(x), B(y)\right] = 0$ is a gauge invariant or physical field.
Now, we can proceed with the derivation of . We start by writing the equation of motion as follows $$\Box A^{\nu} = - J^{\nu} + (1-\alpha)\partial^{\nu}B - m\epsilon^{\nu\mu\beta}\partial_{\mu}A_{\beta}.$$ From this it follows, by straightforward computation, that $$\begin{aligned}
\left[ J^{\alpha}(x), J^{\beta}(y) \right] &= \big[-\Box^x A^{\alpha}(x) + (1-\alpha)\partial_x^{\alpha}B(x) - m\epsilon^{\alpha\mu\nu}\partial^x_{\mu}A_{\nu}(x), -\Box^y A^{\beta}(y) \nonumber \\
&\quad+ (1-\alpha)\partial_y^{\beta}B(y) - m\epsilon^{\beta\rho\sigma}\partial^ y_{\rho}A_{\sigma}(y) \big] \nonumber \\
&= \left[ \Box^x A^{\alpha}(x), \Box^y A^{\beta}(y) \right] \nonumber \\
&\quad- (1-\alpha)\Box^x\partial_y^{\beta} \big[A^{\alpha}(x), B(y) \big] - (1-\alpha)\Box^y\partial_x^{\alpha} \left[B(x), A^{\beta}(y) \right] \nonumber \\
&\quad- m(1-\alpha)\epsilon^{\beta\mu\nu}\partial_x^{\alpha}\partial_{\mu}^y \left[B(x),A_{\nu}(y) \right] - m(1-\alpha)\epsilon^{\alpha\mu\nu}\partial_y^{\beta}\partial_{\mu}^x \left[A_{\nu}(y),B(x) \right] \nonumber \\
&\quad+ m\epsilon^{\beta\mu\nu}\Box^x\partial^y_{\mu} \left[A^{\alpha}(x),A_{\nu}(y) \right] + m\epsilon^{\alpha\mu\nu}\Box^y\partial^x_{\mu} \left[A_{\nu}(x),A^{\beta}(y) \right] \nonumber \\
&\quad+ m^2\epsilon^{\alpha\mu\nu}\epsilon^{\beta\rho\sigma}\partial_{\mu}^x\partial^y_{\rho} \left[ A_{\nu}(x), A_{\sigma}(y) \right].\end{aligned}$$ After using together with $\Box\Delta(x-y;0) = 0$ and the fact that $\epsilon^{\alpha\mu\nu}\partial_{\mu}\partial_{\nu}\Psi$ vanishes for any appropriate function $\Psi$, the second and third line vanishes. Thus, we obtain $$\begin{aligned}
\left[ J^{\alpha}(x), J^{\beta}(y) \right] &= \Box^x\Box^y \left[ A^{\alpha}(x), A^{\beta}(y) \right] + m\epsilon^{\beta\mu\nu}\Box^x\partial^y_{\mu} \left[A^{\alpha}(x),A_{\nu}(y) \right] \nonumber \\
&\quad+ m\epsilon^{\alpha\mu\nu}\Box^y\partial^x_{\mu} \left[A_{\nu}(x),A^{\beta}(y) \right]
+ m^2\epsilon^{\alpha\mu\nu}\epsilon^{\beta\rho\sigma}\partial_{\mu}^x\partial^y_{\rho} \left[ A^{\nu}(x), A_{\sigma}(y) \right],\end{aligned}$$ or more precisely, $$\bigg( \Box^x\eta^{\alpha\nu} + m\epsilon^{\alpha\mu\nu}\partial_{\mu}^x \bigg) \bigg( \Box^y\eta^{\beta\sigma} + m\epsilon^{\beta\mu\sigma}\partial_{\mu}^y \bigg) \big[ A_{\nu}(x), A_{\sigma}(y) \big] = \left[ J^{\alpha}(x), J^{\beta}(y) \right].$$
[99]{}
R. Acharya and P.N. Swamy, Int. Journal Mod. Phys., A9 (1994) 861.
E. Witten, Nuovo Cimento, 39 (2016) 313.
T. Uehlinger, G. Jotzu, M. Messer, D. Greif, W. Hofstetter, U. Bissbort, and T. Esslinger, Phys. Rev. Lett., 111 185307 (2013).
I. Herbut and B. Seradjeh, Phys. Rev. Lett. 91, 171601 (2003). G. Grignani, G. Semenoff and P. Sodano, Phys. Rev. D53 (1996) 7157. G. Grignani, G. Semenoff, P. Sodano and O. Tirkkonen, Nucl. Phys. B473 (1996) 143.
N. Seiberg and E. Witten, PTEP (2016) 12C101.
V. Borokhov, A. Kapustin and X. Wu, JHEP 0211 (2002) 049.
J. Polchinski, String Theory II, (Cambridge Univ. Press, Cambridge, 1998), Appendix A.
X. Chen, Z-C. Gu, Z-X. Liu and X-G Wen, Phys. Rev. B87, 155114 (2013).
G. Dunne, arxiv:hep-th/9902115.
R. D. Pisarski, Phys. Rev. D29 (1984) 2423. H. Gies and J. Jaeckel, Phys. Rev. Lett., 93 (2004) 110405.
E. Witten, Rev. Mod. Phys. 88, 35001 (2016).
L. Alvarez-Gaumé, D. Della Pietra and G. Moore, Annals Phys. 163 (1985) 288. A. Niemi and G. Semenoff, Phys. Rev. Lett. 51 (1983) 2077. A. Redlich, Phys. Rev. D29 (1984) 2366.
H. Kramers and G. Wannier, Phys. Rev. 60 (1941) 252. M. Peskin, Annals Phys. 113 (1978) 122.
N. Seiberg, T. Senthil, C. Wang and E. Witten, Annals Phys. 374 (2016) 395. A. Karch and D. Tong, Phys. Rev. X6, 031043 (2016).
T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. 66 (1979) 1. N. Nakanishi and I. Ojima, *Covariant Operator Formalism of Gauge Theories and Quantum Gravity*, (World Scientific Lecture Notes in Physics, Vol. 27, 1990). N. Nakanishi, Prog. Theor. Phys. 35, (1966) 1111. B. Lautrup, Mat. Fys. Medd. Dan. Vid. Selsk. 35, No. 11 (1967).
E. C. Marino, Nucl. Phys. B 408 (1993) 551. D. Tong, arXiv:1606.06687.
B. M. Pimentel and J. L. Tomazelli, Prog. Theor. Phys. 95 (1996) 1217.
G. Scharf, W. F. Wrezinski, B. M. Pimentel and J. L. Tomazelli, Annals Phys. 231, (1994) 185.
K. Johnson, Nucl. Phys. 25, (1961) 435.
S. Deser, R. Jackiw and S. Templeton, Annals Phys. 140, (1982) 372.
N. Nakanishi, Prog. Theor. Phys. 49, (1973) 640.
N. Imai, K. Ichikawa and I. Tanaka, Prog. Theor. Phys. 81, (1989) 758.
R. Jackiw and S. Templeton, Phys. Rev. D23, (1981) 2291.
N. Nakanishi, Int. J. Mod. Phys. A4, (1989) 1055.
G. B. de Gracia, B. M. Pimentel and L. Rabanal, [*Duality and Self-duality of the Spin-1 Model in the Covariant Operator Formalism*]{}, submitted to publication.
[^1]: gb9950@gmail.com
[^2]: b.m.pimentel@gmail.com
[^3]: luis.rabanal@unesp.br
[^4]: In fact, computations of the IR properties of the theory using Schwinger-Dyson equations show that they might not be non-trivial CFTs [@pisarski]. However, it is common to argue that these kind of conclusions are based on truncation methods. A similar argument could be pointed out against the functional renormalization group (FRG) method [@gies].
[^5]: We will write $\mathfrak{F}$ for both spaces in the assumption of asymptotic completeness, i.e., no bound states will emerge in the asymptotic region.
[^6]: The ellipsis is the result of the unequal-time commutator between the interacting Abelian gauge field and an arbitrary matter current. Although, it is not known, we do not need the explicit result to derive equation because charged fields commute with the Abelian gauge field at equal-time.
|
---
abstract: 'We argue that the account of Coulomb-nuclear interference (CNI) in the differential cross-section of elastic $ pp $ scattering may be easily treated without introduction of intermediate IR regularization (“photon mass”). We also indicate that the parametrization used earlier misses some terms of the second order in $ \alpha $ while it contains a superfluous term of the first order.'
author:
- 'Vladimir A. Petrov [^1]'
title: ' Coulomb-Nuclear Interference: the Latest Modification[^2] '
---
A. A. Logunov Institute for High Energy Physics
NRC “Kurchatov Institute”, Protvino, RF
Introduction
============
The basis of the modern theory of strong interactions is Quantum Chromodynamics, a gauge quantum theory of quark and gluon fields which Professor A. A. Slavnov has made fundamental contributions [@Sla] to.
Our present view of high energy scattering of hadrons is dominated by the idea of a leading Regge trajectory, the Pomeron, which embodies colourless gluon exchanges and leads asymptotically to the hypothesis (ascendant to the celebrated Pomeranchuk theorem and later pushed forward by V. N. Gribov in early 1970s) of universal C-even (“C” means “crossing”) behaviour of cross-sections independent of flavours of colliding hadrons. At low energy this is violated by “usual” quarkic reggeons which, however, die off with energy.
Afterwards, it was argued that besides quarkic C-odd reggeons one can admit a C-odd partner of the Pomeron, “the Odderon”, which can , potentially, violate the above said universality even at high energies [@Luk] .
Recent measurements by the LHC TOTEM Collaboration at 13 TeV [@TOT] caused a vivid discussion (more than 60 publications by now) of a strikingly small value of the parameter $ \rho = {Re T_{N}(s,0)}/{Im T_{N}(s,0)}$ (here $T_{N}(s,t)$ stands for the $ pp $ scattering amplitude) which lies (with some variations) near $ 0.10 $. It was considered in Ref.[@Nic] as manifestation of so-called “maximal Odderon” which is to violate the strong interaction universality in a maximal possible way.
The extraction of this $ \rho $-parameter (which, let us recall, is inherently model dependent) from the data depends decisively on how the Coulomb contributions are taken into account in the full scattering amplitude.
From Bethe to CKL
=================
During quite a long time the Bethe formula [@Be] for the total amplitude $ T_{C+N} $ has been widely applied for extraction of the parameter $ \rho $ from the data (which is defined by $\mid T_{C+N}\mid^{2}$, see Eq.(2)) : $$T_{C+N} = \frac{8\pi s \alpha \mathcal{F}^{2}(t)}{t} + e^{i\alpha\Phi (s,t)} T_{N}(t)$$ where $ \mathcal{F} $ is the proton e.m. form factor and $ \Phi (s,t) $ is the Bethe phase usually in the form given to it by West and Yenni [@We] (or some later modifications of it).
However, over recent years the general practice in the TOTEM publications on this subject is based, instead of Eq.(1), on the use of the Cahn-Kundrát-Lokajíček (CKL) formula [@Cahn] for account of CNI which is more general (e.g., it does not imply the $ t $ independence [^3] of the nuclear phase $ Arg T_{N} (s,t) $ ) than the Bethe formula.
The CKL approximation used in [@TOT] (in a bit different normalization) has the form [^4] $$\frac{d\sigma_{C+N}}{dt}= \frac{(\hbar c)^{2}}{16\pi s^{2}} \mid T_{C+N}\mid^{2}= \frac{(\hbar c)^{2}}{16\pi s^{2}} \mid \frac{8\pi\alpha s}{t}\mathcal{F}^{2}(t) + T_{N} [1-i\alpha G(t)]\mid^{2}$$ with $$G(t)= \int dt^{'} log (\frac{t^{'}}{t}) \frac{d}{dt^{'}} \mathcal{F}^{2}(t^{'})-\int dt^{'}(\frac{T_{N}(t^{'})}{T_{N}(t)} - 1) \frac{I(t,t^{'})}{2\pi}$$ where $ \mathcal{F}(t) $ is the proton electric form factor and
$ I(t,t^{'})= \int_{0}^{2\pi}d\phi\: \mathcal{F}^{2}(t^{''})/t^{''},\: t^{''}=t+t^{'} + 2\sqrt{t t^{'}} \cos\phi . $
It is clear, however, that for proper accounting of powers of $ \alpha $ in perturbative QED expansion used in Eq.(2) one has to retain not only order $ \alpha^{1} $ terms but also terms $\sim \alpha^{2} $ . Otherwise, we will miss some terms $\sim \alpha^{2} $ in the differential section.
Eqs. $(2)\:-\:(3)$ were obtained as a result of rather questionable manipulations [@Cahn] with the IR regulator mass prior it could be finally eliminated. Eq.(2) was criticized in Ref. [@Petr] where it was argued, in particular, that the term $ \int dt^{'} log (\frac{t^{'}}{t}) \frac{d}{dt^{'}} \mathcal{F}^{2}(t^{'}) $ is superfluous.
Modified form of the CNI account
================================
To proceed further we have to notice that many problems can be overcome much easier if we realize that the square of the amplitude is *free from Coulombic IR divergences*.
Below we will use, instead of $ t $, a more convenient variable
$ q^{2} \equiv q^{2}_{\perp} = ut/4k^{2} = k^{2}sin^{2}\theta, \; s= 4k^{2}+ 4m^{2}, $
which reflects the $ t-u $ symmetry of the $ pp $ scattering. At $ \theta\rightarrow 0\;\quad q^{2} \approx -t $ while at $ \theta\rightarrow \pi\;\;\quad q^{2} \approx -u $. We will use the same notation $ q $ both for 2-dimensional vectors $ \textbf{q} $ and their modules $ \vert \textbf{q} \vert\ $. In the latter case, the limits of integration are indicated explicitly. As we deal with high energies and have in the integrands fast decreasing nuclear amplitudes and form factors we can (modulo vanishingly small corrections) extend the integration in $ \textbf{q} $ (kinematically limited by $ \mid \textbf{q} \mid \leq \sqrt{s}/2 $ ) over the whole 2D space. The benefit is the possibility to freely use direct and reversed 2D Fourier transforms.
Thus, based on the same premises as CKL ( the additivity of the eikonal w.r.t. strong and electromagnetic interactions) we have obtained the following expression for the *modulus squared* of the full amplitude (i.e. for the *observed* quantity) which from the very beginning is free from IR regulators (e.g. “photon mass” or $ 2\rightarrow 2+\varepsilon $ regularization or else) and is well defined mathematically: $$\mid T_{C+N}\mid_{q\neq0}^{2} = 4s^{2} S^{C} (q,q) + \int\frac{d^{2}q^{'}}{(2\pi)^{2}}\frac{d^{2}q^{''}}{(2\pi)^{2}} S^{C} (q^{'},q^{''})T_{N} (q-q^{'})T_{N}^{\ast} (q-q^{''})$$ $$+4s \int\frac{d^{2}q^{'}}{(2\pi)^{2}} Im[S^{C} (q,q^{'})T_{N}^{\ast} (q-q^{'})]$$ where $$S^{C} (q^{'},q^{''})= \int d^{2}b^{'}d^{2}b^{''} e^{i{q}^{'}{b}^{'}-i{q}^{''}{b}^{''}} e^{2i\alpha \Delta_{C} ( b^{'},\, b^{''})}$$ and $$\Delta_{C} ( b^{'},\, b^{''})= \frac{1}{2\pi}\int d^{2}k \frac{\mathcal{F}^{2}(k^{2})}{k^{2}}(e^{-ib^{''}k} - e^{-ib^{'}k})=$$ $$=\int_{0}^{\infty}\frac{dk}{k}\mathcal{F}^{2}(k^{2})[J_{0}(b^{''}k)-J_{0}(b^{'}k)].$$
In Eq.(4) we explicitly indicate the condition $ q\neq0 $ which corresponds to real experimental conditions (the scattered proton cannot be detected arbitrarily close to the beam axis). The “forward” observables , e.g. $ \sigma_{tot}(s) = Im T_{N}(s,0)/s $, are understood as the result of extrapolation $ t\rightarrow 0 $. However , this does not concern expressions appearing as integrands and able to contain terms like $ \delta (\textbf{q}) $.
In Eq.(6) the Coulomb singularity at $ k \rightarrow 0 $ is safely cured by the exponential (Bessel function) difference. Note that
$ S^{C} (q^{'},q^{''})\mid_{\alpha=0} = (2\pi)^{2} \delta (\textbf{q}^{'})(2\pi)^{2} \delta (\textbf{q}^{''})$
while
$\int S^{C} (q^{'},q^{''}) d^{2}q^{'}d^{2}q^{''}/(2\pi)^{4} = 1, \; \forall \alpha .$
In principle, when applying Eq.(4) to the data analysis, one could deal directly with Eq.(5) which is all-order (in $ \alpha $) exact expression free of singularities.
In unrealistic case of “electrically point like” nucleons, i.e. if $ \mathcal{F} = 1 $, we would have a compact explicit expression for the Coulomb function $ S^{C} (q^{'},q^{''}) $ expressed in terms of the well known generalized functions described, e.g., in [@Vla]: $$S^{C} (q^{'},q^{''}) = (4\pi\alpha)^{2} \frac{(q^{''2}/q^{'2})^{i\alpha}}{q^{'2}q^{''2}}.$$ However, it is hardly possible to obtain an explicit and “user friendly” expressions for arbitrary $ T_{N} $ and $ \mathcal{F} $.
Thus, in practice we have to use perturbative expansions in $ \alpha $. Let us notice, however, that it would be a bit rash to limit to zero and first orders in $ \alpha $ because, e.g., the pure Coulomb contribution ($ \sim\alpha^{2} $) to the observed $ d\sigma^{C+N}/dt $ at $ -t = \mathcal{O}(10^{-3} GeV^{2}) $ and $ \sqrt{s} = 13 TeV $ reaches near 30 %. We notice that in relevant publications ( see e.g. Refs.[@Cahn] ) only the terms up to the first order in $ \alpha $ are retained in the amplitude, so when passing to the cross-section some terms are missing. This can lead to wrong estimation of parameters like $ \rho $ and so to wrong physical conclusions.
The basic kernel $ S^{C} (q^{'},q^{''}) $ has the following expansion in $ \alpha $ up to $ \alpha^{2} $ inclusively: $$S^{C} (q^{'},q^{''})= (2\pi)^{2}\delta (\textbf{q}')(2\pi)^{2}\delta (\textbf{q}'')+2i\alpha (2\pi)^{3} [\hat{{\delta}_{C}}(q')\delta (\textbf{q}'')+ \hat{{\delta}_{C}}(q'')\delta (\textbf{q}')]+$$ $$+ 2\alpha^{2}\pi^{2}\lbrace 2\hat{{\delta}_{C}}(q')\hat{{\delta}_{C}}(q'')-\delta (\textbf{q}')X(q'')- \delta (\textbf{q}'')X(q'))\rbrace + ...$$ where $$\hat{{\delta}_{C}}(q)\doteq \int \frac{d\textbf{k}}{k^{2}}\mathcal{F}^{2}(k^{2})[\delta (\textbf{q})-\delta(\textbf{q}-\textbf{k})]$$ and $$X(q) = \int\frac{d\textbf{k}}{k^{2}}\mathcal{F}^{2}(k^{2})\int \frac{d\textbf{p}}{p^{2}}\mathcal{F}^{2}(p^{2})[\delta (\textbf{q}-\textbf{k}-\textbf{p}) -\delta (\textbf{q}-\textbf{k})-\delta (\textbf{q}-\textbf{p})+\delta(\textbf{q})].$$ Quantities (9) and (10) are generalized functions which are defined on the space of appropriate test functions $ \phi (\textbf{q}) $. Normally infintely differentiable functions decreasing at infinity faster than any inverse power are used (Schwartz class $ S $) though in our case just differentiable and bounded at infinity functions would be fairly suitable. Generalized functions (9) and (10) are defined as linear functionals $ (...,\phi) $ with $$(\hat{{\delta}_{C}},\phi) =\int \frac{d\textbf {k}}{k^{2}}\mathcal{F}^{2}(k^{2})( \phi(\textbf {k})-\phi(0)),$$ and $$(X,\phi) = \int\frac{d\textbf{k}}{k^{2}}\mathcal{F}^{2}(k^{2})\int \frac{d\textbf{p}}{p^{2}}\mathcal{F}^{2}(p^{2})[\phi(\textbf{k}+\textbf{p}) -\phi(\textbf{k})-\phi (\textbf{p})+\phi(\textbf{0})].$$ Distribution $ X $ can be expressed as a convolution of the distribution $ \hat{{\delta}_{C}} $ with itself: $$X(\textbf{q}) = (\hat{{\delta}_{C}}\star\hat{{\delta}_{C}})(\textbf{q})$$ and in terms of local values we get $$X(q)\mid _{q\neq 0} = \frac{1}{q^{2}}\int \frac{dk^{2}dp^{2}}{k^{2}p^{2}} (-\lambda (q^{2},k^{2},p^{2}))_{+}^{-1/2}\times$$ $$\times[q^{2}\mathcal{F}^{2}(k^{2})\mathcal{F}^{2}(p^{2})- (k^{2}\mathcal{F}^{2}(p^{2}) + p^{2}\mathcal{F}^{2}(k^{2}))\mathcal{F}^{2}(q^{2}) ]$$ where
$ \lambda (q^{2},k^{2},p^{2})=q^{4}+k^{4}+p^{4} -2q^{2}k^{2} -2q^{2}p^{2} -2k^{2}p^{2} .$
and $ x_{+}^{\nu} \doteq x^{\nu},x\geq 0; = 0, x <0. $
One can readily see that the integrals in Eqs.(12)and (14) are well convergent at $ {k}^{2},{p}^{2} \rightarrow 0 $ . UV convergence is provided by the form factors as $ \mathcal{F}^{2}(k^{2})\sim k^{-8}$ at $ k^{2}\rightarrow \infty $.
Now we are able to write down the approximate ( up to $ \sim \alpha^{2} $ inclusively) expression(in units $ GeV^{-4}$) for the observed cross-section for pp scattering with account of Coulomb-nuclear interference ( $ t\approx -\textbf{q}^{2} $ and we do not explicitly indicate the $ s $-dependence in the amplitude):
$$16\pi s^{2} \frac{d\sigma_{C+N}^{pp}}{dt} = \mid T_{N}(q^{2})\mid^{2} + \alpha J_{1}+\alpha^{2} J_{2} + \mathcal{O}(\alpha^{3}).$$
Here $$J_{1} = \lbrace \frac{16\pi s\mathcal{F}^{2}(q^{2})}{q^{2}} ReT_{N}({q}^{2})+\frac{2}{\pi}\int\frac{dk^{2}\mathcal{F}^{2}(k^{2})}{k^{2}} dq'^{2}(- \lambda (q^{2}, q'^{2}, k^{2}))^{-1/2}_{+} Im [T_{N}(q^{2})T^{\ast}_{N}(q'^{2})]\rbrace,$$ and then we break $ J_{2} $, in its turn, into three terms : $$J_{2} = J^{CC} _{2} + J^{CN}_{2} + J^{CNN}_{2},$$ where $ J^{CC}_{2} $ is the term independent on the nuclear amplitude, $ J^{CN}_{2} $ the term linear in the nuclear amplitude, $ J^{CNN}_{2}$ the term quadratic in the nuclear amplitude:
$$J^{CC} _{2} = [\frac{8\pi s\mathcal{F}^{2}({q}^{2})}{q^{2}}]^{2},$$
$$J^{CN} _{2} = \frac{2sImT_{N}({q}^{2})}{q^{2}}\int \frac{dk^{2}dp^{2}}{k^{2}p^{2}}[q^{2}\mathcal{F}^{2}(k^{2})\mathcal{F}^{2}(p^{2})- (k^{2}\mathcal{F}^{2}(p^{2}) + p^{2}\mathcal{F}^{2}(k^{2}))\mathcal{F}^{2}(q^{2}) ]\times$$
$$(-\lambda (q^{2},k^{2},p^{2}))_{+}^{-1/2}+\frac{4 s\mathcal{F}^{2}({q}^{2})}{q^{2}}\int\frac{dk^{2}dq'^{2}\mathcal{F}^{2}(k^{2})}{k^{2}} \times$$ $$\times(-\lambda (q^{2},k^{2},q'^{2}))_{+}^{-1/2}Im(T_{N} (q'^{2}) -T_{N} (q^{2})),$$
$$J^{CNN}_{2} = \: \mid\int \frac{d{k}^{2}\mathcal{F}^{2}((k^{2})}{2\pi k^{2}} dq'^{2}(-\lambda (q^{2},k^{2},p^{2}))_{+}^{-1/2}[T_{N} (q'^{2}) -T_{N} ({q^{2}})] \mid^{2}$$ $$-\frac{1}{(2\pi)^{2}}\int\frac{d\textbf{k}\mathcal{F}^{2}(k^{2})}{k^{2}}\frac{d\textbf{p}\mathcal{F}^{2}(p^{2})}{p^{2}}[ReT_{N}(\textbf{q})( ReT_{N}(\textbf{q}-\textbf{p}-\textbf{k})$$ $$-ReT_{N}(\textbf{q}-\textbf{p})-ReT_{N}(\textbf{q}-\textbf{k})+ReT_{N}(\textbf{q}))+ ImT_{N}(\textbf{q})(ImT_{N}(\textbf{q}-\textbf{p}-\textbf{k})$$ $$-ImT_{N}(\textbf{q}-\textbf{p})-ImT_{N}(\textbf{q}-\textbf{k})+ImT_{N}(\textbf{q}))]\rbrace.$$
In order not to make Eq.(15) too unwieldy we have kept vector arguments in integration and in the scattering amplitudes in the last expression of the $ \alpha^{2} $ term. To pass to invariant variables the integration measure $ d\textbf{k} d\textbf{p} $ is to be changed for $ dk^{2}dp^{2}dq'^{2}dq''^{2} (-\lambda (q^{2},q'^{2},k^{2} ))_{+}^{-1/2}(-\lambda (q^{2},q"^{2},p^{2} ))_{+}^{-1/2}$ and the following substitutions should be made: $$T_{N}(\textbf{q})\rightarrow T_{N}(q^{2}),T_{N}(\textbf{q}-\textbf{p}-\textbf{k})$$ $$\rightarrow T_{N} (\frac{(q'^{2}-k^{2}-q^{2})(q''^{2}-p^{2}-q^{2})+(-\lambda (q^{2},q'^{2},k^{2} ))_{+}^{1/2}(-\lambda (q^{2},q''^{2},p^{2} ))_{+}^{1/2}}{2q^{2}} +$$ $$+ \: q'^{2}+q''^{2}-q^{2}); \: T_{N}(\textbf{q}-\textbf{k})\rightarrow T_{N}(q'^{2}),\: T_{N}(\textbf{q}-\textbf{p})\rightarrow T_{N}(q''^{2}).$$
This expression is certainly quite bulky but we cannot avoid it if we keep $ \mathcal{O}(\alpha^{2}) $ terms which are important at low enough $ q^{2} $ characteristic for the region of CNI. Plain fact is that it significantly differs from the expression that one obtains by taking the square of the CKL amplitude modulus (2),(3) used in Ref.[@TOT] for extraction of the $ \rho $ - parameter from the data. We believe that the application of our expression (15) given above can lead to essentially different values of $ \rho $ and, consequently, to different both numerical and conceptual conclusions.
Conclusion and outlook
=======================
In this note we have exhibited a new, relatively simple but mathematically consisted, formula to deal with the Coulomb-nuclear interference which minimizes the use of IR regularizations and modifies the previously applied formula for $ T_{C+N} $. We also have shown that the usual retaining only the $ \mathcal{O}(\alpha) $ terms in the QED perturbative expansion of the amplitude $ T_{C+N} $ leads to loss of terms which can be important when passing to the cross-section and have explicitly calculated these terms. Their influence is potentially capable to change the values of the parameter $ \rho $ and, hence, the physical interpretation of the elastic proton-proton scattering at the LHC. Phenomenological application of the results presented here is the subject of a special publication [@Ezh] .
Acknowledgements
================
I am grateful to Vladimir Ezhela, Anatolii Likhoded, Jan Kašpar, Vojtech Kundrát, Per Grafström, Roman Ryutin and Nikolai Tkachenko for their interest to this work and inspiring conversations and correspondence. I am particularly indebted to Anatolii Samokhin for very fruitful discussions of some peculiar details of the paper as well as to the reviewer whose comments were helpful for improvement of the presentation.
This work is supported by the RFBR Grant 17-02-00120.
[99]{} A. A. Slavnov,
Nucl.Phys. **B97** (1975) 155-164
L. Lukaszuk and B. Nicolescu,
Lett. Nuovo Cim.**8**(1973) 405.
G. Antchev et al. TOTEM Collaboration.
Eur.Phys.J. **C79** (2019) no.9, 785,
arXiv:1812.04732; CERN-EP-2017-335-v3.
E. Martynov and B. Nicolescu,
EPJ Web Conf. **206** (2019) 06001. H. Bethe, Ann. Phys. **3** (1958) 190.
G.B. West and D. R. Yennie, Phys. Rev. **172**, 1413(1968).
R. Cahn, Z. Phys. **C15**, 253 (1982);
V. Kundrát and M. Lokajíček, Z. Phys. **C63**, 619(1994).
V. A. Petrov, arXiv:1906.038; arXiv:2001.04844 \[hep-ph\]
V. A. Petrov, Eur.Phys.J. **C78** (2018) no.3, 221;
Erratum: Eur.Phys.J. **C78** (2018) no.5, 414.
V. S. Vladimirov, Generalized Functions In Mathematical Physics.
Mir Publishers, 1979;
I.M. Gel’fand and G. E. Shilov. Generalized Functions.
AMS Chelsea Publishing. Volume 1, 1964.
V. V. Ezhela, V.A. Petrov and N. P. Tkachenko, preprint IHEP 2019-9 (in Russian).
[^1]: e-mail: Vladimir.Petrov@ihep.ru
[^2]: To be published in Vol.309 of the Proceedings of the Steklov Institute of Mathematics. (Collected papers. On the occasion of the 80th birth of Academician Andrei Alekseevich Slavnov.)
[^3]: Problems with $ t $ dependence of the nuclear phase were analyzed in [@Pet].
[^4]: The damping factors due to the soft and virtual photons are well known but negligible in the region of CNI.
|
---
abstract: 'We present a study of the instanton size and spatial distributions in pure SU(3) gauge theory using under-relaxed cooling. We also investigate the low-lying eigenmodes of the (improved) Wilson-Dirac operator, in particular, the appearance of zero-modes and their space-time localisation with respect to instantons in the underlying gauge field.'
address:
- 'Dept of Physics and Astronomy, University of Edinburgh, Edinburgh, Scotland'
- 'DESY-Zeuthen, Platanenallee 6, D-15738 Zeuthen,Germany'
- 'Dept of Physics, University of Oxford, Oxford, U.K.'
author:
- 'D.Smith, H.Simma, and M.Teper ([*UKQCD Collaboration*]{})'
title: 'Topological structure of the SU(3) vacuum and exceptional eigenmodes of the improved Wilson-Dirac operator[^1]'
---
Instanton content of the SU(3) vacuum [@teper] {#instanton-content-of-the-su3-vacuum .unnumbered}
==============================================
The importance of the instanton content of $SU(3)$ gauge theory comes through both the intrinsic importance of understanding the ground state of QCD and the role instantons are conjectured to play in light hadron structure. Cooling is a technique for removing the high-frequency non-topological excitations of the gauge-field. However, during cooling instantons are also removed; either if they are very small (lattice artifacts) or through $I \bar{I}$ annihilation. We use under-relaxed cooling to reduce the latter problem. Also, on the cooled configurations there is still the problem of extracting the instanton properties; for this we have developed pattern-recognition algorithms. We present results for 20 configurations at $\beta=6.0$ $(16^348)$ and $\beta=6.2$ $(24^348)$ lattices.
The gauge update for under-relaxed cooling [@michael] is implemented in each Cabibbo-Marinari subgroup as $$U_{new} = c (U_{min} + \alpha U_{old})$$ where $U_{min}$ is the gauge link that minimises the action, $U_{old}$ the original link, $\alpha$ is the under-relaxation parameter and $c$ a normalisation constant. Under-relaxed cooling increases the number of [*calibrated*]{} sweeps needed to annihilate an $I \bar{I}$ pair; for a given value of $\alpha$ a calibrated sweep is the number of sweeps needed to destroy a $\rho = 2$ instanton. With no under-relaxation one occasionally finds a very narrow instanton broadening out under cooling (presumably because of its environment). We have not observed this with (significant) under-relaxation. We chose $\alpha = 1$ and our measurements were carried out between 23 and 46 cooling sweeps (corresponding to between 10 and 20 cooling sweeps at $\alpha=0$).
On the cooled configurations we first find all the local extrema of the symmetrised topological charge density, $Q(x)$, relative to the $3^4$ block surrounding each point. (We do not consider the action, $S(x)$, as it clearly records less structure.) Each peak is treated as a linear superposition of the topological charge of the object at that point, calculated from a lattice-corrected formula, plus a contribution from every other object on the lattice, calculated from the continuum formula. A self-consistent set of widths is then found by iteration. These are our candidate instantons.
Summing up $Q(x)$ over the lattice and comparing it to $n_I - n_{\bar I}$ shows a discrepancy. We define $$\delta = < |Q - (n_I - n_{\bar I})| >$$ and impose filters on our candidate instantons. The parameters of the filters are chosen to minimise $\delta$. We have a “spatial” filter to remove spurious peaks due to ripples on large objects and a “width” filter. The latter compares the width calculated above with the width calculated from the charge within a radius 2 (or 3) of the peak using a lattice-corrected formula; a peak is only included if the various widths are in sufficiently good agreement. Full details will appear elsewhere [@teper].
In Figure 1 we show the instanton size distribution for $\beta=6.2$ at 23 sweeps.
\[figure1\]
=6.0cm =6.0cm
The distribution is peaked around $\rho \approx
\frac{1}{\sqrt K}$. The best fit to the large-$\rho$ tail of the distribution is $D(\rho) \propto \rho^{-\alpha}$ with $\alpha \approx 10$. As one would expect, the total number of instantons is found to vary rapidly with the amount of cooling. However the average size and the form of the small/large $\rho$ tails varies much less. We note that our results are consistent with those of [@zurich] but not with those of [@boulder].
The fact that the impact of a cooling sweep does not scale complicates the scaling analysis. Figure 2 shows that we can tune the number of cooling sweeps so as to get scaling when comparing $\beta=6.2$ and $\beta=6.0$.
\[figure2\]
=6.0cm =6.0cm
Examining the number of unlike charges a distance $R$ away from each peak (normalised by the volume of the shell) gives the distribution that is shown in Fig 3 (for $\beta=6.2$ after 23 sweeps).
\[figure3\]
=6.0cm =6.0cm
It is uniform at long distances and amplified at short distances. The corresponding distribution for like charges is uniform at long distances and suppressed at short distances. This implies some screening of instantons by anti-instantons in the vacuum – as expected.
Calculating $<\frac{Q}{|Q|}\frac{q(\rho)}{n(\rho)}>$ where $n(\rho)$ is the number of objects of size $\rho$ and $q(\rho)$ is the charge carried by objects of size $\rho$ shows that the charge carried by small (large) instantons is correlated (anti-correlated) with the sign of $Q$ (Figure 4). Indeed our results suggest over-screening of large instantons by small anti-instantons.
\[figure4\]
=6.0cm =6.0cm
Zero modes of the improved Wilson-Dirac Operator [@simma] {#zero-modes-of-the-improved-wilson-dirac-operator .unnumbered}
=========================================================
The Wilson-Dirac operator, or equivalently ${\bf Q}(\kappa)=\gamma_5M$, may have vanishing or almost-zero eigenvalues for certain values of $\kappa$. Moreover, it has been shown that the lowest eigenvalues of ${\bf Q}$ are strongly localised in space-time [@jansen].
“Exceptional” configurations are assumed to be related to the appearance of (almost-)zero eigenvalues at some $\kappa_0 < \kappa_{crit}$. They seem to be more frequent at smaller $\beta$ and with SW improvement. On the other hand, it is unclear if and how these zero modes are related to the topology of the underlying gauge field, in particular because the chirality $\chi = \langle \psi\vert\gamma_5\vert\psi\rangle$ of the corresponding eigenvectors $\psi$ of ${\bf Q}(\kappa_0)$ is typically much smaller than one. To clarify this relation we investigate the low-lying eigenmodes of ${\bf Q}(\kappa)$ using a modified conjugate gradient method [@cg]. We verified that the index theorem is realized on single instanton configurations (generated as in ref. [@smit]), for both improved and unimproved Wilson Fermions, with one right-handed (4 right-handed plus 3 left-handed) zero modes in the range $\kappa < 0.2$ for anti-periodic (periodic) boundary conditions. This is in complete analogy to the results for 2-dimensional Wilson and for staggered fermions [@smit].
$\rho$ $c_{sw}$ $\kappa_0$ $\chi$
-------- ---------- --------------- --------------
2 0 (1) 0.135 (0.126) 0.66 (0.986)
3 0 (1) 0.129 (0.125) 0.79 (0.999)
4 0 (1) 0.127 (0.125) 0.90 (0.999)
: Position $\kappa_0$ and chirality $\chi$ of the zero-mode for single instantons without (with) improvement.
For anti-periodic boundary conditions, the eigenmodes are localised and centered on the instanton. Moreover, we find considerable effects from discretization errors and SW improvement which are summarized in table 1.
We also investigated the localisation of the lowest eigenmodes on four exceptional configurations encountered by UKQCD at $\beta=6.0$ on $16^348$ and $32^364$ lattices. In all cases the (almost-)zero mode is localized close to a small ($\rho=2a \ldots 3a$) instanton (less than $\sqrt{2}a$ away). The chirality has the same sign as the topological charge of the topological object but is not correlated with the overall topological charge of the configuration. This provides some evidence that exceptional configurations are related to small instantons in the underlying gauge field. More detailed results will be presented elsewhere [@simma].
Acknowledgements
================
We acknowledge financial support by PPARC grant GR/K41663. D.S. was funded by the Carnegie Trust for the Universities of Scotland.
[8]{} D. Smith and M. Teper, in preparation. H. Simma and D. Smith, in preparation. C. Michael et al., Phys.Rev.[**D**]{}52 (1995) 4691. P. de Forcrand et al., these proceedings. T. DeGrand et al., these proceedings. K. Jansen et al., Nucl.Phys.[**B**]{} (Proc.Suppl.) 53 (1997) 262. T. Kalkreuter and H. Simma, Comp. Phys. Comm. 93 (1996) 33. Lauritsen et al., Nucl.Phys.[**B**]{}343 (1990) 522.
[^1]: presented by Douglas Smith
|
---
abstract: 'Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher-Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modelling scenarios.'
address:
- 'Departamento de Matemáticas, E. T. S. I. Industriales and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain ([juan.belmonte@uclm.es]{}). '
- 'Departamento de Matemáticas, E. T. S. I. Caminos, Canales y Puertos and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain ([gabriel.fernandez@uclm.es]{}).'
- 'Departamento de Matemáticas, E. T. S. I. Industriales and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain ([victor.perezgarcia@uclm.es]{}).'
author:
- 'Juan Belmonte-Beitia'
- 'Gabriel F. Calvo'
- 'Víctor M. Pérez-García'
title: 'Effective Particle Methods for Fisher-Kolmogorov Equations: Theory and Applications to Brain Tumor Dynamics'
---
Fisher-Kolmogorov equations, brain tumors, effective particle methods
Introduction
============
Many partial differential equations of relevance in applied sciences have robust localized solutions displaying particle-like behavior. A wealth of distinct behaviors are encountered and in general they are referred to as coherent structures and/or solitary waves[@General1; @General2; @General3; @General4].
In a limited number of prominent cases the equations are known to be integrable. When that happens, there is an infinite number of conserved quantities and the solution of the initial value problem can be constructed using different mathematical methods such as the inverse scattering transform. In integrable systems initial data can be rigorously decomposed into solitons plus radiation (linear modes) and a complete analysis of the asymptotic dynamics can be made.
While there are several physically relevant systems ruled by integrable partial differential equations, there is a vast majority of problems that are nonintegrable. Remarkably, some of them consist of small perturbations to integrable problems. In those cases one can still construct a rigorous theory for the dynamics of solitons that allows for an analytical description of the dynamics [@General4; @Kivshar1]. However, in many other instances the perturbations are not “small" and/or the basic underlying problem is not integrable but coherent structures still persist and constitute a basic elemente in the dynamics.
In many of those problems a variational formulation can be written and then a very popular method is the, so-called, effective Lagrangian method, collective coordinate method or effective particle method. This method assumes the profile of the solution to be given by a specific ansatz depending on a small number of time dependent parameters. The specific choice of the ansatz depends on the equation under study and in many cases is suggested by physical considerations. The names “effective particle" and “collective coordinates" come from the fact that, in the framework of this approach, one simplifies the dynamics of an extended field with spatio-temporal dependencies, i.e. having “infinite“ degrees of freedom” to a finite (small) number of time-dependent quantities (coordinates). The name comes from analogy with classical mechanics that provides a simple description (coordinates) of a typically extended object.
While the ansatz does not provide an exact solution of the PDE it is tipically used through the variational formulation as a test function and equations are obtained for the evolution of the solitary wave parameters. This approach works remarkably well for many relevant problems having Hamiltonian structure and provides a way to describe the infinite-dimensional dynamics in a simple form when the dynamics is dominated by coherent structures. This method has been exploited extensively in applied sciences in a large number of works for a broad variety of equations having a variational formulation (see e.g. Refs. [@General1; @Var1; @Var2; @Var3; @Var4; @Var5; @Var6; @Var7] and references therein). When coherent structures are robust it furnishes a simple description of the dynamics. The main weaknesses of the effective particle method are that: (i) it requires some experience to select appropriate ansatzes that capture adequately the dynamics, and (ii) the reduction to finite dimensions is provided without a measure of the error of the approximation that is estimated a posteriori on the basis of numerical simulations of the parent PDE. While the method has been used in hundreds of papers dealing with the dynamics of nonlinear waves in non-integrable systems, to our knowledge there are no papers obtaining a priori error bounds for such types of approximation.
Of particular interest are those equations that cannot be derived from a variational principle as it happens e.g. in dissipative systems. In that context there has been a great interest on different types of finite-dimensional descriptions of the dynamics of systems ruled by PDEs in a variety of contexts (see e.g. [@Red4; @Red1; @Red2; @Red3] and references therein). However a simple procedure such as the one provided by effective particle methods that allows applied scientists to reduce the dynamics of a partial differential equation with solitary waves to a set of finite dimensional simple equations for the solitary wave parameters is not available yet. In this paper we present a very simple methodology that allows to obtain those types of approximations for the Fisher-Kolmogorov (FK) and related reaction-diffusion equations. The simplest version of the FK equation is $$\label{dimensionalFK}
u_{t}=Du_{xx}+ \rho u(1-u),$$ and describes the evolution of a population density $u(x,t)$ measured in units of a maximal population $u_*$ on a given spatial domain. This equation is the simplest reaction diffusion model incorporating two effects: dispersion with a dispersal rate $D>0$ and proliferation or population growth with rate $\rho>0$. In Eq. (\[dimensionalFK\]) the population growth $g(u) = \rho u (1-u)$ is of the so-called logistic type although other terms $g(u)$ with similar qualitative form have been used in the literature. Eq. (\[dimensionalFK\]) is written here in dimensional form in order to connect better with applications, although one can rescale the spatial and temporal variables to get rid of the coefficients $D$ and $\rho$.
The FK equation and its extensions are a family of ubiquous reaction-diffusion models arising in population dynamics problems [@Murray; @Shigesada; @PP], most prominently in cancer modelling [@Swanson1; @Swanson2; @PG1], in the description of propagating crystallization/polymerization fronts [@Genzer2007], chemical kinetics [@Gen3], geochemistry [@Gen4] and many other fields (see e.g. [@Gen1; @Gen2] and references therein). These equations do not admit a Lagrangian density depending on the field $u$ [@General2] and thus the variational formulation for the effective particle parameters cannot be written in the usual way.
In this paper we get a small set of ordinary differential equations mimicking, not only the asymptotic dynamics of fronts arising in the Fisher-Kolmogorov (FK) equation but also describing their transient evolution towards the asymptotic regime. The plan of the paper is as follows: First, in section \[method\], we present the theoretical approach and discuss its application to the find the evolution of kink-like initial data classical FK equation. A comparison of the results of the effective particle method with the numerical solution of the FK equation is made. Next, in section \[localized\_method\], we apply the method to get effective equations for the dynamics of initially localized solutions to the FK equation. We identify different dynamical regimes for three relevant quantities associated to the spatio-temporal evolution of such localized profiles of the FK equation which are not apparent from the usual numerical solution. Secs. \[sec-apl\] and \[sec-apl2\] are devoted to several applications of the method relevant for the understanding of the growth dynamics of certain types of brain tumors. Next, in Sec. \[sec-V\] we present an example of applications to models beyond the FK equation by adding a spatial dependence to the diffusion coefficient. Finally, in section \[discussion\], we discuss the implications of our results and summarize the conclusions.
Effective-particle method description of front-type solutions {#method}
=============================================================
Derivation of effective equations for the front parameters
----------------------------------------------------------
Nonnegative single-front-type travelling wave solutions $u=u(z=x-ct)$ of Eq. satisfying the boundary conditions $$\label{bc}
\lim_{z\rightarrow-\infty}u(z)=1,\quad \lim_{z\rightarrow+\infty}u(z)=0,$$ obey the ODE $$\label{TW}
Du''(z)+cu'(z)+\rho u(1-u)=0 ,$$ and have been studied in detail [@Murray]. It is well known that such fronts can be constructed whenever $c \geq 2\sqrt{\rho D}$ and a celebrated result by Kolmogorov and coworkers [@KKK] states that compact support initial data decay asymptotically into this type of waves with $c_\textrm{min}\equiv 2\sqrt{\rho D}$. However, less is known on the transient dynamics until the asymptotic regime is reached (see e.g. [@Sherrat] and references therein).
Following the basic idea behind effective particle methods, our aim is to find an approximation for the dynamics of Eq. by means of a simple finite-dimensional expression of the form $$\label{fp}
u(x,t)=A(t)f\!\left(\frac{x-X(t)}{w(t)}\right) .$$
The “effective particle" describing the front is parametrized by three quantities depending only on time, namely, the wave amplitude $A=A(t)$, the front position $X=X(t)$ and the width $w(t)$.
A key point of the method is the choice of a suitable profile function $f$ in Eq. approximating the spatial profile of the solution. In our case, we will take the profile in to be inspired by the Ablowitz solution [@Ablowitz] $$\label{solFK}
u(z)=\frac{1}{(1+e^{z/\sqrt{6}})^2}.$$ The solution given by Eq. (\[solFK\]) is the only simple explicit solution known for the Fisher-Kolmogorov equation (in its adimensional version, i.e. with $\rho = D=1$), but corresponds to the specific speed $c=5/\sqrt{6}$, slightly larger than the minimal speed solution to the FK equation. Thus, a natural choice for our front profile is $$\label{profile}
u(x,t)=\frac{A(t)}{\left[1+e^{(x-X(t))/w(t)}\right]^2} \, ,$$ that has the expected asymptotic exponential decay for large values of $x$. In what follows we will try to obtain equations for the dynamics of the “effective particle" defined by the parameters $A(t), w(t), X(t)$.
To proceed with the method, let us define the integral quantities:
\[Is\] $$\begin{aligned}
I_{1}(t)&=&\int_{-\infty}^{\infty}u_{x}dx,\label{norma}\\
I_{2}(t)&=&\frac{\int_{-\infty}^{\infty}xu_{x}dx}{I_{1}(t)},\label{cm}\\
I_{3}(t)&=&\frac{\int_{-\infty}^{\infty}(x-I_{2}(t))^{2}u_{x}dx}{I_{1}(t)},\label{anchura}\end{aligned}$$
which are related to the $L_{1}$-norm (number of particles), center of mass and width of the gradient of the density $u$, respectively.
Then, introducing in integral , it follows that $I_{1}(t)=-A(t)$. The evolution of $I_1(t)$ can be obtained by a direct formal calculation $$\begin{gathered}
\frac{dI_{1}}{dt} = \frac{d}{dt}\left( \int_{-\infty}^{\infty} u_{x}dx \right)= \int_{-\infty}^{\infty} u_{xt}dx = \int_{-\infty}^{\infty}\left(Du_{xxx}+\rho u_{x}-2\rho uu_{x}\right)dx \\
= -\rho A(t)+4\rho A^{2}(t)\int_{-\infty}^{\infty}\frac{e^{z}}{(1+e^{z})^5}dz=-\rho A(t)\left[1-A(t)\right],\end{gathered}$$ where we have used Eq. and the fact that $\displaystyle{\int_{-\infty}^{\infty}e^{z}/(1+e^{z})^5dz=1/4}$.
We may calculate $I_{2}(t)$ in a similar way to get $$\label{I2}
I_{2}(t)=X(t)-w(t) .$$ On the other hand, differentiating $I_{2}(t)$ with respect to time and using Eq. , we find $$\begin{aligned}
\frac{dI_{2}}{dt}&=& -\frac{I_{2}(t)}{I_{1}(t)}\frac{dI_{1}}{dt} + \frac{1}{I_{1}(t)}\int_{-\infty}^{\infty} x\left(Du_{xxx}+\rho u_{x}-2\rho uu_{x}\right)dx = \frac{5}{6}\rho A(t)w(t),
\label{dI2dt}\end{aligned}$$ where we have taken into account that $\displaystyle{\int_{-\infty}^{\infty}ze^{z}/(1+e^{z})^5dz=-11/24}$. Thus, differentiating and combining it with , it follows that $$\label{evcm}
\frac{dX}{dt} = \frac{dw}{dt} + \frac{5}{6}\rho A(t)w(t).$$ To get an equation for the time evolution of the width $w(t)$ we can first use Eq. (\[anchura\]) to obtain $$\label{I3}
I_{3}(t)=\left( \frac{\pi^2}{3} - 1\right)\!w^{2}(t).$$
The time evolution of the width can be derived once more from the FK equation $$\begin{gathered}
\label{dI3dt}
\frac{dI_{3}}{dt} = -\frac{I_{3}(t)}{I_{1}(t)}\frac{dI_{1}}{dt} + \frac{1}{I_{1}(t)}\int_{-\infty}^{\infty} \left[ x- I_{2}(t)\right]^{2}\left(Du_{xxx}+\rho u_{x}-2\rho uu_{x}\right)dx \\
= 2D - \frac{1}{3}\rho A(t)w^{2}(t) .\end{gathered}$$
Therefore, using expression for $I_{3}(t)$ together with Eq. , the time evolution of the width $w(t)$ easily follows.
Summarizing, we get the following set of differential equations for the evolution of the front parameters as described by Eq. (\[fp\])
\[ODEs\] $$\begin{aligned}
\frac{dA}{dt} & = & \rho A(t)\left[1-A(t)\right]\! , \label{Ampl} \\
\frac{dX}{dt} - \frac{dw}{dt}& = & \frac{5}{6}\rho A(t)w(t), \label{Veloc}\\
\frac{dw^{2}}{dt} & = & \frac{6D}{\pi^{2}-3}-\frac{\rho}{\pi^{2}-3}A(t)w^{2}(t). \label{Width}\end{aligned}$$
Thus, our method provides a simple set of differential equations governing the evolution of a few (but very noteworthy) quantities describing the propagation of the front. As it will be described in Sec. \[analy\], explicit solutions for Eqs. (\[ODEs\]) can be found what means that the dynamics of the reduced system can be easily found in closed form. Since Eq. (\[dimensionalFK\]) is time invariant and consequently Eqs. (\[ODEs\]) autonomous we will choose without loss of generality $t_0 = 0$ in what follows without loss of generality. Since “a priori" error estimates are not available for our approach, we will later verify the quality of the approximation by resorting to numerical simulations to compare the results obtained from the PDE with the reduced ODE model of Eqs. .
Analytical solutions {#analy}
--------------------
Although Eqs. (\[ODEs\]) are a set of coupled nonlinear evolution equations their exact solutions can be found in closed form. First, Eq. (\[Ampl\]) is a logistic equation whose solution is given by $$\label{Adet}
A(t) = \frac{A_0 e^{\rho t}}{1 + A_0 \left( e^{\rho t}-1\right)},$$ where $A_0 = A(0)$. The expression for $A(t)$ given by Eq. (\[Adet\]) can be inserted into Eq. , which is linear in $w^{2}(t)$, to obtain the explicit solution for $w(t)$ which reads (with $w(0) = w_0$)
$$\begin{aligned}
w^{2}(t)&=&\frac{w_0^{2} - \frac{6D}{\rho A_0}g(0)}{\left( 1 - A_0 + A_0e^{\rho t}\right)^{\frac{1}{\pi^{2}-3}}} + \frac{6D}{\rho A_0} \left( 1 - A_0+ A_0e^{\rho t}\right) e^{-\rho t} g(t)\, ,
\label{Solwidth}\end{aligned}$$
where $$\begin{aligned}
\label{Solg}
g(t) = \sum_{n=0}^{\infty} \frac{\Gamma(n+1)\Gamma\left(1-\frac{1}{\pi^{2}-3}\right)}{\Gamma\left(n+1-\frac{1}{\pi^{2}-3}\right)} \left( 1 -\frac{1}{A_0}\right)^{n}e^{-n\rho t} \, ,\end{aligned}$$ is the Gaussian hypergeometric function $F(\alpha,\beta;\gamma;\eta)$ with $\alpha=\beta=1$, $\gamma=1-\frac{1}{\pi^{2}-3}$, $\eta = \left( 1 -\frac{1}{A_0}\right)e^{-\rho t}$ and $\Gamma(z)=\int_{0}^{\infty} x^{z-1}e^{-x}dx$ the Euler gamma function. Notice that $g(t)\to1$ for $t\to\infty$.
Finally, we can also obtain the analytical expression for the velocity $v(t)$ by combining and to get $$\begin{aligned}
v(t) = \frac{3D}{(\pi^{2}-3)w(t)} + \left[\frac{5}{6} -\frac{1}{2(\pi^{2}-3)}\right] \frac{\rho A_0w(t)}{A_0+(1-A_0)e^{-\rho t}} .
\label{Solveloc}\end{aligned}$$ Thus Eqs. (\[Adet\]-\[Solveloc\]) provide the full dynamics of the front for all times. We can easily find the asymptotic behavior of these solutions to be
\[asymptoticparameters\] $$\begin{aligned}
A(t) & \underset{t \to \infty}{\longrightarrow} & 1, \\
w(t) & \underset{t \to \infty}{\longrightarrow} & \sqrt{\frac{6D}{\rho}} \simeq 2.45 \sqrt{\frac{D}{\rho}}, \\
v(t) & \underset{t \to \infty}{\longrightarrow} & 5\sqrt{\frac{\rho D}{6}} \simeq 2.04\sqrt{\rho D}.
\end{aligned}$$
Also, it is worth mentioning that despite the crudeness of the approximation of the effective particle method, i.e. that the front maintains its basic shape during the evolution, the velocity $v(t)$ in the asymptotic limit $t\to\infty$ is very close to the exact one $c_\textrm{min}=2\sqrt{\rho D}$, with the percentual relative error being of the order of 2%, that is the difference between the real asymptotic speed and the one of the Ablowitz solution. However, the main strength of the method is that it provides qualitative information on the transient evolution of the front parameters.
Comparison with the FK equation
-------------------------------
Due to the approximate nature of Eqs. (\[ODEs\]) and the lack of a priori error bounds, it is necessary to validate the predictions of the effective particle description through a direct comparison with the numerical solution to Eq. . To do so, we have solved numerically the Eq. with initial data given by and a particular set of initial parameters $A_0, w_0, X_0$. From the solution $u(x,t)$ and using Eqs. , we obtain the soliton parameters numerically in terms of integral quantities, i.e.
\[paramsPDE\] $$\begin{aligned}
A_{\textrm{PDE}}(t) & = & -I_{1}(t), \\
w_\textrm{PDE}(t) & = & \sqrt{\frac{3I_{3}(t)}{\pi^{2}-3}},\\
X_{\textrm{PDE}}(t) &= & w_{\textrm{PDE}}(t) + I_{2}(t).
\end{aligned}$$
These values are to be compared with the solutions of Eqs. (\[ODEs\]), i.e. with Eqs. (\[Adet\]-\[Solveloc\]).
Figure \[figura1\] displays typical results of the comparative evolution of fronts according to our reduced model and from the Eq. (\[dimensionalFK\]) through . The agreement between the reduced ODE set and the full PDE is excellent. To exclude the possibility of this result being the consequence of a fortunate choice of the initial data, we have explored different sets of initial conditions in the range $0<A_0<1$, $0.5 \leq w_0\leq 5$, and $0.5 \leq X_0\leq 5$ and model parameters in the range $0.1 \leq D \leq 10$ and $0.1\leq \rho\leq 10$. In all cases a very good quantitative agreement among the three sets of curves is observed for all times, the percent relative errors being always smaller than $10\%$. The small (and expected) discrepancy of the asymptotic values for the speed and the width is always present in our calculations and is a result of our ansatz choice.
![\[Color Online\]. Comparison of the evolution of front solutions of the Fisher-Kolmogorov equation described by Eq. (\[dimensionalFK\]) with the analytical solutions of the effective particle method given by Eqs. (\[Adet\]-\[Solveloc\]). The FK equation is solved numerically using a standard second order in space and time finite difference method with zero derivative boundary conditions and constants $D=1$ and $\rho=1$. The initial data is given by Eq. (\[profile\]) with $A_0 = 0.5$, $w_0 =1$, and $X_0 =2$. In all subplots (a)-(c) the solid lines correspond to the parameters extracted from the numerical solution of the FK equation. The dashed lines correspond to the analytical solutions of the ODEs. The subplots show the: (a) amplitude $A(t)$ (dashed) versus $A_{\textrm{PDE}}(t)$ (solid), (b) velocity of the front $v(t)$ (dashed) versus $v_{\text{PDE}}(t)$ (solid), (c) width of the solution $w(t)$ (dashed) versus $w_{\textrm{PDE}}(t)$ (solid). \[figura1\]](Figure_BCP_1.eps)
Effective particle methods for localized solutions {#localized_method}
==================================================
Motivation
----------
While front solutions of the FK equation have relevance in many practical scenarios, there are cases where the solutions are initially localized. A typical example are models related to the propagation of tumors, that start from the onset as localized low amplitude cell densities and extend through the healthy tissue as localized solutions.
To derive finite-dimensional simple models able to tackle these questions we may extend the effective particle method to obtain approximate localized solutions of the Fisher-Kolmogorov equation. The procedure, however, is less straightforward. The first aspect to be addressed is the choice of a proper ansatz.
The choice of the ansatz
------------------------
In contrast with the profile given by Eq. , we now look for a nonnegative [*two-front wave*]{} $u=u(x,t)$ satisfying the boundary conditions $$\label{bclocalized}
\lim_{x\rightarrow\pm\infty}u(x,t)=0,\quad \forall t>0 .$$
Our simple finite-dimensional approximation will be of the form $$\label{fplocalized}
u(x,t)=A(t)\!\left[ f\!\left(\frac{x-X(t)}{w(t)}\right) - f\!\left(\frac{x+X(t)}{w(t)}\right) \right]^{2}\! ,$$ with $f$ representing a single front. As before, the two counter-propagating fronts are parameterized by three quantities; the amplitude $A=A(t)$, the right-front position $X=X(t)$ and the front widths $w=w(t)$. We will further assume that the resulting profile is spatially symmetric $u(-x,t)=u(x,t)$ although this restriction can be lifted in systems without spatial symmetries. To this end, we resort to an extension of our previous ansatz $$\label{profilelocalized}
u(x,t)= A(t)\!\left[\frac{1}{1+e^{(x-X(t))/w(t)}} - \frac{1}{1+e^{(x+X(t))/w(t)}}\right]^2 .$$
The above ansatz satisfies the following properties: first, if $X(t)=0$, then $u(x,t)=0$; next $u(-x,t)=u(x,t)$, for $t>0$; also $\lim_{x\rightarrow\pm\infty}u(x,t)=0$, for all $t>0$. Finally the amplitude at $x=0$ is given by $u(0,t)=A(t)\tanh^{2}\left[X(t)/2w(t)\right]$, for all $t>0$.
Evolution equations for the parameters of the effective particle
----------------------------------------------------------------
We now proceed to define the integral quantities:
$$\begin{aligned}
n(t)&=&\int_{-\infty}^{\infty}u\, dx,\label{number}\\
\sigma^{2}(t)&=&\frac{1}{n(t)}\int_{-\infty}^{\infty}x^{2}u\, dx,\label{variance}\\
\gamma(t)&=&-\int_{0}^{\infty}u_{x}dx,\label{gamma}\end{aligned}$$
These integral quantities are different from the ones ($I_1,I_2,I_3$) used to characterize the front solutions due to the fact that now we are dealing with localized solutions. The parameter $n(t)$ represents the total “mass" and in population dynamics applications represents the normalized number of individuals. $\sigma^{2}(t)$ gives the variance of the density distribution having the biological meaning of spatial width or spatial extension occupied by the population. Finally, the parameter $\gamma(t)$ provides the right-front size, giving an estimate of the size of the infiltration region in applications.
Upon substitution of Eq. in Eqs. -, and after integration, we get
$$\begin{aligned}
n(t) &=& 2A(t)\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right) - w(t) \right],\label{numberint}\\
\sigma^{2}(t)&=& \frac{1}{3}X^{2}(t) + \frac{\pi^{2}}{3}w^{2}(t) - \frac{2X^{2}(t)w(t)}{3\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right) - w(t)\right]}\, ,\label{varianceint}\\
\gamma(t)&=&A(t)\tanh^{2}\!\left(\frac{X(t)}{2w(t)}\right).\label{gammaint}\end{aligned}$$
![\[Color Online\]. Comparison of the evolution of front solutions of the Fisher-Kolmogorov equation described by Eq. (\[dimensionalFK\]) (solid curves) with that provided by the ansatz (dasher curves) with parameters given by Eqs. - for various times: $t=0, 3, 10, 20$ from the innermost to the outermost profiles. The diffusion coefficient is $D=1$ and the growth rate $\rho=1$. The initial data is given by Eq. (\[profilelocalized\]) with (a) $A_0 = 0.2$, $w_0 =1$, and $X_0 =3$; (b) $A_0 = 0.9$, $w_0 =3$, and $X_0 =1$. \[figura2\]](Figure_BCP_2.eps)
The evolution of $n(t)$, $\sigma^{2}(t)$ and $\gamma(t)$ is obtained via the FK equation as in the case of single-fronts. For $n(t)$, we find $$\begin{aligned}
\hspace*{-3mm}
\frac{dn}{dt} &=& \frac{d}{dt}\left(\int_{-\infty}^{\infty} u\, dx \right) = \int_{-\infty}^{\infty} u_{t} dx = \int_{-\infty}^{\infty} \left[ Du_{xx}+ \rho u(1-u) \right]dx \nonumber\\
&=& \rho n(t) - \rho\int_{-\infty}^{\infty} u^{2}dx = 2\rho A(t)\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right) - w(t) \right] \nonumber\\
&-& \rho A^{2}(t)\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\!\left[ 2 + 5\textrm{csch}^{2}\!\left(\frac{X(t)}{w(t)}\right)\!\right] - \frac{w(t)}{3}\!\left[ 11 + 15\textrm{csch}^{2}\!\left(\frac{X(t)}{w(t)}\right)\!\right]\!\right]\! ,
\label{dndt}\end{aligned}$$ where we have used the fact that $\int_{-\infty}^{\infty} u_{xx} dx=0$. Let us now consider $\sigma^{2}(t)$, for which we get $$\begin{aligned}
\frac{d\sigma^{2}}{dt} &=& \frac{d}{dt}\left(\frac{1}{n(t)}\int_{-\infty}^{\infty} x^{2}u\, dx \right) = -\frac{1}{n^{2}(t)}\frac{dn}{dt}\int_{-\infty}^{\infty} x^{2}u\, dx + \frac{1}{n(t)}\int_{-\infty}^{\infty} x^{2}u_{t} dx \nonumber\\
&=& -\frac{\sigma^{2}(t)}{n(t)}\frac{dn}{dt} + \frac{1}{n(t)}\int_{-\infty}^{\infty} x^{2}\left[ Du_{xx}+ \rho u(1-u) \right]dx \nonumber\\
&=& -\frac{\sigma^{2}(t)}{n(t)}\frac{dn}{dt} + 2D + \rho\sigma^{2}(t) - \frac{\rho}{n(t)}\int_{-\infty}^{\infty} x^{2}u^{2}\, dx \nonumber\\
&=& 2D + \rho A(t)\frac{\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\!\left[ \frac{5X^{2}(t)}{3w(t)} + 7w(t) - 6X(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\right] - w^{2}(t)\right]}{3\!\left[ \frac{X(t)}{w(t)}\coth\!\left(\frac{X(t)}{w(t)}\right) - 1 \right]^{2}},
\label{dsigma2dt}\end{aligned}$$ where we have made use of Eqs. , and .
Finally, for $\gamma(t)$, we obtain $$\begin{gathered}
\frac{d\gamma}{dt} = \frac{d}{dt}\left(-\int_{0}^{\infty} u_{x} dx \right) = -\int_{0}^{\infty} u_{xt} dx = -\int_{0}^{\infty} \left[ Du_{xxx}+ \rho u_{x} - 2\rho u\, u_{x}) \right]dx \\
= A(t)\tanh^{2}\left(\frac{X(t)}{2w(t)}\right)\!\left[ \rho - \rho A(t)\tanh^{2}\left(\frac{X(t)}{2w(t)}\right) - \frac{D}{w^{2}(t)}\textrm{sech}^{2}\left(\frac{X(t)}{2w(t)}\right)\right]\! .
\label{dgammadt}\end{gathered}$$
Now, in order to get a system of ordinary differential equations for the dynamically relevant quantities $A(t)$, $X(t)$ and $w(t)$ we can differentiate Eqs. - and use Eqs. -. Long calculations lead to a final closed set of ordinary differential equations that are written in Appendix \[ApA\].
![\[Color Online\]. Comparison of the evolution of front solutions of the Fisher-Kolmogorov equation described by Eq. (\[dimensionalFK\]) with the solutions of the effective particle method given by Eqs. -. The FK equation is solved numerically using a standard second order in time finite difference method with zero derivative boundary conditions and constants $D=1$ and $\rho=1$. The initial data is given by Eq. (\[profilelocalized\]) with $A_0 = 0.5$, $w_0 =1$, and $X_0 =2$. The corresponding ODEs are solved taking the later as initial values for $A(t), w(t), X(t)$. In all subplots (a)-(c) the solid lines correspond to the parameters $A_{\text{PDE}}(t)$, $v_\text{PDE}(t)$ and $w_\text{PDE}(t)$, extracted from the numerical solution of the FK equation and Eqs. -. The dashed lines correspond to the results of the ODEs. The subplots shown are: (a) amplitude $A(t)$ (dashed) versus $A_{\text{PDE}}(t)$ (solid), (b) velocity of the front $v(t)$ (dashed) versus $v_{\text{PDE}}(t)$ (solid), (c) width of the solution $w(t)$ (dashed) versus $w_{\textrm{PDE}}(t)$ (solid). \[figura3\]](Figure_BCP_3.eps)
![\[Color Online\]. Comparison of the evolution of front solutions of the Fisher-Kolmogorov equation described by Eq. (\[dimensionalFK\]) with the solutions of the effective particle method given by Eqs. -. The FK equation is solved numerically using a standard second order in time finite difference method with zero derivative boundary conditions and constants $D=1$ and $\rho=1$. The initial data is given by a Gaussian profile $u(x,0)=0.2e^{-0.5x^{2}}$. In all subplots (a)-(c) the solid lines correspond to the parameters $A_{\text{PDE}}(t)$, $v_\text{PDE}(t)$ and $w_\text{PDE}(t)$, extracted from the numerical solution of the FK equation and Eqs. -. The dashed lines correspond to the results of the ODEs. The figures shown are: (a) amplitude $A(t)$ (dashed) versus $A_{\text{PDE}}(t)$ (solid), (b) velocity of the front $v(t)$ (dashed) versus $v_{\text{PDE}}(t)$ (solid), (c) width of the solution $w(t)$ (dashed) versus $w_{\textrm{PDE}}(t)$ (solid). \[figura4\]](Figure_BCP_4.eps)
Comparison of the effective particle dynamics with the FK equation
------------------------------------------------------------------
Let $X_0$, $A_0$ and $w_0$ as in Sec. denote the initial values of the right-front position, amplitude and width, respectively. To test the validity of Eqs. (\[EDOn\]-\[EDOgamma\]) as approximations to the full PDE dynamics for localized initial data, we have run extensive series of simulations for different parameter values. As in the case of the front solutions discussed in Sec. \[method\] we have found that the effective particle model approximates with a very good accuracy the dynamics of Eqs. (\[dimensionalFK\]).
As an example in Fig. \[figura2\] we compare the profiles of the numerical solution to the FK equation (\[dimensionalFK\]) with the dynamics provided by the ansatz (\[profilelocalized\]) together with Eqs. (\[EDOn\]-\[EDOgamma\]) for various times and different initial conditions. Figure \[figura2\](a) corresponds to the case where $X_0>w_0$, whereas in Fig. \[figura2\](b) $X_0<w_0$. Notice that in Fig. \[figura2\](b) the right and left fronts advance at a faster pace when compared to Fig. \[figura2\](a) despite the fact that the initial profile is much smaller. In both cases we get an excellent agreement since the profile chosen closely resembles the one arising spontaneously from the partial differential equation (\[dimensionalFK\]).
To get a more direct comparison between the parameter evolution computed from the ODEs (\[EDOn\]-\[EDOgamma\]) and the PDE (\[dimensionalFK\]) we have compared also the evolution of the parameters in many simulations. Figures \[figura3\] and \[figura4\] provide two typical examples. In Fig. \[figura3\] the initial condition is given by the ansatz (\[profilelocalized\]), whereas in Fig. \[figura4\] the initial condition is a Gaussian profile that does not have initially the expected exponential decay of $u(x,t)$ for large values of $x$. Despite this deviation of the initial data there is generally a very good agreement of the approximate effective particle equations with the simulations of the PDEs.
Asymptotic regime
-----------------
Despite Eqs. (\[EDOn\]-\[EDOgamma\]) are ordinary differential equations allowing to get the evolution of the parameters in a more direct way than the PDE, the fact that they have a complicated structure originated in the interactions between the two fronts used to approximate the solution, makes its qualitative analysis complicated. However, there are several limits in which it is possible to elucidate the dynamics of $u(x,t)$ in terms of simpler expressions.
Let us consider the case when the population density $u(x,t)$ is much more extended than the infiltrative zone, i.e. $X(t)\gg w(t)$, a situation that is always verified for long enough times. In that case, the corresponding system of ordinary differential equations reduces to
\[ODEslocalized\] $$\begin{aligned}
\frac{dA}{dt} & = & \rho A(t)\left[1-A(t)\right]\! , \label{Amplocalized} \\
\frac{dX}{dt} - \frac{dw}{dt}& = & \frac{5}{6}\rho A(t)w(t), \label{widthlocalized} \\
\frac{dX}{dt} - \frac{\pi^{2}}{3}\frac{dw}{dt}& = & \rho A(t)w(t) - \frac{D}{w(t)}\, . \label{positionlocalized} \end{aligned}$$
Notice that Eqs. and are exactly the same as the first two equations in . Combining Eqs. and we arrive at the third equation in . This is consistent with the intuitively expected fact that if $u(x,t)$ becomes very broad then the left and right fronts no longer interact and behave as independent single-propagating particles whose time evolution is described by Eqs. . It is clear also that in the limit $t\to\infty$, the parameters $A(t)$, $w(t)$ and $v(t)$ tend to the asymptotic values given by Eqs. . Therefore, if $X(t)\gg w(t)$ the dynamics of the localized profile is the same as the simpler single front wave.
Applications to brain tumor dynamics (I): Transition to malignancy and time of birth of low grade gliomas {#sec-apl}
=========================================================================================================
Motivation
----------
Low grade glioma (LGG) is a term used to describe World Health Organization grade II primary brain tumors of astrocytic and/or oligodendroglial origin [@WHO]. These tumors are highly infiltrative and generally incurable but have a median survival time of $> 5$ years because of low proliferation [@Pignatti2002; @Ruiz2009; @Pouratian2010]. While most patients remain clinically asymptomatic besides seizures, the tumor transformation to aggressive high grade glioma is eventually seen in most patients.
Management of LGG has historically been controversial because these patients are typically young, with few, if any, neurological disorders. Historically, a wait and see approach was often favored in most cases of LGG, due to the lack of symptoms in these mostly young and otherwise healthy adults. The support for this practice came from several retrospective studies showing no difference in outcome (survival, quality of life) if therapy was deferred [@Olson2000; @Batchelor2006]. Other investigations have suggested a prolonged survival through surgery [@Smith2008]. In absence of a randomized controlled trial, recently published studies may provide the most convincing evidence in support of an early surgery strategy [@Jakola2012] and waiting for the use of other therapeutical options such as radiotherapy and chemotherapy. However, the decision on the individual treatment strategy is based on a number of factors including patient preference, age, performance status, and location of tumor [@Ruiz2009; @Pouratian2010].
The FK equation arises as a basic model of the dynamics of low grade brain tumors, accounting in a simple way for the two main features of tumor cells: infiltration and proliferation (this type of tumors do not metastasize to other organs). In this context the density $u(x,t)$ describes a wave of invasive tumor cells as it has been studied in many papers [@SS1; @SS2; @SS3; @Badoual]. One of the main characteristics of those tumors is their low cellular density. However, when the cellular density becomes too high, hipoxia arises triggering the hypoxic response [@SWCR2; @PGRT] with a cascade of metabolic alterations in the cell, including genetic instability, and has a potential effect on the acceleration of the progression which may have an impact in what is called the transition to malignancy.
Estimates for the time of transition to malignancy
--------------------------------------------------
Assuming that one of the events determining the transition to malignancy is the high local tumor density, we can get an estimate of the time taken by the tumor cell density to progress from an initial maximal density $A_0$ to a critical threshold density $A_*$ beyond which hypoxia and tissue damage trigger the cell changes leading to the malignant transformation.
The exact analytical solutions provided by the effective particle method allow us to provide an estimate for the time required for that process. Finding this time $t_{*}$ from the FK generally requires numerical computation. In contrast, it is straightforward to get it from Eq. .
$$\label{TTm}
T_{*}=\frac{1}{\rho}\log\left[\frac{(1-A_0)A_{*}}{(1-A_{*})A_0}\right].$$
It is interesting to note that according to Eq. (\[TTm\]), the time of transition to malignancy does not depend on the diffusion coefficient $D$. Eq. (\[TTm\]) should be taken as an estimate or order of magnitude upper bound since tumor density profiles are not expected to be as smooth as our ansatz functions (e.g. Eq. ) and the transition to malignancy may depend on the highest initial cell density present throughout the tumor.
To use in clinical scenarios it is necessary to estimate the parameters $A_0$, $A_*$ and $\rho$. Firstly, $A_0$ would correspond to the detection amplitude and has been discussed in several papers (see e.g. [@Swanson1] and references therein), $A_*$ is less well known but can be estimated to be in the range between 0.5 and 0.8. However more work with real data is necessary to confirm this choice. Finally the estimate of $\rho$ for individual patients is very difficult in the case of low-grade gliomas because of the slow and irregular growth of these tumors. Recent work has suggested that this number can be obtained from the response of the tumor to radiotherapy [@PGRT] and it seems reasonable that those estimates may correlate also with the histology results for proliferation markers when available (e.g. MIB/Ki-67 labelling index).
Estimates for the time of birth of the tumor
--------------------------------------------
Running the equations (\[EDOn\]-\[EDOgamma\]) backwards in time it is also possible to propose an estimate for the time of birth of the tumor, i.e. the time for which the tumor maximum density corresponds to a single cell $A_s$ (assuming that the tumor starts from one mutated cell). In our case, since astrocytic cells have a typical size of 10 $\mu$m and thus the maximal linear density is about 100 cells/mm, we would get $A_s \simeq 0.01$. The equation $$\label{TTi}
T_{s}=\frac{1}{\rho}\log\left[\frac{(1-A_s)A_{0}}{(1-A_{0})A_s}\right],$$ gives the tumor time of birth. As with Eq. (\[TTm\]), Eq. (\[TTi\]) depends inversely on $\rho$ and thus, assessing this parameter from the available patient’s data may result in a value dependent on each patient. Getting estimates for $T_s$ is very relevant clinically in order to correlate the prediction with the patient’s clinical history and to extract information on possible causes for the origin of this type of tumors. Up to now, they are thought to be sporadic, but there is a strong interest among clinicians to investigate possible causes for these tumors, for which Eq. (\[TTi\]) may be helpful. In fact, this problem has been considered in the framework of simulations of the full FK equation in [@Badoual]. Our approach complements that work providing an equation that allows to circumvent the direct simulation of the PDE.
Applications to brain tumor dynamics (II): Simple description of the response to radiotherapy {#sec-apl2}
=============================================================================================
Radiotherapy of low grade gliomas
---------------------------------
The effectiveness of radiation therapy against low grade gliomas was proven in the clinical trial of Ref. [@trial6]. However, the timing of radiotherapy after biopsy or debulking is debated. It is now well known that immediate radiotherapy after surgery increases the time of response to the therapy known as progression-free survival, but does not seem to improve the overall survival time. At the same time, radiotherapy may contribute to the observed neurological deficit of this patients as a result of the damage to the normal brain [@VandenBent2005]. This is why radiotherapy is usually deferred in time until an increase in the tumor grade is diagnosed or, else, offered to selected patients with a combination of low risk factors such as age, sub-total resection, and diffuse astrocytoma pathology [@trial5].
It is interesting that slight modifications of the radiotherapy protocol have been found to have little or no effect on overall survival [@Karim]. Mathematical modelling has the potential to select patients that may benefit from early radiotherapy. Also, it may help in developing specific optimal fractionation schemes for selected patient subgroups. However, despite its enormous potential, mathematical modelling has had a very limited use with strong focus on some aspects of radiation therapy (RT) for high-grade gliomas [@Powatil2007; @Rockne2010; @BondiauRT; @Konokoglu2010; @Kirkby2010; @Stamatakos2006]. Up to now, no ideas coming from mathematical modelling have been found useful for clinical application in any of these contexts.
There is thus a need for models accounting for the fundamental features of low-grade glioma dynamics and their response to radiation therapy without involving excessive details on the -often unknown- specific processes. The increasing availability of systematic and quantitative measurements of LGG growth rates provides key information for the development and validation of such models [@Pallud1; @Pallud2].
A simple mathematical model of tumor response to therapy
--------------------------------------------------------
We will assume that the radiation doses $d_k$ (Gy) are given instantaneously at times $t_k$ for integer $k=1,...,n$, what implies assuming that the total irradiation time is very short in comparison with the tumor response times. This is typically the case in external beam radiotherapy where treatment times are in the order of minutes while tumor response times are in the range of weeks or months. We will also assume that the radiation is spatially uniform through the tumor what is also a good approximation to clinical practice whenever possible. Following the standard practice in radiotherapy, we will take the damaged fraction of tumor cells as given by the classical linear-quadratic (LQ) model [@Joiner2009], i.e. the fraction of cells that are not lethally damaged by a dose $d_k$, to be given by $$\label{LQ}
\text{SF}_{d_k}=e^{\displaystyle{-\alpha_t d_k -\beta_t d_k^{2}}},$$ where $\alpha_t$ $(\text{Gy}^{-1})$ and $\beta_t$ $(\text{Gy}^{-2})$ are respectively the linear and quadratic coefficients for *tumor* cell damage of the LQ model. The precise values of the parameters remain unknown despite many studies, because what is clinically more relevant is the ratio $\alpha_t/\beta_t$ which for glioma cells is around 10. The full treatment consists of a total dose $D$ split in a series of -typically equal- $n$ doses $d_j$ delivered at times $t_j$. For instance, for high grade gliomas the standard protocol consists of $n=30$ doses of $d = 2$ Gy for a total of $D=$ 60 Gy, administered once per day (except for the weekends) during 6 weeks. For LGGs typical doses range from 45 to 54 Gy with $d=1.8 $ Gy during 5 or 6 weeks of treatment.
Taking into account all this information, and depending on the cell death mechanism, different models can be written. In Ref. [@PGRT] a model has been proposed including delayed response to radiation. However the simplest possible model can be based on the assumption that damaged cells die instantaneously after the therapy (or in times very short compared with the characteristic natural history of the tumor. This leads to the simplest possible model where the evolution of the tumor cell density is given by Eq. (\[dimensionalFK\]) or its ODE effective particle reduction described by ) for each interval between doses $[t_k,t_{k+1}]$. The initial data for each subinterval will be take then as given by $$\begin{aligned}
u(0,x) & = & u_0(x), \\
u(t_k^+,x) & = & \text{SF}_{d_k} u(t_k,x),\end{aligned}$$ where $u(t_k^+,x)$ is the density after the dose $k$ is given.
This allows us to write a simple model for the response of the tumor amplitude to radiation given by Eq. (\[LQ\]) together with Eq. (\[Adet\]). For instance, starting from an initial amplitude $A(t_0)$ at time $t_0$, we get that before and after the first irradiation at time $t_1$ the tumor amplitudes will be given by
$$\begin{aligned}
A(t^-_1) & = & \frac{A(t_0) e^{\rho (t_1-t_0)}}{1+ A(t_0) \left[e^{\rho (t_1-t_0)}-1\right]}, \\
A(t^+_1) & = & \frac{\text{SF}_k A(t_0) e^{\rho (t_1-t_0)}}{1+ A(t_0) \left[e^{\rho (t_1-t_0)}-1\right]}, \end{aligned}$$
Thus, defining $\hat{A}_{k} \equiv A(t_k^+)$ and taking by definition $\hat{A}_0 = A(t_0)$ we arrive to the following recursive formula for the tumor amplitudes after each irradiation cycle $$\label{recur}
\hat{A}_{k+1} = \frac{\text{SF}_{k} \hat{A}_k e^{\rho (t_{k+1}-t_k)}}{1 + \hat{A}_k \left[ e^{\rho (t_{k+1}-t_k)}-1\right]}.$$ Thus, the amplitude $A(t)$ of the tumor after a sequence of radiation sessions administered at times $(t_1, ..., t_n)$ with survival fractions $(\text{SF}_1, ..., \text{SF}_n)$, in the framework of our simple model, is given for all times by the equations
\[At\] $$\begin{aligned}
A(t) & = & \frac{\hat{A}_k e^{\rho (t-t_k)}}{1 + \hat{A}_k \left[ e^{\rho (t-t_k)}-1\right]}, \ \ t \in (t_k,t_{k+1}], \\
A(t) & = & \frac{\hat{A}_n e^{\rho (t-t_n)}}{1 + \hat{A}_n \left[ e^{\rho (t-t_n)}-1\right]}, \ \ t \geq t_n,\end{aligned}$$
where the constants $ \hat{A}_k$, $k=1,..., n$ are obtained recursively using Eq. (\[recur\]).
Validation and implications
---------------------------
Eqs. (\[recur\]) and (\[At\]) provide a very simple model of the response of a low grade glioma to radiation therapy. However these simple formulae have been obtained in the framework of the simple effective particle method. To test if the approximation provided by this approach is acceptable when describing the solutions of Eq. (\[dimensionalFK\]) together with an instantaneous response to radiation given by Eq. we have numerically simulated these equations, computed the amplitude using $A(t) = \| u(x,t) \|_{\infty}$. The evolution of this “exact" amplitude has been compared with the approximation provided by Eqs. (\[recur\]) and (\[At\]) obtained in the framework of the effective particle approximation to the dynamics.
We have compared the evolution in different scenarios. The first one corresponds to the standard radiation fractionation of a total of 54 Gy in 30 fractions of 1.8 Gy over a time range of 6 weeks (5 sessions per week from monday to friday). Radiation is started 1 week after the initial simulation time ($t=0$) and the evolution is followed for six years. The comparison between the approximation based on the effective particle method and the full Fisher-Kolmogorov equation is displayed in Fig. \[RTT\](a). A second example is provided in Fig. \[RTT\](b) where a faster growing tumor ($\rho = 0.007$ day$^{-1}$ versus $\rho = 0.00356$ day$^{-1}$ in the previous case) is irradiated following a non-standard radiotherapy scheme consisting of five consecutive sessions of 1.8 Gy every two months with the intention to control the tumor rather than focusing on minimizing the tumor mass. The agreement between both approaches is excellent.
We have explored several parameter regimes finding a very good accuracy in the approximation provided by the effective particle method meaning that one can follow the evolution of the tumor amplitude using the simple equations Eqs. (\[recur\]) and (\[At\]). This approach allows us to reduce the problem of solving a partial differential equation to computing a finite number of iterations of a simple nonlinear mapping. This fact has very interesting implications since one may then pose optimization problems for finding optimal fractionation schemes in a much simpler context. One example is optimizing radiation for delaying the transition to malignancy, a problem that can be reformulated completely in terms of amplitudes assuming that there is a critical amplitude $A_*$ beyond which the transition to malignancy is triggered. Thus the problem becomes a discrete optimization one with a very simple non-differential model that is equivalent to finding the extrema of a function of several variables $(t_j, D_j)$ with several restrictions. While the complete analysis of this problem is beyond the scope of this paper and will be considered elsewhere, we want to highlight the potential of the effective particle method to obtain simple equations amenable to a complete analysis.
Extension to related Fisher-Kolmogorov equations {#sec-V}
================================================
![\[Color Online\]. Comparison of the evolution of front solutions of the Fisher-Kolmogorov equation with space-dependent diffusion described by Eq. (\[FKDx\]) with the analytical solutions of the effective particle method given by Eqs. . The FK equation is solved numerically using a standard second order in time finite difference method with zero derivative boundary conditions with $D(x)$ as , where $D_{\text{w}}=1$, $D_{\text{g}}=0.5$, $\alpha=0$ and $\rho=1$. The initial data is given by Eq. (\[profile\]) with $A_0 = 0.5$, $w_0 =1$, and $X_0 =0.5$. In all subplots (a)-(c) the solid lines correspond to the parameters extracted from the numerical solution of the FK equation. The dashed lines correspond to the analytical solutions of the ODEs. The subplots show the: (a) amplitude $A(t)$ (dashed) versus $A_{\textrm{PDE}}(t)$ (solid), (b) velocity of the front $v(t)$ (dashed) versus $v_{\text{PDE}}(t)$ (solid), (c) width of the solution $w(t)$ (dashed) versus $w_{\textrm{PDE}}(t)$ (solid). \[figura6\]](Figure_BCP_6.eps "fig:") ![\[Color Online\]. Comparison of the evolution of front solutions of the Fisher-Kolmogorov equation with space-dependent diffusion described by Eq. (\[FKDx\]) with the analytical solutions of the effective particle method given by Eqs. . The FK equation is solved numerically using a standard second order in time finite difference method with zero derivative boundary conditions with $D(x)$ as , where $D_{\text{w}}=1$, $D_{\text{g}}=0.5$, $\alpha=0$ and $\rho=1$. The initial data is given by Eq. (\[profile\]) with $A_0 = 0.5$, $w_0 =1$, and $X_0 =0.5$. In all subplots (a)-(c) the solid lines correspond to the parameters extracted from the numerical solution of the FK equation. The dashed lines correspond to the analytical solutions of the ODEs. The subplots show the: (a) amplitude $A(t)$ (dashed) versus $A_{\textrm{PDE}}(t)$ (solid), (b) velocity of the front $v(t)$ (dashed) versus $v_{\text{PDE}}(t)$ (solid), (c) width of the solution $w(t)$ (dashed) versus $w_{\textrm{PDE}}(t)$ (solid). \[figura6\]](Figure_BCP_7.eps "fig:") ![\[Color Online\]. Comparison of the evolution of front solutions of the Fisher-Kolmogorov equation with space-dependent diffusion described by Eq. (\[FKDx\]) with the analytical solutions of the effective particle method given by Eqs. . The FK equation is solved numerically using a standard second order in time finite difference method with zero derivative boundary conditions with $D(x)$ as , where $D_{\text{w}}=1$, $D_{\text{g}}=0.5$, $\alpha=0$ and $\rho=1$. The initial data is given by Eq. (\[profile\]) with $A_0 = 0.5$, $w_0 =1$, and $X_0 =0.5$. In all subplots (a)-(c) the solid lines correspond to the parameters extracted from the numerical solution of the FK equation. The dashed lines correspond to the analytical solutions of the ODEs. The subplots show the: (a) amplitude $A(t)$ (dashed) versus $A_{\textrm{PDE}}(t)$ (solid), (b) velocity of the front $v(t)$ (dashed) versus $v_{\text{PDE}}(t)$ (solid), (c) width of the solution $w(t)$ (dashed) versus $w_{\textrm{PDE}}(t)$ (solid). \[figura6\]](Figure_BCP_8.eps "fig:")
The effective particle method can be extended to other types of reaction-diffusion equations having fronts as asymptotic attractors of the dynamics. For instance, for the cases where the diffusion coefficient either density dependent $D=D(u)$, space and/or time dependent $D=D(x,t)$ and/or the growth rates also have extra dependencies $\rho=\rho(x,t)$, as well as for other situations where the system is multicomponent (e.g. in cancer modelling, there are several phenotypes). These examples are of interest in a broad range of biological processes [@Shigesada] and particularly in cancer modelling problems [@Maini; @Ayache2; @graywhite; @Ayache1; @JB]. Here, as an example of how this methodology is robust and can be extended beyond the most basic FK equation, we consider the equation with space-dependent diffusion $$\label{FKDx}
u_{t}=[D(x)u_{x}]_{x}+\rho u(1-u).$$ Thus, we are considering Fickian diffusion but with a diffusion coefficient dependent on the space coordinates [@Murray]. Then, we can find the set of effective particle equations for the amplitude $A$, the center of mass $X$ and the width $w$ of the travelling wave , for Eq. following the same procedure of Section \[method\]. The result is
\[ODEsDx\] $$\begin{aligned}
\frac{dA}{dt} & = & \rho A(t)\left[1-A(t)\right]\! , \label{coco} \\
\frac{dX}{dt} - \frac{dw}{dt}& = & \frac{5}{6}\rho A(t)w(t), \label{cece} \\
\frac{dw^2}{dt}&=&-\frac{6}{(\pi^{2}-3)A(t)}\int_{-\infty}^{\infty}D(x)u_{x}-\frac{\rho}{\pi^{2}-3}A(t)w^{2}(t). \label{widthdx}
%\frac{dw^2}{dt}=\frac{2}{(1-2c_{2})A(t)}\int_{-\infty}^{\infty}D(x)u_{x}+\frac{2}{1-2c_{2}}\left[2c_{3}+4c_{4}+1-c_{2}\right]A(t)w^{2}(t)
%\frac{dw^{2}}{dt} & = & \frac{6D}{\pi^{2}-3}-\frac{\rho}{\pi^{2}-3}A(t)w^{2}(t). \label{Width}\end{aligned}$$
To test the limits of the method we will use a piecewise constant discontinuous diffusion coefficient given by $$\label{Dwg}
D(x)=\begin{cases}
D_{\text{g}} & \mbox{if $x\leq \alpha$},\\
D_{\text{w}} & \mbox{if $x>\alpha$}.
\end{cases}$$ Effective particle methods have tipically difficultities in dealing with discontinuous or very fast varying coefficients, and other effects that influence strongly the shape of the solution thus providing asymmetric deformations of the initial ansatz. However as we will see in what follows, in this case the method is able to follow qualitatively the details of the evolution of the deformed front solution.
The explicit choice for the diffusion coefficient given by Eq. (\[Dwg\]) has also some interest in applications. Specifically, in brain tumor modelling it arises in situations in which the tumor invades the gray matter from the white matter. In that case Eq. (\[FKDx\]) has been proposed as a toy model in which the diffusion coefficient is a piecewise constant function, corresponding to different tumor cell motilities in the white and grey matter [@SS3; @Ayache2; @Murray]. Analogous mathematical problems arise in other application scenarios (see e.g. [@Gen2] and references therein). The explicit form of $D(x)$ given by Eq. (\[Dwg\]) allows us to compute explicitly the integral in Eq. to get $$\label{cucu}
\frac{dw^2}{dt}=\frac{6}{(\pi^{2}-3)} \left[ D_{\text{g}}-\frac{D_{\text{g}}-D_{\text{w}}}{\left[1+e^{(\alpha-X(t))/w(t)}\right]^{2}}\right]-\frac{\rho}{\pi^2-3}A(t)w^2(t).$$ Eq. (\[cucu\]) together with Eqs (\[coco\]) and (\[cece\]) is again a closed system of ODEs ruling the dynamics of the front in the framework of the effective particle method. We have run extensive simulations to compare the dynamics of Eqs. with the numerical solution calculated for Eq. and the ansatz . Despite the potential problems that might be expected coming from the discontinuity of the diffusion coefficient the agreement is very in all of the cases studied. A typical example is shown in Fig. \[figura6\] where it is seen how the amplitude dynamics is fully captured by the effective particle method and the width and speed dynamics have only quantitative transient differences with the asymptotic behavior been again correctly described by the approximation method.
Conclusions {#discussion}
===========
In this paper we have presented an extension of effective particle methods to deal with a non-Hamiltonian problem of relevance in applied science: the Fisher-Kolmogorov (FK) equation. The method provides a very simple picture in terms of ordinary differential equations of the behavior of both a single-front and a localized travelling wave. It yields direct information on three relevant parameters: the amplitude, the front position and the width of the wave, which turn out to be parameters more easily accessible to experimental measurement in application scenarios than the entire profile $u(x,t)$, yet furnishing sufficient insight on the characteristic dynamics of the dynamics of the partial differential equation in certain regimes.
In addition to presenting the method and quantifying its accuracy, we have also discussed, through the specific application to problems arising in the description of brain tumors, how it can be used to get very simple estimates useful for applied scientists. Specifically we have developed explicit formulae to estimate the times of transition to malignancy and of birth of a low grade glioma. We have also provided a way to transform the problem of optimizing radiation delivery on the PDE to a finite-dimensional problem involving only a discrete map.
The method presented in this paper has very broad implications and potential uses. As an example we have shown its appropriateness to deal with a problem with spatially dependent diffusion with discontinuous diffusion coefficient. However, it can be extended to many other reaction diffusion equations in order to get a simple qualitative understanding of the dynamics of coherent structures. Secondly extending it to higher dimensions, may allow to get simple models in situations were theoretical results are much more scarce and numerical simulations more difficult. In that case approximate front profiles may be used as tentative test functions for the method [@SIAMTW1]. Finally, the set of ODEs provided by the effective particle method also allow simplifying optimal control problems, such as those involved in finding the optimal combinations of different therapies, that are much more difficult to cope within the framework of partial differential equations.
We hope that this paper would further stimulate the application of the method to get useful information for the many applications of the Fisher-Kolmogorov and related equations.
Acknowledgements {#acknowledgements .unnumbered}
================
This work has been supported by grants MTM2009-13832 and MTM2012-31073 (Ministerio de Economía y Competitividad, Spain). We would like to acknowledge Alicia Martínez (Universidad de Castilla-La Mancha, Spain) and Philip Maini (Oxford University, UK) for discussions.
Full form of the effective particle equations for localized initial data {#ApA}
========================================================================
For completeness, we detail the full expressions of the system of ordinary differential equations for $A(t)$, $X(t)$ and $w(t)$ which follow by differentiating Eqs. - and equating them to Eqs. -.
The first equation corresponds to the total number $n(t)$ $$\begin{aligned}
%\hspace*{-3mm}
&2&\!\!A(t)\!\left[ \coth\!\left(\frac{X(t)}{w(t)}\right)\!\frac{dX}{dt} - \frac{dw}{dt} - \textrm{csch}^{2}\!\left(\frac{X(t)}{w(t)}\right)\!\left(\frac{X(t)}{w(t)}\frac{dX}{dt}- \frac{X^{2}(t)}{w^{2}(t)}\frac{dw}{dt}\right)\right] \nonumber\\
&&+\, 2\left(\frac{dA}{dt} - \rho A(t)\right)\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right) - w(t)\right] \nonumber\\
&=& \rho A^{2}(t)\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\!\left[ 2 + 5\textrm{csch}^{2}\!\left(\frac{X(t)}{w(t)}\right)\!\right] - \frac{w(t)}{3}\!\left[ 11 + 15\textrm{csch}^{2}\!\left(\frac{X(t)}{w(t)}\right)\!\right]\!\right]\! .
\label{EDOn}\end{aligned}$$
The second equation corresponds to the variance $\sigma^{2}(t)$ $$\begin{aligned}
&& \hspace*{-5mm}\frac{2X^{2}(t)\!\left[ w(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\frac{dX}{dt} - X(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\frac{dw}{dt} -X(t)\textrm{csch}^{2}\!\left(\frac{X(t)}{w(t)}\right)\!\left( \frac{dX}{dt} - \frac{X(t)}{w(t)}\frac{dw}{dt}\right)\right]}{3\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right) - w(t)\right]^{2}}\nonumber\\
\hspace*{5mm} &+& \frac{2}{3}X(t)\frac{dX}{dt} + \frac{2\pi^{2}}{3}w(t)\frac{dw}{dt} - \frac{4X(t)w(t)}{3\!\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right) - w(t)\right]}\frac{dX}{dt}\nonumber \\
\hspace*{5mm} &=& 2D + \rho A(t)\frac{\left[ X(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\!\left[ \frac{5X^{2}(t)}{3w(t)} + 7w(t) - 6X(t)\coth\!\left(\frac{X(t)}{w(t)}\right)\!\right] - w^{2}(t)\right]}{3\!\left[ \frac{X(t)}{w(t)}\coth\!\left(\frac{X(t)}{w(t)}\right) - 1 \right]^{2}} \, .
\label{EDOsigma2}
\end{aligned}$$
Finally, the third equation corresponding to the right-front size $\gamma(t)$ is $$\begin{gathered}
\tanh^{2}\!\left(\frac{X(t)}{2w(t)}\right)\!\frac{dA}{dt} + \frac{A(t)}{w(t)}\,\textrm{sech}^{2}\!\left(\frac{X(t)}{2w(t)}\right)\!\tanh\!\left(\frac{X(t)}{2w(t)}\right)\!\left(\frac{dX}{dt}- \frac{X(t)}{w(t)}\frac{dw}{dt}\right) \\
= A(t)\tanh^{2}\!\left(\frac{X(t)}{2w(t)}\right)\!\left[ \rho - \rho A(t)\tanh^{2}\!\left(\frac{X(t)}{2w(t)}\right) - \frac{D}{w^{2}(t)}\textrm{sech}^{2}\!\left(\frac{X(t)}{2w(t)}\right)\right]\! .
\label{EDOgamma}\end{gathered}$$
[1]{}
T. Dauxois, M. Peyrard, *Physics of Solitons*, Cambridge University Press, 2006.
A. Scott, *Nonlinear Science: Emergence and Dynamics of Coherent Structures*, Oxford University Press, 2003.
P. G. Drazin, R. S. Johnson, *Solitons: An introduction*, Cambridge University Press, 1989.
J. Yang, *Nonlinear Waves in Integrable and Non-integrable Systems*, SIAM, 2010.
Y. S. Kivshar, B. A. Malomed, *Dynamics of solitons in nearly integrable systems*, Rev. Mod. Phys. 61, 763-915 (1989).
B. A. Malomed, *Variational methods in nonlinear fiber optics and related fields*, *Prog. Opt.* 43, 7-193 (2002).
N. R. Quintero, E. Zamorano-Sillero, *Lagrangian formalism in perturbed nonlinear Klein-Gordon equations*, Physica D 197, 63 (2004).
V. M. Pérez-García, *Self-similar solutions and collective coordinate methods for nonlinear Schrödinger equations*, Physica D 191, 211-218 (2004).
R. Carretero-González, D. J. Frantzeskakis and P. G. Kevrekidis, *Nonlinear waves in BoseÐEinstein condensates: physical relevance and mathematical techniques*, Nonlinearity 21, R139 (2008).
V. M. Pérez-García, P. Torres, G. D. Montesinos, *The method of moments for Nonlinear ScheÁrödinger equations: Theory and Applications*, SIAM J. Appl. Math. 67, 990 (2007).
S. Cuenda. A. Sánchez, *Length scale competition in nonlinear Klein-Gordon models: A collective coordinate approach*, Chaos 15, 023502 (2005).
N. R. Quintero, F. G. Mertens, A. R. Bishop, *Generalized traveling-wave method, variational approach, and modified conserved quantities for the perturbed nonlinear Schršdinger equation*, Phys. Rev. E 82, 016606 (2010).
C. Foias, B. Nicolaenko, R. Temman (Eds.), *The connection between infinite dimensional and finite dimensional dynamical systems*, American Mathematical Society (1987).
T. Bohr, M. H. Jensen, G. Paladin, A. Vulpiani, *Dynamical systems approach to turbulence*, Cambridge University Press, 1998.
R. Joly, G. Raugel, *A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations*, Confluentes Mathematici **3**, 471-493 (2011).
A. N. Carvalho, J. W. Cholewa, G. Lozada-Cruz, M. R. T. Primo, *Reduction of infinite dimensional systems to finite dimensions: Compact convergence approach*, Cuadernos de Matemática **11**, 281-330 (2010).
J. Murray, [*Mathematical biology*]{}, Third Edition, Springer (2007).
N. Shigesada, K. Kawasaki, *Biological Invasions: Theory and Practice*, Oxford University Press (1997).
V. Volpert, S. Petrovskii, *Reaction-difusion waves in biology*, Physics of Life Reviews **6**, 267Ð310 (2009).
K. R. Swanson, R. Rostomily, E. C. Alvord, Jr, *Predicting Survival of Patients with Glioblastoma by Combining a Mathematical Model and Pre-operative MR imaging Characteristics: A Proof of Principle*, Br. J. Cancer, **98**, 113-119 (2008).
C. Wang, J. K. Rockhill, M. Mrugala, D.L. Peacock, A. Lai, K. Jusenius, J. M. Wardlaw, T. Cloughesy, A. M. Spence, R. Rockne, E. C. Alvord Jr., K. R. Swanson, *Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model*, Cancer Res. **69**, 9133-40 (2009).
V. M. Pérez-García, G. F. Calvo, J. Belmonte-Beitia, D. Diego, L. A. Pérez-Romasanta, *Bright solitons in malignant gliomas*, Phys. Rev. E **84**, 021921 (2011).
J.F. Douglas, K. Efimenko, D. A. Fischer, F. R. Phelan, J. Genzer, *Propagating waves of self-assembly in organosilane monolayers*, Proc. Nat. Acad. Sci. **104**, 10324 (2007).
I. Epstein and J. A. Pojman, *An Introduction to Nonlinear Chemical Dynamics*, Oxford University Press, New York (1998).
P. Grindrod, *The Theory and Applications of Reaction-Diffusion Equations*, Oxford University Press, New York (1996).
J. Xin, *Front Propagation in Heterogeneous Media*, SIAM Rev., **42**, 161-230 (2000).
C. W. Curtis, D. M. Bortz, *Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion*, Phys. Rev. E **86**, 066108 (2012).
A. Kolmogoroff, I. Petrovsky, and N. Piscounoff, *Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique*. Moscow University, Bull. Math. **1**, 1-25 (1937).
J. A. Sherrat, *On the transition from initial data to travelling waves in the Fisher-KPP equation*, Dynamics and Stability of Systems **13**, 167-174 (1998).
M. J. Ablowitz, A. Zeppetella, *Explicit solutions of Fisher’s equation for a special wave speed*, Bull. Math. Biol. **41**, 835 (1979).
Louis, D. N., Ohgaki, H., Wiestler, O. D., Cavenee, W. K., Burger, P. C., Jouvet, A., Scheithauer, B. W. & Kleihues P, *World health organization classification of tumours of the central nervous system*, 4th ed., Renouf Publishing Co. Ltd., Geneva. pp. 33-46 (2007)
Pignatti, F., Van den Bent, M., Curran, D., Debruyne, C., Sylvester, R., Therasse, P., Afra, D., Cornu, P., Bolla, M., Vecht, C. & Karim, A.B., *Prognostic factors for survival in adult patients with cerebral low- grade glioma*, J. Clin. Oncol., **20**, 2076-84 (2002).
Ruiz, J., & Lesser, G. J., *Low-Grade Gliomas*, *Curr. Treat. Opt. Oncol.*, **10**, 231-242 (2009).
Pouratian, N., & Schiff, D. *Management of low-grade glioma*, Curr. Neurol. Neurosci. Rep. **10**, 224-231 (2010)
Olson, J.D., Riedel E., & DeAngelis, L. M. (2000) Long-term outcome of low-grade oligodendroglioma and mixed glioma. *Neurology* **54**, 1442-1448.
Grier, J. T. & Batchelor, T. (2006) Low-Grade Gliomas in Adults. *The Oncologist*, **11**, 681-693.
<span style="font-variant:small-caps;">Smith, J.S., Chang, E.F., Lamborn, K.R., Chang, S.M., Prados, M.D., Cha, S., Tihan, T., Vandenberg, S., McDermott, M.W. & Berger, M.S.</span> *Role of extent of resection in the long-term outcome of low-grade hemispheric gliomas.*, *J. Clin. Oncol.* **26**, 1338-1345 (2008).
Jakola, A.S., Myrmel, K.S., Kloster, R., Torp, S.H., Lindal, S., Unsgard, G., Solheim, O., *Comparison of a strategy favoring early surgical resection vs a strategy favoring watchful waiting in low-grade gliomas*, JAMA **308**, 1881-1888 (2012).
J. D. Murray, *Mathematical Biology I and II*, 3rd ed, Springer (2003).
E. Mandonnet, J-Y Delattre, M-L Tanguy, K. R. Swanson, A. F. Carpentier, H. Duffau, P. Cornu, R. Van Effenterre, E. C. Alvord Jr., and L. Capelle, *Continuous growth of mean tumor diameter in a subset of WHO grade II gliomas*, Annals of Neurology, **53**, 524-528 (2003).
S. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. R. Swanson, M. Pelegrini-Issac, R. Guillevin, H. Benali, *Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging*, Magnetic Resonance in Medicine, **54**, 616-624 (2005).
C. Gerin, J. Pallud, B. Grammaticos, E. Mandonnet, C. Deroulers, P. Varlet, L. Capelle, L. Taillandier, L. Bauchet, H. Duffau, M. Badoual, *Improving the time-machine: estimating date of birth of grade II gliomas.* Cell. Prolif. **45**, 76-90 (2012).
K. R. Swanson, R. C. Rockne, J. Claridge, M. A. Chaplain, E. C. Alvord Jr, A. R. Anderson, *Quantifying the role of angiogenesis in malignant progression of gliomas: in silico modeling integrates imaging and histology.*, Cancer Research **71**, 7366-7375 (2011).
V. M. Pérez-García, M. Bogdanska, A. Martínez-González, J. Belmonte, P. Schucht, L. A. Pérez-Romasanta, *Delay effects in the response of low grade gliomas to radiotherapy: A mathematical model and its therapeutical implications*, Mathematical Medicine and Biology (submitted) (2013).
Garcia, D.M., Fulling, K.H. & Marks, J.E., *The value of radiation therapy in addition to surgery for astrocytomas of the adult cerebrum*, Cancer, **55**, 919-927 (1985).
Van den Bent, M.J., Afra, D., de Witte, O., Ben Hassel, M., Schraub, S., Hoang-Xuan, K., Malmström, P.O., Collette, L., Piérart, M., Mirimanoff, R. & Karim, A.B., *Long-term efficacy of early versus delayed radiotherapy for low-grade astrocytoma and oligodendroglioma in adults: the EORTC 22845 randomised trial.*, Lancet **366**, 985-990 (2005).
Higuchi Y., Iwadate Y. & Yamaura A., *Treatment of low-grade oligodendroglial tumors without radiotherapy.*, Neurology **63**, 2384-2386 (2004).
A. B. M. Karim, B. Maat, et al., *A randomized trial on dose-response in radiation therapy of low-grade cerebral glioma: EORTC study 22844*, Int. J. Rad. Oncol. Biol. Phys. **36**, 549-556 (1996).
G Powathil, M Kohandel, S Sivaloganathan, A Oza and M Milosevic, *Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy*, Phys. Med. Biol. **52**, 3291-3306 (2007)
Rockne, R., Hendrickson, K., Lai, A., Cloughesy, T., Alvord, E.C. Jr & Swanson, K.R. *Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach*, Phys. Med. Biol. **55**, 3271-3285 (2010).
Bondiau, P.Y., Frenay, M. & Ayache, N., *Biocomputing: numerical simulation of glioblastoma growth using diffusion tensor imaging*, Phys. Med. Biol. **53**, 879-893 (2008).
Konukoglu, E., Clatz, O., Bondiau, P.Y., Delingette, H. & Ayache, N., *Extrapolating glioma invasion margin in brain magnetic resonance images: suggesting new irradiation margins*, Med. Image Anal. **14**, 111-125 (2010).
Barazzuol, L., Burnet, N. G., Jena, R., Jones, B., Jefferies, S. J. & Kirkby, N. F., *A mathematical model of brain tumour response to radiotherapy and chemotherapy considering radiobiological aspects*, Journal of Theoretical Biology, **262**, 553-565 (2010).
Stamatakos, G. S., Antipas, V. P. & Uzunoglu, N. K. (2006) Simulating chemotherapeutic schemes in the individualized treatment context: The paradigm of glioblastoma multiforme treated by temozolomide in vivo. [Comp. Biol. Med.]{} **36**, 1216-1234.
J. Pallud, L. Taillandier, L. Capelle, D. Fontaine, M. Peyre, F. Ducray, H. Duffau, & E. Mandonnet, *Quantitative Morphological MRI Follow-up of Low-grade Glioma: A Plead for Systematic Measurement of Growth Rates*, Neurosurgery **71**, 729-740 (2012).
J. Pallud, J. F. Llitjos, F. Dhermain, P. Varlet, E. Dezamis, B. Devaux, R. Souillard-Scemama, N. Sanai, M. Koziak, P. Page, M. Schlienger, C. Daumas-Duport, J. F. Meder, C. Oppenheim, & F. X. Roux, *Dynamic imaging response following radiation therapy predicts long-term outcomes for diffuse low-grade gliomas*, Neuro-Oncology, **14**, 496-505 (2012).
Van der Kogel, A., & Joiner, M., *Basic clinical radiobiology*, Oxford University Press, 2009
K. R. Swanson, E. C. alvord, J. D. Murray, *Dynamics of a model for brain tumors reveals a small window for therapeutic intervention*, Disc. Cont. Dyn. Sys. B **4**, 289-295 (2004).
S. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. Swanson, M. Pelegrini- Issac, R. Guillevin, H. Benali, *Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging*, Magnetic Reson. in Med. **54**, 616Ð624 (2005).
E. Konukoglu, O. Clatz, P.Y. Bondiau, H. Delingette, N. Ayache, *Extrapolating glioma invasion margin in brain magnetic resonance images: suggesting new irradiation margins*, Med. Image Anal. **14**, 111-125 (2010).
F. Sánchez-Garduño, P. K. Maini, *Travelling wave phenomena in some degenerate reaction-diffusion equations*, J. Diff. Eqs. [**117**]{}, 281-319 (1995).
J. Belmonte-Beitia, T. E. Wooley, J. G. Scott, P. K. Maini, E. A. Gaffney, *Modelling biological invasions: Individual to population scales at interfaces*, Journal of Theoretical Biology (submitted)
P. K. Braznik, J. J Tyson, *On travelling wave solutions of Fisher’s equation in two spatial dimensions*, SIAM J. Appl. Math. **60**, 371-391 (1999).
|
---
abstract: 'The purpose of the present article is to study and characterize several types of symmetries of generalized Robertson-Walker space-times. Conformal vector fields, curvature and Ricci collineations are studied. Many implications for existence of these symmetries on generalied Robertson-Walker spacetimes are obtained. Finally, Ricci solitons on generalized Robertson-Walker space-times admitting conformal vector fields are investigated.'
address:
- 'Mathematics Department, Faculty of Science, Tanata University, Tanta, Egypt'
- 'Modern Academy for engineering and Technology, Maadi, Egypt'
author:
- 'H. K. El-Sayied'
- 'S. Shenawy'
- 'N. Syied'
title: 'On symmetries of generalized Robertson-Walker space-times and applications'
---
An introduction
===============
Robertson-Walker spacetimes have been extensively studied in both mathematics and physics for a long time[@Besse2008; @Chen2008; @Ivancevic2007; @Sanchez2000; @Sanchez1998; @Sanchez1999]. This family of spacetimes is a very important family of cosmological models in general relativity[@Chen2008]. A generalized $\left( n+1\right) -$dimensional Robertson-Walker (GRW) spacetime is a warped product manifold $I\times _{f}M$ where $M$ is an $n-$dimensional Riemannian manifold without any additional assumptions on its fiber. The family of generalized Robertson-Walker spacetimes widely extends the classical Robertson-Walker spacetimes $I\times
_{f}S_{k}$ where $S_{k}$ is a $3-$dimensional Riemannian manifold with constant curvature.
The study of spacetime symmetries is of great interest in both mathematics and physics. The existence of some symmetries in a spacetime is helpful in solving Einstein field equation and in providing further insight to conservative laws of dynamical systems(see [@Hall2004] one of the best references for $4-$dimensional spacetime symmetries). Conformal vector fields have been played an important role in both mathematics and physics[@Deshmokh2012; @Deshmokh20141; @Deshmokh20142; @Kuhnel1997; @Nomizo1960; @Yorozu1982]. The existence of a nontrivial conformal vector field is a symmetry assumption for the metric tensor. This assumption has been widely used in relativity to obtain exact solutions of the Einstein field equation[@Caballero2011]. Similarly, collineations display some tensors symmetry properties of spacetimes. They are vector fields which preserve certain feature of a spacetime(physical or geometric quantities such as matter and curvature tensors) along their local flow lines. In this sense, the Lie derivative of such quantities vanishes in direction of collineation vector fields. Matter, curvature and Ricci collineations have been extensively studied on spacetimes because of their essential role in general relativity. Moreover, these collineations help to describe the geometry of spacetimes. In the last two decates, an extensive work has been done studying collineations and their generalizations such as Ricci inheritance collineations on classical spacetimes. Among those, there are many authors who studied these symmetries on classical Robertson-Walker spacetimes(for instance see[@Duggala2005; @Sanchez1999; @Steller2006; @Unal2012] and references therein).
However, as far as we know, there is no study on generalized Robertson-Walker spacetimes investigating neither conformal vector fields nor different types of collineations up to this paper in which we intend to fill this gab by providing many answers of the following questions: Under what conditions is a vector field on $I\times _{f}M$ a conformal vector field or a certain collineation? What does the fibre $M$ inherit from a generalized Robertson-Walker spacetime $I\times _{f}M$ admitting a collineation or a conformal vector field? The main purpose of the current article is to study and explore both conformal vector fields and collineations on generalized Robertson-Walker spacetimes. We gave a special attention to two disjoint classes of conformal vector fields, namely, Killing vector fields of constant length and concircular vector fields. Finally, Ricci solitons on generalized Robertson-Walker spacetime admitting either Killing or concircular vector fields are considered.
This article is organized as follows. The next section presents some connection and curvature related formulas of generalized Robertson-Walker spacetimes that are needed. Then basic definitions of conformal vector fields and collineations are considered. Most of these results are well-known and so proofs are omitted. Section 3 presents a study of conformal vector fields and collineations on generalized Robertson-Walker spacetimes. Finally, in section $4$, we study Ricci solitons on generalized Robertson-Walker spacetimes admitting either Killing or concircular vector fields.
Preliminaries
=============
First, we want to fix some definitions and concepts. The warped product $%
M_{1}\times _{f}M_{2}$ of two Riemannian manifolds $\left(
M_{1},g_{1}\right) $ and $\left( M_{2},g_{2}\right) $ is the product manifold $M_{1}\times M_{2}$ equipped with the metric tensor$$g=\pi _{1}^{\ast }\left( g_{1}\right) \oplus \left( f\circ \pi _{1}\right)
^{2}\pi _{2}^{\ast }\left( g_{2}\right)$$with a smooth function $f:M_{1}\rightarrow \left( 0,\infty \right) $ where $%
\pi _{i}:M_{1}\times _{f}M_{2}\rightarrow M_{i}$ is the natural projection map of the Cartesian product $M_{1}\times M_{2}$ onto $M_{i},$ $i=1,2$ and $%
\ast $ denotes the pull-back operator on tensors. The factors $\left(
M_{1},g_{1}\right) $ and $\left( M_{2},g_{2}\right) $ are usually called the base manifold and fiber manifold respectively while $f$ is called as the warping function[@Bishop1969; @Oneill1983]. In particular, if $f=1$, then $M_{1}\times _{1}M_{2}=M_{1}\times M_{2}$ is the usual Cartesian product manifold. It is clear that the submanifold $M_{1}\times \{q\}$ is isometric to $M_{1}$ for every $q\in M_{2}$. Moreover, $\{p\}\times M_{2}$ is homothetic to $M_{2}$ for every $p\in M_{1}$. Throughout this article we use the same notation for a vector field and for its lift to the product manifold.
Generalized Robertson-Walker spacetimes are well-known examples of warped product spaces. A generalized Robertson-Walker spacetime is the warped product $\bar{M}=I\times _{f}M$ with fiber $\left( M,g\right) $ any $n-$dimensional Riemannian manifold and base an open connected subinterval $%
\left( I,-\mathrm{d}t^{2}\right) $ of the real line $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ endowed with the metric$$\bar{g}=-\mathrm{dt}^{2}\oplus f^{2}g$$where $\mathrm{d}t^{2}$ is the Euclidean metric on $I$. The family of generalized Robertson-Walker spacetimes $\bar{M}=I\times _{f}M$ widely extends the classical Robertson-Walker spacetimes $I\times _{f}S_{k}$ where $%
M=S_{k}$ is a $3-$dimensional Riemannian manifold of constant sectional curvature $k$. The warping function $f\left( t\right) $ is sometimes called the scale factor. This factor tells us how big is the space-like slice at sometime $t$.
For example the $\left( n+1\right) -$dimensional spherically symmetric Friedmann–Robertson–Walker metric is given by$$ds^{2}=-dt^{2}+f^{2}\left( t\right) \left( \frac{dr^{2}}{1-kr^{2}}%
+r^{2}d\Omega _{n-1}^{2}\right)$$where the spherical sector is given by $d\Omega _{n-1}^{2}=d\theta
_{1}^{2}+\sin ^{2}\theta _{1}d\theta _{2}^{2}+...+\sin ^{2}\theta
_{n-2}d\theta _{n-1}$ [@Garcia:2007]. The Einstein field equation for $%
\left( n+1\right) -$dimensional spacetime is given by$$\mathrm{Ric}-\frac{r}{2}g=k_{n}T$$where $k_{n}$ is the multidimensional gravitational constant[Garcia:2007]{}.
The following results are special cases of similar results on warped product manifolds[@Bishop1969; @Oneill1983; @Shenawy:2016; @Shenawy:2015]. Let $\bar{M%
}=I\times _{f}M$ be a generalized Robertson-Walker spacetime equipped with the metric tensor $\bar{g}=-\mathrm{d}t^{2}\oplus f^{2}g$. Then the Levi-Civita connection $\bar{D}$ on $\bar{M}$ is
$$\begin{tabular}{lll}
$\bar{D}_{\partial _{t}}\partial _{t}=0$ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ & $\bar{D}_{\partial _{t}}X=\bar{D}_{X}\partial _{t}=\frac{\dot{f}}{f}X$
\\
\multicolumn{3}{l}{$\bar{D}_{X}Y=D_{X}Y-f\dot{f}g\left( X,Y\right) \partial
_{t}$}%
\end{tabular}
\label{Connection}$$
for any vector fields $X,Y\in \mathfrak{X}(M)$ where $D$ is the Levi-Civita connection on $M$ and dots indicate differentiation with respect to $t$. The curvature tensor of $\bar{M}$ is given by$$\begin{tabular}{lll}
\multicolumn{3}{l}{$\mathrm{\bar{R}}\left( \partial _{t},\partial
_{t}\right) \partial _{t}=\mathrm{\bar{R}}\left( \partial _{t},\partial
_{t}\right) X=\mathrm{\bar{R}}\left( X,Y\right) \partial _{t}=0$} \\
$\mathrm{\bar{R}}\left( X,\partial _{t}\right) \partial _{t}=-\frac{\ddot{f}%
}{f}X$ & & $\mathrm{\bar{R}}\left( \partial _{t},X\right) Y=f\ddot{f}%
g\left( X,Y\right) \partial _{t}$ \\
\multicolumn{3}{l}{$\mathrm{\bar{R}}\left( X,Y\right) Z=\mathrm{R}\left(
X,Y\right) Z+\dot{f}^{2}\left[ g\left( X,Z\right) Y-g\left( Y,Z\right) X%
\right] $}%
\end{tabular}
\label{Curvature}$$
where $\mathrm{R}$ is curvature tensor of $M$. Finally, the Ricci curvature tensor $\mathrm{\bar{R}ic}$ on $\bar{M}$ is$$\begin{tabular}{lll}
$\mathrm{\bar{R}ic}\left( \partial _{t},\partial _{t}\right) =\frac{n\ddot{f}%
}{f}$ & \ \ \ \ \ \ \ \ \ \ \ \ \ & $\mathrm{\bar{R}ic}\left( X,\partial
_{t}\right) =0$ \\
\multicolumn{3}{l}{$\mathrm{\bar{R}ic}\left( X,Y\right) =$ \textrm{$Ric$}$%
\left( X,Y\right) -f^{\diamond }g\left( X,Y\right) $}%
\end{tabular}
\label{Ricci}$$where $f^{\diamond }=-f\ddot{f}-\left( n-1\right) \dot{f}^{2}$.
Now, we will recall the definitions of conformal vector fields and some collineations on an arbitrary pseudo-Riemannian manifold. Let $\left(
M,g,D\right) $ be a pseudo-Riemannian manifold with metric $g$ where $D$ is the Levi-Civita connection on $M$. A vector field $\zeta \in \mathfrak{X}%
\left( M\right) $ is called a Killing vector field if$$\mathcal{L}_{\zeta }g=0$$It is easy to show that
$$\left( \mathcal{L}_{\zeta }g\right) (X,Y)=g(D_{X}\zeta ,Y)+g(X,D_{Y}\zeta )$$
for any $X,Y\in \mathfrak{X}\left( M\right) $. By using symmetry of this equation, we get that $\zeta $ is a Killing vector field if and only if$$g(D_{X}\zeta ,X)=0$$for any vector field $X\in \mathfrak{X}\left( M\right) $. A pseudo-Riemannian $n-$dimensional manifold has at most $n\left( n+1\right)
/2 $ independent Killing vector fields. The symmetry generated by Killing vector fields is called isometry. A pseudo-Riemannian manifold which admits a maximum such symmetry has a constant sectional curvature.
A vector field $\zeta $ is called a conformal vector field if$$\mathcal{L}_{\zeta }g=\rho g$$for some smooth function $\rho :M\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$. $\zeta $ is called homothetic if $\rho $ is constant and Killing if $\rho
=0$. Also, $\zeta $ is called a concircular vector field if$$D_{X}\zeta =\rho X$$for any vector field $X\in \mathfrak{X}\left( M\right) $[@Chen2014]. Let $\zeta \in \mathfrak{X}\left( M\right) $ be a concircular vector field on $M$, then$$\mathcal{L}_{\zeta }g(X,Y)=2\rho g(X,Y)$$i.e. $\zeta $ is a conformal vector field with conformal factor $2\rho $. A concircular vector field is a parallel vector field if $\rho =0$. Moreover, for a constant factor $\rho $, we have$$R\left( X,Y\right) \zeta =0$$
A Riemannian manifold $M$ is said to admit a curvature collineation if the Lie derivative of the curvature tensor $\mathrm{R}$ vanishes in the direction of a vector field $\zeta \in \mathfrak{X}\left( M\right) $, that is$$\mathcal{L}_{\zeta }\mathrm{R}=0$$where $\mathrm{R}$ is the Riemann curvature tensor. Likewise, $M$ is said to admit a Ricci curvature collineation if there is a vector field $\zeta \in
\mathfrak{X}\left( M\right) $ such that$$\mathcal{L}_{\zeta }\mathrm{Ric}=0$$where $\mathrm{Ric}$ is the Ricci curvature tensor. It is clear that every Killing vector field is a curvature collineation and every curvature collineation is a Ricci curvature collineation. The converse is not generally true. Finally, a spacetime $M$ is said to admit a matter collineation if there is a vector field $\zeta \in \mathfrak{X}\left(
M\right) $ such that$$\mathcal{L}_{\zeta }\mathrm{T}=0$$where $\mathrm{T}$ is the energy-momentum tensor. The Einstein’s field equation (with cosmological constant) is given by$$\mathrm{Ric}-\frac{r}{2}g=\kappa \mathrm{T}-\lambda g$$where $r$ is the scalar curvature and $\lambda $ is the cosmological constant. Suppose that $\zeta $ is a Killing vector field, then$$\mathcal{L}_{\zeta }T=0$$i.e.$\zeta $ is a matter collineation field. Note that a matter collineation need not be a Killing vector field.
Symmetries of generalized Robertson-Walker spacetimes
=====================================================
In this section, we investigate several types of symmetries of generalized Robertson-Walker spacetimes. Necessary and sufficient conditions are derived for a generalized Robertson-Walker spacetime to admit a conformal vector field or a collineation.
We begin this section with the following well-known proposition[Shenawy:2015]{}. Let $\bar{M}=I\times _{f}M$ be a generalized Robertson-Walker spacetime equipped with the metric tensor $\bar{g}%
=-dt^{2}\oplus f^{2}g$.
Suppose that $h\partial _{t},x\partial _{t},y\partial _{t}\in \mathfrak{X}%
(I) $ and $\zeta ,X,Y\in \mathfrak{X}(M)$, then$$\left( \mathcal{\bar{L}}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},\bar{Y}%
\right) =-2\dot{h}xy+f^{2}\left( \mathcal{L}_{\zeta }g\right) \left(
X,Y\right) +2hf\dot{f}g\left( X,Y\right) \label{e1}$$where $\bar{\zeta}=h\partial _{t}+\zeta ,$ $\bar{X}=x\partial _{t}+X$ and $%
\bar{Y}=y\partial _{t}+Y$.
An important consequence of this proposition is the following.
Let $\bar{M}=I\times _{f}M$ be a generalized Robertson-Walker spacetime equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Then,
1. a time-like vector field $\bar{\zeta}=h\partial _{t}\in \mathfrak{X}(%
\bar{M})$ is a conformal vector field on $\bar{M}$ if and only if $h=af$ where $a$ is constant. Moreover, the conformal factor is $2\dot{h}.$
2. a space like $\bar{\zeta}=\zeta \in \mathfrak{X}(\bar{M})$ is a Killing vector field on $\bar{M}$ if and only if $\zeta \in \mathfrak{X}(M)$ is Killing vector field on $M$.
3. a space like $\bar{\zeta}=\zeta \in \mathfrak{X}(\bar{M})$ is a Matter collineation on $\bar{M}$ if $\zeta \in \mathfrak{X}(M)$ is a Killing vector field on $M$.
Suppose that $h=af$. If $h=0$, then $a=0$ and the result is obvious. Now, we assume that $h\neq 0$. Using equation (\[e1\]) from the above proposition we get that$$\begin{aligned}
\left( \mathcal{\bar{L}}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},\bar{Y}%
\right) &=&-2\dot{h}xy+f^{2}\left( \mathcal{L}_{\zeta }g\right) \left(
X,Y\right) +2hf\dot{f}g\left( X,Y\right) \\
&=&2\dot{h}\bar{g}\left( \bar{X},\bar{Y}\right)\end{aligned}$$i.e.$\bar{\zeta}=h\partial _{t}$ is a conformal vector field with conformal factor $\rho =2\dot{h}$. Conversely, suppose that $\bar{\zeta}=h\partial
_{t}\in \mathfrak{X}(\bar{M})$ is a conformal vector field with factor $\rho
,$ then$$\left( \mathcal{\bar{L}}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},\bar{Y}%
\right) =\rho \bar{g}\left( \bar{X},\bar{Y}\right)$$for any vector fields $\bar{X},\bar{Y}\in \mathfrak{X}(\bar{M})$. Now, by equation (\[e1\]), we get that$$-\rho xy+\rho f^{2}g\left( X,Y\right) =-2\dot{h}xy+2hf\dot{f}g\left(
X,Y\right)$$Let $X=Y=0$, we get that $\rho =2\dot{h}$. Now, let us put $x=y=0$. This yields $\rho f=2h\dot{f}$. These two differential equations imply that $h=0$ or$$\dot{h}f=h\dot{f}$$and so $h=af$ where $a$ is constant. The second and third assertions are direct from equation (\[e1\]) when $h=0$.
It is well-known that if $X$ and $Y$ are conformal vector fields on $\bar{M}$ with $X=\mu Y$ for some smooth function $\mu $, then $\mu $ is constant. Thus the above result represents a good characterization to both time-like conformal vector fields and space-like Killing vector fields on $\bar{M}$.
A vector field $\bar{\zeta}=h\partial _{t}+\zeta $ on $\bar{M}=I\times
_{f}M $ is conformal if and only if $\zeta $ is a conformal vector field on $%
M$ with factor $\rho =2\left( \dot{h}-\frac{h\dot{f}}{f}\right) $. Moreover, the conformal factor of $\bar{\zeta}$ is $2\dot{h}$.
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a conformal vector field on $\bar{%
M}=I\times _{f}M$ with factor $\bar{\rho}$, then$$-\bar{\rho}xy+\bar{\rho}f^{2}g\left( X,Y\right) =-2\dot{h}xy+f^{2}\left(
\mathcal{L}_{\zeta }g\right) \left( X,Y\right) +2hf\dot{f}g\left( X,Y\right)$$for any vector fields $\bar{X}=x\partial _{t}+X$ and $\bar{Y}=y\partial
_{t}+Y$. This equation yields $\bar{\rho}=2\dot{h}$ and hence$$\left( \mathcal{L}_{\zeta }g\right) \left( X,Y\right) =\left( \bar{\rho}-%
\frac{2h\dot{f}}{f}\right) g\left( X,Y\right)$$i.e. $\zeta $ is a conformal vector field on $M$ with conformal factor $%
\rho =2\dot{h}-\frac{2h\dot{f}}{f}$. The converse is direct.
These two results reveal that the dimension of the conformal algebra $%
C\left( \bar{M}\right) $ of $\bar{M}$ is at least $r+1$ where $r$ is the dimension of $C\left( M\right) $. Suppose that $M$ is maximally symmetric, then$$\frac{n^{2}+3n+4}{2}\leq \dim C\left( \bar{M}\right) \leq \frac{\left(
n+2\right) \left( n+3\right) }{2}$$
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a Killing vector field on a generalized Robertson-Walker spacetime $\bar{M}$ and let $r=\left(
1/2\right) \bar{g}\left( \bar{\zeta},\bar{\zeta}\right) $. Then$$\bar{g}\left( \bar{\nabla}r,\bar{X}\right) =-\bar{g}\left( \bar{D}_{\bar{%
\zeta}}\bar{\zeta},\bar{X}\right)$$for any vector field $\bar{X}\in \mathfrak{X}\left( \bar{M}\right) $ i.e. $%
\bar{\nabla}r=-\bar{D}_{\bar{\zeta}}\bar{\zeta}$. Thus the Hessian definition with some computations yield$$\bar{H}^{r}\left( \bar{X},\bar{X}\right) =-\mathrm{\bar{R}}\left( \bar{\zeta}%
,\bar{X},\bar{\zeta},\bar{X}\right) +\bar{g}\left( \bar{D}_{\bar{X}}\bar{%
\zeta},\bar{D}_{\bar{X}}\bar{\zeta}\right)$$Taking the trace of both sides imply$$\bar{\Delta}r=-\mathrm{\bar{R}ic}\left( \bar{\zeta},\bar{\zeta}\right) +\bar{%
g}\left( \bar{D}\bar{\zeta},\bar{D}\bar{\zeta}\right)$$Assume that $\dot{f}=\dot{h}=0$. Then$$\bar{\Delta}r=-\mathrm{Ric}\left( \zeta ,\zeta \right) +f^{2}g\left( D\zeta
,D\zeta \right)$$
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a Killing vector field on a generalized Robertson-Walker spacetime $\bar{M}$ and let $r=\frac{1}{2}\bar{g%
}\left( \bar{\zeta},\bar{\zeta}\right) $. Then$$\bar{\Delta}r=-\mathrm{\bar{R}ic}\left( \bar{\zeta},\bar{\zeta}\right) +\bar{%
g}\left( \bar{D}\bar{\zeta},\bar{D}\bar{\zeta}\right)$$Moreover, if $\dot{f}=\dot{h}=0$, then$$\bar{\Delta}r=-\mathrm{Ric}\left( \zeta ,\zeta \right) +f^{2}g\left( D\zeta
,D\zeta \right)$$
The following result is an analogue of a similar result in Riemannian manifolds.
A Killing vector field $\bar{\zeta}=h\partial _{t}+\zeta $ on a generalized Robertson-Walker spacetime $\bar{M}=I\times _{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$ has a constant length if and only if $%
\zeta $ satisfies$$D_{\zeta }\zeta +\dfrac{2h\dot{f}}{f}\zeta =0\text{ and }h\dot{h}-f\dot{f}%
g\left( \zeta ,\zeta \right) =0 \label{e8}$$
Suppose that $\bar{\zeta}$ is a Killing vector field. Then$$\bar{g}\left( \bar{D}_{\bar{X}}\bar{\zeta},\bar{Y}\right) +\bar{g}\left(
\bar{X},\bar{D}_{\bar{Y}}\bar{\zeta}\right) =0$$for any vector fields $\bar{X},\bar{Y}\in \mathfrak{X}\left( \bar{M}\right) $. Let $\bar{X}=\bar{\zeta}$ in the above equation, then$$\bar{g}\left( \bar{D}_{\bar{\zeta}}\bar{\zeta},\bar{Y}\right) =-\bar{g}%
\left( \bar{\zeta},\bar{D}_{\bar{Y}}\bar{\zeta}\right)$$But Equations (\[Connection\]) and (\[e8\]) yield $\bar{D}_{\bar{\zeta}}%
\bar{\zeta}=0$ and so$$\bar{g}\left( \bar{\zeta},\bar{D}_{\bar{Y}}\bar{\zeta}\right) =0$$i.e. $\bar{g}\left( \bar{\zeta},\bar{\zeta}\right) $ is constant and so $%
\bar{\zeta}$ has a constant length. Conversely, if $\bar{\zeta}$ has constant length, then$$\bar{g}\left( \bar{\zeta},\bar{D}_{\bar{Y}}\bar{\zeta}\right) =-\bar{g}%
\left( \bar{D}_{\bar{\zeta}}\bar{\zeta},\bar{Y}\right) =0$$for any vector field $\bar{Y}\in \mathfrak{X}\left( \bar{M}\right) $ i.e. $%
\bar{D}_{\bar{\zeta}}\bar{\zeta}=0$. Now, we can use Equation ([Connection]{}) to get the result.
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a Killing vector field $of$ constant length on a generalized Robertson-Walker spacetime $\bar{M}=I\times
_{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Then the flow lines of $\zeta $ are geodesics on $M$ if and only if $f$ is constant or $h=0$.
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a Killing vector field on a generalized Robertson-Walker spacetime $\bar{M}=I\times _{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$ and $\alpha (s)$, $s\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, be a geodesic on $(\bar{M},\bar{g})$ with tangent vector field $\bar{X}%
=x\partial _{t}+X$. Assume that $\dot{f}=0$. Then $h$ is constant and $\zeta
\in \mathfrak{X}\left( M\right) $ is a Jacobi vector field along the integral curves of $X$.
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a conformal vector field along a curve $\alpha (s)$ with unit tangent vector $\bar{V}=v\partial _{t}+V$ on a generalized Robertson-Walker spacetime $\bar{M}=I\times _{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Then the conformal factor $%
\rho $ of $\bar{\zeta}$ is given by$$\rho =2\left[ v^{2}\dot{h}+hf\dot{f}\left\Vert V\right\Vert
^{2}+f^{2}g\left( D_{V}\zeta ,V\right) \right]$$
Let $\bar{\zeta}$ be a conformal vector field with conformal factor $\rho $. Then$$\left( \mathcal{\bar{L}}_{\bar{\zeta}}\bar{g}\right) (\bar{X},\bar{Y})=\rho
\bar{g}\left( \bar{X},\bar{Y}\right)$$Let us put $\bar{X}=\bar{Y}=\bar{V}$, then the conformal factor $\rho $ is given by$$\rho =2\bar{g}(\bar{D}_{\bar{V}}\bar{\zeta},\bar{V})$$Thus$$\begin{aligned}
\rho &=&2\bar{g}(\bar{D}_{\bar{V}}\bar{\zeta},\bar{V}) \\
&=&2g\left( v\dot{h}\partial _{t}+\frac{v\dot{f}}{f}\zeta +\frac{h\dot{f}}{f}%
V+D_{V}\zeta -f\dot{f}g(\zeta ,V)\partial _{t},V\right) \\
&=&2\left[ v^{2}\dot{h}+hf\dot{f}\left\Vert V\right\Vert ^{2}+f^{2}g\left(
D_{V}\zeta ,V\right) \right] \end{aligned}$$
In the sequential, we study the structure of concircular vector fields on generalized Robertson-Walker spacetimes.
\[concircular1\]Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a vector field on a generalized Robertson-Walker spacetime $\bar{M}=I\times _{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Then $\bar{%
\zeta}$ is a concircular vector field on $\bar{M}$ if and only if one of the following conditions holds:
1. $h=af$ and $\zeta =0$, or
2. $\zeta $ is a concircular vector field on $M$ with factor $\rho =\dot{h%
}$ and $f$ is constant.
Let $\bar{X}=x\partial _{t}+X$ be any vector field on $\bar{M}$. Then for any scalar function $\rho $ we get that$$\bar{D}_{\bar{X}}\bar{\zeta}-\rho \bar{X}=\left( x\dot{h}-f\dot{f}g\left(
X,\zeta \right) -x\rho \right) \partial _{t}+\frac{x\dot{f}}{f}\zeta +\frac{h%
\dot{f}}{f}X+D_{X}\zeta -\rho X$$
Suppose that $\bar{\zeta}$ is concircular on $\bar{M}$, then$$\begin{aligned}
x\dot{h}-f\dot{f}g\left( X,\zeta \right) -x\rho &=&0 \\
\frac{x\dot{f}}{f}\zeta +\frac{h\dot{f}}{f}X+D_{X}\zeta -\rho X &=&0\end{aligned}$$If $\dot{f}$ does not vanish, then $g\left( X,\zeta \right) =0$ for all $X$ i.e. $\zeta =0$. Thus$$\begin{aligned}
x\left( \dot{h}-\rho \right) &=&0 \\
\left( \frac{h\dot{f}}{f}-\rho \right) X &=&0\end{aligned}$$and therefore $\dot{h}=\frac{h\dot{f}}{f}$ i.e. $h=af$ for some constant.
However, if $\dot{f}=0$, then we get that$$\begin{aligned}
x\left( \dot{h}-\rho \right) &=&0 \\
D_{X}\zeta -\rho X &=&0\end{aligned}$$i.e. $\zeta $ is a concircular vector field on $M$ with factor $\rho =\dot{h}
$.
Conversely, we have$$\bar{D}_{\bar{X}}\bar{\zeta}=\left[ x\dot{h}-f\dot{f}g\left( X,\zeta \right) %
\right] \partial _{t}+D_{X}\zeta +\left( x\frac{\dot{f}}{f}\right) \zeta
+\left( h\frac{\dot{f}}{f}\right) X$$Finally, any one of the above conditions implies that $\bar{D}_{\bar{X}}\bar{%
\zeta}=\dot{h}\bar{X}$ and consequently $\bar{\zeta}$ is concircular on $%
\bar{M}$.
\[concircular2\]Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a concircular vector field on a generalized Robertson-Walker spacetime $\bar{M}=I\times
_{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Then $${\mathrm{Ric}}\left( \zeta ,\zeta \right) =0\text{ \ \ \ and\ \ \ }\kappa
\left( X,\zeta \right) =0$$for any vector field $X\in \mathfrak{X}(M)$.
Using the above theorem we get that $\zeta $ is zero or concircular with factor $\dot{h}$. If $\zeta =0$, then [$\mathrm{Ric}$]{}$\left( \zeta ,\zeta
\right) =0$ and $\kappa \left( X,\zeta \right) =0$. Now suppose that $\zeta $ is concircular on $M$ i.e.$$D_{X}\zeta =\dot{h}X$$for any vector field $X\in \mathfrak{X}(M)$. Let $e\in \mathfrak{X}(M)$ be a unit vector field, then$$\begin{aligned}
R\left( \zeta ,e,\zeta ,e\right) &=&g\left( -D_{\zeta }D_{e}\zeta
+D_{e}D_{\zeta }\zeta +D_{\left[ \zeta ,e\right] }\zeta ,e\right) \\
&=&g\left( -D_{\zeta }\left( \dot{h}e\right) +D_{e}\left( \dot{h}\zeta
\right) +\dot{h}\left[ \zeta ,e\right] ,e\right) \\
&=&\dot{h}g\left( -D_{\zeta }e+D_{e}\zeta +\left[ \zeta ,e\right] ,e\right)
\\
&=&0\end{aligned}$$Thus [$\mathrm{Ric}$]{}$\left( \zeta ,\zeta \right) =0$ and $\kappa \left(
X,\zeta \right) =0$.
Now, let us turn to curvature collineations. First, we present the following important proposition.
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a vector field on a generalized Robertson-Walker spacetime $\bar{M}=I\times _{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Then$$\begin{aligned}
\left( \mathcal{\bar{L}}_{h\partial _{t}}\mathrm{\bar{R}}\right) \left(
\partial _{t},\partial _{t},\partial _{t},\partial _{t}\right) &=&\left(
\mathcal{\bar{L}}_{h\partial _{t}}\mathrm{\bar{R}}\right) \left( X,\partial
_{t},\partial _{t},\partial _{t}\right) =0 \\
\left( \mathcal{\bar{L}}_{h\partial _{t}}\mathrm{\bar{R}}\right) \left(
X,Y,\partial _{t},\partial _{t}\right) &=&\left( \mathcal{\bar{L}}%
_{h\partial _{t}}\mathrm{\bar{R}}\right) \left( X,Y,Z,\partial _{t}\right) =0
\\
\left( \mathcal{\bar{L}}_{h\partial _{t}}\mathrm{\bar{R}}\right) \left(
X,Y,Z,W\right) &=&0\end{aligned}$$$$\left( \mathcal{\bar{L}}_{h\partial _{t}}\mathrm{\bar{R}}\right) \left(
Y,\partial _{t},\partial _{t},W\right) =-\left[ h\dot{f}\ddot{f}+hf\dddot{f}%
+2\dot{h}f\ddot{f}\right] g\left( Y,W\right)$$$$\begin{aligned}
\left( \mathcal{\bar{L}}_{\zeta }\mathrm{\bar{R}}\right) \left( \partial
_{t},\partial _{t},\partial _{t},\partial _{t}\right) &=&\left( \mathcal{%
\bar{L}}_{\zeta }\mathrm{\bar{R}}\right) \left( X,\partial _{t},\partial
_{t},\partial _{t}\right) =0 \\
\left( \mathcal{\bar{L}}_{\zeta }\mathrm{\bar{R}}\right) \left( X,Y,\partial
_{t},\partial _{t}\right) &=&\left( \mathcal{\bar{L}}_{\zeta }\mathrm{\bar{R}%
}\right) \left( X,Y,Z,\partial _{t}\right) =0\end{aligned}$$$$\left( \mathcal{\bar{L}}_{\zeta }\mathrm{\bar{R}}\right) \left( Y,\partial
_{t},\partial _{t},W\right) =-f\ddot{f}\left( \mathcal{L}_{\zeta }g\right)
\left( Y,W\right)$$$$\begin{aligned}
\left( \mathcal{\bar{L}}_{\zeta }\mathrm{\bar{R}}\right) \left(
X,Y,Z,W\right) &=&f^{2}\left( \mathcal{L}_{\zeta }R\right) \left(
X,Y,Z,W\right) \\
&&+f^{2}\dot{f}^{2}\left[ \left( \mathcal{L}_{\zeta }g\right) \left(
X,Z\right) g\left( Y,W\right) +g\left( X,Z\right) \left( \mathcal{L}_{\zeta
}g\right) \left( Y,W\right) \right] \\
&&-f^{2}\dot{f}^{2}\left[ \left( \mathcal{L}_{\zeta }g\right) \left(
Y,Z\right) g\left( X,W\right) +g\left( Y,Z\right) \left( \mathcal{L}_{\zeta
}g\right) \left( X,W\right) \right]\end{aligned}$$where $X,Y,Z,W\in \mathfrak{X}(M)$.
The following theorem represents a characterization of curvature collineations on generalized Robertson-Walker spacetimes.
Let $\bar{M}=I\times _{f}M$ be a generalized Robertson-Walker spacetime equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Then,
1. $\bar{\zeta}=h\partial _{t}$ is a curvature collineation on $\bar{M}$ if and only if $h\dot{f}\ddot{f}+hf\dddot{f}+2\dot{h}f\ddot{f}=0$.
2. $\bar{\zeta}=\zeta $ is a curvature collineation on $\bar{M}$ if $%
\zeta $ is a Killing vector field on $M$.
Let $\bar{\zeta}=h\partial _{t}+\zeta $ be a curvature collineation on a generalized Robertson-Walker spacetime $\bar{M}=I\times _{f}M$ equipped with the metric tensor $\bar{g}=-dt^{2}\oplus f^{2}g$. Assume that $\Delta f\neq
0 $. Then $\zeta $ is a Killing vector field on $M$.
Now we will do the same job for Ricci collineations on generalized Robertson-Walker spacetimes
Let $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $, then$$\begin{aligned}
\left( \mathcal{\bar{L}}_{\bar{\zeta}}\mathrm{\bar{R}ic}\right) \left( \bar{X%
},\bar{Y}\right) &=&\frac{nxy}{f^{2}}\left( hf\dddot{f}-h\dot{f}\ddot{f}+2%
\dot{h}f\ddot{f}\right) +h\left( f\dddot{f}+\left( 2n-1\right) \dot{f}\ddot{f%
}\right) g\left( X,Y\right) \\
&&+\left( \mathcal{L}_{\zeta }\mathrm{Ric}\right) \left( X,Y\right)
-f^{\diamond }\left( \mathcal{L}_{\zeta }g\right) \left( X,Y\right)\end{aligned}$$for any $X,Y\in \mathfrak{X}(M)$
The following results are immediate consequences of the above proposition.
Let $\bar{\zeta}=h\partial _{t}\in \mathfrak{X}\left( \bar{M}\right) $ be a vector field on a generalized Robertson-Walker spacetime $\bar{M}=I\times
_{f}M$. Assume that $H^{f}=0$. Then, $\bar{\zeta}$ is a Ricci collineation on $\bar{M}$.
Now, we consider the converse of the above result.
Let $\bar{\zeta}=h\partial _{t}\in \mathfrak{X}\left( \bar{M}\right) $ be a Ricci collineation on a generalized Robertson-Walker spacetime $\bar{M}%
=I\times _{f}M$. Then one of the following conditions holds
1. $H^{f}=0$, or
2. $h=af^{n}$ for some constant $a$.
Suppose that $\bar{\zeta}$ is a Ricci collineation on $\bar{M}$. Then$$\begin{aligned}
\left( \mathcal{\bar{L}}_{\bar{\zeta}}\mathrm{\bar{R}ic}\right) \left(
x\partial _{t},y\partial _{t}\right) &=&\frac{nxy}{f^{2}}\left( hf\dddot{f}-h%
\dot{f}\ddot{f}+2\dot{h}f\ddot{f}\right) =0 \\
\left( \mathcal{\bar{L}}_{\bar{\zeta}}\mathrm{\bar{R}ic}\right) \left(
X,Y\right) &=&h\left( f\dddot{f}+\left( 2n-1\right) \dot{f}\ddot{f}\right)
g\left( X,Y\right) =0\end{aligned}$$The second equation yields$$f\dddot{f}=-\left( 2n-1\right) \dot{f}\ddot{f}$$and so the first equation implies$$2\ddot{f}\left[ nh\dot{f}-\dot{h}f\right] =0$$which proves the result.
Let $\bar{\zeta}=\zeta \in \mathfrak{X}\left( \bar{M}\right) $ be a vector field on a generalized Robertson-Walker spacetime $\bar{M}=I\times _{f}M$. Assume that $f^{\diamond }=0$. Then, $\bar{\zeta}$ is a Ricci collineation on $\bar{M}$ if and only if $\zeta $ is a Ricci collineation on $M$.
A vector field $\zeta \in \mathfrak{X}\left( M\right) $ is called a conformal Ricci collineation if$$\left( \mathcal{L}_{\zeta }\mathrm{Ric}\right) \left( X,Y\right) =\rho
g\left( X,Y\right)$$for some smooth function $\rho $ on $M$.
Let $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $ be a Ricci collineation on a generalized Robertson-Walker spacetime $\bar{M}%
=I\times _{f}M$. Then, $\zeta $ is a conformal Ricci collineation on $M$ if and only if $\zeta $ is a conformal vector field on $M$.
Ricci Soliton
=============
A smooth vector field $\zeta $ on a Riemannian manifold $\left( M,g\right) $ is said to define a Ricci soliton $\left( M,g,\zeta ,\lambda \right) $ if it satisfies the soliton equation$$\frac{1}{2}\mathcal{L}_{\zeta }g+\mathrm{Ric}=\lambda g
\label{Ricci soliton}$$where $\mathcal{L}_{\zeta }$ is the Lie-derivative with respect to $\zeta $, is the Ricci tensor and $\lambda $ is a constant. If $\zeta =%
\mathrm{grad}u$, for a smooth function $u$ on $M$, the Ricci soliton $\left(
M,g,\zeta ,\lambda \right) =\left( M,g,u,\lambda \right) $ is called a gradient Ricci soliton and the function $u$ is called the potential function. The study of Ricci solitons was first introduced by Hamilton as fixed or stationary points of the Ricci flow in the space of the metrics on $%
M$ modulo diffeomorphism and scaling. Gradient Ricci solitons are natural generalizations of Einstein manifolds[@Barros:2012; @Barros:2013; @Bernstein:2015; @Fernandez:2011; @Peterson:2009; @Munteanu:2013].
Let us take the Lie derivative of both sides of Equation (\[Ricci soliton\]) in direction of $\zeta $, then we have$$\frac{1}{2}\mathcal{L}_{\zeta }\mathcal{L}_{\zeta }g+\mathcal{L}_{\zeta }%
\mathrm{Ric}=\lambda \mathcal{L}_{\zeta }g$$A vector field $\zeta $ is called $2-$Killing if $\mathcal{L}_{\zeta }%
\mathcal{L}_{\zeta }g=0$. Thus the above equation reveals the following result.
Let $\left( M,g,\zeta ,\lambda \right) $ be a Ricci soliton where $\zeta $ is a $2-$Killing vector field. Then,
1. $\zeta $ is Killing if and only if $\left( M,g\right) $ is Einstein. Moreover, the Einstein factor is $\lambda $.
2. $\zeta $ is Killing if and only if $\zeta $ is a Ricci collineation.
Now, we consider a Ricci soliton structure on a generalized Robertson-Walker spacetime.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\bar{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $. Then
1. $h\partial _{t}$ is conformal on $I$ with factor $2\left( \lambda +n%
\frac{\ddot{f}}{f}\right) $, and
2. $\left( M,g,f^{2}\zeta ,\lambda f^{2}+f^{\diamond }-hf\dot{f}\right) $ is a Ricci soliton whenever $\lambda f^{2}+f^{\diamond }-hf\dot{f}$ is constant.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton, then$$\frac{1}{2}\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},%
\bar{Y}\right) +\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) =\lambda
\bar{g}\left( \bar{X},\bar{Y}\right)$$where $\bar{X}=x\partial _{t}+X$ and $\bar{Y}=y\partial _{t}+Y$ are vector fields on $\bar{M}$. Let $X=Y=0$, then$$\begin{aligned}
\frac{1}{2}\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( x\partial
_{t},y\partial _{t}\right) +\mathrm{\bar{R}ic}\left( x\partial
_{t},y\partial _{t}\right) &=&\lambda \bar{g}\left( x\partial _{t},y\partial
_{t}\right) \\
\frac{1}{2}\left( \mathcal{L}_{h\partial _{t}}^{I}g_{I}\right) \left(
x\partial _{t},y\partial _{t}\right) +{\mathrm{Ric}}^{I}\left( x\partial
_{t},y\partial _{t}\right) -\frac{n}{f}H^{f}\left( x\partial _{t},y\partial
_{t}\right) &=&\lambda g_{I}\left( x\partial _{t},y\partial _{t}\right) \\
\frac{1}{2}\left( \mathcal{L}_{h\partial _{t}}^{I}g_{I}\right) (x\partial
_{t},y\partial _{t}) &=&\left( \lambda +\frac{n}{f}\ddot{f}\right)
g_{I}\left( x\partial _{t},y\partial _{t}\right)\end{aligned}$$Thus, $h\partial _{t}$ is a conformal vector field on $I$ with factor $%
2\left( \lambda +n\frac{\ddot{f}}{f}\right) $. Now, let $x=y=0$, then$$\frac{1}{2}\left( f^{2}\left( \mathcal{L}_{\zeta }g\right) (X,Y)+2hf\dot{f}%
g(X,Y)\right) +{\mathrm{Ric}}\left( X,Y\right) -f^{\diamond }g\left(
X,Y\right) =\lambda f^{2}g\left( X,Y\right)$$where $f^{\diamond }=-f\ddot{f}-\left( n-1\right) \dot{f}^{2}$. Simply, we may rewrite it as follows,$$\frac{1}{2}f^{2}\left( \mathcal{L}_{\zeta }g\right) (X,Y)+{\mathrm{Ric}}%
\left( X,Y\right) =\left( f^{\diamond }-hf\dot{f}+\lambda f^{2}\right)
g\left( X,Y\right)$$Thus $\left( M,g,f^{2}\zeta ,\mu \right) $ is a Ricci soliton where $\mu
=\lambda f^{2}+f^{\diamond }-hf\dot{f}$.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\overline{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M%
}\right) $ be a conformal vector field on $\bar{M}$. Then $\left( M,g\right)
$ is an Einstein manifold with factor $$\mu =-\left[ \left( n+1\right) f\ddot{f}+\left( n-1\right) \dot{f}^{2}\right]$$Moreover, the conformal factor is $\lambda +\frac{n\ddot{f}}{f}$.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\overline{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M%
}\right) $ be a conformal vector field on $\overline{M}$. Then$$\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) =\left( \lambda -\rho
\right) \bar{g}\left( \bar{X},\bar{Y}\right)$$Then$$\begin{aligned}
&&{\mathrm{Ric}}^{I}\left( x\partial _{t},y\partial _{t}\right) -\frac{n}{f}%
H^{f}\left( x\partial _{t},y\partial _{t}\right) +{\mathrm{Ric}}\left(
X,Y\right) -f^{\diamond }g\left( X,Y\right) \\
&=&\left( \lambda -\rho \right) \left( g_{I}\left( x\partial _{t},y\partial
_{t}\right) +f^{2}g\left( X,Y\right) \right)\end{aligned}$$Let $X=Y=0$, then$$\mathrm{\bar{R}ic}\left( x\partial _{t},y\partial _{t}\right) =\left(
\lambda -\rho \right) \bar{g}\left( x\partial _{t},y\partial _{t}\right)$$Using Proposition (\[Ricci\]), we get that$$\begin{aligned}
{\mathrm{Ric}}^{I}\left( x\partial _{t},y\partial _{t}\right) -\frac{n}{f}%
H^{f}\left( x\partial _{t},y\partial _{t}\right) &=&\left( \lambda -\rho
\right) g_{I}\left( x\partial _{t},y\partial _{t}\right) \\
-\frac{n}{f}\ddot{f}g_{I}\left( x\partial _{t},y\partial _{t}\right)
&=&\left( \lambda -\rho \right) g_{I}\left( x\partial _{t},y\partial
_{t}\right)\end{aligned}$$Thus$$\lambda -\rho =-\frac{n\ddot{f}}{f} \label{E10}$$Now, we let $x=y=0$, then$$\begin{aligned}
\mathrm{\bar{R}ic}\left( X,Y\right) &=&\left( \lambda -\rho \right) \bar{g}%
\left( X,Y\right) \\
{\mathrm{Ric}}\left( X,Y\right) -f^{\diamond }g\left( X,Y\right) &=&\left(
\lambda -\rho \right) f^{2}g\left( X,Y\right) \\
{\mathrm{Ric}}\left( X,Y\right) &=&\left[ \left( \lambda -\rho \right)
f^{2}+f^{\diamond }\right] g\left( X,Y\right)\end{aligned}$$where $$f^{\diamond }=-f\ddot{f}-\left( n-1\right) \dot{f}^{2} \label{E11}$$By using equations (\[E10\]) and (\[E11\]), we get that$${\mathrm{Ric}}\left( X,Y\right) =-\left[ \left( n+1\right) f\ddot{f}+\left(
n-1\right) \dot{f}^{2}\right] g\left( X,Y\right) \notag$$Then $\left( M,g\right) $ is an Einstein manifold with factor $\mu =-\left[
\left( n+1\right) f\ddot{f}+\left( n-1\right) \dot{f}^{2}\right] $.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\bar{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $ be a conformal vector field on $\bar{M}$. Then $\left( M,g\right) $ is Ricci flat if $f$ is constant.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\bar{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $ be a Killing vector field on $\bar{M}$. Then $\lambda =-\frac{n\ddot{f}}{f}$.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\bar{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $. Assume that $H^{f}=0$ and $\left( M,g\right) $ is Einstein with factor $%
-\left( n-1\right) \dot{f}^{2}$. Then $\bar{\zeta}$ is conformal with factor $2\lambda $.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\bar{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $. Then$$\frac{1}{2}\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},%
\bar{Y}\right) +\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) =\lambda
\bar{g}\left( \bar{X},\bar{Y}\right)$$for any vector fields $\bar{X},\bar{Y}\in \mathfrak{X}\left( \bar{M}\right) $. Equation (\[Ricci\]) implies that$$\begin{aligned}
\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) &=&\mathrm{\bar{R}ic}\left(
x\partial _{t},y\partial _{t}\right) +\mathrm{\bar{R}ic}\left( X,Y\right) \\
&=&-\frac{n}{f}H^{f}\left( x\partial _{t},y\partial _{t}\right) +\mathrm{Ric}%
\left( X,Y\right) -f^{\diamond }g\left( X,Y\right) \\
&=&\mathrm{Ric}\left( X,Y\right) -f^{\diamond }g\left( X,Y\right) \\
&=&\mathrm{Ric}\left( X,Y\right) +\left( n-1\right) \dot{f}^{2}g\left(
X,Y\right) =0\end{aligned}$$Thus$$\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},\bar{Y}\right)
=2\lambda \bar{g}\left( \bar{X},\bar{Y}\right)$$i.e. $\bar{\zeta}$ is a conformal vector field with factor $2\lambda $.
Let $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ be a Ricci soliton where $\bar{M}=I\times _{f}M$ is a generalized Robertson-Walker spacetime and $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $ be a concircular vector field on $\overline{M}$ with factor one. Then $%
\left( M,g\right) $ is Ricci flat if $-\left[ \left( n+1\right) f\ddot{f}%
+\left( n-1\right) \dot{f}^{2}\right] $ is constant.
A concircular vector field is conformal, and so the above theorem implies that$${\mathrm{Ric}}\left( X,Y\right) =-\left[ \left( n+1\right) f\ddot{f}+\left(
n-1\right) \dot{f}^{2}\right] g\left( X,Y\right)$$for any vector fields $\bar{X},\bar{Y}\in \mathfrak{X}\left( \bar{M}\right) $. Since $\bar{\zeta}$ is concircular, Theorems (\[concircular1\]) and ([concircular2]{}) yield$$\mathrm{Ric}\left( \zeta ,\zeta \right) =0$$Therefore, for a constant factor $\mu =-\left[ \left( n+1\right) f\ddot{f}%
+\left( n-1\right) \dot{f}^{2}\right] $ we have$$\begin{aligned}
{\mathrm{Ric}}\left( X,Y\right) &=&\mu g\left( X,Y\right) \\
\mathrm{Ric}\left( \zeta ,\zeta \right) &=&0 \\
\mu &=&0\end{aligned}$$i.e. $\left( M,g\right) $ is Ricci flat.
Let $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $ be a vector field on a generalized Robertson-Walker spacetime $\overline{M}%
=I\times _{f}M$ Then $\left( \bar{M},\bar{g},\bar{\zeta},\lambda \right) $ is a Ricci soliton if
1. $\zeta $ is a conformal vector field with conformal factor $2\rho $,
2. $h\partial _{t}$ is a conformal vector field with conformal factor $%
2\sigma $,
3. $\left( M,g\right) $ is Einstein with factor $\mu $, and
4. $\left( \sigma -\rho \right) f^{2}=\mu +h\dot{f}f+\left( n+1\right) f%
\ddot{f}+\left( n-1\right) \dot{f}^{2}$
Moreover, $\lambda =\sigma -\frac{n\ddot{f}}{f}$.
Let $\bar{\zeta}=h\partial _{t}+\zeta \in \mathfrak{X}\left( \bar{M}\right) $, then$$\frac{1}{2}\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},%
\bar{Y}\right) =\frac{1}{2}\left( \left( \mathcal{L}_{h\partial
_{t}}g\right) (x\partial _{t},y\partial _{t})+f^{2}\left( \mathcal{L}_{\zeta
}g\right) (X,Y)+2hf\dot{f}g(X,Y)\right)$$and$$\begin{aligned}
\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) &=&{\mathrm{Ric}}\left(
X,Y\right) -\frac{n}{f}H^{f}(x\partial _{t},y\partial _{t})-f^{\diamond
}g\left( X,Y\right) \\
&=&{\mathrm{Ric}}\left( X,Y\right) -\frac{n\ddot{f}}{f}g_{I}(x\partial
_{t},y\partial _{t})-f^{\diamond }g\left( X,Y\right)\end{aligned}$$Since $\zeta $ is a conformal vector field with conformal factor $2\rho $ and $h\partial _{t}$ is a conformal vector field with conformal factor $%
2\sigma $,$$\begin{aligned}
\frac{1}{2}\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},%
\bar{Y}\right) &=&\frac{1}{2}\left( \left( \mathcal{L}_{h\partial
_{t}}g\right) (x\partial _{t},y\partial _{t})+f^{2}\left( \mathcal{L}_{\zeta
}g\right) (X,Y)+2hf\dot{f}g(X,Y)\right) \\
&=&\sigma g_{I}(x\partial _{t},y\partial _{t})+\rho f^{2}g(X,Y)+hf\dot{f}%
g(X,Y) \\
&=&\sigma g_{I}(x\partial _{t},y\partial _{t})+\left( \rho f^{2}+hf\dot{f}%
\right) g(X,Y)\end{aligned}$$and$$\begin{aligned}
\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) &=&\mu g\left( X,Y\right) -%
\frac{n\ddot{f}}{f}g_{I}(x\partial _{t},y\partial _{t})-f^{\diamond }g\left(
X,Y\right) \\
&=&-\frac{n\ddot{f}}{f}g_{I}(x\partial _{t},y\partial _{t})+\left( \mu
-f^{\diamond }\right) g\left( X,Y\right)\end{aligned}$$where $f^{\diamond }=-f\ddot{f}-\left( n-1\right) \dot{f}^{2}$. Thus
$$\begin{aligned}
&&\frac{1}{2}\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},%
\bar{Y}\right) +\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) \\
&=&\left( \sigma -\frac{n\ddot{f}}{f}\right) g_{I}(x\partial _{t},y\partial
_{t})+\left( \rho f^{2}+hf\dot{f}+\mu -f^{\diamond }\right) g(X,Y) \\
&=&\left( \sigma -\frac{n\ddot{f}}{f}\right) g_{I}(x\partial _{t},y\partial
_{t})+f^{2}\left( \rho +\frac{h\dot{f}}{f}+\frac{\mu -f^{\diamond }}{f^{2}}%
\right) g(X,Y)\end{aligned}$$
The last condition implies that$$\sigma -\frac{n\ddot{f}}{f}=\rho +\frac{h\dot{f}}{f}+\frac{\mu -f^{\diamond }%
}{f^{2}}=\lambda$$and so$$\begin{aligned}
\frac{1}{2}\left( \mathcal{L}_{\bar{\zeta}}\bar{g}\right) \left( \bar{X},%
\bar{Y}\right) +\mathrm{\bar{R}ic}\left( \bar{X},\bar{Y}\right) &=&\lambda %
\left[ g_{I}(x\partial _{t},y\partial _{t})+f^{2}g(X,Y)\right] \\
&=&\lambda \bar{g}\left( \bar{X},\bar{Y}\right)\end{aligned}$$and the proof is complete.
[99]{} A. Barros and E. Ribeiro Jr., *Some characterizations for compact almost Ricci solitons*, Proc. Amer. Math. Soc. **140**(2012), 1033-1040.
A. Barros, José N. Gomes, E. Ribeiro Jr. *A note on rigidity of the almost Ricci soliton,* Archiv der Mathematik, **100(**2013), no. 5, 481-490
V. N. Berestovskii, Yu. G. Nikonorov, *Killing vector fields of constant length on Riemannian manifolds*, Siberian Mathematical Journal, **49**(2008), Issue 3 , pp 395-407.
Jacob Bernstein and Thomas Mettler, *Two-Dimensional Gradient Ricci Solitons Revisited*, Int Math Res Not, **2015**(2015), no. 1, Pp. 78-98
A. L. Besse**,** *Einstein Manifolds*, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
R. L. Bishop and B. O’Neill, *Manifolds of negative curvature*, Trans. Amer. Math. Soc. **145** (1969), 1-49.
M. Caballero, A. Romero and R. M. Rubio, *Constant Mean Curvature Spacelike Hypersurfaces in Lorentzian Manifolds with a Timelike Gradient Conformal Vector Field*, Classical and Quantum Gravity, **28**(2011), no. 14, 145009(1-14).
B.-Y. Chen and S. W. Wei, *Differential Geometry of Submanifolds of Warped Product Manifolds* $I\times _{f}S^{m-1}\left(
k\right) $, Journal Geometry, **91**(2008), 21–42.
B.-Y. Chen, *A Simple Characterization Of Generalized Robertson-Walker Spacetimes*, General Relativity and Gravitation, **46**(2014), 1833-9.
S. Deshmukh , *Conformal vector fields and eigen vectors of the Laplacian operator,* Math. Phys. Anal. Geo., **15**(2012), pp. 163-172.
S. Deshmukh and F. R. Al-Solamy, *Conformal vector fields on a Riemannian manifold,* Balkan Journal of Geometry and Its Applications, **19**(2014), No.2, pp. 86-93.
S. Deshmukh and F. R. Al-Solamy, *A note on conformal vector fields on a Riemannian manifold,* Colloq. Math. **136** (2014), 65-73.
F. Dobarro, B. Unal, *Characterizing killing vector fields of standard static space-times*, J. Geom. Phys. **62** (2012), 1070–1087.
K. L. Duggala and R. Sharma,*Conformal killing vector fields on spacetime solutions of Einstein’s equations and initial data*, Nonlinear Analysis **63** (2005), 447-454.
M. Fernández-López, Eduardo García-Río, *Rigidity of shrinking Ricci solitons*, Mathematische Zeitschrift, 2011, Volume 269, Issue 1-2, pp 461-466
J. L. Flores and M. Sánchez, * Geodesic Connectedness and Conjugate Points in GRW Space-times*, J. Geom. Phys., **36** (2000), no.3-4, 285-314.
Alberto A. Garcia, Steve Carlip, $n-$*dimensional generalizations of the Friedmann–Robertson–Walker cosmology*, Physics Letters B 645 (2007) 101–107.
G. S. Hall, *Symmetries and Curvature Structure in General Relativity*, World Scientific Publishing Co. Ltd, London, 2004.
V.** **G. Ivancevic and T. T. Ivancevic, *Applied Differential Geometry: A Modern Introduction*, World Scientific Publishing Co. Ltd, London, 2007.
P. Petersen and W. Wylie, *Rigidity of gradient Ricci solitons*, Pacific Journal of Mathematics, **241**(2009), no. 2, 329-345.
W. Kuhnel and H. Rademacher, *Conformal vector fields on pseudo-Riemannian spaces*, Journal of Geometry and its Applications, **7**(1997), 237–250.
Ovidiu Munteanu, Natasa Sesum, *On Gradient Ricci Solitons*, Journal of Geometric Analysis, **23(**2013), no. 2, pp 539-561
K. Nomizu, *On local and global existence of Killing vector fields*, Ann. of Math. **72**(1960), no. 1, 105–120.
B. O’Neill, *Semi-Riemannian Geometry with Applications to Relativity*, Academic Press Limited, London, 1983.
M. Sánchez, * On the Geometry of Generalized Robertson-Walker Spacetimes: geodesics*, Gen. Relativ. Gravitation, **30** (1998), no.6, 915-932.
M. Sánchez, *On the Geometry of Generalized Robertson-Walker Spacetimes: Curvature and Killing fields*, J. Geom. Phys., **31** (1999), No.1, 1-15.
S. Shenawy and B. Unal. *The* $W_{2}-$*curvature tensor on warped product manifolds and applications*, International Journal of Geometric methods in Mathematical Physics, to appear.
S. Shenawy and B. Unal. $2-$*Killing vector fields on warped product manifolds*, International Journal of Mathematics, **26**(2015), no. 8, 1550065(17 pages).
M. Steller, *Conformal vector fields on spacetimes,* Ann Glob Anal Geom **29**(2006), 293–317.
S. Yorozu, *Killing Vector Fields on Non-Compact Riemannian Manifolds with Boundary*, Kodai Math. J. **5**(1982), 426–433.
|
---
abstract: 'We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.'
address:
- 'School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK'
- 'Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany'
- 'Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany'
author:
- Denis Denisov
- Elena Perfilev
- Vitali Wachtel
title: 'Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments'
---
Introduction and statement of results
=====================================
Let $\{S_n; n\geq 1\}$ be a random walk with i.i.d. increments $\lbrace X_k; k\geq 1\rbrace$. We shall assume that the increments have negative expected value, ${\mathbf E}X_1=-a$. Let $\overline F(x)={\mathbf P}(X_1>x)$ be the tail distribution function of $X_1$. Let $$\tau:=\min\lbrace n\geq1: S_n\leq 0 \rbrace$$ be the first time the random walk exits the positive half-line. We consider the area under the random walks excursion $\{S_1,S_2,\ldots,S_{\tau-1}\}$: $$A_\tau:=\sum_{k=0}^{\tau-1}S_k.$$ Since $\tau$ is finite almost surely, the area $A_\tau$ is finite as well. In this note we will study asymptotics for ${\mathbf P}(A_\tau>x),$ as $x\to \infty$, in the case when distribution of increments is heavy-tailed. This paper continues the research of [@PW17], where the light-tailed case has been considered.
The heavy-tailed asymptotics for ${\mathbf P}(A_\tau>x)$ was studied previously by Borovkov, Boxma and Palmowski [@BBP03]. They considered the case when the increments of the random walk have a distribution with regularly varying tail, that is $\overline F(x)=x^{-\alpha }L(x),$ where $L(x)$ is a slowly varying function. For $\alpha>1$ they showed $$\label{reg.tail}
{\mathbf P}(A_\tau>x)\sim{\mathbf E}\tau\overline F(\sqrt{2ax}),\quad x\to\infty.$$ These asymptotics can be explained by a traditional heavy-tailed one big jump heuristics. In order to have a huge area, the random walk should have a large jump, say $y$, at the very beginning of the excursion. After this jump the random walk goes down along the line $y-an$ according to the Law of Large Numbers. Thus, the duration of the excursion should approximately be around $y/a$. As a result, the area will be of order $y^2/2a$. Now, from the equality $x=y^2/2a$ one infers that a jump of order $\sqrt{2ax}$ is needed. Since the same strategy is valid for the maximum $M_\tau:=\max_{n<\tau}S_n$ of the first excursion, one can rewrite in the following way: $$\mathbf{P}(A_\tau>x)\sim\mathbf{P}(M_\tau>\sqrt{2ax}),\quad x\to\infty.$$
However, the class of regularly varying distributions does not include all subexponential distributions and excludes, in particular, log-normal distribution and Weibull distribution with parameter $\beta<1$. The asymptotics for these remaining cases have been put as an open problem in [@KP11 Conjecture 2.2] for a strongly related workload process. We will reformulate this conjecture as follows $$\label{eq:conj}
{\mathbf P}(A_\tau>x)\sim {\mathbf P}\left(\tau>\sqrt{\frac{2x}{a}}\right), \quad x\to\infty,$$ when $F\in\mathcal S$ and $\mathcal S$ is a sublclass of subexponential distributions. Note that using the asymptotics for $$\label{eq:asymp.tau}
{\mathbf P}(\tau>x)\sim {\mathbf E}\tau\overline F(a x)$$ from [@DS13] for Weibull distributions with parameter $\beta<1/2$, one can see that in this case asymptotics is equivalent to . In this note we partially settle . It is not difficult to show that the same arguments hold for the workload process and to prove the same asymptotics for the area of the workload process, thus settling the original [@KP11 Conjecture 2.2]. In passing we note that it is doubtful that holds in full. The reason for that is that for both $\tau$ and $A_\tau$ the asymptotics and are no longer valid for Weibull distributions with parameter $\beta>1/2$. The analysis for $\beta>1/2$ involves more complicated optimisation procedure leading to a Cramer series and it is unlikely that the answers will be the same for the area and for the exit time.
Main results
------------
We will now present the results. We will start with the regularly varying case. In this case the connection between the tails of $A_\tau$ and $M_\tau$ is strong and we will be able to use the asymptotics for ${\mathbf P}(M_\tau>x)$ found in [@FPZ], see also a short proof in [@D2005], to find the asymptotics for ${\mathbf P}(A_\tau>x)$.
\[prop:joint\] We have the following two statements.
- If $\overline{F}(x):=\mathbf{P}(X_1>x)=x^{-\alpha}L(x)$ with some $\alpha\ge1$ and $\mathbf{E}|X_1|<\infty$ then, uniformly in $y\in[\varepsilon\sqrt{x},\sqrt{2ax}]$, $$\label{joint.1}
\mathbf{P}(A_\tau>x,M_\tau>y)\sim \mathbf{E}\tau\overline{F}(\sqrt{2ax}).$$
- If $\overline{F}(x)\sim x^{-\varkappa} e^{-g(x)}$, where $g(x)$ is a monotone continuously differentiable function satisfying $\frac{g(x)}{x^\beta}\downarrow$ for $\beta\in(0,1/2)$, and $\mathbf{E}|X_1|^\varkappa<\infty$ for some $\varkappa>1/(1-\beta)$ then holds uniformly in $y\in\left[\sqrt{2ax}-\frac{R\sqrt{2ax}}{g(\sqrt{2ax})},\sqrt{2ax}\right].$
This statement implies obviously the following lower bound for the tail of $A_\tau$: $$\label{lower.bound}
\liminf_{x\to\infty}\frac{{\mathbf P}(A_\tau>x)}{\overline{F}(\sqrt{2ax})}\ge1.$$ Furthermore, using this proposition, one can give an alternative proof of under the assumption of the regular variation of $\overline F$, which is much simpler than the original one in [@BBP03]. We first split the event $\lbrace A_\tau>x\rbrace$ into two parts $$\begin{aligned}
\lbrace A_\tau>x\rbrace=\lbrace A_\tau>x, M_\tau>y\rbrace\cup\lbrace A_\tau>x, M_\tau\leq y\rbrace.\end{aligned}$$ Clearly, $$\lbrace A_\tau>x, M_\tau\leq y\rbrace\subseteq\lbrace\tau>x/y\rbrace,$$ and, therefore, $$\begin{aligned}
\label{atausplit}
{\mathbf P}(A_\tau>x, M_\tau>y)\leq{\mathbf P}(A_\tau>x)\leq{\mathbf P}(A_\tau>x, M_\tau>y)+{\mathbf P}(\tau>x/y).\end{aligned}$$ When $\alpha>1$, according to Theorem I in Doney [@Doney89] or [@DS13 Theorem 3.2], $${\mathbf P}(\tau>t)\sim{\mathbf E}\tau\bar{F}\left(a t\right)\quad\text{as }t\to\infty.$$ Choosing $y=\varepsilon \sqrt{x}$ and recalling that $\overline{F}$ is regularly varying, we get $$\label{tau_tail}
{\mathbf P}(\tau>x/y)={\mathbf P}(\tau>\sqrt{x}/\varepsilon)\sim \varepsilon^\alpha\mathbf{E}\tau\overline{F}(\sqrt{x}).$$ It follows from the first statement of Proposition \[prop:joint\] that $$\mathbf{P}(A_\tau>x,M_\tau>\varepsilon \sqrt{x})\sim\mathbf{E}\tau\overline{F}(\sqrt{2ax}).$$ Plugging this and into , we get, as $x\to\infty$, $$\mathbf{E}\tau\overline{F}(\sqrt{2ax})(1+o(1))
\le\mathbf{P}(A_\tau>x)\le
\mathbf{E}\tau\overline{F}(\sqrt{2ax})\left(1+\frac{\varepsilon^\alpha}{(2a)^{\alpha/2}}+o(1)\right).$$ Letting $\varepsilon\to0$, we arrive at .
The case of semi-exponential distributions is more complicated. In particular it seems that in this case there is a regime when the asymptotics are no longer valid. We will treat this case by using the exponential bounds similar to Section 2.2 in [@PW17] and asymptotics for ${\mathbf P}(\tau>x)$ from [@DS13] and [@DDS08].
First we will introduce a sublclass of subexponential distributions that we will consider. We will assume that ${\mathbf E}[X_1^2]=\sigma^2<\infty$. Without loss of generality we may assume that $\sigma=1$. Let $$\label{sc1}
\overline F(x) \sim e^{-g(x)}x^{-2}, \quad x\to \infty,$$ where $g(x)$ is an eventually increasing function such that eventually $$\label{sc2}
\frac{g(x)}{x^{\gamma_0}}\downarrow 0, \quad x\to\infty,$$ for some $\gamma_0\in(0,1)$. Due to the asymptotic nature of equivalence in without loss of generality we may assume that $g$ is continuously differentiable and that hold for all $x>0$. Clearly, monotonicity in implies $$\label{sc3}
g'(x)\le \gamma_0 \frac{g(x)}{x}$$ for all sufficiently large $x$. Using the Karamata representation theorem one can show that this class of subexponential distributions includes regularly varying distributions $\overline F(x)\sim x^{-r}L(x),$ for $r>2$. Also, it is not difficult to show that lognormal distributions and Weibull distributions ($\overline F(x) \sim e^{-x^\beta},\beta\in(0,1)$) belong to our class of distributions. Previously this class appeared in [@R93] for the analysis of large deviations of sums of subexponential random variables on the whole axis.
Now we are able to give rough(logarithmic) asymptotics for $\gamma_0\le 1$.
\[cor:logarithmic.upper.bound\] Let ${\mathbf E}[X_1]=-a<0$ and $\mbox{Var}(X_1)<\infty$. Assume that the distribution function $F$ of $X_j$ satisfies and that holds with $\gamma_0=1$. Then, there exits a constant $C>0$ such that $${\mathbf P}(A_\tau>x)\le Cx^{1/4} \exp
\left\{
-g(\sqrt{2ax})\sqrt{
1-\frac{2Cg(\sqrt{2ax})}{a\sqrt{2ax}}
}
\right\}.$$ Furthermore, for any $\varepsilon>0$ there exist $C>0$ such that, $$\liminf_{x\to\infty}\frac{{\mathbf P}(A_\tau>x)}{\overline F(\sqrt{2ax}+Cx^{1/4+\varepsilon})}\ge {\mathbf E}\tau.$$ In, particular, if $\gamma_0<1$ then $$\lim_{x\to\infty}\frac{\ln {\mathbf P}(A_\tau>x)}{\ln \overline F(\sqrt{2ax})}=1.$$
To obtain the exact asymptotics we will impose a further assumption $$\label{sc4}
xg'(x)\to \infty, \quad x\to \infty.$$ This assumption implies that $$\label{sc5}
\frac{g(x)}{\log x}\to \infty.$$ In particular, it excludes all regularly varying distributions.
\[thm:exact.asymptotics\] Let ${\mathbf E}[X_1]=-a<0$ and $\mathbf{Var}(X_1)<\infty$. Assume that the distribution function $F$ of $X_j$ satisfies , that holds with $\gamma_0<1/2$ and that holds. Then, $${\mathbf P}(A_\tau>x) \sim {\mathbf E}\tau \overline F(\sqrt{2ax}), \quad x\to \infty.$$
Discussion and organisation of the paper
----------------------------------------
In this note we provided exact asymptotics for the case $\gamma_0<1/2$. We believe that this restriction is not technical and the asymptotics for $\gamma_0\ge 1/2$ is different. This boundary is well-known, for example, the same bound appears in the analysis of the exact asymptotics for ${\mathbf P}(\tau>n)$ and ${\mathbf P}(S_n>an)$, see, correspondingly [@DS13] and [@DDS08].
The conjecture in [@KP11] was formulated for the workload process of a single-server queue rather than the area under the random walk excursion. However, one can prove analogous results for the Lévy processes by essentially the same arguments. It is well-known that workload of the M/G/1 queue can be represent as a Lévy process and thus our results can be transferred to this setting almost immediately. We believe that the treatment of the workload of the general G/G/1 queue is not that different as well.
The paper is organised as follows. We will start by proving Proposition \[prop:joint\] in Section \[sec:area.via.maximum\]. Then we will derive a useful exponential bound and prove Theorem \[cor:logarithmic.upper.bound\] in Section \[sec:logarithmic\]. Finally we derive exact asymptotics for ${\mathbf P}(A_\tau>x)$ and thus prove Theorem \[thm:exact.asymptotics\] in Section \[sec:asymp.beta.12\].
Proof of Proposition \[prop:joint\] {#sec:area.via.maximum}
===================================
Before giving the proof we will collect some known results that we will need in this and the following Sections. We will require the following statement, the first part of which follows from Theorem 2 in Foss, Palmowski and Zachary [@FPZ] (see also [@D2005] for a short proof), and the second part from [@DS13 Theorem 3.2].
\[prop:ds13\] Let ${\mathbf E}[X_1]=-a$ and either (a) $\overline{F}(x):=\mathbf{P}(X_1>x)=x^{-\alpha}L(x)$ with some $\alpha>1$ or (b) $\overline{F}(x)\sim x^{-\varkappa} e^{-g(x)}$, where $g(x)$ is a monotone continuously differentiable function satisfying $\frac{g(x)}{x^\beta}\downarrow$ for $\beta\in(0,1/2)$, and $\mathbf{E}|X_1|^\varkappa<\infty$ for some $\varkappa>1/(1-\beta)$ then for any fixed $k$, $$\begin{aligned}
\label{eq:Sk-MK}
{\mathbf P}(M_k>y)&\sim {\mathbf P}(S_k>y)\sim k\overline F(y), \quad y\to \infty\\
\label{Mtau-k}
\mathbf{P}\left(\max_{n\le\tau\wedge k}S_n>y\right)&\sim \mathbf{E}(\tau\wedge k)\overline{F}(y),\quad y\to \infty\\
\label{Mtau}
{\mathbf P}(M_\tau>y)&\sim{\mathbf E}\tau\bar{F}(y),\quad y\to\infty\end{aligned}$$ and $$\label{eq:ds13}
{\mathbf P}(\tau>n)\sim {\mathbf E}[\tau]\overline F(an), \quad n\to \infty.$$
To prove , and , by Theorem 2 of [@FPZ] it is sufficient to show that (a) or (b) implies that $F\in\mathcal S^*$, that is $\int_0^\infty \overline F(y)<\infty$ and $$\int_0^x\overline F(y)\overline F(x-y)dy \sim 2\overline F(x) \int_0^\infty \overline F(y)dy , \quad x\to \infty.$$ The fact that (a) implies $F\in\mathcal S^*$ is well-known and follows immediately from the dominated confergence theorem, since $ \overline F(x)\sim \overline F(x-y)$ for all fixed $y$ and $$\int_0^x \frac{\overline F(y)\overline F(x-y)}{\overline F(x) }dy
=2\int_0^{x/2} \frac{\overline F(y)\overline F(x-y)}{\overline F(x) }dy$$ and $\overline F(x-y)\le C\overline F(x)$ for some $C>0$ when $y\le x/2$.
Now, assume that (b) holds and show that $F\in\mathcal S^*$. Consider now $$\begin{aligned}
2\int_0^{x/2} \frac{\overline F(y)\overline F(x-y)}{\overline F(x) }dy.
\end{aligned}$$ Uniformly in $y\in [\ln x ,x/2]$ we have $$\begin{aligned}
\frac{\overline F(y)\overline F(x-y)}{\overline F(x) }
&\le C e^{g(x)-g(x-y)-g(y)}
=C e^{\int_{x-y}^x g'(t)dt -g(y)}
\le C e^{\beta \int_{x-y}^x \frac{g(t)}{t}dt -g(x-y)} \\
&\le C e^{\beta y\frac{g(x-y)}{x-y} -g(x-y)} \le
Ce^{(\beta-1)g(x-y)} \to 0, \quad x\to\infty,
\end{aligned}$$ and, therefore, $$\begin{aligned}
2\int_{\ln x }^{x/2} \frac{\overline F(y)\overline F(x-y)}{\overline F(x) }dy\to 0.
\end{aligned}$$ Next for $y\in [0, \ln x]$, $$\begin{aligned}
1&\le \frac{\overline F(x-y)}{\overline F(x)}\le
\frac{\overline F(x-\ln x)}{\overline F(x)}
\sim e^{g(x)-g(x-\ln x)} =
e^{\int_{x-\ln x}^x g'(t)dt}\\
&\le e^{\beta \int_{x-\ln x}^x \frac{g(t)}{t}dt}\le
e^{\beta \frac{g(x-\ln x)}{(x-\ln x)^\beta}\int_{x-\ln x}^x t^{\beta-1}dt}
\le e^{C \frac{g(x-\ln x)}{(x-\ln x)^\beta}\frac{\ln x}{x^{1-\beta}}}\to 1,\end{aligned}$$ which implies that $F\in\mathcal S^*$.
The proof of is very similar and can be done by straightforward verification that and imply that conditions of Theorem 3.1 (and hence of Theorem 3.2) of [@DS13] hold.
Define $$\sigma_y=\inf\lbrace n<\tau:S_n>y\rbrace.$$ Then, for every $k\geq 1$, $$\begin{aligned}
{\mathbf P}(\sigma_y=k\vert M_\tau>y)&=\frac{{\mathbf P}(\sigma_y=k)}{{\mathbf P}(M_\tau>y)}\\
&=\frac{\mathbf{P}\left(\max_{n\le\tau\wedge k}S_n>y\right)-\mathbf{P}\left(\max_{n\le\tau\wedge(k-1)}S_n>y\right)}{\mathbf{P}(M_\tau>y)}.\end{aligned}$$ It wollows from and that $$\begin{aligned}
\label{limprob1}
\nonumber
\lim_{y\rightarrow\infty}{\mathbf P}(\sigma_y=k\vert M_\tau>y)
&=\frac{\mathbf{E}\tau\wedge k-\mathbf{E}\tau\wedge(k-1)}{{\mathbf E}\tau}\\
&=\frac{\mathbf{P}(\tau>k-1)}{{\mathbf E}\tau}=: q_k, \hspace{0,5cm}k\geq 1.\end{aligned}$$ It is clear that $$\begin{aligned}
\sum_{k=1}^\infty q_k=\frac{1}{{\mathbf E}\tau}\sum_{k=0}^\infty{\mathbf P}(\tau>k-1)=1.\end{aligned}$$
For every fixed $N\geq1$ we have $$\label{ThS1}
\begin{split}
{\mathbf P}(A_\tau&>x, M_\tau>y)\\
&=\sum_{k=1}^N{\mathbf P}(A_\tau>x, \sigma_y=k, M_\tau>y)+{\mathbf P}(A_\tau>x, \sigma_y>N, M_\tau>y).
\end{split}$$ For the last term on the right hand side we have $$\begin{aligned}
{\mathbf P}(A_\tau>x, \sigma_y>N, M_\tau>y)&\leq{\mathbf P}(\sigma_y>N, M_\tau>y)\\
&={\mathbf P}(M_\tau>y){\mathbf P}(\sigma_y>N\vert M_\tau>y).\end{aligned}$$ It follows from that ${\mathbf P}(\sigma_y>N\vert M_\tau>y)\rightarrow\sum_{j=N+1}^\infty q_j$, as $y\rightarrow\infty$. Then, using , we get $$\begin{aligned}
\label{ataumtau}
{\mathbf P}(A_\tau>x,\sigma_y>N, M_\tau>y)\leq \varepsilon_N\bar{F}(y),\end{aligned}$$ where $\varepsilon_N\rightarrow 0$ as $N\rightarrow\infty$.
For every fixed $k$ we have $$\begin{aligned}
{\mathbf P}(A_\tau>x, \sigma_y=k, M_\tau>y)={\mathbf P}(A_\tau>x, \sigma_y=k).\end{aligned}$$ Since $S_j\in(0,y)$ for all $j<k$, we obtain $$\begin{aligned}
{\mathbf P}(A_\tau>x, \sigma_y=k)\leq{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>x-(k-1)y, \sigma_y=k\right)\end{aligned}$$ and $$\begin{aligned}
{\mathbf P}(A_\tau>x, \sigma_y=k)\geq{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>x, \sigma_y=k\right).\end{aligned}$$ By the Markov property, for every $z>0$, $$\begin{aligned}
{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>z, \sigma_y=k\right)=\int_y^\infty{\mathbf P}(S_k\in dv, \sigma_y=k){\mathbf P}(A_\tau>z\vert S_0=v).\end{aligned}$$
Let $\varkappa\in(1/(1-\beta),2)$ if $\overline{F}$ satisfies the conditions of the part (b) and let $\varkappa=1$ in the case when $\overline{F}$ is regularly varying. Fix some $\delta>0$ and consider the set $$B_v:=\left\{v-\delta v^{1/\varkappa}\le S_n+na\le v+\delta v^{1/\varkappa}
\mbox{ for all }n\le\frac{v+\delta v^{1/\varkappa}}{a}\right\}.$$ Since ${\mathbf E}|X_1|^\varkappa<\infty$, it follows from the Marcinkiewicz-Zygmund Law of Large Numbers that $$\label{refined_lln}
\mathbf{P}(B_v|S_0=v)\to 1\quad\text{as }v\to\infty.$$ This implies that, as $y\rightarrow\infty$, $$\begin{aligned}
{\mathbf P}&\left(\sum_{j=k}^{\tau-1}S_j>z, \sigma_y=k\right)\\
&=\int_y^\infty{\mathbf P}\left(S_k\in dv,\sigma_y=k\right){\mathbf P}\left(\lbrace A_\tau>z\rbrace\cap B_v\vert S_0=v\right)+o\left({\mathbf P}(\sigma_y=k)\right).\end{aligned}$$ On the event $B_v$ one has $$\frac{(v-\delta v^{1/\varkappa})^2}{2a}\leq
A_\tau\leq\frac{(v+\delta v^{1/\varkappa})^2}{2a}.$$ In other words, $${\mathbf P}\left(\lbrace A_\tau>z\rbrace\cap B_v\vert S_0=v\right)={\mathbf P}(B_v)
\quad\text{if}\quad v-\delta v^{1/\varkappa}\ge\sqrt{2az}$$ and $${\mathbf P}\left(\lbrace A_\tau>z\rbrace\cap B_v\vert S_0=v\right)=0
\quad\text{if}\quad v+\delta v^{1/\varkappa}<\sqrt{2az}.$$ Therefore, for all $v$ large enough, $$\begin{aligned}
{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>z, \sigma_y=k\right)&\leq\int_{\sqrt{2az}-\delta(2az)^{1/2\varkappa}}^\infty {\mathbf P}(S_k\in dv, \sigma_y=k)+o({\mathbf P}(\sigma_y=k))\\
&={\mathbf P}\left(S_{\sigma_y}>\sqrt{2az}-\delta(2az)^{1/2\varkappa}, \sigma_y=k\right)+o({\mathbf P}(\sigma_y=k))\end{aligned}$$ and $$\begin{aligned}
{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>z, \sigma_y=k\right)&\leq\int_{\sqrt{2az}+2\delta(2az)^{1/2\varkappa}}^\infty {\mathbf P}(S_k\in dv, \sigma_y=k)\mathbf{P}(B_v)+o({\mathbf P}(\sigma_y=k))\\
&={\mathbf P}\left(S_{\sigma_y}>\sqrt{2az}+2\delta(2az)^{1/2\varkappa}, \sigma_y=k\right)+o({\mathbf P}(\sigma_y=k)).\end{aligned}$$
For every fixed $k$, $$\begin{aligned}
\sup_{v>y}\Bigg\vert\frac{{\mathbf P}(S_k>v,\sigma_y=k)}{\overline{F}(v)}-{\mathbf P}(\tau>k-1)\Bigg\vert\rightarrow 0\hspace{0,5cm}\text{as } y\rightarrow\infty\end{aligned}$$
Fix some $N>0$ and define the events $$D_{k,N}=\cup_{j=1}^k\left\lbrace X_j>v+kN, \vert X_l\vert \leq N\hspace{0,3cm} \text{for all } l\neq j, l\le k\right\rbrace.$$ It is clear that $D_{k,N}\subseteq \lbrace S_k>v\rbrace.$ Therefore, $$\begin{aligned}
{\mathbf P}(S_k>v,\sigma_y=k)&={\mathbf P}(D_{k,N},\sigma_y=k)+{\mathbf P}(S_k>v, D^c_{k,N},\sigma_y=k)\\
&={\mathbf P}(X_k>v+kN,\vert X_l\vert\leq N, \text{for all } l<k, \sigma_y>k-1)\\
&\hspace{1cm}+{\mathbf P}(S_k>v, D^c_{k,N},\sigma_y=k).\end{aligned}$$ For the first term we have $(y>(k-1)N)$ $$\label{A}
\begin{split}
{\mathbf P}&(X_k>v+kN,\vert X_l\vert\leq N, \text{for all } l>k, \sigma_y>k-1)\\
&={\mathbf P}(\tau>k-1,\vert X_l\vert\leq N, l<k)\overline{F}(v+kN)\\
&={\mathbf P}(\tau>k-1)\overline{F}(v)-\varepsilon_N^{(1)}\overline{F}(v)+o(\overline{F}(v)),\hspace{0,3cm} \text{uniformly in } v>y,
\end{split}$$ where $$\varepsilon_N^{(1)}:=
{\mathbf P}(\tau>k-1,\vert X_l\vert> N \text{ for some }l<k)\to0
\quad N\to\infty.$$
Furthermore, $$\label{B}
\begin{split}
{\mathbf P}&(S_k>v, D_{k,N}^c,\sigma_y=k)\leq{\mathbf P}(S_k>v, D_{k,N}^c)={\mathbf P}(S_k>v)-{\mathbf P}(D_{k,N})\\
&={\mathbf P}(S_k>v)-k{\mathbf P}(X_1>v+kN)({\mathbf P}(\vert X_1\vert\leq N))^{k-1}\\
&=\varepsilon_N^{(2)}\overline{F}(v)+o(\overline{F}(v)),
\end{split}$$ where $$\varepsilon_N^{(2)}:=
k\left(1-({\mathbf P}(\vert X_1\vert\leq N))^{k-1}\right)\to0,
\quad N\to\infty.$$
Combining and and letting $N\rightarrow\infty$ we set the desired relation.
Since with the previous lemma $${\mathbf P}(S_{\sigma_y}>v,\sigma_y=k)\sim\bar{F}(v){\mathbf P}(\tau>k-1),\hspace{0,5cm} v,y\rightarrow\infty$$ for $v\geq y$, we infer that $$\begin{aligned}
&{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>z, \sigma_y=k\right)\leq\bar{F}\left(\sqrt{2az}-\delta(2az)^{1/2\varkappa}\right)({\mathbf P}(\tau>k-1)+o(1))\\
&\hspace{4cm}+o(\mathbf{P}(\sigma_y=k))\end{aligned}$$ and $$\begin{aligned}
&{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>z, \sigma_y=k\right)\geq\overline{F}\left(\sqrt{2az}+2\delta(2az)^{1/2\varkappa}\right)({\mathbf P}(\tau>k-1)+o(1))\\
&\hspace{4cm}+o(\mathbf{P}(\sigma_y=k)).\end{aligned}$$ Under our assumptions on $\overline{F}$ one has $$\lim_{\delta\to0}\lim_{z\to\infty}\frac{\overline{F}\left(\sqrt{2az}+2\delta(2az)^{1/2\varkappa}\right)}
{\overline{F}\left(\sqrt{2az}-\delta(2az)^{1/2\varkappa}\right)}=1.$$ Therefore, $$\begin{aligned}
&{\mathbf P}\left(\sum_{j=k}^{\tau-1}S_j>z, \sigma_y=k\right)
=\overline{F}\left(\sqrt{2az}\right)({\mathbf P}(\tau>k-1)+o(1))+o(\mathbf{P}(\sigma_y=k)).\end{aligned}$$ Consequently,
$$\begin{aligned}
{\mathbf P}(A_\tau>x,\sigma_y=k)=\bar{F}(\sqrt{2ax}){\mathbf P}(\tau>k-1)+o\left({\mathbf P}(\sigma_y=k)\right).\end{aligned}$$
Combining and , one gets $$\mathbf{P}(\sigma_y=k)\sim q_k{\mathbf E}\tau\overline{F}(y).$$ Therefore, $$\begin{aligned}
{\mathbf P}(A_\tau>x,\sigma_y=k)=\bar{F}(\sqrt{2ax})({\mathbf P}(\tau>k-1)+o(1))+o(\overline{F}(y)).\end{aligned}$$ Consequently, $$\begin{aligned}
\label{sumpr}
\nonumber
&\sum_{k=1}^N{\mathbf P}(A_\tau>x, \sigma_y=k, M_\tau>y)\\
&\hspace{1cm}=(\overline{F}(\sqrt{2ax})+o(1))\sum_{k=1}^N{\mathbf P}(\tau>k-1)
+o(\overline{F}(y)).\end{aligned}$$ Plugging and into and letting $N\rightarrow\infty$, we obtain $$\begin{aligned}
{\mathbf P}(A_\tau>x, M_\tau>y)=({\mathbf E}\tau+o(1))\bar{F}(\sqrt{2ax})+o(\overline{F}(y)).\end{aligned}$$ Thus, it remains to show that $\overline{F}(y)=O(\overline{F}(\sqrt{2ax}))$. This is obvious for regularly varying tails and $y\ge \varepsilon\sqrt{x}$.
Assume now that $\overline{F}$ satisfies the conditions of part (b). To simplify notation put $y_*=\sqrt{2ax}-\frac{R\sqrt{2ax}}{g(\sqrt{2ax})}$. Then, $$1\le \frac{\overline F(y_*)}{\overline F(\sqrt{2ax})}
\le (1+o(1))e^{g(\sqrt{2ax})-g\left(y_*\right)}.$$ Since $\frac{g(x)}{x^\beta}$ is monotone decreasing and $g$ is differentiable then clearly $$g'(x)\le \beta \frac{g(x)}{x}.$$ Then, $$\begin{aligned}
g(\sqrt{2ax})-g\left(y_*\right)
&=\int^{\sqrt{2ax}}_{y_*} g'(t) dt
\le \beta \int^{\sqrt{2ax}}_{y_*} \frac{g(t)}{t} dt
\le \beta \frac{g(y_*)}{(y_*)^\beta}\int^{\sqrt{2ax}}_{y_*} \frac{dt}{t^{1-\beta}} \\
&=\frac{g(y_*)}{(y_*)^\beta}((2ax)^{\beta/2}-(y_*)^\beta)
\le \frac{g(y_*)}{(y_*)^{\beta}}\frac{\beta}{(y_*)^{1-\beta}}
C\frac{\sqrt{2ax}}{g(\sqrt{2ax})}\\
&\le \beta C \frac{\sqrt{2ax}}{y_*}\le (1+o(1))\beta C.\end{aligned}$$ Therefore, $$\begin{aligned}
\overline{F}(y)\leq C\overline{F}(x),\hspace{0,5cm}\forall y\in \left[\sqrt{2ax}-\frac{R\sqrt{2ax}}{g(\sqrt{2ax})},\sqrt{2ax}\right].\end{aligned}$$
Proof of Theorem \[cor:logarithmic.upper.bound\] {#sec:logarithmic}
================================================
We start by proving an exponential estimate for the area $A_n$ when random variables $X_j$ are truncated. Let $$\overline X_n=\max(X_1,\ldots,X_n).$$ The next result is our main technical tool to investigate trajectories without big jumps.
\[lem:chebyshev\] Let ${\mathbf E}[X_1]=-a$ and $\sigma^2:=\mathbf{Var}(X_1)<\infty$. Assume that the distribution function $F$ of $X_j$ satisfies and that holds with $\gamma_0=1$. Then, there exists a constant $C>0$ such that $${\mathbf P}(A_n>x,\overline X_n\le y) \le \exp\left\{
-\lambda \frac{x}{n} - \lambda\frac{a n}{2} +C\lambda^2 n
\right\},$$ where $\lambda = \frac{g(y)}{y}$.
We will prove this lemma by using the exponential Chebyshev inequality. For that we need to obtain estimates for the moment generating function of $A_n$. First, $${\mathbf E}\left[ e^{\frac{\lambda}{n}A_n};\overline X_n\le y\right]=
{\mathbf E}\left[e^{\frac{\lambda}{n}\sum_1^n(n-j+1)X_j};\overline X_n\le y\right]
=\prod_{j=1}^n\varphi_y\left(\lambda_{n,j}\right),$$ where $$\varphi_y(t) := {\mathbf E}[e^{t X_j};X_j\le y]$$ and $$\lambda_{n,j} := \lambda\frac{(n-j+1)}{n}.$$ Then, $$\begin{aligned}
\varphi_y(\lambda_{n,j})&= {\mathbf E}[e^{\lambda_{n,j}X_j}; X_j\le 1/\lambda_{n,j}]+
{\mathbf E}[e^{\lambda_{n,j}X_j};1/\lambda_{n,j} < X_j\le y]\\
&=: E_1+E_2.\end{aligned}$$ Using the elementary bound $e^x\le 1+x+x^2$ for $x\le 1$ we obtain, $$\begin{aligned}
E_1\le 1+\lambda_{n,j}{\mathbf E}[X_j]
+\lambda_{n,j}^2 {\mathbf E}[X_j^2]
=1-a\lambda_{n,j}+(a^2+\sigma^2)\lambda_{n,j}^2. \end{aligned}$$ Next, using the integration by parts and the assumption , $$\begin{aligned}
E_2&=\int_{1/\lambda_{n,j}}^{y} e^{\lambda_{n,j}t}dF(t)
=-\overline F(t)e^{\lambda_{n,j}t}\biggl |_{t=1/\lambda_{n,j}}^{t=y}
+\lambda_{n,j}\int_{1/\lambda_{n,j}}^{y} e^{\lambda_{n,j}t}\overline F(t) dt\\
&\le e \overline F(1/\lambda_{n,j})+C \lambda_{n,j}\int_{1/\lambda_{n,j}}^{y} e^{\lambda_{n,j}t-g(t)}t^{-2}dt. \end{aligned}$$ Now note that for $t\le y$, $$\lambda_{n,j} t - g(t) = t \left(\lambda_{n,j}-\frac{g(t)}{t}\right)
\le t \left(\lambda_{n,j}-\frac{g(y)}{y}\right),$$ due to the condition . Then, $$\lambda_{n,j}-\frac{g(y)}{y} \le
\lambda - \frac{g(y)}{y}=0$$ and, therefore, $$E_2\le e \overline F(1/\lambda_{n,j})+C \lambda_{n,j}\int_{1/\lambda_{n,j}}^{y} t^{-2}dt \le (C+e)\lambda_{n,j}^2,$$ where we also used the Chebyshev inequality. As a result, for some constant $C$, $$\varphi_y(t)= E_1+E_2\le 1-a\lambda_{n,j} +C\lambda_{n,j}^2.$$ Consequently, $$\begin{aligned}
{\mathbf E}\left[e^{\frac{\lambda}{n}A_n};\overline X_n\le y\right]&\le \prod_{j=1}^n \left(1-a\lambda_{n,j} +C\lambda_{n,j}^2\right) \\
&=
\exp\left\{
\sum_{j=1}^n\ln \left(1-a\lambda_{n,j} +C\lambda_{n,j}^2\right)
\right\}\\
&\le
\exp\left\{
\sum_{j=1}^n\left(-a\lambda_{n,j} +C\lambda_{n,j}^2\right)
\right\}\\
&=\exp\left\{
\sum_{j=1}^n\left(-a\lambda\frac{n-j+1}{n} +C\left(\lambda\frac{n-j+1}{n} \right)^2\right)
\right\}\\
&\le \exp\left\{
-\frac{a \lambda}{2} n +C\lambda^2 n
\right\}.\end{aligned}$$ Finally, $${\mathbf P}(A_n>x,\overline X_n\le y)\le e^{-\lambda\frac{x}{n}}{\mathbf E}\left[e^{\frac{\lambda}{n}A_n};\overline X_n\le y\right]
\le
\exp\left\{
-\lambda \frac{x}{n} -\frac{a \lambda}{2} n +C\lambda^2 n
\right\}.$$
We can now obtain a rough upper bound using the exponential bound in Lemma \[lem:chebyshev\].
\[lem:logarithmic.upper.bound\] Let ${\mathbf E}[X_1]=-a<0$ and $\mathbf{Var}(X_1)<\infty$. Assume that the distribution function $F$ of $X_j$ satisfies and that holds with $\gamma_0=1$. Then, there exists a constant $C>0$ such that $${\mathbf P}(A_\tau>x)\le Cx^{1/4} \exp
\left\{
-g(\sqrt{2ax})\sqrt{
1-\frac{2Cg(\sqrt{2ax})}{a\sqrt{2ax}}
}
\right\}$$
Clearly, $${\mathbf P}(A_\tau>x)\le {\mathbf P}(A_\tau>x, \overline X_\tau\le \sqrt{2ax})
+{\mathbf P}(A_\tau>x, \overline X_\tau>\sqrt{2ax})=:P_1+P_2.$$ First, using Lemma \[lem:chebyshev\] with $y=\sqrt{2ax}$ we obtain, $$\begin{aligned}
P_1&\le \sum_{n=0}^\infty\mathbf{P}(A_n\geq x, \overline X_n \le \sqrt{2ax},\tau=n+1)\\
&\le \sum_{n=1}^\infty
\exp\left\{
-\lambda \frac{x}{n} -\frac{a \lambda}{2} n +C\lambda^2 n
\right\}
=
\sum_{n=1}^\infty
\exp\left\{
-\lambda \frac{x}{n} - \lambda I n
\right\},
\end{aligned}$$ where $\lambda = \frac{g(\sqrt {2ax} )}{\sqrt {2ax}}$ and $I=\frac{a}{2}-C\lambda. $ With formula (25) at page 146 of Bateman [@BTIT] we have, $$\begin{aligned}
\sum_{n=1}^\infty\exp\left\lbrace-\lambda\frac{x}{n}-\lambda I n\right\rbrace&\leq \int_0^\infty\exp\left\lbrace-\lambda \frac{x}{y}-\lambda I(y+1)\right\rbrace dy\\
&=e^{-\lambda I}\sqrt{\frac{4x}{I}}K_1(2\lambda\sqrt{Ix}).\end{aligned}$$ Now using the asymptotics for the modified Bessel function $$K_1(z)\sim \sqrt{\frac{\pi}{2z}}e^{-z}$$ we obtain $$\begin{aligned}
\sum_{n=1}^\infty\exp\left\lbrace-\lambda\frac{x}{n}-\lambda I n\right\rbrace\leq Cx^{1/4}\exp\{-2\lambda\sqrt{Ix}\}.\end{aligned}$$ Therefore, $$\begin{aligned}
\label{eq:p1}
P_1&\le Cx^{1/4}\exp\{-2\lambda\sqrt{Ix}\}\\
\nonumber
&\le Cx^{1/4} \exp
\left\{
-g(\sqrt{2ax})\sqrt{
1-\frac{2Cg(\sqrt{2ax})}{a\sqrt{2ax}}
}
\right\}.\end{aligned}$$ Next, $$\begin{aligned}
P_2&\le \sum_{n=0}^\infty\mathbf{P}(A_\tau\geq x, M_n \le \sqrt{2ax}, X_{n+1}>\sqrt{2ax} ,\tau>n)\\
&\le \sum_{n=0}^\infty\mathbf{P}( X_{n+1}>\sqrt{2ax}) {\mathbf P}(\tau>n) \le
{\mathbf E}[\tau] \overline F(\sqrt {2a x}) = o(P_1).\end{aligned}$$ Then, the claim follows.
Now we will give a lower bound.
\[lem:lower.bound\] Let ${\mathbf E}[X_1]=-a<0$ and $\mathbf{Var}(X_1)<\infty$. Then, for any $\varepsilon>0$ there exists $C>0$ such that, $$\liminf_{x\to\infty}\frac{{\mathbf P}(A_\tau>x)}{\overline F(\sqrt{2ax}+Cx^{1/4+\varepsilon})}\ge {\mathbf E}\tau.$$
Fix $N\ge 1$. Put $y^+ =\sqrt{2ax}+Cx^{1/2-\varepsilon},$ where $C$ will picked later. Since ${\mathbf E}[X_1^2]<\infty$, by the Strong Law of Large Numbers, $$\frac{S_{l}+al}{l^{1/2+\varepsilon}}\to 0, \quad l\to \infty \mbox{ a.s.}$$ Hence, for any $\delta>0$ we can pick $R>0$ such that $${\mathbf P}\left( \min_{l\le \sqrt{2x/a}} (S_{l}+al +R + l^{1/2+\varepsilon})>0\right)>(1-\delta)$$ Now note that there exists a sufficiently large $C$ such that, for every $k\le N$, $$\left\{\min_{l\le \sqrt{2x/a}} (S_{k+l}-S_k+al +R + l^{1/2+\varepsilon})>0,\tau>k, S_k>y^+\right\}\subset
\{A_\tau>x\}$$ Hence, $$\begin{aligned}
&{\mathbf P}(A_{\tau}>x) \ge \sum_{k=0}^N{\mathbf P}(A_\tau>x, \overline X_{k-1}\le y^+, X_k>y^+, \tau>k)\\
&\ge
\sum_{k=0}^N{\mathbf P}\left(\overline X_{k-1}\le y^+,\tau>k-1, X_k>y^+, \min_{l\le \sqrt{2x/a}} (S_{l+k}-S_k +R + j^{1/2+\varepsilon})>0\right)\\
&\ge (1-\delta)\sum_{k=0}^N {\mathbf P}\left(\overline X_{k-1}\le y^+,\tau>k-1\right)\overline F(y^+).
\end{aligned}$$ For every fixed $k$ we have $${\mathbf P}\left(\overline X_{k-1}\le y^+,\tau>k-1\right)
\to {\mathbf P}\left(\tau>k-1\right),\quad x\to\infty.$$ Furthermore, $\sum_{k=0}^N {\mathbf P}(\tau>k)\to{\mathbf E}\tau$ as $N\to\infty$. Therefore, we can pick sufficiently large $N$ such that $$\liminf_{x\to\infty}\sum_{k=0}^N {\mathbf P}\left(\overline X_{k-1}\le y^+,\tau>k-1\right)\ge (1-\delta){\mathbf E}\tau.$$ Then, for all $x$ sufficiently large, $${\mathbf P}(A_{\tau}>x)\ge (1-\delta)^2{\mathbf E}\tau\overline F(y^+).$$ As $\delta>0$ is arbitratily small we arrive at the conclusion.
[*Completion of the proof of Theorem \[cor:logarithmic.upper.bound\].*]{} The upper bound follows from Lemma \[lem:logarithmic.upper.bound\]. The lower bound follows from Lemma \[lem:lower.bound\]. The rough asymptotics follows immediately from the lower and upper bounds and from the observation that $$\label{observation}
\sup_{|y|\le x\rho(x)}\left|\frac{\log \overline F(x)}{\log \overline F(x+y)}-1\right|\to0,$$ where $\rho(x)\to0$.
To prove we note that by and $$\begin{aligned}
\label{eq:insensitivity}
g(x+y)-g(x)&=\int_x^{x+y} g'(t) dt \le
\gamma_0 \int_x^{x+y} \frac{g(t)}{t} dt
\le \gamma_0 \frac{g(x)}{x^{\gamma_0}}\int_x^{x+y} \frac{1}{t^{1-\gamma_0}} dt\\
\nonumber
&\le \gamma_0\frac{g(x)}{x^{\gamma_0}}\frac{y}{x^{1-\gamma_0}}
=\gamma_0g(x)\frac{y}{x},\quad y>0.
\end{aligned}$$ This implies that, as $x\to\infty$, $$\label{g-ratio}
\sup_{|y|\le x\rho(x)}\left|\frac{g(x+y)}{g(x)}-1\right|\to0.$$ Recalling that $$\log\overline{F}(x)\sim-g(x)-2\log x,$$ one obtains easily .
Proof of Theorem \[thm:exact.asymptotics\] {#sec:asymp.beta.12}
==========================================
Set $$h(x):= \frac{\sqrt{2ax}}{g(\sqrt{2ax})}$$ and $$\label{eq:y-}
y=\sqrt{2ax} - Ch(x) \log x ,$$ where $C>\frac{5/4}{1-\gamma_0}$. First we will split the probability ${\mathbf P}(A_{\tau}>x)$ as follows $$\begin{aligned}
{\mathbf P}(A_{\tau}>x)&={\mathbf P}(A_{\tau}>x, \overline X_\tau\le y)
+{\mathbf P}\left(A_{\tau}>x, \overline X_\tau> \sqrt{2ax}-\frac{1}{\log x}h(x)\right)\\
&+{\mathbf P}\left(A_{\tau}>x, \overline X_\tau\in \left[y, \sqrt{2ax}-\frac{1}{\log x}h(x))\right]\right)=:P_1+P_2+P_3.\end{aligned}$$ The first term will be estimated using the exponential bound proved in Lemma \[lem:chebyshev\].
\[lem:p1\] Let ${\mathbf E}[X_1]=-a$ and $\mathbf{Var}(X_1)<\infty$. Assume that and hold for some $\gamma_0<1/2$ together with . Then, $$P_1 = o(\overline F(\sqrt{2ax})).$$
According to , $$\begin{aligned}
P_1&\le Cx^{1/4}\exp\{-2\lambda\sqrt{Ix}\},
\end{aligned}$$ where $I=\frac{a}{2}-C\lambda$ and $\lambda = g(y)/y$. Since holds for some $\gamma_0<1/2$, $g^2(y)/y\to 0$ and hence $$P_1 \le Cx^{1/4}\exp\left\{-\frac{g(y)}{y}\sqrt{2ax}\right\}.$$ Then, $$\frac{P_1}{\overline F(\sqrt{2ax})}\le C x^{5/4}
\exp\left\{g(\sqrt{2ax})-\frac{g(y)}{y}\sqrt{2ax}\right\}.$$ To finish the proof it is sufficient to show that $$\label{eq:p1.intermediate}
g(\sqrt{2ax})-\frac{g(y)}{y}\sqrt{2ax} + \frac{5}{4}\log x \to -\infty,\quad x\to \infty.$$ We first note that $$\begin{aligned}
d(x)&:=g(\sqrt{2ax})-\frac{g(y)}{y}\sqrt{2ax} =
g(\sqrt{2ax}) - \frac{g(y)}{1-C\frac{\log x}{g(\sqrt{2ax})}}\\
&= g(\sqrt{2ax}) - g(y) + (C+o(1)) \log x \frac{g(y)}{g(\sqrt{2ax})}.\end{aligned}$$ Using and one can see that $$\begin{aligned}
\label{eq:bound.g}
g(\sqrt{2ax}) - g(y) &= \int_y^{\sqrt{2ax}} g'(z) dz
\le \gamma_0\int_y^{\sqrt{2ax}} \frac{g(z)}{z} dz
\le \gamma_0 \frac{g(y)}{y} (\sqrt{2ax}-y)\\
&=\gamma_0 C \frac{g(y)}{y}\log x \frac{\sqrt{2ax}}{g(\sqrt{2ax})}.
\nonumber\end{aligned}$$ Hence, $$d(x)\le \left(\gamma_0\frac{\sqrt{2ax}}{y}-1 \right) (C+o(1)) \frac{g(y)}{g(\sqrt{2ax})}\log x.$$ According to , $g(y)\sim g(\sqrt{2ax})$. Therefore, is valid for any $C$ satisfying $C(\gamma_0-1)+\frac{5}{4}<0$.
Next lemma gives the term with the main contribution.
\[lem:p2\] Under the assumptions of Lemma \[lem:p1\] we have the following estimate $$P_2\le (1+o(1))\overline F(\sqrt{2ax}), \quad x\to \infty.$$
Put $$y^*=\sqrt{2ax}-\frac{h(x)}{\log x}.$$
By the total probability formula, $$\begin{aligned}
P_2&\le \sum_{n=0}^\infty\mathbf{P}(A_\tau\geq x, \overline{X}_n \le y^*, X_{n+1}>y^* ,\tau>n)\\
&\le \sum_{n=0}^\infty\mathbf{P}( X_{n+1}>y^*) {\mathbf P}(\tau>n)
= {\mathbf E}[\tau] \overline F(y^*).
\end{aligned}$$ Now note that by and $$\begin{aligned}
\frac{\overline F(y^*)}{\overline F(\sqrt{2ax})}
&\le (1+o(1)) e^{g(\sqrt{2ax})-g(y^*)}
\le (1+o(1)) e^{\frac{\gamma_0 g(y^*)}{y^*}(\sqrt{2ax}-y^*)}\\
&\le (1+o(1)) e^{\frac{\gamma_0 g(y^*)}{y^*}\frac{1}{\log x}\frac{\sqrt{2ax}}{g(\sqrt{2ax})}} = 1+o(1). \end{aligned}$$ Then the statement immediately follows.
We will proceed to the analysis of $P_3$. Fix some $\delta>0$ and set $$z=\frac{1}{a}\left(\sqrt{2ax} + \delta\sqrt{x} \right).$$ We will split $P_3$ further as follows, $$\begin{aligned}
P_3\le P_{31}+P_{32}+P_{33} &:=
{\mathbf P}\left(A_{\tau}>x, \overline X_\tau\in \left[y, \sqrt{2ax}-R(x)h(x)\right];J_1;\tau\le
z\right )\\
&+
{\mathbf P}\left(A_{\tau}>x, \overline X_\tau\in \left[y, \sqrt{2ax}-R(x)h(x)\right];J_{\ge 2}, \tau\le z\right)\\
&+{\mathbf P}(\tau>z),\end{aligned}$$ where $$J_1 =\left\{
\mbox{there exists $k\in (1, \tau)$ such that } X_k>y \mbox{ and }
\max_{1\le i\le \tau, i\neq k} X_i \le y
\right\}$$ and, correspondingly, $$J_{\ge 2} =\left\{
\mbox{there exist $k, l\in (1, \tau)$ such that } X_k>y \mbox{ and } X_l>y
\right\}$$
We will start with easier terms $P_{32}$ and $P_{33}$. To deal with these terms we will use Proposition \[prop:ds13\]. One can see then
\[lem:p33\] Let the assumptions , and hold for $\gamma_0<1/2$. Then, $$P_{33} = o(\overline F(\sqrt{2ax})), \quad x\to\infty.$$
We have, by Proposition \[prop:ds13\], $$P_{33}\le {\mathbf P}(\tau>z)\le (\mathbf{E}\tau+o(1))
\overline F(az) =
O\left(\overline F(\sqrt{2ax}+\delta \sqrt{x})\right).$$ Therefore, $$\begin{aligned}
\frac{P_{33}}{\overline F(\sqrt{2ax})}
&\le C
e^{g(\sqrt{2ax})-g(\sqrt{2ax}+\delta \sqrt{x})}.
\end{aligned}$$ By the mean value theorem and by the assumption , $$g(cx)-g(x)\to\infty,\quad x\to\infty$$ for every $c>1$. This completes the proof.
\[lem:p32\] Let the conditions of Lemma \[lem:p2\] hold. Then, $$\label{eq:p32}
P_{32} = o(\overline F(\sqrt{2ax})).$$
We can use the formula of total probability to write $$P_{32} \le \sum_{k=1}^z {\mathbf P}(\tau>k, J_{\ge 2})
\le \sum_{k=1}^z \frac{k^2}{2}\overline F(y)^2.$$ Then, $$\frac{P_{32}}{\overline F(\sqrt{2ax})}
\le C x^{3/2}\frac{\overline F(y)^2}{\overline F(\sqrt{2ax})}
\le C x^{1/2} e^{g(\sqrt{2ax})-2g(y)}.$$ Using now one can see that $$\begin{aligned}
\frac{P_{32}}{\overline F(\sqrt{2ax})}
\le C x^{1/2} e^{C\ln x -g(y)}\to 0,\end{aligned}$$ in view of .
We are left to analyse $P_{31}$. For that introduce $$\mu(y):=\min\{n\ge 1: X_k>y\}.$$ Now we will complete the proof with the following Lemma.
\[lem:q1\] Let the assumptions , and hold for $\gamma_0<1/2$. Then, $$P_{31} = o(\overline F(\sqrt{2ax})), \quad x\to\infty.$$
First represent event $J_1=J_{11}\cup J_{12}$, where $$\begin{aligned}
J_{11}&:=
\{
\mbox{$X_k>y$ for exactly one $k\in(0,\tau)$ and
$X_i\le x^\varepsilon$ for all other
$i<\tau$ }
\}\\
J_{12}&:=
\{
\mbox{$X_k>y$ for exactly one $k\in(0,\tau)$ and
$X_i>x^\varepsilon $ for some
$i\neq k, i<\tau$}
\}. \end{aligned}$$ Then, $$\begin{aligned}
Q_{2}&:={\mathbf P}\left(A_{\tau}>x, \overline X_\tau\in \left[y, \sqrt{2ax}-\frac{1}{\log x}h(x)\right];J_{12},
\tau\le
z\right )\\
&\le \sum_{j=1}^z {\mathbf P}(\tau=j, J_{12})
\le \sum_{j=1}^z \frac{j^2}{2}\overline F(y)\overline F(x^\varepsilon)
\le z^3 \overline F(y)\overline F(x^\varepsilon).
\end{aligned}$$
Then, $$\begin{aligned}
\frac{Q_{2}}{\overline F(\sqrt{2ax})}\le
Cx^{3/2+2\varepsilon} e^{g(\sqrt{2ax})-g(y)-g(x^{\varepsilon}))}\end{aligned}$$ By , $$g(\sqrt{2ax})-g(y)\le C\ln x.$$ Then, in view of the relation we have $$g(\sqrt{2ax})-g(y)-g(x^{\varepsilon}))\le -4\ln x,$$ which implies that $Q_{2} = o(\overline F(\sqrt{2ax}))$.
To estimate $$Q_{1}:={\mathbf P}\left(A_{\tau}>x, \overline X_\tau\in \left[y, \sqrt{2ax}-\frac{1}{\log x}h(x)\right];J_{11},
\tau\le
z\right )$$ we make use of the exponential bound given in Lemma \[lem:chebyshev\]. Put putting $$x^+(k)=x-k\left(\sqrt{2ax}-\frac{h(x)}{\log x}\right).$$ Then, we have, $$\begin{aligned}
Q_{1}&=
\sum_{k=0}^{z-1} \sum_{j=1}^k {\mathbf P}\left(A_k>x, \max_{i\neq j, i\le k} X_i\le x^\varepsilon, X_j\in\left[y, \sqrt{2ax}-\frac{h(x)}{\log x}\right] , \tau=k+1\right)\\
&\le
\sum_{k=1}^z (k+1){\mathbf P}(A_k>x^+(k), \overline X_k\le x^\varepsilon)\overline F(y)\\
&\le Cx^{1/2}\overline F(y)
\sum_{k=1}^z
\exp\left\{
-\lambda \frac{x^+(k)}{k} -\frac{a \lambda}{2} k +C\lambda^2 k
\right\},\end{aligned}$$ where $\lambda = \frac{g(x^\varepsilon)}{x^\varepsilon}$. Now note that $$-\lambda \frac{x^+(k)}{k} -\frac{a \lambda}{2} k
=-\lambda\left(-\sqrt{2ax}+\frac{h(x)}{\log x} +\frac{x}{k}+\frac{ak}{2}\right).$$ Since $$\frac{x}{k}+\frac{ak}{2}\ge \sqrt{2ax},\quad k\ge1,$$ we obtain, $$-\lambda \frac{x^+(k)}{k} -\frac{a \lambda}{2} k
\le -\lambda \frac{h(x)}{\log x},\quad k\ge1.$$ Thus, $$Q_{1} \le Cx e^{-\lambda h(x)/\log x+\lambda^2 z}
\overline{F}(y).$$ Next, we can pick $\varepsilon = \frac{1}{4(1-\gamma_0)} $ to achieve $$\begin{aligned}
\lambda^2 z &\le C \left(\frac{g(x^\varepsilon)}{x^\varepsilon}\right)^2x^{1/2}
=C
\left(\frac{g(x^\varepsilon)}{x^{\varepsilon(1-1/(4\varepsilon))}}\right)^2
=C
\left(\frac{g(x^\varepsilon)}{x^{\gamma_0\varepsilon}}\right)^2\\
&<C\sup_t\left(\frac{g(t)}{t^{\gamma_0}}\right)^2<\infty,\end{aligned}$$ by the condition . Note that since $\gamma_0<1/2$, the picked $\varepsilon<1/2$ as well. Then, $$\begin{aligned}
\frac{Q_{1}}{\overline F(\sqrt{2ax})}\le
Cx^{2} e^{g(\sqrt{2ax})-g(y)-\lambda h(x)/\log x},\end{aligned}$$ and using , $$\begin{aligned}
\frac{Q_{1}}{\overline F(\sqrt{2ax})}\le
Cx^{C} e^{-\lambda h(x)/\log x}.\end{aligned}$$ Finally noting that $$\begin{aligned}
\lambda h(x) = \frac{g(x^\varepsilon)}{x^{\varepsilon}} \frac{\sqrt{2ax}}{g(\sqrt{2ax})}\end{aligned}$$ is decreasing polynomially we obtain required convergence to $0$. The polynomial decay can be immediately seen for $g(x)=x^{\gamma_0}$. However, a proper proof goes as follows, $$\begin{aligned}
g(C\sqrt{x}) &= g(x^\varepsilon)+\int_{x^\varepsilon}^{C\sqrt{x}} g'(t)dt
\le g(x^\varepsilon)+\gamma_0\int_{x^\varepsilon}^{C\sqrt{x}} \frac{g(t)}{t}dt\\
&\le g(x^\varepsilon)+
\gamma_0\int_{x^\varepsilon}^{C\sqrt{x}} \frac{g(t)}{t^{\gamma_0}} t^{\gamma_0-1}dt
\le g(x^\varepsilon)+\frac{g(x^\varepsilon)}{x^{\varepsilon\gamma_0}}\int_{x^\varepsilon}^{C\sqrt{x}} t^{\gamma_0-1}dt\\
&\le g(x^\varepsilon)+C\frac{g(x^\varepsilon)}{x^{\varepsilon \gamma_0}} x^{\gamma_0/2}
\le C g(x^\varepsilon)x^{\gamma_0(1/2-\varepsilon)}\end{aligned}$$ Therefore, $$\lambda h(x)\ge x^{1/2-\varepsilon} x^{-\gamma_0(1/2-\varepsilon)}$$
[*Completion of the proof of Theorem \[thm:exact.asymptotics\]*]{} Combination of the preceding Lemmas give us the upper bound. The lower bound has been shown in under even weaker conditions.
[99]{} Bateman, H. Tables of Integral Transforms. Vol.1 McGraw-Hill Book Company, INC. 1954.
Borovkov, A.A., Boxma, O.J., and Palmowski, Z. On the integral of the workload process of the single server queue. **40**:200–225, 2003.
Denisov, D. A note on the asymptotics for the maximum on a random time interval of a random walk. 165-–169, 2005.
Denisov, D., Dieker, A. B. and Shneer, V. Large deviations for random walks under subexponentiality: The big-jump domain. 1946–1991, 2008.
Denisov, D. and Shneer, V. Asymptotics for the first passage times of Lévy processes and random walks. 64-–84, 2013.
Foss, S., Palmowski, Z. and Zachary, S. , [**15**]{}(3):1936–1957, 2005.
Doney, R.A. On the asymptotic behaviour of first passage times for transient random walks. , [**81**]{}, 239–246, 1989.
Kulik, R. and Palmowski, Z. Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations. :275–284, 2011.
Perfilev, A. and Wachtel, V. Local asymptotics for the area under the random walk excursion. , [**50**]{}(2):600–620, 2018.
Rozovskii, L.V. (1993) Probabilities of large deviations on the whole axis. , [**38**]{}(1), 53–79.
|
---
abstract: |
Broadcast is one of the fundamental network communication primitives. One node of a network, called the [*source*]{}, has a message that has to be learned by all other nodes. We consider broadcast in radio networks, modeled as simple undirected connected graphs with a distinguished source. Nodes communicate in synchronous rounds. In each round, a node can either transmit a message to all its neighbours, or stay silent and listen. At the receiving end, a node $v$ hears a message from a neighbour $w$ in a given round if $v$ listens in this round and if $w$ is its only neighbour that transmits in this round. If more than one neighbour of a node $v$ transmits in a given round, we say that a [*collision*]{} occurs at $v$. We do not assume collision detection: in case of a collision, node $v$ does not hear anything (except the background noise that it also hears when no neighbour transmits).
We are interested in the feasibility of deterministic broadcast in radio networks. If nodes of the network do not have any labels, deterministic broadcast is impossible even in the four-cycle. On the other hand, if all nodes have distinct labels, then broadcast can be carried out, e.g., in a round-robin fashion, and hence $O(\log n)$-bit labels are sufficient for this task in $n$-node networks. In fact, $O(\log \Delta)$-bit labels, where $\Delta$ is the maximum degree, are enough to broadcast successfully. Hence, it is natural to ask if very short labels are sufficient for broadcast. Our main result is a positive answer to this question. We show that every radio network can be labeled using 2 bits in such a way that broadcast can be accomplished by some universal deterministic algorithm that does not know the network topology nor any bound on its size. Moreover, at the expense of an extra bit in the labels, we can get the following additional strong property of our algorithm: there exists a common round in which all nodes know that broadcast has been completed. [Finally, we show that 3-bit labels are also sufficient to solve both versions of broadcast in the case where the labeling scheme does not know which node is the source.]{}
[**keywords:**]{} broadcast, radio network, labeling scheme, feasibility
author:
- Faith Ellen
- Barun Gorain
- Avery Miller
- Andrzej Pelc
bibliography:
- 'labeling.bib'
title: |
[**Constant-Length Labeling Schemes for\
Deterministic Radio Broadcast**]{}
---
|
---
abstract: 'CoGeNT employs p-type point-contact (PPC) germanium detectors to search for Weakly Interacting Massive Particles (WIMPs). By virtue of its low energy threshold and ability to reject surface backgrounds, this type of device allows an emphasis on low-mass dark matter candidates ($m_{\chi}\sim10$ GeV/c$^{2}$). We report on the characteristics of the PPC detector presently taking data at the Soudan Underground Laboratory, elaborating on aspects of shielding, data acquisition, instrumental stability, data analysis, and background estimation. A detailed background model is used to investigate the low energy excess of events previously reported, and to assess the possibility of temporal modulations in the low-energy event rate. Extensive simulations of all presently known backgrounds do not provide a viable background explanation for the excess of low-energy events in the CoGeNT data, or the previously observed temporal variation in the event rate. Also reported on for the first time is a determination of the surface (slow pulse rise time) event contamination in the data as a function of energy. We conclude that the CoGeNT detector technology is well suited to search for the annual modulation signature expected from dark matter particle interactions in the region of WIMP mass and coupling favored by the DAMA/LIBRA results.'
author:
- 'C.E. Aalseth'
- 'P.S. Barbeau'
- 'J. Colaresi'
- 'J.I. Collar'
- 'J. Diaz Leon'
- 'J.E. Fast'
- 'N.E. Fields'
- 'T.W. Hossbach'
- 'A. Knecht'
- 'M.S. Kos'
- 'M.G. Marino'
- 'H.S. Miley'
- 'M.L. Miller'
- 'J.L. Orrell'
- 'K.M. Yocum'
title: 'CoGeNT: A Search for Low-Mass Dark Matter using $p$-type Point Contact Germanium Detectors'
---
Introduction
============
CoGeNT (Coherent Germanium Neutrino Technology) is a program aiming to exploit the characteristics of p-type point-contact germanium detectors in areas as diverse as the search for low-mass dark matter candidates, coherent neutrino-nucleus elastic scattering, and $^{76}$Ge double-beta decay [@jcap].
Data collected from a first CoGeNT detector at a shallow underground location demonstrated sensitivity to low-mass ($<10$ GeV/c$^{2}$) dark matter particles [@Aal08]. In particular, it appeared CoGeNT was particularly well suited to address the DAMA/LIBRA [@DAMA] modulation result. Following the identification of several sources of internal background in this prototype, a second CoGeNT detector was installed in the Soudan Underground Laboratory (SUL) during 2009 with the goal of improving upon the dark matter sensitivity reach of the 2008 result [@Aal08]. The first 56-days of operation of the CoGeNT detector at SUL showed an unexpected excess of events [@Aal11] above the anticipated backgrounds for ionization energies below 2 keV. Further data collection from this detector continued until an interruption imposed by a fire in the access shaft to the laboratory halted the initial run in March of 2011. Analysis of the accumulated data set [@Aal11b], spanning 442 live days over the period 4 December 2009 to 6 March 2011, showed a $\sim2.8\sigma$ significance modulation of the monthly event rate in the low-energy region that is compatible with the dark matter signature described in [@andrzej]. The fitting procedure generating this low-significance modulation result used unconstrained phase, period, and amplitude variables. Time-stamped data have been made publicly available, allowing for a number of independent analyses and interpretations.
In this paper we provide a more in-depth description of the apparatus and data analysis, concentrating on aspects of instrument stability, data cuts, uncertainties, and background estimation. The data set employed for this discussion is the same as in [@Aal11b], and all energies are in keVee (keV electron equivalent, i.e., ionization energy), unless otherwise stated. Following the three-month outage resulting from the Soudan fire, this detector has taken data continuously, starting 7 June 2011. An additional body of data is to be released in the near future. The design and expectations for CoGeNT-4 (C-4), a planned expansion aiming at an increase in active mass by a factor of ten, featuring four large PPC detectors with a reduced energy threshold and lower background, are discussed in a separate publication [@inprep].
Description of the Apparatus
============================
The present CoGeNT detector is located at the Soudan Underground Laboratory (Soudan, Minnesota, USA) at a vertical depth of 2341 feet (689 feet below sea level), providing 2090 meters of water equivalent (m.w.e.) overburden as shielding against cosmic rays and associated backgrounds. The detector shield is placed on a floor built on top of base I-beams that once supported the Soudan-2 proton decay experiment [@All96]. The detector element is a single modified BEGe germanium diode. BEGe (Broad Energy Germanium) is the commercial denomination used by the manufacturer (CANBERRA Industries) for their line of PPC detectors. The technical characteristics of this PPC are shown in Table \[tab:BEGeCharacteristics\]. The detector is contained within an OFHC copper end cap cryostat, and mounted in an OFHC copper inner can connected to an OFHC copper cold finger. Internal detector parts were custom manufactured in either OFHC copper or PTFE. All internal parts were etched to remove surface contaminations using ultra-pure acids in class 100 clean room conditions, following procedures similar to those described in [@etch]. A commercial stainless steel horizontal cryostat encloses the rear of the assembly, providing electrical feed-through to a side-mounted CANBERRA DPRP pulse-reset preamplifier typically used in high-resolution X-ray detectors (figure \[fig:BEGeInnerShield\]).
Property Value
------------------------- --------------------------
Manufacturer CANBERRA (modified BEGe)
Total Mass 443 gram
Estimated Fiducial Mass $\sim$330 gram
Outer Diameter 60.5 mm
Length 31 mm
Capacitance 1.8 pF (at 3000 V bias)
: \[tab:BEGeCharacteristics\] Characteristics of the CoGeNT high purity PPC germanium detector at SUL.
Shield design
-------------
![\[fig:BEGeInnerShield\] Partially disassembled shield of the CoGeNT detector at SUL, showing the cylindrical OFHC end cap and innermost 5 cm of ancient 0.02 Bq $^{210}$Pb/kg lead, characteristically oxidized following etching. The preamplifier is visible at the top right (black box). A minimum of 7 cm of lead thickness shields the detector from the naturally occuring radioactivity in the preamplifier’s electronic components.](shieldlayers.eps){height="0.27\textheight"}
The lead shield involves three categories of lead bricks. The innermost 5 cm layer is composed of acid-etched ultra-low background ancient lead having a $^{210}$Pb content of approximately 0.02 Bq $^{210}$Pb/kg, measured using radiochemical extraction followed by alpha spectroscopy at PNNL [@SMiley]. This layer provides shielding against the $^{210}$Pb bremsstrahlung continuum from external contemporary lead, resulting in a negligible low-energy background from this source of less than 0.01 counts / keVee / kg-Ge / day [@pb210]. OFHC copper bricks are used to provide mechanical support around the stainless steel horizontal cryostat body (figure \[fig:BEGeInnerShield\]). A middle 10 cm thick layer of contemporary ($\sim$100 Bq $^{210}$Pb/kg) lead bricks is also chemically etched and cleaned. The outer 10 cm thick layer is composed of stock bricks not chemically etched. A minimum of 25 cm of lead surround the detector element in all directions. The assembly of the lead shield was performed inside a temporary soft-wall clean room, to avoid excess dust.
![\[fig:SoudanOuterShield\] Layout of the complete shield for the CoGeNT detector. The outermost component is a layer of recycled HDPE, used to moderate neutrons. Next towards the interior, a 1 inch thick layer of borated polyethylene captures moderated neutrons. Three layers of lead are indicated by the three different inner shaded regions. The outermost lead is composed of stock bricks, not chemically etched, the middle layer is chemically etched and cleaned, and the innermost layer consists of ultra-low background ancient lead. An automated liquid nitrogen transfer system refills the detector Dewar every 48 hours, maintaining the germanium crystal at a near constant temperature. See text for a full description of these components.](soudandetectorrev8.eps){height="0.35\textheight"}
Exterior to the lead shield is a 2.5 cm thick layer of 30% borated polyethylene, intended to act as a thermal neutron absorber. The borated polyethylene panels are sealed using heavy vinyl tape as a barrier against radon ingress. The inner lead shield and the borated polyethylene are contained inside of an aluminum sheet-metal box (table base, four walls, and top). All edges are once again sealed using heavy vinyl tape. Shielding materials internal to this radon-exclusion volume are supported by an aluminum extrusion table approximately 66 cm above the floor. This volume is continuously flushed with boil-off nitrogen gas from a dedicated pressurized Dewar, at a rate of 2 liters per minute. An extruded aluminum structural frame provides mechanical rigidity to the sealed aluminum box. The detector Dewar rests on a layer of vibration absorbing foam aiming at reducing microphonic events (Sec.IV). Finally, an external layer of recycled high-density polyethylene (HDPE) deck planking is used to enclose the entire assembly, acting as a neutron moderator. The HPDE is 18.3 cm thick, with nearly complete $4\pi$ coverage (the only breach being the table legs supporting the lead cave). These elements can be seen in figure \[fig:SoudanOuterShield\].
Not visible in figure \[fig:SoudanOuterShield\] is an active muon veto composed of 10 flat panels surrounding the HDPE shield, with six 120 cm $\times$ 120 cm panels on the sides and four 100 cm $\times$ 100 cm panels covering the top with considerable overlap and overhang. The veto panels are 1 cm thick and read-out via a single PMT located at the center of each panel. The light collection efficiency was measured at a grid of positions in the panels using a low-energy gamma source, observing a minimum yield at all locations better than 50% of the central maximum. A $\sim$90% geometric coverage of the shield is estimated for this muon veto. Further discussion of its efficiency is provided in Sec.IV-A.
Data acquisition
----------------
Figure \[fig:electronics\] shows an schematic of the data acquisition (DAQ) system used in the present CoGeNT installation at SUL [@phil]. It combines analog amplification of detector pulses with digitization of raw preamplifier traces, the second permitting the rejection of events taking place near the surface of the germanium crystal via rise time cuts [@Aal11]. An initial data taking period from the end of August 2009 to 1 December 2009 did not include preamplifier trace digitization. This period allowed for the decay of short-lived cosmogenic isotopes (e.g., $^{71}$Ge with $t_{1/2} = 11.4$ d). In early December 2009 a third National Instruments PCI-5102 digitizer card was installed to collect preamplifier traces. During this initial period a parallel DAQ system based on the GRETINA Mark IV digitizer [@gretina] was also tested, but found to provide limited information for low energy analysis [@mike].
![\[fig:electronics\] Schematic of the data acquisition system for the CoGeNT detector at SUL (see text). ](electronics.eps){width="1.\textwidth"}
A pulse-reset preamplifier, typically employed for silicon X-ray detectors, is used in combination with a field-effect transistor (FET) specially selected to match the PPC’s small ($\sim$2 pF) capacitance. This allows for the lowest possible electronic noise and energy threshold [@jcap]. The preamplifier generates two equivalent signal outputs, an inhibit logic signal when the pulse reset circuitry of the preamplifier is active, and accepts a test input (electronic pulser). The test input is normally disconnected, terminated, and isolated to avoid spurious noise injections. While the ORTEC 671 and 672 shaping amplifiers utilize the inhibit logic signal to protect against distortions caused by the preamplifier reset, the amplifier outputs are sufficiently altered to initiate the DAQ, which is set to trigger on very low energy (300 eVee) shaped pulses. Even with the very long reset period ($\sim$320 ms) achieved in this detector — a result of its sub-pA leakage current — this would generate an unacceptable $\sim300$ Gbyte/day of pulse reset induced traces streaming to disk. The triggering output of the 671 shaping amplifier is therefore further inhibited by use of a linear gate operated in blocking mode. The gate is observed to add a negligible amount of noise to the already sufficiently amplified pulses. The duration of the inhibit logic pulse is set to its maximum (650 $\mu$s) in order to ensure a complete restoration of the amplifier baseline following resets (achieved within $\sim100$ $\mu$s), while generating a negligible 0.2% dead time. The frequency of the preamplifier resets, which is directly proportional to the leakage current of the detector and in turn to the germanium crystal temperature, has been periodically measured and shown to have remained constant thus far. Any significant alteration of this leakage current would also appear as a measurable increase in the white parallel component of the detector noise [@pullia], dominant for the channel used in noise monitoring (shaping time $\tau=10$ $\mu$s). The detector noise is observed to be very stable over the detector’s operational period (figure \[fig:stability\]). Further discussion on DAQ stability is provided in Sec.III-E.
The readout system is composed of three hardware-synchronized PCI-based National Instruments digitizers totalling 6 channels, sampling at 20 MSamples/s, each with a resolution of 8 bits. The acquisition software is a Windows-based LabVIEW program, also responsible for liquid nitrogen auto-refills and electronic pulser control. Raw preamplifier traces are amplified prior to digitization using a low-noise Phillips Scientific 777 fast amplifier (200 MHz bandwidth), using a DC-blocking capacitor at its input to yield a $\sim$50 $\mu$s preamplifier pulse decay time, noticeable in figure \[fig:CoGeNT-Traces\].
![\[fig:CoGeNT-Traces\] Example digitized traces from the six CoGeNT DAQ read-out channels, corresponding to an event with energy $\sim2.5$ keVee. Preamplifier traces are DC-offset at the Phillips Scientific 777 amplifier to allow for rise time measurements of pulses in the range 0-12 keVee, following offline wavelet denoising [@Aal11] (not yet applied to these traces).](CoGeNT-Traces-cropped.eps){width="46.00000%"}
Following gain-matching bias adjustments, the PMT outputs from all muon veto panels are daisy-chained and reduced to one single channel, which is linearly amplified, discriminated with a threshold set at single photo-electron level, and further conditioned using a gate generator, the output of which is digitized by the DAQ [@phil]. Traces captured for an example event are shown in Figure \[fig:CoGeNT-Traces\]. Digitized trace lengths are an intentionally long at 400 $\mu$s, with 80% pre-trigger content. Pre-trigger information allows for pulse diagnostics (Sec.IV), monitoring of detector noise and trigger threshold stability (Sec.III-E), and is also used in pulse simulations (Sec.IV-B).
The PC housing the digitizer cards maintains an internal buffer to store a set of events. After 20 events are stored, data from the digitizer buffer is written to disk. File names are cycled (open file closed, saved, and new file opened) every 3 hours. Data are automatically backed-up to a second PC, from which they are transferred to a remote server.
Detector Characterization
=========================
Several aspects of detector and DAQ characterization are described in this section.
Energy Calibration
------------------
The existing DAQ system was developed with an emphasis on instrumental stability, minimization of electronic noise, and on providing a maximum of information about low-energy events. It is however limited in its energy range, 0-16 keVee. While it is possible to increase this range during dedicated background characterization runs (figure \[fig:data300gammas\]), this can be done only at the expense of valuable information used for data selection cuts at lower energies. During normal operation, no viable external gamma sources exist for low-energy calibration. This is due to the thickness of the OFHC cryostat parts and germanium dead layer surrounding the active bulk of the detector, which dramatically attenuate external low-energy photons. Fortunately, a number of internal peaks arising from cosmogenic isotopes decaying via electron capture (EC) are visible in the region 1-10 keVee. These are used to extract an accurate energy calibration and to characterize the energy resolution as a function of energy. The reader is referred to [@Aal08; @Aal11; @Aal11b] for additional details.
Quenching Factor
----------------
The quenching factor, defined as the measurable fraction of the energy deposited by a nuclear recoil in a detecting medium, is a quantity of particular relevance for WIMP dark matter studies. For PPCs and conventional germanium detectors, its characterization involves a measurement of the ionization generated by a discrete recoil energy, typically induced in a neutron calibration. The CoGeNT PPC described in [@Aal08] was exposed to a custom-built monochromatic 24 keV filtered neutron beam at the Kansas State University research reactor. This PPC crystal is nearly identical to that operating in SUL [@Aal11; @Aal11b] (BEGe contact geometry, similar 160 eVee FWHM electronic noise and 0.5 keVee threshold, 83.4 cc [vs.]{} 85 cc crystal volume, and the same nominal Li diffusion depth in the outer contact). Triggering on the neutron capture peak of the $^{6}$LiI scintillator used to detect the scattered neutrons [@phil] allowed the measurement of sub-keV quenching factors, found to be in good agreement with other available data (figure \[fig:quenching\]). Details on neutron beam design and characterization, and on the analysis of these data are provided in [@reactor] and [@phil], respectively.
![\[fig:quenching\] Neutron scattering measurements of the low-energy quenching factor for nuclear recoils in germanium, compared to Lindhard theory predictions. CoGeNT adopts the expression relating ionization and recoil energy $E_{i}$(keVee) $= 0.2 \times E_{r}^{1.12}$(keVr), valid for the range 0.2 keVr $<E_{r}<$ 10 keVr, and essentially indistinguishable from the Lindhard case plotted.](quenching.eps){width="46.00000%"}
Dead Layer
----------
PPC detectors feature an inert outer contact layer over most of their surface. The depth of this dead layer can be tuned during the manufacturing process, by controlling the amount of lithium diffused into this region. CoGeNT detectors are built with the maximum diffusion depth possible during BEGe fabrication, nominally a $\sim1$ mm dead layer over all surfaces except for a small (3.8 cm$^{2}$) intra-contact passivated area. This dead layer acts as a passive barrier against external low-energy radiation (X-rays, betas, etc.). Events taking place in the region immediately below this dead layer (“transition layer,” figure \[fig:deadlayer\]) generate pulses with a characteristically slow rise time, and a partial charge collection efficiency [@Aal11; @sakai; @ryan]. The surface structure of the CoGeNT PPC in [@Aal08] was characterized using uncollimated $^{241}$Am 59.5 keV gammas impinging on the top surface of the germanium crystal, opposite to the central contact. Following a MCNP-Polimi simulation [@polimi] of interaction depth [vs.]{} energy deposition including all internal cryostat parts, and assuming a sigmoid description of charge collection efficiency as a function of depth into the crystal, we find a best-fit profile quantitatively and qualitatively similar to that described in [@sakai] ($\sim\!1$ mm dead layer, $\sim\!1$ mm transition layer, figure \[fig:deadlayer\] inset). This characterization was unfortunately not possible for the PPC at SUL [@Aal11; @Aal11b] prior to installation within its shield. Due to the aforementioned very similar characteristics for these two PPCs, we adopt the same surface structure when calculating the fiducial (bulk) volume following rise time cuts [@Aal11], while cautiously assigning a $\sim$10% uncertainty to its value. Additional tests are planned following removal of the PPC at SUL from its shielding.
While the passive shielding provided by the deepest possible lithium diffusion is useful for low-energy background reduction in a dark matter search, it is clearly detrimental to the fiducial mass of a relatively small PPC crystal (Table \[tab:BEGeCharacteristics\]). This fiducial mass loss due to deep lithium diffusion for background reduction creates a contrast to the requirements of $^{76}$Ge double-beta decay experiments like <span style="font-variant:small-caps;">Majorana</span> [@majorana] and GERDA [@gerda], where a maximization of the active enriched germanium mass is preferable. Surface characterization studies using a PPC featuring a shallower lithium diffusion can be found in [@ryan] and support the notional model of energy depositions in the transition layer resulting in pulses of partial charge collection and slowed rise times.
![\[fig:deadlayer\] Characterization of surface structure on the external n+ contact of a PPC (see text). The two free sigmoid parameters are fit via comparison of calibration data and Monte Carlo simulation. Energy depositions taking place in the transition layer near its boundary with the dead layer lead to large signal rise times, i.e., slow pulses. On the opposite side of the transition layer, rise times progressively approach the small values typical of a fast (bulk) event.](deadlayer.eps){width="46.00000%"}
Trigger Efficiency
------------------
The PPC detector in [@Aal11; @Aal11b] and its DAQ were operated for a year at a depth of 30 m.w.e., up to a few weeks before installation at SUL. During that time (and the cosmogenic activation “cooling” period August-December 2009 at SUL) automatic pulser calibrations were performed for a minute every two hours, revealing an excellent trigger rate stability (better than 0.1%) for electronic pulses with energy equivalent to 1.85 keVee [@phil]. To avoid the injection of any noise or spurious pulses through the preamplifier test input during dark matter search runs, these automatic calibrations were suspended in December of 2009, isolating and terminating that input. However, trigger efficiency calibrations using an electronic pulser have been performed thus far four times, during each interruption to physics runs, yielding reproducible results (figure \[fig:triggering\]). These calibrations allow us to calculate triggering efficiency corrections to the energy spectrum near threshold, as well as to determine the energy-dependent signal acceptance for fast rise time pulses, representative of ionization events occurring in the bulk of the crystal [@Aal11; @Aal11b]. In addition to these pulser calibrations, the trigger threshold level is monitored continuously, as described in the following section.
![\[fig:triggering\] Trigger efficiency vs. energy equivalent for 10 Hz tailed electronic pulses generated with a 814FP CANBERRA pulser. Inset: gain shift stability monitored through the centroid of a Gaussian fit to the 10.3 keV cosmogenic peak. The count rate under this peak decayed from roughly 500 to 150 events per month over the time span plotted.](triggering.eps){width="46.00000%"}
Overall Stability
-----------------
No significant changes in gain have been observed for the PPC at SUL over more than two years of continuous operation, as monitored by the position of the 10.37 keV $^{68}$Ge decay peak (inset figure \[fig:triggering\]) and of the energy threshold, immutable at 0.5 keVee. The long (320 $\mu$s) pre-trigger segment of the traces collected by the DAQ allows us to monitor both the electronic noise of the detector and the small fluctuations in trigger threshold level induced by fluctuations of the CH0 baseline with respect to the constant (i.e., digitally-set) threshold level (figure \[fig:stability\]). These baseline fluctuations do not result in a smearing of the energy resolution, given that the zero-energy level is recomputed for each individual pulse from its pre-trigger baseline. They result instead in small shifts by a maximum of $\pm$20 eVee in the sigmoid-like threshold efficiency curve in figure \[fig:triggering\]. As a result, they produce correlated changes in trigger rate below the 0.5 keVee threshold, but their effect is negligible above $\sim$0.55 keVee, an energy for which the triggering efficiency reaches 100%. It is possible to calculate the effect of these baseline fluctuations on the counting rate above the analysis threshold for an exponentially decreasing spectrum like that observed [@Aal11b]: this is $\pm$0.1% for the region 0.5-0.9 keVee (figure \[fig:stability\]), and smaller for any energy range extending beyond 0.9 keVee, which is negligible from the point of view of a search for a few percent annual modulation.
Much interest has been traditionally placed on investigating modulated backgrounds having an origin in natural radioactivity (underground muons, radon emanations, etc., see Sec.V), but little discussion can be found in the literature on the specific details of possible instrumental instabilities affecting the DAMA/LIBRA experiment. Searches for a dark matter annual modulation signature need to be concerned about these, in view of the small (few percent) fluctuations in rate expected, the low energies involved, and the unfortunate seasonally correlated phase, having a maximum in summer and minimum in winter, similar to so many unrelated natural processes. As mentioned, it is possible to exclude gain shifts, variations in detector noise and threshold position, and trigger threshold level fluctuations as sources of a significant modulation in CoGeNT rates. The trigger rate is very low (few per hour, including noise triggers), precluding trigger saturation effects. Interference from human activity also seems to be absent (figure \[fig:diurnal\] and discussion in [@neal]). However, an arbitrarily long list of other possibilities can be examined. For instance, the performance of the linear gate present in the triggering channel (figure \[fig:electronics\]) can be considered. Fluctuations in detector leakage current could in principle alter the preamplifier reset period to the point of creating sufficiently large changes in the 0.2% trigger dead time induced by the inhibit logic signal (Sec.II-B). For these to mimic a modulation in rate of the $\sim$16% amplitude reported in [@Aal11b], the detector leakage current and reset period would have to inadvertently vary by a factor of $\sim80$. This would induce changes to the FWHM white parallel electronic noise, dominant for the channel monitored in figure \[fig:stability\], by a factor $\sim\sqrt{80}$ [@pullia]. These are clearly excluded. In addition to this, linear gate blocking circuitry fluctuations having any other origin would affect all pulses independently of their energy or rise time, an effect not observed [@Aal11b].
![\[fig:stability\] Daily average electronic noise and trigger threshold in the CoGeNT PPC at SUL. The small jump in electronic noise post-fire has a negligible effect on the detector threshold. It is the result of either temperature cycling of the crystal (leading to known processes capable of altering the detector leakage current, minimally in this case) or a displacement of cables during emergency post-fire interventions. The fluctuations in trigger threshold agree well with expectations based on manufacturer specifications for the ORTEC 672 shaping amplifier and NI PCI-5102 digitizers, and the observed $\pm1$ $^\circ$C environmental temperature changes measured at SUL.](stability.eps){width="46.00000%"}
![\[fig:diurnal\] Diurnal stability of CoGeNT at SUL. Periods of human presence at SUL are $\sim$7 am - 5 pm.](diurnal.eps){width="50.00000%"}
An additional example of an instrumental effect able, in principle, to generate event rate fluctuations is the pulse rise time dependence on crystal temperature described in [@bela] for n-type germanium detectors. For the CoGeNT detector, these changes would translate into anti-correlated modulations in surface and bulk event rates, which are not observed, and only for very large seasonal swings in detector temperature of $>\!10$ degrees Celsius. These temperature swings are not expected, given the precautionary 48 hour automatic refills of the Dewar, and the constant LN2 consumption through the year. Ambient temperature at the location of the CoGeNT detector (20.5 $^{\circ}$C) is monitored to be constant within $\pm1$ $^\circ$C, the expected maximum yearly temperature variation in detector and DAQ. In addition to this, the effect is expected to be less noticeable for p-type diodes, which feature considerably better charge mobility than n-type detectors. However, it is worth emphasizing the existence of such subtle instrumental effects, in order to fully appreciate the difficulties involved in obtaining convincing evidence for a dark matter annual modulation signature from any single experiment. A pragmatic approach to this issue is to redesign as much of the DAQ and electronics as possible in all future searches, as planned for the C-4 experiment [@inprep].
Data Selection Cuts
===================
![\[fig:analysis\] Steps in data selection through the UC analysis pipeline: a) All data including microphonics-intensive periods of LN2 Dewar filling. b) Following removal of LN2 transfer periods and ensuing 10 minutes (boiling in the Dewar lasts a few minutes). No correlated excess of events is observed to extend beyond this 10 min cut. c) Following application of cuts intended to remove anomalous electronic pulses (see text). The boundaries for a final cut using the CH0/CH1 amplitude method in [@julio] are shown as horizontal lines. These boundaries are selected to minimize the effect of this cut for both radiation-induced and pulser events, with the exception of a distinct family of residual microphonic events visible as a diagonal band in this panel. d) Fast electronic pulser events prior to any cuts (only the CH0/CH1 amplitude criterion is seen to minimally affect these).](analysis.eps){height=".8\textheight"}
![\[fig:bursts\] Distribution of time span between consecutive events passing microphonic cuts (see text). A small deviation from a Poisson distribution is observed at t$<$12 s. A large fraction of events in the first bin correspond to the decay of cosmogenic $^{73}$As, involving a short-lived (t$_{1/2}$=0.5 s) excited state [@Aal11; @phil].](bursts.eps){height=".22\textheight"}
![\[fig:grayscale\] Grayscale plot showing the distribution of rise time vs. energy for events passing all other cuts, collected over a 27 month live period for the detector at SUL. Fast bulk events appear highly concentrated around a $\sim$325 ns rise time, their distribution becoming progressively slower towards zero energy by the effect of electronic noise in preamplifier traces (Sec.IV-B), already visibly affecting the cosmogenic peaks around 1.3 keV. The dotted red line corresponds to the 90% acceptance boundary for fast electronic pulse events, used for rise time cuts in [@Aal11; @Aal11b].](grayscale.eps){height=".2\textheight"}
![\[fig:EnergyEstimatorCompare\] Event-by-event comparison of energy estimators from UC and UW data analysis pipelines (442 day dataset, [@Aal11b]). The top panel shows that the two energy estimators are very well correlated. The bottom panel indicates that the maximum difference between energy estimators is $<$ 4% above the analysis threshold of 0.5 keV.](Ecomps_UW_UC.eps){height="0.3\textheight"}
![\[fig:RisetimeEstimatorCompare\] Event-by-event comparison of rise time estimators from UC and UW data analysis pipelines (442 day dataset, [@Aal11b]) in the region 0.5-3.0 keVee. The top panel (a) shows the correlation between the two estimators. The fractional difference between the two estimators is shown in (b). The two rise time estimators are fairly well correlated, with a disagreement in the classification as fast (bulk) or slow (surface) for only 11% of the events in the 0.5-3.0 keVee analysis region. In the region of 0.5-1.0 keVee this disagreement affects 16% of the events.](risetimeupdateUCUW.eps){height="0.3\textheight"}
![\[fig:RisetimeEstimatorCompare2\] Comparison of energy spectra and overall event rate from the UC and UW analysis pipelines. Panel (a) shows the similar energy spectra obtained following independent data selection and rise time cuts. Panel (b) displays the daily rates in the region 0.5-3.0 keVee for events passing all cuts.](rateenergyupdateUCUW.eps){height="0.3\textheight"}
![\[fig:modulation\] Comparison between irreducible monthly rates in two different energy regions, for the UC (black) and UW (circles) analysis pipelines. The correction for low-energy cosmogenics present in these regions [@Aal11b] is applied, and calculated independently for each pipeline.](modupdateUCUW.eps){height="0.3\textheight"}
The data acquisition system described in Sec.II-B is designed to exploit a technique detailed in [@julio], able to provide efficient discrimination against low-energy microphonic pulses arising from acoustic or mechanical disturbances to the detector. In this method, any anomalous preamplifier trace characteristic of a microphonic event is assigned markedly different amplitudes when processed through amplifiers set to dissimilar shaping times (CH0 and CH1 here, figure \[fig:analysis\]). An alternative approach to microphonic rejection based on wavelet analysis [@igor] was tested. It was found to offer no advantage over that in [@julio] for these data, while imposing a considerable penalty on the analysis CPU time. In addition to this microphonic cut, preamplifier traces are screened against deviations from the pattern of a normal radiation-induced pulse (rise time of less than a few $\mu$s, decay time $\sim$50 $\mu$s): several custom data cuts discriminate against sporadic characteristic electronic noise signals (ringing, spikes, reverse polarity pulses from HV micro-discharges, “telegraph” noise). These cuts are observed to remove a majority of microphonic pulses on their own, even prior to CH0/CH1 amplitude ratio cuts (figure \[fig:analysis\]). As in [@julio], we observe a very small number of microphonic events escaping amplitude ratio cuts. These can be identified by their time correlation, appearing in bunches around times of disturbance. They are removed with an additional time cut (vertical line in figure \[fig:bursts\]) that imposes a negligible dead time.
A final cut selects fast rise time preamplifier pulses, identified with those taking place in the fiducial bulk volume of the crystal, i.e., rejecting the majority of slow, partial charge collection pulses originating in the surface transition layer (Sec.III-C, [@Aal11]). This cut is defined by the energy-dependent boundary for 90% acceptance of fast electronic pulser signals (figure \[fig:grayscale\], [@Aal11]). Pulser scans are used to build an efficiency curve in passing all analysis cuts, used in combination with the trigger efficiency (figure \[fig:triggering\]) to generate a modest correction to the energy spectrum [@Aal11; @Aal11b] (top panel in figure \[fig:steps\]).
Two parallel schemes were developed for CoGeNT data analysis. Both employ independent methods of wavelet denoising on preamplifier traces previous to rise time determination, which also follows separate algorithms. Custom cuts against electronic noise are also independently designed, as well as those for microphonic rejection. Emphasis was placed on avoiding mutual influence between the teams developing these analysis pipelines. The first one, developed at University of Chicago (“UC”) was employed in [@Aal08; @Aal11; @Aal11b]. The second, developed at University of Washington [@mike] (“UW”) was used in cross-checking the results in [@Aal11; @Aal11b]. There is good event overlap between the two analysis pipelines, with roughly 90% of the events passing one set of cuts also passing the other. Figures \[fig:EnergyEstimatorCompare\], \[fig:RisetimeEstimatorCompare\], \[fig:RisetimeEstimatorCompare2\], and \[fig:modulation\] display several of the cross-checks performed prior to publication of a search for an annual modulation [@Aal11b]. Slighty more events pass the UW risetime cut than the UC risetime cut, by 6.7%. Both pipelines generate remarkably close irreducible energy spectra and temporal evolution (figures \[fig:RisetimeEstimatorCompare2\] and \[fig:modulation\]). In particular, the possible modulation investigated in [@Aal11b] is visible in both lines of analysis (figure \[fig:modulation\]). The parameters used for data selection cuts for both pipelines are constant in time, and in the case of the UC pipeline, they were frozen prior to the publication of [@Aal11], implementing a [*de facto*]{} blind analysis for the larger dataset in [@Aal11b].
Cosmic ray veto cuts
--------------------
While the CoGeNT detector at SUL incorporates an active muon veto system, no veto cuts are applied to the data in [@Aal11; @Aal11b]. This is done to avoid introducing any artificial modulation to the event rates arising from fluctuations in the efficiency of this veto or its electronics (recall its setting to single photo-electron detection, which makes it particularly sensitive to such effects). As discussed in this section, it is however possible to make use of this veto to demonstrate that only a negligible fraction of the low-energy events arise from muon-induced radiations, rendering this cut superfluous. This negligible contribution is confirmed by the ($\mu$,n) and ($\mu$,$\gamma$) simulations discussed in Sec.V-A.
Operation at single photo-electron sensitivity is required to ensure good efficiency for muon detection from thin (1 cm) scintillator panels, for which a discriminator setting able to separate muon passage from environmental gamma interactions with the veto is not possible. This good efficiency is confirmed by the agreement between the rate of true veto-germanium coincidences (figure \[fig:muon1\]) and that predicted by the simulations (Sec.V-A). Specifically, 0.67$\pm$0.12 true coincidences per day were observed during the 442 d of data analyzed in [@Aal11b], whereas 0.77$\pm$0.15 coincidences per day are expected from ($\mu$,n) and ($\mu,\gamma$) simulations. The price to pay for this good muon-detection efficiency is a high veto triggering rate ($\sim$5,000 Hz), which would result in a $\sim$14% dead time from dominant spurious coincidences were a veto cut applied to the data. It is however evident that the application of the veto coincidence cut would effectively remove a majority of muon-induced events in the germanium detector.
The inset in figure \[fig:muon1\] displays as a function of energy the fraction of events that are removed by application of this cut with a conservative 20 $\mu$s coincidence window. No deviation from the $\sim$14% rate reduction expected from spurious coincidences is noticeable at low energy, indicating that at maximum a few percent of the spectral rise at low energy observed in [@Aal11; @Aal11b] can be due to muon-induced events. A similar conclusion is derived from the simulations in Sec.V-A. As expected, the application of the veto cut simply decreases the irreducible event rate by this $\sim$14% fraction, not altering the possible modulation investigated in [@Aal11b] (figure \[fig:muon2\]). In Sec.V-A we will conclude that the muon-induced modulation amplitude expected for CoGeNT at SUL is of O(0.1)%. Separately, the MINOS collaboration finds a three-sigma inconsistency between the phases of their measured modulation in muon flux at SUL, and that observed in CoGeNT data [@minosmod].
![\[fig:muon1\] True coincidences between muon veto and PPC appear as an excess above spurious coincidences, displaying the typical delay by a few $\mu$s characteristic of fast neutron straggling. See text for a discussion on the comparison of their rate with that predicted by simulations. Inset: fraction of events removed by a muon veto cut (see text).](muon1.eps){height="0.31\textheight"}
![\[fig:muon2\] Effect of the application of a veto coincidence cut on the monthly irreducible event rate (see text). White circles incorporate this cut following all other data cuts, as opposed to the inset of figure \[fig:muon1\], where it is applied directly on uncut data. This leads to minor differences in the obtained reduction in event rate. The energy range for this figure is 0.5-3.0 keVee.](muon2.eps){width="0.34\textheight"}
Uncertainties in the rejection of surface events
------------------------------------------------
As discussed in [@Aal11; @Aal11b] and visible in figure \[fig:grayscale\], the ability to discriminate between fast rise time (bulk) and slow rise time (surface) events is progressively diminished for energies approaching the 0.5 keVee threshold. When the amplitude of a preamplifier pulse becomes close to the amplitude of the circuit’s electronic noise variations, an accurate measurement of rise time becomes more difficult to perform, even after wavelet denoising. Determining the bulk-event signal acceptance (SA) is straightforward when electronic pulser signals are identified to be a close replica of fast radiation-induced events in the bulk of the crystal [@Aal11]. In the analysis described in this section this SA is kept at an energy-independent 90% (red dotted line in figure \[fig:grayscale\]), as in [@Aal11; @Aal11b]. Using an additional 12 months of exposure beyond the dataset in [@Aal11b], we can finally attempt the exercise of calculating the surface event background rejection (BR) as a function of energy. It must be emphasized that the resulting correction (the true fraction of bulk events in those passing all cuts, figure \[fig:surface\_correction\]) can only be applied to the irreducible energy spectrum, and not to individual pulses on an event-by-event basis, similar to the case of low-energy nuclear and electron recoil discrimination in sodium iodide detectors [@smith].
![\[fig:surface\_simulations\] Simulated preamplifier pulses with an initial rise time of 325 ns, representing ideal fast (bulk) events, are convoluted with electronic noise and treated with the same wavelet denoising and rise time measurement algorithms applied to real events. This electronic noise is grafted directly from pre-trigger preamplifier traces taken from real detector events, leading to perfect modeling of the noise frequency spectrum. The resulting rise time distributions are represented by red curves, labelled by their energy equivalent. The same is repeated for typical slow (surface) pulses with a rise time of 2 $\mu$s, generating the blue curves. Each simulation contains 35k events. These simulations provide a qualitative understanding of the behavior observed in figure \[fig:grayscale\].](surface_simulations.eps){width="0.36\textheight"}
![\[fig:surface\_fits\] Example rise time distributions for events falling within discrete energy bins, from a 27 month exposure of the CoGeNT detector at SUL. These are fitted by two log-normal distributions with free parameters, corresponding to slow rise time surface events (blue) and fast rise time bulk events (red). Small vertical arrows point at the location of the 90% C.L. fast signal acceptance boundary dictated by electronic pulser calibrations (dotted red line in figure \[fig:grayscale\]). A contamination of the events passing this cut by unrejected surface events progresses as energy decreases (see text). ](surface_fits.eps){width="0.35\textheight"}
![\[fig:surface\_correction\] Fraction of events passing the 90% fast signal acceptance cut (pulser cut, dotted red line in figure \[fig:grayscale\]) identified as true bulk events via the analysis discussed in Sec.IV-B. Alternatively defined, its complement is the fraction of events passing the pulser cut that are in actuality misidentified surface events (see figure \[fig:surface\_fits\]). The dotted line is a fit with functional form $1-e^{-a\cdot E(keVee)}$, with $a=1.21\pm0.11$. Error bars are extracted from the uncertainties in fits like those exemplified in figure \[fig:surface\_fits\].](surface_correction.eps){width="0.35\textheight"}
![\[fig:steps\] Steps in the treatment of a low-energy CoGeNT spectrum. a) Spectrum following data selection cuts (Sec.IV), including 90% fast signal acceptance cuts from pulser calibrations (dotted red line in figure \[fig:grayscale\]) [@Aal11; @Aal11b]. This spectrum is nominally composed by a majority of bulk events. Overimposed is the combined trigger and background cut efficiency. This efficiency is derived from high-statistics pulser runs (figures \[fig:triggering\] and \[fig:analysis\]), resulting in a negligible associated uncertainty. b) Spectrum following this trigger plus cut efficiency correction. Overimposed is the residual surface event correction. This correction and its associated uncertainty can be found in figure \[fig:surface\_correction\]. c) Spectrum following this surface event contamination correction. Overimposed is the predicted cosmogenic background contribution, reduced by 10% as in [@Aal11b]. The modest uncertainties associated to this prediction, dominated by present knowledge of L/K shell electron capture ratios, are discussed in [@Aal11b]. d) Irreducible spectrum of bulk events, now devoid of surface and cosmogenic contaminations [@consistency; @gerbier1]. Overimposed is the expected signal from a m$_{\chi} = 8.2$ GeV/c$^{2}$, $\sigma_{SI} = 2.2 \times 10^{-41}$ cm$^{2}$ WIMP, corresponding to the best-fit to a possible nuclear recoil excess in CDMS germanium detector data [@ourcdms]. A bump-like feature around 0.95 keVee is absent in the alternative UW analysis shown in figure \[fig:RisetimeEstimatorCompare2\] and is therefore likely merely a fluctuation.](steps.eps){width="0.35\textheight"}
![\[fig:overlay\] Irreducible spectrum of bulk events (points) showing cumulative uncertainties from the corrective steps discussed in figure \[fig:steps\]. The simulated total background spectrum from Sec.V is shown as a histogram, scaled to the larger exposure in this figure, and corrected for the combined trigger and background cut efficiency.](erroroverlay.eps){width="0.35\textheight"}
![\[fig:roi\] 90% C.L. WIMP limits extracted from the irreducible bulk event spectrum in figure \[fig:overlay\], placed in the context of other low-threshold detectors. A Maxwellian galactic halo is assumed, with local parameters $v_{0}=$220 km/s, $v_{esc}=$550 km/s, $\rho=$0.3 GeV/c$^{2}$cm$^{3}$. A ROI (red solid 90% C.L., red dashed 99% C.L.) can be extracted if a WIMP origin is assigned to the rise in the spectrum. This ROI includes the cumulative uncertainties shown in figure \[fig:overlay\], and allows for a flat background component, independent of energy, in addition to a WIMP signal. The reader is referred to [@kelso] for a discussion on astrophysical uncertainties not included here (see also [@Aal11b; @gerbier1]). This ROI partially overlaps with another one, not shown here for clarity, extracted from a possible excess of low-energy nuclear recoils in CDMS germanium data [@ourcdms]. A best-fit to that possible excess is shown in the bottom panel of figure \[fig:steps\]. Recent low-mass WIMP limits from CDMS-Ge [@prevcdms], EDELWEISS [@edelweiss], TEXONO [@texono], MALBEK [@malbek], and CDMS-Si [@cdmssi] are indicated. A blue asterisk indicates the centroid within a large ROI generated by an excess of three nuclear-recoils in CDMS silicon detector data [@cdmssi].](roi.eps){width="0.35\textheight"}
In the ideal situation where all radiation sources affecting the detector were known in intensity, radioisotope and location, including surface activities, it might be possible to consider a simulation able to predict the exact distribution of pulse rise times as a function of measured energy. This simulation would also require a precise knowledge of the surface layer structure estimated in Sec.III-C (charge collection efficiency and pulse rise time should correlate within the transition region [@sakai]), and modeling of the ensuing processes of charge transport and electronic signal generation. This approach is particularly unrealistic when dealing with few keVee energy depositions. Calibrations using external gamma sources are of value in understanding the structure and effect of the transition layer [@Aal11], but cannot replicate the exact distribution of events in rise time vs. energy during physics runs, which is specific of the particular environmental radiation field affecting a PPC.
An alternative route ensues from a study of simulated preamplifier pulses, as described in figure \[fig:surface\_simulations\]. These provide a qualitative understanding of the blending in rise time of surface and bulk events as energy decreases. It is also observed that all simulated rise time distributions can be described by log-normal probability distributions. A next step is to divide the large (27 month) dataset accumulated up to June 2012 into discrete energy bins for events passing all cuts, but prior to any discrimination based on rise time (figure \[fig:surface\_fits\]). This large exposure allows study of the evolution of these two families of events as a function of energy. Surface and bulk events are observed to form two distinct distributions for energies above a few keVee (top panel in figure \[fig:surface\_fits\]), where the impact of the electronic noise on rise time measurements is minimal (figure \[fig:surface\_simulations\]). A progressive mixing of the two distributions, expected qualitatively from the simulations, is observed to take place at lower energies (figure \[fig:surface\_fits\]). This results in a contamination with unrejected surface (slow) events of the energy spectrum of pulses passing the 90% C.L. fast signal acceptance cut derived from electronic pulser calibrations (figure \[fig:grayscale\]). The magnitude of this contamination (figure \[fig:surface\_correction\]) can be derived from the fits to the rise time distributions shown in figure \[fig:surface\_fits\], and to others like them. The electronic pulser cut (vertical arrows in figure \[fig:surface\_fits\]) correctly approximates the $\sim$90% boundary to the fitted fast pulse distributions (shown in red), confirming that bulk event SA can be correctly estimated using the electronic pulser method.
These fits reveal two significant trends, both visible in figure \[fig:surface\_fits\]: first, the mean of the slow pulse distribution is seen to drift towards slower rise times with decreasing energy, an effect already observed in surface irradiations of PPCs using $^{241}$Am gammas [@Aal11; @ryan]. Second, the standard deviation of the fitted fast pulse distribution (i.e., its broadening towards slower rise times) is noticed to increase with decreasing energy, in good qualitative agreement with the behavior expected from simulated pulses (figure \[fig:surface\_simulations\]).
Figure \[fig:steps\] summarizes the steps necessary in the treatment of CoGeNT low-energy data, leading to an irreducible spectrum of events taking place within the bulk of the crystal, devoid of surface events and cosmogenic backgrounds [@consistency]. As discussed in the following section, the exponential excess observed at low energy is hard to understand based on presently known radioactive backgrounds. Figure \[fig:overlay\] shows the irreducible spectrum of bulk events including the uncertainties discussed in figure \[fig:steps\], overlayed with the total background estimate from Sec.V, pointing at an excess of events above the background estimate. Figure \[fig:roi\] displays WIMP exclusion limits that can be extracted from this irreducible spectrum, compared to those from other low-threshold detectors. The figure includes a region of interest (ROI) generated when assuming a WIMP origin for the low-energy exponential excess.
Best-fit distributions like those in figure \[fig:surface\_fits\] point at the possibility of obtaining $\sim$45% BR of surface events for a 90% SA of bulk events at 0.5 keVee threshold, rapidly rising to $\sim$90% BR at 1.0 keV, for the same 90% SA. A pragmatic approach to improving this event-by-event separation between surface and bulk events, is to tackle the origin of the issue, i.e., to further improve the electronic noise of PPCs. A path towards achieving this within the C-4 experiment is delineated in [@inprep]. In the mean time, the large exposure collected by the PPC at SUL should allow a refined weighted likelihood annual modulation analysis, in which the rise time of individual events provides a probability for their belonging to the surface or bulk categories (figure \[fig:surface\_fits\]). This analysis is in preparation.
Background Studies
==================
The present understanding of backgrounds affecting the CoGeNT detector at SUL is described in this section, including contributions from neutrons, both muon-induced and also for those arising from natural radioactivity in the SUL cavern. Early calculations for these made use of MCNP-Polimi [@polimi] simulations, NJOY-generated germanium cross-section libraries, muon-induced neutron yields and emission spectra exclusively from the (dominant) lead-shielding target as in [@ming; @spectra], and SUL cavern neutron fluxes from [@cdmsn]. These are shown in figure \[fig:mcnp\]. Fair agreement (better than 50% overall) was found between these and subsequent GEANT [@G] simulations, which however include muon-induced neutron production in the full shield assembly and cavern walls, and are able to track the (subdominant) electromagnetic component from muon interactions. The rest of this chapter describes these more comprehensive GEANT simulations.
![\[fig:mcnp\] MCNP-Polimi neutron simulations compared with an early spectrum from the CoGeNT detector at SUL (see text).](mcnp.eps){height="0.24\textheight"}
Neutrons
--------
### Muon-Induced Neutrons
The muon-induced neutron background can be broken up into two components: those produced by muon interactions in the cavern walls, and those generated by interactions in the CoGeNT shielding materials. The energy spectrum of external ($\mu$,n) cavern neutrons was taken from [@Araujo]. Figure \[fig:neutronfraction\] shows the fraction of these neutrons making it through the shielding and depositing energy in the germanium detector, as a function of incident neutron energy. The same figure shows the input neutron energy distribution taken from [@Araujo] in units of neutrons / $\mu$ / MeV. Convolving the two distributions, taking into account the muon flux at SUL, and integrating over all neutron energies gives an upper limit of 1.4 external muon-induced neutrons depositing energy in the 0.5-3.0 keVee window for the entire 442 day CoGeNT dataset in [@Aal11b].
![\[fig:neutronfraction\] Fraction of external ($\mu$,n) cavern neutrons giving rise to energy depositions in the 0.5-3.0 keVee energy window of the CoGeNT detector at SUL, as a function of incident neutron energy, derived from a Monte Carlo simulation (open circles). Also shown, using the right-hand scale, is the emission energy spectrum for these neutrons, taken from [@Araujo] (histogram).](neutronFraction.eps){height="0.23\textheight"}
The largest contribution from neutrons to CoGeNT events arises from spallation neutrons produced by muons traversing the CoGeNT shielding. Their simulation uses as input the energy and angular distribution given by [@ming]. This simulation also keeps track of electrons, positrons, and gammas produced along the muon track through pair production, subsequent positron annihilation, and bremsstrahlung. Figure \[fig:muoninduced\] shows the simulated energy deposition of these muon-induced events (blue band) compared to CoGeNT data. The estimated number of muon-induced events in the 0.5-3.0 keVee region for the 442 day CoGeNT dataset is 339 $\pm$ 68. Only about 8% of these events involve electron or gamma interactions with the detector, the rest being mediated by neutrons.
![\[fig:muoninduced\] Deposited energy spectra from all known backgrounds in the CoGeNT detector at SUL, compared to the 442 d of data in [@Aal11b]. An unidentified low-energy excess and L-shell EC cosmogenic contributions are visible [@Aal11; @Aal11b]. The corrections in figure \[fig:steps\] reduce this excess by $\sim$30% at 0.5 keVee. The blue band represents the sum of muon-induced backgrounds (Sec.V-A1), the green hatched band is a conservative upper limit to the background from cosmogenic $^{3}$H (Sec.V-B), and the red band is from ($\alpha$,n) natural radioactivity in cavern walls (Sec.V-A2). The solid line represents the background distribution from the $^{238}$U and $^{232}$Th chains as well as $^{40}$K contamination in the front-end resistors, estimated in Sec.V-D2. The dashed line is the sum of all background contributions. Contributions from bremsstrahlung from $^{210}$Pb in the inner lead shield (Sec.II-A) and radioactivity from cryostat parts (Sec.V-D1) are found to contribute negligibly. ](bkgs_05_3keV.eps){height="0.27\textheight"}
Both MCNP-Polimi and GEANT simulations point at less than 10% of the irreducible rate at threshold in CoGeNT having an origin in ($\mu$,n) sources, an estimate confirmed by the separate muon-veto considerations discussed in Sec.IV-A. The MINOS experiment at the same location provides an accurate measurement of the magnitude of seasonal fluctuations in underground muon flux, limited to less than $\pm$1.5% [@goodm; @minos]. Any muon-induced modulation is therefore expected to be of a negligible O(0.1)% for the present CoGeNT detector. Muons at SUL exhibit a maximum rate on July 9th [@minos], in tension with the best-fit modulation phase found in [@Aal11b]. The reader is referred to recent studies [@muonstudies] pointing at similar conclusions. Of special relevance is work recently performed by the MINOS collaboration [@minosmod], leading to conclusions similar to those presented here.
### Fission and ($\alpha$,n) neutrons
The flux of ($\alpha$,n) neutrons from radioactivity in the cavern rock is much higher than that of neutrons produced through muon spallation in the rock. Cavern ($\alpha$,n) neutrons were simulated using the energy distribution and flux in [@ming]. The contribution of these cavern ($\alpha$,n) neutrons to the low-energy CoGeNT spectrum is shown in figure \[fig:muoninduced\] (red band).
The high-density polyethylene (HDPE) in the outer layer of the CoGeNT shielding is known to have relatively high levels of $^{238}$U and $^{232}$Th contamination. These $^{238}$U and $^{232}$Th concentrations were measured for HDPE samples at SNOLAB, finding 115$\pm$5 mBq/kg and 80$\pm$4 mBq/kg, respectively. $^{238}$U has a small spontaneous fission (SF) branching ratio with an average multiplicity per fission of 2.07 [@Axton]. Neutrons from this source depositing energy in the 0.5-3.0 keVee region of the spectrum are estimated to be just 17.7$\pm$7.2 for the entire 442 day data set. An isotope of carbon, $^{13}$C, has a 1.07% natural abundance and a non-negligible cross-section for the ($\alpha$,n) reaction at $\alpha$ energies emitted by the U and Th decay chains. The HDPE is therefore a weak source of ($\alpha$,n) neutrons. The neutron production from ($\alpha$,n) in HDPE was scaled from a SOURCES [@sources] calculation for plastic material [@Perry]. The number of ($\alpha$,n) neutron-induced events in the CoGeNT data set from $^{238}$U and $^{232}$Th in HDPE was determined to be a negligible $<$ 0.02 and $<$ 0.01, respectively. Table \[tab:neutronsources\] summarizes the contributions from the various sources of neutrons in the 442 day CoGeNT data set. The lead surrounding the detector is also a weak source of fission neutrons. The $^{238}$U concentration in lead has been measured at SNOLAB to be 0.41$\pm$0.17 mBq/kg. This results in $<$ 0.5 events from $^{238}$U fission in lead for the entire CoGeNT data set.
------------------------------------------------- --------------
Cavern muon-induced neutrons $<$1.4
Cavern ($\alpha$,n) neutrons $<$54
Muon-induced events in shielding 339$\pm$68
$^{238}$U fission in HDPE 17.7$\pm$7.2
($\alpha$,n) from $^{238}$U in HDPE $<$0.02
($\alpha$,n) from $^{232}$Th in HDPE $<$0.01
$^{3}$H in the Ge detector $<$150
$^{238}$U and $^{232}$Th in Cu shield $\sim$9
$^{238}$U,$^{232}$Th, and $^{40}$K in resistors $\sim$324
------------------------------------------------- --------------
: \[tab:neutronsources\] Summary of backgrounds in a 442 day CoGeNT data set, from various sources investigated.
Cosmogenic Backgrounds in Germanium
-----------------------------------
Tritium can be produced via neutron spallation of the various natural germanium isotopes. Most of the $^{3}$H production occurs at the surface of the Earth where the fast neutron flux is much higher than underground. Tritium has a half-life of 12.3 years, which means its reduction over the lifetime of the experiment is small. Its beta decay is a potential background for CoGeNT, given its modest end-point energy of 18.6 keV. Using the $^{3}$H production rate in [@Elliott] and [@Morales] and assuming an overly conservative two years of sea-level exposure for the crystal, an upper limit of $<$150 $^{3}$H decay events was extracted for the CoGeNT data set. While this number would present a significant background, the energy spectrum of the $^{3}$H events is relatively flat over the 0.5-3.0 keVee analysis region and does not provide for the excess observed at low energies. Figure \[fig:muoninduced\] shows the upper limit to the contribution from $^{3}$H decays (shaded green) in the analysis region, compared to the data.
All other sufficiently long-lived cosmogenic radioisotopes of germanium produce monochromatic energy depositions at low energy [@Aal11; @Aal11b; @collarthesis], or have endpoints large enough not to be able to contribute significantly in the few keVee region. The fraction of these taking place in the transition surface layer might however lead to an accumulation of partial charge depositions at energies below the cosmogenic peaks, even if most of these events should in principle be rejected by the rise time cut. That this accumulation is indeed negligible can be ascertained by the lack of correlation between the relatively constant rates shown in figure \[fig:modulation\] and the much larger change under the dominant 10.3 keV cosmogenic peak, which reduced its activity from $\sim$500 counts/month to $\sim$150 counts/month over the same period of time.
An episode of intense thermal neutron activation of $^{71}$Ge in a PPC with identical characteristics to that operating at SUL, related in [@Aal11], provides additional confirmation that this possible source of background is small. Figure \[fig:ge71\] shows the spectrum acquired during the first few days following this thermal neutron activation. The data were taken at the San Onofre nuclear plant at a depth of 30 m.w.e., inside a large passive shield and triple active veto. The initial $^{71}$Ge decay rate under the 10.3 keV peak was very high, at $\sim$0.3 Bq. The low-energy $^{71}$Ge spectral template shown in the figure was therefore entirely dominated by the response to this activation, with the counting rate below 10 keVee dropping by several orders of magnitude over the ensuing weeks, to stabilize at a factor of just a few above the rate observed at SUL. Once the $^{71}$Ge activation template is normalized to the same rate under the 10.3 keV peak as that observed at SUL, as is done in figure \[fig:ge71\], less than 10% of the low-energy spectral excess at SUL can be assigned to partial energy depositions from $^{68}$Ge activation (both radioisotopes undergo the same decay). This $<$10% is a conservative upper limit, given that the DAQ used in San Onofre did not feature the digitization of preamplifier traces necessary for rise time cuts (i.e., the low energy component of the $^{71}$Ge template in figure \[fig:ge71\] would be further reduced by those).
![\[fig:ge71\] Negligible upper-limit to the contribution from cosmogenic activity in the near-threshold energy region of the CoGeNT detector at SUL (see text).](ge71.eps){height="0.26\textheight"}
Environmental radon and radon daughter deposition on detector surfaces
----------------------------------------------------------------------
Sec.II-A describes active measures against penetration of radon into the detector’s inner shielding cavity. External gamma activity from this source is efficiently blocked by the minimum of 25 cm of lead shielding around the detector (the attenuation length in lead for the highest-energy radon associated gamma emission is $\sim\!2$ cm). These measures include precautions such as automatic valving off of the evaporated nitrogen purge gas lines during replacement of the dedicated Dewar. A time analysis of the low-energy counting rate looking for signatures of radon injection (a surge followed by a decay with t$_{1/2}$=3.8 d) revealed no such instances. Radon levels at SUL are continuously measured by the MINOS experiment, showing a large seasonal variation (a factor of $\sim\!\pm$2) [@goodm; @minos]. Figure \[fig:radon\] displays a comparison between these measurements and the germanium counting rate, showing an evident lack of correlation (see also [@minosmod]). While we have not requested access to information regarding diurnal changes in radon level at SUL, these are commonly observed in underground sites, and seemingly absent from CoGeNT data (figure \[fig:diurnal\]). A modulated radon signature would appear at all energies in CoGeNT spectra, an effect not observed, due to partial energy deposition from Compton scattering of gamma rays emitted by this radioactive gas and its progeny [@radon].
![\[fig:radon\] Counts per 30 day bins from the 0.5-3.0 keVee CoGeNT energy window (black dots) compared to the MINOS radon data at SUL (dashed), averaged over the period 2007-2011, exhibiting a peak on August 28th [@goodm; @minos]. The solid curve represents a sinusoidal fit to CoGeNT data. An analysis by the MINOS collaboration finds a three-sigma inconsistency between the phase of their measured seasonal modulation in radon concentration at SUL and CoGeNT data [@minosmod].](modradon.eps){height="0.22\textheight"}
Additional sources of radon-related backgrounds are the delayed emissions from $^{222}$Rn daughters deposited on detector surfaces during their fabrication. The dominant low-energy radiations of concern are a beta decay with 17 keV endpoint from $^{210}$Pb, and 102 keVr lead alpha-recoils from the decay of $^{210}$Po. These radiations are known to produce a low-energy spectral rise in germanium detectors lacking sufficiently-thick protective inert surface layers [@edelweissthesis]. The PPC detector considered here is insulated against these over most of its surface by the $\sim$1 mm dead layer discussed in Sec.III-C. Only its intra-contact surface (3.8 cm$^{2}$) is partially sensitive to these. An inert 150 nm thick SiO$_{x}$ layer is deposited there during manufacture in order to passivate this surface, reducing leakage current across the contacts. Its thickness is almost four times the projected range of $^{210}$Po alpha recoils, effectively blocking their possible contribution. We calculate the contribution from $^{210}$Pb betas via MCNP simulation, taking as input the 90 % C.L. upper limit to the activity of their accompanying 46.5 keV gamma emission from the spectrum in figure \[fig:data300gammas\]. This upper limit translates into $<$2.8 $^{210}$Pb decays per day, conservatively assumed to take place in their entirety on the intra-contact surface. The resulting degraded beta energies reaching the surface of the active germanium are spectrally very different from the residual background observed, not exhibiting an abrupt rise near threshold, and contribute only a maximum of 5% to the rate in the 0.5-1.5 keVee region of the irreducible spectrum in figure \[fig:overlay\]. We consider this upper limit to be overly conservative.
Backgrounds from radioactivity in cryostat materials
----------------------------------------------------
Materials surrounding the CoGeNT detector are selected for their low radioactivity (Sec.II-A). However, due to the proximity of these materials to the detector, even small activities could potentially be a background to a possible dark matter signal. We have therefore performed simulations of these backgrounds to determine their contribution to the low-energy spectrum.
### Backgrounds from OFHC Copper and PTFE
The CoGeNT detector is contained within OFHC copper parts, etched to reduce surface contaminations (Sec.II-A). Gamma counting of large samples of OFHC copper at Gran Sasso yield $^{238}$U and $^{232}$Th concentrations of 18 $\mu$Bq/kg and 28 $\mu$Bq/kg, respectively [@EHoppe]. We have simulated the $^{238}$U and $^{232}$Th decay chains in the copper shield, including gamma emission, betas and their associated bremsstrahlung. The simulation also includes the alpha-decays in both chains, since alpha-induced X-ray emission is potentially a background. The number of events within the 0.5-3.0 keVee region is estimated as a negligible $\sim$9 events for the entire 442 day data set in [@Aal11b]. A similar calculation for the 0.5 mm PTFE liner surrounding the crystal, also chemically etched, yields only 1.5 events for the same energy region and time period, using a conservative activity of 15 mBq/kg ($^{238}$U) and 7 mBq/kg ($^{232}$Th) [@SNOLAB]. In addition to this, we calculate an absence of measurable contribution from standard concentrations of $^{40}$K and $^{14}$C in the PTFE crystal liner ($<$85 mBq/kg and $\sim$60 Bq/kg, respectively).
### Backgrounds from resistors in front-end electronics
The front-end FET capsule, fabricated in PTFE, contains two small resistors in close proximity (within $\sim\!2$ cm) to the germanium crystal. Resistors are known to have relatively high levels of radioactive contaminants, and their location make them a primary candidate for the source of a large fraction of events. Table \[tab:resbkgs\] summarizes measured levels of $^{238}$U, $^{232}$Th, and $^{40}$K concentrations in various resistors from the ILIAS database [@ilias]. The ceramic in most resistors is the largest contributor to the radioactivity. The type of resistors used in CoGeNT are metal film on ceramic, with an approximate mass of 50 mg each. Table \[tab:resbkgs\] also summarizes the number of background events in the 0.5-3.0 keVee region of the 442 day data set, determined from a simulation scaled to the various activity measurements. These range from 324$\pm$165 to 4509$\pm$352, the dominant contributions being gammas in the $^{238}$U and $^{232}$Th chains. The spectrum of energy deposition is shown in Figs. \[fig:muoninduced\] and \[fig:resbkgs\]. These figures specifically show results for a metal film resistor, the same type of resistor in CoGeNT, without any scaling. Since we have not assayed the specific resistors used in CoGeNT, we cannot be certain that most of the flat background component observed in CoGeNT data is due to this source, but the agreement with this flat component of the spectrum is suggestive. A scheme to eliminate these resistors in the C-4 design [@inprep] has been developed.
--------------------------- ------------- ---------------- ------------- ---------------- ------------- ----------------
Description Rate(Bq/kg) Events in data Rate(Bq/kg) Events in data Rate(Bq/kg) Events in data
carbon film resistor 4.3 269$\pm$74 12.7 687$\pm$95 21.9 16.5$\pm$4.3
metal film resistor 1 4.3 269$\pm$126 0.5 27$\pm$104 37.5 28.2$\pm$7.5
metal film resistor 2 5.1 319$\pm$99 16.1 870$\pm$125 24.7 18.6$\pm$5.7
ceramic core resistor 5.9 369$\pm$99 4.6 249$\pm$85 34.3 25.8$\pm$6.0
metal on ceramic resistor 28 1750$\pm$193 40.7 2740$\pm$294 25.7 19.4$\pm$4.7
--------------------------- ------------- ---------------- ------------- ---------------- ------------- ----------------
![\[fig:resbkgs\] Similar to figure \[fig:muoninduced\], with expanded ranges: energy spectrum of the simulated $^{238}$U, $^{232}$Th, and $^{40}$K resistor background (dotted line) compared to CoGeNT data (solid). In the energy range displayed the estimated resistor backgrounds are by far dominant. The resistor background spectrum is for metal film resistors, the same used in the CoGeNT front end. Also shown are other background contributions and their sum. Contributions from $^{210}$Pb bremsstrahlung and radioactivity in PTFE and OFHC cryostat parts are comparatively negligible.](bkgs_05_12keV.eps){height="0.27\textheight"}
As a further consistency check we examined the existing CoGeNT data out to an energy of 300 keVee. The statistics in this range are limited (5 days of dedicated exposure, see Sec.III-A). Figure \[fig:data300gammas\] shows possible 238 keV $^{212}$Pb ($^{232}$Th chain) and 295 keV ($^{238}$U chain) gamma lines. Due to their relatively-low energy, their source would be near the crystal, within the inner lead cavity. If they are considered as a measure of the $^{238}$U and $^{232}$Th chain contamination in front-end resistors, a 14$\pm$7 Bq/kg for $^{238}$U contamination and 1.6$\pm$0.7 Bq/kg for $^{232}$Th contamination is obtained for the resistors. This activity would provide $\sim$937 events in the 0.5-3.0 keVee region, in good agreement with the measured flat component of the spectrum. The statistical evidence for these lines is however slim, and their presence is seen to be mutually exclusive when examining the uncertainties associated to the energy scale extrapolation used for this short run.
![\[fig:data300gammas\] Existing CoGeNT data in the range up to 300 keV, with possible weak $^{212}$Pb (238 keV) and $^{214}$Pb (295 keV) gamma lines indicated by arrows. The extrapolated energy scale can only be considered approximate. The energy binning corresponds to the approximate FWHM resolution for these two lines. See text for a discussion on a possible origin for these putative lines in the front-end resistors. Notoriously absent are a $^{210}$Pb peak at 46.5 keV and excess lead x-rays, a result of the radiopurity of the inner lead layers in the shield (Sec.II-A) and detector surfaces (Sec.V-C).](gammalines300kev.eps){height="0.17\textheight"}
Backgrounds from neutrino scattering
------------------------------------
While the smallness of neutrino cross-sections indicate that their contribution to the CoGeNT spectrum should be negligible, the signal from coherent neutrino-nucleus scattering [@freedman] from several sources (e.g. solar, atmospheric, diffuse supernova, and geo-neutrinos) would be highly concentrated at low energies. We engage here in the exercise of providing a few estimated upper limits for these contributions. Inferring from a recent analysis on solar and atmospheric neutrinos [@Gutlein], a germanium detector with 0.33 kg active mass and a $\sim$2 keV nuclear recoil threshold (as in the present CoGeNT detector) would observe a rate of just $\sim$0.012 counts / year from coherent neutrino-nucleus scattering from $^{8}$B and $^{3}$He-proton fusion (HEP) solar neutrinos, the only solar sources able to produce a signal above threshold. Diffuse supernova background neutrinos and atmospheric neutrinos might also contribute, however their rate is reduced by factors of $>$ 10$^{4}$ [@Strigari] and $>$ 10$^{5}$ [@Gutlein], respectively. Geoneutrinos, having energies less than 4.5 MeV [@Monroe], cannot produce nuclear recoil energies above the CoGeNT threshold. Each of these sources may also induce direct electron scattering. However, the neutrino-electron scattering rate is suppressed by $\sim$ 10$^{5}$ relative to the neutrino-nucleus coherent scattering rate [@Cabrera]. Therefore this other channel cannot significantly contribute even taking into account the factor of 32 increase in scattering targets, the absence of a quenching factor, and the higher electron recoil energies. We notice however that interaction rates large enough to be of interest can be generated by solar neutrinos with enhanced baryonic currents [@maxim]. Additional mechanisms [@joachim] are able to generate a phenomenology involving diurnal and yearly modulations in rates.
Conclusions
===========
CoGeNT is the first detector technology specifically designed to look for WIMP candidates in the low mass range around 10 GeV/c$^{2}$, an area of particular interest in view of existing anomalies in other dark matter experiments, recent phenomenological work in particle physics, and possible signals using indirect detection methods [@indirect]. However, investigation of the largely unexplored $\sim$few keV recoil energy range brings along new challenges in the understanding of low-energy backgrounds. The experience accumulated during the ongoing CoGeNT data-taking at SUL demonstrates that PPC detectors have excellent properties of long-term stability, simplicity of design, and ease of operation. This makes them highly suitable in searches for the annual modulation signature expected from dark matter particles forming a galactic halo.
Besides their excellent energy resolution, low energy threshold and ability to reject surface backgrounds, PPCs compare well to other solid-state detectors under several criteria: a) the relative simplicity of CoGeNT’s data analysis results in comparable irreducible spectra regardless of analysis pipeline, b) the response to nuclear recoils is satisfactorily understood, resulting in a reliable nuclear recoil energy scale, c) uninterrupted stable operation of PPC detectors can be expected over very long (several year) timescales. We plan to continue improving this technology and our understanding of low-energy backgrounds within the framework of a CoGeNT expansion, the C-4 experiment [@inprep].
We are indebted to Jeffrey de Jong and Alec Habig (MINOS collaboration) for sharing with us information on radon and muon rates at SUL, and to all SUL personnel for their constant support in operating the CoGeNT detector. Work sponsored by NSF grants PHY-0653605 and PHY-1003940, The Kavli Foundation, and the PNNL Ultra-Sensitive Nuclear Measurement Initiative LDRD program (Information Release number PNNL-SA-90298). N.E.F. and T.W.H. are supported by the DOE/NNSA Stewardship Science Graduate Fellowship program (grant number DE-FC52-08NA28752) and the Intelligence Community (IC) Postdoctoral Research Fellowship Program, respectively.
[9]{}
P.S. Barbeau, J.I. Collar and O. Tench, JCAP [**09**]{} (2007) 009.
C.E. Aalseth [*et al.*]{}, Phys. Rev. Lett. [**101**]{} (2008) 251301; Erratum [*ibid*]{} [**102**]{} (2009) 109903.
R. Bernabei [*et al.*]{}, Eur. Phys. J. [**C67**]{} (2010) 39.
C.E. Aalseth [*et al.*]{}, Phys. Rev. Lett. [**106**]{} (2011) 131301.
C.E. Aalseth [*et al.*]{}, Phys. Rev. Lett. [**107**]{} (2011) 141301.
A.K. Drukier, K. Freese and D.N. Spergel, Phys. Rev. [**D33**]{} (1986) 3495.
R. Bonicalzi [*et al.*]{}, Nucl. Instrum. Meth. [**A712**]{} (2013) 27.
W.W.M. Allison [*et al.*]{}, Nucl. Instrum. Meth. [**A376**]{} (1996) 36.
E.W. Hoppe [*et al.*]{}, Nucl. Instrum. Meth. [**A579**]{} (2007) 486.
S.M. Miley [*et al.*]{}, [J Radioanal Nucl Chem]{} [**282**]{} (2009) 869.
P. Vojtyla, Nucl. Instrum. Meth. [**B117**]{} (1996) 189.
P.S. Barbeau, Ph.D. Diss., University of Chicago (2009).
J. Anderson [*et al.*]{}, IEEE TNS [**56**]{} (2009) 258.
M.G. Marino, PhD Diss., Univ. of Washington (2010).
G. Bertuccio, A. Pullia and G. De Geronimo, Nucl. Instrum. Meth. [**A380**]{} (1996) 301.
P.S. Barbeau, J.I. Collar and P.M. Whaley, Nucl. Instrum. Meth. [**A574**]{} (2007) 385.
E. Sakai, IEEE TNS [**18**]{} (1971) 208 and refs. therein.
E. Aguayo [*et al.*]{}, [Nucl. Instr. Meth.]{} [**A701**]{} (2013) 176.
S.A. Pozzi [*et al.*]{}, [Nucl. Instr. Meth.]{} [**A513**]{} (2003) 550.
C.E. Aalseth [*et al.*]{}, J. Phys. Conf. Ser. [**203**]{} (2010) 012057.
M. Agostini [*et al.*]{}, JINST [**6**]{} (2011) P03005.
P.J. Fox [*et al.*]{}, [Phys. Rev.]{} [**D85**]{} (2012) 036008.
I. Abt [*et al.*]{}, Eur. Phys. J. Appl. Phys. [**56**]{} (2011) 10104.
J. Morales [*et al.*]{}, [Nucl. Instr. Meth.]{} [**A321**]{} (1992) 410; I. Stekl [*et al.*]{} [Czech. J. Phys.]{} [**52**]{} (2002) 541.
I.G. Irastorza [*et al.*]{}, [Astropart. Phys.]{} [**20**]{} (2003) 247.
P. Adamson [*et al.*]{}, Phys. Rev. [**D87**]{} (2013) 032005.
P.F. Smith [*et al.*]{}, [Phys. Lett.]{} [**B379**]{} (1996) 299.
As a consistency test, the same corrected spectrum at the bottom panel of figure \[fig:steps\] can be recovered by ignoring the 90% C.L. fast pulser cut, and applying in lieu of it the overall fraction of fast events derived from fits like those in figure \[fig:surface\_fits\]. Following a sufficiently long exposure, the distribution in rise time of surface and bulk events becomes self-evident, making possible to calculate bulk SA and surface BR for any choice of rise time cuts. This supersedes the electronic pulser method, which is limited by only providing a bulk SA, but no surface BR information.
C. Kelso, D. Hooper, M.R. Buckley, Phys. Rev. [**D85**]{} (2012) 043515.
J.I. Collar and N. Fields, [arXiv:1204.3559]{}, submitted to Phys. Rev. Lett.
The reduction in best-fit WIMP signal rate following the surface event correction discussed in Sec.IV-B is a factor $\sim$2 [@ucla; @kelso]. An order of magnitude reduction is incorrectly claimed in [@gerbier2].
J.I. Collar, UCLA Dark Matter 2012, presentation at\
[https://hepconf.physics.ucla.edu/dm12/]{}
Z. Ahmed [*et al.*]{}, Phys. Rev. Lett. [**106**]{} (2011) 131302.
G. Armengaud [*et al.*]{}, Phys. Rev. [**D86**]{} (2012) 051701(R).
H.B. Li [*et al.*]{}, [arXiv:1303.0925]{}.
P.D. Finnerty, Ph.D. dissertation, U. of North Carolina, 2013.
R. Agnese [*et al.*]{}, [arXiv:1304.4279]{}.
M. Dress and G. Gerbier, PDG 2012 dark matter review, [arXiv:1204.2373]{}.
D.-M. Mei and A. Hime, [Phys. Rev.]{} [**D73**]{} (2006) 053004.
A. Da Silva [*et al.*]{}, [Nucl. Instr. Meth.]{} [**A354**]{} (1995) 553.
S. Eichblatt, CDMS internal note 97-01-25.
H.M. Araujo [*et al.*]{}, [Nucl. Instr. Meth.]{} [**A545**]{} (2005) 398.
M. Goodman, Procs. of the 26th Intl. Cosmic Ray Conf., Aug. 17-25, Salt Lake City, Utah.
Jeffrey de Jong and Alec Habig (MINOS collaboration), private communication.
E. Fernandez-Martinez and R. Mahbubani, [JCAP]{} [**1207**]{} (2012) 029; R. Bernabei [*et al.*]{} Eur. Phys. J. [**C72**]{} (2012) 2064; S. Chang, J. Pradler and I. Yavin, [Phys. Rev.]{} [**D85**]{} (2012) 063505; J. Pradler, [arXiv:1205.3675]{}.
E.J. Axton, [Nucl. Stand. Ref. Data]{} (1985) 214.
W.B. Wilson [*et al.*]{}, [Rad. Prot. Dosim.]{} [**115**]{} (2005) 117.
R.T. Perry, ANS Radiation Protection and Shielding Topical Meeting, 2002.
D.M. Mei, Z.B. Yin and S.R. Elliott, [Astropart. Phys.]{} [**31**]{} (2009) 417.
A. Morales [*et al.*]{}, [Phys. Lett.]{} [**B532**]{} (2002) 8.
J.I. Collar, Ph.D. Diss., Univ. of South Carolina (1992); F.T. Avignone [*et al.*]{}, [Nucl. Phys. B (Proc. Suppl.)]{} [**28A**]{} (1992) 280; S. Cebrian [*et al.*]{}, [Astropart. Phys.]{} [**33**]{} (2010) 316.
G. Heusser [*et al.*]{}, Appl. Radiat. Isot. [**43**]{} (1992) 9.
J. Domange, PhD dissertation, Université de Paris Sud XI, 2011.
Eric Hoppe (PNNL), private communication 2012.
SNOLAB Gamma Assay 2006 [ www.snolab.ca/users/services/gamma-assay/HPGe\_samples\_pub.html]( www.snolab.ca/users/services/gamma-assay/HPGe_samples_pub.html)
D.Z. Freedman, [Phys. Rev.]{} [**D9**]{} (1974) 1389.
A. Gutlein [*et al.*]{}, [Astropart. Phys.]{} [**34**]{} (2010) 90.
L.E. Strigari, [New J. of Phys.]{} [**11**]{} (2009) 105011.
J. Monroe and P. Fisher, [Phys. Rev.]{} [**D76**]{} (2007) 033007.
B. Cabrera [*et al.*]{}, [Phys. Rev. Lett.]{} [**55**]{} (1985) 25.
M. Pospelov and J. Pradler, [Phys. Rev.]{} [**D85**]{} (2012) 113016.
R. Harnik, J. Kopp and P.A.N. Machado, JCAP [**07**]{} (2012) 026.
reviewed in D. Hooper, [arXiv:1201.1303]{}.
|
---
abstract: 'We show that a minimal local $B-L$ symmetry extension of the standard model can provide a unified description of both neutrino mass and dark matter. In our model, $B-L$ breaking is responsible for neutrino masses via the seesaw mechanism, whereas the real part of the $B-L$ breaking Higgs field (called $\sigma$ here) plays the role of a freeze-in dark matter candidate for a wide parameter range. Since the $\sigma$-particle is unstable, for it to qualify as dark matter, its lifetime must be longer than $10^{25}$ seconds implying that the $B-L$ gauge coupling must be very small. This in turn implies that the dark matter relic density must arise from the freeze-in mechanism. The dark matter lifetime bound combined with dark matter relic density gives a lower bound on the $B-L$ gauge boson mass in terms of the dark matter mass. We point out parameter domains where the dark matter mass can be both in the keV to MeV range as well as in the PeV range. We discuss ways to test some parameter ranges of this scenario in collider experiments. Finally, we show that if instead of $B-L$, we consider the extra $U(1)$ generator to be $-4I_{3R}+3(B-L)$, the basic phenomenology remains unaltered and for certain gauge coupling ranges, the model can be embedded into a five dimensional $SO(10)$ grand unified theory.'
author:
- '**Rabindra N. Mohapatra$^a$**'
- Nobuchika Okada$^b$
title: 'Freeze-in Dark Matter from a Minimal B-L Model and Possible Grand Unification'
---
1. Introduction
===============
If small neutrino masses arise via the seesaw mechanism [@seesaw1; @seesaw2; @seesaw3; @seesaw4; @seesaw5], the addition of a local $B-L$ symmetry [@marshak1; @marshak2] to the standard model (SM) provides a minimal scenario for beyond the standard model (BSM) physics to achieve this goal. There are two possible classes of $B-L$ models: one where the $B-L$ generator contributes to the electric charge [@marshak1; @marshak2; @davidson] and another where it does not [@BL1; @BL2; @BL3]. In the first case, the $B-L$ gauge coupling $g_{BL}$ has a lower limit whereas in the second case it does not and therefore can be arbitrarily small. There are constraints on the allowed ranges of $g_{BL}$ from different observations [@heeck; @bauer] in the second case depending on whether there is or is not a dark matter particle in the theory.
In Refs. [@nobu; @heeba], it was shown that if we added a $B-L$ charge carrying vector-like fermion to the minimal $B-L$ model and want it to play the role of dark matter, new constraints emerge. In this note, we discuss an alternative possibility with the following new results. First is that the minimal version of the $B-L$ model itself, without any extra particles, can provide a dark matter (DM) candidate. The DM turns out to be the real part (denoted here as $\sigma$) of the complex $B-L=2$ Higgs field, that breaks $B-L$ and gives mass to the right handed neutrinos in the seesaw formula. Even though this particle is not stable, there are certain allowed parameter ranges of the model, where its lifetime can be so long that it can play the role of a decaying dark matter. We isolate this parameter range and show that in this case, the freeze-in mechanism [@hall] can generate its relic density. We find this possibility to be interesting since it unifies both neutrino masses and dark matter in a single minimal framework. We show how a portion of the parameter range of the model suggested by the dark matter possibility, can be probed by the recently approved FASER experiment at the LHC [@faser] and other Lifetime Frontier experiments.
We then show that if we replace the $B-L$ symmetry by $\tilde{I}\equiv -4I_{3R}+3(B-L)$ (where $I_{3R}$ is the right handed weak isospin), the dark matter phenomenology remains largely unchanged and the model can be embedded into the $SO(10)$ grand unified theory in five space-time dimensions. Such a symmetry breaking of $SO(10)$ to $SU(5)\times U(1)_{\tilde{I}}$ has already been shown to arise from a symmetry breaking by a particular alignment for a vacuum expectation value (VEV) of a [**45**]{}-dimensional Higgs field [@diluzio].
This paper is organized as follows: in Sec. 2 after briefly introducing the model, we discuss the lifetime of the $\sigma$ dark matter and its implications. In Sec. 3, we discuss the small gauge coupling $g_{BL}$ range where the dark matter lifetime is long enough for it to play the role of dark matter. In Sec. 4, we show how freeze in mechanism determines the relic density of dark matter and its implications for the allowed parameter range of the model. We also discuss how to test this model at the FASER and other Lifetime Frontier experiments. In Sec. 5, we show that this model can also accommodate a PeV dark matter. In Sec. 6, we discuss the $SO(10)$ embedding of the closely allied model and in Sec. 7, we conclude with some comments and other implications of the model.
2. Brief overview of the model
==============================
Our model is based on the $U(1)_{B-L}$ extension of the SM with gauge quantum numbers under $U(1)_{B-L}$ determined by the baryon or lepton number of the particles. The gauge group of the model is $SU(3)_c \times SU(2)_L\times U(1)_Y\times U(1)_{B-L}$, where $Y$ is the SM hypercharge. We need three right handed neutrinos (RHNs) with $B-L=-1$ to cancel the $B-L$ anomaly. The RHNs being SM singlets do not contribute to SM anomalies. The electric charge formula in this case is same as in the SM i.e. $Q=I_{3L}+\frac{Y}{2}$.
We break $B-L$ symmetry by giving a VEV to a $B-L=2$ SM neutral complex Higgs field $\Delta$ i.e. $\langle \Delta\rangle =v_{BL}/\sqrt{2}$. This gives Majorana masses to the right handed neutrinos ($N$) via the coupling $f N N \Delta$. The real part of $\Delta$ (denoted by $\sigma$) is a physical field. Our goal in this paper is to show that $\sigma$ has the right properties to play the role of a dark matter of the universe. There are three challenges to achieving this goal:
\(i) The $\sigma$ field has couplings to the RHNs which in turn couples to SM particles providing a way for $\sigma$ to decay. Also, the $\sigma$ field has couplings to two $B-L$ gauge bosons ($Z_{BL}$) which in turn couple to SM fields providing another channel for $\sigma$ to decay. In the next section, we show that there are parameter regions of the model where these decay modes give a long enough lifetimes for $\sigma$, so that it can be a viable unstable dark matter in the universe.
\(ii) The second challenge is that for $\sigma$ to be a sole dark matter, it must account for the total observed relic density of the universe $\Omega_{DM}h^2\simeq 0.12$ [@Planck2018]. We show in Sec. 4 that in the same parameter range, that gives rise to the long lifetime of $\sigma$, can also explain the observed relic density of dark matter via the freeze-in mechanism.
\(iii) The $\sigma$ field could mix with the standard model Higgs field $h$ via the potential term $\lambda^\prime H^\dagger H \Delta^\dagger \Delta$ after symmetry breaking. However, it turns out that if we set $\lambda^\prime =0$ at the tree level, it can be induced at the one-loop level by fermion contributions and at the two-loop level from the top loop as shown in Ref. [@BL3]. These induced couplings can be so small that they still lead to very long lifetimes for $\sigma$ in the parameter range of interest to us.
3. Dark matter lifetime
========================
As noted earlier in Sec. 2, the $\sigma$ field has couplings which could make it unstable and thereby disqualify it from being a dark matter. However, we will show that there is a viable parameter range of the model where this decay lifetime is longer than $10^{25}$ sec. [@farinaldo] so that it can be a dark matter candidate. We discuss these two modes now:
\(i) Decay mode [**$\sigma \to NN \to \ell f\bar{f}\ell f\bar{f}$**]{}: the decay width for this process is estimated as $$\begin{aligned}
\Gamma_{NN}\simeq \frac{(f \, h_\nu^2 \, h^2_{SM})^2}{(4 \pi)^8} \frac{m^{13}_\sigma}{M^4_N \, m_h^8},\end{aligned}$$ where $h_\nu$ is a neutrino Dirac Yukawa coupling, $h_{SM}$ is a Yukawa coupling of an SM fermion $f$, and $m_h=125$ GeV is the SM Higgs boson mass. For a GeV mass $\sigma$ and TeV mass RHN, the lifetime of $\sigma $ turns out to be $\tau_\sigma[{\rm sec}] \sim 10^{37}/(f^2 \, h_{SM}^4)$, which is quite consistent with the requirement for it to be a dark matter. Here, we have used the seesaw formula $h_\nu^2 v_{EW}^2/M_N \simeq m_\nu$ with $v_{EW}=246$ GeV and a typical neutrino mass scale $m_\nu \simeq 0.1$ eV. (ii) Decay mode [**$\sigma\to Z_{BL}Z_{BL}\to f\bar{f}f\bar{f}$**]{}: the decay width for this process is $$\begin{aligned}
\Gamma_{Z_{BL}Z_{BL}}\simeq \frac{(2 g_{BL})^4 \, v_{BL}^2 \, g_{BL}^4 \, m_{\sigma}^7}{(4 \pi)^5 M^8_{Z_{BL}}}
=\frac{g_{BL}^6}{ 256 \pi^5} \frac{m_\sigma^7}{M^6_{Z_{BL}}}. \end{aligned}$$ This mode is sensitive to the values of $g_{BL}$ as well as $M_{Z_{BL}}$. The estimate of $\tau_\sigma$ due to this decay mode is given by $$\begin{aligned}
\tau_\sigma\simeq 5.2 \times 10^{-20} \left(\frac{1}{g_{BL}}\right)^6\left(\frac{{\rm 1 \, GeV}}{m_\sigma}\right)^7 \left(\frac{M_{Z_{BL}}}{{\rm 1\, GeV}}\right)^6~~{\rm sec.}
\label{LF}\end{aligned}$$ Imposing $\tau_\sigma > 10^{25}$ sec., this puts an upper bound on the $g_{BL}$ as a function of $M_{Z_{BL}}$ and $m_\sigma$: $$\begin{aligned}
\label{lifetime}
g_{BL} \leq 4.2 \times 10^{-8} \left(\frac{M_{Z_{BL}}}{{\rm 1 \, GeV}}\right) \left(\frac{{\rm 1 \,GeV}}{m_\sigma}\right)^{7/6}
\label{LF2}\end{aligned}$$ We find that the allowed regions where the $\sigma$ field can be a dark matter correspond to a very small $g_{BL}$ coupling. For instance, for $m_\sigma \sim 1$ GeV and $M_{Z_{BL}} \sim 1$ TeV, we find that $g_{BL} \lesssim 4 \times 10^{-5}$.
\(iii) We now comment on the $\sigma$-Higgs mixing effect on the DM lifetime. To keep the lifetime above limit $\tau_\sigma > 10^{25}$ sec., we set the tree-level $H$-$\Delta$ coupling in the Higgs potential to zero so that $\sigma$ and the SM Higgs field $h$ do not mix at the tree level. This will, for example be true if the model becomes supersymmetric at a high scale. The $\sigma$-Higgs mixing in this case is loop induced as shown in Ref. [@BL3] and for the parameter range of interest to us, can be small enough to satisfy the DM lifetime constraint as we show below.
For the case when $m_\sigma \leq m_h$, the dominant contribution to the loop induced mixing comes from a RHN fermion box diagram and the mixing angle can be estimated to be $\theta\sim \frac{f^2 h^2_\nu}{16\pi^2}\frac{v_{EW} v_{BL}}{m^2_h}
\sim \frac{1}{16\pi^2} \frac{m_\nu M_N^3}{v_{EW} m_h^2} \frac{2 g_{BL}}{M_{Z_{BL}}} $. Through this mixing, the DM particle can decay to a pair of SM fermions with a partial decay width of $\Gamma_{\sigma \to f \bar{f}} \sim \frac{\theta^2}{4 \pi} \left(\frac{m_f}{v_{EW}} \right)^2 m_\sigma$. The lifetime constraint then translates to a limit on $g_{BL}$ as follows: $$\begin{aligned}
g_{BL} < 2.8 \times 10^{-6} \left(\frac{v_{EW}}{m_f} \right)
\left( \frac{1 \,{\rm GeV}}{m_\sigma} \right)^{1/2}
\left(\frac{1 \,{\rm GeV}}{M_N}\right)^{3} \left(\frac{M_{Z_{BL}}}{1 \, {\rm GeV}}\right). \end{aligned}$$ With a suitable choice of $M_N (> m_\sigma)$, we can see that this limit is quite compatible with our results shown in the right panel of Figs. \[Fig:1\], Fig. \[Fig:2\] . For the case when $m_\sigma > m_h$, on the other hand, the DM particle can decay to a pair of Higgs doublets through the mixing, and we find that the loop induced mixing is not small enough to be consistent with the results shown in the right panels of Figs. \[Fig:1\] and \[Fig:3\]. In this case, we consider a cancellation of the mixing between the tree and loop levels contriburions. We will now explore whether for such small parametric values for $g_{BL}$, we can generate the observed dark matter relic density of the universe.
4. Relic density
================
4.1 Allowed range of $g_{BL}$ from pre-conditions to freeze-in
--------------------------------------------------------------
First point to notice is that for GeV scale DM ($\sigma$), for values of $g_{BL}$ that satisfy the lifetime constraint, the $\sigma$ field is out of equilibrium from the SM particles. Therefore, the standard thermal freeze-out mechanism for creation of DM relic density does not apply and one has to explore the freeze-in mechanism. For this to work, we need the $Z_{BL}$ field, whose annihilation will produce the DM, to be in equilibrium with the SM fields. This question was explored in Ref. [@nobu] and it was pointed out that the most efficient process for $Z_{BL}$ to be in equilibrium with SM particles is via the process $f\bar{f}\to Z_{BL}+\gamma$. The condition on $g_{BL}$ for this to happen is $g_{BL} > 2.7\times 10^{-8}\left(\frac{M_{Z_{BL}}}{{\rm 1\, GeV}}\right)^{1/2}$.
An upper bound on $g_{BL}$ comes from the fact that the DM particle $\sigma$ is out of equilibrium in the early universe. The first process to consider is $Z_{BL} Z_{BL} \leftrightarrow \sigma \sigma$ for which the out-of-equilibrium condition is given by $n_\sigma \langle \sigma v \rangle < H$,. Here $n_\sigma \sim T^3$ is the number density of the DM $\sigma$, $\langle \sigma v \rangle \sim g_{BL}^4/(4 \pi T^2)$, and the Hubble parameter $H =\sqrt{\frac{\pi^2}{90} g_*} T^2/M_P$ with the reduced Planck mass $M_P=2.43 \times 10^{18}$ GeV and the effective total number of relativistic degrees of freedom $g_*$ (we set $g_*=106.75$ for the SM particle plasma in our analysis throughout this paper). Requiring that this inequality is satisfied until $T \sim M_{Z_{BL}}$, we find that $g_{BL} < 6.4 \times 10^{-5} \left(\frac{M_{Z_{BL}}}{\rm 1 \, GeV}\right)^{1/4}$. Combining with the equilibrium condition for $Z_{BL}$, we find that we have to work in the range of $g_{BL}$ values $$\begin{aligned}
2.7\times 10^{-8}\left(\frac{M_{Z_{BL}}}{{\rm 1\, GeV}}\right)^{1/2} < g_{BL} < 6.4 \times 10^{-5} \left(\frac{M_{Z_{BL}}}{\rm 1 \, GeV}\right)^{1/4}.
\label{TH}\end{aligned}$$ to generate the relic density.
There is another upper bound on $g_{BL}$ that arises from the fact that the process $NN\to \sigma\sigma$ should also out of equilibrium. The reason is that in the early universe, the right handed neutrinos are always in equilibrium with SM particles via processes such as $N+t \leftrightarrow \nu+t$ etc. and $N \leftrightarrow H \ell$ for $M_N > m_h$. If $NN \leftrightarrow \sigma\sigma$ is also in equilibrium, the freeze-in mechanism for relic density generation of $\sigma$ will not work. To get this upper bound on $g_{BL}$ using this condition, we use $n_\sigma \langle \sigma_{NN\to \sigma\sigma}v \rangle < H$ at $T \sim M_N$ and find $$\begin{aligned}
\frac{1}{4\pi} \left(\frac{M^5_N}{v^4_{BL}} \right) < \sqrt{\frac{\pi^2 }{90} g_*}\frac{M_N^2}{M_P}\end{aligned}$$ Using $M_{Z_{BL}}=2g_{BL}v_{BL}$, this leads to $$\begin{aligned}
%g_{BL}\leq \frac{1}{2}\left(4\pi\sqrt{\frac{\pi^2 g^*}{90}}\right)^{1/4}\frac{M_{Z_{BL}}}{M_N}\left(\frac{M_N}{M_P}\right)^{1/4}\\\nonumber
%\simeq 3.24\times 10^{-5} \left( \frac{M_{Z_{BL}}}{{\rm 1 \, TeV}} \right) \left(\frac{{\rm 1 \, GeV}}{M_N}\right)^{3/4}
%
g_{BL} < 3.2 \times 10^{-5} \left(\frac{M_{Z_{BL}}}{\rm 1 \, GeV}\right)^{1/4} \left(\frac{M_{Z_{BL}}}{M_N}\right)^{3/4}.
\label{THN}\end{aligned}$$ Note that for $M_N \sim M_{Z_{BL}}$, this upper limit is about the same level as in Eq. (\[TH\]) so that indeed the freeze-in mechanism is called for in creating the relic density build-up. In the following, we consider $M_N < M_{Z_{BL}}$, for which the upper bound is determined by the $B-L$ gauge interaction. Incidentally, we note that If $M_N < m_h$, the interactions of the RHNs with the SM particles are too week for them to be in thermal equilibrium, and the above discussion is not applicable.[^1]
4.2 Relic density build-up
--------------------------
In order to calculate the relic density build-up via the freeze-in mechanism, we solve the following Boltzmann equation (defining $x=\frac{m_\sigma}{T}$): $$\begin{aligned}
\frac{dY}{dx}\simeq \frac{ \langle \sigma v \rangle}{x^2}\frac{s(m_\sigma)}{H(m_\sigma)} Y^2_{eq},
\label{Boltzmann}\end{aligned}$$ where $Y$ is the yield of the DM $\sigma$, $Y_{eq}$ is $Y$ if the DM $\sigma$ is in thermal equilibrium, and $s(m_\sigma)$ and $H(m_\sigma)$ are the entropy density and the Hubble parameter, respectively, evaluated at $T=m_\sigma$. For the DM particle creation process $Z_{BL} Z_{B} \to \sigma \sigma$, we approximate $\langle \sigma v \rangle \simeq \frac{g_{BL}^4}{4 \pi T^2}= \frac{g^4_{BL}}{4 \pi} \frac{x^2}{m^2_\sigma}$. Note that this formula is applicable for $T\geq M_{Z_{BL}}\gg m_\sigma$. The reason for this is that for $T \leq M_{Z_{BL}}$, the number density of $Z_{BL}$ is Boltzmann suppressed and $\sigma$ particle creation stops. Using $\frac{S(m_\sigma)}{H(m_\sigma)}\simeq 14 \, m_\sigma M_P$ and $Y_{eq}\simeq 2.2 \times 10^{-3}$ and integrating the above equation from $x_{RH}$ to $x$ (where $x_{RH}= m_\sigma/T_{RH}$ with the reheating temperature after inflation $T_{RH} \gg M_{Z_{BL}}$), we obtain $$\begin{aligned}
Y(x)-Y(x_{RH})\simeq 5.1\times 10^{-6} \, g^4_{BL} \left( \frac{M_P}{m_\sigma} \right) \, (x-x_{RH}). \end{aligned}$$ Then taking $Y(\infty)\simeq Y(x_{BL}=m_\sigma/M_{Z_{BL}})$, we estimate the DM relic density, $$\begin{aligned}
\Omega_{DM}h^2\simeq\frac{m_\sigma s_0 Y(\infty)}{\rho_0/h^2}\simeq
3.4 \times 10^{21} \, g^4_{BL} \, \left(\frac{m_\sigma}{{\rm 1 \, GeV}}\right) \left(\frac{{\rm 1 \, GeV}}{M_{Z_{BL}}}\right),
\label{Omega}\end{aligned}$$ where $s_0=2890/{\rm cm}^3$ is the entropy density of the present universe, and $\rho_c/h^2=1.05 \times 10^{-5} \, {\rm GeV}{{\rm cm}^3}$ is the critical density. This leads to the following expression for $g_{BL}$: $$\begin{aligned}
\label{gBL}
g_{BL}\simeq 2.4 \times 10^{-6}\left(\frac{M_{Z_{BL}}}{{\rm 1 \, GeV}}\right)^{1/4}\left(\frac{{\rm 1 \, GeV}}{m_{\sigma}}\right)^{1/4}\end{aligned}$$ to reproduce the observed DM relic density $\Omega_{DM}h^2 = 0.12$.
Using Eq. (\[gBL\]) in Eq. (\[LF\]), we show the lifetime for various values of $m_\sigma$ in Fig. \[Fig:1\] (Left Panel). The diagonal lines from left to right correspond to $m_\sigma=1$ MeV, 10 MeV, 100 MeV and 1 GeV, respectively, along which $\Omega_{DM}h^2 = 0.12$ is reproduced. The horizontal dashed line indicates the astrophysical bound on $\tau_\sigma > 10^{25}$ sec. Combining Eqs. (\[LF2\]) and (\[gBL\]), we obtain a lower bound on $M_{Z_{BL}}$: $$\begin{aligned}
M_{Z_{BL}} \gtrsim 210 \, \left(\frac{m_\sigma}{{\rm 1 \, GeV}}\right)^{11/9}~{\rm GeV}.
\label{Lmass} \end{aligned}$$
Considering all the constraints from Eqs. (\[TH\]), (\[gBL\]) and (\[Lmass\]), we show the allowed parameter region in Fig. \[Fig:1\] (Right Panel). The region between two diagonal black lines satisfies the condition of Eq. (\[TH\]), and the horizontal black line corresponds to Eq. (\[Lmass\]). The observed $\Omega_{DM}h^2 = 0.12$ is reproduced along the red lines each of which corresponds to a fixed $m_\sigma$ value. In the right panel, the region for $M_{Z_{BL}} \lesssim 10$ MeV and $g_{BL} \sim 10^{-5}$ is excluded by the long-lived $Z_{BL}$ boson search results. See Fig. \[Fig:2\] for details.
![ [**Left Panel**]{}: The dark matter $\sigma$ lifetime as a function of $M_{Z_{BL}}$. The diagonal solid lines correspond to $m_\sigma$ =1 MeV, 10 MeV, 100 MeV, and 1 GeV from left to right, along which the observed DM relic density of $\Omega_{DM} h^2= 0.12$ is reproduced. [**Right Panel**]{}: The $g_{BL}$ values as a function of $M_{Z_{BL}}$ from the requirement of relic density build-up. Different red lines correspond to different DM masses ($m_\sigma$ starting with 10 keV at the top and as we go below, we go in steps of a factor of 10 to 100 keV, 1 MeV, etc. till 100 GeV) that satisfy the relic density constraint i.e. $\Omega_{DM} h^2= 0.12$. Two diagonal black lines denote the condition of Eq. (\[TH\]), and the horizontal black line corresponds to Eq. (\[Lmass\]). []{data-label="Fig:1"}](Fig1-1.pdf "fig:"){width="0.46\linewidth"} ![ [**Left Panel**]{}: The dark matter $\sigma$ lifetime as a function of $M_{Z_{BL}}$. The diagonal solid lines correspond to $m_\sigma$ =1 MeV, 10 MeV, 100 MeV, and 1 GeV from left to right, along which the observed DM relic density of $\Omega_{DM} h^2= 0.12$ is reproduced. [**Right Panel**]{}: The $g_{BL}$ values as a function of $M_{Z_{BL}}$ from the requirement of relic density build-up. Different red lines correspond to different DM masses ($m_\sigma$ starting with 10 keV at the top and as we go below, we go in steps of a factor of 10 to 100 keV, 1 MeV, etc. till 100 GeV) that satisfy the relic density constraint i.e. $\Omega_{DM} h^2= 0.12$. Two diagonal black lines denote the condition of Eq. (\[TH\]), and the horizontal black line corresponds to Eq. (\[Lmass\]). []{data-label="Fig:1"}](Fig1-2.pdf "fig:"){width="0.46\linewidth"}
In the right panel of Fig. \[Fig:1\], we can see that there is an allowed parameter region for $g_{BL} ={\cal O}(10^{-5})$ and $M_{Z_{BL}}=1$ MeV$-1$ GeV. For the parameter region, $Z_{BL}$ boson can be long-lived and such a long-lived neutral particle can be explored in the near future by the Lifetime Frontier experiments, such as FASER [@faser], SHiP [@SHiP], LDMX [@LDMX], Belle II [@B2], and LHCb [@LHCb1; @LHCb2]. The $Z_{BL}$ boson search of the FASER experiment at the LHC is summarized in Ref. [@faser] along with the search reaches of other experiments as well as the current excluded region [@Bauer:2018onh]. In Fig. \[Fig:2\], we show our results of the right panel of Fig. \[Fig:2\] along with the summary plot in Ref. [@faser]. The red lines correspond to $m_\sigma=10$ keV, 100 keV, 1 MeV, and 10 MeV from top to bottom, respectively. The parameter region of 10 keV $\lesssim m_\sigma \lesssim 1$ MeV and 10 MeV $\lesssim M_{Z_{BL}} \lesssim$ a few GeV can be tested by various Lifetime Frontier experiments in the near future.
![FASER reachable region of the parameter space of our model. The black lines at the top and bottom denote the upper and lower limits on the $g_{BL}$ (Eq. (\[TH\])). The red lines correspond to $m_\sigma=10$ keV, 100 keV, 1 MeV, and 10 MeV from top to bottom, respectively, along which $\Omega_{DM}=0.12$ is satisfied. The parameter region of 10 keV $\lesssim m_\sigma \lesssim 1$ MeV and 10 MeV$\lesssim M_{Z_{BL}} \lesssim$ a few GeV can be tested by various Lifetime Frontier experiments in the near future. []{data-label="Fig:2"}](Fig2.pdf){width="0.6\linewidth"}
Before moving on to the next section, we comment on the dark matter production processes involving the RHN. If the RHN is in thermal equilibrium, the DM particles can also be created through $NN \to \sigma \sigma$. The estimate of $Y(\infty)$ from this process is analogous to the process $Z_{BL} Z_{BL} \to \sigma \sigma$, and resultant density is roughly given by Eq. (\[Omega\]) with replacing $g_{BL} \to f$ and $M_{Z_{BL}} \to M_N$. Thus, we take $M_N < M_{Z_{BL}}$, or equivalently $f < g_{BL}$, so that the RHN mediated DM production becomes subdominant. Calculations for other processes such as $NN \to Z_{BL} \sigma$ and $N Z_{BL} \to N \sigma$ are also analogous, and we can arrive at the same conclusion. We can also consider DM production processes through Dirac Yukawa couplings ($h_{SM}$) such as $N \, H \to \ell \, \sigma$ and $H \, \ell \to N \, \sigma$, where $H$ and $\ell$ are the Higgs and lepton doublets, respectively. The DM productions can be subdominant if $h_{SM}$ is sufficiently small, in other words, through the seesaw formula, $N$ is sufficiently light. The discussion for the DM production process of $H \, \ell \to N \, \sigma$ is applicable even if the RHN is not in thermal equilibrium.
5. PeV dark matter from $B-L$ breaking
======================================
So far we have explored the lower mass range of the dark matter. In this section, we explore the possibility that the $\sigma$ mass is in the PeV range so that one could attempt to explain the 100 TeV to PeV neutrinos observed in IceCube Neutrino Observatory [@Aartsen:2013bka] by using $\sigma$ decay. We do not attempt to explain the IceCube signal here but simply to raise the possibility that a PeV mass $\sigma$ can also qualify as the dark matter in our model in a different parameter range. For this purpose, let us go through all the constraints on the model discussed above for this case.
5.1 Lifetime constraint
-----------------------
This constraint is same as in the case of light $\sigma$ in Eq. (\[lifetime\]) except that in the right-hand side, the masses of $\sigma$ and $Z_{BL}$ are now higher and the new constraint can be written as $$\begin{aligned}
\label{lifetime1}
g_{BL} \leq 4.2 \times 10^{-8} \, \left(\frac{M_{Z_{BL}}}{{\rm 1 \, PeV}}\right) \left(\frac{{\rm 1 \, PeV}}{m_\sigma}\right)^{7/6}\end{aligned}$$ If we restrict the $B-L$ breaking VEV $v_{BL}\leq 10^{16}$ GeV, then the lifetime constraint can be translated to $M_{Z_{BL}}\sim 10^{10}$ GeV for $g_{BL}$ as large as $10^{-5}$.
We note that the one-loop $\sigma-h$ mixing contribution in this case leads to a very strong upper limit on the $g_{BL}$ value and much too small to generate enough relic density for the dark matter. In this case tyherefore, we fine tune the tree-level and one-loop $\sigma$-Higgs coupling to zero.
5.2 Relic density constraints
-----------------------------
We next explore the constraints of relic density on the heavy DM case. For such low $g_{BL}$ values, a heavy PeV scale DM and the $10^{10}$ GeV or higher mass $Z_{BL}$ would never have been in equilibrium. The relic density must arise as in the first case via the freeze-in mechanism. Since $Z_{BL}$ is not in thermal equilibrium, the production takes place via the process $f\bar{f} \to Z_{BL} \sigma$ through the SM fermion pair annihilations in the thermal plasma. In this case, the Boltzmann equation is given by $$\begin{aligned}
\frac{dY}{dx} \simeq \frac{ \langle \sigma v \rangle}{x^2}\frac{s(m_\sigma)}{H(m_\sigma)} Y_{eq} Y_{eq}^{BL}, \end{aligned}$$ where $Y_{eq}^{BL}$ is the yield of $Z_{BL}$ in thermal equilibrium and the cross section for the process $f\bar{f}\to Z_{BL}\sigma$ is estimated as $$\begin{aligned}
\langle \sigma v \rangle = \frac{g^4_{BL}}{4\pi}\frac{M^2_{Z_{BL}}}{m^4_\sigma} x^4. \end{aligned}$$ Recall that the DM production stops at $T\simeq M_{Z_{BL}}$ due to kinematics. Using $Y_{eq}^{BL} \simeq 2 Y_{eq}$ for $T \gtrsim M^2_{Z_{BL}} \gg m_\sigma$, we integrate the Boltzmann equation from $x_{RH}$ to $x_{BL}=\frac{m_\sigma}{M_{Z_{BL}}}$ and obtain $$\begin{aligned}
Y(x_{BL}) &\simeq & 3.4 \times 10^{-6} \, g^4_{BL}
\left(\frac{M_{Z_{BL}}}{m_\sigma} \right)^2
\left( \frac{M_P}{m_\sigma} \right) \, (x_{BL}^3-x_{RH}^3) \nonumber \\
& \simeq &
3.4 \times 10^{-6} \, g^4_{BL} \left( \frac{M_P}{M_{Z_{BL}}} \right), \end{aligned}$$ where we have used $Y(x_{RH})=0$ and $x_{RH} \gg x_{BL}$. We now use, as before, $Y(\infty)\simeq Y(x_{BL})$ and estimate the DM relic density, $$\begin{aligned}
\Omega_{DM}h^2\simeq\frac{m_\sigma s_0 Y(\infty)}{\rho_0/h^2}\simeq
2.3 \times 10^{21} \, g^4_{BL} \, \left(\frac{m_\sigma}{{\rm 1 \, GeV}}\right) \left(\frac{{\rm 1 \, GeV}}{M_{Z_{BL}}}\right). \end{aligned}$$ In order to reproduce $\Omega_{DM} h^2=0.12$, we find $$\begin{aligned}
g_{BL}\simeq 2.7\times 10^{-6}\left(\frac{M_{Z_{BL}}}{m_\sigma}\right)^{1/4}.
\label{PeVDM} \end{aligned}$$ We require that the $Z_{BL}$ is not in equilibrium which gives the consistency condition $$\begin{aligned}
g_{BL} < 2.7\times 10^{-8}\left(\frac{M_{Z_{BL}}}{{\rm GeV}}\right)^{1/2}.
\label{out_eq} \end{aligned}$$
![ [**Left Panel**]{}: The red line corresponds to DM mass $m_\sigma= 1$ PeV with $\Omega_{DM} h^2 = 0.12$. This corresponds to the case where the DM is produced by $f \, \bar{f}\to Z_{BL} \, \sigma$. The left of dashed line corresponds to the DM being in thermal equilibrium and therefore is not the area for freeze-in case. The left of black solid line corresponds to $\tau_\sigma < 10^{25}$ sec. and is excluded. [**Right Panel**]{}: The red lines represent the DM masses from top 100 keV, 10 MeV, 1 GeV (jump of 100 times) till 100 PeV being the lowest red line. Along the red line $\Omega_{DM} h^2 =0.12$ is satisfied. The lower black line comes from the DM lifetime lower limit. Upper black line corresponds to $Z_{BL}$ not being in equilibrium. The condition of $v_{BL} \leq M_P$ is depicted by the right diagonal black line. []{data-label="Fig:3"}](Fig3-1.pdf "fig:"){width="0.46\linewidth"} ![ [**Left Panel**]{}: The red line corresponds to DM mass $m_\sigma= 1$ PeV with $\Omega_{DM} h^2 = 0.12$. This corresponds to the case where the DM is produced by $f \, \bar{f}\to Z_{BL} \, \sigma$. The left of dashed line corresponds to the DM being in thermal equilibrium and therefore is not the area for freeze-in case. The left of black solid line corresponds to $\tau_\sigma < 10^{25}$ sec. and is excluded. [**Right Panel**]{}: The red lines represent the DM masses from top 100 keV, 10 MeV, 1 GeV (jump of 100 times) till 100 PeV being the lowest red line. Along the red line $\Omega_{DM} h^2 =0.12$ is satisfied. The lower black line comes from the DM lifetime lower limit. Upper black line corresponds to $Z_{BL}$ not being in equilibrium. The condition of $v_{BL} \leq M_P$ is depicted by the right diagonal black line. []{data-label="Fig:3"}](Fig3-2.pdf "fig:"){width="0.46\linewidth"}
In Fig. \[Fig:3\] (Left Panel), we show our result for $m_\sigma=1$ PeV. The dashed line denotes the upper bound on $g_{BL}$ from the out-of-equilibrium condition of Eq. (\[out\_eq\]). The diagonal black line shows the lifetime constraint of Eq. (\[LF2\]). Along the red line, the observed DM relic density is reproduced (see Eq. (\[PeVDM\])). In the figure, we find the lower bound on $M_{Z_{BL}}=4.5 \times 10^9$ GeV. In the right panel of Fig. \[Fig:3\], we show the results for various values of $m_\sigma$. The red lines from top to bottom correspond to the results for $m_\sigma=100$ keV, 10 MeV, 1 GeV, 100 GeV, 10 TeV, 1 PeV, and 100 PeV, respectively. The left diagonal black line denotes the out-of-equilibrium condition of Eq. (\[out\_eq\]), while the horizontal line depicts the lower bound on $M_{Z_{BL}}$ from the lifetime constraint for various fixed values of $m_\sigma$. We also impose a condition of $v_{BL} \leq M_P$, which is depicted by the right diagonal black line. We thus see that there is enough parameter range in the model for the dark matter to be in the PeV range so that it can be relevant to the PeV neutrinos observed in IceCube experiment. This is possible for $M_{Z_{BL}} \gtrsim 10^{10}$ GeV and $v_{BL}\gtrsim 10^{14}$ GeV.
6. Prospects for SO(10) embedding
=================================
In this section, we like to point out that a slight variation of the model leads to its possible embedding into $SO(10)$ grand unified theory (GUT), which we believe should add to its theoretical appeal as a minimal GUT model that unifies neutrino masses and dark matter. The starting point of this discussion is the observation that the hypercharge generator $Y$ is a linear combination of the $I_{3R}$ and the normalized $B-L$ generators $I_{BL}$ of $SO(10)$ as follows: $$\begin{aligned}
Y=I_{3R}+\sqrt{\frac{2}{3}}I_{BL}\end{aligned}$$ where $I_{BL}=\sqrt{\frac{3}{2}} \frac{B-L}{2}$. The $B-L$ generator in the main body of the paper is not orthogonal to the $Y$ generator defined above. Therefore, it cannot emerge from $SO(10)$ breaking since $I_{BL}$ is not orthogonal to $Y$ defined above. Instead if we consider the generator $\tilde{I}\equiv -4I_{3R}+3(B-L)$, we get ${\rm Tr}(\tilde{I}Y)=0$ (i.e. they are orthogonal) for any irreducible representation of $SO(10)$ and can therefore emerge from $SO(10)$ breaking. This generator was also identified in Ref [@Nobu1] as the generator $U(1)_X$ for $x_H= -4/5$. Indeed, it has been shown in Ref. [@diluzio] that such a generator emerges out of $SO(10)$ breaking by a [**45**]{} Higgs field. To see this note that [**45**]{} Higgs under $SU(3)\times SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ group has multiplets $(1,1,1,0)$ and $(1,1,3,0)$ which can take VEVs $\omega_Y$ and $\omega_{BL}$, respectively. If we fine-tune the parameters of the Higgs potential, we can get $\omega_Y=\omega_{BL}$ in which case the unbroken generators are $U(1)_Y\times U(1)_{\tilde{I}}$. The normalized $\tilde{I}=\frac{1}{2\sqrt{10}}(-4I_{3R}+3(B-L))$.
As it turns out, the dark matter phenomenology discussed above remains unchanged if we use the Higgs field $\sigma$ to break the $U(1)_{\tilde{I}}$ symmetry. The $\sigma$ field then emerges from the [**126**]{}-dimensional representation of $SO(10)$ and our dark matter field $\sigma$ has $\tilde{I}=\frac{\sqrt{10}}{2}$ and therefore has all the properties required above for our dark matter.
Our scenario for $SO(10)$ breaking is as follows: we use [**45**]{}-dimensional Higgs field to break $SO(10)$ down to $SU(5)\times U(1)_{\tilde{I}}$ by choosing the vacuum with $\omega_Y=\omega_{BL}$, as noted above. The $\tilde{I}$ quantum numbers of fermions are then given by $\tilde{I}({\bf 10})= \frac{1}{2\sqrt{10}}$, $\tilde{I}({\bar{\bf 5}})= \frac{-3}{2\sqrt{10}}$ and $\tilde{I}({\bf 1})= \frac{5}{2\sqrt{10}}$, where [**10**]{}, ${\bar{\bf 5}}$ and [**1**]{} are the $SU(5)$ representations in $SO(10)$ spinor [**16**]{}. For a [**10**]{}-representation Higgs field in $SO(10)$, which is decomposed into ${\bf 5}+{\bar{\bf5}}$ under $SU(5)$ and includes the SM Higgs doublet, the $\tilde{I}$ quantum numbers are given by $\tilde{I}({\bf 5})= \frac{-2}{2\sqrt{10}}$ and $\tilde{I}({\bf 5})= \frac{2}{2\sqrt{10}}$.
Let us now discuss the evolution of the $\tilde{I}$ gauge coupling. The evolution of $U(1)_{\tilde{I}}$ gauge coupling ($g_{\tilde I}$) is given by $$\begin{aligned}
\mu \frac{d \alpha^{-1}_{\tilde I}}{d \mu}=\frac{b_{\tilde I}}{2\pi},
\end{aligned}$$ where $b_{\tilde I}=-49/10$ at a scale $\mu$ below the $SU(5)$ unification while $b_{\tilde I}=-5$ in $SU(5) \times U(1)_{\tilde{I}}$ theory by considering that the SM Higgs doublet is embedded into a [**5**]{}-representation in $SU(5)$. For simplicity, we have assumed that in each step of the gauge symmetry breaking, $SO(10) \to SU(5) \times U(1)_{\tilde{I}} \to SU(3)_c \times SU(2)_L\times U(1)_Y \times U(1)_{\tilde{I}}
\to SU(3)_c \times SU(2)_L\times U(1)_Y$, only the minimal sets of Higgs fields are light.
To see our coupling unification strategy in this model, we first discuss the $SU(5)$ unification without supersymmetry. As is clear, in this case, we will need extra fields beyond the SM fields below the $SU(5)$ unification scale. For this purpose, we introduce $n_3$ real scalar $SU(2)_L$ triplets with $Y=0$ and $n_8$ real scalar color octets with $Y=0$. The coupling evolution equations in this case are the following: $$\begin{aligned}
\mu \frac{d\alpha^{-1}_1}{d \mu}&=&-\frac{1}{2\pi} \left( \frac{41}{10} \right), \nonumber\\
\mu \frac{d\alpha^{-1}_2}{d \mu}&=& \frac{1}{2\pi} \left( \frac{19}{6} - \frac{n_3}{3}\theta(\mu-M_3) \right), \nonumber\\
\mu \frac{d\alpha^{-1}_3}{d \mu}&=& \frac{1}{2\pi} \left( 7 - \frac{n_8}{2}\theta(\mu-M_8) \right), \end{aligned}$$ where $M_{3,8}$ stand for the masses of the triplet $({\bf 1}, {\bf 3},1)$ and octet $({\bf 8}, {\bf 1}, 0)$ fields, respectively. Solving these equations with $n_3=5$ with mass $M_3=5$ TeV and $n_8=3$ with mass $M_8=200$ TeV, we find that the $SU(5)$ gauge coupling unification is achieved at $M_U=6.8 \times 10^{15}$ GeV.
Let us now proceed to $SO(10)$ unification i.e. the running of the $g_{\tilde I}$ coupling from its breaking scale (which does not affect very much) to where it unifies with the $SU(5)$ coupling evolving after the $SU(5)$ unification scale. We see that due to the small value of $g_{\tilde I}$ required to get the relic density from the freeze-in mechanism, the $SO(10)$ gauge coupling unification in 4-dimensions is hard to obtain. We therefore assume that above the $SU(5)$ GUT scale, the model becomes five dimensional [@keith] with the fifth dimension compactified on $S^1/Z_2$ orbifold with a radius $R={M_U}^{-1}$. In that case if we assume that the gauge fields are in the bulk while all the matter and Higgs fields are on a brane at an orbifold fixed point, their Kaluza-Klein (KK) modes contribute to the running of the $SU(5)$ coupling whereas $U(1)_{\tilde{I}}$ being abelian its coupling running does not get any extra contribution from the opening of fifth dimension. The evolution of the $SU(5)$ gauge coupling ($\alpha_5$) obeys $$\begin{aligned}
\mu \frac{d\alpha^{-1}_5}{d \mu}= \frac{1}{2\pi}
\left( \frac{43}{3} - \frac{1}{6} -\frac{5}{6} \left(1+n_3+n_8 \right)
+ \frac{55}{3} \sum_{n=1} \theta(\mu-\sqrt{1+n^2} M_U)
\right). \end{aligned}$$ Here, in the parenthesis of the right-hand side, $43/3$ is the contribution from the zero-mode $SU(5)$ gauge boson and the SM fermions, $-1/6$ from the ${\bf 5}$-representation Higgs field, and $-\frac{5}{6} \left(1+n_3+n_8 \right)$ from one adjoint Higgs to break the $SU(5)$ symmetry and $n_3+n_8$ adjoint Higgs field into which the triplet and octet scalars are embedded, and the last term is the contribution from the $SU(5)$ gauge boson KK modes. For the KK mode mass spectrum, we have simply added the contribution from the $SU(5)$ symmetry breaking. Once the extra dimension opens, the contribution from the KK modes changes the scale dependence of the running gauge coupling from a log to a power [@keith]. Thus it is possible to unify the $SU(5)$ and $U(1)_{\tilde{I}}$ couplings into $SO(10)$ coupling as desired. This is shown in Fig. \[Fig:GCU\]. In the figure, the $SO(10)$ gauge coupling unification is achieved at $M_P$ with a unified coupling $g_{SO(10)} \simeq 0.1$. This result corresponds to an allowed parameter set, $m_\sigma \simeq 100$ keV and $M_{Z_{BL}}=10^{14}$ GeV, in the right panel of Fig. \[Fig:3\].
![ Unification of gauge couplings in the presence of one extra dimension. The horizontal blue line denotes $\alpha_{\tilde I}^{-1}$ while solid black lines from top to bottom denote $\alpha_1^{-1}$, $\alpha_2^{-1}$ and $\alpha_3^{-1}$, respectively. Here, we have set the $U(1)_{\tilde I}$ gauge boson mass (corresponding to $M_{Z_{BL}}$ in the previous sections) to be $10^{14}$ GeV as an example. The red curve represents the running of $\alpha_5^{-1}$ in the presence of the gauge boson KK modes. For a comparison with 4-dimensional theory, we show the dashed line for the $SU(5)$ without the KK mode contributions. []{data-label="Fig:GCU"}](GCU.pdf){width="0.7\linewidth"}
As far as proton decay is concerned, the primary mode is $p\to e^++\pi^0$ mediated by the $SU(5)$ gauge boson. The proton decay amplitude gets contribution from all the KK excitations of the SU(5) gauge fields, and we estimate the modification of a coefficient of the 4-Fermi operator to be $$\begin{aligned}
\frac{1}{M_U^2} \to \frac{1}{M_U^2} \left(1 + \sum_{n=1}^\infty \frac{1}{1+n^2} \right)
\simeq \frac{2.08}{M_U^2} \equiv \frac{1}{\Lambda^2}. \end{aligned}$$ Then, (ignoring threshold effects) the proton lifetime is estimated as $$\begin{aligned}
\tau_p \simeq \frac{\Lambda^4}{\alpha_U^2 m_p^5}, \end{aligned}$$ where $m_p=0.938$ GeV. Using $\alpha_5(M_U) \simeq 0.026$ and $M_U \simeq 6.8 \times 10^{15}$ GeV from Fig. \[Fig:GCU\], we find that $\tau_p\simeq 2.1 \times 10^{34}$ years, which is consistent with the lower bound $\tau_p \geq1.6 \times 10^{34}$ years from the Super-Kamionkande results [@Miura:2016krn]. More importantly, we would expect that $p\to e^+\pi^0$ should be observable in the next round of proton decay searches at Hyper-Kamiopkande [@Abe:2011ts] or the model will be ruled out.
7. Concluding remarks
=====================
We have presented a minimal model based on a $U(1)_{B-L}$ extension of the standard model where the $B-L$ breaking Higgs field plays the role of a decaying dark matter. We discuss two regions of the DM masses: one light mass region in the keV to MeV range and another where the DM mass is in the PeV range. In both cases, due to the stability requirement of the Dark matter, the freeze-in mechanism is required to understand the observed relic density of DM. We then discuss how the model can be tested in the FASER and other Lifetime Frontier experiments. Finally, we show how the model can emerge from an $SO(10)$ GUT model. Coupling unification in this case requires that the model be part of a five dimensional space-time with the compactification radius being of the order of the inverse of the $SU(5)$ unification scale $M_U$. This embedding reflects itself in an enhanced decay rate for the proton due to extra gauge KK mode contributions, which we have estimated. The model may have TeV scale hypercharge neutral weak iso-triplet and color octet scalars, which have interesting LHC phenomenology [@triplet; @octet]. Discussion of this phenomenology is beyond the scope of this paper. There are also ranges for the RHN masses in the model where resonant leptogenesis can generate the baryon asymmetry of the universe. This will be the subject of a forthcoming publication.
Acknowledgement {#acknowledgement .unnumbered}
===============
The work of R.N.M. is supported by the National Science Foundation grant No. PHY-1620074 and PHY-1914631, and the work of N.O. is supported by the US Department of Energy grant No. DE-SC0012447.
[99]{} P. Minkowski, Phys. Lett. B [**67**]{}, 421 (1977).
R. N. Mohapatra and G. Senjanović, Phys. Rev. Lett. [**44**]{}, 912 (1980).
T. Yanagida, Conf. Proc. C [**7902131**]{}, 95 (1979).
M. Gell-Mann, P. Ramond and R. Slansky, Conf. Proc. C [**790927**]{}, 315 (1979) \[arXiv:1306.4669 \[hep-th\]\].
S. L. Glashow, NATO Sci. Ser. B [**61**]{}, 687 (1980).
R. E. Marshak and R. N. Mohapatra, Phys. Lett. [**91B**]{}, 222 (1980).
R. N. Mohapatra and R. E. Marshak, Phys. Rev. letters [**44**]{}, 1316 (1980).
A. Davidson, Phys. Rev. D [**20**]{}, 776 (1979).
Lorenzo Basso, Alexander Belyaev, Stefano Moretti, and Claire H. Shepherd-Themistocleous. Phys. Rev. [**D 80**]{}, 055030, (2009).
A. A. Abdelalim, A. Hammad and S. Khalil, Phys. Rev. D [**90**]{}, no. 11, 115015 (2014)
S. Iso, N. Okada and Y. Orikasa, Phys. Lett. B **676** (2009), 81-87; Phys. Rev. D **80** (2009), 115007.
J. Heeck, Phys. Lett. B [**739**]{}, 256 (2014)
M. Bauer, P. Foldenauer and J. Jaeckel, JHEP [**1807**]{}, 094 (2018).
R. N. Mohapatra and N. Okada, arXiv:1908.11325 \[hep-ph\].
S. Heeba and F. Kahlhoefer, arXiv:1908.09834 \[hep-ph\].
L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, JHEP [**1003**]{}, 080 (2010)
J. L. Feng, I. Galon, F. Kling and S. Trojanowski, Phys. Rev. D [**97**]{}, no. 3, 035001 (2018). S. Bertolini, L. Di Luzio and M. Malinsky, Phys. Rev. D [**81**]{}, 035015 (2010).
N. Aghanim [*et al.*]{} \[Planck Collaboration\], arXiv:1807.06209 \[astro-ph.CO\].
M. G. Baring, T. Ghosh, F. S. Queiroz and K. Sinha, Phys. Rev. D [**93**]{}, no. 10, 103009 (2016).
S. Alekhin [*et al.*]{}, Rept. Prog. Phys. [**79**]{}, no. 12, 124201 (2016).
T. Akesson [*et al.*]{} \[LDMX Collaboration\], arXiv:1808.05219 \[hep-ex\].
M. J. Dolan, T. Ferber, C. Hearty, F. Kahlhoefer, and K. Schmidt-Hoberg, JHEP [**12**]{}, 094 (2017).
P. Ilten, J. Thaler, M. Williams, and W. Xue, Phys. Rev. D92, no. 11, 115017 (2015).
P. Ilten, Y. Soreq, J. Thaler, M. Williams, and W. Xue, Phys. Rev. Lett. [**116**]{} no. 25, 251803 (2016).
M. Bauer, P. Foldenauer and J. Jaeckel, JHEP [**1807**]{}, 094 (2018) \[JHEP [**2018**]{}, 094 (2020)\].
M. Aartsen *et al.* \[IceCube\], Phys. Rev. Lett. **111**, 021103 (2013).
N. Okada, S. Okada and D. Raut, Phys. Lett. B **780**, 422-426 (2018). K. R. Dienes, E. Dudas and T. Gherghetta, Phys. Lett. B **436** (1998), 55-65; Nucl. Phys. B **537** (1999), 47-108. K. Abe *et al.* \[Super-Kamiokande\], Phys. Rev. D **95** (2017) no.1, 012004
K. Abe [*et al.*]{}, arXiv:1109.3262 \[hep-ex\].
See for example, J. Gunion, R. Vega and J. Wudka, Phys. Rev. D **42** (1990), 1673-1691; H. E. Logan and M. Roy, Phys. Rev. D **82** (2010), 115011; C. Englert, E. Re and M. Spannowsky, Phys. Rev. D **88** (2013), 035024.
T. Han, I. Lewis and Z. Liu, JHEP **12** (2010), 085.
[^1]: Note also that, as a general possibility, if $M_N$ is greater than the reheating temperature after inflation ($T_{RH}$), the RHN is irrelevant to our DM physics discussion.
|
---
abstract: 'The new multi-wavelength monitoring campaign on NGC 5548 shows clearly that the variability of the UV/optical lightcurves lags by progressively longer times at longer wavelengths, as expected from reprocessing of an optically thick disk, but that the timescales are longer than expected for a standard Shakura-Sunyaev accretion disc. We build a full spectral-timing reprocessing model to simulate the UV/optical lightcurves of NGC 5548. We show that disc reprocessing of the observed hard X-ray lightcurve produces optical lightcurves with too much fast variability as well as too short a lag time. Supressing the fast variability requires an intervening structure preventing the hard X-rays from illuminating the disc. We propose this is the disc itself, perhaps due to atomic processes in the UV lifting the photosphere, increasing the scale-height, making it less dense and less able to thermalise, so that it radiates low temperature Comptonised emission as required to produce the soft X-ray excess. The outer edge of the puffed-up Comptonised disc region emits FUV flux, and can illuminate the outer thin blackbody disc but while this gives reprocessed variable emission which is much closer to the observed UV and optical lightcurves, the light travel lags are still too short to match the data. We reverse engineer a solution to match the observations and find that the luminosity and temperature of the lagged emission is not consistent with material at the light travel lag distance responding to the irradiating flux (either FUV or X-ray) from the AGN. We conclude that the UV/optical lags of NGC 5548 are not the light travel time from X-ray reprocessing, nor the light travel time from FUV reprocessing, but instead could be the timescale for the outer blackbody disc vertical structure to respond to the changing FUV illumination.'
author:
- |
Emma Gardner and Chris Done\
Centre for Extragalactic Astronomy, Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK\
date: Submitted to MNRAS
title: 'The origin of the UV/optical lags in NGC 5548'
---
=1
= -0.5cm
\[firstpage\]
Black hole physics, accretion, X-rays: galaxies, galaxies: Seyfert, galaxies: individual: NGC 5548.
Introduction {#sec:introduction}
============
The emission from Active Galactic Nuclei (AGN) is typically variable, with faster variability seen at shorter wavelengths. This variability can be used as a tool to probe the surrounding structures, with reverberation mapping of the broad line region being an established technique. However, the same techniques can be used to probe the structure of the accretion flow itself. Hard X-ray illumination of the disc should produce a lagged and smeared thermal reprocessing signal. Larger radii in the disc produce lower temperature emission, so this disc reprocessing picture predicts longer lags at longer wavelengths. Such differential lags are now starting to be seen (Sergeev et al. 2005; McHardy et al. 2014; Edelson et al. 2015; Fausnaugh et al. 2015; McHardy et al. 2016) confirming qualitatively that we are indeed seeing reprocessing from radially extended, optically thick material, as expected from a disc.
However, quantitatively, the picture runs into difficulties. It has long been known that the implied size-scales are larger by a factor of a few compared to the expected sizes from a Shakura-Sunyaev disc (e.g. Cackett, Horne & Winkler 2007). Independent size-scale estimates from microlensing also imply that the optical/UV emission region is larger than expected by a similar factor (e.g. Morgan et al. 2010). Yet the optical/UV spectrum shows a strong rise to the blue, and can be fairly well fit by the emission expected from the outer disc regions in a Shakura-Sunyaev model (e.g. Jin et al. 2012; Capellupo et al. 2015), though there are discrepancies in detail (e.g. Davis, Woo & Blaes 2007).
Thus qualitatively the disc reprocessing picture appears sound, yet quantitatively it fails to match the data. This is perhaps not surprising for several reasons. Irradiation can change the structure of the disc e.g. by flaring it, as well as by changing the local heating (Cunningham 1976). Secondly, the spectra of AGN are clearly not simply a disc. The hard X-ray corona itself must be powered by accretion, pointing to a change from a pure Shakura-Sunyaev disc structure in the inner regions (e.g. Done et al. 2012). There is also the generic additional component seen in AGN, the soft X-ray excess, which again points to some change in disc structure which is not captured by the simple Shakura-Sunyaev equations (e.g. Gierlinski & Done 2004; Porquet et al. 2004).
Here we use the unprecedented SWIFT and HST lightcurves collected by the 2014 campaign on NGC 5548, which spans 120 days with sampling of 0.5 days, across 9 continuum bands from V to hard X-rays (Edelson et al. 2015). On long (month-year) timescales, the optical and X-ray lightcurves are well correlated, but the optical lightcurves show more variability than the hard X-rays, ruling out the simplest reprocessing models as accounting for all the optical variability in this source (Uttley et al. 2003, see also Arevalo et al 2008; 2009). However on day timescales the optical lightcurves lag behind the X-rays, with lag times increasing with wavelength as expected of disc reprocessing, and this is sampled in unprecedented detail by the 2014 campaign lightcurves. We use these to quantitatively test disc reprocessing models using a full model of illumination and reprocessing, with the aim to reproduce both the spectrum and variability of the source.
We first examine traditional disc reprocessing models and confirm that these cannot explain the lag timescales of the optical/UV lightcurves of NGC 5548. However, these quantitative models reveal another more fundamental conflict, which is that the UV and optical lightcurves cannot be produced by reprocessing of the observed hard X-ray flux. Disc reprocessing smears the variability by a similar timescale to the lag. The UV and optical are lagged by 1-2 days, but are much smoother than the hard X-ray lightcurve, with no sign of the rapid 1-2 day variability seen in the X-rays (see also Arevalo et al. 2008 and the discussion of variability in Lawrence 2012). This clearly shows that the fast hard X-ray variability is not seen by the outer disc, so not only are the observed X-rays not the driver for reprocessing, but the outer disc must be shielded from the observed X-rays. Instead, the FUV (represented by the HST lightcurve) is a much better match to the observed smoothness of the optical/UV lightcurves of NGC 5548. We incorporate these two aspects together in a model where the soft X-ray excess is produced from the inner regions of a moderately thickened disc which emits optically thick Compton (hence we name it the Comptonised disc) and which shields the outer blackbody (BB) disc from direct hard X-ray illumination from the central corona. The observed difference between the FUV and soft X-ray lightcurves clearly shows that this is not a single component, so we assume that the outer regions of the thickened Comptonised disc structure produce the FUV which can illuminate the outer thin BB disc. However, this still predicts light travel time lags which are shorter than observed. Increased flaring of the outer BB disc does not help because these large radii regions with the required long lag times are too cool to contribute significant optical flux due to their large area.
We explore the suggestion that the longer than expected lags come from the contribution of the classic BLR (H$\beta$ line) to the optical and UV emission (Korista & Goad 2001), but this does not work either as these do not contribute enough lagged flux. Since all known models fail, we reverse engineer a geometry which can fit both spectral and variability constraints. We use the observed optical lags in the different wavelength bands to constrain the luminosity and temperature of blackbody components at different lag times. We find the observations can be well matched by a single blackbody component lagged by 6 days behind the FUV irradiation, consistent with reprocessing on a population of clouds interior to the classic BLR as suggested by Lawrence (2012). However, the derived area of the reprocessor at this light travel time distance is far too small to intercept enough of the AGN luminosity (either FUV, X-ray or total) to give the observed luminosity of the lagged component. We conclude that the reprocessing timescale is not set by the light travel time. Interestingly, the temperature of the required lagged component is close to $10^4$ K, which is the trigger for the onset of the dramatic disc instability connected to hydrogen ionisation (see e.g. Lasota 2001) so it could instead be linked to the changing structure of the disc at this point.
Fundamentally, the time lags give us the wrong answers because the reverberation signal is not from a thin BB disc responding on the light travel time to illumination of either X-rays or FUV. We suggest it is instead from the inner edge of the thin BB disc changing its structure in response to an increase in FUV illumination and expanding on the vertical timescale to join the larger scale height Comptonised disc region.
Energetics of Disk Illumination and Reprocessing
================================================
For all models we fix $M=3.2\times 10^7M_\odot$ (as used by Edelson et al. 2015, from Pancoast et al. 2014: see also their Erratum 2015). This is similar to the Denny et al. (2010) estimate of $4.4\times
10^7M_\odot$, though a factor of $\sim 2$ smaller than the Bentz et al. (2010) estimate of $7.8_{-2.7}^{+1.9}\times 10^7M_\odot$. We also fix distance $D=75$Mpc, and spin $a=0$, and assume an inclination angle of $45^\circ$.
The unabsorbed, dereddened broadband spectrum of NGC 5548 from Summer 2013 is shown in Mehdipour et al. (2015). The X-ray flux is very hard, with photon index $\Gamma=1.6$, and the X-rays dominate the energy output of the source, peaking in $\nu F_\nu$ at $\sim 8\times 10^{-11}$ ergs cm$^{-2}$ s$^{-1}$ at 100 keV. The simultaneous optical/UV spectrum looks similar to that expected from an outer thin BB disc (though its shape is subtly different: Mehdipour et al. 2015). The flux in UVW1 (at around 5 eV) has $\nu F_\nu\approx 5\times
10^{-11}$ ergs cm$^{-2}$ s$^{-1}$.
We use the [optxagnf]{} model in [xspec]{} to find reasonable physical parameters for the accretion flow. This assumes that the mass accretion rate is constant with radius, with fully relativistic Novikov-Thorne emissivity per unit area, $L_{NT}(r)$, with dimensionless radius $r=R/R_g$ for $R_g=GM/c^2$, but that this energy is dissipated in a (colour temperature corrected) BB disc only down to some radius $r_{cor}$, with the remainder split between powering an optically thick Compton component (which provides the soft X-ray excess emission) and an optically thin Compton component, which models the hard X-ray coronal emission. This code calculates the angle averaged spectrum, so we boost the normalisation by a factor $\cos i/\cos 60=1.41$ to roughly account for our assumed inclination angle. If we assume that all the power within $r_{cor}=70$ goes to make the hard X-ray corona ($f_{cor}=1$), we find that we can match the UVW1 and X-ray flux for $\log L/L_{Edd}=-1.4$ with $r_{cor}=70$.
This does not necessarily mean that the geometrically thin BB disk itself is not present below $70\,R_g$, only that the accretion power is not dissipated within this structure (Svensson & Zdziarski 1994; Petrucci et al 2010). However, there is additional information in the hard X-ray spectrum which does point to this conclusion. An optically thick BB disc cannot be present underneath an isotropically emitting corona in this object, as such a disc will intercept around half of the hard X-ray flux, giving a strong Compton hump which is not present in the NuSTAR data (see spectrum in Mehdipour et al. 2015 and Ursini et al. 2015). Thermalisation of the non-reflected emission also produces too many seed photons for the X-ray source to remain hard (Haardt & Maraschi 1991; 1993; Stern et al. 1995; Malzac et al. 2005; Petrucci et al. 2013). Together, these imply that either the X-ray source is very anisotropic, or the disc truly truncates for sources with hard X-ray spectra.
Black hole binaries similarly show hard X-ray spectra and small Compton hump in their low/hard state, but here there are clear limits on the possible anisotropy from comparing sources with different binary inclination angles (Heil, Uttley & Klein-Wolt 2015). This argues strongly for true truncation of the BB accretion disc, as does the currently popular Lense-Thirring precession model for the origin of the low frequency QPO’s seen in high inclination binary systems (Ingram, Done & Fragile 2009). Hence we assume the BB disc is truly truncated in these low luminosity AGN (see also Petrucci et al. 2013; Noda 2016).
The irradiation pattern strongly depends on the relative geometry of the hard X-ray source and BB disc. In the black hole binaries, there is evidence from the complex pattern of energy dependent lags that the hard X-ray source is somewhat radially extended (Kotov, Churazov & Gilfanov 2001; Ingram & Done 2012), as is also expected if it is some form of hot, radiatively inefficient accretion flow such as an advection dominated accretion flow (Narayan & Yi 1995). We assume that the extended hard X-ray source has volume emissivity $\propto
L_{NT}(r)/r$ and neglect light-bending and red/blueshifts to work out the irradiating flux as in the appendix of Zycki et al (1999), i.e. $$F_{rep}(r) = \frac{f_{irr}L_{cor}\cos(n)}{4\pi (\ell R_g)^2}$$ where $\ell$ is the distance from the hard X-ray source to the disc surface element, $n$ is the angle between the source and normal to the disc, and $f_{irr}$ is the fraction of coronal hard X-ray luminosity ($L_{cor}$) which thermalises. We set $f_{irr}=1$ in this section in order to see the maximum irradiation flux. We then assume that $$T_{eff}(r) = T_{grav}(r) \left(\frac{F_{rep}(r)+F_{grav}(r)}{F_{grav}(r)}\right)^{1/4}$$ and model the resulting optically thick disc emission at this radius as a (colour temperature corrected) blackbody. However, the colour temperature correction makes very little difference for the black hole mass, mass accretion rate and $r_{cor}$ used here, as the disc peaks at 5.5 eV ($\sim 2500$Å) so is too cool for the colour temperature correction to be significant.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illuminating flux as a function of BB disc radius. Green and magenta dashed lines show flat ($h=0.1$) and flared ($h=h_0 (r/r_0)^{9/7}$, where $h/r=0.1$ at $r=660$) discs, respectively, irradiated by an extended spherical source with $r=70$. Cyan and blue dotted lines show same flat disc and flared discs, respectively, illuminated by a central point source with height $h_x=10$. Black dotted line shows flared disc illuminated by a central point source with $h_x=0$. Red solid line shows gravitational flux dissipation of BB disc. Black solid line shows self-gravity radius.[]{data-label="fig0"}](NewIlluminationProfile.pdf "fig:"){width="8cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We first assume the extended X-ray source illuminates a flat BB disc (formally, we give this disc constant height of $h=H/R_g=0.1$). The green dashed line in Fig.\[fig0\] shows the resulting irradiation flux per unit area on the equatorial plane. This is $\propto r^{-3}$ at large $r$, similar to the intrinsic gravitational flux dissipation (red), and is around 10-20% of the intrinsic flux at each radius. We compare this to the mathematically simpler form of a lamppost at height $h_x=10$ (cyan dotted line, where $h=H/R_g$) illuminating the BB disc, and show that the two are comparable at all disc radii. This is important as it shows that using lamppost illumination is not necessarily the same as assuming that the source is a lamppost (compact source on the spin axis). The more physical extended source geometry is identical in its illumination properties to the mathematically simpler lamppost.
It is unlikely that the BB disc remains flat under illumination. Cunningham (1976) shows that the disc structure responds to illumination, and can form a flared disc with height $h=h_0 (r/r_0)^{9/7}$, where $h_0$ is the disc height at its outer radius $r_0$. We show the resulting illumination for an extended X-ray source (Fig.\[fig0\], magenta dashed line), and a lamppost source of height $h_x=10$ (blue dotted line) and a central source ($h_x=0$: black dotted line) for a flared disc with $h/r=0.1$ at $r=660$. Clearly, irradiation can dominate over gravitational energy release for such a flared disc, but only at large radii.
Crucially, irradiation changes the predicted lag-wavelength profile of the BB disc. The standard argument for a lag time $\tau\propto
\lambda^{4/3}$ comes from assuming that the wavelength at which the disc peaks at each radius is $\lambda_{max}\propto 1/T\propto
L_{NT}(r)^{-1/4}\propto (M\dot{M})^{-1/4} R^{3/4}$. Hence $R\propto\tau\propto (M\dot{M})^{1/3}\lambda_{max}^{4/3}$. However, in the irradiation dominated region, the emissivity $L(r)\propto
r^{-12/7}$, so the lags are no longer expected to go as $\tau\propto
\lambda^{4/3}$ but as $\propto \lambda^{7/3}$.
Fig.\[fig0\] shows irradiation only gives $L(r)$ which is substantially different to $r^{-3}$ at $r>2000$, so all our irradiation models predict $\tau\propto
\lambda_{max}^{4/3}$ for $r<1000$. The self-gravity radius for a disc with these parameters is only 660$R_g$ (black vertical line: Laor & Netzer 1989, using Shakura-Sunyaev $\alpha=0.1$), so that truncating the BB disc at this point means that irradiation never dominates, and there is only a factor of $\sim 2$ between the flared and flat disc illumination fluxes at $r=660$.
In the following reprocessing models we wish to maximise irradiation. We therefore use the lamppost at $h_x=10$ illuminating a flared BB disc (Fig.\[fig0\], blue dotted line; illumination pattern identical to an extended source spherical source with $r=70$) in all subsequent models.
Calculation of Cross Correlation Functions
------------------------------------------
Throughout the paper we compare lags between lightcurves by calculating the cross correlation function (CCF). For two lightcurves $x(t)$ and $y(t)$, which are evenly sampled on time $\Delta t$ so $t=t_0+i\Delta t$, the CCF as a function of lag time $\tau=j\Delta t$ is defined as:
$$CCF(\tau) = \frac{\sum (x(i)-\bar{x})(y(i-j)-\bar{y})}
{[(\sigma_x^2-\sigma_{ex}^2) (\sigma_y^2-\sigma_{ey}^2)]^{1/2} }$$
where the sum is over all data which contribute to the lag measurement, so there are a smaller number of points for longer lags. The averages, ($\bar{x},\bar{y}$), and measured variances ($\sigma^2_x,\sigma^2_y$), and error bar variances ($\sigma^2_{ex},\sigma^2_{ey}$), are also recalculated for each $\tau$ over the range of data used. With this definition, then $CCF(\tau)=1$ implies complete correlation.
However, real data are not exactly evenly sampled. Interpolation is often used to correct for this, but this can be done in multiple ways (Gaskell & Peterson 1987). The red solid lines in Fig.\[figccfs\] shows the interpolated CCF, computed by linearly interpolating both lightcurves onto a grid of $0.1$d spacing and then resampling to produce evenly sampled lightcurves with $dt=0.5$d. This interpolation scheme introduces correlated errors, and it is not simple to correct for these so we first set $\sigma_{ex}=\sigma_{ey}=0$. The correlation is very poor between the hard X-ray and FUV lightcurve (Fig.\[figccfs\]a), while the FUV and UVW1 are consistent with almost perfect correlation with a lag of $\sim 0.5$ days (Fig.\[figccfs\]b). The correlation without considering the error bar variance is slightly worse between FUV and V band, while the lag is somewhat longer (Fig.\[figccfs\]c).
The neglect of the error bar variance can suppress the correlation, so we first investigate how much of the lack of correlation in Fig.\[figccfs\]a-c is due to this. We assess the size of this effect by calculating the autocorrelation function (ACF). For an evenly sampled lightcurve with independent errors, the error bars are correlated only at zero lag, giving an additional spike at zero on top of the intrinsic ACF shape. The interpolated lightcurves have correlated errors on timescales of the interpolation, so instead of a spike at zero lag, these form a component with width $\sim 0.5~d$ on top of the intrinsic ACF. We show the ACF of the interpolated UVW1 and V band lightcurves as the solid black lines in Fig.\[figccfs\]b&c, respectively. We do not show the ACF of the FUV lightcurve as this has such small errors and such good sampling that there is negligible error bar variance in this lightcurve. We fit the ACFs with two Gaussians, one broad to model the intrinsic ACF, and one narrow to represent the correlated errors, both of which should peak at zero lag. This gives the value of $\sigma_e^2$ that must be subtracted in order for the broad Gaussian to peak at unity. For UVW1 we find that $\sigma_e^2 = 0.022
\sigma_{UVW1}^2$, and for the V band we find $\sigma_e^2 = 0.095 \sigma^2_V$. We subtract this correlated error variance to get the error corrected interpolated correlation function (Fig.\[figccfs\]b&c, dashed red lines). The effect is quite small for the (already very good) FUV–UVW1 correlation, but makes a noticeable difference to the FUV–V band cross-correlation. The intrinsic variability in the V band lightcurve is then consistent with an almost perfect correlation with the variable FUV lightcurve, but with a $\sim 2$ day lag (see also McHardy et al. 2014; Edelson et al. 2015; Fausnaugh et al. 2015).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Solid red lines show cross correlation functions calculated from the observed lightcurves of NGC 5548: (a) FUV with respect to hard X-rays. (b) UVW1 with respect to FUV. (c) V band with respect to FUV. Solid black lines in a, b & c show the autocorrelation functions of the hard X-ray, UVW1 and V bands, respectively. The narrow peaks at zero lag in b & c are due to correlated errors introduced by interpolating the lightcurves. Red dashed lines in b & c show the cross correlation functions (UVW1 w.r.t. FUV and V band w.r.t. FUV, respectively) after correcting for these correlated errors.[]{data-label="figccfs"}](HXFUV.pdf "fig:"){width="8cm"}
![Solid red lines show cross correlation functions calculated from the observed lightcurves of NGC 5548: (a) FUV with respect to hard X-rays. (b) UVW1 with respect to FUV. (c) V band with respect to FUV. Solid black lines in a, b & c show the autocorrelation functions of the hard X-ray, UVW1 and V bands, respectively. The narrow peaks at zero lag in b & c are due to correlated errors introduced by interpolating the lightcurves. Red dashed lines in b & c show the cross correlation functions (UVW1 w.r.t. FUV and V band w.r.t. FUV, respectively) after correcting for these correlated errors.[]{data-label="figccfs"}](FUVUVW1.pdf "fig:"){width="8cm"}
![Solid red lines show cross correlation functions calculated from the observed lightcurves of NGC 5548: (a) FUV with respect to hard X-rays. (b) UVW1 with respect to FUV. (c) V band with respect to FUV. Solid black lines in a, b & c show the autocorrelation functions of the hard X-ray, UVW1 and V bands, respectively. The narrow peaks at zero lag in b & c are due to correlated errors introduced by interpolating the lightcurves. Red dashed lines in b & c show the cross correlation functions (UVW1 w.r.t. FUV and V band w.r.t. FUV, respectively) after correcting for these correlated errors.[]{data-label="figccfs"}](FUVV.pdf "fig:"){width="8cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In Fig.\[figccfs\]a we show the hard X-ray ACF (black line). It is not possible to apply the Gaussian fitting method to the hard X-ray ACF as the fast varying hard X-rays show an intrinsic peak of correlated variability on $\sim0.5$d timescales (Noda 2016) that cannot easily be separated from the effects of any correlated errors. However, we calculate the error bar variance of the hard X-ray lightcurve from the data and find it is roughly $\sigma^2_e\sim0.013\sigma^2_X$. This is negligible and cannot explain the poor correlation shown by the hard X-ray–FUV CCF, indicating that the poor correlation between the hard X-ray and FUV lightcurves is intrinsic to the process.
Lightcurves and Spectra from Blackbody Disc Reprocessing Hard X-ray Emission
============================================================================
Not all the irradiating flux will thermalize, as some part will be reflected. The reflection albedo depends on the ionization state of the disc, but for such a hard spectrum it varies only from $0.3$ (neutral) to $0.5$ (completely ionised), giving $f_{irr}=0.7-0.5$. We choose $f_{irr}=0.5$ in all subsequent models.
-------------------------------------------------------- ----------------------------------------------------
{width="8cm"} {width="8cm"}
{width="8cm"} {width="8cm"}
-------------------------------------------------------- ----------------------------------------------------
We first explore the scenario in Section 2, where a flared BB disc with scale-height 0.1 on its outer edge extends inwards from the self gravity radius at $r_{out}=660$ down to $70$, with the flow then forming a hot corona whose illumination can be approximated by a point source at height $h_x=10$ above the black hole. The solid red line in Fig.\[fig1\]a shows the intrinsic BB disc emission that results from gravitational heating alone. The dashed red line shows the total disc emission including additional heating by the illuminating corona. For illustrative purposes we also show the emission from two individual annuli in black, with the lower energy example corresponding to the emission from $r_{out}=660$ and the higher energy example corresponding to the emission from the innermost disc radius at $70$. Again solid lines show the intrinsic emission from gravitational heating alone and dashed lines show the total emission including reprocessing.
--------------------------------------------------- -----------------------------------------------
{width="8cm"} {width="8cm"}
{width="8cm"} {width="8cm"}
--------------------------------------------------- -----------------------------------------------
The dashed red line in Fig.\[fig1\]a shows that reprocessing makes very little difference to the total BB disc luminosity. This is because the disc is dominated everywhere by the intrinsic emission (red solid line in Fig.\[fig1\]b) rather than by reprocessing (blue dashed line in Fig.\[fig1\]b), and even increasing $f_{irr}$ to its maximum plausible value of $0.7$ cannot overcome this. Reprocessing does have slightly more effect at larger radii (because the BB disc is flared), but the disc emission is dominated by the smallest radii.
So far we have considered the steady state or time-averaged spectrum. In order to know how fluctuations in the hard X-ray flux will produce changes in the UV/optical emission from the BB disc, we must quantify how well each disc radius can respond to and reproduce changes in the illuminating continuum. For each annulus in the disc we calculate its transfer function following Welsh & Horne (1991). This accounts for light travel time distances to different radii within the annulus and different azimuths within each radius. The transfer function, $T(r,\tau)$, for a given radius, $r$, describes what fraction of the reprocessed flux from that radius has a given time delay, $\tau$, with respect to the illuminating continuum. The fluctuations in the reprocessed flux from a given annulus is then:
$$F_{rep}(r,t) = \frac{f_{irr}\cos(n)}{4\pi (rR_g)^2}\int_{\tau_{min}}^{\tau_{max}} T(r,\tau) L_{cor}(t) \, d\tau$$
This causes the effective temperature of the annulus to vary as:
$$T_{eff}(r,t) = T_{grav}(r) \left(\frac{F_{rep}(r,t)+F_{grav}(r)}{F_{grav}(r)}\right)^{1/4}$$
The fluctuations in a given spectral band (e.g. UVW1) are then the sum of the fluctuations in the emission from each annulus contributing flux to that band. Fluctuations in reprocessed flux change the relative contributions of individual annuli to the total band flux. An increase in reprocessed flux increases the temperature of the annulus. This both increases the luminosity of the annulus and shifts the peak of its blackbody spectrum to higher energies. This may shift the peak emission from smaller radii out of the bandpass and shift more of the emission from larger radii, which usually peak below the bandpass, to higher energies and so increase their contribution to the total band flux. As such it is not appropriate to assume a band is always dominated by emission from any one radius. By calculating the effective temperature and corresponding blackbody spectrum from each annulus at each timestep our code accounts for this. We neglect any fluctuations in the intrinsic BB disc emission so that the only source of disc variability is the reprocessed fluctuations.
-------------------------------------------------- ---------------------------------------------------
{width="8cm"} {width="8cm"}
{width="8cm"} {width="8cm"}
-------------------------------------------------- ---------------------------------------------------
For our coronal power law fluctuations we use the hard X-ray light curve of NGC 5548 presented by Edelson et al. (2015), interpolated as described in the previous subsection to produce an evenly sampled input lightcurve with $dt=0.5$d. We input this into our disc reprocessing model and this allows us to calculate a model UVW1 light curve, which we can then compare to the observed data. We show our results in Fig.\[fig1\]c (bottom panel), where the red line shows the observed UVW1 lightcurve and the blue line shows our predicted UVW1 lightcurve using this model. For reference we also show the input hard X-ray lightcurve (top panel). Our simulated UVW1 light curve clearly fails to reproduce both the amplitude of fast variability (much more in the model than in the data) and the overall long term shape of the observed UVW1 lightcurve (especially the dip in the observed lightcurve at 18 days and the rise at 110 days).
In Fig.\[fig1\]d we show the cross correlation function (CCF) of the simulated UVW1 lightcurve with respect to the hard X-ray lightcurve (blue), compared to the CCF of the observed UVW1 lightcurve with respect to the hard X-ray lightcurve (red). A positive lag indicates the UVW1 band lagging the hard X-rays. The predicted CCF is strongly peaked with almost perfect correlation at close to zero lag, and is quite symmetric. The observed CCF has none of the well correlated, narrow component at lags $<1$ day, but is instead quite poorly correlated, and asymmetric with a peak indicating that the UVW1 band lags behind the hard X-rays by $\sim0.5-2$d (Edelson et al. 2015).
Clearly, this is not a viable model of the UVW1 lightcurve. The data require more reprocessing at longer lags, and less reprocessing at shorter lags. Certainly the inner disc produces the shortest time lags, so truncating the BB disc at a larger radius could supress some of the fast variability. We explore this in the next section.
Increasing Disc Truncation
--------------------------
We rerun our model with the BB disc truncated at a much larger radius, such that $r_{cor}=200$. Such a large truncation radius for such a low mass accretion rate AGN is consistent with the observed trend in local AGN for $r_{cor}$ to anticorrelate with $L/L_{Edd}$ (Jin et al 2012; Done et al 2012). We show the results of using this larger truncation radius in Fig.\[fig2\]. The key difference in the spectrum is that the hottest parts of the BB disc are no longer present, with the energy instead giving a slight increase in the normalisation of the hard X-ray Comptonisation component. The small disc component gives a significantly lower UVW1 flux, so the variable components from both the direct Comptonisation component and its reprocessed UV flux now contribute a higher fraction of the UVW1 band.
The model can now better reproduce the observed amplitude of UVW1 band long term flux variations (blue lightcurve in Fig.\[fig2\]c, especially the dip at 18 days). However our simulated UVW1 lightcurve still has much more fast variability than is seen in the real UVW1 lightcurve. The simulated UVW1 lightcurve clearly looks like the hard X-ray light curve from which it was produced. Yet the observed UVW1 lightcurve looks quite different to the observed hard X-ray lightcurve.
This is shown clearly in Fig.\[fig2\]d where we compare the CCFs. The CCF peak from our simulated lightcurve has shifted to $\sim0.5$d rather than the close to zero-peaked CCF of the previous model, but the lag is still not as long as in the observed CCF, and the model UVW1 lightcurve is much more correlated with the hard X-ray lightcurve. This is seen at all lags, but the problem is especially evident on short lag times, showing quantitatively that the model UVW1 lightcurve has much more of the fast variability seen in the hard X-ray lightcurve than the real data.
Edelson et al. (2015) commented that the lags they measure are much longer than expected from reprocessing on a standard BB disc, i.e. the radii that should show peak emission in the UVW1 band are much smaller than the radii implied by the light travel time delayed response of emission in that band. This would suggest that the accretion disc around NGC 5548 is not a standard BB disc. Somehow the same emission is produced at a larger radius than standard BB disc models predict.
We test this by altering our disc transfer functions such that the reprocessing effectively occurs at twice (Fig.\[fig2.1\]a&b) and then four times (Fig.\[fig2.1\]c&d) the radius at which a standard BB disc would produce that emission. In theory this should improve our simulated lightcurves, as reprocessing at larger radii smooths out fast fluctuations so should reduce the amount of high frequency power in the lightcurve.
----------------------------------------------------- ------------------------------------------------------
{width="9cm"} {width="9cm"}
----------------------------------------------------- ------------------------------------------------------
However, Fig.\[fig2.1\]a shows that an effective radius twice that of a standard BB disc does not smooth the simulated lightcurve enough. An effective radius four times that of a standard BB disc does a better job (Fig.\[fig2.1\]c), but comparison of the CCFs (Fig.\[fig2.1\]d) illustrates that this is still not a good match to the data. An effective radius four times that of a standard BB disc may be required to sufficiently reduce the high frequency power in the lightcurve, but this then gives light travel time lags that are too long. The peak lag of the observed CCF is roughly $0.5-2$d, while the peak lag from the model lightcurve extends from $\sim1-3$d. The observed smoothing timescale is much longer than the lag timescale, and this cannot be replicated by light travel time smoothing, as light travel time effectively ties the smoothing to the lag timescale. For this reason, doubling the black hole mass to match the Bentz et al. (2010) estimate does not solve the problem of transmitting too much high frequency power into the optical. Increasing the disc inclination angle increases the amount of smoothing, however this effect is negligible even when setting $i=75^\circ$ (an unreasonably large angle for a Seyfert 1 such as NGC 5548) and likewise cannot reduce the high frequency power in the model optical lightcurves.
Moreover, the model lightcurves are all far more correlated with the hard X-ray lightcurve, on all timescales, than the observed lightcurve is. Even in Fig.\[fig2.1\]d the peak correlation coefficient between the hard X-rays and the model UVW1 lightcurve is $\sim0.8$, while between the hard X-rays and the observed lightcurve it is only $\sim0.3$.
Fig.\[fig3\] further illustrates this. In the left panel we show the observed lightcurves — from top to bottom, hard X-rays, soft X-rays, FUV, UVW1 and V band. In the right panel we again show the observed hard and soft X-ray lightcurves, followed by our model FUV, UVW1 and V band lightcurves using the standard BB disc model. It is clear that reprocessing the hard X-ray lightcurve off a standard BB disc size-limited by the self-gravity radius produces UV and optical lightcurves that look like the original hard X-ray lightcurve. The hard X-ray flares (e.g. at 42 days) are slightly more smoothed in the longer wavelength bands, but they are still clearly recognisable.
By contrast the observed UV and optical lightcurves lack any short term flares and show additional longer timescale variability that is not present in the hard X-ray lightcurve (e.g. the dip between $\sim10-30$d). The FUV to V band lightcurves are clearly well correlated with one another, with peak CCFs with respect to the FUV of $0.7-0.9$ (Edelson et al. 2015) and the increasing lags with increasing wavelength suggest reprocessing is a key linking factor. But they are much less well correlated with the hard X-rays (with a peak CCF of only $\sim0.3$). The left panel of Fig.\[fig3\] shows there is a clear break in properties between the observed X-ray and UV–optical lightcurves. The right panel shows that, if the hard X-rays are the source of illuminating flux, this cannot occur.
There are really only two ways out of this impasse. Either the hard X-ray lightcurve is not a good tracer of the illuminating flux, or the BB accretion disc is shielded from being illuminated by the hard X-rays. The former is plainly a possibility, as the hard X-ray lightcurve from 0.8-10 keV is not at the peak of the hard X-ray emission and Noda et al. (2011) show that there can be a fast variable steep power law component. Mehdipour et al. (2015) and Ursini et al. (2015) show that the hard X-ray variability encompasses both a change in normalisation [*and*]{} in spectral index, with the spectrum softening as the source brightens. A hard X-ray lightcurve at 100 keV would be a better direct tracer of the total hard X-ray flux, but until this is available, we estimate this using the intrinsic power law spectral index and normalisation derived by Mehdipour et al. (2015). These are binned on 10 day intervals to get enough signal to noise, so it is not a sensitive test of the model, but assuming that the spectral cut-off remains fixed at 100 keV this still does not give a good match to the FUV lightcurve. We conclude that it is more likely that the reprocessing region is shielded from the hard X-ray illumination, and consider below how this might also shed light on the origin of the soft X-ray excess.
Lightcurves and Spectra from Blackbody Disc Reprocessing FUV Emission
=====================================================================
----------------------------------------------------- -----------------------------------------------------
{width="8cm"} {width="8cm"}
{width="8cm"} {width="8cm"}
{width="8cm"} {width="8cm"}
----------------------------------------------------- -----------------------------------------------------
The broadband spectrum of NGC 5548 presented by Mehdipour et al. (2015) shows that a two component BB disc + hard power law model is clearly not sufficient to fit its spectrum. This source shows a strong soft X-ray excess above the $2-10$keV power law which can be well fit with an additional low temperature, optically thick Compton component, though this is not a unique interpretation. It can also be well fit in the 0.3-10 keV bandpass with highly smeared, partially ionized reflection (Crummy et al. 2006). However, with the advent of NuSTAR and other high energy instruments, it is now clear that the reflection interpretation does not give such a good fit to the data up to 50-100 keV for AGN with hard X-ray spectra (e.g. Matt et al. 2014; Boissay et al. 2014). Hence we assume that the optically thick Compton component, which produces the soft X-ray excess, is an additional intrinsic continuum component.
To include this, we now assume that the BB disc truncates at $r_{cor}$ as before, but that the remaining gravitational power inwards of this radius is split between two coronal components: the hard power law and the optically thick Compton component. We fix the temperature and optical depth of the optically thick Compton component to $kT_e =
0.17$keV and $\tau=21$ (Mehdipour et al. 2015). This component then peaks in the UV as required by the spectrum. By reducing the fraction of coronal energy in the hard power law to $f_{pl}=0.75$, i.e. $0.25$ of the coronal energy goes instead into powering the optically thick Compton component, we are again able to match the ratio of $F_{UVW1}/F_{10{\rm{\,keV}}}\sim1.7$ found by Mehdipour et al. (2015), with $r_{cor}=200$, which we could not do previously with a substantially truncated disc and no optically thick Compton component. In Fig.\[fig4\]a we show this new model spectrum, with the optically thick Compton component shown in green.
While the origin of this emission is not well understood, it is clearly not from a standard disc. This component takes over from the standard BB disc in the UV, which may not be a coincidence as the substantial atomic opacity in the UV can cause changes in the disc structure compared to a Shakura-Sunyaev disc which incorporates only plasma opacities of electron scattering and free-free absorption. In particular, UV line driving has the potential to lift the disc photosphere (e.g. Laor & Davis 2014). The copious hard X-ray emission in this object should quickly over-ionise any potential UV line driven wind, which would result in the material falling back down again without being expelled from the system. This scenario has the potential to effectively increase the scale-height of the disc, decreasing its density (Jiang, Davis & Stone 2016). This decreases its true opacity, hence increasing the effective colour temperature correction (e.g. Done et al. 2012). Alternatively, UV temperatures are also linked to the onset of the dramatic disc instability connected to hydrogen ionisation (Lasota 2001; Hameury et al. 2009; Coleman et al 2016). Whatever the origin, the total optical depth of the disc to electron scattering at the UV radii is probably of order 10-100 (Laor & Netzer 1989), so the optically thick Compton emission in this picture is coming from the disc itself, with the emission not quite able to thermalise to standard BB emission because of the increased scale-height of the disc due to the UV radiation pressure and/or the onset of the hydrogen ionisation instability. We therefore refer to the optically thick Compton component as the ‘Comptonised disc’, to distinguish it from the outer BB disc.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illuminating FUV flux as a function of BB disc radius (for a flat disc) assuming the FUV flux is emitted by a wall of material of height $10\,R_g$ located at $200\,R_g$ (green line). Red line shows gravitational flux dissipation of BB disc.[]{data-label="fig4_0"}](WallIlluminationProfile.pdf "fig:"){width="8cm"}
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The break in properties between the soft X-ray and FUV lightcurve show clearly that the Comptonised disc is itself stratified rather than being a single spectral component as in Mehdipour et al. (2015). We show our potential geometry in Fig.\[fig4\]b, where the soft X-ray emission comes from the inner regions of this large scale-height flow, which still cannot illuminate the outer BB disc, while the FUV is produced at larger radii and can illuminate the outer BB disc.
We model this FUV illumination by assuming a cylinder of material located at a particular radius ($r_{irr}>0$), with a particular height ($h_{max}$). This changes the illumination pattern, so we can no longer use the approximation of an on-axis point source. We calculate the reprocessed flux at a given BB disc radius by dividing the ‘surface’ of the wall into elements (azimuthally — $d\phi$ — and vertically — $dh$) and summing the flux contribution from each element:
$$F_{rep}(r) = \int_0^{h_{max}}\int_0^{2\pi} \frac{f_{irr}L_{cor}}{2\pi^2h_{max}R_g^2} \frac{h \,dh \,d\phi}{(r^2+r_{irr}^2+h^2-2r_{irr}r\cos\phi)^{3/2}}$$
This new illumination pattern intensifies the illumination on radii close to $r_{irr}$ but for $r\gg r_{irr}$ it becomes indistinguishable from the case of a lamppost point source. We set $r_{irr}=r_{cor}$ and $h_{max}=10$, implying a scale-height for the outer edge of the Comptonised disc of $h/r=0.05$, which should be sufficient to obscure the hard X-ray emission. The resulting illumination profile is shown in Fig.\[fig4\_0\] (green line).
In Fig.\[fig4\]c&d we compare the resulting model UVW1 lightcurve with the observations. We now find a much better match to the behaviour of the observed UVW1 lightcurve, matching the amplitude of fluctuations and reproducing the shape of the UVW1/hard X-ray CCF.
In Fig.\[fig4\]e&f we also compare our model V band lightcurve with the data. We find a good match to the amplitude of the observed fluctuations, but the model V band CCF with respect to the hard X-rays (blue) is not lagging by as much as the real V band lightcurve (red). This mismatch is more evident when comparing the model UVW1 and V band CCFs with respect to the FUV. Fig.\[fig5a\] shows that both the model UVW1 (blue dashed) and V band (blue dotted) CCFs with respect to the FUV peak at close to zero lag, while the real UVW1 (red dashed) and V band (red dotted) CCFs are significantly shifted away from zero lag.
Thus while reprocessing the FUV gives lightcurves which are a much better match to the data, the response of our model lightcurves is too fast. The observed V band lag behind the FUV lightcurve is $\sim2$d (Edelson et al 2015). For a black hole mass of $3.2\times10^7\,M_\odot$, this means the reprocessed V band flux must be emitted roughly $1080\,R_g$ away from wherever the FUV emission occurs. The observed UVW1 lag is $\sim0.5$d, implying it is emitted at a distance of $\sim270\,R_g$ from the FUV emission. In our model we assume the FUV emission is supplied by the outer edge of the Comptonised disc at $\sim200\,R_g$. However, regardless of the exact location of the FUV emission, the observed lightcurves imply that the reprocessed V band flux must be emitted $\sim700\,R_g$ further away from the FUV continuum than the reprocessed UVW1 flux. So far we have used a BB disc truncated at the self-gravity radius of $660\,R_g$ as our reprocessor. Clearly this does not provide a large enough span of reprocessing radii. We now rerun our model with a larger outer radius for the BB disc, to see if we can reproduce the observed length of the V band lag.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Comparison of cross correlation functions with respect to the FUV for the standard BB disc plus Comptonised disc FUV reprocessing model. Dashed lines show CCF of UVW1 with respect to FUV, and dotted lines show CCF of V band with respect to FUV. Red lines show CCFs calculated using the observed lightcurves, while blue lines show the corresponding CCFs calculated using the simulated lightcurves.[]{data-label="fig5a"}](WallIlluminationFUVCCFs.pdf "fig:"){width="8cm"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Disc spectrum from the standard BB disc reprocessing FUV emission model with $r_{cor}=200$ and $r_{out}$ increased to $2000$. Solid red line shows total intrinsic BB disc emission; dashed red line shows total BB disc emission including reprocessing; pairs of black lines from left to right show emission from outermost radius, an intermediate radius ($r=1000$) and innermost radius of the BB disc, with solid and dashed lines for intrinsic and intrinsic plus reprocessed emission respectively. Grey shaded regions, from left to right, show location of V band, UVW1 band and FUV band, respectively.[]{data-label="fig5"}](NSWallIlluminationSpectrum.pdf "fig:"){width="8cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
------------------------------------ -----------------------------------------
{width="8cm"} {width="8cm"}
------------------------------------ -----------------------------------------
Increasing Outer Disc Radius
----------------------------
We increase the outer radius of the BB disc to $2000\,R_g$ and rerun our FUV reprocessing simulation. We find that increasing the outer radius of the BB disc makes no difference to the length of our simulated V band lag. Fig.\[fig5\] illustrates why.
A $2\,$d V band lag requires the V band to be dominated by reprocessed flux emitted from $R=2\,ld>1000\,R_g$ (for a $3.2\times10^7\,M_\odot$ like NGC 5548). Fig.\[fig5\] shows that BB disc annuli at these large radii are simply too cool to contribute significant flux to the V band. This is due to their large area. Illumination by the FUV continuum simply cannot raise their temperature enough to make them contribute significantly to the V band, let alone dominate its flux. The dashed lines in Fig.\[fig5\] show how little the illumination increases the total BB disc flux at these very large radii.
The amount of FUV flux intercepted by large disc radii can be increased by increasing the BB disc flaring, i.e. by increasing $h_{out}/r_{out}$. The larger the outer disc scale-height, the more illuminating flux the annulus intercepts, the greater the heating and the more V band flux it contributes. We set $h_{out}/r_{out}=0.5$ but this produces a negligible increase. The heating flux is simply spread over too large an area.
In order for a reprocessor located at $2\,ld$ ($R>1000\,R_g$) to contribute significant V band flux it must have a small area. Clearly disc annuli are not suitable for this since the area of the annulus is constrained to scale with its radius as $A(dr)\sim 2\pi
R dr$. An obvious source of small area reprocessors at large radii are the broad line region (BLR) clouds. In the next section we investigate the possibility that the observed optical lags are due to reprocessing of the FUV emission, not by a BB disc, but by BLR clouds.
Broad Line Region Clouds Reprocessing FUV Emission
--------------------------------------------------
--------------------------------------------- --------------------------------------------
{width="8cm"} {width="8cm"}
--------------------------------------------- --------------------------------------------
BLR clouds absorb UV continuum emission and re-emit the energy as optical lines/recombination continua. Clearly these do contribute to the observed flux, and are lagged by the size-scale of the BLR. We first explore if this contamination by the BLR can influence the lags, as suggested by Korista & Goad (2001).
Mehdipour et al. (2015) show the UV/optical spectrum taken during the campaign. We reproduce this in Fig.\[fig6\]a (M. Mehdipour, private communication), with the continuum bands superimposed. The UVW1 band contains a substantial amount of blended Fe and Balmer continuum (Fig.\[fig6\]a, magenta line) as well as a broad Mg emission line (Fig.\[fig6\]a, blue line). The strongest line contribution to V band comes from the narrow \[O\] emission line (Fig.\[fig6\]a, blue line), with a small contribution from the wing of H$\beta$.
We conduct a simple test to determine whether this BLR line contamination could explain the observed optical lags. The FUV is dominated by continuum emission, so we assume the continuum component varies as the FUV lightcurve, so that there are no real continuum lags. H$\beta$ lags the continuum in this source by roughly 15 days (Peterson et al. 2002), though this does change with flux, spanning 4-20 days (Cackett & Horne 2005; Bentz et al. 2010). We assume that all the BLR emission components (eg. Fe/Balmer blend, Mg and H$\beta$) are a lagged and smoothed version of the FUV lightcurve, where the lag and smoothing timescale is $15$d, while the narrow \[O\] emission line is constant on the timescale of our observations. Finally, we dilute this by the required amount of constant host galaxy component (Mehdipour et al. 2015) to get the full spectrum as a function of time. Integrating this over the UVW1 and V bands gives the simulated lightcurves for this model. Fig.\[fig6\]b shows the CCFs of these with respect to the FUV band lightcurve (blue), in comparison with the CCFs of the real lightcurves (red), with the UVW1 band as the dashed lines, and the V band as dotted. The red solid line shows the CCF of the FUV lightcurve with itself, i.e. the FUV autocorrelation function.
Both model CCFs peak at zero, whereas the observed CCFs have peaks offset from zero. This is because the flux contribution from broad lines is simply not large enough to shift the CCF peaks away from zero in either UVW1 or V band. Furthermore, UVW1 contains more broad line contamination than V band, and as a result the UVW1 model CCF (blue dashed line) is more positively skewed than the V band CCF, which is almost identical to the FUV ACF (compare blue dotted and red solid lines). This is in clear contrast to the data, where the V band lightcurve contains less line contamination than UVW1 and yet shows a longer lag than UVW1. The UVW1 and V band lags therefore cannot be explained through contamination by lagged broad line emission.
Thermal Reprocessing
====================
So far, we have worked forwards from a geometric model of the spectrum and its reprocessed emission, then calculated the resulting timing properties and compared these to the observations. Now we take the opposite approach. We begin by matching the timing properties of the source (specifically the lightcurve amplitudes and CCFs) and use these to infer the spectral components and then the geometry.
We begin by matching the shape of the UVW1 and V band CCFs. The peak lag (i.e. the lag at which the CCF is a maximum) of the observed UVW1 lightcurve with respect to the observed FUV lightcurve is $\sim0.5-1$d. The CCF of the FUV lightcurve with respect to itself (i.e. the FUV autocorrelation) peaks at zero, as there is no lag between the two lightcurves. The CCF of the FUV lightcurve lagged and smoothed by one day with respect to the original FUV lightcurve will peak at $1$d. This is not the only way a CCF peak at $1$d can be produced. If a lightcurve consists of equal amplitude contributions from two lightcurves, one of which is the original un-lagged FUV lightcurve and the other of which is the FUV lightcurve lagged and smoothed by $2$d, then the CCF of this composite lightcurve with respect to the original FUV lightcurve will be the sum of the two CCFs — the un-lagged FUV with respect to itself and the $2$d-lagged FUV with respect to the unlagged-FUV — so that the resulting CCF will peak, not at $0$d or $2$d, but at $1$d. More generally, any composite lightcurve CCF will be the sum of the component lightcurve CCFs weighted by the fraction of the total band flux coming from each lightcurve that is correlated with the reference lightcurve. In this way, we can constrain the flux contributions of variable components with different lags to a given band by matching the peak and shape of the CCF.
We use sixteen variable component lightcurves: the original FUV lightcurve, the FUV lightcurve lagged and smoothed by $1$d, the FUV lightcurve lagged and smoothed by $2$d, the FUV lightcurve lagged and smoothed by $3$d, e.t.c, up to a maximum lag of $15$d. We then combine these component lightcurves to simulate model UVW1 and V band lightcurves and then compare the CCFs of these model lightcurves to the observed CCFs. We systematically adjust the component lightcurve contributions and select the fractional contributions which produce the smallest difference ($\Delta$) between model and observed CCFs, where $\Delta=\sum_{\tau =-20d}^{\tau =+20d}\lvert CCF_{UVW1,obs}(\tau)-CCF_{UVW1,model}(\tau)\rvert+\sum_{\tau =-20d}^{\tau= +20d}\lvert CCF_{V,obs}(\tau)-CCF_{V,model}(\tau)\rvert$. Since UVW1 and V band are spectrally close (and the blackbodies we will fit to the components are broad in comparison), we require both bands to contain the same lagged components (although the fractional contributions of these components in each band will differ).
We find the best fitting model under these constraints requires UVW1 and V band to contain a contribution from the original FUV lightcurve, plus a contribution from the FUV lightcurve lagged and smoothed by $6$d. Table \[table2\] lists the corresponding fractional contributions ($f_{FUV}$ and $f_{FUV-6d}$), while Fig.\[fig7\]a shows the resulting model CCFs (blue), compared to the observed CCFs (red), where UVW1 CCFs are shown with dashed lines and V band CCFs with dotted lines. The model V band CCF peaks at $1.5-2$d, in agreement with the data. The UVW1 CCF peaks at $0.5$d, again in agreement with the data.
Matching the CCFs allows us to constrain the lagged, i.e. variable, flux contributions to the UVW1 and V bands. If the UVW1 and V bands contained only these variable components then they would have equal amplitude fluctuations. The observed FUV, UVW1 and V band lightcurves shown in Fig.\[fig7\]b in solid, dashed and dotted red lines, respectively, clearly do not have equal amplitude. The amplitude decreases with increasing wavelength, implying the fluctuations are being increasingly diluted by a constant component. Matching the amplitude of our simulated UVW1 and V band lightcurves to the observations allows us to constrain the fractional contribution of this constant component ($f_c$) to each band.
In reality this constant component has two possible sources: intrinsic BB disc flux and flux from the host galaxy. The spectral decomposition in Mehdipour et al. (2015) (Fig.\[fig6\]a) shows that the contribution of the host galaxy to the UVW1 band is negligible, hence we can assume all the constant flux in the UVW1 band is supplied by intrinsic BB disc emission (i.e. $f_d=f_c$, $f_g=0$). In contrast, the spectrum in Mehdipour et al. (2015) shows that the flux contributions of the intrinsic BB disc continuum and the host galaxy are roughly equal in the V band, i.e. $f_d=f_g=f_c/2$. Each band therefore has contributions from a maximum of six spectral components, with the proportions of each listed in Table \[table2\].
Having established the flux contributions of each temporal component to each band, we can now begin finding spectral components that can provide these flux levels in each band.
UVW1 V band
-------------- ------- --------
$f_{FUV}$ 0.395 0.189
$f_{FUV-6d}$ 0.105 0.111
$f_c$ 0.500 0.700
$f_d$ 0.500 0.350
$f_g$ 0.000 0.350
: Lightcurve fractions from model fits to the observed UVW1 and V band CCFs shown in Fig.\[fig7\]. $f_{FUV}$ is the fraction of total band flux contributed by un-lagged FUV lightcurve. $f_{FUV-6d}$ is the fraction of total band flux contributed by the FUV lightcurve lagged and smoothed by 6 days. $f_c$ is the fraction of total band flux that is constant. There are two possible sources of constant flux: intrinsic BB disc emission ($f_d$) and the host galaxy ($f_g$), such that $f_d+f_g=f_c$.[]{data-label="table2"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Spectral decomposition inferred from the component fractions listed in Table \[table2\], which are shown to reproduce the observed timing properties of NGC 5548 in Fig.\[fig7\]. Red line shows intrinsic (constant) emission from a BB disc truncated at $r_{in}=200$, with $r_{out}=300$. Black points, from left to right, show total flux in the V, UVW1 and FUV bands respectively. Orange star shows galaxy flux contribution to V band. Grey crosses mark the inferred flux levels of the spectral component varying as the FUV lightcurve. Magenta squares mark the inferred flux levels of the variable component with a $6\,$d lag with respect to the FUV lightcurve, while the magenta line shows the corresponding blackbody spectrum that matches these flux levels. The distance, temperature and covering factor of this blackbody component are: $R=3240\,R_g$, $T=9600$K and $f=0.002$.[]{data-label="fig8"}](BandfractionSpectrum.pdf "fig:"){width="8cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
We begin with intrinsic BB disc emission. We introduce a standard BB disc (red line, Fig.\[fig8\]), truncated at $r_{in}=200$ with $\log L/L_{Edd}=-1.4$, as is required by the energetically constrained spectral decomposition shown in Fig.\[fig4\]a. This is the sole provider of constant flux in the UVW1 band. Since we know the fractional contribution of constant flux to the UVW1 band ($f_c=f_d=0.5$; Table \[table2\]), this tells us the total UVW1 band flux is $F_{UVW1}=F_d(v_{UVW1})/f_d$ (central black point, Fig.\[fig8\]). From Fig.6a&b in Mehdipour et al. (2015) we can derive the ratio of dereddened FUV flux – UVW1 flux ($vF(v)_{FUV}
\sim 0.8vF(v)_{UVW1}$), and the ratio of (deredenned but still including host galaxy) V band flux — UVW1 flux ($vF(v)_{Vband} \sim
1.1vF(v)_{UVW1}$). From our total UVW1 band flux we can therefore calculate the total FUV and total V band flux (right hand and left hand black points, Fig. \[fig8\]). We know from Table \[table2\] that the fraction of intrinsic BB disc flux in the V band is $f_d=0.35$, i.e. $F_d(v_{V band})=f_dF_{V band}$, which requires $r_{out}=300$, in order to not over-predict the V band disc flux.
Using the total band fluxes and the fractional contributions in Table \[table2\] we can similarly constrain the UVW1 and V band flux levels of the $6$d lagged component (magenta squares, Fig.\[fig8\]). We then construct a blackbody spectrum that can supply these flux levels, shown in Fig.\[fig8\] by the magenta line. A blackbody spectral component has two parameters: its temperature and its area. The temperature defines the frequency of the blackbody peak, while the temperature and emitting area together determine the luminosity of the blackbody. We adjust the temperature until the peak is placed such that we can match the ratio of that component’s UVW1 to V band flux. We then adjust the area of the blackbody reprocessor to give the blackbody the appropriate luminosity.
The grey crosses in Fig.\[fig8\] show the flux level of the component which varies as the observed FUV lightcurve (the flux level in the FUV band is found by subtracting the flux contributions of the other components from the total FUV flux). The spectral shape of this component is qualitatively similar to the Comptonised disc component shown in Fig.\[fig4\]a. Although the un-lagged component dominates in the FUV band, there is some contamination from the lagged blackbodies. We recalculate the FUV lightcurve including this contamination to check that it does not affect the CCF peak lags and find its effect is negligible.
Separating the variability of the source into its different components and constraining the contribution of these components to different bands, has allowed us to constrain the spectrum of the reprocessor. Assuming this reprocessor is a blackbody emitter, allowed us to constrain its temperature ($T=9600$K) and emitting area ($A=3.04\times 10^{30}$ cm$^2$ implying a luminosity $L_{BB}=1.37\times 10^{42}$ ergs s$^{-1}$). The radial location of the reprocessor is constrained by its lag time i.e. it must be located six light days from the central source, which is $3240\,R_g$ for NGC 5548 with $M=3.2\times10^7\,M_\odot$. Hence we can derive the covering factor, $f_{cov}$, of the reprocessor, since $f_{cov}=A/(2\pi R^2)=0.002$. This is tiny and further underlines why the reprocessor cannot be a disc, which has a huge area at these radii.
A possible source of small area reprocessors at large radii could be dense clumps originating in a dust-driven disc wind just like the broad line emitting clouds (Czerny et al. 2015), which are too dense to emit broad lines so instead reprocess the illuminating flux as thermal emission (see also Lawrence 2012). Knowing the covering fraction, it is clear that this material only intercepts 0.2% of the illuminating flux. This is far too small to produce the observed luminosity required for the lagged component, given the observed bolometric luminosity of the source ($\sim2\times10^{44}$; Mehdipour et al. 2015).
Hence we can rule out the lag originating from the reprocessing of irradiating flux, where the reprocessing makes blackbody radiation and the lag is from the light travel time. Such a model fails on energetic grounds. A blackbody reprocessor at large radii ($\sim2000-3000\,R_g$, as required by the lag times) must have a small area (ie. covering fraction), if it is to emit at a high enough temperature to contribute significant flux to UVW1 and V band, which means it cannot intercept sufficient illuminating flux to reprocess and heat it to this required temperature.
It seems more likely that the lag is not simply from light travel time delays, but is instead lengthened by some response of the disc structure to the changing illuminating flux.
An Alternative Explanation for the UV/Optical Lags
--------------------------------------------------
So far we have assumed the FUV regions of the puffed-up, optically thick, Comptonised disc region illuminate some separate reprocessor — either a standard BB disc or optically thick BLR clouds. Having shown that neither reprocessor can produce the observed lags, we suggest that perhaps light travel time lags from an illuminating source to an external reprocessor are not involved at all; perhaps the lags instead represent the lag time for the BB disc vertical structure to respond to changes in the FUV illumination.
In this scenario, the hard X-rays heat the inner (soft X-ray emitting) edge of the puffed-up Comptonised disc and this causes a heating wave, which dissipates outwards. From examination of the observed lightcurves (Fig.\[fig3\]) we know this heating wave must quickly lose the high frequency power of the hard X-rays, and it must also include some intrinsic fluctuations produced within the Comptonised disc itself. We speculate that an increase in the hard X-ray flux produces a stronger heating wave, which dissipates outwards through the Comptonised disc. When this heating wave reaches the outer edge of the Comptonised disc, this increases the FUV illumination of the surrounding BB disc. This illumination is concentrated on the innermost BB disc radii adjacent to the Comptonised disc. These BB disc radii respond to the increase in illumination by expanding upwards, becoming less dense and less able to thermalise, so they may switch from emitting BB radiation to emitting via optically thick Compton — the Comptonised disc region has essentially expanded outwards. When the hard X-ray flux decreases, there is less heating of the Comptonised disc region and perhaps its outer radii can then cool and return to emitting BB. The Comptonised disc region is effectively breathing in and out in response to the X-ray heating of its inner edge. We suggest this expanding and contracting of the puffed-up Comptonised disc region, ie. this movement of the transition radius between Comptonised disc and BB disc, is then the cause of the interband UV–optical lags. The lag times should therefore reflect the response time of the disc vertical structure to changing irradiation.
The fastest response timescale of the disc is the dynamical timescale. This sets both the orbital timescale, and the timescale on which the vertical structure responds to loss of hydrostatic equilibrium. The latter seems more appropriate for the physical mechanism we envisage, but this is 32 days for a mass of $3.2\times 10^7M_\odot$ at $200\,R_g$. It is only as short as 6 days for $70R_g$. Hameury et al. (2009) calculate the effect of the disc instability for an AGN, but their illumination geometry is for a central source rather than the larger scale height FUV illumination envisaged here. More detailed simulations are required to see if such a mechanism is feasible and if so, how to determine the radius more precisely from the timescale.
Conclusions
===========
We have built a full disc reprocessing model in an attempt to simulate the simultaneous multi-wavelength lightcurve data presented by Edelson et al. (2015), assuming the UV/optical variability seen in NGC 5548 is due to reprocessing of higher energy radiation by a BB accretion disc. This higher energy radiation is traditionally assumed to be the hard X-ray power law, produced at small accretion flow radii. We find that reprocessing of the hard X-rays by a standard BB accretion disc cannot replicate the observed UV/optical lightcurves or their lags with respect to the illuminating X-rays. Specifically the simulated lightcurves reproduce too much of the hard X-ray high frequency power and the light travel lag times are too short.
One obvious answer to increase the amount of smoothing and increase the light travel lag times would appear to be to make the BB accretion disc larger. The first reason this is not an option is a constraint from the spectral energy distribution of NGC 5548. We have limited the size of our model accretion disc to the self-gravity radius. We could relax this condition and allow the BB disc to extend to larger radii, but this would cause the disc spectrum to extend to lower energies. By contrast, the observed spectrum of NGC 5548, as shown by Mehdipour et al. (2015), clearly does not show such emission. The peak UV/optical emission occurs in the UVW1 bandpass and the emission below this energy rapidly drops off. With an outer disc radius at the self-gravity radius, we already slightly overpredict the lowest energy fluxes. However, we modelled this directly, and find that the radii required to smooth the reprocessed hard X-ray lightcurve to a level matching the data is so large that the resulting lag times are too long to be compatible with the observations.
We are then driven to a scenario where the reprocessor cannot directly see the hard X-rays. This requires some source of material with sufficient scale-height that it can block the optical emitting regions’ view of the hard X-rays. The obvious candidate for this is the extra component required to fit the soft X-ray excess.
There is much debate over the physical origin of this component. We envisage a scenario where it is produced in an optically thick Comptonising region at larger radii than the central hard X-rays. UV/FUV emission produced in this region lifts material out of the plane of the accretion flow, where it is illuminated by the hard X-ray flux, which overionises the material. The resultant loss of UV opacity means it falls back down, resulting in a region with scale-height sufficient to prevent the hard X-rays illuminating the outer BB disc. The soft X-ray emission could come from the inner radii of this Comptonised disc region, while the lowest energy FUV emission could come from its largest radii.
One problem with complete shielding is that the observed FUV and hard X-ray lightcurves [*are*]{} significantly correlated, though the correlation is quite poor. One possibility to incorporate some feedback between the two regions is if hard X-rays illuminate the inner edge of the Comptonised disc region and these fluctuations then dissipate outwards through the Comptonised disc until they reach the outer FUV emitting regions, which then illuminate the surrounding BB disc. It is then dissipation of the hard X-ray fluctuations through the Comptonised disc component that causes the loss of high frequency power, not light travel time smoothing. This dissipation process has to be extremely fast, however the viscous timescale in the Comptonised section of the disc should be faster than in a standard Shakura-Sunyaev BB disc.
We show that a model where the FUV emission (represented by the Hubble band lightcurve) provides the illumination gives a much better match to the shape of the observed optical lightcurves. However, our model response is still too fast at the longest wavelengths. The observed V band lag in NGC 5548 is $\sim2$d behind the FUV emission, which requires the reprocessed V band flux to be emitted at radii $R>1000\,R_g$ for this source. However BB disc annuli at these large radii are too cool to contribute significant flux to the V band, which is instead dominated by hotter emission from smaller BB disc radii with shorter lag times. The heating effect of the illuminating flux makes very little difference, as large disc annuli have enormous area. The illuminating flux is simply spread over too large an area, so barely changes the temperature of the annulus and certainly cannot heat the annulus enough for it to contribute significant V band flux. In order for the illuminating flux to heat a reprocessor at these large radii enough to contribute to the V band requires the reprocessor to have a small area.
We use the UV/optical lightcurves and cross correlation functions to constrain the amount of reprocessed flux with different lag times in the UVW1 and V bands and find a combination of unlagged FUV lightcurve and FUV lightcurve lagged by 6d can fit both bands. Assuming this 6d-lag reprocessed flux is blackbody emission, we then estimated the temperature and covering factor of the blackbody reprocessor and find $T=9600$K and $f_{cov}=0.002$. This tiny covering factor rules out the BB disc as the source of the reprocessed emission. We consider the possibility that this emission may arise from optically thick BLR clouds that are too dense to emit via line emission so instead reprocess the FUV flux as thermal blackbody emission. However this scenario is ruled out on energetic grounds — the inferred covering factor is too small for the reprocessor to intercept sufficient illuminating flux to heat it to the required temperature.
We conclude that the UV/optical lightcurves of NGC 5548 are not consistent with reprocessing of the hard X-rays by a BB accretion disc, but can instead be explained by reprocessing of the FUV emission where the lag is not a light travel time. We propose the continuum lags of NGC 5548 are entirely generated by the ‘puffed-up’ Comptonised disc region of the accretion flow: the inner (soft X-ray emitting) edge of this region is heated by the hard X-rays, producing heating waves which dissipate outwards. The outer (UV/optical emitting) edge of the Comptonised disc then expands and contracts, in both radius and height, in response to the passage of these heating waves and it is this behaviour which produces the continuum lags. The model still requires the presence of a standard outer Shakura-Sunyaev BB accretion disc, but this is a mostly constant component.
Ultimately, whatever the origin of the lags in NGC 5548, the datasets now available contain much more information than is encapsulated in a single lag time measurement. We urge full spectral–timing modelling of these data in order to extract all the new physical information on the structure and geometry of the accretion flow which is now within reach.
Acknowledgements
================
EG and CD acknowledge funding from the UK STFC grants ST/I001573/1 and ST/L00075X/1. CD acknowledges Missagh Mehdipour for multiple very helpful conversations and data, and Hagai Netzer for discussions about thermalisation. We thank the referee for their detailed report which made us assess the energetics of our original reverse engineered model with small clouds, and hence change the conclusions of our paper.
[99]{}
Ar[é]{}valo, P., Uttley, P., Kaspi, S., et al. 2008, , 389, 1479
Bentz, M. C., Walsh, J. L., Barth, A. J., et al. 2010, , 716, 993
Boissay, R., Paltani, S., Ponti, G., et al. 2014, , 567, A44
Cackett, E. M., Horne, K., & Winkler, H. 2007, , 380, 669
Capellupo, D. M., Netzer, H., Lira, P., Trakhtenbrot, B., & Mej[í]{}a-Restrepo, J. 2015, , 446, 3427
Coleman, M. S. B., Kotko, I., Blaes, O., Lasota, J.-P., & Hirose, S. 2016, , 462, 3710
Crummy, J., Fabian, A. C., Gallo, L., & Ross, R. R. 2006, , 365, 1067
Cunningham C., 1976, ApJ, 208, 534
Czerny B., et al., 2015, AdSpR, 55, 1806
Davis, S. W., Woo, J.-H., & Blaes, O. M. 2007, , 668, 682
Denney, K. D., Peterson, B. M., Pogge, R. W., et al. 2010, , 721, 715
Done C., Davis S. W., Jin C., Blaes O., Ward M., 2012, MNRAS, 420, 1848
Edelson R., et al., 2015, ApJ, 806, 129
Fausnaugh, M. M., Denney, K. D., Barth, A. J., et al. 2015, arXiv:1510.05648
Gaskell, C. M., & Peterson, B. M. 1987, , 65, 1
Gierli[ń]{}ski, M., & Done, C. 2004, , 349, L7
Haardt, F., & Maraschi, L. 1991, , 380, L51
Haardt, F., & Maraschi, L. 1993, , 413, 507
Hameury, J.-M., Viallet, M., & Lasota, J.-P. 2009, , 496, 413
Heil, L. M., Uttley, P., & Klein-Wolt, M. 2015, , 448, 3348
Ingram, A., Done, C., & Fragile, P. C. 2009, , 397, L101
Ingram, A., & Done, C. 2012, , 419, 2369
Jiang, Y.-F., Davis, S. W., & Stone, J. M. 2016, , 827, 10
Jin C., Ward M., Done C., 2012, MNRAS, 425, 907
Korista, K. T., & Goad, M. R. 2001, , 553, 695
Kotov, O., Churazov, E., & Gilfanov, M. 2001, , 327, 799
Laor A., Netzer H., 1989, MNRAS, 238, 897
Lasota, J.-P. 2001, , 45, 449
Lawrence, A. 2012, , 423, 451
Malzac, J., Dumont, A. M., & Mouchet, M. 2005, , 430, 761
Matt, G., Marinucci, A., Guainazzi, M., et al. 2014, , 439, 3016
McHardy I. M., et al., 2014, MNRAS, 444, 1469
McHardy, I., Connolly, S., Peterson, B., et al. 2016, arXiv:1601.00215
Mehdipour M., et al., 2015, A&A, 575, A22
Morgan, C. W., Kochanek, C. S., Morgan, N. D., & Falco, E. E. 2010, , 712, 1129
Narayan, R., & Yi, I. 1995, , 452, 710
Noda, H., Makishima, K., Uehara, Y., Yamada, S., & Nakazawa, K. 2011, , 63, 449
Noda, H. 2016, X-ray Studies of the Central Engine in Active Galactic Nuclei with Suzaku, Springer Theses. ISBN 978-981-287-720-8. Springer Science+Business Media Singapore, 2016.,
Novikov I. D., Thorne K. S., 1973, blho.conf, 343
Pancoast, A., Brewer, B. J., Treu, T., et al. 2014, , 445, 3073
Pancoast, A., Brewer, B. J., Treu, T., et al. 2015, , 448, 3070
Petrucci, P. O., Ferreira, J., Henri, G., Malzac, J., & Foellmi, C. 2010, , 522, A38
Petrucci, P.-O., Paltani, S., Malzac, J., et al. 2013, , 549, A73
Porquet, D., Reeves, J. N., O’Brien, P., & Brinkmann, W. 2004, , 422, 85
Sergeev, S. G., Doroshenko, V. T., Golubinskiy, Y. V., Merkulova, N. I., & Sergeeva, E. A. 2005, , 622, 129
Stern, B. E., Poutanen, J., Svensson, R., Sikora, M., & Begelman, M. C. 1995, , 449, L13
Svensson, R., & Zdziarski, A. A. 1994, , 436, 599
Ursini, F., Boissay, R., Petrucci, P.-O., et al. 2015, , 577, A38
Welsh W. F., Horne K., 1991, ApJ, 379, 586 Zycki, P. T., Done, C., & Smith, D. A. 1999, , 305, 231
\[lastpage\]
|
---
abstract: |
For $\sigma \in S_n$, let $D(\sigma) = \{ i : \sigma_{i} > \sigma_{i+1}\}$ denote the descent set of $\sigma$. The length of the permutation is the number of inversions, denoted by $inv(\sigma) = \big | \{ (i,j) : i<j,
\sigma_i > \sigma_j\} \big |$. Define an unusual quadratic statisitic by $baj(\sigma) = \sum_{i \in D(\sigma)} i ( n-i)$. We present here a bijective proof of the identity $\sum_{{\sigma \in S_n}
\atop {\sigma(n) = k}} q^{baj(\sigma) -
inv(\sigma)} = \prod_{i=1}^{n-1} {{1-q^{i (n-i)}} \over {1-q^i}}$ where $k$ is a fixed integer.
address: 'Mathematics and Statistics, TEL Building, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada'
author:
- Mike Zabrocki
title: A bijective proof of an unusual symmetric group generating function
---
The following identity was presented as a special case of a Weyl group generating function in a seminar talk at UCSD in November 1996 by John Stembridge. We present here a bijective proof of the identity.
Define a statistic on the permutations on $n$ letters $$\begin{aligned}
baj(\sigma) = \sum_{i=1}^{n-1} i (n-i) \chi ( \sigma_{i+1}
< \sigma_{i} ) \nonumber\end{aligned}$$ where $\chi$ is an indicator function
$$\begin{aligned}
\nonumber
\chi(A) = \Bigg\{ {{1 \sp if \sp A \sp true} \atop
{0 \sp if \sp A \sp false}} \end{aligned}$$
The number of inversions of a permutation may be expressed as $$\begin{aligned}
\nonumber
inv(\sigma) = \sum_{i=1}^{n-1} \sum_{j>i} \chi( \sigma_j < \sigma_i )\end{aligned}$$
A special case of the main result presented in [@SW] when the root system is $A_{n-1}$ is the following
$$\nonumber
\sum_{\sigma \in S_n} q^{baj(\sigma) - inv(\sigma)} =
n \prod_{i=1}^{n-1} {{1 - q^{i ( n-i )}} \over {1 - q^i}}$$
A slightly stronger statement can be made for the specialization of this formula to this root system. For a fixed $k \in \{ 1 \ldots n \}$ the following equation also holds
$$\nonumber
\sum_{{\sigma \in S_n} \atop {\sigma_n = k}} q^{baj(\sigma) - inv(\sigma)} =
\prod_{i=1}^{n-1} {{1 - q^{i ( n-i )}} \over {1 - q^i}}$$
where $k$ is an integer between $1$ and $n$.
Notice that the right hand side of this equation can be expressed as a product of sums by the formula
$$\begin{aligned}
\nonumber
\prod_{i=1}^{n-1} {{1 - q^{i ( n-i )}} \over {1 - q^i}}
&=& \prod_{i=1}^{n-1} {{1 - q^{i ( n-i )}} \over {1 - q^{n-i}}}
= \prod_{i=1}^{n-1} \sum_{r_i = 0}^{i-1} q^{(n-i) r_i} \\
&=& \sum_{{{(r_1, r_2, \ldots, r_{n-1})} \atop {0 \leq r_i < i}}}
q^{\sum_{i=1}^{n-1} (n-i) r_i} \nonumber\end{aligned}$$
The object of this proof will be to find a bijection from the permutations, $\sigma$, of $\{1 \ldots n \}$ with $\sigma_n = k$ ($k$ fixed) to sequences of integers $(r_1, r_2, \ldots r_{n-1})$ with the additional property that $$\begin{aligned}
\nonumber
baj(\sigma) - inv(\sigma) = \sum_{i=1}^{n-1} (n-i) r_i\end{aligned}$$
Let $\sigma$ be a permutation of $\{1 \ldots n\}$. $\sigma$ can be represented by a sequence of integers $(v_1, v_2, \ldots, v_n)$ where $v_i$ is $1 \leq v_i \leq i$ and is given by $v_i =
\sum_{j \leq i} \chi( \sigma_j \leq \sigma_i ) $.
Given such a sequence of $v_i$, it is possible to recover the permutation that it corresponds to by first constructing $\sigma' \in S_{n-1}$ for the sequence $(v_1, v_2, \ldots, v_{n-1})$ and then defining the permutation $\sigma \in S_n$ by $\sigma_n = v_n$, $\sigma_i = \sigma_i'$ if $\sigma_i' < v_n$, and $\sigma_i = \sigma_i' +1$ if $\sigma_i \geq v_n$. This construction gives that the number of $i \in \{ 1 \ldots n \}$ such that $\sigma_i \leq \sigma_n$ is $v_n$. This quantity does not change by building a larger permutation in the same manner.
Say $v = (1,1,3,1,2,5,1)$\
$\bspp v_1 = 1 \bsp \sigma^{(1)} = 1$\
$\bspp v_2 = 1 \bsp \sigma^{(2)} = 21$\
$\bspp v_3 = 3 \bsp \sigma^{(3)} = 213$\
$\bspp v_4 = 1 \bsp \sigma^{(4)} = 3241$\
$\bspp v_5 = 2 \bsp \sigma^{(5)} = 43512$\
$\bspp v_6 = 5 \bsp \sigma^{(6)} = 436125$\
$\bspp v_7 = 1 \bsp \sigma^{(7)} = 5472361 = \sigma$
Notice that $\sigma_{i+1} < \sigma_i$ if and only if $v_{i+1} \leq
v_i$. This is because if $v_{i+1} \leq v_i$ then $\sigma^{(i+1)}$ will have a descent in the $i^{th}$ position, and this descent will remain for all $\sigma^{(k)}$ with $k>i$ (and in particular $\sigma^{(n)} = \sigma$).
Define the bijection from permutations $\sigma$ with $\sigma_n = k$ by first computing the sequence of $v_i$ and then setting $r_i = i \chi(v_{i+1} \leq v_i) + v_{i+1} - v_i -1$.
This defines a map from such permutations to sequences of integers $(r_1, r_2, \ldots, r_{n-1})$ with $0 \leq r_i < i$. Note that if $v_i \geq v_{i+1}$ then $i-1 \geq v_i - v_{i+1} \geq 0$ so that $0 \leq r_i = i-1 -(v_i - v_{i+1}) \leq i-1$. If $v_i < v_{i+1}$ then $0 < v_{i+1} - v_i \leq i$, hence $0 \leq r_i = v_{i+1} - v_i - 1 \leq i-1$.
Given a sequence $(r_1, r_2, \ldots, r_{n-1})$ that is the image of some permutation and assume that the values of $v_{i+1}, \ldots, v_{n}$ are known, then $v_i - i \chi(v_{i+1} \leq v_i) =
v_{i+1} - r_i + 1$. If the right hand side of the equation less than or equal to $0$ then it must be that $\chi(v_{i+1} \leq v_i) =1$ and so $v_i = i + v_{i+1} - r_i + 1$. Otherwise $\chi(v_{i+1} \leq v_i) =0$ and then $v_i = v_{i+1} - r_i + 1$. The whole sequence of $v_i$ can be recovered, and thus, the original permutation also.
This map is 1-1 since it is possible to recover the permutation if the sequence $(r_1, r_2, \ldots, r_{n-1})$ is given and the value of $v_n = k$ is known. There are the same number of permutations with the last element fixed as there are such sequences of numbers, hence this map is a bijection.
It remains to show the result $$\begin{aligned}
\nonumber
baj(\sigma) - inv(\sigma) = \sum_{i=1}^{n-1} (n-i) r_i\end{aligned}$$
Note that $v_i$ can be expressed by the formula $v_i = \sum_{j \leq i} \chi( \sigma_j \leq \sigma_i )$. Because $\big({{n+1} \atop {2}} \big) =
\sum_{i=1}^{n} \sum_{j \leq i} (\chi( \sigma_j \leq \sigma_i ) +
\chi(\sigma_j > \sigma_i) ) = \sum_{i=1}^{n} v_i + inv(\sigma)$, the number of inversions of the permutation $\sigma$ is given by the formula $$\begin{aligned}
\nonumber
inv(\sigma) = \bigg({{n+1} \atop {2}} \bigg) - \sum_{i=1}^n v_i\end{aligned}$$
The statistic $baj$ can be given in terms of the $v_i$’s because of the remark that $v_{i+1} \leq v_i$ if and only if $\sigma_{i+1} < \sigma_i$. $$\begin{aligned}
\nonumber
baj( \sigma) = \sum_{i=1}^{n-1} i ( n-i) \chi(v_{i+1} \leq v_i)\end{aligned}$$
Thus, $$\begin{aligned}
\nonumber
\sum_{i=1}^{n-1} r_i(n-i) &=& \sum_{i=1}^{n-1} (i \chi(
v_{i+1}
\leq v_i) + v_{i+1} - v_i -1)(n-i) \nonumber \\
&=& \sum_{i=1}^{n-1} i(n-i) \chi( v_{i+1} \leq v_i) + \sum_{i=1}^{n-1}
v_{i+1} (n-i) - \sum_{i=1}^{n-1} (v_i+1)(n-i)
\nonumber \\
&=& \sum_{i=1}^{n-1} i(n-i) \chi( v_{i+1} \leq v_i) +
\sum_{i=2}^{n} v_{i} (n-i+1) - \sum_{i=1}^{n-1} v_i (n-i) -
\sum_{i=1}^{n-1} i \nonumber \\
&=& \sum_{i=1}^{n-1} i(n-i) \chi( v_{i+1}
\leq v_i) + v_n +
\sum_{i=2}^{n-1} v_{i} - v_1 (n-1) - \bigg({{n} \atop {2}} \bigg)
\nonumber \\
&=& \sum_{i=1}^{n-1} i(n-i) \chi( v_{i+1} \leq v_i) +
\sum_{i=1}^{n} v_{i} - v_1 n - \bigg({{n} \atop {2}} \bigg)
\nonumber \\
&=& \sum_{i=1}^{n-1} i ( n-i) \chi(v_{i+1} \leq v_i) +
\sum_{i=1}^n v_i - \bigg({{n+1} \atop {2}} \bigg)
\nonumber \\
&=& \, baj(\sigma) - inv(\sigma) \nonumber\end{aligned}$$
This shows the last property of the bijection and hence the theorem.
[9]{}
N. Iwahori and H. Matsumato, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, [*Inst. Hautes Étudies Sci. Publ. Math.*]{} [**25**]{} (1965) 5–48.
J. Stembridge and D. Waugh, A Weyl group generating function that ought to be better known, [*Indag. Math.*]{} 9 (1998), 451–457.
|
---
abstract: 'In many physical systems, inputs related by intrinsic system symmetries are mapped to the same output. When inverting such systems, i.e., solving the associated inverse problems, there is no unique solution. This causes fundamental difficulties for deploying the emerging end-to-end deep learning approach. Using the generalized phase retrieval problem as an illustrative example, we show that careful symmetry breaking on the training data can help get rid of the difficulties and significantly improve the learning performance. We also extract and highlight the underlying mathematical principle of the proposed solution, which is directly applicable to other inverse problems.'
author:
- 'Kshitij Tayal[^1]'
- 'Chieh-Hsin Lai[^2]'
- 'Vipin Kumar[^3]'
- 'Ju Sun[^4]'
bibliography:
- 'DL4INV.bib'
title: 'Inverse Problems, Deep Learning, and Symmetry Breaking'
---
[^1]: Department of Computer Science and Engineering, University of Minnesota, Twin Cities. Email: [`tayal007@umn.edu`](mailto:tayal007@umn.edu)
[^2]: School of Mathematics, University of Minnesota, Twin Cities. Email: [`laixx313@umn.edu`](mailto:laixx313@umn.edu)
[^3]: Department of Computer Science and Engineering, University of Minnesota, Twin Cities. Email: [`kumar001@umn.edu`](mailto:kumar001@umn.edu)
[^4]: Department of Computer Science and Engineering, University of Minnesota, Twin Cities. Email: [`jusun@umn.edu`](mailto:jusun@umn.edu)
|
---
author:
- |
Marcus Sperling, Dominik Stöckinger, Alexander Voigt\
[*Institut für Kern- und Teilchenphysik, TU Dresden, Dresden, Germany*]{}
title: '**Renormalization of vacuum expectation values in spontaneously broken gauge theories: Two-loop results**'
---
|
---
abstract: 'General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$, that is the Galois group of the second Hilbert $p$-class field $\mathrm{F}_p^2(K)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $\mathcal{G}(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group $\mathrm{Cl}_p(K)$, the position of $G$ is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second $p$-class groups $G$ for various series of number fields $K$ having $p$-class groups $\mathrm{Cl}_p(K)$ of fixed type and $p\in\lbrace 2,3,5,7\rbrace$.'
address: |
Naglergasse 53\
8010 Graz\
Austria
author:
- 'Daniel C. Mayer'
date: 'December 25, 2012'
title: |
The distribution of second $p$-class groups\
on coclass graphs
---
[^1]
Introduction {#s:Intro}
============
Let $p$ denote a prime and let $K$ be an algebraic number field. By the Hilbert $p$-class field $\mathrm{F}_p^1(K)$ of $K$ we understand the maximal abelian unramified $p$-extension of $K$. The Hilbert $p$-class field tower, briefly $p$-*tower*, of $K$, $\mathrm{F}_p^0(K)\le\mathrm{F}_p^1(K)\le\mathrm{F}_p^2(K)\le\ldots$, is defined recursively by $\mathrm{F}_p^0(K)=K$ and $\mathrm{F}_p^n(K)=\mathrm{F}_p^1\left(\mathrm{F}_p^{n-1}(K)\right)$, for $n\ge 1$. According to the uniqueness theorem of class field theory, all members of the $p$-tower are Galois extensions of $K$, and their union $\mathrm{F}_p^\infty(K)=\cup_{n=0}^\infty\mathrm{F}_p^n(K)$ is the maximal unramified pro-$p$ extension of $K$. Let $\mathrm{Cl}_p(K)$ be the $p$-class group of $K$, that is the Sylow $p$-subgroup of the class group $\mathrm{Cl}(K)$. If $\mathrm{Cl}_p(K)=1$ is trivial, then $\mathrm{F}_p^1(K)=K$ and the $p$-tower of $K$ has *length* $\ell_p(K)=0$. If $\mathrm{Cl}_p(K)\ne 1$ and $\mathrm{F}_p^{n-1}(K)<\mathrm{F}_p^n(K)=\mathrm{F}_p^{n+1}(K)$, for some $n\ge 1$, the $p$-tower is finite of length $\ell_p(K)=n$. Otherwise the $p$-tower of $K$ is infinite and $\mathrm{F}_p^\infty(K)$ is an infinite Galois extension of $K$, having a pro-$p$ group $\mathrm{G}_p^\infty(K)=\mathrm{Gal}(\mathrm{F}_p^\infty(K)\vert K)$ as Galois group, endowed with the Krull topology. For each $n\ge 1$, the finite quotient $\mathrm{G}_p^n(K)=\mathrm{Gal}(\mathrm{F}_p^n(K)\vert K)
=\mathrm{G}_p^\infty(K)/\left(\mathrm{G}_p^\infty(K)\right)^{(n)}$ of the $p$-*tower group* $\mathrm{G}_p^\infty(K)$ by the closed subgroup $\left(\mathrm{G}_p^\infty(K)\right)^{(n)}=\mathrm{Gal}(\mathrm{F}_p^\infty(K)\vert\mathrm{F}_p^n(K))$ is called the $n$th $p$-*class group* of $K$, in analogy to $\mathrm{G}_p^1(K)=\mathrm{Gal}(\mathrm{F}_p^1(K)\vert K)$, which is isomorphic to the (first) $p$-class group $\mathrm{Cl}_p(K)$ of $K$, by Artin’s reciprocity law. All these higher $p$-class groups $\mathrm{G}_p^n(K)$ of $K$ with $n\ge 2$ are usually non-abelian and share two essential common invariants, as the following theorem shows. The germs of these general concepts are contained in Artin’s famous papers [@Ar1; @Ar2].
\[thm:TTTandTKT\]
Suppose that $n\ge 2$ and $G=\mathrm{G}_p^n(K)$. For any subgroup $H\le G$ which contains the commutator subgroup $G^\prime$ of $G$, there exists a unique intermediate field $K\le L\le\mathrm{F}_p^1(K)$ such that $H=\mathrm{Gal}(\mathrm{F}_p^n(K)\vert L)$, and the following statements hold.
1. The abelianization $H/H^\prime$ is isomorphic to the $p$-class group $\mathrm{Cl}_p(L)$. In particular,\
$G/G^\prime\simeq\mathrm{Cl}_p(K)$ and $G^\prime/G^{\prime\prime}\simeq\mathrm{Cl}_p(\mathrm{F}_p^1(K))$.
2. The kernel $\ker(T_{G,H})$ of the transfer $T_{G,H}:G/G^\prime\to H/H^\prime$ is isomorphic to the $p$-principalization kernel of $K$ in $L$, that is, the kernel $\ker(j_{L\vert K})$ of the natural class extension homomorphism $j_{L\vert K}:\mathrm{Cl}_p(K)\to\mathrm{Cl}_p(L)$. In particular,\
$\ker(T_{G,G})\simeq\ker(j_{K\vert K})=1$ and $\ker(T_{G,G^\prime})=G/G^\prime\simeq\ker(j_{\mathrm{F}_p^1(K)\vert K})=\mathrm{Cl}_p(K)$.
Since $H$ contains $G^\prime$, $H$ is a normal subgroup of $G=\mathrm{Gal}(\mathrm{F}_p^n(K)\vert K)$. The intermediate field $K\le L\le\mathrm{F}_p^1(K)$ of degree $\lbrack L:K\rbrack=(G:H)$, such that $H=\mathrm{Gal}(\mathrm{F}_p^n(K)\vert L)$, is determined uniquely as the fixed field $\mathrm{Fix}(H)$ within $\mathrm{F}_p^n(K)$, by the Galois correspondence. From the viewpoint of class field theory, the norm class group $\mathrm{N}_{L\vert K}(\mathrm{Cl}_p(L))$ of $L\vert K$ is isomorphic to $H/G^\prime=\mathrm{Gal}(\mathrm{F}_p^n(K)\vert L)/\mathrm{Gal}(\mathrm{F}_p^n(K)\vert\mathrm{F}_p^1(K))
\simeq\mathrm{Gal}(\mathrm{F}_p^1(K)\vert L)$ and thus of index $\lbrack L:K\rbrack$ in $\mathrm{Cl}_p(K)\simeq\mathrm{Gal}(\mathrm{F}_p^1(K)\vert K)$, as required.
1. We have $H/H^\prime=\mathrm{Gal}(\mathrm{F}_p^n(K)\vert L)/\mathrm{Gal}(\mathrm{F}_p^n(K)\vert \mathrm{F}_p^1(L))
\simeq\mathrm{Gal}(\mathrm{F}_p^1(L)\vert L)\simeq\mathrm{Cl}_p(L)$, by the Galois correspondence and Artin’s reciprocity law [@Ar1].
2. The isomorphism $\ker(T_{G,H})\simeq\ker(j_{L\vert K})$ is a consequence of the commutativity of the diagram in Table \[tbl:ExtensionAndTransfer\], which was proved by Artin [@Ar2] and investigated in more detail by Miyake [@My]. The special case $\ker(T_{G,G^\prime})=G/G^\prime$ is the principal ideal theorem [@Fw].
------------------- -------------------- ------------------- -------------------- -------------------
$j_{L\vert K}$
$\mathrm{Cl}_p(K)$ $\longrightarrow$ $\mathrm{Cl}_p(L)$
Artin isomorphism $\updownarrow$ $///$ $\updownarrow$ Artin isomorphism
$G/G^\prime$ $\longrightarrow$ $H/H^\prime$
$T_{G,H}$
------------------- -------------------- ------------------- -------------------- -------------------
: Class extension homomorphism $j_{L\vert K}$ and transfer $T_{G,H}$[]{data-label="tbl:ExtensionAndTransfer"}
Since each finite quotient $G=\mathrm{G}_p^n(K)$ of the $p$-tower group $\mathrm{G}_p^\infty(K)$ of $K$, which is isomorphic to the inverse limit $\lim\limits_{\longleftarrow}{}_{n\ge 1}\,\mathrm{G}_p^n(K)$, behaves in the same manner with respect to the kernels $\ker(T_{G,H})$ and targets $H/H^\prime$ of the transfers $T_{G,H}:G/G^\prime\to H/H^\prime$, for $G^\prime\le H\le G$, we define two invariants $\tau(K)=\tau(G)$ and $\varkappa(K)=\varkappa(G)$ either of the entire $p$-tower of $K$ or of the individual $n$th $p$-class group $G$.
Let $p$ be a prime and $K$ be a number field.
1. The family $\tau(K)=(\mathrm{Cl}_p(L))_{K\le L\le\mathrm{F}_p^1(K)}$ of $p$-class groups of all intermediate fields $L$ between $K$ and $\mathrm{F}_p^1(K)$ is called *transfer target type*, briefly TTT, of the $p$-tower of $K$.
2. The family $\varkappa(K)=(\ker(j_{L\vert K}))_{K\le L\le\mathrm{F}_p^1(K)}$ of $p$-principalization kernels of $K$ in all intermediate fields $L$ between $K$ and $\mathrm{F}_p^1(K)$ is called *transfer kernel type*, briefly TKT, of the $p$-tower of $K$.
In general, third and higher $p$-class groups $\mathrm{G}_p^n(K)$ of $K$, $n\ge 3$, are non-metabelian with rather complex structure. For this reason, we focus our investigation on the *second* $p$-class group $G=\mathrm{G}_p^2(K)=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$ which is *metabelian* with commutator subgroup $$G^\prime=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert\mathrm{F}_p^1(K))\simeq\mathrm{Cl}_p(\mathrm{F}_p^1(K)).$$ It is the simplest group admitting the calculation of the TTT, $\tau(K)=(H/H^\prime)_{G^\prime\le H\le G}$, and the TKT, $\varkappa(K)=(\ker(T_{G,H}))_{G^\prime\le H\le G}$, of $K$ by means of the transfers from $G$ to the subgroups $H\le G$ containing $G^\prime$.
Identifying $\mathrm{G}_p^2(K)$ via $\varkappa(K)$ and $\tau(K)$ {#ss:IdSndPeClsGrp}
----------------------------------------------------------------
First, we illustrate that second $p$-class groups $\mathrm{G}_p^2(K)$ of number fields $K$ can frequently but not always be identified uniquely by means of TKT and TTT.
For $p=5$, we apply our recently calculated TKTs of §\[ss:StemPhi6\] to prove the following Theorem. It gives criteria for second $5$-class groups of number fields in terms of TKTs which were unknown up to now.
\[thm:Snd5ClsGrp\] Let $K$ be an arbitrary number field with $5$-class group $\mathrm{Cl}_5(K)$ of type $(5,5)$. In the following four cases, the second $5$-class group $\mathrm{G}_5^2(K)$ of $K$ is determined uniquely by the TKT and TTT of $K$.
1. $\varkappa(K)=(1,2,3,4,5,6)$ (identity), $\tau(K)=\left((5,5,5)^6\right)$ $\Longrightarrow$\
$\mathrm{G}_5^2(K)\simeq\langle 3125,14\rangle$.
2. $\varkappa(K)=(1,2,5,3,6,4)$ ($4$-cycle), $\tau(K)=\left((5,25)^4,(5,5,5)^2\right)$ $\Longrightarrow$\
$\mathrm{G}_5^2(K)\simeq\langle 3125,11\rangle$.
3. $\varkappa(K)=(5,1,2,6,4,3)$ ($6$-cycle), $\tau(K)=\left((5,25)^6\right)$ $\Longrightarrow$\
$\mathrm{G}_5^2(K)\simeq\langle 3125,12\rangle$.
4. $\varkappa(K)=(3,1,2,5,6,4)$ (two $3$-cycles), $\tau(K)=\left((5,25)^6\right)$ $\Longrightarrow$\
$\mathrm{G}_5^2(K)\simeq\langle 3125,9\rangle$.
In one case, there are two possibilities for $\mathrm{G}_5^2(K)$.\
$\varkappa(K)=(6,1,2,4,3,5)$ ($5$-cycle), $\tau(K)=\left((5,25)^5,(5,5,5)\right)$ $\Longrightarrow$\
either $\mathrm{G}_5^2(K)\simeq\langle 3125,8\rangle$ or $\mathrm{G}_5^2(K)\simeq\langle 3125,13\rangle$.
The powers in TTTs denote iteration and the $5$-groups are identified by their numbers in the SmallGroups library [@BEO].
In section §\[ss:StemPhi6\], we prove that the metabelianization $\mathrm{G}_5^2(K)=\mathrm{Gal}(\mathrm{F}_5^2(K)\vert K)=G/G^{\prime\prime}$ of the $5$-tower group $G=\mathrm{G}_5^\infty(K)$ of any algebraic number field $K$ with $5$-class group $\mathrm{Cl}_5(K)$ of type $(5,5)$, having one of the five pairs of TKT and TTT in Theorem \[thm:Snd5ClsGrp\], is one of the six terminal top vertices $\langle 3125,8\ldots 9\rangle$ and $\langle 3125,11\ldots 14\rangle$ of the coclass graph $\mathcal{G}(5,2)$ in Figure \[fig:Typ55Cocl2\].
We conjecture that the TKT alone suffices for the characterization of $\mathrm{G}_5^2(K)$ in Theorem \[thm:Snd5ClsGrp\].
\[exm:Snd5ClsGrp\] Discriminants $D$ with smallest absolute values of complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$ having one of the five pairs of TKT and TTT in Theorem \[thm:Snd5ClsGrp\] are given by $-89751$, $-37363$, $-11199$, $-17944$, $-12451$, in the same order. They were computed by means of MAGMA [@MAGMA]. According to section §\[sss:StatScnd5ClgpCocl2\], the vertices $\langle 3125,8\rangle$ and $\langle 3125,13\rangle$ of coclass graph $\mathcal{G}(5,2)$ in Figure \[fig:Typ55Cocl2\], corresponding to the last case of Theorem \[thm:Snd5ClsGrp\], are populated by $167$ occurrences $(17.4\%)$ of $959$ complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$ with $-2\,270\,831\le D<0$ and $\mathrm{Cl}_5(K)$ of type $(5,5)$. This shows that even the last case alone occurs with rather high density.
For $p=3$, we use four TKTs which occurred repeatedly in the literature [@SoTa; @HeSm; @BrGo]. These TKTs define infinite sequences, in fact periodic coclass families (§\[s:CoclGrph\]), of possible groups $\mathrm{G}_3^2(K)$, and neither Heider and Schmithals [@HeSm] nor Brink and Gold [@Br; @BrGo] have been aware that the TTT is able to identify a unique member of the sequences, as we proved in [@Ma1; @Ma3].
\[thm:SectionE\] Let $K$ be an arbitrary number field with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$. In the following four cases, the second $3$-class group $\mathrm{G}_3^2(K)$ of $K$ is either determined uniquely or up to the sign of the relational exponent $\gamma$ by the TKT and the parametrized TTT of $K$, for each integer $j\ge 2$.
1. $\varkappa(K)=(1,3,1,3)$, $\tau(K)=\left((3^j,3^{j+1}),(3,9)^2,(3,3,3)\right)$ $\Longrightarrow$\
$\mathrm{G}_3^2(K)\simeq G_0^{2j+2,2j+3}(1,-1,1,1)$.
2. $\varkappa(K)=(2,3,1,3)$, $\tau(K)=\left((3^j,3^{j+1}),(3,9)^2,(3,3,3)\right)$ $\Longrightarrow$\
$\mathrm{G}_3^2(K)\simeq G_0^{2j+2,2j+3}(0,-1,\pm 1,1)$.
3. $\varkappa(K)=(1,2,3,1)$, $\tau(K)=\left((3^j,3^{j+1}),(3,9)^3\right)$ $\Longrightarrow$\
$\mathrm{G}_3^2(K)\simeq G_0^{2j+2,2j+3}(1,0,-1,1)$.
4. $\varkappa(K)=(2,2,3,1)$, $\tau(K)=\left((3^j,3^{j+1}),(3,9)^3\right)$ $\Longrightarrow$\
$\mathrm{G}_3^2(K)\simeq G_0^{2j+2,2j+3}(0,0,\pm 1,1)$.
The groups in the first two cases are located on the coclass tree $\mathcal{T}(\langle 243,6\rangle)$ in Fig. \[fig:TreeQTyp33Cocl2\], the groups in the last two cases on the coclass tree $\mathcal{T}(\langle 243,8\rangle)$ in Fig. \[fig:TreeUTyp33Cocl2\]. $3$-groups of order $3^n$ and index $m$ of nilpotency are identified by their parametrized presentations given in the form $G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)$ in §\[sss:PrmPres2\].
We proved that the second derived quotient $\mathrm{G}_3^2(K)=G/G^{\prime\prime}$ of the $3$-tower group $G=\mathrm{G}_3^\infty(K)$ of any algebraic number field $K$ with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$, transfer kernel type E.6, $\varkappa(K)=(1,3,1,3)$, resp. E.14, $\varkappa(K)=(2,3,1,3)$ [@Ma2 Tbl. 6, p. 492], and parametrized transfer target type $\tau(K)=\left((3^j,3^{j+1}),(3,9)^2,(3,3,3)\right)$, $j\ge 2$, is isomorphic to the unique group $G_0^{2j+2,2j+3}(1,-1,1,1)$, resp. to one of the two groups $G_0^{2j+2,2j+3}(0,-1,\pm 1,1)$, of the coclass tree $\mathcal{T}(\langle 243,6\rangle)$ in Figure \[fig:TreeQTyp33Cocl2\] [@Ma3 Thm. 4.4, Tbl. 8].\
Similarly, we proved for transfer kernel type E.8, $\varkappa(K)=(1,2,3,1)$, resp. E.9, $\varkappa(K)=(2,2,3,1)$, and parametrized transfer target type $\tau(K)=\left((3^j,3^{j+1}),(3,9)^3\right)$, $j\ge 2$, that $\mathrm{G}_3^2(K)$ is isomorphic to the unique group $G_0^{2j+2,2j+3}(1,0,-1,1)$, resp. to one of the two groups $G_0^{2j+2,2j+3}(0,0,\pm 1,1)$, of the coclass tree $\mathcal{T}(\langle 243,8\rangle)$ in Figure \[fig:TreeUTyp33Cocl2\].
\[exm:SectionE\] By [@Ma1 Thm. 5.2, p. 492], the complex quadratic field $K=\mathbb{Q}(\sqrt{-9748})$ is a number field having the TKT and TTT of the last case in Theorem \[thm:SectionE\], actually with smallest absolute discriminant. It was first mentioned by Scholz and Taussky [@SoTa p. 25]. Among the $93$ complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$ with discriminants $-6\cdot 10^4<D<0$ and $\mathrm{Cl}_3(K)$ of type $(3,3)$, there are $11$ cases $(12\%)$ having the TKT and TTT of the last case in Theorem \[thm:SectionE\]. So even the last case alone occurs quite frequently.
In contrast, we can also prove that certain metabelian $p$-groups are excluded as second $p$-class groups for special base fields. The following negative result for $p=3$ gives an exact justification for a particular instance of our *weak leaf conjecture*, Cnj. \[cnj:WeakLeafCnj\].
\[thm:SpecWeakLeaf\] The $3$-group $\langle 243,4\rangle$, resp. $\langle 243,9\rangle$, cannot occur as second $3$-class group $\mathrm{G}_3^2(K)$ for a complex quadratic field $K=\mathbb{Q}(\sqrt{D})$, $D<0$, whose TKT and TTT are given by $\varkappa(K)=(4,4,4,3)$ and $\tau(K)=\left((3,9),(3,3,3)^3\right)$, resp. $\varkappa(K)=(2,1,4,3)$ and $\tau(K)=\left((3,9)^4\right)$.
The assumption that one of the groups $g\in\lbrace\langle 243,4\rangle,\langle 243,9\rangle\rbrace$ were the second $3$-class group $g=\mathrm{G}_3^2(K)$ of a complex quadratic field $K$ implies two contradictory consequences. On the one hand, both groups $g$ are of class $\mathrm{cl}(g)=3$, whence the fourth lower central $\gamma_4(g)=1$ is trivial. According to Heider and Schmithals [@HeSm p. 20], any number field $K$ whose second $p$-class group $g=\mathrm{G}_p^2(K)$ has trivial $\gamma_4(g)=1$ possesses a $p$-tower of length $\ell_p(K)=2$. On the other hand, since $K$ is complex quadratic, its $3$-tower group $G$ must be a Schur $\sigma$-group [@Sh], [@KoVe p. 58]. The Schur multiplier $\mathrm{H}_2(g,\mathbb{Z})$ of both groups $g$ is non-trivial of order $3$, as can be verified by means of GAP [@GAP]. Hence they cannot be Schur $\sigma$-groups [@BBH p. 6]. Therefore, the $3$-tower of $K$ cannot stop at the second stage, $G\ne\mathrm{G}_3^2(K)$, and $G$ must be a non-metabelian group of derived length at least $3$, that is, the $3$-tower of $K$ has length $\ell_3(K)\ge 3$.
Length of $p$-towers {#ss:TowerLength}
--------------------
As the following Theorems \[thm:3TowerLength2\]–\[thm:3TowerLengthAtLeast3\] show, the length $\ell_p(K)$ of the $p$-tower of $K$ can either be determined exactly or at least be estimated by a lower bound, once the second $p$-class group $\mathrm{G}_p^2(K)$ of $K$ and its properties are known in sufficient detail.
A criterion for $3$-towers of exact length $2$ was proved in three independent ways by Scholz and Taussky [@SoTa], by Heider and Schmithals [@HeSm], and by Brink and Gold [@Br; @BrGo]. With our new methods, we can give a short proof of this criterion.
\[thm:3TowerLength2\] Let $K$ be an arbitrary number field with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$. In the following two cases, the second $3$-class group $\mathrm{G}_3^2(K)$ of $K$ and the TTT of $K$ are determined uniquely by the TKT of $K$.
1. $\varkappa(K)=(2,2,4,1)$ $\Longrightarrow$ $\mathrm{G}_3^2(K)\simeq\langle 243,5\rangle$, $\tau(K)=\left((3,9)^3,(3,3,3)\right)$.
2. $\varkappa(K)=(4,2,2,4)$ $\Longrightarrow$ $\mathrm{G}_3^2(K)\simeq\langle 243,7\rangle$, $\tau(K)=\left((3,9)^2,(3,3,3)^2\right)$.
In both cases, the $3$-class field tower of $K$ has exact length $\ell_3(K)=2$.
The metabelianization $\mathrm{G}_3^2(K)\simeq G/G^{\prime\prime}$ of the $3$-tower group $G=\mathrm{G}_3^\infty(K)$ of any algebraic number field $K$ with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$ having transfer kernel type D.10, $\varkappa(K)=(2,2,4,1)$, resp. D.5, $\varkappa(K)=(4,2,2,4)$ [@Ma2 Tbl. 6, p. 492], is isomorphic to the terminal top vertex $\langle 243,5\rangle$, resp. $\langle 243,7\rangle$, of the sporadic part $\mathcal{G}_0(3,2)$ of the coclass graph $\mathcal{G}(3,2)$ in Figure \[fig:Typ33Cocl2\], according to Nebelung [@Ne1 Thm. 6.14, p. 208]. In [@Ma3 Thm. 4.2, Tbl. 4] it is shown that the corresponding transfer target type is given by $\tau(K)=\left((3,9)^3,(3,3,3)\right)$, resp. $\tau(K)=\left((3,9)^2,(3,3,3)^2\right)$.\
According to the proof of [@BBH Thm. 4.2, p. 14], $\langle 243,5\rangle$ and $\langle 243,7\rangle$ are Schur $\sigma$-groups.\
However, independently from $K$ being complex quadratic or not, when the second derived quotient $G/G^{\prime\prime}\simeq\mathrm{G}_3^2(K)$ of $G$ is a Schur $\sigma$-group, then the $3$-tower group $G$ of $K$ must be isomorphic to it, $G\simeq\mathrm{G}_3^2(K)$, by the argument given in [@BoEl Lem. 4.10]. Consequently, the $3$-tower of $K$ stops at the second stage and has length $\ell_3(K)=2$.
We point out that the figure in [@BBH p. 10] is not a coclass graph in our sense (§ \[ss:CoclGrph\]), since it contains vertices of four different coclass graphs $\mathcal{G}(3,r)$, $1\le r\le 4$, partially connected by edges of depth $2$. The top level of this figure, where $\langle 243,5\rangle$ and $\langle 243,7\rangle$ are emphasized by surrounding circles, coincides with the top vertices of our Figure \[fig:Typ33Cocl2\].
\[exm:3TowerLength2\] Discriminants $D$ with smallest absolute values of complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$ having one of the two TKTs in Theorem \[thm:3TowerLength2\] are given by $-4027$, $-12131$, in the same order. The first was communicated by Scholz and Taussky [@SoTa p. 22], the second by Heider and Schmithals [@HeSm p. 19]. Corresponding minimal discriminants of real quadratic fields are $422573$, $631769$ [@Ma1 Tbl. 4, p. 498]. Among the $2020$ complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$ with discriminants $-10^6<D<0$ and $\mathrm{Cl}_3(K)$ of type $(3,3)$, there are $936$ cases $(46.3\%)$ having one of the two pairs of TKT and TTT in Theorem \[thm:3TowerLength2\] [@Ma1 Tbl. 3, p. 497]. So these types of fields are definitely among the high-champs with respect to density of population.
In the next Theorem, the second $3$-class group $\mathrm{G}_3^2(K)$ is not at all determined by the TKT $\varkappa(K)$ alone. Furthermore, we must restrict ourselves to an estimate of the $3$-tower length $\ell_3(K)\ge 3$.
\[thm:3TowerLengthAtLeast3\] Let $K=\mathbb{Q}(\sqrt{D})$, $D<0$, be a complex quadratic field with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$. In the following two cases, the second $3$-class group $\mathrm{G}_3^2(K)$ of $K$ is determined uniquely by the TKT and the TTT of $K$.
1. $\varkappa(K)=(4,4,4,3)$, $\tau(K)=\left((3,9),(3,3,3)^3\right)$ $\Longrightarrow$ $\mathrm{G}_3^2(K)\simeq\langle 729,45\rangle$.
2. $\varkappa(K)=(2,1,4,3)$, $\tau(K)=\left((3,9)^4\right)$ $\Longrightarrow$ $\mathrm{G}_3^2(K)\simeq\langle 729,57\rangle$.
In both cases, $K$ has a $3$-class field tower of length $\ell_3(K)\ge 3$.
We proved that the second derived quotient $\mathrm{G}_3^2(K)=G/G^{\prime\prime}$ of the $3$-tower group $G=\mathrm{G}_3^\infty(K)$ of a complex quadratic field $K$ with $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$, transfer kernel type H.4, $\varkappa(K)=(4,4,4,3)$, resp. G.19, $\varkappa(K)=(2,1,4,3)$ [@Ma2 Tbl. 6, p. 492], and transfer target type $\tau(K)=\left((3,9),(3,3,3)^3\right)$, resp. $\tau(K)=\left((3,9)^4\right)$ [@Ma3 Thm. 4.3, Tbl. 6], is isomorphic to the unique vertex $\langle 729,45\rangle$, resp. $\langle 729,57\rangle$, of the sporadic part $\mathcal{G}_0(3,2)$ of coclass graph $\mathcal{G}(3,2)$ in Figure \[fig:Typ33Cocl2\]. For an arbitrary number field $K$, several other candidates for $\mathrm{G}_3^2(K)$ are possible. However, for a complex quadratic field $K$, $\langle 243,45\rangle$, resp. $\langle 243,57\rangle$, are discouraged by Theorem \[thm:SpecWeakLeaf\], and the siblings $\langle 729,44\rangle$ and $\langle 729,46\ldots 47\rangle$, resp. $\langle 729,56\rangle$, of $\langle 729,45\rangle$, resp. $\langle 729,57\rangle$, do not admit the mandatory automorphism of order $2$ acting as inversion on the abelianization.\
Since $K$ is complex quadratic, its $3$-tower group $G$ must be a Schur $\sigma$-group [@Sh], [@KoVe p. 58]. However, neither $\langle 729,45\rangle$ nor $\langle 729,57\rangle$ is a Schur $\sigma$-group [@BBH p. 6], because the Schur multiplier is non-trivial of type $(3,3)$, as can be verified with the aid of GAP [@GAP]. Therefore, the $3$-tower of $K$ cannot stop at the second stage, $G\ne\mathrm{G}_3^2(K)$, and $G$ must be a non-metabelian group of derived length at least $3$, that is, the $3$-tower has length $\ell_3(K)\ge 3$.
\[exm:3TowerLengthAtLeast3\] Discriminants $D$ with smallest absolute values of complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$ having one of the two pairs of TKT and TTT in Theorem \[thm:3TowerLengthAtLeast3\] are given by $-3896$, $-12067$, in the same order. They were communicated by Heider and Schmithals [@HeSm p. 19]. Among the $2020$ complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$ with discriminants $-10^6<D<0$ and $\mathrm{Cl}_3(K)$ of type $(3,3)$, there are $391$ cases $(19.4\%)$ having one of the two pairs of TKT and TTT in Theorem \[thm:3TowerLengthAtLeast3\] [@Ma1 Tbl. 3, p. 497]. This shows that fields with $3$-towers of at least three stages occur quite frequently.
Note that the proofs of the preceding Theorems \[thm:3TowerLength2\], and \[thm:3TowerLengthAtLeast3\] are very brief. This is the beginning of powerful new methods of research concerning the maximal unramified pro-$p$ extensions of number fields by joining the coclass theory of finite $p$-groups and suitable generalizations of Schur $\sigma$-groups [@BBH2]. We optimistically expect further prolific impact of these new foundations on the investigation of $p$-towers and $p$-principalization, although Artin called the capitulation problem hopeless.
Overview {#ss:Overview}
--------
In §§\[ss:ScndClgpCocl1\] and \[ss:ScndClgpTyp33CoclGe2\], we analyze number fields $K$ with $p$-class group $\mathrm{Cl}_p(K)$ of type $(p,p)$. Based on [@Ma1], we prove that the $p$-class numbers $\mathrm{h}_p(L_i)=\lvert\mathrm{Cl}_p(L_i)\rvert$ of *two distinguished* intermediate fields $L_i$, $1\le i\le 2$, lying strictly between $K$ and $\mathrm{F}_p^1(K)$, and the $p$-class number $\mathrm{h}_p(\mathrm{F}_p^1(K)$ of the Hilbert $p$-class field of $K$, that is, the orders of three special members of the TTT $\tau(K)$, contain sufficient information for determining the order $\lvert G\rvert=3^n$, class $c=\mathrm{cl}(G)$, coclass $r=\mathrm{cc}(G)$, and the so-called *defect of commutativity* $k=k(G)$ of the second $p$-class group $G=\mathrm{G}_p^2(K)$ of $K$. These invariants are related by the equation $n=\mathrm{cl}(G)+\mathrm{cc}(G)$ and restrict $G$ to the *finite* subset of groups of equal order $3^n$ of the coclass graph $\mathcal{G}(p,r)$. If the TKT $\varkappa(K)$ is known additionally, the position of $G$ can be restricted further, either to a branch $\mathcal{B}$ of a coclass tree $\mathcal{T}$, forming a subgraph of $\mathcal{G}(p,r)$, or even to a unique isomorphism type of metabelian $p$-groups.
Group theoretic foundations concerning coclass graphs and their mainlines, parametrized presentations, polarization, and defect are provided in preliminary sections §§\[s:CoclGrph\], \[ss:MtabCocl1\], and \[ss:MtabTyp33CoclGe2\].
In [@Ma3], it was shown for number fields $K$ of type $(p,p)$, that the abelian type invariants of the $p$-class groups $\mathrm{Cl}_p(L_i)$ of *all* intermediate fields $K<L_i<\mathrm{F}_p^1(K)$, $1\le i\le p+1$, that is, the structures of the *first layer* of the TTT $\tau(K)$, usually determine the TKT $\varkappa(K)$, at least in the case that $\mathrm{G}_p^2(K)$ is one of the most densely populated metabelian $p$-groups.
The density of population of a metabelian $p$-group $G$ by second $p$-class groups $\mathrm{G}_p^2(K)$ of certain base fields $K$ can be calculated explicitly from a purely group theoretic probability measure by non-abelian generalizations of the Cohen-Lenstra-Martinet asymptotic, as developed recently by Boston, Bush, Hajir [@BBH], and also by Bembom [@Bm], resp. Boy [@By], under supervision by Mihailescu, resp. Malle. The heuristic is in good accordance with our extensive computational results for quadratic base fields in [@Ma1]. These results have in fact actually been used in [@Bm pp. 5, 126]. Further, they eliminate all incomplete IPADs (index-$p$ abelianization data), which coincide with the first layer of our TTTs, and correct the frequencies given in [@BBH Tbl. 1–2, pp. 17–18], which are uniformly slightly too low.
It is to be expected that similar strategies, exploiting the interplay between TTT and TKT, but now extended to the *higher layers* of these invariants, can be used to identify the isomorphism type of the second $p$-class group $G=\mathrm{G}_p^2(K)$ of number fields $K$ with more complicated $p$-class group $\mathrm{Cl}_p(K)$, for example of type $(p^2,p)$ or $(p,p,p)$. Extensions in this direction will be presented in subsequent papers [@Ma4; @AZTM]. An outlook is given in section §\[s:DoubleLayer\].
Visualizing finite $p$-groups on coclass graphs {#s:CoclGrph}
===============================================
Periodic patterns {#ss:Periodicity}
-----------------
An important purpose of this paper is to emphasize that coclass graphs $\mathcal{G}(p,r)$ are particularly well suited for visualizing periodic properties [@dS; @EkLg] of infinite sequences of finite $p$-groups $G$, such as parametrized power-commutator presentations [@Bl1; @Ne1], automorphism groups $\mathrm{Aut}(G)$, Schur multipliers $\mathrm{H}_2(G,\mathbb{Z}_p)$ and other cohomology groups of $G$, transfer kernel types $\varkappa(G)$ [@Ma2], transfer target types $\tau(G)$ [@Ma3], and defect of commutativity $k(G)$ expressed by the depth $\mathrm{dp}(G)$ (Corollaries \[cor:DpthCocl1\] and \[cor:DpthCoclGe2\]). In number theoretic applications, selection rules for second $p$-class groups $G=\mathrm{G}_p^2(K)$ of special base fields $K$ [@Ma1] are additional periodic properties. Computational results on the density of distribution of second $p$-class groups can also be represented very clearly on coclass graphs.
Coclass graphs {#ss:CoclGrph}
--------------
For a given prime $p$, Leedham-Green and Newman [@LgNm] have defined the structure of a directed graph $\mathcal{G}(p)$ on the set of all isomorphism classes of finite $p$-groups. Two vertices are connected by a directed edge $H\to G$ if $G$ is isomorphic to the last lower central quotient $H/\gamma_c(H)$ of $H$, where $c$ denotes the nilpotency class $\mathrm{cl}(H)$ of $H$.\
If the condition $\lvert H\rvert=p\lvert G\rvert$ is imposed on the edges, $\mathcal{G}(p)$ is partitioned into countably many disjoint subgraphs $\mathcal{G}(p,r)$, $r\ge 0$, called *coclass graphs* of $p$-groups $G$ of coclass $r=\mathrm{cc}(G)=n-\mathrm{cl}(G)$, where $\lvert G\rvert=p^n$ [@LgMk p. 155, 166]. A coclass graph $\mathcal{G}(p,r)$ is a forest of finitely many coclass trees $\mathcal{T}_i$, each with a single infinite mainline having a pro-$p$ group of coclass $r$ as its inverse limit, and additionally contains a set $\mathcal{G}_0(p,r)$ of finitely many sporadic groups outside of coclass trees, $\mathcal{G}(p,r)=\left(\cup_i\,\mathcal{T}_i\right)\cup\mathcal{G}_0(p,r)$.
The terminology concerning the structure of coclass graphs $\mathcal{G}(p,r)$ with a prime $p\ge 2$ and an integer $r\ge 0$ must be recalled briefly. We adopt the most recent view of coclass graphs, which is given by Eick and Leedham-Green [@EkLg], and by Dietrich, Eick, Feichtenschlager [@DEF p. 46].
- The *coclass* $\mathrm{cc}(G)$ of a finite $p$-group $G$ of order $\lvert G\rvert=p^n$ and nilpotency class $\mathrm{cl}(G)$ is defined by $n=\mathrm{cl}(G)+\mathrm{cc}(G)$.
- By a *vertex* of the coclass graph $\mathcal{G}(p,r)$ we understand the isomorphism class of a finite $p$-group $G$ of coclass $\mathrm{cc}(G)=r$.
- The vertex $H$ is an *immediate descendant* of the vertex $G$, if $G$ is isomorphic to the last lower central quotient $H/\gamma_c(H)$ of $H$, where $c=\mathrm{cl}(H)$ denotes the nilpotency class of $H$, and $\gamma_c(H)$ is cyclic of order $p$, that is, $\mathrm{cl}(H)=1+\mathrm{cl}(G)$ and $\lvert H\rvert=p\lvert G\rvert$. In this case, $H$ and $G$ are connected by a *directed edge* $H\to G$ of the coclass graph and $G$ is called the *parent* $G=\pi(H)$ of $H$.
- A *capable vertex* has at least one immediate descendant, whereas a *terminal vertex* has no immediate descendants.
- The vertex $G_m$ is a *descendant* of the vertex $G_0$, if there is a *path* $(G_j\to G_{j-1})_{m\ge j\ge 1}$ of directed edges from $G_m$ to $G_0$. In particular, the vertex $G_0$ is descendant of itself, with empty path.
- The *tree* $\mathcal{T}(G)$ with root $G$ consists of all descendants of the vertex $G$.
- A *coclass tree* is a maximal rooted tree containing exactly one infinite path.
- The *mainline* $(M_{j+1}\to M_j)_{j\ge n}$ of a coclass tree $\mathcal{T}(M_n)$ with root $M_n$ of order $\lvert M_n\rvert=p^n$ is its unique maximal infinite path. The projective limit $S=\lim\limits_{\longleftarrow}{}_{j\ge n}\,M_j$, is an infinite pro-$p$ group, whose finite quotients by closed subgroups return the mainline vertices $M_j$.
- For $i\ge n$, the *branch* $\mathcal{B}(M_i)$ of a coclass tree $\mathcal{T}(M_n)$ with tree root $M_n$ and mainline $(M_{j+1}\to M_j)_{j\ge n}$ is the difference set $\mathcal{T}(M_i)\setminus\mathcal{T}(M_{i+1})$. The branch $\mathcal{B}(M_i)$ is briefly denoted by $\mathcal{B}_i$ and we assume that the order of the branch root $M_i$ is $\lvert M_i\rvert=p^i$.
- The *depth* $\mathrm{dp}(G)=m-j$ of a vertex $G$ of order $\lvert G\rvert=p^m$ on a branch $\mathcal{B}(M_j)$ of a coclass tree is its distance from the branch root $M_j$ of order $\lvert M_j\rvert=p^j$ on the mainline. For $d\ge 1$, $\mathcal{B}_d(M_j)$ denotes the *pruned branch of bounded depth* $d$ with root $M_j$.
- The *periodic sequence* $\mathcal{S}(G)$ of a vertex $G\in\mathcal{B}_d(M_i)$ of order $\lvert G\rvert=p^m$, $i\le m\le i+d$, on a coclass tree of $\mathcal{G}(p,r)$, where $M_i$ denotes the vertex of order $p^i$ on the mainline and $i$ is sufficiently large so that periodicity has set in already [@EkLg], is the infinite sequence $(G_{m+j\ell})_{j\ge 0}$ of vertices defined recursively by $G_m=G$ and $G_{m+j\ell}=\varphi_{i+(j-1)\ell}(G_{m+(j-1)\ell})$, for $j\ge 1$, using the periodicity isomorphisms of graphs $\varphi_{i+(j-1)\ell}:\mathcal{B}_d(M_{i+(j-1)\ell})\to\mathcal{B}_d(M_{i+j\ell})$ with period length $\ell$, which is a divisor of $p^{r+1}(p-1)$.
$p$-Groups with single layered metabelianization of type $(p,p)$ {#s:SingleLayer}
================================================================
Metabelian $p$-groups $G$ of coclass $\mathrm{cc}(G)=1$ {#ss:MtabCocl1}
-------------------------------------------------------
For an arbitrary prime $p\ge 2$, let $G$ be a metabelian $p$-group of order $\lvert G\rvert=p^n$ and nilpotency class $\mathrm{cl}(G)=n-1$, where $n\ge 3$. In the terminology of Blackburn [@Bl1] and Miech [@Mi], $G$ is of maximal class, that is, of coclass $\mathrm{cc}(G)=1$, whence the commutator factor group $G/G^\prime$ of $G$ is of type $(p,p)$. The converse is only true for $p=2$: A $2$-group $G$ with $G/G^\prime\simeq(2,2)$ is of coclass $1$, a fact which is usually attributed to Taussky [@Ta1]. The lower central series of $G$ is defined recursively by $\gamma_1(G)=G$ and $\gamma_j(G)=\lbrack\gamma_{j-1}(G),G\rbrack$ for $j\ge 2$. Nilpotency of $G$ is expressed by $\gamma_{n-1}(G)>\gamma_n(G)=1$.
### Polarization and defect {#sss:PlrzDfct}
The *two-step centralizer* $\chi_2(G)
=\lbrace g\in G\mid\lbrack g,u\rbrack\in\gamma_4(G)\text{ for all }u\in\gamma_2(G)\rbrace$ of the two-step factor group $\gamma_2(G)/\gamma_4(G)$, which can also be defined by $$\chi_2(G)/\gamma_4(G)
=\mathrm{Centralizer}_{G/\gamma_4(G)}(\gamma_2(G)/\gamma_4(G))\,,$$ is the largest subgroup of $G$ such that $\lbrack\chi_2(G),\gamma_2(G)\rbrack\le\gamma_4(G)$. It is characteristic, contains the commutator subgroup $\gamma_2(G)$, and coincides with $G$ if and only if $n=3$. For $n\ge 4$, $\chi_2(G)$ is one of the maximal subgroups $(H_i)_{1\le i\le p+1}$ of $G$ and causes a *polarization* among them, which will be standardized in Definition \[dfn:NatOrdCocl1\]. Let the isomorphism invariant $k=k(G)$ of $G$ be defined by $$\lbrack\chi_2(G),\gamma_2(G)\rbrack=\gamma_{n-k}(G)\,,$$ where $k=0$ for $n=3$, $0\le k\le n-4$ for $n\ge 4$, and $0\le k\le\min\lbrace n-4,p-2\rbrace$ for $n\ge p+1$, according to Miech [@Mi p. 331]. $k(G)$ provides a measure for the deviation from the maximal degree of commutativity $\lbrack\chi_2(G),\gamma_2(G)\rbrack=1$ and will be called *defect of commutativity* of $G$.
### Parametrized presentation {#sss:PrmtPres}
Suppose that generators of $G=\langle x,y\rangle$ are selected such that $x\in G\setminus\chi_2(G)$, if $n\ge 4$, and $y\in\chi_2(G)\setminus\gamma_2(G)$, and define the main commutator by $s_2=\lbrack y,x\rbrack\in\gamma_2(G)$ and the higher commutators by $s_j=\lbrack s_{j-1},x\rbrack=s_{j-1}^{x-1}\in\gamma_j(G)$ for $j\ge 3$. We use identifiers $s_j$ to emphasize those elements of $G$ for which addition of symbolic exponents $f_1,f_2$ in the group ring $\mathbb{Z}\lbrack G\rbrack$ is commutative, $s_j^{f_1+f_2}=s_j^{f_2+f_1}$. Nilpotency of $G$ is expressed by $s_n=1$ and a *power-commutator presentation* of $G$ with generators $x,y,s_2,\ldots,s_{n-1}$ is given as follows. There are two relations for $p$th powers of the generators $x$ and $y$ of $G$,
$$\label{eqn:PwrRelCocl1}
x^p=s_{n-1}^w\quad\text{ and }\quad
y^p\prod_{\ell=2}^p\,s_\ell^{\binom{p}{\ell}}=s_{n-1}^z
\quad \text{ with exponents }\quad 0\le w,z\le p-1\,,$$
according to Miech [@Mi p. 332, Thm. 2, (3)]. Blackburn uses the notation $\delta=w$ and $\gamma=z$ for these relational exponents [@Bl1 p. 84, (36), (37)].
Additionally, the group $G$ satisfies relations for $p$th powers of the higher commutators, $$s_{j+1}^p\prod_{\ell=2}^p\,s_{j+\ell}^{\binom{p}{\ell}}=1\quad\text{ for }1\le j\le n-2\,,$$ and the commutator relation of Miech [@Mi p. 332, Thm. 2, (2)], containing the defect $k=k(G)$,
$$\label{eqn:CmtRelCocl1}
\lbrack y,s_2\rbrack=\prod_{\ell=1}^k s_{n-\ell}^{a(n-\ell)}
\in\lbrack\chi_2(G),\gamma_2(G)\rbrack=\gamma_{n-k}(G)\,,$$
with exponents $0\le a(n-\ell)\le p-1$ for $1\le\ell\le k$, and $a(n-k)>0$, if $k\ge 1$. Blackburn restricts his investigations to $k\le 2$ and uses the notation $\beta=a(n-1)$ and $\alpha=a(n-2)$ [@Bl1 p. 82, (33)].
By $G_a^n(z,w)$ we denote the representative of an isomorphism class of metabelian $p$-groups $G$ of coclass $\mathrm{cc}(G)=1$ and order $\lvert G\rvert=p^n$, which satisfies the relations (\[eqn:PwrRelCocl1\]) and (\[eqn:CmtRelCocl1\]) with a fixed system of exponents $a=(a(n-k),\ldots,a(n-1))$,$w$, and $z$. We have $a=0$ if and only if $k=0$.
### A distinguished maximal subgroup {#sss:DstgMaxSbgp1}
Since the maximal normal subgroups $H_i$, $1\le i\le p+1$, of $G$ contain the commutator subgroup $G^\prime$ as a normal subgroup of index $p$, they are of the shape $H_i=\langle g_i,G^\prime\rangle$ with suitable generators $g_i$, and we can arrange them in a fixed order.
\[dfn:NatOrdCocl1\] The *polarization* or *natural order* of the maximal subgroups $(H_i)_{1\le i\le p+1}$ of $G$ is given by the *distinguished first generator* $g_1=y\in\chi_2(G)$ and the other generators $g_i=xy^{i-2}\notin\chi_2(G)$ for $2\le i\le p+1$, provided that $\lvert G\rvert\ge p^4$. Then, in particular $\chi_2(G)=H_1=\langle y,G^\prime\rangle$.
### Parents of CF groups {#sss:PrntCocl1}
Together with group counts in Blackburn’s theorems [@Bl1 p. 88, Thm. 4.1–4.3], Theorem \[thm:PrntCocl1\] describes the structure of the *metabelian skeleton* of the unique coclass tree $\mathcal{T}(C_p\times C_p)$ [@Dt1 § 1, p. 851] of the coclass graph $\mathcal{G}(p,1)$ with an arbitrary prime $p\ge 2$. The graph consists of all isomorphism classes of CF *groups* (with cyclic factors) [@AHL § 4, p. 264] of coclass $1$.
\[thm:PrntCocl1\] Let $p\ge 2$ be an arbitrary prime, and $G$ be a metabelian $p$-group of coclass $\mathrm{cc}(G)=1$ having defect of commutativity $k=k(G)$, such that $G\simeq G_a^n(z,w)$ with parameters $n\ge 3$, $a=(a(n-k),\ldots,a(n-1))$, $0\le a(n-k),\ldots,a(n-1),w,z<p$, where $a(n-k)>0$, if $k\ge 1$, that is, $G$ is of order $\lvert G\rvert=p^n$ and nilpotency class $\mathrm{cl}(G)=n-1$. Then the parent $\pi(G)$ of $G$ on the coclass tree $\mathcal{T}(C_p\times C_p)$ is given by $$\pi(G)\simeq
\begin{cases}
C_p\times C_p, & \text{ if } n=3 \text{ (and thus } k=0), \\
G_0^{n-1}(0,0), & \text{ if } n\ge 4,\ k=0, \\
G_0^{n-1}(0,0), & \text{ if } n\ge 5,\ k=1 \text{ (and thus } p\ge 3), \\
G_{\tilde a}^{n-1}(0,0), & \text{ where }\tilde a=(a(n-k),\ldots,a(n-2)),\text{ if } n\ge 6,\ k\ge 2 \text{ (and thus } p\ge 5).
\end{cases}$$
\[rmk:PrntCocl1\] The various cases of Theorem \[thm:PrntCocl1\] can be described as follows.
1. In the first case, $n=3$, where $G\simeq G_0^3(0,w)$ with $0\le w\le 1$ is an extra-special $p$-group of order $p^3$, the parent $\pi(G)$ is the abelian root $C_p\times C_p$ of the tree $\mathcal{T}(C_p\times C_p)$, which can formally be viewed as $G_0^2(0,0)$.
2. In the second and third case of a group $G$ of defect $k\le 1$, the parent $\pi(G)$ is a mainline group.
3. In the last case of a group $G$ of higher defect $k\ge 2$, which can occur only for $p\ge 5$, the parent $\pi(G)$ lies outside of the mainline and the defect $\tilde k$ and family $\tilde a$ of relational exponents of $\pi(G)$ are given by $\tilde k=k-1$ and $\tilde a=(\tilde a((n-1)-(k-1)),\ldots,\tilde a((n-1)-1))=(a(n-k),\ldots,a(n-2))$, where $\tilde a((n-1)-(k-1))>0$. We point out that the parent is always characterized by parameters $\tilde z=0$ and $\tilde w=0$.
For $n=3$, $G$ is an extra special $p$-group of nilpotency class $\mathrm{cl}(G)=n-1=2$ having the commutator subgroup $\gamma_2(G)$ as its last (non-trivial) lower central $\gamma_{n-1}(G)$. In this special case, the definition of the parent $\pi(G)=G/\gamma_{n-1}(G)$ of $G$ yields the abelianization $\pi(G)=G/\gamma_2(G)$ of type $(p,p)$, which is isomorphic to the root $C_p\times C_p$ of $\mathcal{T}(C_p\times C_p)$.
For $n\ge 4$, $G$ can be assumed to be isomorphic to a group $G\simeq G_a^n(z,w)$ with pc-presentation consisting of the relations (\[eqn:PwrRelCocl1\]) and (\[eqn:CmtRelCocl1\]) for the two generators $x,y$, $$x^p=s_{n-1}^w,\qquad y^p\prod_{\ell=2}^{p}\,s_{\ell}^{\binom{p}{\ell}}=s_{n-1}^z,\qquad \lbrack y,s_2\rbrack=\prod_{\ell=1}^k\,s_{n-\ell}^{a(n-\ell)}.$$ Since the parent $\pi(G)=G/\gamma_{n-1}(G)$ of $G$ is defined as the last lower central quotient, we denote the left coset of an element $g\in G$ with respect to $\gamma_{n-1}(G)$ by $\bar{g}=g\cdot\gamma_{n-1}(G)$ and we obtain $\bar{s}_{n-1}=1$, because $\gamma_{n-1}(G)=\langle s_{n-1}\rangle$. Therefore, the nilpotency class of the parent is $\mathrm{cl}(\pi(G))=\mathrm{cl}(G)-1=n-2$ and a pc-presentation of $\pi(G)$ is given by $$\bar{x}^p=\bar{s}_{n-1}^w=1,\qquad \bar{y}^p\prod_{j=2}^{p}\,\bar{s}_{j}^{\binom{p}{j}}=\bar{s}_{n-1}^z=1,\qquad \lbrack\bar{y},\bar{s}_2\rbrack=\prod_{\ell=1}^k\,\bar{s}_{n-\ell}^{a(n-\ell)},$$ where the last product equals $1$, if $k\le 1$, and $\prod_{\ell=2}^k\,\bar{s}_{n-\ell}^{a(n-\ell)}\ne 1$, if $k\ge 2$, because $\bar{s}_{n-k}^{a(n-k)}\ne 1$. Since the order of the parent is $\lvert\pi(G)\rvert=\lvert G\rvert:\lvert\gamma_{n-1}(G)\rvert=p^n:p=p^{n-1}$, the coclass remains the same $\mathrm{cc}(\pi(G))=n-1-\mathrm{cl}(\pi(G))=n-1-(n-2)=1=\mathrm{cc}(G)$.
The following principle, that the kernel $\varkappa(1)$ of the transfer from $G$ to the first distinguished maximal subgroup $H_1=\chi_2(G)$ decides about the relation between depth $\mathrm{dp}(G)$ and defect $k=k(G)$ of $G$, will turn out to be crucial for metabelian $p$-groups $G$ of coclass $\mathrm{cc}(G)\ge 2$, too.
\[cor:DpthCocl1\] For a metabelian $p$-group $G$ of coclass $\mathrm{cc}(G)=1$ with defect of commutativity $k=k(G)$, the depth $\mathrm{dp}(G)$ of $G$ on the coclass tree $\mathcal{T}(C_p\times C_p)$ of $\mathcal{G}(p,1)$ is given by $$\mathrm{dp}(G)=
\begin{cases}
k+1, & \text{ if } \varkappa(1)\ne 0, \\
k, & \text{ if } \varkappa(1)=0,
\end{cases}$$ with respect to the natural order of the maximal subgroups of $G$.
Theorem \[thm:PrntCocl1\] shows that $(G_0^n(0,0))_{n\ge 2}$ is the mainline of the coclass tree $\mathcal{T}(C_p\times C_p)$, consisting of all groups $G$ of depth $\mathrm{dp}(G)=0$ and defect $k=0$, because each of these vertices occurs as a parent and possesses infinitely many descendants, whereas the groups $G_a^n(0,0)$ with $a\ne 0$, $k\ge 1$ can only have finitely many descendants, due to the bound $k\le p-2$ by Miech [@Mi]. Since the defect of any group $G_a^n(z,w)$ with parameter $a=0$ is given by $k=0$, all the other groups $G=G_0^n(z,w)$, $(z,w)\ne (0,0)$, which contain $H_1$ as an abelian maximal subgroup, must be located as terminal vertices at depth $\mathrm{dp}(G)=1=k+1$, because they never occur as a parent. On the other hand, the third and fourth case of Theorem \[thm:PrntCocl1\] show that the relation between the defects of parent $\pi(G)$ and immediate descendant $G$ is given by $\tilde k=k-1$ for any group $G=G_a^n(z,w)$, $a\ne 0$, with positive defect $k\ge 1$, whence the depth, being the number of steps required to reach the mainline by successive construction of parents, $(G,\pi(G),\pi^2(G),\ldots)$, is given by $\mathrm{dp}(G)=k$. Finally, the groups $G=G_0^n(z,w)$, $(z,w)\ne (0,0)$, containing the abelian maximal subgroup $H_1$, are characterized uniquely by a partial transfer $\varkappa(1)\ne 0$ to the distinguished maximal subgroup $H_1$, according to [@Ma2 Thm. 2.5–2.6].
We conjecture that the following property of mainline groups of $\mathcal{G}(p,1)$ might be true for mainline groups on any coclass tree of $\mathcal{G}(p,r)$, $r\ge 1$.
\[cor:MainLineCocl1\] Mainline groups of $\mathcal{G}(p,1)$, that is, groups of depth $\mathrm{dp}(G)=0$, must have a total transfer $\varkappa(1)=0$ to the distinguished maximal subgroup $H_1=\chi_2(G)$. The converse is only true for $p=2$: A $2$-group $G\in\mathcal{T}(C_2\times C_2)$ having $\varkappa(1)=0$ is mainline.
The statement for $p\ge 2$ is an immediate consequence of Corollary \[cor:DpthCocl1\] and it only remains to prove the converse for $p=2$. This, however, is contained in [@Ma2 Thm. 2.6, p. 481].
Concerning the transfer kernel type $\varkappa(G)$ of a $p$-group $G$ of coclass $1$ we can state:
\[cor:TKTCocl1\]
The transfer kernel types of groups on the unique coclass tree $\mathcal{T}(C_p\times C_p)$ of coclass graph $\mathcal{G}(p,1)$ are given by the following rules.
1. The root $C_p\times C_p$ is of TKT $\mathrm{a}.1$ $(0^{p+1})$ for any prime $p\ge 2$. The extra-special group $G_0^3(0,1)$ is of TKT $\mathrm{A}.1$ $(1^{p+1})$ for odd $p\ge 3$, and of TKT $\mathrm{Q}.5$ $(123)$ for $p=2$. In the sequel, these exceptions are excluded.
2. Mainline groups are of TKT $\mathrm{a}.1$ $(0^{p+1})$ for odd $p\ge 3$, and of TKT $\mathrm{d}.8$ $(032)$ for $p=2$.
3. Groups of depth $1$ and defect $0$ are of TKT either $\mathrm{a}.2$ $(1,0^p)$ or $\mathrm{a}.3$ $(2,0^p)$ for $p\ge 3$, and of TKT either $\mathrm{Q}.6$ $(132)$ or $\mathrm{S}.4$ $(232)$ for $p=2$.
4. Groups of positive defect $1\le k\le p-2$ are exclusively of TKT $\mathrm{a}.1$ $(0^{p+1})$.
This is a result of combining Theorem \[thm:PrntCocl1\] with Theorems 2.5 and 2.6 in [@Ma2].
Second $p$-class groups $G=\mathrm{G}_p^2(K)$ of coclass $\mathrm{cc}(G)=1$ {#ss:ScndClgpCocl1}
---------------------------------------------------------------------------
### Weak transfer target type $\tau_0(G)$ expressed by $p$-class numbers {#sss:wTTTCocl1}
The group theoretic information on the second $p$-class group $G=\mathrm{G}_p^2(K)$, that is, order, class, coclass, and defect, is contained in the $p$-class numbers of the distinguished extension $L_1$ and of the Hilbert $p$-class field $\mathrm{F}_p^1(K)$. Additionally, the principalization $\kappa(1)$ of $K$ in the distinguished extension $L_1$ determines the connection between defect and depth of $G$.
\[thm:wTTTCocl1\]
Let $K$ be an arbitrary number field with $p$-class group $\mathrm{Cl}_p(K)$ of type $(p,p)$. Suppose that the second $p$-class group $G=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$ is abelian or metabelian of coclass $\mathrm{cc}(G)=1$ with defect $k=k(G)$, order $\lvert G\rvert=p^n$, and class $\mathrm{cl}(G)=n-1$, where $n\ge 2$. With respect to the natural order among the maximal subgroups of $G$, the weak transfer target type $\tau_0(G)$ of $G$, that is, the family of $p$-class numbers of the multiplet $(L_1,\ldots,L_{p+1})$ of unramified cyclic extension fields of $K$ of relative prime degree $p\ge 2$ is given for the first layer by
$$\begin{aligned}
\tau_0(G) = (\mathrm{h}_p(L_1),\mathrm{h}_p(L_2),\ldots,\mathrm{h}_p(L_{p+1}) =
\begin{cases}
(\overbrace{p\ldots,p}^{p+1\text{ times}}), & \text{ if }n=2,\\
(p^{\mathrm{cl}(G)-k},\overbrace{p^2,\ldots,p^2}^{p\text{ times}}), & \text{ if }n\ge 3,
\end{cases}\end{aligned}$$
where defect $k$ and depth $\mathrm{dp}(G)$ of $G$ are related by $$k=
\begin{cases}
\mathrm{dp}(G)-1, & \text{ if } \varkappa(1)\ne 0, \\
\mathrm{dp}(G), & \text{ if } \varkappa(1)=0,
\end{cases}$$
and for the single member of the second layer by $$\mathrm{h}_p(\mathrm{F}_p^1(K)) = p^{\mathrm{cl}(G)-1}.$$
The statement is a succinct version of [@Ma1 Thm. 3.2], expressed by concepts more closely related to the position of $G$ on the coclass graph $\mathcal{G}(p,1)$ and to the transfer kernel type $\varkappa(G)$ of $G$, using Corollary \[cor:DpthCocl1\].
Whereas $\mathrm{h}_p(L_2),\ldots,\mathrm{h}_p(L_{p+1})$ only indicate that $\mathrm{cc}(G)=1$, the $p$-class number $\mathrm{h}_p(\mathrm{F}_p^1(K))$ of the Hilbert $p$-class field of $K$ determines the order $p^n$, $n=\mathrm{cl}(G)+1$, and class of $G$, and the distinguished $\mathrm{h}_p(L_1)$ gives the defect $k$ of $G$.\
With respect to the mainline $(M_j)_{j\ge 2}$ of the coclass tree $\mathcal{T}(C_p\times C_p)$, the order $\lvert M_i\rvert=3^i$ of the branch root $M_i$ of $G$ is given by $i=n-\mathrm{dp}(G)=\mathrm{cl}(G)+1-\mathrm{dp}(G)$, where $$\mathrm{dp}(G)=
\begin{cases}
k, & \text{ if } \varkappa(1)=0, \\
k+1, & \text{ if } \varkappa(1)\ne 0.
\end{cases}$$
### The complete coclass graph $\mathcal{G}(2,1)$ {#sss:Distr2Cocl1}
We start our investigation of special cases by showing that the distribution of second $2$-class groups $\mathrm{G}_2^2(K)$ of complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $D<0$, with $\mathrm{Cl}_2(K)\simeq(2,2)$ on $\mathcal{G}(2,1)$ is not restricted by selection rules. This distribution will only be given qualitatively, without exact counts.
\[thm:2Cocl1\] The diagram in Figure \[fig:TKT2Cocl1\] visualizes the complete coclass graph $\mathcal{G}(2,1)$ up to order $2^8=256$. It is periodic with length $1$. The first period consists of branch $\mathcal{B}_3$, whereas branch $\mathcal{B}_2$ is irregular and forms the pre-period.
$\mathcal{G}(2,1)$ begins with two abelian groups of order $2^2$, the isolated cyclic group $C_4$, having different abelianization, and Klein’s four group $V_4$, that is the bicyclic root $C_2\times C_2$ of the unique coclass tree $\mathcal{T}(C_2\times C_2)$.\
As immediate descendants of the root, $\mathcal{G}(2,1)$ contains the capable mainline group $D(8)$ and the terminal group $Q(8)$, both of order $2^3$.
Applying Blackburn’s results [@Bl1] on counts of metabelian $p$-groups of maximal class and order $p^n$ with $n\ge 4$, to the special case $p=2$, we only need to consider metabelian groups containing an abelian maximal subgroup, characterized by defect $k=0$. They consist of the capable mainline group $D(2^n)=G_0^n(0,0)$, the terminal group $Q(2^n)=G_0^n(0,1)$, and the terminal group $S(2^n)=G_0^n(1,0)$, which is expressed by specialization of [@Bl1 p. 88, Thm. 4.3] to $p=2$. The count is independent from $n$, yielding the constant number $2+(n-2,p-1)=3$.
We recall from [@Ma2] that the transfer kernel types $\varkappa(G)$ for $p$-groups of coclass $\mathrm{cc}(G)=1$ are exceptional in the case $p=2$, compared to the uniform standard case of odd primes $p\ge 3$.
\[thm:TKT2Cocl1\] Table \[tab:TKT2Cocl1\] gives the transfer kernel type $\varkappa(G)$ of all non-isolated vertices $G$, having abelianization $G/G^\prime\simeq(2,2)$, on the coclass graph $\mathcal{G}(2,1)$. The $2$-groups $G$ are identified by their Blackburn invariants $\lvert G\rvert=2^n$ and $a,w,z$ as exponents in the relations (\[eqn:PwrRelCocl1\]) and (\[eqn:CmtRelCocl1\]). The graph information gives the depth $\mathrm{dp}(G)$ and the location of each $2$-group $G$ with respect to the unique coclass tree $\mathcal{T}(C_2\times C_2)$ of $\mathcal{G}(2,1)$.\
The mainline, consisting of the dihedral $2$-groups $D(2^n)=G_0^n(0,0)$ including the abelian root $C_2\times C_2=D(4)=G_0^2(0,0)$, is characterized by the total transfer $\varkappa(1)=0$ to the distinguished maximal subgroup $H_1=\chi_2(G)$. Total transfers $\varkappa(i)=0$ are counted by $\nu(G)$.
----------------- ------------------ --------- ----- ----- ----- ----- ------------------ ------------------- ----- ---------------- ----------
$G$ $\mathrm{cl}(G)$ $n$ $a$ $z$ $w$ $k$ $\mathrm{dp}(G)$ tree position TKT $\varkappa(G)$ $\nu(G)$
$C_2\times C_2$ $1$ $2$ $0$ $0$ $0$ $0$ $0$ root a.1 $(000)$ $3$
$Q(8)$ $2$ $3$ $0$ $0$ $1$ $0$ $1$ pre-period Q.5 $(123)$ $0$
$D(2^n)$ $\ge 2$ $\ge 3$ $0$ $0$ $0$ $0$ $0$ mainline d.8 $(032)$ $1$
$Q(2^n)$ $\ge 3$ $\ge 4$ $0$ $0$ $1$ $0$ $1$ periodic sequence Q.6 $(132)$ $0$
$S(2^n)$ $\ge 3$ $\ge 4$ $0$ $1$ $0$ $0$ $1$ periodic sequence S.4 $(232)$ $0$
----------------- ------------------ --------- ----- ----- ----- ----- ------------------ ------------------- ----- ---------------- ----------
: $\varkappa(G)$, $\nu(G)$ in dependence on non-isolated $G\in\mathcal{G}(2,1)$[]{data-label="tab:TKT2Cocl1"}
See [@Ma2 Thm. 2.6, Tbl. 2–3] for the technique of determining kernels of transfers and the definition of transfer kernel types as orbits of integer triplets $\lbrack 0,3\rbrack^3$ under the action of the symmetric group of degree $3$.
The statements of Theorem \[thm:TKT2Cocl1\] can be visualized very conveniently by the diagram of a finite part of the coclass graph $\mathcal{G}(2,1)$, which is shown in Figure \[fig:TKT2Cocl1\]. It contains the isolated vertex $C_4$, the root $C_2\times C_2$ and branches $\mathcal{B}_j$, $2\le j\le 7$, of the coclass tree $\mathcal{T}(C_2\times C_2)$. Branch $\mathcal{B}_2$ consists of two initial exceptions, the elementary abelian bicyclic $2$-group $C_2\times C_2$ with TKT a.1, $\varkappa=(000)$, and the quaternion group $Q(8)$ with TKT Q.5, $\varkappa=(123)$. Periodicity of length $\ell=1$ sets in with branch $\mathcal{B}_3$ which consists of the starting vertices of three periodic sequences, $\mathcal{S}(D(8))$, the mainline of *dihedral* groups, $\mathcal{S}(Q(16))$, the sequence of *generalized quaternion* groups, and $\mathcal{S}(S(16))$, the sequence of *semi-dihedral* groups. Transfer kernel types (TKT) in the bottom rectangle concern all vertices in the periodic sequence located vertically above. Large contour squares $\square$ denote abelian groups and big full discs [$\bullet$]{} denote metabelian groups with defect $k=0$. A number in angles gives the identifier of a group in the SmallGroups Library [@BEO]. The symbols $\Gamma_s$ denote isoclinism families given by Hall and Senior [@HaSn]. The population of each vertex is indicated by a surrounding circle labelled by the discriminant $D<0$ of a suitable complex quadratic field $K=\mathbb{Q}(\sqrt{D})$ of type $(2,2)$. There are no selection rules for $p=2$ and the numerical results suggest the conjecture that the tree $\mathcal{T}(V_4)$ is covered entirely by second $2$-class groups $G_2^2(K)$ of complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $D<0$. Ground states are due to Kisilevsky [@Ki2 p. 277–278]. All excited states have been determined with the aid of Theorem \[thm:wTTTCocl1\]. See [@Ma1 § 9]. Here we refrain from giving the exact distribution up to some bound for $\lvert D\rvert$ and we do not claim that the given examples have minimal absolute discriminants.
### Selection Rule for quadratic base fields {#sss:SelRuleCocl1}
Let $K=\mathbb{Q}(\sqrt{D})$ be a quadratic number field with discriminant $D$ and $p$-class group $\mathrm{Cl}_p(K)$ of type $(p,p)$, where $p\ge 3$ denotes an odd prime. Then the $p+1$ unramified cyclic extension fields $(L_1,\ldots,L_{p+1})$ of $K$ of relative prime degree $p$ have dihedral absolute Galois groups $\mathrm{Gal}(L_i\vert K)$ of degree $2p$, according to [@Ma1 Prop. 4.1].
\[thm:SelRuleCocl1\]
Let $G=\mathrm{Gal}(\mathrm{F}_p^2(K)\vert K)$ be the second $p$-class group of $K$. If $G\in\mathcal{G}(p,1)$, then $K$ must be real quadratic, $D>0$, and, with respect to the natural order of the maximal subgroups of $G$, the family of $p$-class numbers of the non-Galois subfields $K_i$ of $L_i$ is given by
$$\begin{aligned}
(\mathrm{h}_p(K_1),\mathrm{h}_p(K_2),\ldots,\mathrm{h}_p(K_{p+1})) =
(p^{\frac{\mathrm{cl}(G)-\mathrm{dp}(G)}{2}},\overbrace{p,\ldots,p}^{p\text{ times}}),\end{aligned}$$
where depth and defect $k$ of $G$ are related via the first component of the TKT $\varkappa(K)$ by
$$\mathrm{dp}(G)=
\begin{cases}
k, & \text{ if } \varkappa(1)=0, \\
k+1, & \text{ if } \varkappa(1)\ne 0.
\end{cases}$$
Consequently, the order $\lvert M_i\rvert=p^i$ of the branch root $M_i$ of $G\in\mathcal{B}(M_i)$ on the unique coclass tree of $\mathcal{G}(p,1)$ with mainline $(M_j)_{j\ge 2}$ must have *odd* exponent $$i=n-\mathrm{dp}(G)=\mathrm{cl}(G)+1-\mathrm{dp}(G)\equiv 1\pmod{2}.$$
Whereas $\mathrm{h}_p(K_2),\ldots,\mathrm{h}_p(K_{p+1})$ do not give any information, the distinguished $p$-class number $\mathrm{h}_p(K_1)$ enforces the congruence $\mathrm{cl}(G)-\mathrm{dp}(G)\equiv 0\pmod{2}$.
The statement is a compact version of [@Ma1 Thm. 4.1], expressed by the depth $\mathrm{dp}(G)$, and thus more closely related to the position of $G$ on the coclass graph $\mathcal{G}(p,1)$ and to the transfer kernel type $\varkappa(G)$ of $G$, using Corollary \[cor:DpthCocl1\].
\[thm:TKTpCocl1\] Table \[tab:TKTpCocl1\] gives the transfer kernel type (TKT) $\varkappa(G)$ of all non-isolated metabelian vertices $G$ on the coclass graph $\mathcal{G}(p,1)$, for odd $p\ge 3$. The $p$-groups $G$ are identified by their Blackburn-Miech invariants $\lvert G\rvert=p^n$ and $a,w,z$ as exponents in the relations (\[eqn:PwrRelCocl1\]) and (\[eqn:CmtRelCocl1\]). The graph information gives the depth $\mathrm{dp}(G)$ and the location of each $p$-group $G$ with respect to the unique coclass tree $\mathcal{T}(C_p\times C_p)$ of $\mathcal{G}(p,1)$.\
The mainline, consisting of the $p$-groups $G_0^n(0,0)$ including the abelian root $C_p\times C_p=G_0^2(0,0)$, and all groups of positive defect $k\ge 1$ are characterized by the total transfer $\varkappa(1)=0$ to the distinguished maximal subgroup $H_1=\chi_2(G)$.
----------------- ------------------ --------- --------- --------- ----- --------- ------------------ -------------------- ----- ---------------------------------------------- ----------
$G$ $\mathrm{cl}(G)$ $n$ $a$ $z$ $w$ $k$ $\mathrm{dp}(G)$ tree position TKT $\varkappa(G)$ $\nu(G)$
$C_p\times C_p$ $1$ $2$ $0$ $0$ root a.1 $(\overbrace{0\ldots 0}^{p+1\text{ times}})$ $p+1$
$G_0^3(0,1)$ $2$ $3$ $0$ $0$ $1$ $0$ $1$ pre-period A.1 $(\overbrace{1\ldots 1}^{p+1\text{ times}})$ $0$
$G_0^n(0,0)$ $\ge 2$ $\ge 3$ $0$ $0$ $0$ $0$ $0$ mainline a.1 $(\overbrace{0\ldots 0}^{p+1\text{ times}})$ $p+1$
$G_0^n(0,1)$ $\ge 3$ $\ge 4$ $0$ $0$ $1$ $0$ $1$ periodic sequences a.2 $(1\overbrace{0\ldots 0}^{p\text{ times}})$ $p$
$G_0^n(z,0)$ $\ge 3$ $\ge 4$ $0$ $\ne 0$ $0$ $0$ $1$ periodic sequences a.3 $(2\overbrace{0\ldots 0}^{p\text{ times}})$ $p$
$G_a^n(z,w)$ $\ge 4$ $\ge 5$ $\ne 0$ $ $ $ $ $\ge 1$ $\ge 1$ periodic sequences a.1 $(\overbrace{0\ldots 0}^{p+1\text{ times}})$ $p+1$
----------------- ------------------ --------- --------- --------- ----- --------- ------------------ -------------------- ----- ---------------------------------------------- ----------
: $\varkappa(G)$, $\nu(G)$ in dependence on non-isolated $G\in\mathcal{G}(p,1)$ for $p\ge 3$[]{data-label="tab:TKTpCocl1"}
See [@Ma2 Thm. 2.5, Tab. 1].
### The complete coclass graph $\mathcal{G}(3,1)$ {#sss:3Cocl1}
This section and the following sections §§\[sss:TKTFromCoarseTTTCocl1\]–\[sss:7Cocl1\] will show, that second $p$-class groups $\mathrm{G}_p^2(K)$ of real quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $D>0$, with $\mathrm{Cl}_p(K)\simeq(p,p)$ are only distributed on odd branches of the metabelian skeleton of $\mathcal{G}(p,1)$, for an odd prime $p\ge 3$, in contrast to the complete population of the coclass graph $\mathcal{G}(2,1)$. The effect is due to the number theoretic selection rule in Theorem \[thm:SelRuleCocl1\]. The quantitative distribution for $p\in\lbrace 3,5,7\rbrace$ reveals a dominant population of ground states and decreasing frequency of hits of excited states.
\[thm:3Cocl1\] The diagram in Figure \[fig:Distr3Cocl1\] visualizes the complete coclass graph $\mathcal{G}(3,1)$ up to order $3^8=6\,561$. It is periodic with length $2$. The period consists of branches $\mathcal{B}_j$ with $4\le j\le 5$, whereas branches $\mathcal{B}_j$ with $2\le j\le 3$ are irregular and form the pre-period.
The top of $\mathcal{G}(3,1)$ consists of two abelian groups of order $3^2$, the isolated cyclic group $C_9$ and the bicyclic root $C_3\times C_3$ of the unique coclass tree $\mathcal{T}(C_3\times C_3)$.\
Immediate descendants of the root are the two well-known extra special groups, the capable mainline group $G_0^3(0,0)$ of exponent $3$ and the terminal group $G_0^3(0,1)$ of exponent $9$, both of order $3^3$.
Blackburn’s results [@Bl1] on counting metabelian $p$-groups of maximal class and order $p^n$ with $n\ge 4$ can now be applied to the special case $p=3$, which is entirely metabelian [@Bl2 p. 26, Thm. 6].\
We start with metabelian groups containing an abelian maximal subgroup, which are characterized by defect $k=0$. They consist of the capable mainline group $G_0^n(0,0)$, the terminal group $G_0^n(0,1)$ and $(n-2,p-1)$ terminal groups of the form $G_0^n(z,0)$. Specialization of [@Bl1 p. 88, Thm. 4.3] to $p=3$ in dependence on $n$ yields their number $$2+(n-2,p-1)=
\begin{cases}
2+1=3 & \text{ for } 5\le n\equiv 1\pmod{2}, \\
2+2=4 & \text{ for } 4\le n\equiv 0\pmod{2}.
\end{cases}$$ Further, the number of metabelian groups with defect $k=1$, which are terminal and of the form $G_1^n(z,w)$ with $a=(a(n-1))=(1)$, is given, independently from $n\ge 5$, by $3$ [@Bl1 p. 88, Thm. 4.2].
Vertices $G$ of coclass graph $\mathcal{G}(3,1)$ in Figure \[fig:Distr3Cocl1\] are classified according to their defect $k(G)$ by using different symbols:
1. large contour squares $\square$ denote abelian groups,
2. big full discs [$\bullet$]{} denote metabelian groups with abelian maximal subgroup and $k=0$,
3. small full discs [$\bullet$]{} denote metabelian groups with defect $k=1$.
The actual distribution of the $2576$ second $3$-class groups $G_3^2(K)$ of real quadratic number fields $K=\mathbb{Q}(\sqrt{D})$ of type $(3,3)$ with discriminant $0<D<10^7$ is represented by underlined boldface counters of hits of vertices surrounded by the adjacent oval. See [@Ma1 § 6, Tab. 2] and [@Ma3 § 6, Tab. 11]. The results verify the selection rule, Theorem \[thm:SelRuleCocl1\], for groups $G_3^2(\mathbb{Q}(\sqrt{D}))$, $D>0$, and underpin the *weak leaf conjecture* \[cnj:WeakLeafCnj\] that mainline vertices are forbidden for second $3$-class groups of quadratic fields. A remarkably different behavior is revealed by certain biquadratic fields in Figure \[fig:WimanBlackburn1\] of section §\[ss:NewRsltESR\].
\[cnj:WeakLeafCnj\] A vertex $G$ on the metabelian skeleton $\mathcal{M}(p,r)$ of a coclass graph $\mathcal{G}(p,r)$, with an odd prime $p\ge 3$ and $r\ge 1$, cannot be realized as second $p$-class group $G_p^2(K)$ of a quadratic field $K=\mathbb{Q}(\sqrt{D})$, if it possesses a metabelian immediate descendant $H$ having the same transfer kernel type $\varkappa(H)=\varkappa(G)$ and a higher defect of commutativity $k(H)>k(G)$.
### Separating TKTs on $\mathcal{G}(p,1)$ via first TTT {#sss:TKTFromCoarseTTTCocl1}
For increasing odd primes $p\ge 5$, the structure $\tau(1)$ of the $p$-class group $\mathrm{Cl}_p(L_1)$ of the distinguished first unramified extension $L_1$ of degree $p$ of an arbitrary base field $K$ with second $p$-class group $G=\mathrm{G}_p^2(K)$ of coclass $\mathrm{cc}(G)=1$ admits the separation of more and more excited states of the TKTs $\mathrm{a}.2$, with fixed point $\varkappa(1)=1$, and $\mathrm{a}.3$, without fixed point $\varkappa(1)\in\lbrace 2,\ldots,p+1\rbrace$. Further, the order of the exceptional $p$-group $G\simeq\mathrm{Syl}_p A_{p^2}$ becomes increasingly larger.
\[thm:TKTFromCoarseTTTCocl1\] Let $p\ge 3$ be an odd prime and $G\in\mathcal{G}(p,1)$ a $p$-group of order $\lvert G\rvert=p^n$, $n\ge 4$, depth $\mathrm{dp}(G)=1$, and defect $k(G)=0$.
1. The exceptional case of TKT $\mathrm{a}.3^\ast$, having an elementary abelian first TTT $\tau(1)$ of elevated $p$-rank $\mathrm{r}_p=p$, occurs if and only if $n=p+1$ and $G\simeq G_0^{p+1}(1,0)=\mathrm{Syl}_p A_{p^2}$
2. The regular cases of the TKTs $\mathrm{a}.2$ and $\mathrm{a}.3$, having a first TTT $\tau(1)$ of usual $p$-rank $\mathrm{r}_p\le p-1$, can be separated by the structure $\tau(1)$ of the distinguished first maximal subgroup $H_1=\chi_2(G)$ if and only if $n\le p$. In this case,
1. $G$ is of TKT $\mathrm{a}.2$ if and only if $\tau(1)=(\overbrace{p\ldots,p}^{n-1\text{ times}})$ is elementary abelian of rank $\mathrm{r}_p=n-1\le p-1$,
2. $G$ is of TKT $\mathrm{a}.3$ if and only if $\tau(1)=(p^2,\overbrace{p\ldots,p}^{n-3\text{ times}})$ is of rank $\mathrm{r}_p=n-2\le p-2$, neither nearly homocyclic nor elementary abelian.
All groups $G\in\mathcal{G}(p,1)$ of order $\lvert G\rvert\ge p^n$, $n\ge 4$, depth $\mathrm{dp}(G)=1$, and defect $k(G)=0$ are metabelian and contain the abelian distinguished maximal subgroup $A=H_1=\chi_2(G)$, having $A^\prime=1$. Thus, all statements are a consequence of [@HeSm Thm. 7, p. 11], where $G$ is of TKT $\mathrm{a}.2$ if and only if $G\simeq G_0^n(0,1)$, and $G$ is of TKT $\mathrm{a}.3$ if and only if $G\simeq G_0^n(z,0)$, $z\notin\lbrace 0,1\rbrace$.
Table \[tbl:TKTFromCoarseTTTCocl1\] displays the possibilities for the first TTT $\tau(1)$ in dependence on the ground state (GS) and excited states (ES) of TKTs, as stated in Theorem \[thm:TKTFromCoarseTTTCocl1\] for the smallest odd primes $p\in\lbrace 3,5,7\rbrace$. Here, we assume a quadratic base field $K=\mathbb{Q}(\sqrt{D})$, taking into account the selection rule, Theorem \[thm:SelRuleCocl1\], for odd branches.
----- ------- ------------------------------ ------------------- ------------------- ---------------------
$p$ state branch of
$\mathcal{T}(C_p\times C_p)$ $\mathrm{a}.2$ $\mathrm{a}.3$ $\mathrm{a}.3^\ast$
$3$ GS $\mathcal{B}_3$ $(3^2,3)$ $(3^2,3)$ $(3,3,3)$
ES 1 $\mathcal{B}_5$ $(3^3,3^2)$ $(3^3,3^2)$ —
ES 2 $\mathcal{B}_7$ $(3^4,3^3)$ $(3^4,3^3)$ —
$5$ GS $\mathcal{B}_3$ $(5,5,5)$ $(5^2,5)$ —
ES 1 $\mathcal{B}_5$ $(5^2,5,5,5)$ $(5^2,5,5,5)$ $(5,5,5,5,5)$
ES 2 $\mathcal{B}_7$ $(5^2,5^2,5^2,5)$ $(5^2,5^2,5^2,5)$ —
$7$ GS $\mathcal{B}_3$ $(7,7,7)$ $(7^2,7)$ —
ES 1 $\mathcal{B}_5$ $(7,7,7,7,7)$ $(7^2,7,7,7)$ —
ES 2 $\mathcal{B}_7$ $(7^2,7,7,7,7,7)$ $(7^2,7,7,7,7,7)$ $(7,7,7,7,7,7,7)$
----- ------- ------------------------------ ------------------- ------------------- ---------------------
: Separating TKT $\mathrm{a}.2$, $\mathrm{a}.3$ and $\mathrm{a}.3*$ on $\mathcal{G}(p,1)$[]{data-label="tbl:TKTFromCoarseTTTCocl1"}
### Metabelian $5$-groups $G$ of coclass $\mathrm{cc}(G)=1$ {#sss:5Cocl1}
\[thm:5Cocl1\] The diagram in Figure \[fig:Distr5Cocl1\] visualizes the metabelian skeleton $\mathcal{M}(5,1)$ of coclass graph $\mathcal{G}(5,1)$ up to order $5^{11}=48\,828\,125$. This subgraph of $\mathcal{G}(5,1)$ is periodic with length $4$. The period consists of the branches $\mathcal{B}_j$ with $5\le j\le 8$, whereas the branches $\mathcal{B}_j$ with $2\le j\le 4$ are irregular and form the pre-period.
Vertices of coclass graph $\mathcal{G}(5,1)$ in Figure \[fig:Distr5Cocl1\] are classified according to their defect $k$ by using different symbols:
1. large contour squares $\square$ represent abelian groups,
2. big full discs [$\bullet$]{} represent metabelian groups with defect $k=0$,
3. big contour circles [$\circ$]{} represent metabelian groups with $k=1$,
4. small full discs [$\bullet$]{} represent metabelian groups with $k=2$,
5. small contour circles [$\circ$]{} represent metabelian groups with $k=3$.
The symbol $n\ast$ adjacent to a vertex denotes the multiplicity of a batch of $n$ immediate descendants sharing a common parent. The selection rule, Theorem \[thm:SelRuleCocl1\], for second $5$-class groups $G_5^2(K)$ of real quadratic number fields $K=\mathbb{Q}(\sqrt{D})$, $D>0$, is indicated by ovals surrounding admissible vertices.
The actual distribution of the $377$ second $5$-class groups $G_5^2(\mathbb{Q}(\sqrt{D}))$ with discriminant $0<D\le 26\,695\,193$, discussed in section \[sss:StatScnd5ClgpCocl1\], is represented by underlined boldface counters of hits of vertices in the adjacent oval. The $13$ cases of TKT $\mathrm{a}.1$, $\varkappa=(000000)$, underpin the weak leaf conjecture \[cnj:WeakLeafCnj\].
$\mathcal{G}(5,1)$ starts with two abelian groups of order $5^2$, the isolated cyclic group $C_{25}$ and the bicyclic root $C_5\times C_5$ of the unique coclass tree $\mathcal{T}(C_5\times C_5)$.\
As immediate descendants of the root, $\mathcal{G}(5,1)$ contains the two well-known extra special groups, the capable mainline group $G_0^3(0,0)$ of exponent $5$ and the terminal group $G_0^3(0,1)$ of exponent $25$, both of order $5^3$.
Now we use Blackburn’s results [@Bl1] on counting metabelian $p$-groups of maximal class and order $p^n$ with $n\ge 4$, for the special case $p=5$.\
First, we consider the metabelian groups containing an abelian maximal subgroup, which are characterized by the defect $k=0$. They consist of the capable mainline group $G_0^n(0,0)$, the terminal group $G_0^n(0,1)$ and $(n-2,p-1)$ terminal groups of the form $G_0^n(z,0)$. Specialization of [@Bl1 p. 88, Thm. 4.3] for $p=5$ in dependence on $n$ yields their number $$2+(n-2,p-1)=
\begin{cases}
2+1=3 & \text{ for } 5\le n\equiv 1\pmod{2}, \\
2+2=4 & \text{ for } 4\le n\equiv 0\pmod{4}, \\
2+4=6 & \text{ for } 6\le n\equiv 2\pmod{4}. \\
\end{cases}$$ Next, the number of metabelian groups with defect $k=1$, which contain exactly one capable group $G_1^n(0,0)$ with $a=(a(n-1))=(1)$ for every $n\ge 5$, is given by [@Bl1 p. 88, Thm. 4.2]: $$1+(2n-6,p-1)+(n-2,p-1)=
\begin{cases}
1+4+1=6 & \text{ for } 5\le n\equiv 1\pmod{2}, \\
1+2+2=5 & \text{ for } 8\le n\equiv 0\pmod{4}, \\
1+2+4=7 & \text{ for } 6\le n\equiv 2\pmod{4}. \\
\end{cases}$$ Finally, the number of metabelian groups with defect $k=2$, containing exactly two capable groups $G_a^n(0,0)$ with $a=(1,\pm 1)$ for every $n\ge 7$ [@LgMk3 § 3, ramification level], but only one capable group $G_{(1,-1)}^n(0,0)$ for $n=6$, is given by [@Bl1 p. 88, Thm. 4.1]:
$$\begin{aligned}
p+(2n-7,p-1)+(n-2,p-1) &=& 5+1+4=10 \text{ for } n=6,\\
2p+(2n-7,p-1)+(n-2,p-1) &=&
\begin{cases}
10+1+1=12 & \text{ for } 7\le n\equiv 1\pmod{2}, \\
10+1+2=13 & \text{ for } 8\le n\equiv 0\pmod{4}, \\
10+1+4=15 & \text{ for } 10\le n\equiv 2\pmod{4}. \\
\end{cases}\end{aligned}$$
Since Blackburn restricts his investigations to defects $k\le 2$, we need a supplementary count of metabelian groups with defect $k=3$. The results for the head of the $4$ virtually periodic branches $\mathcal{B}_j$ with $14\le j\le 17$ given by Dietrich, Eick, Feichtenschlager [@DEF Fig. 7–10, p. 57–60] and by Dietrich [@Dt2 Fig. 4–5, p. 1086] are accumulated counts of metabelian and non-metabelian groups, whereas the collars and tails entirely consist of non-metabelian groups. According to private communications by H. Dietrich, one of the two capable groups $G_{(1,\pm 1)}^{n-1}(0,0)$ at depth two has always $25$ metabelian descendants, which are all terminal, independently from $n\ge 8$, and the other has $15$, resp. $13$, metabelian descendants, which are all terminal, for even $n\ge 8$, resp. odd $n\ge 9$. The count of metabelian groups with defect $k=3$ is also given by Miech [@Mi Thm. 6–7, p. 336–337].
### Distribution of $\mathrm{G}_5^2(K)$ on $\mathcal{G}(5,1)$ {#sss:StatScnd5ClgpCocl1}
In Table \[tbl:RealQuad5x5\], we list the $8$ variants of second $5$-class groups $\mathrm{G}_5^2(K)$ for the $377$ real quadratic fields $K=\mathbb{Q}(\sqrt{D})$ of type $(5,5)$ with discriminant $0<D\le 26\,695\,193$, mainly on the coclass graph $\mathcal{G}(5,1)$, but modestly also on $\mathcal{G}(5,2)$. $\tau(0)$ denotes the $5$-class group of $\mathrm{F}_5^1(K)$. Schur $\sigma$-groups are starred.
$D$ $\tau(K);\tau(0)$ Type $\varkappa(K)$ $G$ $\mathrm{cc}(G)$ $\#$ $\%$
---------------- --------------------------------- ----------------- ---------------- ----------------------------------------- ------------------ ------- --------
$244\,641$ $(5,5^2),(5,5)^5;(5,5)$ a.3 $(2,0^5)$ $\langle 625,9\vert 10\rangle$ $1$ $292$ $77.5$
$1\,167\,541$ $(5,5,5),(5,5)^5;(5,5)$ a.2 $(1,0^5)$ $\langle 625,8\rangle$ $1$ $55$ $14.6$
$1\,129\,841$ $(5,5,5,5),(5,5)^5;(5,5,5,5)$ a.1 $(0^6)$ $\langle 15625,637\ldots 642\rangle$ $1$ $13$ $3.4$
$3\,812\,377$ $(5,5,5,5^2),(5,5)^5;(5,5,5,5)$ a.2,3$\uparrow$ $(?,0^5)$ $\langle 15625,631\ldots 635\rangle$ $1$ $3$ $ $
$4\,954\,652$ $(5,5^2)^6;(5,5,5)$ $(B^6)$ $\langle 3125,9\vert 10\vert 12\rangle$ $2$ $7$ $1.9$
$10\,486\,805$ $(5,5,5)^2,(5,5^2)^4;(5,5,5)$ $(A^2,B^4)$ $\langle 3125,7\vert 11\rangle$ $2$ $2$ $ $
$18\,070\,649$ $(5,5,5),(5,5^2)^5;(5,5,5)$ $(A,B^5)$ $\langle 3125,8\vert 13\rangle*$ $2$ $1$ $ $
$7\,306\,081$ $(5,5,5,5),(5,5,5),(5,5^2)^4$ $(0,A,B^4)$ $\langle 3125,4\rangle$ $2$ $4$ $1$
: $8$ variants of $G=\mathrm{G}_5^2(K)$ for $377$ $K=\mathbb{Q}(\sqrt{D})$, $0<D\le 26\,695\,193$[]{data-label="tbl:RealQuad5x5"}
There occur $13$ cases $(3.4\%)$ of TKT $\mathrm{a}.1$, $\varkappa=(000000)$, starting with $D=1\,129\,841$, $3$ cases of the first excited state of TKT $\mathrm{a}.2\uparrow$, $\varkappa=(100000)$, or $\mathrm{a}.3\uparrow$, $\varkappa=(200000)$, for $D\in\lbrace 3\,812\,377,19\,621\,905,21\,281\,673\rbrace$, and $55$ cases $(14.6\%)$ of the ground state of TKT $\mathrm{a}.2$, $\varkappa=(100000)$, starting with $D=1\,167\,541$. The remaining $292$ cases $(77.5\%)$ of the ground state of TKT $\mathrm{a}.3$, $\varkappa=(200000)$, starting with $D=244\,641$, are clearly dominating. The TKTs were identified by means of Theorem \[thm:TKTFromCoarseTTTCocl1\], taking into account the selection rule for quadratic base fields as given in Table \[tbl:TKTFromCoarseTTTCocl1\]. The distribution of the corresponding second $5$-class groups $\mathrm{G}_5^2(K)$ on the coclass graph $\mathcal{G}(5,1)$, resp. $\mathcal{G}(5,2)$, is shown in Figure \[fig:Distr5Cocl1\], resp. \[fig:Typ55Cocl2\]. See also section §\[sss:StatScnd5ClgpCocl2\].
### Metabelian $7$-groups $G$ of coclass $\mathrm{cc}(G)=1$ {#sss:7Cocl1}
Figure \[fig:Distr7Cocl1\] visualizes the lowest range of the distribution of second $7$-class groups $\mathrm{G}_7^2(K)$ for the $17$ real quadratic fields $K=\mathbb{Q}(\sqrt{D})$ of type $(7,7)$ with discriminant $0<D<10^7$ on the coclass graph $\mathcal{G}(7,1)$. With the aid of MAGMA [@MAGMA], we found $13$ cases, $76\%$, of TKT $\mathrm{a}.3$, $\varkappa=(20000000)$, starting with $D=1\,633\,285$, and $3$ occurrences, $18\%$, of TKT $\mathrm{a}.2$, $\varkappa=(10000000)$, for $D\in\lbrace 2\,713\,121,6\,872\,024,9\,659\,661\rbrace$. These two TKTs can be separated by means of Theorem \[thm:TKTFromCoarseTTTCocl1\]. There were no cases of excited states, but for the single discriminant $D=6\,986\,985$, $\mathrm{G}_7^2(K)$ is a top vertex of $\mathcal{G}(7,2)$ without total $7$-principalization and of Taussky’s coarse transfer kernel type $\kappa=(BBBBBBBB)$ [@Ta2]. Table \[tbl:RealQuad7x7\] shows the corresponding TTT $\tau(K)=(\mathrm{Cl}_7(L_i))_{1\le i\le 8}$ using power notation for repetitions and including $\tau(0)=\mathrm{Cl}_7(\mathrm{F}_7^1(K))$, separated by a semicolon. $7$-groups $G^n_a(z,w)$ of positive defect $k\ge 1$ appear in higher branches and are invisible in Figure \[fig:Distr7Cocl1\].
$D$ $\tau(K);\tau(0)$ Type $\varkappa(K)$ $G$ $\mathrm{cc}(G)$ $\#$ $\%$
--------------- ------------------------- ------ ---------------- -------------------------------------------- ------------------ ------ ------
$1\,633\,285$ $(7,7^2),(7,7)^7;(7,7)$ a.3 $(2,0^7)$ $\langle 2401,9\vert 10\rangle$ $1$ $13$ $76$
$2\,713\,121$ $(7,7,7),(7,7)^7;(7,7)$ a.2 $(1,0^7)$ $\langle 2401,8\rangle$ $1$ $3$ $18$
$6\,986\,985$ $(7,7^2)^8;(7,7,7)$ $(B^8)$ $\langle 16807,10\vert 14\ldots 16\rangle$ $2$ $1$ $6$
: $3$ variants of $G=\mathrm{G}_7^2(K)$ for $17$ $K=\mathbb{Q}(\sqrt{D})$, $0<D<10^7$[]{data-label="tbl:RealQuad7x7"}
In the next section, we proceed to $p$-groups $G$ of coclass $\mathrm{cc}(G)\ge 2$.
Metabelian $3$-groups $G$ of coclass $\mathrm{cc}(G)\ge 2$ with $G/G^\prime\simeq (3,3)$ {#ss:MtabTyp33CoclGe2}
----------------------------------------------------------------------------------------
### Non-CF groups {#sss:NonCF}
In contrast to CF groups of coclass $1$, metabelian $3$-groups $G$ of coclass $\mathrm{cc}(G)\ge 2$ with abelianization $G/G^\prime$ of type $(3,3)$ must have at least one bicyclic factor $\gamma_3(G)/\gamma_4(G)$ [@Ne1], and are therefore called *non-CF groups*. They are characterized by an isomorphism *invariant* $e=e(G)$, defined by $e+1=\min\lbrace 3\le j\le m\mid 1\le\lvert\gamma_j(G)/\gamma_{j+1}(G)\rvert\le 3\rbrace$. This invariant $2\le e\le m-1$ indicates the first cyclic factor $\gamma_{e+1}(G)/\gamma_{e+2}(G)$ of the lower central series of $G$, except $\gamma_2(G)/\gamma_3(G)$, which is always cyclic. We can calculate $e$ from order $\lvert G\rvert=3^n$ and nilpotency class $\mathrm{cl}(G)=m-1$, resp. index $m$ of nilpotency, of $G$ by the formula $e=n-m+2$. Since the coclass of $G$ is given by $\mathrm{cc}(G)=n-\mathrm{cl}(G)=n-m+1$, we have the relation $e=\mathrm{cc}(G)+1$. CF groups are characterized by $e=2$ and non-CF groups by $e\ge 3$.
### Bipolarization and defect {#sss:BiplrzDfct}
For a group $G$ of coclass $\mathrm{cc}(G)\ge 2$ we need a generalization of the group $\chi_2(G)$. Denoting by $m$ the index of nilpotency of $G$, we let $\chi_j(G)$ with $2\le j\le m-1$ be the centralizers of two-step factor groups $\gamma_j(G)/\gamma_{j+2}(G)$ of the lower central series, that is, the biggest subgroups of $G$ with the property $\lbrack\chi_j(G),\gamma_j(G)\rbrack\le\gamma_{j+2}(G)$. They form an ascending chain of characteristic subgroups of $G$, $\gamma_2(G)\le\chi_2(G)\le\ldots\le\chi_{m-2}(G)<\chi_{m-1}(G)=G$, which contain the commutator subgroup $\gamma_2(G)$, and $\chi_j(G)$ coincides with $G$ if and only if $j\ge m-1$. We characterize the smallest *two-step centralizer* different from the commutator subgroup by an isomorphism *invariant* $s=s(G)=\min\lbrace 2\le j\le m-1\mid\chi_j(G)>\gamma_2(G)\rbrace$. Again, CF groups are characterized by $s=2$ and non-CF groups by $s\ge 3$.
Now we can generalize the *defect of commutativity* $k=k(G)$ to any metabelian $3$-group $G$ with $G/G^\prime$ of type $(3,3)$ by defining $0\le k\le 1$ such that $\lbrack\chi_s(G),\gamma_e(G)\rbrack=\gamma_{m-k}(G)$.
The following assumptions for a metabelian $3$-group $G$ of coclass $\mathrm{cc}(G)\ge 2$ with abelianization $G/\gamma_2(G)$ of type $(3,3)$ can always be satisfied, according to Nebelung [@Ne1 Thm. 3.1.11, p. 57, and Thm. 3.4.5, p. 94].
Let $G$ be a metabelian $3$-group of coclass $\mathrm{cc}(G)\ge 2$ with abelianisation $G/\gamma_2(G)$ of type $(3,3)$. Assume that $G$ has order $\lvert G\rvert=3^n$, class $\mathrm{cl}(G)=m-1$, and invariant $e=n-m+2\ge 3$, where $4\le m<n\le 2m-3$. Let generators of $G=\langle x,y\rangle$ be selected such that the bicyclic factor $\gamma_3(G)/\gamma_4(G)$ is generated by their third powers, $\gamma_3(G)=\langle y^3,x^3,\gamma_4(G)\rangle$, and that $x\in G\setminus\chi_s(G)$, if $s<m-1$, and $y\in\chi_s(G)\setminus\gamma_2(G)$. This causes a *bipolarization* among the four maximal subgroups $H_1,\ldots,H_4$ of $G$, which will be standardized in Definition \[dfn:NatOrdCoclGe2\].
### Parametrized presentation {#sss:PrmPres2}
Let the main commutator of $G$ be declared by $s_2=t_2=\lbrack y,x\rbrack\in\gamma_2(G)$ and higher commutators recursively by $s_j=\lbrack s_{j-1},x\rbrack$, $t_j=\lbrack t_{j-1},y\rbrack\in\gamma_j(G)$ for $j\ge 3$. Starting with the powers $\sigma_3=y^3$, $\tau_3=x^3\in\gamma_3(G)$, which generate $\gamma_3(G)$ modulo $\gamma_4(G)$, let $\sigma_j=\lbrack\sigma_{j-1},x\rbrack$, $\tau_j=\lbrack\tau_{j-1},y\rbrack\in\gamma_j(G)$ for $j\ge 4$. Nilpotency of $G$ is expressed by $\sigma_{m-1}=1$ and $\tau_{e+2}=1$. According to Nebelung [@Ne1], the group $G$ satisfies the following relations with certain exponents $-1\le\alpha,\beta,\gamma,\delta,\rho\le 1$ as parameters.
$$\label{eqn:PwrCmtPres}
s_2^3=\sigma_4\sigma_{m-1}^{-\rho\beta}\tau_4^{-1},\quad
\ s_3\sigma_3\sigma_4=\sigma_{m-2}^{\rho\beta}\sigma_{m-1}^\gamma\tau_e^\delta,\quad
\ t_3^{-1}\tau_3\tau_4=\sigma_{m-2}^{\rho\delta}\sigma_{m-1}^\alpha\tau_e^\beta,\quad
\ \tau_{e+1}=\sigma_{m-1}^{-\rho}.$$
By $G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)$ we denote the representative of an isomorphism class of metabelian $3$-groups $G$, having $G/G^\prime$ of type $(3,3)$, of coclass $\mathrm{cc}(G)=n-m+1\ge 2$, class $\mathrm{cl}(G)=m-1$, and order $\lvert G\rvert=3^n$, which satisfies the relations (\[eqn:PwrCmtPres\]) with a fixed system of exponents $(\alpha,\beta,\gamma,\delta,\rho)$. We have $\rho=0$ if and only if $k=0$.
### Two distinguished maximal subgroups {#sss:DstgMaxSbgp2}
The maximal normal subgroups $H_1,\ldots,H_4$ of $G$ contain the commutator subgroup $G^\prime$ as a normal subgroup of index $3$ and are thus of the shape $H_i=\langle g_i,G^\prime\rangle$ with suitable generators $g_i$. We want to arrange them in a fixed order.
\[dfn:NatOrdCoclGe2\] The *bipolarization* or *natural order* of the maximal subgroups $(H_i)_{1\le i\le 4}$ of $G$ is given by the *distinguished first generator* $g_1=y\in\chi_s(G)$, the *distinguished second generator* $g_2=x\notin\chi_s(G)$, both satisfying $y^3,x^3\in\gamma_3(G)\setminus\gamma_4(G)$, and the other generators $g_i=xy^{i-2}\notin\chi_s(G)$ for $3\le i\le 4$, provided that $s<m-1$. Then, in particular, $\chi_s(G)=H_1=\langle y,G^\prime\rangle$.
### Parents of core and interface groups {#sss:PrntCoclGe2}
\[dfn:CoreAndIntf\] For an arbitrary prime $p$, let $G$ be a finite $p$-group of nilpotency class $c=\mathrm{cl}(G)$. We call $G$ a *core group*, resp. an *interface group*, if its last lower central $\gamma_c(G)$ is of order $p^d$ with $d=1$, resp. $d\ge 2$.
If $G$ is of order $p^n$, the last lower central quotient $Q=G/\gamma_c(G)$ of $G$ is of order $\lvert Q\rvert=\lvert G\rvert/p^d=p^{n-d}$ and of class $\mathrm{cl}(Q)=\mathrm{cl}(G)-1$. Therefore, the coclass of $Q$ is given by $${cc}(Q)=n-d-\mathrm{cl}(Q)=n-d-\mathrm{cl}(G)+1=\mathrm{cc}(G)-(d-1).$$ Consequently, the last lower central quotient $Q$ of a core group $G$ is of the same coclass as $G$, whereas the last lower central quotient $Q$ of an interface group $G$ is of lower coclass than $G$. Obviously, a CF group must necessarily be a core group.
Now we apply these new concepts to the case $p=3$ and investigate the parent $\pi(G)$ of a metabelian $3$-group $G$ with $G/G^\prime$ of type $(3,3)$. Since the invariant $e=e(G)=\mathrm{cc}(G)+1$ indicates the first cyclic quotient $\gamma_{e+1}(G)/\gamma_{e+2}(G)$, $G$ is an interface group if and only if $e=\mathrm{cl}(G)=m-1$, where $m$ denotes the index of nilpotency of $G$. This maximal possible value of $e$ enforces a special relation between order $\lvert G\rvert=3^n$ and class $\mathrm{cl}(G)=m-1$ of $G$, $$n=e+m-2=m-1+m-2=2m-3.$$
Together with group counts in Nebelung’s theorem [@Ne1 p. 178, Thm. 5.1.16], the following two theorems describe the structure of the *metabelian skeleton* of those subgraphs of the coclass graphs $\mathcal{G}(3,r)$, $r\ge 2$, which are formed by isomorphism classes of metabelian $3$-groups $G$ having abelianization $G/G^\prime\simeq (3,3)$. This restriction concerns both, the coclass trees $\mathcal{T}$ and the sporadic part $\mathcal{G}_0(3,r)$ of each coclass graph $\mathcal{G}(3,r)$. We distinguish *core* groups and *interface* groups and begin with the former.
\[thm:CoclGe2Core\] Let $G$ be a metabelian $3$-group of coclass $r=\mathrm{cc}(G)\ge 2$ with $G/G^\prime\simeq (3,3)$, such that $G\simeq G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)\in\mathcal{G}(3,r)$ with parameters $-1\le\alpha,\beta,\gamma,\delta,\rho\le 1$, that is, $G$ is of order $\lvert G\rvert=3^n$, class $\mathrm{cl}(G)=m-1$, $4\le m<n\le 2m-3$, coclass $2\le r=n-m+1\le m-2$, and invariant $3\le e=n-m+2\le m-1$. Assume additionally that $G$ is a core group with cyclic last lower central $\gamma_{m-1}$ of order $3$, thus having $5\le m<n\le 2m-4$ and $e\le m-2$. Then the parent $\pi(G)$ of $G$ is generally given by $\pi(G)\simeq G_0^{m-1,n-1}(\rho\delta,\beta,\rho\beta,\delta)\in\mathcal{G}(3,r)$, and in particular, $$\pi(G)\simeq
\begin{cases}
G_0^{m-1,n-1}(0,\beta,0,\delta), \text{ if } \rho=0, \\
G_0^{m-1,n-1}(\rho\delta,\beta,\rho\beta,\delta), \text{ if } \rho=\pm 1,\ (\beta,\delta)\ne (0,0), \\
G_0^{m-1,n-1}(0,0,0,0), \text{ if } \rho=\pm 1,\ (\beta,\delta)=(0,0).
\end{cases}$$
The various cases of Theorem \[thm:CoclGe2Core\] can be described as follows.
1. If $G$ is a group with parameter $\rho=0$, or equivalently with defect $k=0$, then the parent $\pi(G)\simeq G_0^{m-1,n-1}(0,\beta,0,\delta)$ is a mainline group on one of the coclass trees, since these groups are characterized uniquely by $\alpha=0$, $\gamma=0$, $\rho=0$ [@Ne2]. A summary is given in Table \[tbl:MainLines\].
2. However, if $G$ is a group with $\rho=\pm 1$ or equivalently $k=1$, then the parent $\pi(G)$ has defect $\tilde{k}=0=k-1$ but lies outside of any mainline, either on a branch of a coclass tree $\mathcal{T}$ or on the sporadic part $\mathcal{G}_0(3,r)$.
3. The only exception is the very special case that $G$ with $\rho=\pm 1$ has the parameters $\beta=0$ and $\delta=0$. According to [@Ne2], this uniquely characterizes groups $G$ of transfer kernel type $\mathrm{b}.10$, $\varkappa=(0043)$, outside of mainlines, having mainline parent $\pi(G)$ of the same TKT.
Table \[tbl:MainLines\] summarizes parametrized power-commutator presentations $G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)$ with parameters $\alpha=\gamma=0$, $-1\le\beta,\delta\le 1$, $\rho=0$, $4\le m<n\le 2m-3$, $n=r+m-1$, and transfer kernel types $\varkappa$ of all metabelian mainline groups on coclass trees $\mathcal{T}$ of the coclass graphs $\mathcal{G}(3,r)$ with given coclass $r\ge 2$. In any case, the metabelian root $G_i$ of a tree $\mathcal{T}=\mathcal{T}(G_i)$ is given by the top vertex $G_i=G_0^{r+2,2r+1}(0,\beta,0,\delta)$, for which $e=m-1$, $r=m-2$, and thus $n=2r+1=2(r+2)-3=2m-3$. For the sake of comparison, the mainline of $\mathcal{G}(3,1)$ is also included. Total transfers $\varkappa(i)=0$ are counted by the invariant $\nu(G)$, cfr. [@Ma1 Dfn. 4.2, p. 488].
----------------------- ------------------ ------------ ------------- ---------------------- ---------------- ----------
$G$ $\mathrm{cc}(G)$ $m\ge r+2$ $n\ge 2r+1$ TKT $\varkappa(G)$ $\nu(G)$
$G_0^n(0,0)$ $r=1$ $\ge 3$ $\ge 3$ $\mathrm{a}.1$ $(0000)$ $4$
$G_0^{m,n}(0,0,0,0)$ $r\ge 2$ $\ge 4$ $\ge 5$ $\mathrm{b}.10$ $(0043)$ $2$
$G_0^{m,n}(0,-1,0,1)$ $r=2$ $\ge 4$ $\ge 5$ $\mathrm{c}.18$ $(0313)$ $1$
$G_0^{m,n}(0,0,0,1)$ $r=2$ $\ge 4$ $\ge 5$ $\mathrm{c}.21$ $(0231)$ $1$
$G_0^{m,n}(0,1,0,1)$ $r\ge 3$ $\ge 5$ $\ge 7$ $\mathrm{d}^\ast.19$ $(0443)$ $1$
$G_0^{m,n}(0,-1,0,1)$ $r\ge 4$ even $\ge 6$ $\ge 9$ $\mathrm{d}^\ast.19$ $(0343)$ $1$
$G_0^{m,n}(0,0,0,1)$ $r\ge 3$ $\ge 5$ $\ge 7$ $\mathrm{d}^\ast.23$ $(0243)$ $1$
$G_0^{m,n}(0,1,0,0)$ $r\ge 3$ $\ge 5$ $\ge 7$ $\mathrm{d}^\ast.25$ $(0143)$ $1$
$G_0^{m,n}(0,-1,0,0)$ $r\ge 4$ even $\ge 6$ $\ge 9$ $\mathrm{d}^\ast.25$ $(0143)$ $1$
----------------------- ------------------ ------------ ------------- ---------------------- ---------------- ----------
: Metabelian mainline groups of $\mathcal{G}(3,r)$, $r\ge 1$, sharing $\varkappa(1)=0$[]{data-label="tbl:MainLines"}
The assumption $5\le m<n\le 2m-4$, and thus $e=n-m+2\le m-2$, ensures that $G$ is not a top vertex of the coclass graph $\mathcal{G}(3,r)$. Therefore, the last lower central $\gamma_{m-1}(G)=\langle\sigma_{m-1}\rangle$ of $G$ is cyclic of order $3$. Since $G\simeq G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)$, $G$ is defined by the relations (\[eqn:PwrCmtPres\]), $$s_3\sigma_3\sigma_4=\sigma_{m-2}^{\rho\beta}\sigma_{m-1}^{\gamma}\tau_e^{\delta},\quad
t_3\tau_3^{-1}\tau_4^{-1}=\sigma_{m-2}^{-\rho\delta}\sigma_{m-1}^{-\alpha}\tau_e^{-\beta},\quad
\tau_{e+1}=\sigma_{m-1}^{-\rho},$$ and the relations for the parent $\pi(G)=G/\gamma_{m-1}(G)$ of $G$ are $$\bar{s}_3\bar{\sigma}_3\bar{\sigma}_4=\bar{\sigma}_{m-2}^{\rho\beta}\bar{\sigma}_{m-1}^{\gamma}\bar{\tau}_e^{\delta},\quad
\bar{t}_3\bar{\tau}_3^{-1}\bar{\tau}_4^{-1}=\bar{\sigma}_{m-2}^{-\rho\delta}\bar{\sigma}_{m-1}^{-\alpha}\bar{\tau}_e^{-\beta},\quad
\bar{\tau}_{e+1}=\bar{\sigma}_{m-1}^{-\rho},$$ where the left coset of an element $g\in G$ with respect to $\gamma_{m-1}(G)$ is denoted by $\bar{g}=g\cdot\gamma_{m-1}(G)$. In particular, we have $\bar{\sigma}_{m-1}=1$. Since the order of the parent is $\lvert\pi(G)\rvert=\lvert G\rvert:\lvert\gamma_{m-1}(G)\rvert=3^n:3=3^{n-1}$ and the nilpotency class is $\mathrm{cl}(\pi(G))=\mathrm{cl}(G)-1=m-2$, the coclass $r$ and the invariant $e$ remain the same, and we can view the relations as $$\bar{s}_3\bar{\sigma}_3\bar{\sigma}_4=\bar{\sigma}_{m-3}^{0}\bar{\sigma}_{m-2}^{\rho\beta}\bar{\tau}_e^{\delta},\quad
\bar{t}_3\bar{\tau}_3^{-1}\bar{\tau}_4^{-1}=\bar{\sigma}_{m-3}^{0}\bar{\sigma}_{m-2}^{-\rho\delta}\bar{\tau}_e^{-\beta},\quad
\bar{\tau}_{e+1}=1.$$ Consequently, $\pi(G)\simeq G_0^{m-1,n-1}(\rho\delta,\beta,\rho\beta,\delta)$, that is $\pi(G)\simeq G_0^{m-1,n-1}(\tilde\alpha,\tilde\beta,\tilde\gamma,\tilde\delta)$ with $\tilde\alpha=\rho\delta$, $\tilde\gamma=\rho\beta$, but $\tilde\beta=\beta$, $\tilde\delta=\delta$ remain unchanged.
The following principle, that the kernel $\varkappa(1)$ of the transfer from $G$ to the first distinguished maximal subgroup $H_1=\chi_s(G)$ decides about the relation between depth $\mathrm{dp}(G)$ and defect $k=k(G)$ of $G$, is already known from metabelian $p$-groups $G$ of coclass $\mathrm{cc}(G)=1$.
\[cor:DpthCoclGe2\] For a metabelian $3$-group $G$ of coclass $r=\mathrm{cc}(G)\ge 2$ having abelianization $G/G^\prime\simeq(3,3)$ and defect of commutativity $k=k(G)$, which does not belong to the sporadic part $\mathcal{G}_0(3,r)$, the depth $\mathrm{dp}(G)$ of $G$ on its coclass tree $\mathcal{T}$, as a subset of $\mathcal{G}(3,r)$, is given by $$\mathrm{dp}(G)=
\begin{cases}
k+1, & \text{ if } \varkappa(1)\ne 0, \\
k, & \text{ if } \varkappa(1)=0,
\end{cases}$$ with respect to the natural order of the maximal subgroups of $G$.
This follows immediately from Theorem \[thm:CoclGe2Core\] and the remark thereafter: The system of all groups $G_0^{m,n}(0,\beta,0,\delta)$ with arbitrary $4\le m<n\le 2m-3$, $-1\le\beta,\delta\le 1$, but $\alpha=\gamma=\rho=0$, consists of all mainline groups on coclass trees $\mathcal{T}$ of $\mathcal{G}(3,r)$, $r=n-m+1$, that is, of all groups $G$ with depth $\mathrm{dp}(G)=0=k$ equal to the defect $k$. According to Table \[tbl:MainLines\], all these mainline groups have a total transfer $\varkappa(1)=0$ to the first distinguished maximal subgroup $H_1$.
Since the defect of a group $G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)$ with parameter $\rho=0$ is $k=0$, all the other groups $G=G_0^{m,n}(\alpha,\beta,\gamma,\delta)$, $(\alpha,\gamma)\ne (0,0)$, with defect $k=0$ must be located at depth $\mathrm{dp}(G)=1=k+1$ on a coclass tree $\mathcal{T}$ or as a top vertex on the sporadic part $\mathcal{G}_0(3,r)$ of $\mathcal{G}(3,r)$. According to [@Ne1 Thm. 6.14, pp. 208 ff], supplemented by [@Ma2 Thm. 3.3], all these groups have a partial transfer $\varkappa(1)\ne 0$ to the first distinguished maximal subgroup $H_1$.
On the other hand, Theorem \[thm:CoclGe2Core\] shows that the relation between the defects of parent $\tilde\pi(G)$ and descendant $G$ is given by $\tilde k=0=k-1$ for any group $G=G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)$, $\rho=\mp 1$, with positive defect $k=1$, whence the depth, that is the number of steps required to reach the mainline by successive construction of parents, is given by $$\mathrm{dp}(G)=
\begin{cases}
2=k+1, \text{ if }\varkappa(1)\ne 0, \\
1=k, \text{ if }\varkappa(1)=0.
\end{cases}$$ The groups $G$ with positive defect $k=1$ are characterized by a partial transfer $\varkappa(1)\ne 0$ to the first distinguished maximal subgroup $H_1$, according to [@Ne1 Thm. 6.14, pp. 208 ff]. The only exception are the groups $G$ with parameters $\beta=0$ and $\delta=0$, that is, those with transfer kernel type $\mathrm{b}.10$, $\varkappa=(0043)$, $\varkappa(1)=0$, outside of mainlines.
We conjecture that the following property of mainline groups of $\mathcal{G}(3,r)$ might be true for mainline groups on any coclass tree of $\mathcal{G}(p,r)$, $p\ge 3$ prime, $r\ge 1$.
\[cor:MainLineCoclGe2\] Mainline groups on a coclass tree $\mathcal{T}$ of $\mathcal{G}(3,r)$, $r\ge 1$, that is, groups of depth $\mathrm{dp}(G)=0$, must have a total transfer $\varkappa(1)=0$ to the distinguished maximal subgroup $H_1=\chi_s(G)$.
See Table \[tbl:MainLines\].
Only the groups of TKT a.1, $\varkappa=(0000)$, and b.10, $\varkappa=(0043)$, outside of mainlines, prohibit that the converse of Corollary \[cor:MainLineCoclGe2\] is also true.
Top vertices $G$ on coclass trees $\mathcal{T}$ and on the sporadic part $\mathcal{G}_0(3,r)$ of a coclass graph $\mathcal{G}(3,r)$ are groups of minimal class within their coclass $r\ge 2$. They are BF *groups* with bicyclic factors, except $\gamma_2(G)/\gamma_3(G)$, in particular having a bicyclic last lower central $\gamma_{m-1}(G)$ of type $(3,3)$, and consequently they do not possess a parent $\pi(G)$ on the same coclass graph. They form the *interface* between the coclass graphs $\mathcal{G}(3,r)$ and $\mathcal{G}(3,r-1)$. We call the last lower central quotient $G/\gamma_{m-1}(G)$ of $G$ the *generalized parent* $\tilde\pi(G)$ of $G$ but we point out that there is no directed edge of depth $1$ from $\tilde\pi(G)$ to $G$. However, in the complete graph $\mathcal{G}(3)$ of all finite $3$-groups as defined by Leedham-Green and Newman [@LgNm p. 194], there is a directed edge of depth $2$ from $\tilde\pi(G)$ to $G$. This supergraph $\mathcal{G}(3)$ is the disjoint union of all coclass graphs $\mathcal{G}(3,r)$, $r\ge 0$.
\[thm:CoclGe2Intf\] Let $G$ be a metabelian $3$-group of coclass $r=\mathrm{cc}(G)\ge 2$ with $G/G^\prime\simeq (3,3)$, such that $\lvert G\rvert=3^n$, $\mathrm{cl}(G)=m-1$, $4\le m<n=2m-3$, $r=n-m+1=m-2$, $e=n-m+2=m-1$, and consequently $k=0$, that is, $G$ is an interface group with bicyclic last lower central $\gamma_{m-1}$ of type $(3,3)$. Then the generalized parent $\tilde\pi(G)\in\mathcal{G}(3,r-1)$ of $G\in\mathcal{G}(3,r)$ is given by $$\tilde\pi(G)\simeq
\begin{cases}
G_0^3(0,0)\in\mathcal{G}(3,1), & \text{ if } m=4 \text{ (and thus } n=5,\ r=2), \\
G_0^{m-1,n-2}(0,0,0,0)\in\mathcal{G}(3,r-1), & \text{ if } m\ge 5 \text{ (and thus } n\ge 7,\ r\ge 3).
\end{cases}$$
First, we consider the very special transition from second maximal to maximal class. The assumption $m=4$ implies $n=2m-3=5$. The last lower central $\gamma_{3}(G)=\langle\sigma_{3},\tau_{3}\rangle$ is bicyclic of order $3^2$, and the generalized parent $\tilde\pi(G)=G/\gamma_{3}(G)$ is of order $3^3$, of nilpotency class $2$ and of coclass $1$. The group $G$ of type $G\simeq G_0^{4,5}(\alpha,\beta,\gamma,\delta)$ satisfies the following special form of Nebelung’s relations (\[eqn:PwrCmtPres\]), $$s_2^3=1,\quad
s_3\sigma_3=\sigma_3^{\gamma}\tau_3^{\delta},\quad
t_3\tau_3^{-1}=\sigma_3^{-\alpha}\tau_3^{-\beta},$$ and since $\bar{\sigma}_3=1$ and $\bar{\tau}_3=1$, the relations for the generalized parent $\tilde\pi(G)$ can be written as Blackburn’s relations (\[eqn:PwrRelCocl1\]) and (\[eqn:CmtRelCocl1\]), $$\bar{x}^3=\bar{\tau_3}=1,\quad
\bar{y}^3\bar{s}_2^3\bar{s}_3=\bar{\sigma}_3\cdot 1\cdot\bar{\sigma}_3^{\gamma-1}\bar{\tau}_3^\delta=1,\quad
\lbrack\bar{s_2},\bar{y}\rbrack=\bar{t_3}=\bar{\sigma}_3^{-\alpha}\bar{\tau}_3^{1-\beta}=1,$$ which imply that $\tilde\pi(G)\simeq G_0^3(0,0)$.
Now, let $m\ge 5$. Since $e=m-1$, and the last lower central $\gamma_{m-1}(G)=\langle\sigma_{m-1},\tau_{m-1}\rangle$ is bicyclic of type $(3,3)$, the order of the generalized parent is $\lvert\tilde\pi(G)\rvert=\lvert G\rvert:\lvert\gamma_{m-1}(G)\rvert=3^n:3^2=3^{n-2}$, the nilpotency class is $\mathrm{cl}(\tilde\pi(G))=\mathrm{cl}(G)-1=m-2$, and the coclass $\tilde r=n-2-(m-2)=2m-3-m=m-3=r-1$ and the invariant $\tilde e=\tilde r+1=r=e-1$ decrease by $1$. Since $k=0$, $\rho=0$, the group $G$ of type $G\simeq G_0^{m,n}(\alpha,\beta,\gamma,\delta)$, is defined by a special form of the relations (\[eqn:PwrCmtPres\]), $$s_3\sigma_3\sigma_4=\sigma_{m-1}^{\gamma}\tau_e^{\delta},\quad
t_3\tau_3^{-1}\tau_4^{-1}=\sigma_{m-1}^{-\alpha}\tau_e^{-\beta},\quad
\tau_{e+1}=1.$$ Since $\bar{\sigma}_{m-1}=1$ and $\bar{\tau}_{m-1}=1$, the relations for the generalized parent $\tilde\pi(G)$ are $$\bar{s}_3\bar{\sigma}_3\bar{\sigma}_4=\bar{\sigma}_{m-1}^{\gamma}\bar{\tau}_{m-1}^{\delta}=1,\quad
\bar{t}_3\bar{\tau}_3^{-1}\bar{\tau}_4^{-1}=\bar{\sigma}_{m-1}^{-\alpha}\bar{\tau}_{m-1}^{-\beta}=1,\quad
\bar{\tau}_e=\bar{\tau}_{m-1}=1,$$ and therefore we have $\tilde\pi(G)\simeq G_0^{m-1,n-2}(0,0,0,0)$.
Second $3$-class groups $G=\mathrm{G}_3^2(K)$ of coclass $\mathrm{cc}(G)\ge 2$ with $G/G^\prime\simeq (3,3)$ {#ss:ScndClgpTyp33CoclGe2}
------------------------------------------------------------------------------------------------------------
### Weak transfer target type $\tau_0(G)$ expressed by $3$-class numbers {#sss:wTTTCoclGe2}
The group theoretic information on the second $3$-class group $G=\mathrm{G}_3^2(K)$, that is, its class, coclass, and defect, is contained in the $3$-class numbers of the two distinguished extensions $L_1,L_2$ and of the Hilbert $3$-class field $\mathrm{F}_3^1(K)$. Additionally, the principalization $\kappa(1)$ of $K$ in the first distinguished extension $L_1$ determines the connection between defect and depth of $G$.
\[thm:wTTTCoclGe2\]
Let $K$ be an arbitrary number field having $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$. Suppose the second $3$-class group $G=\mathrm{Gal}(\mathrm{F}_3^2(K)\vert K)$ of $K$ is of coclass $\mathrm{cc}(G)\ge 2$ with defect $k=k(G)$, order $\lvert G\rvert=3^n$, and class $\mathrm{cl}(G)=m-1$, where $4\le m<n\le 2m-3$. With respect to the natural order of the maximal subgroups $(H_i)_{1\le i\le 4}$ of $G$, fixed in Definition \[dfn:NatOrdCoclGe2\], the weak transfer target type $\tau_0(G)=(\lvert H_i/H_i^{\prime}\rvert)_{1\le i\le 4}$, that is the family of $3$-class numbers of the quadruplet $(L_1,\ldots,L_4)$ of unramified cyclic cubic extension fields of $K$, forming the first layer, is given by
$$\begin{aligned}
\tau_0(G) = (\mathrm{h}_3(L_1),\ldots,\mathrm{h}_3(L_4)) =
(3^{\mathrm{cl}(G)-k},3^{\mathrm{cc}(G)+1},3^3,3^3),\end{aligned}$$
where, in the case of a non-sporadic group $G$ on some coclass tree, defect $k$ and depth $\mathrm{dp}(G)$ are related by $$k=
\begin{cases}
\mathrm{dp}(G)-1, & \text{ if } \varkappa(1)\ne 0, \\
\mathrm{dp}(G), & \text{ if } \varkappa(1)=0.
\end{cases}$$
For the second layer, consisting of the Hilbert $3$-class field $\mathrm{F}_3^1(K))$ only, the $3$-class number is given by $$\mathrm{h}_3(\mathrm{F}_3^1(K)) = 3^{\mathrm{cl}(G)+\mathrm{cc}(G)-2}.$$
The statement is a succinct version of [@Ma1 Thm. 3.4], expressed by concepts more closely related to the position of $G$ on the coclass graphs $\mathcal{G}(3,r)$, $r\ge 2$, and to the transfer kernel type $\varkappa(G)$ of $G$, using Corollary \[cor:DpthCoclGe2\].
Whereas $\mathrm{h}_3(L_3)$ and $\mathrm{h}_3(L_4)$ only indicate that $\mathrm{cc}(G)\ge 2$, the second distinguished $\mathrm{h}_3(L_2)$ gives the precise coclass $r$ of $G$, $\mathrm{h}_3(\mathrm{F}_3^1(K))$ determines the order $3^n$, $n=\mathrm{cl}(G)+\mathrm{cc}(G)$, and class of $G$, and the first distinguished $\mathrm{h}_3(L_1)$ yields the defect $k$ of $G$.\
With respect to the mainline $(M_j)_{j\ge 2r+1}$ of the coclass tree $\mathcal{T}$ containing $G$, the order $\lvert M_i\rvert=3^i$ of the branch root $M_i$ of a non-sporadic group $G$ is given by $i=n-\mathrm{dp}(G)=\mathrm{cl}(G)+\mathrm{cc}(G)-\mathrm{dp}(G)$, where $$\mathrm{dp}(G)=
\begin{cases}
k, & \text{ if } \varkappa(1)=0, \\
k+1, & \text{ if } \varkappa(1)\ne 0.
\end{cases}$$
### Selection Rules for quadratic base fields {#sss:SelRuleCoclGe2}
Let $K=\mathbb{Q}(\sqrt{D})$ be a quadratic number field with discriminant $D$ and $3$-class group $\mathrm{Cl}_3(K)$ of type $(3,3)$. Then the $4$ unramified cyclic cubic extension fields $(L_1,\ldots,L_4)$ of $K$ have dihedral absolute Galois groups $\mathrm{Gal}(L_i\vert K)$ of degree $6$, according to [@Ma1 Prop. 4.1]. Consequently each sextic field $L_i$ contains a cubic subfield $K_i$, whose invariants can be computed easier than those of $L_i$ and are also sufficient to determine complete information on the group $G=\mathrm{G}_3^2(K)$.
\[thm:SelRuleCoclGe2\]
Let $G=\mathrm{Gal}(\mathrm{F}_3^2(K)\vert K)$ be the second $3$-class group of $K=\mathbb{Q}(\sqrt{D})$. If $G\in\mathcal{G}(3,r)$ for some coclass $r\ge 2$, then the family of $3$-class numbers of the non-Galois subfields $K_i$ of $L_i$, with respect to the natural order fixed in Definition \[dfn:NatOrdCoclGe2\] is given by
$$\begin{aligned}
(\mathrm{h}_3(K_1),\ldots,\mathrm{h}_3(K_4)) =
\begin{cases}
(3^{\frac{\mathrm{cl}(G)-(k+1)}{2}},3^{\frac{\mathrm{cc}(G)}{2}},3,3) & \text{ for sporadic }G,\text{ (where always }\varkappa(2)\ne 0,) \\
(3^{\frac{\mathrm{cl}(G)-\mathrm{dp}(G)}{2}},3^{\frac{\mathrm{cc}(G)+1}{2}},3,3) & \text{ otherwise, if }\varkappa(2)=0, \\
(3^{\frac{\mathrm{cl}(G)-\mathrm{dp}(G)}{2}},3^{\frac{\mathrm{cc}(G)}{2}},3,3) & \text{ otherwise, if }\varkappa(2)\ne 0.
\end{cases} \\\end{aligned}$$
The order $\lvert M_i\rvert=3^i$ of the root $M_i$ for a non-sporadic group $G$ on branch $\mathcal{B}(M_i)$ of some coclass tree is given by $$i=\mathrm{cl}(G)+\mathrm{cc}(G)-\mathrm{dp}(G)\equiv
\begin{cases}
1\pmod{2}, & \text{ if }\varkappa(2)=0, \\
0\pmod{2}, & \text{ if }\varkappa(2)\ne 0.
\end{cases}$$
While $\mathrm{h}_3(K_3)$ and $\mathrm{h}_3(K_4)$ do not provide any information, the second distinguished $\mathrm{h}_3(K_2)$ indicates the coclass of $G$ and enforces the parity $$\mathrm{cc}(G)\equiv
\begin{cases}
1\pmod{2}, & \text{ if }\varkappa(2)=0, \\
0\pmod{2}, & \text{ if }\varkappa(2)\ne 0,
\end{cases}$$ in dependence on the principalization $\kappa(2)$ of $K$ in the second distinguished extension $L_2$, and the first distinguished $\mathrm{h}_3(K_1)$ demands $\mathrm{cl}(G)-\mathrm{dp}(G)\equiv 0\pmod{2}$, for non-sporadic $G$.
See [@Ma1 Thm. 4.2.]. For the branch root order $\lvert M_i\rvert=3^i$ of a non-sporadic vertex $G$ of order $\lvert G\rvert=3^n$, we use the relations $n=\mathrm{cl}(G)+\mathrm{cl}(G)$ and $i=n-\mathrm{dp}(G)$, the depth being the number of successive steps on the path between $G$ and $M_i$, each decreasing order and class by $1$ and keeping the coclass constant.
### Identifying densely populated vertices by fast algorithms {#sss:DensePopulation}
The top vertices $G$ on $\mathcal{G}(3,2)$ with $G/G^\prime$ of type $(3,3)$ in Figure \[fig:Typ33Cocl2\] can be identified by the fast algorithm given in [@Ma3 § 5.2–5.3], using the TTT and the counter $\varepsilon$ of $\tau(i)=(3,3,3)$, $1\le i\le 4$, in Table \[tbl:TttTop33\].
------------------------- ------------- ----------------- ------------- ------------- ----------- ----------- ----------- ----------- ---------------
Id of isoclinism Hilbert
$3$-group $G$ family TKT $\varkappa$ TTT
$\tau(0)$ $\tau(1)$ $\tau(2)$ $\tau(3)$ $\tau(4)$ $\varepsilon$
$\langle 243,5\rangle$ $\Phi_6$ $\mathrm{D}.10$ $(2241)$ $(3,3,3)$ $(9,3)$ $(9,3)$ $(3,3,3)$ $(9,3)$ $1$
$\langle 243,7\rangle$ $\Phi_6$ $\mathrm{D}.5$ $(4224)$ $(3,3,3)$ $(3,3,3)$ $(9,3)$ $(3,3,3)$ $(9,3)$ $2$
$\langle 243,9\rangle$ $\Phi_6$ $\mathrm{G}.19$ $(2143)$ $(3,3,3)$ $(9,3)$ $(9,3)$ $(9,3)$ $(9,3)$ $0$
$\langle 729,57\rangle$ $\Phi_{43}$ $\mathrm{G}.19$ $(2143)$ $(3,3,3,3)$ $(9,3)$ $(9,3)$ $(9,3)$ $(9,3)$ $0$
$\langle 243,4\rangle$ $\Phi_6$ $\mathrm{H}.4$ $(4443)$ $(3,3,3)$ $(3,3,3)$ $(3,3,3)$ $(9,3)$ $(3,3,3)$ $3$
$\langle 729,45\rangle$ $\Phi_{42}$ $\mathrm{H}.4$ $(4443)$ $(9,3,3)$ $(3,3,3)$ $(3,3,3)$ $(9,3)$ $(3,3,3)$ $3$
$\langle 243,3\rangle$ $\Phi_6$ $\mathrm{b}.10$ $(0043)$ $(3,3,3)$ $(9,3)$ $(9,3)$ $(3,3,3)$ $(3,3,3)$ $2$
$\langle 243,6\rangle$ $\Phi_6$ $\mathrm{c}.18$ $(0313)$ $(3,3,3)$ $(9,3)$ $(9,3)$ $(3,3,3)$ $(9,3)$ $1$
$\langle 729,49\rangle$ $\Phi_{23}$ $\mathrm{c}.18$ $(0313)$ $(9,3,3)$ $(9,9)$ $(9,3)$ $(3,3,3)$ $(9,3)$ $1$
$\langle 243,8\rangle$ $\Phi_6$ $\mathrm{c}.21$ $(0231)$ $(3,3,3)$ $(9,3)$ $(9,3)$ $(9,3)$ $(9,3)$ $0$
$\langle 729,54\rangle$ $\Phi_{23}$ $\mathrm{c}.21$ $(0231)$ $(9,3,3)$ $(9,9)$ $(9,3)$ $(9,3)$ $(9,3)$ $0$
------------------------- ------------- ----------------- ------------- ------------- ----------- ----------- ----------- ----------- ---------------
: TTT and $\varepsilon$ of the top vertices of type $(3,3)$ on $\mathcal{G}(3,2)$[]{data-label="tbl:TttTop33"}
For quadratic fields $K=\mathbb{Q}(\sqrt{D})$, the following metabelian $3$-groups $G\in\Phi_6$ cannot be realized as second $3$-class groups $\mathrm{G}_3^2(K)$.
- $\langle 243,9\rangle$, $\langle 243,4\rangle$, since they are not Schur $\sigma$-groups (Thm. \[thm:SpecWeakLeaf\]).
- $\langle 243,6\rangle$, $\langle 243,8\rangle$, due to the selection rule for branches in Theorem \[thm:SelRuleCoclGe2\].
- $\langle 243,3\rangle$, according to the remark after Theorem \[thm:SelRuleCoclGe2\], since $\varkappa(2)=0$ enforces odd coclass.
### Top vertices of type $(3,3)$ on $\mathcal{G}(3,2)$ {#sss:Typ33Cocl2}
Figure \[fig:Typ33Cocl2\] shows the interface between the coclass graphs $\mathcal{G}(3,1)$ and $\mathcal{G}(3,2)$. The extra special group $G_0^3(0,0)$ of order $27$ and exponent $3$, which is the second member of the unique mainline of $\mathcal{G}(3,1)$, is the generalized parent $\tilde\pi(G)$ of all top vertices $G$ of $\mathcal{G}(3,2)$. We point out that the connecting edges of depth $2$ neither belong to $\mathcal{G}(3,1)$ nor to $\mathcal{G}(3,2)$. The metabelian skeleton of this graph is also shown in [@Ne1 p. 189 ff] and the complete graph, including the non-metabelian leaves, was first drawn in [@AHL Tbl. 1–2, pp. 265–266] and [@As1 Fig. 4.6–4.7, p. 74]. Among the non-CF groups $G$ with abelianization $G/G^\prime$ of type $(3,3)$ at the top of coclass graph $\mathcal{G}(3,2)$, which form the stem of isoclinism family $\Phi_6$, we have, from the left to the right:
- two terminal vertices $\langle 243,5\rangle$, $\langle 243,7\rangle$, the only Schur $\sigma$-groups of order $3^5$ [@BBH Thm. 4.2, p. 14],
- two roots $\langle 243,9\rangle$ and $\langle 243,4\rangle$ of finite trees of depth $2$, whose metabelian descendants belong to the stem of the isoclinism families $\Phi_{43}$ and $\Phi_{42}$,
- a root $\langle 243,3\rangle$ of an infinite tree, which is not a coclass tree and is shown in Figure \[fig:TreeBTyp33Cocl2\],
- and two roots $\langle 243,6\rangle$, $\langle 243,8\rangle$ of coclass trees, shown in detail in Figures \[fig:TreeQTyp33Cocl2\] and \[fig:TreeUTyp33Cocl2\].
The sporadic part $\mathcal{G}_0(3,2)$ of $\mathcal{G}(3,2)$ consists of the terminal vertices $\langle 243,5\rangle$ and $\langle 243,7\rangle$, the finite trees $\mathcal{T}(\langle 243,9\rangle)$ and $\mathcal{T}(\langle 243,4)\rangle$, and a certain finite subset of the difference $\mathcal{T}(\langle 243,3\rangle)\setminus\mathcal{T}(\langle 729,40\rangle)$.
The vertices of the coclass graph $\mathcal{G}(3,2)$ in Figure \[fig:Typ33Cocl2\] are classified by using different symbols:
1. a large contour square $\square$ represents an abelian group,
2. a big contour circle [$\circ$]{} represents a metabelian group containing an abelian maximal subgroup, all other metabelian groups do not possess abelian subgroups,
3. big full discs [$\bullet$]{} represent metabelian groups with bicyclic centre of type $(3,3)$ and defect $k=0$,
4. small full discs [$\bullet$]{} represent metabelian groups with cyclic centre of order $3$ and defect $k=1$,
5. small contour squares [$\square$]{} represent non-metabelian groups.
Groups of particular importance are labelled by a number in angles. This is the identifier in the SmallGroups library [@BEO] of GAP [@GAP] and [@MAGMA], where we omit the order, which is given on the left hand scale.
The actual distribution of the $2020$, resp. $2576$, second $3$-class groups $G_3^2(K)$ of complex, resp. real, quadratic number fields $K=\mathbb{Q}(\sqrt{D})$ of type $(3,3)$ with discriminant $-10^6<D<10^7$ is represented by underlined boldface counters (in the format complex/real) of the hits of vertices surrounded by the adjacent oval.
It is illuminating to compare these frequencies, which we have computed in [@Ma1 § 6, Tbl. 3–5] and [@Ma3 § 6, Tbl. 13–15,17] with the non-abelian generalization of the asymptotic Cohen-Lenstra-Martinet probability, which is given for complex quadratic fields by [@BBH p. 18, Tbl. 2] with respect to all $3190$ discriminants of $3$-rank $2$ instead to the $2020$ discriminants of type $(3,3)$ in the range $-10^6<D<0$. In three cases of Table \[tbl:NonAbCohenLenstra\], the actual percentage exceeds the conjectural asymptotic probability. In the first case the excess is significant. A possible interpretation is that the population of vertices of higher order will become more probable in ranges of considerably bigger absolute values of discriminants so that the percentage of hits of the low order vertices in the table will decrease.
Id of $G$ frequency percentage probability
------------------------- ----------- ------------ -------------
$\langle 243,5\rangle$ $667$ $20.91$ $17.56$
$\langle 243,7\rangle$ $269$ $8.43$ $8.78$
$\langle 729,57\rangle$ $94$ $2.95$ $2.19$
$\langle 729,45\rangle$ $297$ $9.31$ $8.78$
: Asymptotic probability for densely populated vertices of $\mathcal{G}(3,2)$[]{data-label="tbl:NonAbCohenLenstra"}
Identification of the vertex $\langle 729,57\rangle$, resp. $\langle 729,45\rangle$, among two, resp. four, closely related vertices in isoclinism family $\Phi_{43}$, resp. $\Phi_{42}$, was possible by means of the following *Artin criterion* for second $p$-class groups of quadratic base fields, which can be verified by testing for a suitable automorphism of order $2$.
Let $p\ge 3$ be an odd prime. The second $p$-class group $G=\mathrm{G}_p^2(K)$ of a quadratic field $K=\mathbb{Q}(\sqrt{D})$ admits an extension by the cyclic group $C_2$, $1\to G\to H\to C_2\to 1$, such that $H^\prime\simeq G$.
See the letter of E. Artin to H. Hasse from November 19, 1928 [@FRL].
### Coclass trees of type $(3,3)$ and $(9,3)$ on $\mathcal{G}(3,2)$ {#sss:Trees33Cocl2}
\[d:ForbTrees\] Let $p\ge 3$ be an odd prime. A rooted subtree $\mathcal{T}$ of a coclass graph $\mathcal{G}(p,r)$, $r\ge 1$, is called *forbidden*, if none of its vertices $G\in\mathcal{T}$ can be realized as the second $p$-class group $\mathrm{G}_p^2(K)$ of a quadratic field $K=\mathbb{Q}(\sqrt{D})$. Otherwise $\mathcal{T}$ is called *admissible*.
\[thm:ForbTrees\] Let $\mathcal{T}$ be a coclass tree of a coclass graph $\mathcal{G}(3,r)$, $r\ge 1$, such that all its mainline groups $M$ are metabelian with abelianization of type $(3,3)$.
1. The unique tree of $\mathcal{G}(3,1)$ and the trees of $\mathcal{G}(3,2)$, whose mainline groups $M$ are of transfer kernel type either $\mathrm{c}.18$, $\varkappa=(0313)$, or $\mathrm{c}.21$, $\varkappa=(0231)$, are admissible.
2. If all mainline groups $M$ are of transfer kernel type $\mathrm{b}.10$, $\varkappa=(0043)$, with distinguished second member $\varkappa(2)=0$, then $\mathcal{T}$ is forbidden if and only if the coclass $r\ge 2$ is even.
3. If all mainline groups $M$ are of transfer kernel type either $\mathrm{d}^\ast.23$, $\varkappa=(0243)$, or $\mathrm{d}^\ast.19$, $\varkappa=(0443)$, or $\mathrm{d}^\ast.25$, $\varkappa=(0143)$, with distinguished second member $\varkappa(2)\ne 0$, then $\mathcal{T}$ is forbidden if and only if the coclass $r\ge 3$ is odd.
Referring to [@Ne1 p. 189 ff] we point out the following details.
1. There is a periodic pattern of period length $2$ of rooted subtrees with metabelian mainlines of type $(3,3)$ among the coclass graphs $\mathcal{G}(p,r)$, setting in with $r=3$. The roots of the trees with fixed coclass $r$ are of order $3^{2r+1}$. The metabelian skeletons of the trees with common transfer kernel type and coclass of the same parity are isomorphic as graphs. The same is true for the metabelian skeletons of sporadic groups with coclass of the same parity.
2. For odd coclass $r\ge 3$, there are $4$ trees with mainlines of transfer kernel types $\mathrm{b}.10$, $\varkappa=(0043)$, $\mathrm{d}^\ast.23$, $\varkappa=(0243)$, $\mathrm{d}^\ast.19$, $\varkappa=(0443)$, and $\mathrm{d}^\ast.25$, $\varkappa=(0143)$.
3. For even coclass $r\ge 4$, there are $6$ trees with mainlines of transfer kernel types $\mathrm{b}.10$, $\varkappa=(0043)$, and $\mathrm{d}^\ast.23$, $\varkappa=(0243)$, each occurring only once, and on the other hand $\mathrm{d}^\ast.19$, $\varkappa=(0443)$, and $\mathrm{d}^\ast.25$, $\varkappa=(0143)$, each occurring in two instances, isomorphic as graphs.
Theorem \[thm:ForbTrees\] is an immediate consequence of Theorem \[thm:SelRuleCoclGe2\], due to the second distinguished member $\varkappa(2)$ of the TKT. See the diagrams on the pages without numbers, following [@Ne1 p. 189]. These diagrams were constructed by means of the lists of representatives $G_\rho^{m,n}(\alpha,\beta,\gamma,\delta)$ for isomorphism classes, given in the appendix [@Ne2] of Nebelung’s thesis. The connection with the transfer kernel types $\varkappa$ is established in [@Ne1 Thm. 6.14, pp. 208 ff].
SmallGroups Id of root $G$ $G/G^\prime$ TKT resp. pTKT $\varkappa$ $\varepsilon$ $\eta\in$ $(f,g,h)$ population
---------------------------- -------------- ----------------- ------------- --------------- ---------------------- ----------- ------------
$\langle 729,40\rangle$ $(3,3)$ $\mathrm{b}.10$ $(0043)$ $2$ $\lbrace 1,2\rbrace$ $(0,1,0)$ forbidden
$\langle 243,6\rangle$ $(3,3)$ $\mathrm{c}.18$ $(0313)$ $1$ $\lbrace 0,1\rbrace$ $(0,1,2)$ admissible
$\langle 243,8\rangle$ $(3,3)$ $\mathrm{c}.21$ $(0231)$ $0$ $\lbrace 2,3\rbrace$ $(1,1,2)$ admissible
$\langle 243,17\rangle$ $(9,3)$ $\mathrm{a}.1$ $(000;0)$ $2$ $\lbrace 3,4\rbrace$ $(0,0,1)$ admissible
$\langle 243,15\rangle$ $(9,3)$ $\mathrm{a}.1$ $(000;0)$ $1$ $\lbrace 3,4\rbrace$ $(1,0,0)$ admissible
: Infinite trees of types $(3,3)$, $(9,3)$ with metabelian mainline on $\mathcal{G}(3,2)$[]{data-label="tbl:TreesBQUAG"}
Aside from the single forbidden coclass tree with metabelian mainline of transfer kernel type $\mathrm{b}.10$, $\varkappa=(0043)$, there exist $4$ admissible coclass trees with metabelian mainline on $\mathcal{G}(3,2)$ which are populated quite densely by second $3$-class groups $\mathrm{G}_3^2(K)$ of quadratic fields $K$ with $3$-class group of type $(3,3)$, resp. $(9,3)$. They can be characterized by the number of members of the TTT with $3$-rank bigger than $2$, $\varepsilon=\#\lbrace 1\le i\le 4\mid\mathrm{r}_3(H_i/H_i^\prime)\ge 3\rbrace$, or by the number of members of the TKT, resp. punctured TKT (pTKT), having Taussky’s type A, $\eta=\#\lbrace 1\le i\le 4\mid\kappa(i)=\mathrm{A}\rbrace$, as shown in Table \[tbl:TreesBQUAG\]. Groups along their mainlines arise as quotients of infinite pro-$3$-groups of coclass $2$ having a non-trivial centre, whose pro-$3$ presentations are defined by suitable triplets $(f,g,h)$ of relational exponents in [@ELNO Thm. 4.1].
\[t:TreeQ\] The structure of the complete coclass tree $\mathcal{T}(\langle 243,6\rangle)$ as part of the coclass graph $\mathcal{G}(3,2)$, restricted to $3$-groups $G$ with abelianization $G/G^\prime\simeq (3,3)$, is globally characterized by the tree invariant $\varepsilon(G)=1$ and given up to order $3^{11}=177\,147$ by Figure \[fig:TreeQTyp33Cocl2\]. The branches are of depth $3$ and periodic of length $2$. The pre-period consists of $\mathcal{B}_5,\mathcal{B}_6$, the primitive period of $\mathcal{B}_7,\mathcal{B}_8$
In Figure \[fig:TreeQTyp33Cocl2\], we have $G_3^2(\mathbb{Q}(\sqrt{D}))\in\mathcal{T}(\langle 243,6\rangle)$ for $270$ $(13.4\%)$ of the $2020$ discriminants $-10^6<D<0$ and for $39$ $(1.5\%)$ of the $2576$ discriminants $0<D<10^7$, investigated in [@Ma1 § 6], [@Ma3 § 6].\
Since the TKT $\mathrm{c}.18$, $\varkappa=(0313)$, of the mainline is *total* with $\varkappa(1)=0$, there only occur $G_3^2(K)$ of *real* quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $D>0$, on the mainline.\
Due to the *Selection Rule* in Theorem \[thm:SelRuleCoclGe2\], the $G_3^2(K)$ are distributed on *even branches* only, since the second distinguished transfer kernel $\varkappa(2)\ne 0$.\
Underpinning the weak leaf conjecture, there is no actual hit of the vertices at depth $1$ with TKT $\mathrm{H}.4$, $\varkappa=(3313)$.
\[t:TreeU\] The structure of the complete coclass tree $\mathcal{T}(\langle 243,8\rangle)$ as part of the coclass graph $\mathcal{G}(3,2)$, restricted to $3$-groups $G$ with abelianization $G/G^\prime\simeq (3,3)$, is globally characterized by $\varepsilon(G)=0$ and given up to order $3^{11}=177\,147$ by Figure \[fig:TreeUTyp33Cocl2\]. The branches are of depth $3$ and periodic of length $2$. The pre-period consists of $\mathcal{B}_5,\mathcal{B}_6$, the primitive period of $\mathcal{B}_7,\mathcal{B}_8$
In Figure \[fig:TreeUTyp33Cocl2\], we have $G_3^2(\mathbb{Q}(\sqrt{D}))\in\mathcal{T}(\langle 243,8\rangle)$ for $291$ $(14.4\%)$ of the $2020$ discriminants $-10^6<D<0$ and for $43$ $(1.7\%)$ of the $2576$ discriminants $0<D<10^7$, investigated in [@Ma1 § 6], [@Ma3 § 6].\
Since the TKT $\mathrm{c}.21$, $\varkappa=(0231)$, of the mainline is *total* with $\varkappa(1)=0$, there only occur $G_3^2(K)$ of *real* quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $D>0$, on the mainline.\
Due to the *Selection Rule* in Theorem \[thm:SelRuleCoclGe2\], the $G_3^2(K)$ are distributed on *even branches* only, since the second distinguished transfer kernel $\varkappa(2)\ne 0$.\
Underpinning the weak leaf conjecture, there is no actual hit of the vertices at depth $1$ with TKT $\mathrm{G}.16$, $\varkappa=(4231)$.
The vertices of the coclass trees in both Figures \[fig:TreeQTyp33Cocl2\] and \[fig:TreeUTyp33Cocl2\] are classified by using different symbols:
1. big full discs [$\bullet$]{} represent metabelian groups with bicyclic centre of type $(3,3)$ and defect $k=0$,
2. small full discs [$\bullet$]{} represent metabelian groups with cyclic centre of order $3$ and defect $k=1$,
3. small contour squares [$\square$]{} represent non-metabelian groups.
A number adjacent to a vertex denotes the multiplicity of a batch of immediate descendants sharing a common parent. The groups of particular importance are labelled by a number in angles, which is the identifier in the SmallGroups library [@BEO] of GAP [@GAP] and MAGMA [@MAGMA]. The metabelian skeletons were drawn in [@Ne1 p. 189 ff], the complete trees were given in [@As1 p. 76, Fig. 4.8 and p. 123, Fig. 6.1].
The actual distribution of the $2020$, resp. $2576$, second $3$-class groups $G_3^2(K)$ of complex, resp. real, quadratic number fields $K=\mathbb{Q}(\sqrt{D})$ of type $(3,3)$ with discriminant $-10^6<D<10^7$ is represented by underlined boldface counters (in the format complex/real) of the hits of vertices surrounded by the adjacent oval. See [@Ma1 § 6, tbl. 3–5] and [@Ma3 § 6, tbl. 15–18].\
The realization of mainline vertices with TKT $\mathrm{c}.18$ and $\mathrm{c}.21$ as $G_3^2(K)$ is no violation of the weak leaf conjecture \[cnj:WeakLeafCnj\], since these vertices do not possess metabelian immediate descendants of the same TKT.
When we had completed our extensive investigation of second $3$-class groups $\mathrm{G}_3^2(K)$ of all $4596$ quadratic fields $K=\mathbb{Q}(\sqrt{D})$ of type $(3,3)$ in the range $-10^6<D<10^7$, we wondered whether the distribution of second $3$-class groups $\mathrm{G}_3^2(K)$ for other sequences of base fields $K$ of type $(3,3)$ shows similarities or differences.
Since fields of degree $4$ are still within the reach of numerical computations, we are able to present the results for bicyclic biquadratic fields of Gauss-Dirichlet-Hilbert type $K=\mathbb{Q}\left(\sqrt{\strut -1},\sqrt{\strut d}\right)$ [@Hi] in section §\[ss:NewRsltGDH\]. These fields reveal strong similarities to quadratic fields. In section §\[ss:NewRsltESR\], however, we show that bicyclic biquadratic fields of Eisenstein-Scholz-Reichardt type $K=\mathbb{Q}\left(\sqrt{\strut -3},\sqrt{\strut d}\right)$ [@So; @Re] exhibit a completely different behavior.
$d$ $d^\prime$ $\tau(G)=(\mathrm{Cl}_3(L_i))_{1\le i\le 4}$ Type $\varkappa(G)$ $G$ $\mathrm{cc}(G)$ $\Phi$
--------- ------------ ---------------------------------------------- --------------- ---------------- --------------------------------- ------------------ -------------
$3896$ $-3896$ $(3,3,3),(3,3,3),(9,3),(3,3,3)$ H.4 $(4443)$ $\langle 729,45\rangle$ $2$ $\Phi_{42}$
$5069$ $-20276$ $(3,3,3),(9,3),(3,3,3),(9,3)$ D.5 $(4224)$ $\langle 243,7\rangle$ $2$ $\Phi_{6}$
$10173$ $-40692$ $(3,3,3),(9,3),(9,3),(9,3)$ D.10 $(2241)$ $\langle 243,5\rangle$ $2$ $\Phi_{6}$
$12481$ $-49924$ $(9,3),(9,3),(9,3),(9,3)$ G.19 $(2143)$ $\langle 729,57\rangle$ $2$ $\Phi_{43}$
$2437$ $-9748$ $(27,9),(9,3),(9,3),(9,3)$ E.9 $(2231)$ $G_0^{6,7}(0,0,\pm 1,1)$ $2$
$5417$ $-21668$ $(27,9),(9,3),(3,3,3),(9,3)$ H.4$\uparrow$ $(3313)$ $G_{\pm 1}^{7,8}(0,-1,\pm 1,1)$ $2$
$6221$ $-24884$ $(27,9),(9,3),(9,3),(9,3)$ G.16 $(4231)$ $G_1^{7,8}(0,0,\pm 1,1)$ $2$
$12837$ $-51348$ $(27,9),(9,3),(3,3,3),(9,3)$ E.14 $(2313)$ $G_0^{6,7}(0,-1,\pm 1,1)$ $2$
$15544$ $-15544$ $(27,9),(9,3),(3,3,3),(9,3)$ E.6 $(1313)$ $G_0^{6,7}(1,-1,1,1)$ $2$
$6789$ $-27156$ $(27,9),(27,9),(3,3,3),(3,3,3)$ F.11 $(1143)$ $G_0^{6,9}(0,0,\pm 1,1)$ $4$
$7977$ $-31908$ $(27,9),(27,9),(3,3,3),(3,3,3)$ F.12 $(1343)$ $G_0^{6,9}(\mp 1,0,\pm 1,1)$ $4$
: $11$ inherited variants of $G=\mathrm{G}_3^2(K)$ for $K=\mathbb{Q}\left(\sqrt{\strut -1},\sqrt{\strut d}\right)$[]{data-label="tbl:LiftedVariants"}
$d$ $\tau(G)=(\mathrm{Cl}_3(L_i))_{1\le i\le 4}$ Type $\varkappa(G)$ $G$ $\mathrm{cc}(G)$ $\Phi$
--------- ---------------------------------------------- ------------------------ ---------------- --------------------------------- ------------------ -------------
$473$ $(9,3),(3,3),(3,3),(3,3)$ a.3 $(2000)$ $\langle 81,8\rangle$ $1$ $\Phi_3$
$1937$ $(3,3,3),(3,3),(3,3),(3,3)$ a.3\* $(2000)$ $\langle 81,7\rangle$ $1$ $\Phi_3$
$2993$ $(27,9),(3,3),(3,3),(3,3)$ a.3$\uparrow$ $(2000)$ $\langle 729,97\vert 98\rangle$ $1$ $\Phi_{35}$
$2713$ $(3,3,3),(3,3,3),(9,3),(3,3,3)$ H.4 $(4443)$ $\langle 729,45\rangle$ $2$ $\Phi_{42}$
$3305$ $(9,9),(9,3),(9,3),(9,3)$ c.21 $(0231)$ $\langle 729,54\rangle$ $2$ $\Phi_{23}$
$3941$ $(9,3),(9,3),(9,3),(9,3)$ G.19 $(2143)$ $\langle 729,57\rangle$ $2$ $\Phi_{43}$
$13153$ $(3,3,3),(9,3),(9,9),(9,3)$ c.18 $(0313)$ $\langle 729,49\rangle$ $2$ $\Phi_{23}$
$7665$ $(27,9),(9,3),(9,3),(9,3)$ G.16 $(4231)$ $G_1^{7,8}(0,0,\pm 1,1)$ $2$
$15265$ $(27,9),(9,3),(3,3,3),(9,3)$ H.4$\uparrow$ $(3313)$ $G_{\pm 1}^{7,8}(0,-1,\pm 1,1)$ $2$
$5912$ $(27,9),(27,9),(3,3,3),(3,3,3)$ G.16$\uparrow\uparrow$ $(1243)$ $G_{\pm 1}^{7,10}(0,0,\pm 1,1)$ $4$
$12685$ $(27,9),(27,9),(3,3,3),(3,3,3)$ G.19$\uparrow\uparrow$ $(2143)$ $G_{\pm 1}^{7,10}(0,1,0,0)$ $4$
: $11$ genuine variants of $G=\mathrm{G}_3^2(K)$ for $K=\mathbb{Q}\left(\sqrt{\strut -1},\sqrt{\strut d}\right)$[]{data-label="tbl:GenuineVariants"}
Bicyclic biquadratic Dirichlet fields of type $(3,3)$ {#ss:NewRsltGDH}
-----------------------------------------------------
In the range $0<d<2\cdot 10^4$ of real quadratic discriminants $d$, we discovered $22$ variants of the second $3$-class group $G=\mathrm{G}_3^2(K)$ of bicyclic biquadratic fields $K=\mathbb{Q}\left(\sqrt{\strut -1},\sqrt{\strut d}\right)$ containing the fourth roots of unity and having a $3$-class group of type $(3,3)$. In Table \[tbl:LiftedVariants\], resp. Table \[tbl:GenuineVariants\], we present the smallest discriminants $d$ for which these $22$ variants occur, divided into $11$ *lifted* variants *inherited* from the complex quadratic subfield of $K$, resp. $11$ *intrinsic* or *genuine* variants of $K$ itself.
About $20\%$ of these bicyclic biquadratic fields $K$ are composita of a real quadratic field $k_1=\mathbb{Q}\left(\sqrt{\strut d}\right)$ of $3$-class rank $0$ and its dual complex quadratic field $k_2=\mathbb{Q}\left(\sqrt{\strut -d}\right)$ of $3$-class rank $2$. In this case, the second $3$-class group $\mathrm{G}_3^2(K)\simeq\mathrm{G}_3^2(k_2)$ is *inherited* from the complex quadratic subfield $k_2$ by *lifting* the entire $3$-class field tower isomorphically from $k_2$ to $K$. The discriminants of the quadratic subfields are denoted by $\mathrm{d}(k_1)=d$ and $\mathrm{d}(k_2)=d^\prime$.
Roughly $80\%$ of these bicyclic biquadratic fields $K$ are composita of dual quadratic fields $k_1$ and $k_2$ of equal $3$-class rank $1$. Their second $3$-class groups $\mathrm{G}_3^2(K)$ are intrinsic, genuine invariants of the bicyclic biquadratic fields $K$.
Bicyclic biquadratic Eisenstein fields of type $(3,3)$ {#ss:NewRsltESR}
------------------------------------------------------
In cooperation with A. Azizi, M. Talbi, and A. Derhem [@ATDM], and based on [@AAIT1; @Tb] we have completely determined all possibilities for the isomorphism type of the second $3$-class group $G=\mathrm{G}_3^2(K)$ of a bicyclic biquadratic base field $K=\mathbb{Q}\left(\sqrt{\strut -3},\sqrt{\strut d}\right)$, containing the third roots of unity, of type $(3,3)$ and we are able to draw an impressive resumé in comparison to a quadratic base field $K=\mathbb{Q}(\sqrt{\strut D})$. The possibilities are totally disjoint.
- For odd coclass $\mathrm{cc}(G)=2n+1\equiv 1\pmod{2}$, the groups of biquadratic fields are exclusively mainline vertices of depth $0$, whereas the groups of real quadratic fields are vertices of depth $1$ on branches, and odd coclass is impossible at all for compex quadratic fields. However, in the case of coclass $2n+1\ge 3$ both kinds of fields have their groups on the same tree with mainline of transfer kernel type b.10, $\varkappa=(0,0,4,3)$, and the other three trees of coclass graph $\mathcal{G}(3,2n+1)$ are populated by the groups of neither biquadratic fields nor quadratic fields.
- For even coclass $\mathrm{cc}(G)=2n\equiv 0\pmod{2}$, the separation is even more striking. While the groups of biquadratic fields are restricted to the single tree whose mainline vertices share the transfer kernel type b.10, $\varkappa=(0,0,4,3)$, exactly this tree is entirely forbidden for any quadratic field and the groups of real and complex quadratic fields are located on all the other two, resp. five, trees and on the sporadic part of coclass graph $\mathcal{G}(3,2n)$, where $2n=2$, resp. $2n\ge 4$.
Since the behavior of these biquadratic fields $K$ with respect to second $3$-class groups is totally different from quadratic fields, the following Figures \[fig:WimanBlackburn1\]–\[fig:TreeTyp33Cocl3\] visualize the distribution of their second $3$-class group $G=\mathrm{G}_3^2(K)$ on the coclass graphs $\mathcal{G}(3,r)$, $1\le r\le 3$.
In Figures \[fig:WimanBlackburn1\]–\[fig:TreeTyp33Cocl3\] the actual distribution of the $930$ second $3$-class groups $G=G_3^2(K)$ of bicyclic biquadratic number fields $K=\mathbb{Q}\left(\sqrt{\strut -3},\sqrt{\strut d}\right)$ of type $(3,3)$ with discriminant $0<d<5\cdot 10^4$ is represented by underlined boldface counters of hits of the vertices surrounded by the adjacent oval. Isomorphisms among the extensions $L_i\vert K$, $1\le i\le 4$, cause severe constraints on the group $G$.
We point out that only every other mainline vertex of $\mathcal{G}(3,1)$ is populated by second $3$-class groups $\mathrm{G}_3^2(K)$ of quartic fields $K$ in Figure \[fig:WimanBlackburn1\] in contrast to the distribution of the groups $\mathrm{G}_3^2(K)$ of quadratic fields $K$ in Figure \[fig:Distr3Cocl1\].
Vertices of the tree $\mathcal{T}(\langle 243,3\rangle)$ of coclass graph $\mathcal{G}(3,2)$ in Figure \[fig:TreeBTyp33Cocl2\] are classified according to their defect $k(G)$ by using different symbols:
1. big full discs [$\bullet$]{} denote metabelian groups with defect $k(G)=0$ and centre of type $(3,3)$,
2. small full discs [$\bullet$]{} denote metabelian groups with $k(G)=1$ and cyclic centre of order $3$,
3. small contour squares [$\square$]{} denote terminal non-metabelian groups,
4. large contour squares $\square$ denote infinitely capable non-metabelian groups, giving rise to non-metabelian mainlines [@AHL; @ELNO].
This tree is completely forbidden for quadratic fields. A symbol $n\ast$ adjacent to a vertex denotes the multiplicity of a batch of $n$ immediate descendants of a common parent. Numbers in angles denote identifiers in the SmallGroups library [@BEO; @GAP], where we omit the orders, which are given on the left hand scale. The symbols $\Phi_s$ denote isoclinism families [@Hl; @Ef; @Jm]. Transfer kernel types, briefly TKT, [@Ma2 Thm. 2.5, Tbl. 6–7] in the bottom rectangle concern all vertices located vertically above. The periodicity with length $2$ of branches, $\mathcal{B}(j)\simeq\mathcal{B}(j+2)$ for $j\ge 7$, sets in with branch $\mathcal{B}(7)$, having root of order $3^7$.
Vertices of the metabelian skeleton of tree $\mathcal{T}(G_0^{5,7}(0,0,0,0))$ of coclass graph $\mathcal{G}(3,3)$ in Figure \[fig:TreeTyp33Cocl3\] are classified according to their defect $k(G)$ by using different symbols:
1. big full discs [$\bullet$]{} denote metabelian groups with defect $k(G)=0$ and centre of type $(3,3)$,
2. small full discs [$\bullet$]{} denote metabelian groups with $k(G)=1$ and cyclic centre of order $3$.
The symbol $n\ast$ denotes a batch of $n$ siblings of a common parent. Transfer kernel types, briefly TKT, [@Ma2 Thm. 2.5, Tbl. 6–7] in the bottom rectangle concern all vertices located vertically above. Metabelian periodicity with length $2$ of branches, $\mathcal{B}(j)\simeq\mathcal{B}(j+2)$ for $j\ge 8$, sets in with branch $\mathcal{B}(8)$, having root of order $3^8$.
$d$ $\tau(G)=(\mathrm{Cl}_3(L_i))_{1\le i\le 4}$ Type $\varkappa(G)$ $G$ $\mathrm{cc}(G)$ $\Phi$
--------- ---------------------------------------------- -------------------------- ---------------- ----------------------------- ------------------ -------------
$229$ $(3),(3),(3),(3)$ a.1 $(0000)$ $\langle 9,2\rangle$ $1$ $\Phi_1$
$469$ $(9,3),(3,3),(3,3),(3,3)$ a\*.1 $(0000)$ $\langle 81,9\rangle$ $1$ $\Phi_3$
$7453$ $(27,9),(3,3),(3,3),(3,3)$ a\*.1$\uparrow$ $(0000)$ $\langle 729,95\rangle$ $1$ $\Phi_{35}$
$2177$ $(9,3),(9,3),(3,3,3),(3,3,3)$ b.10 $(0043)$ $\langle 729,37\rangle$ $2$ $\Phi_{41}$
$2589$ $(9,3),(9,3),(3,3,3),(3,3,3)$ b.10 $(0043)$ $\langle 729,34\rangle$ $2$ $\Phi_{40}$
$17609$ $(9,3),(9,9),(3,3,3),(3,3,3)$ b\*.10 $(0043)$ $\langle 729,40\rangle$ $2$ $\Phi_{23}$
$14056$ $(27,9),(9,3),(3,3,3),(3,3,3)$ b.10$\uparrow$ $(0043)$ $G_{\pm 1}^{7,8}(0,0,0,0)$ $2$
$20521$ $(9,3),(27,9),(3,3,3),(3,3,3)$ b.10$\uparrow$ $(0043)$ $G_{\pm 1}^{7,8}(0,0,0,0)$ $2$
$44581$ $(81,27),(9,3),(3,3,3),(3,3,3)$ b.10$\uparrow^2$ $(0043)$ $G_{\pm 1}^{9,10}(0,0,0,0)$ $2$
$4933$ $(27,9),(9,9),(3,3,3),(3,3,3)$ b\*.10$\uparrow\uparrow$ $(0043)$ $G_{0}^{6,8}(0,0,0,0)$ $3$
$47597$ $(27,9),(27,9),(3,3,3),(3,3,3)$ b.10$\uparrow\uparrow^2$ $(0043)$ $G_{\pm 1}^{7,10}(0,0,0,0)$ $4$
: $11$ variants of $G=\mathrm{G}_3^2(K)$ for $930$ $K=\mathbb{Q}\left(\sqrt{\strut -3},\sqrt{\strut d}\right)$[]{data-label="tbl:11Variants"}
In the range $0<d<5\cdot 10^4$ of real quadratic discriminants $d$, we discovered $11$ variants of the second $3$-class group $G=\mathrm{G}_3^2(K)$ of bicyclic biquadratic fields $K=\mathbb{Q}\left(\sqrt{\strut -3},\sqrt{\strut d}\right)$ having $3$-class group of type $(3,3)$. In Table \[tbl:11Variants\] we present the smallest discriminants $d$ for which these $11$ variants occur.
The invariants listed are the discriminant $d$ of the real quadratic subfield $\mathbb{Q}\left(\sqrt{\strut d}\right)$ of $K$, the TTT $\tau(G)$ of $G$, the TKT $\varkappa(G)$ of $G$, with arrows $\uparrow$ denoting excited states, the GAP $4$ identifier of $G$ in the SmallGroups library [@BEO; @GAP], provided that $\lvert G\rvert\le 3^6$, otherwise the symbol $G_\varrho^{m,n}(\alpha,\beta,\gamma,\delta)$ for the isomorphism type of $G$ defined in section §\[sss:PrmPres2\], if $\lvert G\rvert\ge 3^8$, the coclass $\mathrm{cc}(G)$ of $G$, and the isoclinism family $\Phi$ to which $G$ belongs, as far as it is defined in [@Hl; @Ef; @Jm].
Stem of isoclinism family $\Phi_6$ {#ss:StemPhi6}
----------------------------------
In this section, we provide group theoretic foundations for determining second $5$-class groups $\mathrm{G}_5^2(K)$ of coclass $\mathrm{cc}(G)\ge 2$ for quadratic and quartic number fields $K$ of type $(5,5)$. The stem of Hall’s isoclinism family $\Phi_6$ is the key for a deeper understanding of the $5$-principalization of these base fields in their six unramified cyclic quintic extensions $L_1,\ldots,L_6$, which has partially but not completely been investigated by Heider and Schmithals [@HeSm] and by Bembom [@Bm].
The *stem groups* $G$ of Hall’s isoclinism family $\Phi_6$ [@Hl p. 139] are $p$-groups of order $\lvert G\rvert=p^5$ with odd prime $p$, nilpotency class $\mathrm{cl}(G)=3$, and coclass $\mathrm{cc}(G)=2$. They were discovered in 1898 by Bagnera [@Bg pp. 182–183], and were constructed as extensions of $C_p^3$ by $C_p^2$, for $p\ge 5$, in 1926 by Schreier [@Sr2 pp. 341–345]. Bagnera also pointed out that these groups do not have an analog for $p=2$.
Every stem group of isoclinism family $\Phi_6$ is a $2$-generator group $G=\langle x,y\rangle$ with main commutator $s_2=\lbrack y,x\rbrack$ in $\gamma_2(G)$ and higher commutators $s_3=\lbrack s_2,x\rbrack$, $t_3=\lbrack s_2,y\rbrack$ in $\gamma_3(G)$, satisfying the power relations $s_2^p=s_3^p=t_3^p=1$. The lower central series of $G$ is given by
$\gamma_2(G)=\langle s_2,s_3,t_3\rangle$ of type $(p,p,p)$,$\gamma_3(G)=\langle s_3,t_3\rangle$ of type $(p,p)$,$\gamma_4(G)=1$,
and the center by $\zeta_1(G)=\gamma_3(G)$. The central quotient $G/\zeta_1(G)$ is of type $\Phi_2(1^3)\simeq G_0^3(0,0)$, the extra special $p$-group of order $p^3$ and exponent $p$, and the abelianization $G/\gamma_2(G)$ is of type $(p,p)$. Therefore, the lower central structure of these groups uniformly consists of two bicyclic factors, the *head* $G/\gamma_2(G)$, and the *tail* $\gamma_3(G)/\gamma_4(G)$, separated by the cyclic factor $\gamma_2(G)/\gamma_3(G)$.
For any stem group $G$ in $\Phi_6$, there exists a nice $1$-to-$1$ correspondence between the two bicyclic factors, the head and the tail, by taking the derived subgroups.
\[lmm:StemIcl6\] The maximal normal subgroups $H_i$ of $G$ contain the commutator subgroup $G^\prime=\gamma_2(G)$ and are given by
$H_i=\langle g_i,G^\prime\rangle$ with generators $g_1=y$ and $g_i=xy^{i-2}$ for $2\le i\le p+1$.
Their derived subgroups $H_i^\prime=(G^\prime)^{g_i-1}$ are given by
$H_1^\prime=\langle t_3\rangle$ and $H_i^\prime=\langle s_3t_3^{i-2}\rangle$ for $2\le i\le p+1$.
As a consequence of Lemma \[lmm:StemIcl6\], we only have trivial two-step centralizers $G^\prime=\chi_2(G)<\chi_3(G)=G$ and the invariants $e(G)$ and $s(G)$ of section §\[ss:MtabTyp33CoclGe2\] take the same value $e=s=3$.
Individual relations for isomorphism classes by James [@Jm pp. 620–621] are given in Table \[tab:RelStemIcl6\], where $\nu$ denotes the smallest positive quadratic non-residue modulo $p$ and $g$ denotes the smallest positive primitive root modulo $p$.
stem group $x^p$ $y^p$ parameters
---------------------- -------------- ---------------------- ----------------------------------------------------
$\Phi_6(221)_a$ $s_3$ $t_3$
$\Phi_6(221)_{b_r}$ $s_3$ $t_3^k$ $1\le r\le\frac{p-1}{2}$, $k=g^r$
$\Phi_6(221)_{c_r}$ $s_3^rt_3^r$ $s_3^{-\frac{r}{4}}$ $r\in\lbrace 1,\nu\rbrace$
$\Phi_6(221)_{d_0}$ $t_3^\nu$ $s_3$
$\Phi_6(221)_{d_r}$ $s_3t_3$ $s_3^k$ $1\le r\le\frac{p-1}{2}$, $k=\frac{g^{2r+1}-1}{4}$
$\Phi_6(21^3)_a$ $1$ $t_3$ $p\ge 5$
$\Phi_6(21^3)_{b_r}$ $t_3^r$ $1$ $r\in\lbrace 1,\nu\rbrace$, $p\ge 5$
$\Phi_6(1^5)$ $1$ $1$
: Relations for the stem groups of $\Phi_6$[]{data-label="tab:RelStemIcl6"}
These presentations for $7$ isomorphism classes of $3$-groups, resp. $12$ isomorphism classes of $5$-groups, among the stem of $\Phi_6$ are now used to calculate the kernels of all transfers $\mathrm{T}_i:G/G^\prime\to H_i/H_i^\prime$, $1\le i\le p+1$, whose images are given for $p=5$ in very convenient form by Lemma \[lmm:TransferIcl6\], since the expressions for *inner transfers* are well-behaved $p$th powers. *Outer transfers* always map to $p$th powers, anyway.
\[lmm:TransferIcl6\] For any $1\le i\le 6$, the image of an arbitrary element $gG^\prime\in G/G^\prime$ with representation $x^jy^\ell G^\prime$, $0\le j,\ell\le 4$, under the transfer $\mathrm{T}_i$ is given by $\mathrm{T}_i(x^jy^\ell G^\prime)=x^{pj}y^{p\ell}H_i^\prime$.
------------------------ ------------------------ ------------- ------------------------- -------------------------- --------------------- ------------ -------------------- -----------------
TKT $\varkappa$ $\eta$ $\varkappa$ property
$\langle 243,7\rangle$ $G_0^{4,5}(1,1,-1,1)$ D.$5$ $(4224)$ $\langle 3125,14\rangle$ $\Phi_6(221)_a$ 6 $(123456)$ identity
$\langle 243,4\rangle$ $G_0^{4,5}(1,1,1,1)$ H.$4$ $(4443)$ $\langle 3125,11\rangle$ $\Phi_6(221)_{b_1}$ 2 $(125364)$ $4$-cycle
$\langle 3125,7\rangle$ $\Phi_6(221)_{b_2}$ 2 $(126543)$ two transpos.
$\langle 243,8\rangle$ $G_0^{4,5}(0,0,0,1)$ c.$21$ $(0231)$ $\langle 3125,8\rangle$ $\Phi_6(221)_{c_1}$ 1 $(612435)$ $5$-cycle
$\langle 243,5\rangle$ $G_0^{4,5}(0,0,-1,1)$ D.$10$ $(2241)$ $\langle 3125,13\rangle$ $\Phi_6(221)_{c_2}$ 1 $(612435)$ $5$-cycle
$\langle 243,9\rangle$ $G_0^{4,5}(0,-1,-1,0)$ G.$19$ $(2143)$ $\langle 3125,10\rangle$ $\Phi_6(221)_{d_0}$ 0 $(214365)$ three transpos.
$\langle 243,6\rangle$ $G_0^{4,5}(0,-1,0,1)$ c.$18$ $(0313)$ $\langle 3125,12\rangle$ $\Phi_6(221)_{d_1}$ 0 $(512643)$ $6$-cycle
$\langle 3125,9\rangle$ $\Phi_6(221)_{d_2}$ 0 $(312564)$ two $3$-cycles
$\langle 3125,4\rangle$ $\Phi_6(21^3)_a$ 2 $(022222)$ nrl.const.with fp.
$\langle 3125,5\rangle$ $\Phi_6(21^3)_{b_1}$ 1 $(011111)$ nearly constant
$\langle 3125,6\rangle$ $\Phi_6(21^3)_{b_2}$ 1 $(011111)$ nearly constant
$\langle 243,3\rangle$ $G_0^{4,5}(0,0,0,0)$ b.$10$ $(0043)$ $\langle 3125,3\rangle$ $\Phi_6(1^5)$ 6 $(000000)$ constant
------------------------ ------------------------ ------------- ------------------------- -------------------------- --------------------- ------------ -------------------- -----------------
: TKT of corresponding $p$-groups in $\Phi_6$ for $p\in\lbrace 3,5\rbrace$[]{data-label="tab:TrfKerStemIcl6"}
In Table \[tab:TrfKerStemIcl6\], TKTs $\varkappa$ of $3$-groups in the notation of [@Ma2 § 3.3] were determined by Nebelung [@Ne1 p. 208, Thm. 6.14] already, using different presentations in equation (\[eqn:PwrCmtPres\]), section §\[sss:PrmPres2\]. For $5$-groups the TKTs are given here for the first time. The only exception is the group $\langle 3125,14\rangle$, which was discussed in the well-known paper by Taussky [@Ta2 p. 436, Thm.2] as an example to show that the coarse TKT $\kappa=(\mathrm{AAAAAA})$ can occur for $p=5$, and also for primes $p\ge 7$. A convenient partial characterization is provided by counters of fixed point transfer kernels, resp. abelianizations of type $(5,5,5)$, $\eta=\#\lbrace 1\le i\le 6\mid\kappa(i)=\mathrm{A}\rbrace
=\#\lbrace 1\le i\le 6\mid\tau(i)=(5,5,5)\rbrace$, which must coincide, according to [@HeSm Thm. 7, p. 11].
The correspondence between $p=3$ and $p=5$ is due to the formally identical power-commutator presentation. However, it is partially rather shallow, since corresponding groups can have different properties with respect to their role on the coclass graphs $\mathcal{G}(3,2)$ and $\mathcal{G}(5,2)$. For example, the $3$-groups $\langle 243,6\rangle$ and $\langle 243,8\rangle$ are mainline vertices having the mandatory total first transfer kernel $\varkappa(1)=0$ whereas the $5$-groups $\langle 3125,12\rangle$ and $\langle 3125,8\rangle$ are terminal without total transfer.
### Top vertices of type $(5,5)$ on $\mathcal{G}(5,2)$ {#sss:Typ55Cocl2}
Figure \[fig:Typ55Cocl2\] shows the non-CF groups at the top of coclass graph $\mathcal{G}(5,2)$. It was constructed by means of the SmallGroups library [@BEO] of GAP [@GAP] and MAGMA [@MAGMA]. The groups are labelled by a number in angles, which is their identifier in that library. Additional confirmation was obtained by explicit descendant calculation with the aid of the ANUPQ package [@GNO].
The vertices of the coclass graph $\mathcal{G}(5,2)$ in Figure \[fig:Typ55Cocl2\] are classified by using different symbols:
1. a large contour square $\square$ represents an abelian group,
2. a big contour circle [$\circ$]{} represents a metabelian group with abelian maximal subgroup,
3. big full discs [$\bullet$]{} represent metabelian groups with bicyclic centre of type $(5,5)$,
4. small full discs [$\bullet$]{} represent metabelian groups with cyclic centre of order $5$.
The actual distribution of the $959$, resp. $377$, second $5$-class groups $G_5^2(K)$ of complex, resp. real, quadratic number fields $K=\mathbb{Q}(\sqrt{D})$ of type $(5,5)$ with discriminant $-2\,270\,831<D<26\,695\,193$ is represented by underlined boldface counters (in the format complex/real) of the hits of vertices surrounded by the adjacent oval.
### Fixed point principalization problem {#sss:TausskyProblem}
We are pleased to present the solution of a problem posed in 1970 by Taussky [@Ta2 p. 438, Rem. 1]. It concerns the lack of realizations, in the form of second $5$-class groups $\mathrm{G}_5^2(K)$ of number fields $K$, of the unique metabelian $5$-group $\langle 5^5,14\rangle$ with $6$ fixed point transfer kernels, that is with coarse TKT $\kappa=(\mathrm{AAAAAA})$, but without total transfer kernels $\varkappa(i)=0$. Actually, we now have $5$ realizations of this very special TKT $\varkappa=(123456)$ (identity permutation) for quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $D\in\lbrace -89\,751,-235\,796,-1\,006\,931,-1\,996\,091,$ $-2\,187\,064\rbrace$, in Table \[tbl:CompQuad5x5Details\], and $4$ further realizations in Table \[tbl:CyclQrt5x5\], for certain cyclic quartic fields $K=\mathbb{Q}\left((\zeta_5-\zeta_5^{-1})\sqrt{D}\right)$, $D\in\lbrace 581,753,2\,296,4\,553\rbrace$.
### Statistical evaluation of second $5$-class groups $\mathrm{G}_5^2(K)$ {#sss:StatScnd5ClgpCocl2}
The possibilities for $5$-groups of coclass $2$ are more extensive than those for coclass $1$.
For the $377$ real quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $0<D<26\,695\,193$, in Table \[tbl:RealQuad5x5\], there occur $7$ cases of coarse TKT $\kappa=(\mathrm{BBBBBB})$, for $D\in\lbrace 4\,954\,652,7\,216\,401,12\,562\,849,16\,434\,245,18\,434\,456,$ $19\,115\,293,20\,473\,841\rbrace$, a single case of TKT $\varkappa=(612435)$, for $D=18\,070\,649$, $2$ cases of coarse TKT$\kappa=(\mathrm{AABBBB})$, for $D\in\lbrace 10\,486\,805,18\,834\,493\rbrace$, and $4$ cases of the first excited state of TKT $\varkappa=(022222)$, for $D\in\lbrace 7\,306\,081,11\,545\,953,14\,963\,612,22\,042\,632\rbrace$.
$D$ $\tau(K)$ $\tau(0)$ $\kappa(K)$ $G$ $\mathrm{cc}(G)$ $\#$ $\%$
------------- ------------------------------------- ----------- ------------- ----------------------------------------- ------------------ ------- --------
$-11\,199$ $(5,5^2)^6$ $(5,5,5)$ $(B^6)$ $\langle 3125,9\vert 10\vert 12\rangle$ $2$ $301$ $31.4$
$-12\,451$ $(5,5,5),(5,5^2)^5$ $(5,5,5)$ $(A,B^5)$ $\langle 3125,8\vert 13\rangle*$ $2$ $167$ $17.4$
$-30\,263$ $(5,5,5)^2,(5,5^2)^4$ $(5,5,5)$ $(A^2,B^4)$ $\langle 3125,7\vert 11\rangle$ $2$ $283$ $29.5$
$-89\,751$ $(5,5,5)^6$ $(5,5,5)$ $(A^6)$ $\langle 3125,14\rangle*$ $2$ $5$ $ $
$-62\,632$ $(5,5,5,5^2),(5,5,5),(5,5^2)^4$ $(?,A,B^4)$ $\langle 78125,\#\rangle$ $2$ $124$ $12.9$
$-67\,031$ $(5,5,5,5,5),(5,5^2)^5$ $(B^6)$ $\langle 78125,\#\rangle$ $2$ $6$ $ $
$-67\,063$ $(5,5,5,5,5),(5,5,5),(5,5^2)^4$ $(B,A,B^4)$ $\langle 78125,\#\rangle$ $2$ $37$ $3.9$
$-280\,847$ $(5,5,5,5^2),(5,5^2)^5$ $(?,B^5)$ $\langle 78125,\#\rangle$ $2$ $32$ $3.3$
$-181\,752$ $(5,5^2,5^2,5^2),(5,5,5),(5,5^2)^4$ $(?,A,B^4)$ $\langle 1953125,\#\rangle$ $2$ $4$ $ $
: $9$ variants of $G=\mathrm{G}_5^2(K)$ for $959$ $K=\mathbb{Q}(\sqrt{D})$, $-2\,270\,831\le D<0$[]{data-label="tbl:CompQuad5x5"}
Among the $959$ complex quadratic fields $K=\mathbb{Q}(\sqrt{D})$, $-2\,270\,831\le D<0$, in Table \[tbl:CompQuad5x5\], ground states (GS) appear exclusively with sporadic, and mostly terminal, top vertices of $\mathcal{G}(5,2)$. The $5$ cases of TKT $\varkappa=(123456)$ have been presented separately in section §\[sss:TausskyProblem\] as solutions of Taussky’s problem of 1970. Further, there are $167$ cases $(17.4\%)$ of TKT $\varkappa=(612435)$ (5-cycle with coarse TKT $\kappa=(\mathrm{BBBABB})$) starting with $D=-12\,451$, which was attempted but not analyzed completely in 1982 by Heider and Schmithals [@HeSm] and $283$ cases $(29.5\%)$ of coarse TKT $\kappa=(\mathrm{AABBBB})$ starting with $D=-30\,263$. The remaining $301$ cases $(31.4\%)$ of coarse TKT $\kappa=(\mathrm{BBBBBB})$, starting with $D=-11\,199$ are slightly dominating.
For excited states (ES) of coclass $2$ as well as of coclass $1$, the distinguished first $5$-class group $\mathrm{Cl}_5(K_1)$ of the non-Galois absolute quintic subfield $K_1$ of the unramified extension $L_1\vert K$ is of $5$-rank $\mathrm{r}_5(K_1)=2$, which shows impressively that the rank equation for $p=3$, $\mathrm{r}_3(K_i)=\mathrm{r}_3(K)-1$, by Gras [@Gr] and Gerth [@Ge] generalizes to a double inequality for $p\ge 5$, $$\mathrm{r}_p(K)-1\le\mathrm{r}_p(K_i)\le\frac{p-1}{2}\cdot(\mathrm{r}_p(K)-1),$$ as predicted, and partially proved, by Bölling [@Boe] and Lemmermeyer [@Lm].
$D$ $\tau(K)$ $\tau(0)$ $\varkappa(K)$ $G$ $\mathrm{cc}(G)$ $\#$
------------ --------------------------------- ----------- ---------------- ------------------------------- ------------------ ------
$-11\,199$ $(5,5^2)^6$ $(5,5,5)$ $(512643)$ $\langle 3125,12\rangle*$ $2$ $7$
$-17\,944$ $(5,5^2)^6$ $(5,5,5)$ $(312564)$ $\langle 3125,9\rangle*$ $2$ $2$
$-42\,871$ $(5,5^2)^6$ $(5,5,5)$ $(214365)$ $\langle 3125,10\rangle$ $2$ $3$
or $\langle 15625,680\rangle$
$-12\,451$ $(5,5,5),(5,5^2)^5$ $(5,5,5)$ $(612435)$ $\langle 3125,8\rangle*$ $2$ $5$
or $\langle 3125,13\rangle*$
$-30\,263$ $(5,5,5)^2,(5,5^2)^4$ $(5,5,5)$ $(126543)$ $\langle 3125,7\rangle$ $2$ $4$
or $\langle 15625,647\rangle$
$-37\,363$ $(5,5,5)^2,(5,5^2)^4$ $(5,5,5)$ $(125364)$ $\langle 3125,11\rangle*$ $2$ $2$
$-89\,751$ $(5,5,5)^6$ $(5,5,5)$ $(123456)$ $\langle 3125,14\rangle*$ $2$ $5$
$-62\,632$ $(5,5,5,5^2),(5,5,5),(5,5^2)^4$ $(322222)$ $\langle 78125,\#\rangle$ $2$ $1$
$-67\,031$ $(5,5,5,5,5),(5,5^2)^5$ $(211111)$ $\langle 78125,\#\rangle$ $2$ $1$
$-67\,063$ $(5,5,5,5,5),(5,5,5),(5,5^2)^4$ $(322222)$ $\langle 78125,\#\rangle$ $2$ $2$
: $9$ subvariants of $G=\mathrm{G}_5^2(K)$ for $31$ fields $K=\mathbb{Q}(\sqrt{D})$, $-89\,751\le D<0$[]{data-label="tbl:CompQuad5x5Details"}
The transfer target type (TTT) $\tau(G)$ of second $5$-class groups $\mathrm{G}_5^2(K)$ has been computed for all quadratic number fields $K=\mathbb{Q}(\sqrt{D})$, having discriminant $-2\,270\,831<D<26\,695\,193$ and $5$-class group $\mathrm{Cl}_5(K)$ of type $(5,5)$, with the aid of MAGMA [@MAGMA] As a refinement, we calculated the transfer kernel type (TKT) $\varkappa(G)$ for $31$ fields $K=\mathbb{Q}(\sqrt{D})$, $-89\,751\le D<0$, as given in Table \[tbl:CompQuad5x5Details\]. This also refines results of Bembom in [@Bm p. 129]. Observe that Bembom does not give TKTs in our sense and consequently was not able to discover the distinguished role of $D=-89\,751$ with respect to the Taussky problem.
### Statistical evaluation of second $7$-class groups $\mathrm{G}_7^2(K)$ {#sss:StatScnd7Clgp}
Among the $94$ complex quadratic fields $K$ of type $(7,7)$ with discriminants $-10^6\le D<0$, we found $7$ variants of the second $7$-class group $G=\mathrm{G}_7^2(K)$, characterized by different TTT $\tau(K)$ and Taussky’s coarse TKT $\kappa(K)$, which are related by [@HeSm Thm. 7, p. 11].
$D$ $\tau(K)$ $\tau(0)$ $\kappa(K)$ $G$ $\mathrm{cc}(G)$ $\#$ $\%$
------------- ------------------------------------- ----------- ------------- --------------------------------------------------- ------------------ ------ -------
$-63\,499$ $(7,7^2)^8$ $(7,7,7)$ $(B^8)$ $\langle 16807,10\vert 14\vert 15\vert 16\rangle$ $2$ $40$ $43$
$-183\,619$ $(7,7,7)^2,(7,7^2)^6$ $(7,7,7)$ $(A^2,B^6)$ $\langle 16807,11\vert 12\vert 13\rangle$ $2$ $29$ $31$
$-227\,860$ $(7,7,7),(7,7^2)^7$ $(7,7,7)$ $(A,B^7)$ $\langle 16807,8\vert 9\rangle$ $2$ $9$ $9.6$
unknown $(7,7,7)^8$ $(7,7,7)$ $(A^8)$ $\langle 16807,7\rangle$ $2$ $ $ $ $
$-159\,592$ $(7,7,7,7,7),(7,7,7),(7,7^2)^6$ $(A^2,B^6)$ $\langle 823543,\#\rangle$ $2$ $3$ $ $
$-227\,387$ $(7,7,7,7^2),(7,7,7),(7,7^2)^6$ $(B,A,B^6)$ $\langle 823543,\#\rangle$ $2$ $10$ $ $
$-272\,179$ $(7,7,7,7^2),(7,7^2)^7$ $(B^8)$ $\langle 823543,\#\rangle$ $2$ $2$ $ $
$-673\,611$ $(7,7,7,7,7,7^2),(7,7,7),(7,7^2)^6$ $(?,A,B^6)$ $\langle 40353607,\#\rangle$ $2$ $1$ $ $
: $7$ variants of $G=\mathrm{G}_7^2(K)$ for $70$ fields $K=\mathbb{Q}(\sqrt{D})$, $-751\,288\le D<0$[]{data-label="tbl:CompQuad7x7"}
In Table \[tbl:CompQuad7x7\] we present the discriminants with smallest absolute values, corresponding to these variants. $\tau(0)$ denotes the $7$-class group of $\mathrm{F}_7^1(K)$. Using the SmallGroups library [@BEO], we identified $78$ *ground states* $(83\%)$ having their $G$ among the sporadic top vertices of $\mathcal{G}(7,2)$ in the stem of isoclinism family $\Phi_6$. Unfortunately, there didn’t occur a solution of Taussky’s 1970 fixed point capitulation problem for $p=7$, in form of a realization of $\langle 16807,7\rangle$. However, there appeared $15$ *first excited states* $(16\%)$ with $G$ located on coclass trees of $\mathcal{G}(7,2)$, where the non-Galois subfield $K_1$ of the distinguished extension $L_1$ has a $7$-class group of type $(7,7)$, and, particularly remarkable, a single *second excited state* for $D=-673\,611$, where the maximal $7$-rank $3$ in Bölling’s inequality [@Boe; @Lm] is attained in form of $\mathrm{Cl}_7(K_1)\simeq(7,7,7)$.
Cyclic quartic fields of type $(5,5)$ {#ss:NewRslt5Mirror}
-------------------------------------
In cooperation with A. Azizi and M. Talbi [@ATM], and based on the quintic reflection theorem [@Ks], we have computed the isomorphism type of the second $5$-class group $G=\mathrm{G}_5^2(K)$ of $41$ cyclic quartic fields $K=\mathbb{Q}\left((\zeta_5-\zeta_5^{-1})\sqrt{D}\right)$, $\zeta_5=\exp(\frac{2\pi i}{5})$, $-15\,419\le D<5\,000$, $5\nmid D$, of type $(5,5)$. Such a field is the $5$-dual mirror imageof the quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{5D})$. Isomorphisms among the extensions $L_i\vert K$, $1\le i\le 6$, cause severe constraints on the group $G$, as Table \[tbl:CyclQrt5x5\], visualized by Figures \[fig:Distr5Cocl1\] and \[fig:Typ55Cocl2\], shows.
$D$ $\tau(K)$ $\tau(0)$ $\varkappa(K)$ $G$ $\mathrm{cc}(G)$ $\#$ $\%$
------------ --------------------------------- ----------- ---------------- ------------------------------- ------------------ ------ ---------
$-12\,883$ $(5,5)^6$ $(000000)$ $\langle 125,3\rangle$ $1$ $4$ $100\%$
$257$ $(5,5,5)^2,(5,5^2)^4$ $(5,5,5)$ $(022222)$ $\langle 3125,4\rangle$ $2$ $9$ $24\%$
$457$ $(5,5,5)^2,(5,5^2)^4$ $(5,5,5)$ $(125364)$ $\langle 3125,11\rangle*$ $2$ $ $ $ $
$508$ $(5,5,5)^2,(5,5^2)^4$ $(5,5,5)$ $(126543)$ $\langle 3125,7\rangle$ $2$ $ $ $ $
or $\langle 15625,647\rangle$
$581$ $(5,5,5)^6$ $(5,5,5)$ $(123456)$ $\langle 3125,14\rangle*$ $2$ $4$ $11\%$
$1\,137$ $(5,5,5,5^2),(5,5,5),(5,5^2)^4$ $(111111)$ $\langle 78125,\#\rangle$ $2$ $3$ $8\%$
$4\,357$ $(5)^6$ $(000000)$ $\langle 25,2\rangle$ $1$ $3$ $8\%$
: $7$ variants of $G=\mathrm{G}_5^2(K)$ for $41$ fields $K=\mathbb{Q}\left((\zeta_5-\zeta_5^{-1})\sqrt{D}\right)$[]{data-label="tbl:CyclQrt5x5"}
$p$-Groups with double layered metabelianization of type $(p^2,p)$ or $(p,p,p)$ {#s:DoubleLayer}
===============================================================================
For a number field $K$ with $p$-class group $\mathrm{Cl}_p(K)$ of type $(p^2,p)$, resp. $(p,p,p)$, there exist *two layers* of unramified abelian extensions $L\vert K$, each containing $p+1$, resp. $p^2+p+1$, members. Extensions in the *first layer* are of relative degree $p$, those in the *second layer* are of relative degree $p^2$. Consequently, the second layer tends to be out of the scope of actual computations. However, there are some exceptions of modest degree.
Quadratic fields of type $(9,3)$ {#ss:Qdr9x3}
--------------------------------
On the one hand, there is the case $p=3$ for quadratic fields $K=\mathbb{Q}(\sqrt{D})$ with $3$-class group $\mathrm{Cl}_3(K)$ of type $(9,3)$ or $(3,3,3)$, where extensions in the second layer are of absolute degree $18$. From the viewpoint of $3$-towers, there are no open problems for complex quadratic fields $K$ of type $(3,3,3)$, since it is known that $\ell_3(K)=\infty$ [@KoVe; @mL], that is, $\mathrm{G}_3^\infty(K)$ is always an infinite pro-$3$ group.
Thus, we focussed on $3$-class rank $2$ and computed the first layer of the TTT and TKT of all $875$ complex quadratic fields $K$ of type $(9,3)$ with discriminant $-10^6<D<0$ and of all $271$ real quadratic fields $K$ of type $(9,3)$ with discriminant $0<D<10^7$. In [@Ma4], we will show that this information is sufficient to identify the second $3$-class group $\mathrm{G}_3^2(K)$ for $565$ negative discriminants $(65\%)$ and for $188$ positive discriminants $(70\%)$. For the remainder, the second layer of the TTT and TKT is required.
Quadratic and biquadratic fields of type $(2,2,2)$ {#ss:BiQdr2x2x2}
--------------------------------------------------
On the other hand, we have the case $p=2$ for quadratic, resp. quartic, fields $K$ with $2$-class group $\mathrm{Cl}_2(K)$ of type $(4,2)$ or $(2,2,2)$, where extensions in the second layer are of absolute degree $8$, resp. $16$.
We were particularly interested in fields $K$ of $2$-class rank $3$, where the $2$-tower length $\ell_2(K)$ is still an open problem. We found that the coclass tree $\mathcal{T}(\langle 16,11\rangle)$, which is the unique tree of coclass graph $\mathcal{G}(2,2)$ containing groups with abelianization of type $(2,2,2)$, is populated by second $2$-class groups $\mathrm{G}_2^2(K)$ of real quadratic fields $K$ of type $(2,2,2)$. The tree $\mathcal{T}(\langle 16,11\rangle)$ corresponds to the pro-$2$ group $S_5$ in [@Fs; @EkFs] with periodic sequences $K_x^{i}$, $46\le i\le 51$, given by explicit parametrized presentations for $x\ge 0$ in [@Fs]. It also corresponds to the so-called family $\#59$ with explicit pro-$2$ presentation given in [@NmOb].
Further, we obtained deeper results concerning second $2$-class groups $G=\mathrm{G}_2^2(K)$ of complex quadratic fields of type $(2,2,2)$, which have been classified in terms of the smallest non-abelian lower central quotient $G/\gamma_3(G)$, usually coinciding with the root of the coclass tree $\mathcal{T}$ such that $G\in\mathcal{T}$, by E. Benjamin, F. Lemmermeyer, and C. Snyder [@Lm1; @BLS]. Note that these authors use the Hall-Senior classification [@HaSn], whereas we give identifiers of the SmallGroups library [@BEO]. The groups $G$ are mainly, but not exclusively, located at the terminal top vertices $\langle 32,32\rangle$ and $\langle 32,33\rangle$ of coclass graph $\mathcal{G}(2,3)$ and on the coclass trees $\mathcal{T}(\langle 32,29\rangle)$, $\mathcal{T}(\langle 32,30\rangle)$, $\mathcal{T}(\langle 32,35\rangle)$, corresponding to the families $\#75$, $\#76$, $\#79$ with explicit pro-$2$ presentations given in [@NmOb]. These subtrees of $\mathcal{G}(2,3)$ seem to be populated on every branch, with the only exception of the root.
We intend to include these results on complex quadratic fields in [@AZTM], where the principal aim is to investigate bicyclic biquadratic fields $K=\mathbb{Q}\left(\sqrt{\strut -1},\sqrt{\strut d}\right)$, called *special Dirichlet fields* by Hilbert [@Hi], with $2$-class groups of type $(2,2,2)$, based on work by A. Azizi, A. Zekhnini, and M. Taous [@AzTs; @AZT1; @AZT2]. The second $2$-class groups $\mathrm{G}_2^2(K)$ for certain series of radicands $d>0$, for example $d\in\lbrace 170,730,2314\rbrace$, seem to be distributed on every branch of the coclass tree $\mathcal{T}(\langle 64,140\rangle)$ of coclass graph $\mathcal{G}(2,3)$, which corresponds to family $\#73$ with explicit pro-$2$ presentation given in [@NmOb].
Acknowledgements {#s:Thanks}
================
The author is indebted to Nigel Boston, University of Wisconsin, Madison, and Michael R. Bush, Washington and Lee University, Lexington, for intriguing discussions about the length of $p$-towers and Schur $\sigma$-groups in §\[ss:TowerLength\].
Sincere thanks are given to Mike F. Newman, Australian National University, Canberra, for valuable suggestions concerning use of the SmallGroups library [@BEO] and ANUPQ package [@GNO] of GAP 4 [@GAP] and MAGMA [@MAGMA], and for precious aid in identifying finite metabelian $p$-groups, produced by various approaches to the classification problem, in particular, by Blackburn [@Bl1], James [@Jm], Ascione [@As1], and Nebelung [@Ne1].
We thank Abdelmalek Azizi and Mohammed Talbi, Faculté des Sciences, Oujda, and Aïssa Derhem, Casablanca, for our joint investigation of bicyclic biquadratic fields containing third roots of unity in §\[ss:NewRsltESR\].
Further, we gratefully acknowledge helpful advice for constructing class fields [@Fi] with the aid of MAGMA [@MAGMA; @BCP; @BCFS] by Claus Fieker, University of Kaiserslautern.
Concerning the coclass graph $\mathcal{G}(5,1)$ in §\[sss:5Cocl1\], we thank Heiko Dietrich, University of Trento, for making available unpublished details of the tree structure.
[XX]{} M. Arrigoni, On Schur $\sigma$-groups, *Math. Nachr.* **192** (1998), 71–89. E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes, *Abh. Math. Sem. Univ. Hamburg* **5** (1927), 353–363. E. Artin, Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz, *Abh. Math. Sem. Univ. Hamburg* **7** (1929), 46–51. J. A. Ascione, G. Havas, and C. R. Leedham-Green, A computer aided classification of certain groups of prime power order, *Bull. Austral. Math. Soc.* **17** (1977), 257–274, Corrigendum 317–319, Microfiche Supplement p. 320. J. A. Ascione, *On $3$-groups of second maximal class* (Ph.D. Thesis, Australian National University, Canberra, 1979). J. A. Ascione, On $3$-groups of second maximal class, *Bull. Austral. Math. Soc.* **21** (1980), 473–474. A. Azizi, M. Ayadi, M. C. Ismaïli et M. Talbi, Sur les unités des extensions cubiques cyliques non ramifiées sur certains sous-corps de $\mathbb{Q}(\sqrt{d},\sqrt{-3})$, *Ann. Math. Blaise Pascal* **16** (2009), no. 1, 71–82. A. Azizi, M. Talbi, A. Derhem, and D. C. Mayer, *The group $\mathrm{Gal}(k_3^{(2)}\vert k)$ for $k=\mathbb{Q}\left(\sqrt{\strut -3},\sqrt{\strut d}\right)$ of type $(3,3)$* (Preprint, 2012). A. Azizi, M. Talbi, and D. C. Mayer, *The group $\mathrm{Gal}(k_5^{(2)}\vert k)$ for $k=\mathbb{Q}\left((\zeta_5-\zeta_5^{-1})\sqrt{\strut D}\right)$ of type $(5,5)$* (in preparation). A. Azizi et M. Taous, Determination des corps $k=\mathbb{Q}\left(\sqrt{\strut d},i\right)$ dont le $2$-groupe de classes est de type $(2,4)$ ou $(2,2,2)$, *Rend. Istit. Mat. Univ. Trieste* **40** (2008), 93–116. A. Azizi, A. Zekhnini et M. Taous, *Capitulation dans le corps des genres de certain corps de nombres biquadratique imaginaire dont le $2$-groupe des classes est de type $(2,2,2)$* (prépublication, Journées de théorie des nombres, FSO, Oujda, Maroc, Septembre 2010). A. Azizi, A. Zekhnini et M. Taous, *Sur la capitulation des $2$-classes d’idéaux du corps $\mathbb{Q}\left(\sqrt{\strut 2p_1p_2},i\right)$* (prépublication, Workshop International NTCCCS, FSO, Oujda, Maroc, Avril 2012). A. Azizi, A. Zekhnini, M. Taous, and D. C. Mayer, *The group $\mathrm{Gal}(k_2^{(2)}\vert k)$ for $k=\mathbb{Q}\left(\sqrt{\strut -1},\sqrt{\strut d}\right)$ of type $(2,2,2)$* (in preparation). G. Bagnera, La composizione dei gruppi finiti il cui grado è la quinta potenza di un numero primo, *Ann. di Mat.* (Ser. 3) **1** (1898), 137–228. L. Bartholdi and M. R. Bush, Maximal unramified $3$-extensions of imaginary quadratic fields and $\mathrm{SL}_2\mathbb{Z}_3$, *J. Number Theory* **124** (2007), 159–166. T. Bembom, *The capitulation problem in class field theory* (Dissertation, Georg-August-Universität Göttingen, 2012). E. Benjamin, F. Lemmermeyer, C. Snyder, Imaginary quadratic fields with $\mathrm{Cl}_2(k)\simeq(2,2,2)$ *J. Number Theory* **103** (2003), 38–70. H. U. Besche, B. Eick, and E. A. O’Brien, *The SmallGroups Library — a Library of Groups of Small Order*, 2005, an accepted and refereed GAP 4 package, available also in MAGMA. N. Blackburn, On a special class of $p$-groups, *Acta Math.* **100** (1958), 45–92. N. Blackburn, On prime-power groups in which the derived group has two generators, *Proc. Camb. Phil. Soc.* **53** (1957), 19–27. R. Bölling, On ranks of class groups of fields in dihedral extensions over $\mathbb{Q}$ with special reference to cubic fields, *Math. Nachr.* **135** (1988), 275–310. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, *J. Symbolic Comput.* **24** (1997), 235–265. W. Bosma, J. J. Cannon, C. Fieker, and A. Steels (eds.), *Handbook of Magma functions* (Edition 2.18, Sydney, 2012). N. Boston, M. R. Bush and F. Hajir, *Heuristics for $p$-class towers of imaginary quadratic fields* (arXiv: 1111.4679 v1 \[math.NT\] 20 Nov 2011). N. Boston, M. R. Bush and F. Hajir, *Heuristics for $p$-class towers of real quadratic fields* (in preparation). N. Boston and J. Ellenberg, Random pro-$p$ groups, braid groups, and random tame Galois groups, *Groups Geom. Dyn.* **5** (2011), 265–280. M. Boy, *On the second class group of real quadratic number fields* (Dissertation, Technische Universität Kaiserslautern, 2012). J. R. Brink, *The class field tower for imaginary quadratic number fields of type $(3,3)$* (Dissertation, Ohio State University, 1984). J. R. Brink and R. Gold, Class field towers of imaginary quadratic fields, *manuscripta math.* **57** (1987), 425–450. M. R. Bush, *Schur $\sigma$-groups of small prime power order* (in preparation). M. R. Bush and D. C. Mayer, *$3$-class field towers of exact length $3$* (in preparation). H. Dietrich, B. Eick, and D. Feichtenschlager, Investigating $p$-groups by coclass with GAP, *Computational group theory and the theory of groups*, 45–61 (Contemp. Math. **470**, AMS, Providence, RI, 2008). H. Dietrich, Periodic patterns in the graph of $p$-groups of maximal class, *J. Group Theory* **13** (2010) 851–871. H. Dietrich, A new pattern in the graph of $p$-groups of maximal class, *Bull. London Math. Soc.* **42** (2010) 1073–1088. M. du Sautoy, Counting $p$-groups and nilpotent groups, *Inst. Hautes Études Sci. Publ. Math.* **92** (2001) 63–112. T. E. Easterfield, *A classification of groups of order $p^6$* (Ph. D. Thesis, Univ. of Cambridge, 1940). B. Eick and D. Feichtenschlager, *Infinite sequences of $p$-groups with fixed coclass* (arXiv: 1006.0961 v1 \[math.GR\], 4 Jun 2010). B. Eick and C. Leedham-Green, On the classification of prime-power groups by coclass, *Bull. London Math. Soc.* **40** (2) (2008), 274–288. B. Eick, C. R. Leedham-Green, M. F. Newman, and E. A. O’Brien, *On the classification of groups of prime-power order by coclass: The $3$-groups of coclass $2$* (Preprint, 2011). D. Feichtenschlager, *Symbolic computation with infinite sequences of $p$-groups with fixed coclass* (Dissertation, TU Braunschweig, 2010). C. Fieker, Computing class fields via the Artin map, *Math. Comp.* **70** (2001), no. 235, 1293–1303. G. Frei, P. Roquette, and F. Lemmermeyer, *Emil Artin and Helmut Hasse. Their Correspondence 1923–1934*, Universitätsverlag Göttingen, 2008. Ph. Furtwängler, Beweis des Hauptidealsatzes für die Klassenkörper algebraischer Zahlkörper, *Abh. Math. Sem. Univ. Hamburg* **7** (1929), 14–36. G. Gamble, W. Nickel, and E. A. O’Brien, *ANU p-Quotient — p-Quotient and p-Group Generation Algorithms*, 2006, an accepted GAP 4 package, available also in MAGMA. The GAP Group, GAP – Groups, Algorithms, and Programming — a System for Computational Discrete Algebra, Version 4.4.12, Aachen, Braunschweig, Fort Collins, St. Andrews, 2008, `(http://www.gap-system.org)`. F. Gerth III, Ranks of $3$-class groups of non-Galois cubic fields, *Acta Arith.* **30** (1976), 307–322. G. Gras, Sur les $\ell$-classes d’idéaux des extensions non galoisiennes de degré premier impair $\ell$ à la clôture galoisienne diédrale de degré $2\ell$, *J. Math. Soc. Japan* **26** (1974), 677–685. M. Hall and J. K. Senior, *The groups of order $2^n$ ($n\le 6$)* (Macmillan, New York, 1964). P. Hall, The classification of prime-power groups, *J. Reine Angew. Math.* **182** (1940), 130–141. F.-P. Heider und B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, *J. Reine Angew. Math.* **336** (1982), 1–25. D. Hilbert, Ueber den Dirichlet’schen biquadratischen Zahlkörper, *Math. Annalen* **45** (1894), 309–340. R. James, The groups of order $p^6$ ($p$ an odd prime), *Math. Comp.* **34** (1980), no. 150, 613–637. Y. Kishi, The Spiegelungssatz for $p=5$ from a constructive approach, *Math. J. Okayama Univ.* **47** (2005), 1–27. H. Kisilevsky, Number fields with class number congruent to $4$ mod $8$ and Hilbert’s theorem $94$, *J. Number Theory* **8** (1976), 271–279. H. Koch und B. B. Venkov, Über den $p$-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, *Astérisque* **24–25** (1975), 57–67. C. R. Leedham-Green and S. McKay, On the classification of $p$-groups of maximal class, *Q. J. Math. Oxford* **35** (1984), 293–304. C. R. Leedham-Green and S. McKay, *The structure of groups of prime power order*, London Math. Soc. Monographs, New Series, **27**, Oxford Univ. Press, 2002. C. R. Leedham-Green and M. F. Newman, Space groups and groups of prime power order I, *Arch. Math.* **35** (1980), 193–203. F. Lemmermeyer, On $2$-class field towers of some imaginary quadratic number fields, *Abh. Math. Sem. Hamburg* **67** (1997), 205–214. F. Lemmermeyer, Class groups of dihedral extensions, *Math. Nachr.* **278** (2005), no. 6, 679–691. The MAGMA Group, MAGMA Computational Algebra System, Version 2.19-2, Sydney, 2012, `(http://magma.maths.usyd.edu.au)`. D. C. Mayer, Principalization in complex $S_3$-fields, *Congressus Numerantium* **80** (1991), 73–87 (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, Manitoba, Canada, 1990). D. C. Mayer, The second $p$-class group of a number field, *Int. J. Number Theory* **8** (2012), no. 2, 471–505, DOI 10.1142/S179304211250025X. D. C. Mayer, Transfers of metabelian $p$-groups, *Monatsh. Math.* **166** (2012), no. 3–4, 467–495, DOI 10.1007/s00605-010-0277-x. D. C. Mayer, *Principalisation algorithm via class group structure* (Preprint, 2011). D. C. Mayer, *Metabelian $3$-groups with abelianisation of type $(9,3)$* (Preprint, 2011). D. C. Mayer, *The distribution of second $p$-class groups on coclass graphs* (27th Journées Arithmétiques, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, 2011). C. McLeman, $p$-tower groups over quadratic imaginary number fields, *Ann. Sci. Math. Québec* **32** (2008), no. 2, 199–209. R. J. Miech, Metabelian $p$-groups of maximal class, *Trans. Amer. Math. Soc.* **152** (1970), 331–373. K. Miyake, Algebraic investigations of Hilbert’s Theorem $94$, the principal ideal theorem and the capitulation problem, *Expo. Math.* **7** (1989), 289–346. B. Nebelung, *Klassifikation metabelscher $3$-Gruppen mit Faktorkommutatorgruppe vom Typ $(3,3)$ und Anwendung auf das Kapitulationsproblem* (Inauguraldissertation, Band 1, Universität zu Köln, 1989). B. Nebelung, *Anhang zu Klassifikation metabelscher $3$-Gruppen mit Faktorkommutatorgruppe vom Typ $(3,3)$ und Anwendung auf das Kapitulationsproblem* (Inauguraldissertation, Band 2, Universität zu Köln, 1989). M. F. Newman, *Groups of prime-power order*, Groups — Canberra 1989, Lecture Notes in Mathematics, vol. 1456, Springer, 1990, pp. 49–62. M. F. Newman and E. A. O’Brien, Classifying $2$-groups by coclass, *Trans. Amer. Math. Soc.* **351** (1999), 131–169. H. Reichardt, Arithmetische Theorie der kubischen Zahlkörper als Radikalkörper, *Monatsh. Math. Phys.* **40** (1933), 323–350. A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, *J. Reine Angew. Math.* **166** (1932), 201–203. A. Scholz und O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm, *J. Reine Angew. Math.* **171** (1934), 19–41. O. Schreier, Über die Erweiterung von Gruppen II, *Abh. Math. Sem. Univ. Hamburg* **4** (1926), 321–346. I. R. Shafarevich, Extensions with prescribed ramification points, *Inst. Hautes Études Sci. Publ. Math.* **18** (1963), 71–95. M. Talbi, *Capitulation des $3$-classes d’idéaux dans certains corps de nombres* (Thèse, Université Mohammed Premier, Faculté des Sciences, Oujda, 2008). O. Taussky, A remark on the class field tower, *J. London Math. Soc.* **12** (1937), 82–85. O. Taussky, A remark concerning Hilbert’s Theorem $94$, *J. Reine Angew. Math.* **239/240** (1970), 435–438.
[^1]: Research supported by the Austrian Science Fund, Grant Nr. J0497-PHY
|
---
abstract: |
A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable $\omega$-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, $\operatorname{\mathsf P}$, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates *all* relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to pp-interpretations. For the semantic part we introduce a new construction, called [reflection]{}, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height $1$).
As a consequence, the complexity of the CSP of an at most countable $\omega$-categorical structure depends only on the identities of height $1$ satisfied in its polymorphism clone as well as the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures.
Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.
address:
- 'Department of Algebra, MFF UK, Sokolovska 83, 186 00 Praha 8, Czech Republic'
- 'Department of Algebra, MFF UK, Sokolovska 83, 186 00 Praha 8, Czech Republic'
- 'Department of Algebra, MFF UK, Sokolovska 83, 186 00 Praha 8, Czech Republic'
bibliography:
- 'CSPbib.bib'
- 'snek.bib'
- 'global.bib'
title: The Wonderland of Reflections
---
Introduction and Main Results
=============================
The motivation for this work is to resolve some unsatisfactory aspects in the fundamentals of the theory of fixed-template constraint satisfaction problems (CSPs). The CSP over a relational structure ${{\mathbb{A}}}$ in a finite language, denoted $\operatorname{CSP}({{\mathbb{A}}})$, is the decision problem which asks whether a given primitive positive (pp-) sentence over ${{\mathbb{A}}}$ is true. The focus of the theoretical research on such problems is to understand how the complexity of $\operatorname{CSP}({{\mathbb{A}}})$, be it computational or descriptive, depends on the structure ${{\mathbb{A}}}$. We start by briefly reviewing and discussing the basics of the theory, first for structures ${{\mathbb{A}}}$ with a finite universe, and then for those with an infinite one. The pioneering papers for finite structures ${{\mathbb{A}}}$ are [@FV98] and [@JBK], and our presentation is close to the recent survey [@BSL:9956673]. For infinite structures ${{\mathbb{A}}}$, a detailed account of the current state of the theory can be found in [@Bodirsky-HDR], and a compact introduction in [@Pin15]. For the sake of compactness, we will define standard notions only after this introduction, in Section \[sect:prelims\].
The finite case
---------------
For finite relational structures ${{\mathbb{A}}}$ and ${{\mathbb{B}}}$ there are three general reductions which are used to compare the complexity of their CSPs. Namely, we know that $\operatorname{CSP}({\mathbb{B}})$ is at most as hard as $\operatorname{CSP}({\mathbb{A}})$ if
- ${{\mathbb{B}}}$ is pp-interpretable in ${{\mathbb{A}}}$, or
- ${{\mathbb{B}}}$ is homomorphically equivalent to ${{\mathbb{A}}}$, or
- ${{\mathbb{A}}}$ is a core and ${{\mathbb{B}}}$ is obtained from ${{\mathbb{A}}}$ by adding a singleton unary relation.
Item (a) has two algebraic counterparts. The semantic one, item (ii) in Theorem \[thm:old\_finite\] below, follows from the well-known Galois correspondence between relational clones and function clones (see for example [@Szendrei]), while the syntactic one, item (iii) in the same theorem, follows from the Birkhoff’s HSP theorem [@Bir-On-the-structure].
\[thm:old\_finite\] Let ${{\mathbb{A}}}$, ${{\mathbb{B}}}$ be finite relational structures and ${{\mathscr{A}}}$, ${{\mathscr{B}}}$ their polymorphism clones. Then the following are equivalent.
- ${{\mathbb{B}}}$ is pp-interpretable in ${{\mathbb{A}}}$.
- ${{\mathscr{B}}}\in \operatorname{\mathsf{EHSP}_{fin}}{{\mathscr{A}}}$, or equivalently, ${{\mathscr{B}}}\in \operatorname{\mathsf{EHSP}}{{\mathscr{A}}}$; here, $\operatorname{\mathsf E}$ denotes the expansion operator.
- There exists a clone homomorphism from ${{\mathscr{A}}}$ into ${{\mathscr{B}}}$, i.e., a mapping ${{\mathscr{A}}}\to {{\mathscr{B}}}$ preserving identities.
One unsatisfactory feature of this theorem is that it does not cover the other two reductions (b) and ([c]{}), in particular the easiest reduction to homomorphically equivalent structures. The way around this fact is, usually, to assume that those reductions have already been applied. This is the same as saying that we can “without loss of generality” assume that structures are cores containing all unary singleton relations, or equivalently that their polymorphism clones are idempotent, and then only use reductions by pp-interpretations. However, this causes slightly awkward formulations, e.g., of the conjectured condition for polynomial solvability [@JBK] (also see Conjecture \[conj:old\_finite\] below) or of the condition for expressibility in Datalog [@LZ07; @BK14]. Even worse, it results in a loss of power: Example \[ex:hepp\] shows that there are cores ${{\mathbb{A}}}$ and ${{\mathbb{B}}}$ such that ${{\mathbb{B}}}$ is not pp-interpretable in ${{\mathbb{A}}}$, but ${{\mathbb{B}}}$ is homomorphically equivalent to a structure which is pp-interpretable in ${{\mathbb{A}}}$. These considerations bring up the following question: is there a variant of Theorem \[thm:old\_finite\] that covers all three reductions (a), (b), and ([c]{})?
Another question concerns item (iii), which implies that the complexity of $\operatorname{CSP}({{\mathbb{A}}})$ depends only on identities satisfied by operations in the polymorphism clone ${{\mathscr{A}}}$. However, the polymorphism clones of homomorphically equivalent structures need not necessarily satisfy the same identities, with the exception of height $1$ identities. Naturally, the question arises: Is it possible to prove that the complexity of the CSP of a structure depends only on the height $1$ identities that hold in its polymorphism clone?
Finally, and related to the preceding question, the finite tractability conjecture [@JBK] states that, assuming P${}\neq{}$NP, the CSP of a finite core ${{\mathbb{B}}}$ is NP-hard if and only if the idempotent reduct of its polymorphism clone ${{\mathscr{B}}}$ does not satisfy any non-trivial identities; here, we say that identities are non-trivial if they are not satisfiable in the clone of projections on a two-element set, which we denote by $\mathbf 1$. For general finite structures, the conjecture can be formulated as follows.
\[conj:old\_finite\] Let ${{\mathbb{A}}}$ be a finite relational structure and let ${{\mathbb{B}}}$ be its idempotent core, i.e., its core expanded by all singleton unary relations. Then one of the following holds.
- The polymorphism clone ${{\mathscr{B}}}$ of ${{\mathbb{B}}}$ maps homomorphically to $\mathbf{1}$ (and consequently $\operatorname{CSP}({{\mathbb{A}}})$ is NP-complete).
- $\operatorname{CSP}({{\mathbb{A}}})$ is solvable in polynomial-time.
Is it possible to find a criterion on the structure of the polymorphism clone ${{\mathscr{A}}}$ of ${{\mathbb{A}}}$, rather than ${{\mathscr{B}}}$, which divides NP-hard from polynomial-time solvable CSPs for all finite structures ${{\mathbb{A}}}$, without the necessity to consider their cores?
It turns out that Theorem \[thm:old\_finite\] can be generalized to answer all three questions in positive. First, we observe that ${{\mathbb{B}}}$ can obtained from ${{\mathbb{A}}}$ by using any number of the constructions (a), (b), ([c]{}) if and only if ${{\mathbb{B}}}$ is homomorphically equivalent to a pp-power of ${{\mathbb{A}}}$, where pp-power is a simplified version of pp-interpretation which we are going to define. We then introduce a simple algebraic construction, the [reflection]{}[^1], which is in a sense an algebraic counterpart to homomorphic equivalence. This gives us a suitable generalization of item (ii) in Theorem \[thm:old\_finite\]. Finally, we provide an analogue of Birkhoff’s HSP theorem for classes of algebras described by height $1$ identities, by which we obtain syntactic characterization corresponding to item (iii). Altogether, we get the following.
\[thm:new\_finite\] Let ${{\mathbb{A}}}$, ${{\mathbb{B}}}$ be finite relational structures and ${{\mathscr{A}}}$, ${{\mathscr{B}}}$ their polymorphism clones. Then the following are equivalent.
- ${{\mathbb{B}}}$ is homomorphically equivalent to a pp-power of ${{\mathbb{A}}}$, or equivalently, ${{\mathbb{B}}}$ can be obtained from ${{\mathbb{A}}}$ by a finite number of constructions among (a), (b), ([c]{}).
- ${{\mathscr{B}}}\in \operatorname{\mathsf{ERP}_{fin}}{{\mathscr{A}}}$, or equivalently, ${{\mathscr{B}}}\in \operatorname{\mathsf{ERP}}{{\mathscr{A}}}$; here, $\operatorname{\mathsf R}$ denotes the new operator of taking [reflections]{}.
- There exists an h1 clone homomorphism from ${{\mathscr{A}}}$ into ${{\mathscr{B}}}$, i.e., a mapping ${{\mathscr{A}}}\to {{\mathscr{B}}}$ preserving identities of height $1$.
This allows us to rephrase the conjectured sufficient condition for polynomial solvability. In the following theorem, items (i) – (iv) are equivalent by [@T77; @BK12; @Sig10; @KMM14], the primed items are new, core-free versions (see Section \[sec:wrapup\] for more details).
\[thm:equiv\_conditions\] Let ${{\mathbb{A}}}$ be a finite relational structure, let ${{\mathbb{B}}}$ be its idempotent core, and let ${{\mathscr{A}}}$, ${{\mathscr{B}}}$ be the polymorphism clones of ${{\mathbb{A}}}$, ${{\mathbb{B}}}$. Then the following are equivalent.
- there is no clone homomorphism from ${{\mathscr{B}}}$ to $\mathbf{1}$.
- there is no h1 clone homomorphism from ${{\mathscr{B}}}$ to $\mathbf{1}$.
- there is no h1 clone homomorphism from ${{\mathscr{A}}}$ to $\mathbf{1}$.
- ${{\mathscr{B}}}$ contains a cyclic operation, that is, an operation $t$ of arity $n \geq 2$ such that $t(x_1, \dots x_n) \approx t(x_2, \dots, x_n,x_1)$.
- ${{\mathscr{A}}}$ contains a cyclic operation.
- ${{\mathscr{B}}}$ contains a Siggers operation, that is, a $4$-ary operation $t$ such that $t(a,r,e,a) \approx t(r,a,r,e)$.
- ${{\mathscr{A}}}$ contains a Siggers operation.
In particular, the tractability conjecture can be equivalently formulated as follows.
\[conj:new\_finite\] Let ${{\mathbb{A}}}$ be a finite relational structure. Then one of the following holds.
- The polymorphism clone of ${{\mathbb{A}}}$ maps to $\mathbf{1}$ via an h1 clone homomorphism (and consequently $\operatorname{CSP}({{\mathbb{A}}})$ is NP-complete).
- $\operatorname{CSP}({{\mathbb{A}}})$ is solvable in polynomial-time.
The infinite case
-----------------
For countable $\omega$-categorical structures ${{\mathbb{A}}}$ and ${{\mathbb{B}}}$ we have the same general reductions (a), (b), and ([c]{}) – only the notion of a *core* has to be replaced by that of a *model-complete core*. A large part of the research on CSPs of infinite structures investigates when a given finite structure ${{\mathbb{B}}}$ can be obtained from an infinite structure ${{\mathbb{A}}}$ via those constructions [@BPP-projective-homomorphisms]. For what concerns the reduction by pp-interpretations, we have the following theorem from [@Topo-Birk].
\[thm:old\_infinite\] Let ${{\mathbb{A}}}$ be a countable $\omega$-categorical and ${{\mathbb{B}}}$ be a finite relational structure, and let ${{\mathscr{A}}}$, ${{\mathscr{B}}}$ their polymorphism clones. Then the following are equivalent.
- ${{\mathbb{B}}}$ is pp-interpretable in ${{\mathbb{A}}}$.
- ${{\mathscr{B}}}\in \operatorname{\mathsf{EHSP}_{fin}}{{\mathscr{A}}}$.
- There exists a continuous clone homomorphism from ${{\mathscr{A}}}$ into ${{\mathscr{B}}}$, i.e., a continuous mapping ${{\mathscr{A}}}\to {{\mathscr{B}}}$ preserving identities.
Regarding applicability to CSPs, this theorem suffers from the same shortcomings as its finite counterpart, as discussed above. But in the infinite case, two other unsatisfactory features which are not present in the situation for finite structures arise in addition.
Firstly, the class of all infinite structures being too vast to be approached as a whole, research on CSPs of infinite structures focusses on structures with particular properties, such as the Ramsey property or finite boundedness; cf. for example [@BP-reductsRamsey]. By assuming that a structure ${{\mathbb{A}}}$ is a model-complete core one might lose these properties. In other words, if we start with an $\omega$-categorical structure ${{\mathbb{A}}}$ satisfying a certain property such as the Ramsey property, then the unique model-complete core which is homomorphically equivalent to ${{\mathbb{A}}}$ might fail to satisfy this property. This results in a serious technical disadvantage: much of the machinery developed for the investigation of infinite CSPs cannot be applied, for example, in the absence of the Ramsey property.
Secondly, contrary to the situation in the finite, adding constants to a model-complete core does not terminate after a finite number of steps in the infinite case. Hence, while there is an analog of the concept of a core for the infinite, namely that of a model-complete core, there is no analog of the notion of an idempotent core, or an idempotent polymorphism clone, for the $\omega$-categorical setting. This leads to less elegant formulations than in the finite, such as in the following conjecture of Bodirsky and Pinsker (cf. [@BPP-projective-homomorphisms]).
\[conj:old\] Let ${{\mathbb{A}}}$ be a reduct of a finitely bounded homogeneous structure, and let ${{\mathbb{B}}}$ be its model-complete core. Then one of the following holds.
- There exist elements $b_1,\ldots,b_n$ in ${{\mathbb{B}}}$ such that the polymorphism clone of the expansion of ${{\mathbb{B}}}$ by those constants maps homomorphically and continuously to $\mathbf 1$ (and consequently $\operatorname{CSP}({{\mathbb{A}}})$ is NP-complete).
- $\operatorname{CSP}({{\mathbb{A}}})$ is solvable in polynomial-time.
We are going to prove the following theorem which will avoid the issues raised above.
\[thm:new\_infinite\] Let ${{\mathbb{A}}}$ be an at most countable $\omega$-categorical and ${{\mathbb{B}}}$ be a finite relational structure, and let ${{\mathscr{A}}}$, ${{\mathscr{B}}}$ their polymorphism clones. Then the following are equivalent.
- ${{\mathbb{B}}}$ is homomorphically equivalent to a pp-power of ${{\mathbb{A}}}$, or equivalently, ${{\mathbb{B}}}$ can be obtained from ${{\mathbb{A}}}$ by a finite number of constructions among (a), (b), ([c]{}).
- ${{\mathscr{B}}}\in \operatorname{\mathsf{ERP}_{fin}}{{\mathscr{A}}}$, or equivalently, ${{\mathscr{B}}}\in \operatorname{\mathsf{ERP}}{{\mathscr{A}}}$.
- There exists a uniformly continuous h1 clone homomorphism from ${{\mathscr{A}}}$ into ${{\mathscr{B}}}$, i.e., a uniformly continuous mapping ${{\mathscr{A}}}\to {{\mathscr{B}}}$ preserving identities of height $1$.
This allows us, in particular, to formulate the following conjecture, which is implied by Conjecture \[conj:old\] but not necessarily equivalent to it; we refer to Section \[sec:wrapup\] for more details.
\[conj:new\] Let ${{\mathbb{A}}}$ be a reduct of a finitely bounded homogeneous structure, and let ${{\mathscr{A}}}$ be its polymorphism clone. Then one of the following holds.
- ${{\mathscr{A}}}$ maps to $\mathbf{1}$ via a uniformly continuous h1 clone homomorphism (and consequently $\operatorname{CSP}({{\mathbb{A}}})$ is NP-complete).
- $\operatorname{CSP}({{\mathbb{A}}})$ is solvable in polynomial-time.
Coloring of clones by relational structures {#sect:intro_colorings}
-------------------------------------------
It came as a big surprise to us that there is a tight connection between h1 clone homomorphisms and Sequeira’s notion of compatibility with projections [@sequeira01] which he used to attack some open problems concerning Maltsev conditions.
Each Maltsev condition determines a filter in the lattice of interpretability types of varieties [@neumann74; @garcia.taylor84]. The question whether a given Maltsev condition is implied by the conjunction of two strictly weaker conditions translates into the question whether the corresponding filter is join prime (that is, whether the complement of the filter is closed under joins). Garcia and Taylor [@garcia.taylor84] conjectured that two important Maltsev conditions determine join prime filters:
\[conj:taylor\]
- The filter of congruence permutable varieties is join prime.
- The filter of congruence modular varieties is join prime.
The first conjecture was confirmed by Tschantz in an unpublished paper [@tschantz]. His proof is technically extremely difficult and it seems impossible to generalize the arguments to even slightly more complex Maltsev conditions, such as the one characterizing $3$-permutability. In an effort to resolve the second conjecture and similar problems, Sequeira introduced the notion of compatibility with projections [@sequeira01] and used it to prove some interesting partial results. We generalize his concept to “colorability of clones by relational structures” and provide a simple link to h1 clone homomorphisms:
\[thm:coloring-and-h1\] Let ${{\mathscr{A}}}$ be a function clone and let ${{\mathscr{B}}}$ be the polymorphism clone of a relational structure ${{\mathbb{B}}}$. Then the following are equivalent.
- There exists an h1 clone homomorphism from ${{\mathscr{A}}}$ into ${{\mathscr{B}}}$.
- ${{\mathscr{A}}}$ is ${{\mathbb{B}}}$-colorable.
As a corollary, we obtain the following generalization of [@bentz.sequeira14] where it was additionally assumed that the varieties are idempotent.
\[thm:perm-and-modular\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be two varieties defined by identities of height at most 1.
- If ${\mathcal{V}}$, ${\mathcal{W}}$ are not congruence modular, then neither is ${\mathcal{V}} \vee {\mathcal{W}}$.
- If ${\mathcal{V}}$, ${\mathcal{W}}$ are not congruence $n$-permutable for any $n$, then neither is ${\mathcal{V}} \vee {\mathcal{W}}$.
Outline of the article
----------------------
Section \[sect:prelims\] contains the definitions of the standard notions we used in the introduction. In Section \[sec:rel\] we discuss relational constructions, in particular pp-powers. Then follow algebraic constructions, notably [reflections]{} and the operator $\operatorname{\mathsf R}$[^2], in Section \[sec:tra\]. Their syntactic counterpart, h1 clone homomorphisms, is dealt with in Section \[sec:birk\]. The $\omega$-categorical case is discussed together with topological considerations in Section \[sec:cont\]. Section \[sec:snek\] is devoted to the connection between colorings and h1 clone homomorphisms. We conclude this work with Section \[sec:wrapup\] where we prove and discuss the results and conjectures from the introduction.
Preliminaries {#sect:prelims}
=============
We explain the classical notions which appeared in the introduction, and fix some notation for the rest of the article. The new notion of pp-power, the operator $\operatorname{\mathsf R}$ acting on function clones, h1 clone homomorphisms, and colorings of clones by relational structures will be defined in their own sections. For undefined universal algebraic concepts and more detailed presentations of the notions presented here we refer to [@BS81; @Berg11]. For notions from model theory we refer to [@Hodges].
Relational structures and polymorphism clones
---------------------------------------------
We denote relational structures by ${{\mathbb{A}}}, {{\mathbb{B}}}$, etc. When ${{\mathbb{A}}}$ is a relational structure, we reserve the symbol $A$ for its domain. We write ${{\mathscr{A}}}$ for its *polymorphism clone*, i.e., the set of all finitary operations on $A$ which preserve all relations of ${{\mathbb{A}}}$, usually denoted by $\operatorname{Pol}({{\mathbb{A}}})$ in the literature. The polymorphism clone ${{\mathscr{A}}}$ is always a *function clone*, i.e., it is closed under composition and contains all projections.
CSPs
----
For a finite relational signature $\Sigma$ and a $\Sigma$-structure ${\mathbb{A}}$, the *constraint satisfaction problem* of ${\mathbb{A}}$, or $\operatorname{CSP}({\mathbb{A}})$ for short, is the membership problem for the class $$\{ {\mathbb{C}}: {\mathbb{C}} \mbox{ is a finite $\Sigma$-structure and there exists a~homomorphism ${\mathbb{C}} \to {\mathbb{A}}$}\} \enspace.$$ An alternative definition of $\operatorname{CSP}({\mathbb{A}})$ is via primitive positive (pp-) sentences. Recall that a *pp-formula* over ${\mathbb{A}}$ is a first order formula which only uses predicates from ${\mathbb{A}}$, conjunction, equality, and existential quantification. $\operatorname{CSP}({\mathbb{A}})$ can equivalently be phrased as the membership problem of the set of pp-sentences which are true in ${\mathbb{A}}$.
Our results, in particular Theorems \[thm:new\_finite\] and \[thm:new\_infinite\], are purely structural and complexity-free. Still it is worthwhile mentioning that when we say that a computational problem reduces to, or is not harder than, another computational problem, we have log-space reductions in mind (although the claim is true also for other meanings of hardness).
Notions from the infinite {#sect:notions_infinite}
-------------------------
The algebraic theory of the CSP is best developed for finite structures and countable $\omega$-categorical structures. Recall that an at most countable relational structure ${\mathbb{A}}$ is *$\omega$-categorical* if, for every $n \geq 1$, the natural componentwise action of its automorphism group $\operatorname{Aut}({\mathbb{A}})$ on $A^n$ has only finitely many orbits. In particular finite structures are always $\omega$-categorical.
The class of infinite $\omega$-categorical structures which has probably received most attention in the literature are reducts of finitely bounded homogeneous structures, which appear in Conjectures \[conj:old\] and \[conj:new\]. Since we do not need these notions in the present paper, we refrain from defining them and refer to the surveys [@BP-reductsRamsey], [@Bodirsky-HDR], and [@Pin15].
pp definitions and interpretations
----------------------------------
A relation is *pp-definable* in a relational structure ${\mathbb{A}}$ if it is definable with a pp-formula over ${\mathbb{A}}$ without parameters. Let ${\mathbb{A}}$, ${\mathbb{B}}$ be relational structures with possibly different signatures. We say that ${\mathbb{A}}$ *pp-interprets* ${\mathbb{B}}$, or that ${\mathbb{B}}$ is *pp-interpretable* in ${\mathbb{A}}$, if there exists $n \geq 1$ and a mapping $f$ from a subset of $A^n$ onto $B$ such that the following relations are pp-definable in ${\mathbb{A}}$:
- the domain of $f$;
- the preimage of the equality relation on $B$ under $f$, viewed as a $2n$-ary relation on $A$;
- the preimage of every relation in ${\mathbb{B}}$ under $f$, where the preimage of a $k$-ary relation under $f$ is again regarded as a $kn$-ary relation on $A$.
Homomorphic equivalence and cores
---------------------------------
When relational structures ${\mathbb{A}}$ and ${\mathbb{B}}$ have the same signature, then we say that ${\mathbb{A}}$ and ${\mathbb{B}}$ are *homomorphically equivalent* if there exists a homomorphism ${\mathbb{A}} \to {\mathbb{B}}$ and a homomorphism ${\mathbb{B}} \to {\mathbb{A}}$. A relational structure ${\mathbb{B}}$ is called a *model-complete core* if the automorphisms of ${\mathbb{B}}$ are dense in its endomorphisms, i.e., for every endomorphism $e$ of ${\mathbb{B}}$ and every finite subset $B'$ of $B$ there exists an automorphism of ${\mathbb{B}}$ which agrees with $e$ on $B'$. When ${\mathbb{B}}$ is finite, then this means that every endomorphism is an automorphism, and ${\mathbb{B}}$ is simply called a *core*. Every at most countable $\omega$-categorical structure is homomorphically equivalent to a unique model-complete core, which is again $\omega$-categorical [@Cores-journal; @JBK]. A special case of a core is an *idempotent core*, by which we mean a relational structure whose only endomorphism is the identity function on its domain. By adding all unary singleton relations to a finite core (recall reduction ([c]{}) from the introduction) one obtains an idempotent core. Note that on the other hand, a countably infinite $\omega$-categorical structure is never idempotent since its automorphism group is *oligomorphic*, i.e., large in a certain sense; recall Section \[sect:notions\_infinite\].
HSP and E
---------
When ${{\mathscr{A}}}$ is a function clone, then we denote by $\operatorname{\mathsf H}({{\mathscr{A}}})$ all function clones obtained by letting ${{\mathscr{A}}}$ act naturally on the classes of an invariant equivalence relation on its domain. By $\operatorname{\mathsf S}({{\mathscr{A}}})$ we denote all function clones obtained by letting ${{\mathscr{A}}}$ act on an an invariant subset of its domain via restriction. We write $\operatorname{\mathsf P}({{\mathscr{A}}})$ and $\operatorname{\mathsf P_{fin}}({{\mathscr{A}}})$ for all componentwise actions of ${{\mathscr{A}}}$ on powers and finite powers of its domain, respectively. The operator $\operatorname{\mathsf E}({{\mathscr{A}}})$ yields all function clones obtained from ${{\mathscr{A}}}$ by adding functions to it. All these operators are to be understood up to renaming of elements of domains, i.e., we consider two function clones equal if one can obtained from the other via a bijection between their respective domains. We use combinations of these operators, such as $\operatorname{\mathsf{HSP}}({{\mathscr{A}}})$, with their obvious meaning.
We denote algebras by ${\mathbf{A}}$, ${\mathbf{B}}$, etc., and their domains by $A$, $B$, etc. We apply the operators $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, $\operatorname{\mathsf P}$, and $\operatorname{\mathsf P_{fin}}$ also to algebras and to classes of algebras of the same signature as it is standard in the literature. When we apply $\operatorname{\mathsf P}$ to a class of algebras, then we refer to all products of algebras in the class (rather than powers only). Concerning the operator $\operatorname{\mathsf E}$, we shall also apply it to algebras and sets of functions which are not necessarily function clones, and mean that it returns all algebras whose operations contain all operations the original algebra (all sets of functions that contain the original set of functions).
Clone homomorphisms
-------------------
A *clone homomorphism* from a function clone ${{\mathscr{A}}}$ to a function clone ${{\mathscr{B}}}$ is a mapping $\xi\colon {{\mathscr{A}}}{\rightarrow}{{\mathscr{B}}}$ which
- preserves arities, i.e., it sends every function in ${{\mathscr{A}}}$ to a function of the same arity in ${{\mathscr{B}}}$;
- preserves each projection, i.e., it sends the $k$-ary projection onto the $i$-th coordinate in ${{\mathscr{A}}}$ to the same projection in ${{\mathscr{B}}}$, for all $1\leq i\leq k$;
- preserves composition[^3], i.e., $\xi(f(g_1,\ldots,g_n))=\xi(f)(\xi(g_1),\ldots,\xi(g_n))$ for all $n$-ary functions $f$ and all $m$-ary functions $g_1,\ldots,g_n$ in ${{\mathscr{A}}}$.
For all $1\leq i\leq k$ we denote the $k$-ary projection onto the $i$-th coordinate by $\pi^k_i$, in any function clone and irrespectively of the domain of that clone. This slight abuse of notation allows us, for example, to express the second item above by writing $\xi(\pi^k_i)=\pi^k_i$.
The lattice of interpretability types of varieties
--------------------------------------------------
A *variety* is a class of algebras of the same signature closed under homomorphic images, subalgebras, and products. By Birkhoff’s HSP theorem [@Bir-On-the-structure], a class of algebras of the same signature is a variety if and only if it is the class of models of some set of identities (where an *identity* is a universally quantified equation). Each variety ${\mathcal{V}}$ has a *generator*, that is, an algebra ${\mathbf{A}} \in {\mathcal{V}}$ such that ${\mathcal{V}} = \operatorname{\mathsf{HSP}}{{\mathbf{A}}}$. We denote by ${\mathrm{clo}(V)}$ the clone of term operations of any generator of ${\mathcal{V}}$ (the choice of generator is immaterial for our purposes).
We quasi-order the class of all varieties (of varying signatures) by defining ${\mathcal{V}} \leq {\mathcal{W}}$ if there exists a clone homomorphism ${\mathrm{clo}({\mathcal{V}})} \to {\mathrm{clo}({\mathcal{W}})}$. Then we identify two varieties ${\mathcal{V}}$, ${\mathcal{W}}$ if ${\mathcal{V}} \leq {\mathcal{W}} \leq {\mathcal{V}}$. The obtained partially ordered class is a lattice – the *lattice of interpretability types of varieties*. The lattice join can be described by means of the defining identities: the signature of ${\mathcal{V}} \vee {\mathcal{W}}$ is the disjoint union of the signatures of ${\mathcal{V}}$ and ${\mathcal{W}}$ and the set of defining identities of ${\mathcal{V}} \vee {\mathcal{W}}$ is the union of the defining identities of ${\mathcal{V}}$ and ${\mathcal{W}}$.
Topology
--------
Every function clone is naturally equipped with the topology of pointwise convergence: in this topology, a sequence $(f_i)_{i\in\omega}$ of $n$-ary functions converges to an $n$-ary function $f$ on the same domain if and only if for all $n$-tuples $\bar a$ of the domain the functions $f_i$ agree with $f$ on $\bar a$ for all but finitely many $i\in\omega$. Therefore, every function clone gives rise to an abstract topological clone [@Reconstruction]. We imagine function clones always as carrying this topology, which is, in the case of a countable domain, in fact induced by a metric, and in general by a uniformity [@Reconstruction]. Then a mapping $\xi\colon {{\mathscr{A}}}{\rightarrow}{{\mathscr{B}}}$, where ${{\mathscr{A}}}$ and ${{\mathscr{B}}}$ are function clones, is continuous if and only if for all $f\in{{\mathscr{A}}}$ and all finite sets $B'\subseteq B$ there exists a finite set $A'\subseteq A$ such that for all $g\in{{\mathscr{A}}}$ of the same arity as $f$, if $g$ agrees with $f$ on $A'$, then $\xi(g)$ agrees with $\xi(f)$ on $B'$. It is uniformly continuous if and only if for all finite sets $B'\subseteq B$ there exists a finite set $A'\subseteq A$ such that whenever two functions $f, g\in{{\mathscr{A}}}$ of the same arity agree on $A'$, then their images $\xi(f), \xi(g)$ under $\xi$ agree on $B'$.
We remark that the polymorphism clones of relational structures are precisely the function clones which are complete with respect to this topology. Function clones on a finite domain are discrete.
Relational Constructions {#sec:rel}
========================
We first recall the general CSP reductions mentioned in the introduction, there labelled (a), (b), and ([c]{}), and then introduce a weaker variant of a pp-interpretation which we call *pp-power*.
Classical reductions
--------------------
The first reduction (a) from the introduction is the one via primitive positive interpretations, justified by the following proposition [@JBK].
Let ${\mathbb{A}}$, ${\mathbb{B}}$ are relational structures with finite signatures. If ${\mathbb{A}}$ pp-interprets ${\mathbb{B}}$, then $\operatorname{CSP}({\mathbb{B}})$ is log-space reducible to $\operatorname{CSP}({\mathbb{A}})$.
The second reduction, (b) in the introduction, is homomorphic equivalence. We have the following observation, which is easily verified using the fact that homomorphisms preserve pp-formulas.
Let ${\mathbb{A}}$, ${\mathbb{B}}$ be relational structures in the same finite signature which are homomorphically equivalent. Then $\operatorname{CSP}({\mathbb{A}})=\operatorname{CSP}({\mathbb{B}})$.
Let us observe here that for the CSPs of ${\mathbb{A}}$ and ${\mathbb{B}}$ to be equal, we only have to require that every finite substructure of ${\mathbb{A}}$ maps homomorphically into ${\mathbb{B}}$ and vice-versa. This gives us a more general reduction in general; however, in the countable $\omega$-categorical case this condition of “local” homomorphic equivalence is easily seen to be equivalent to full homomorphic equivalence via a standard compactness argument, and so we shall not further consider this notion in the present paper.
The following proposition from [@JBK] states that an $\omega$-categorical model-complete core can be expanded by a singleton relation without making the CSP harder, giving us reduction ([c]{}) from the introduction. Although stated in [@JBK] for finite structures, it holds also in the $\omega$-categorical case (cf. [@Bodirsky-HDR]).
Let ${\mathbb{A}}$ be an at most countable $\omega$-categorical structure which is a model-complete core and let ${\mathbb{B}}$ be a structure obtained from ${\mathbb{A}}$ by adding a singleton unary relation. The $\operatorname{CSP}({\mathbb{A}})$ and $\operatorname{CSP}({\mathbb{B}})$ are log-space equivalent.
The following definition assembles all three reductions.
\[defn:ppconstructible\] Let ${\mathbb{A}}$, ${\mathbb{B}}$ be relational structures. We say that ${\mathbb{B}}$ can be *pp-constructed* from ${\mathbb{A}}$ if there exists a sequence ${\mathbb{A}} = {\mathbb{C}}_1, {\mathbb{C}}_2, \dots, {\mathbb{C}}_k = {\mathbb{B}}$ such that for every $1 \leq i < k$
- ${\mathbb{C}}_i$ pp-interprets ${\mathbb{C}}_{i+1}$, or
- ${\mathbb{C}}_{i+1}$ is homomorphically equivalent to ${\mathbb{C}}_i$, or
- ${\mathbb{C}}_i$ is an at most countable $\omega$-categorical model-complete core, and ${\mathbb{C}}_{i+1}$ is obtained from ${\mathbb{C}}_i$ by adding a singleton unary relation.
\[cor:complexity\] Let ${\mathbb{A}}$, ${\mathbb{B}}$ be relational structures. If ${\mathbb{B}}$ can be pp-constructed from ${\mathbb{A}}$, then $\operatorname{CSP}({\mathbb{B}})$ is log-space reducible to $\operatorname{CSP}({\mathbb{A}})$.
pp-powers
---------
We now introduce a weakening of pp-interpretations which together with homomorphic equivalence will cover all classical reductions. This weakening is obtained by requiring that the partial surjective mapping $f$ from the definition of pp-interpretation in Section \[sect:prelims\] is a bijection with full domain.
Let ${\mathbb{A}}, {\mathbb{B}}$ be relational structures. We say that ${\mathbb{B}}$ is a *pp-power* of ${\mathbb{A}}$ if it is isomorphic to a structure with domain $A^n$, where $n\geq 1$, whose relations are pp-definable from ${\mathbb{A}}$; as before, a $k$-ary relation on $A^n$ is regarded as a $kn$-ary relation on $A$.
The next lemma deals with reductions (a) and (b) and their combinations. We need an auxiliary definition.
For a class ${\mathcal{K}}$ of relational structures, we denote by
- $\operatorname{\mathsf Pp-int}{\mathcal{K}}$ the class of structures which are pp-interpretable in some structure in ${\mathcal{K}}$;
- $\operatorname{\mathsf Ppp}{\mathcal{K}}$ the class of pp-powers of structures in ${\mathcal{K}}$;
- $\operatorname{\mathsf He}{\mathcal{K}}$ the class of structures which are homomorphically equivalent to a member of ${\mathcal{K}}$.
\[lem:pppp\] Let ${\mathcal{K}}$ be a class of relational structures. Then
1. $\operatorname{\mathsf Pp-int}{\mathcal{K}} \subseteq \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathcal{K}}$;
2. $\operatorname{\mathsf He}\operatorname{\mathsf He}{\mathcal{K}} = \operatorname{\mathsf He}{\mathcal{K}}$;
3. $\operatorname{\mathsf Ppp}\operatorname{\mathsf Ppp}{\mathcal{K}} = \operatorname{\mathsf Ppp}{\mathcal{K}}$;
4. $\operatorname{\mathsf Ppp}\operatorname{\mathsf He}{\mathcal{K}} \subseteq \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathcal{K}}$.
To show (i), let ${\mathbb{A}}$, ${\mathbb{B}}$ be relational structures such that ${\mathbb{A}}$ pp-interprets ${\mathbb{B}}$ and let $f\colon A^n \to B$ be a partial surjective mapping witnessing the pp-interpretability. Take the relational structure ${\mathbb{C}}$ with the same signature as ${\mathbb{B}}$ whose universe is $C = A^n$ and whose relations are $f$-preimages of relations in ${\mathbb{B}}$. Let $f'$ be any mapping from $A^n$ to $B$ extending $f$ and let $g$ be any mapping from $B$ to $A^n$ such that $f'\circ g$ is the identity on $B$. Now ${\mathbb{C}}$ is a pp-power of ${\mathbb{A}}$, $f'$ is a homomorphism from ${\mathbb{C}}$ to ${\mathbb{B}}$ and $g$ is a homomorphism from ${\mathbb{B}}$ to ${\mathbb{C}}$. Thus ${\mathbb{B}} \in \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathbb{A}}$, as required.
Item (ii) follows from the fact that the composition of two homomorphisms ${\mathbb{A}} \to {\mathbb{B}} \to {\mathbb{C}}$ is a homomorphism ${\mathbb{A}} \to {\mathbb{C}}$. Item (iii) is readily seen as well.
Finally, let ${\mathbb{A}}$ and ${\mathbb{B}}$ be homomorphically equivalent relational structures of the same signature and let ${\mathbb{C}}$ be an $n$-th pp-power of ${\mathbb{B}}$. We define an $n$-th pp-power ${\mathbb{D}}$ of ${\mathbb{A}}$ of the same signature as ${\mathbb{C}}$ as follows: for each relation in ${\mathbb{C}}$, consider some pp-definition from ${\mathbb{B}}$ and replace all relations in this pp-formula by the corresponding relations of ${\mathbb{A}}$. It is easy to see that the $n$-th power of any homomorphism from ${\mathbb{A}}$ to ${\mathbb{B}}$ (from ${\mathbb{B}}$ to ${\mathbb{A}}$, respectively) is a homomorphism from ${\mathbb{D}}$ to ${\mathbb{C}}$ (from ${\mathbb{C}}$ to ${\mathbb{D}}$, respectively). In particular, ${\mathbb{C}}$ and ${\mathbb{D}}$ are homomorphically equivalent, proving (iv).
The following lemma shows that reduction ([c]{}), namely the adding of singleton unary relations to model-complete cores, is actually already covered by homomorphic equivalence and pp-interpretations; in particular, we could have omitted it in Definition \[defn:ppconstructible\].
\[lem:adding\_const\] Let ${\mathbb{A}}$ be an at most countable $\omega$-categorical structure which is a model-complete core and let ${\mathbb{B}}$ be a structure obtained from ${\mathbb{A}}$ by adding a singleton unary relation. Then ${\mathbb{B}} \in \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathbb{A}}$.
We denote $S = \{s\}$, $s \in A$, the added singleton unary relation. Let $O \subseteq A$ denote the orbit of $s$ under the action of $\operatorname{Aut}({\mathbb{A}})$. Since ${\mathbb{A}}$ is a model-complete core, $O$ is preserved by all endomorphisms of ${\mathbb{A}}$, and since $O$ consists of only one orbit with respect to the action of $\operatorname{Aut}({\mathbb{A}})$, this implies that $O$ is preserved by all polymorphisms of ${\mathbb{A}}$ [@tcsps-journal]. Hence, using $\omega$-categoricity we infer that $O$ is pp-definable in ${\mathbb{A}}$ [@BodirskyNesetrilJLC].
We define a relational structure ${\mathbb{C}}$ over the domain $C = A^2$ with the same signature as ${\mathbb{B}}$. The relation of ${\mathbb{C}}$ corresponding to a $k$-ary relation $R$ in ${\mathbb{A}}$ is defined by $$\overline{R} = \{((a_1,b_1), \dots, (a_k,b_k)) \in (A^2)^k : b_1=b_2=\dots=b_k \in O, (a_1,\dots, a_k) \in R\}$$ and the relation corresponding to the singleton unary relation $S$ is defined as $\overline{S} = \{(a,a): a \in O\}$. Clearly, ${\mathbb{C}}$ is a pp-power of ${\mathbb{A}}$, so it remains to show that ${\mathbb{C}}$ is homomorphically equivalent to ${\mathbb{B}}$.
The mapping $A \to A^2$, defined by $a \mapsto (a,s)$ is a homomorphism from ${\mathbb{B}}$ to ${\mathbb{C}}$.
To define a homomorphism $f\colon {\mathbb{C}} \to {\mathbb{B}}$ we first pick, for each $a \in O$, an automorphism $\alpha_a \in \operatorname{Aut}({\mathbb{A}})$ with $\alpha_a(a) = s$. Now $f$ is defined by $f(a,b) = \alpha_b(a)$ if $b \in O$, and otherwise arbitrarily. We check that this mapping is indeed a homomorphism. If $R$ is a $k$-ary relation in ${\mathbb{A}}$ and $\mathbf{x} = ((a_1,b_1), \dots, (a_k,b_k)) \in \overline{R}$, then $b_1=b_2= \dots=b_k=b \in O$ and we have $f(\mathbf{x}) = (\alpha_b(a_1), \dots, \alpha_b(a_k))$. This $k$-tuple is in $R$ as $(a_1, \dots, a_k) \in R$ and $\alpha_b$ is an automorphism of ${\mathbb{A}}$. Finally, $f$ also preserves the added relation since each $(a,a) \in \overline{S}$ is mapped to $\alpha_a(a) = s \in S$.
A corollary of the last two lemmata is that we can cover all general reductions to a given structure by considering all structures which are homomorphically equivalent to a pp-power of that structure.
\[cor:relational\] The following are equivalent for at most countable $\omega$-categorical relational structures ${\mathbb{A}}, {\mathbb{B}}$.
- ${\mathbb{B}}$ can be pp-constructed from ${\mathbb{A}}$.
- ${\mathbb{B}} \in \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathbb{A}}$; that is, ${\mathbb{B}}$ is homomorphically equivalent to a pp-power of ${\mathbb{A}}$.
Lemma \[lem:pppp\], item (i), and Lemma \[lem:adding\_const\] imply that if ${\mathbb{B}}$ can be pp-constructed from ${\mathbb{A}}$, then it can be constructed using pp-powers and homomorphic equivalence. From items (ii), (iii) and (iv) of Lemma \[lem:pppp\] we can conclude that ${\mathbb{B}} \in \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathbb{A}}$.
The other implication is trivial.
We shall conclude this section with an example of two finite idempotent relational structures ${\mathbb{A}}$, ${\mathbb{B}}$ such that ${\mathbb{B}} \in \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathbb{A}}$ but ${\mathbb{B}} \not\in \operatorname{\mathsf Pp-int}{\mathbb{A}}$. Therefore, as mentioned in the introduction, the common practice of first reducing to cores and then considering only pp-interpretations results in a true loss of power.
\[ex:hepp\] Consider the relational structure ${\mathbb{A}}$ with domain $A = \mathbb{Z}_2^2$ consisting of ternary relations $R_{(a,b)}$, $(a,b) \in \mathbb{Z}_2^2$ defined by $$R_{(a,b)} = \{(\mathbf{x},\mathbf{y},\mathbf{z}) \in (\mathbb{Z}_2^2)^3: \mathbf{x} + \mathbf{y} + \mathbf{z} = (a,b)\},$$ and unary singleton relations $\{(a,b)\}$, $(a,b) \in \mathbb{Z}_2^2$. Let ${\mathbb{A}}'$ be the reduct of ${\mathbb{A}}$ formed by the relations $R_{(0,0)}$, $R_{(1,0)}$, $\{(0,0)\}$, and $\{(1,0)\}$. Trivially, ${\mathbb{A}}'$ is pp-definable from ${\mathbb{A}}$. Finally, take the relational structure ${\mathbb{B}}$ with $B = \mathbb{Z}_2$ and relations $R_a$, $a \in \mathbb{Z}_2$, where $$R_a = \{(x,y,z) \in \mathbb{Z}_2^3: x+y+z = a\},$$ together with singletons $\{0\}$, $\{1\}$.
Note that ${\mathscr{A}}$ consists of idempotent affine operations of the module $\mathbb{Z}_2^2$ over $\operatorname{End}(\mathbb{Z}_2^2)$ and ${\mathscr{B}}$ is formed by idempotent affine operations of the abelian group $\mathbb{Z}_2$.
The mappings $A \to B$, $(x_1,x_2) \mapsto x_1$ and $B \to A$, $x \mapsto (x,0)$ are homomorphisms from ${\mathbb{A}}'$ to ${\mathbb{B}}$ and from ${\mathbb{B}}$ to ${\mathbb{A}}'$, respectively. (In fact, ${\mathbb{B}}$ is the core of ${\mathbb{A}}'$.) Therefore, ${\mathbb{B}} \in \operatorname{\mathsf He}\operatorname{\mathsf Ppp}{\mathbb{A}}$.
Suppose that ${\mathbb{A}}$ pp-interprets ${\mathbb{B}}$ and $f$ is a mapping from $C \subseteq A^n$ onto $B$ witnessing this. Let $\alpha$ be the kernel of $f$ – it is an equivalence relation on $C$ with two blocks. By the definition of pp-interpretation, both $C$ and $\alpha$ are pp-definable from ${\mathbb{A}}$. Since ${\mathbb{A}}$ contains the singleton unary relations, then both blocks of $\alpha$ are pp-definable as well. Thus $C$ is a pp-definable relation which is a disjoint union of two pp-definable relations. This is impossible as it is easily seen that the cardinality of any relation pp-definable from ${\mathbb{A}}$ is a power of $4$.
Algebraic Constructions {#sec:tra}
=======================
In the light of Theorems \[thm:old\_finite\] and \[thm:old\_infinite\], one can say that the algebraic counterpart of pp-interpretations in $\omega$-categorical structures is roughly the $\operatorname{\mathsf{HSP}_{fin}}$ operator. We introduce a new algebraic operator which corresponds to homomorphic equivalence.
Let ${\mathbf{A}}$ be an algebra with signature $\tau$. Let $B$ be a set, and let $h_1\colon B{\rightarrow}A$ and $h_2 \colon A{\rightarrow}B$ be functions. We define a $\tau$-algebra ${\mathbf{B}}$ with domain $B$ as follows: for every operation $f(x_1,\ldots,x_n)$ of ${\mathbf{A}}$, ${\mathbf{B}}$ has the operation $$(x_1,\ldots,x_n)\mapsto h_2(f(h_1(x_1),\ldots,h_1(x_n))).$$ We call ${\mathbf{B}}$ a *[reflection]{}* of ${\mathbf{A}}$. If $h_2\circ h_1$ is the identity function on $B$, then we say that ${\mathbf{B}}$ is a *retraction* of ${\mathbf{A}}$.
For a class ${\mathcal{K}}$ of algebras, we write $\operatorname{\mathsf R}{\mathcal{K}}$ for all [reflections]{} of algebras in ${\mathcal{K}}$, and similarly $\operatorname{\mathsf R_{ret}}{\mathcal{K}}$ for all retractions of algebras in ${\mathcal{K}}$. We also apply this operator to single algebras, writing $\operatorname{\mathsf R}{\mathbf{A}}$, with the obvious meaning.
By viewing the operations of a function clone as those of an algebra, we moreover apply the operator to function clones, similarly to the operators $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, and $\operatorname{\mathsf P}$. Observe, however, that contrary to the situation with the latter operators, the [reflection]{} of a function clone need not be a function clone since it need not contain the projections or be closed under composition.
For a function clone ${{\mathscr{A}}}$, we denote by $\operatorname{\mathsf R}{{\mathscr{A}}}$ all sets of functions obtained as in the above definition.
It is well-known that for any class of algebras ${\mathcal{K}}$, the class $\operatorname{\mathsf{HSP}}{\mathcal{K}}$ is equal to the closure of ${\mathcal{K}}$ under $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$ and $\operatorname{\mathsf P}$ [@Bir-On-the-structure]. We now provide a similar observation which includes the operator $\operatorname{\mathsf R}$. Note that in particular, the first item of the following lemma implies that the operator $\operatorname{\mathsf R}$ is a common generalization of the operators $\operatorname{\mathsf H}$ and $\operatorname{\mathsf S}$.
\[lem:algebraicinclusions\] Let ${\mathcal{K}}$ be a class of algebras of the same signature.
- $\operatorname{\mathsf H}{\mathcal{K}} \subseteq \operatorname{\mathsf R}{\mathcal{K}}$ and $\operatorname{\mathsf S}{\mathcal{K}} \subseteq \operatorname{\mathsf R}{\mathcal{K}}$;
- $\operatorname{\mathsf{RR}}{\mathcal{K}} \subseteq \operatorname{\mathsf R}{\mathcal{K}}$;
- $\operatorname{\mathsf{PR}}{\mathcal{K}} \subseteq \operatorname{\mathsf{RP}}{\mathcal{K}}$ and $\operatorname{\mathsf P_{fin}}\operatorname{\mathsf R}{\mathcal{K}} \subseteq \operatorname{\mathsf{RP}_{fin}}{\mathcal{K}}$.
Analogous statements hold for the operator $\operatorname{\mathsf R_{ret}}$ instead of $\operatorname{\mathsf R}$.
If ${\mathbf{B}}\in \operatorname{\mathsf H}{\mathcal{K}}$, then there exists an algebra ${\mathbf{A}}\in {\mathcal{K}}$ and a surjective homomorphism $h\colon A{\rightarrow}B$. Picking any function $h'\colon B{\rightarrow}A$ such that $h\circ h'$ is the identity function on $B$ and setting $h_1:=h'$ and $h_2:=h$ we then see that ${\mathbf{B}}$ is a [reflection]{}, and indeed even a retraction, of ${\mathbf{A}}$.
Now let ${\mathbf{B}}\in \operatorname{\mathsf S}{\mathcal{K}}$, and let ${\mathbf{A}}\in {\mathcal{K}}$ such that $B\subseteq A$ is an invariant under the operations of ${\mathbf{A}}$ and such that ${\mathbf{B}}$ arises by restricting the operations of ${\mathbf{A}}$ to $B$. Setting $h_1$ to be the identity function on $B$, and $h_2$ to be any extension of $h_1$ to $A$, then shows that ${\mathbf{B}}$ is a retraction of ${\mathbf{A}}$.
We now show (ii). Suppose that ${\mathbf{C}}$ is a [reflection]{} of ${\mathbf{B}}$, witnessed by functions $h_1\colon C{\rightarrow}B$ and $h_2\colon B{\rightarrow}C$, and that ${\mathbf{B}}$ is a [reflection]{} of ${\mathbf{A}}$, witnessed by functions $h_1'\colon B{\rightarrow}A$ and $h_2'\colon A{\rightarrow}B$. Then the functions $h_1'\circ h_1\colon C{\rightarrow}A$ and $h_2\circ h_2'\colon A{\rightarrow}C$ witness that ${\mathbf{C}}$ is a [reflection]{} of ${\mathbf{A}}$. The same proof works for retractions instead of [reflections]{}.
We turn to the proof of (iii). Let $I$ be an arbitrary set, and suppose that ${\mathbf{B}}_i$ is a [reflection]{} of ${\mathbf{A}}_i$ for every $i\in I$, witnessed by functions $g_i\colon B_i{\rightarrow}A_i$ and $h_i\colon A_i{\rightarrow}B_i$. We show that the product $\prod_{i\in I}{\mathbf{B}}_i$ is a [reflection]{} of the product $\prod_{i\in I}{\mathbf{A}}_i$. But this is easy: $g_i$ and $h_i$ induce functions $g\colon \prod_{i\in I}B_i{\rightarrow}\prod_{i\in I} A_i$ and $h\colon \prod_{i\in I}A_i{\rightarrow}\prod_{i\in I} B_i$ by letting them act on the respective components. It is easily verified that $\prod_{i\in I}{\mathbf{B}}_i$ is a [reflection]{} of $\prod_{i\in I}{\mathbf{A}}_i$ via those functions. This proves also the finite product version of the statement.
Let ${\mathcal{K}}$ be a class of algebras of the same signature. Then
- $\operatorname{\mathsf{RP}}{\mathcal{K}}$ is equal to the closure of ${\mathcal{K}}$ under $\operatorname{\mathsf R}$, $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, and $\operatorname{\mathsf P}$;
- $\operatorname{\mathsf{RP}_{fin}}{\mathcal{K}}$ is equal to the closure of ${\mathcal{K}}$ under $\operatorname{\mathsf R}$, $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, and $\operatorname{\mathsf P_{fin}}$;
- $\operatorname{\mathsf R_{ret}}\operatorname{\mathsf P}{\mathcal{K}}$ is equal to the closure of ${\mathcal{K}}$ under $\operatorname{\mathsf R_{ret}}$, $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, and $\operatorname{\mathsf P}$;
- $\operatorname{\mathsf R_{ret}}\operatorname{\mathsf P_{fin}}{\mathcal{K}}$ is equal to the closure of ${\mathcal{K}}$ under $\operatorname{\mathsf R_{ret}}$, $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, and $\operatorname{\mathsf P_{fin}}$.
This is a direct consequence of Lemma \[lem:algebraicinclusions\].
The following proposition establishes relationships between the relational and algebraic constructions discussed to far. Items (i) and (iii) belong to the fundamentals of the algebraic approach to the CSP, and have been observed in [@JBK] and [@BodirskySurvey]; see also [@Topo-Birk]. The other two items are similar statements for our notion of pp-power and the operator $\operatorname{\mathsf R}$.
\[prop:algebraic\] Let ${\mathbb{A}}$, ${\mathbb{B}}$ be at most countable $\omega$-categorical structures. Then
- ${\mathbb{B}}$ is pp-definable from ${\mathbb{A}}$ iff ${\mathscr{B}} \in \operatorname{\mathsf E}{\mathscr{A}}$;
- ${\mathbb{B}}$ is a pp-power of ${\mathbb{A}}$ iff ${\mathscr{B}} \in \operatorname{\mathsf{EP}_{fin}}{\mathscr{A}}$;
- ${\mathbb{B}}$ is pp-interpretable in ${\mathbb{A}}$ iff ${\mathscr{B}} \in \operatorname{\mathsf{EHSP}_{fin}}{\mathscr{A}}$;
- ${\mathbb{B}}$ is homomorphically equivalent to a structure which is pp-definable from ${\mathbb{A}}$ iff ${\mathscr{B}} \in \operatorname{\mathsf{ER}}{\mathscr{A}}$.
For (i) and (iii) we refer to the literature, namely [@JBK] and [@BodirskySurvey]. Item (ii) is a straightforward consequence of (i).
To prove (iv), let ${\mathbb{A}}'$ be pp-definable in ${\mathbb{A}}$, and let $h_1\colon B{\rightarrow}A$ and $h_2\colon A{\rightarrow}B$ be homomorphisms from ${\mathbb{B}}$ into ${\mathbb{A}}'$ and vice-versa. We want to show ${\mathscr{B}} \in \operatorname{\mathsf{ER}}{\mathscr{A}}$. Let ${{\mathscr{C}}}'$ be the set of all functions on $B$ which are obtained by applying a [reflection]{} to ${{\mathscr{A}}}'$ via the mappings $h_1, h_2$. Because the latter mappings are homomorphisms, it follows that all functions in ${{\mathscr{C}}}'$ preserve all relations of ${\mathbb{B}}$, and so ${{\mathscr{C}}}'$ is contained in ${{\mathscr{B}}}$. By (i) we have ${{\mathscr{A}}}'\supseteq {{\mathscr{A}}}$, and so ${{\mathscr{C}}}'$ contains the set ${{\mathscr{C}}}$ of all functions on $B$ which are obtained by applying a [reflection]{} to ${{\mathscr{A}}}$ via the mappings $h_1, h_2$. Hence, ${\mathscr{B}} \in \operatorname{\mathsf{ER}}{\mathscr{A}}$.
For the other direction, let $h_1\colon B{\rightarrow}A$ and $h_2\colon A{\rightarrow}B$ be so that the [reflection]{} ${{\mathscr{C}}}$ of ${{\mathscr{A}}}$ by those functions is contained in ${{\mathscr{B}}}$. For every relation $R$ of ${\mathbb{B}}$ set $$R':=\{f(h_1(r_1),\ldots,h_1(r_n)) : f\in{{\mathscr{A}}},\; r_1,\ldots,r_n\in R\};$$ here, we apply $h_1$ and functions from ${{\mathscr{A}}}$ to tuples componentwise. In other words, $R'$ is the closure of $h_1[R]$ under ${{\mathscr{A}}}$. Let ${\mathbb{A}}'$ be the structure on $A$ whose relations are precisely those of this form. By definition, all relations of ${\mathbb{A}}'$ are invariant under functions in ${{\mathscr{A}}}$, so ${{\mathscr{A}}}'\in \operatorname{\mathsf E}{\mathscr{A}}$ and hence ${\mathbb{A}}'$ is pp-definable in ${\mathbb{A}}$ by (i). Clearly, $h_1$ is a homomorphism from ${\mathbb{B}}$ to ${\mathbb{A}}'$, since ${{\mathscr{A}}}$ contains the projections. If on the other hand $f(h_1(r_1),\ldots,h_1(r_n))$ is any tuple in a relation $R'$ of ${\mathbb{A}}'$, then $h_2(f(h_1(r_1),\ldots,h_1(r_n)))\in R$ because $h_2(f(h_1(x_1),\ldots,h_1(x_n)))\in{{\mathscr{C}}}$ is contained in ${{\mathscr{B}}}$ and because $R$ is invariant under the functions in ${{\mathscr{B}}}$.
The following corollary incorporates all we have obtained so far, characterizing the notion of pp-constructability via algebraic operators.
\[cor:pp\_mut\] Let ${\mathbb{A}}$, ${\mathbb{B}}$ be at most countable $\omega$-categorical structures. Then ${\mathbb{B}}$ can be pp-constructed from ${\mathbb{A}}$ iff ${\mathscr{B}} \in \operatorname{\mathsf{ERP}_{fin}}{\mathscr{A}}$. In this case $\operatorname{CSP}({\mathbb{B}})$ is log-space reducible to $\operatorname{CSP}({\mathbb{A}})$.
By Corollary \[cor:relational\], ${\mathbb{B}}$ can be pp-constructed from ${\mathbb{A}}$ iff ${\mathbb{B}}$ is homomorphically equivalent to a pp-power of ${\mathbb{A}}$. By Proposition \[prop:algebraic\], this is the case iff ${{\mathscr{B}}}$ is contained in $\operatorname{\mathsf{EREP}_{fin}}{\mathscr{A}}$. Clearly, the latter class equals $\operatorname{\mathsf{ERP}_{fin}}{\mathscr{A}}$. The second statement follows by application of Corollary \[cor:complexity\].
Linear Birkhoff {#sec:birk}
===============
We now turn to a syntactic characterization of the operator $\operatorname{\mathsf R}$ together with $\operatorname{\mathsf P}$, similar to the syntactic description of the operators $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, and $\operatorname{\mathsf P}$ in item (iii) of Theorem \[thm:old\_finite\]. Let us recall the notion of a clone homomorphism and introduce two weakenings thereof.
\[defn:clonehomo\] Let ${\mathscr{A}}$ and ${\mathscr{B}}$ be function clones and let $\xi\colon {\mathscr{A}} \to {\mathscr{B}}$ be a mapping that preserves arities. We say that $\xi$ is
- a *clone homomorphism*, or *preserves identities*, if $$\xi(\pi^n_k)=\pi^n_k\quad \text{ and }\quad
\xi(f(g_1, \dots, g_n)) = \xi(f)(\xi(g_1), \dots, \xi(g_n))$$ for any $1\leq k \leq n$, any $m\geq 1$, any $n$-ary operation $f \in {\mathscr{A}}$, and all $m$-ary operations $g_1, \dots, g_m \in {\mathscr{A}}$;
- an *h1 clone homomorphism*, or *preserves identities of height $1$*, if $$\xi(f(\pi^m_{i_1}, \dots, \pi^m_{i_n})) = \xi(f)(\pi^m_{i_1}, \dots, \pi^m_{i_n})$$ for all $n, m\geq 1$, all $i_1, \dots, i_n \in \{1, \dots, m\}$, and any $n$-ary operation $f\in{{\mathscr{A}}}$;
- a *strong h1 clone homomorphism*, or *preserves identities of height at most $1$*, if it is an h1 clone homomorphism and preserves all projections.
In the following, we will give some motivation for our terminology.
\[def:5.2\] Let $\tau$ be a functional signature, and let $t, s$ be $\tau$-terms. An identity $t \approx s$ is said to have *height $1$* (*height at most $1$*, respectively) if both $t$ and $s$ are terms of height $1$ (height at most $1$, respectively).
So a height $1$ identity is of the form $$f(x_{1},\ldots,x_{n}) \approx g(y_{1},\ldots,y_{m})$$ where $f,g$ are functional symbols in $\tau$ and $x_1,\dots,x_n, y_1,\dots, y_m$ are not necessarily distinct variables. Identities of height at most $1$ include moreover identities of the form $$f(x_{1},\ldots,x_{n}) \approx y$$ and $$x \approx y\; .$$ We note that identities of height at most $1$ are also called *linear* in the literature.
Observe that the variants of a clone homomorphism introduced in Definition \[defn:clonehomo\] have the suggested meaning. Indeed, if $\xi\colon {\mathscr{A}} \to {\mathscr{B}}$ is an arity preserving mapping, ${\mathbf{A}}$ is the algebra $(A; (f)_{f\in{{\mathscr{A}}}})$ of signature $\tau = {\mathscr{A}}$, and ${\mathbf{B}}$ is the $\tau$-algebra $(B; (\xi(f))_{f\in{{\mathscr{A}}}})$, then $\xi$ is a clone homomorphism if and only if $s \approx t$ is true in ${\mathbf{B}}$ whenever $s \approx t$ is true in ${\mathbf{A}}$, where $s \approx t$ is an identity in the signature $\tau$; similarly, it is an h1 clone homomorphism if and only if this condition holds for identities of height 1, and a strong h1 clone homomorphism if and only if it holds for identities of height at most 1.
\[prop:abstract\] Let ${\mathscr{A}}$, ${\mathscr{B}}$ be function clones. Then
- ${\mathscr{B}} \in \operatorname{\mathsf{EHSP}}{\mathscr{A}}$ iff there exists a clone homomorphism ${\mathscr{A}} \to {\mathscr{B}}$;
- ${\mathscr{B}} \in \operatorname{\mathsf{ER}_{ret}\mathsf P}{\mathscr{A}}$ iff there exists a strong h1 clone homomorphism ${\mathscr{A}} \to {\mathscr{B}}$;
- ${\mathscr{B}} \in \operatorname{\mathsf{ERP}}{\mathscr{A}}$ iff there exists an h1 clone homomorphism ${\mathscr{A}} \to {\mathscr{B}}$.
In all cases, if $A$ and $B$ are finite, then the operator $\operatorname{\mathsf P}$ can be equivalently replaced by $\operatorname{\mathsf P_{fin}}$.
The implications from left to right follow from the fact that the operators $\operatorname{\mathsf H}$, $\operatorname{\mathsf S}$, and $\operatorname{\mathsf P}$ preserve all identities, that the operator $\operatorname{\mathsf R_{ret}}$ preserves identities of height at most $1$, and that $\operatorname{\mathsf R}$ preserves identities of height $1$.
We show the converses, starting with (i) although this follows from Birkhoff’s theorem. Let $\xi\colon {{\mathscr{A}}}{\rightarrow}{{\mathscr{B}}}$ be a clone homomorphism. For every $b\in B$, let $\pi^B_b\in A^{A^B}$ be the function which projects any tuple in $A^B$ onto the $b$-th coordinate. Let ${{\mathscr{A}}}$ act on the tuples $\{\pi^B_b : b\in B\}$ componentwise; closing the latter subset of $A^{A^B}$ under this action, we obtain an invariant subset $S$ of $A^{A^B}$. The action of ${{\mathscr{A}}}$ thereon is a function clone in $\operatorname{\mathsf{SP}}{{\mathscr{A}}}$. In fact, if we see this action as an algebra with signature ${{\mathscr{A}}}$, it is the free algebra in the variety generated by the algebra $(A; (f)_{f\in {{\mathscr{A}}}})$ with generators $\{\pi^B_b : b\in B\}$. The mapping from $\{\pi^B_b : b\in B\}$ to $B$ which sends every $\pi^B_b$ to $b$ therefore extends to a homomorphism $h\colon
S{\rightarrow}B$ from the free algebra $(S;(f)_{f\in {{\mathscr{A}}}})$ onto the algebra $(B;(\xi(f))_{f\in {{\mathscr{A}}}})$, since the latter algebra is an element of the mentioned variety as $\xi$ preserves identities. Therefore, the function clone $\{\xi(f) : {f\in {{\mathscr{A}}}}\}$ is an element of $\operatorname{\mathsf{HSP}}({{\mathscr{A}}})$, and whence ${\mathscr{B}} \in \operatorname{\mathsf{EHSP}}{\mathscr{A}}$.
With the intention of modifying this proof for (ii) and (iii), let us remark the following. If one wishes to avoid reference to free algebras in the previous proof, then it is enough to define the set $S$ as above and then observe that for all $n,
m\geq 1$, all $n$-ary $f\in{{\mathscr{A}}}$ and all $m$-ary $g\in{{\mathscr{A}}}$, and all $b_1,\ldots,b_n,c_1,\ldots,c_m\in B$ we have that if $f(\pi^B_{b_1},\ldots,\pi^B_{b_n})=g(\pi^B_{c_1},\ldots,\pi^B_{c_m})$, then $\xi(f)(b_1,\ldots,b_n)=\xi(g)(c_1,\ldots,c_m)$. This follows from the fact that $\xi$ preserves, in particular, identities of height $1$, and allows us to define a mapping $h\colon S{\rightarrow}B$ uniquely by sending every tuple in $S$ of the form $f(\pi^B_{b_1},\ldots,\pi^B_{b_n})$ to $\xi(f)(b_1,\ldots,b_n)$. In the case of a clone homomorphism $\xi\colon{{\mathscr{A}}}{\rightarrow}{{\mathscr{B}}}$, this yields a homomorphism from the action of ${{\mathscr{A}}}$ on $S$ onto the action of $\xi[{{\mathscr{A}}}]$ on $B$; in other words, the action of $\xi[{{\mathscr{A}}}]$ on $B$ is isomorphic to the action of ${{\mathscr{A}}}$ on the kernel classes of $h$.
We prove (ii). By the argument above, we obtain a surjective mapping $h\colon
S{\rightarrow}B$ which sends every $\pi^B_b$ to $b$ since $\xi$ preserves projections. Let $h'\colon B{\rightarrow}S$ be the mapping which sends every $b\in B$ to $\pi^B_b$. Then the function clone $\{\xi(f): {f\in {{\mathscr{A}}}}\}$ is a retraction of the action of ${{\mathscr{A}}}$ on $S$ via the functions $h'$ and $h$, and so it is an element of $\operatorname{\mathsf R_{ret}}\operatorname{\mathsf{SP}}{{\mathscr{A}}}$, which equals $\operatorname{\mathsf R_{ret}}\operatorname{\mathsf P}{{\mathscr{A}}}$ by Lemma \[lem:algebraicinclusions\]. Whence, ${\mathscr{B}} \in \operatorname{\mathsf{ER}_{ret}\mathsf P}{\mathscr{A}}$.
The proof of (iii) is identical, with the difference that $h$ does not necessarily send every $\pi^B_b$ to $b$; this is because $\xi$ does not necessarily preserve projections. Defining $h'$ the same way as above, we then get that $\{\xi(f) : {f\in {{\mathscr{A}}}}\}$ is a [reflection]{} of the action of ${{\mathscr{A}}}$ on $S$ via the functions $h'$ and $h$, rather than a retraction.
The additional statement about finite domains follows from the proof above: the power we take is $A^B$.
We remark that in the proof above, the mapping $h'$ was always injective. Hence, one could alter the definition of a [reflection]{} by requiring that the mapping $h_1\colon B{\rightarrow}A$ is injective, and obtain the very same syntactic characterization. In other words, if we introduced an operator $\operatorname{\mathsf R}'$ for such [reflections]{}, from which we shall refrain, then we would have $\operatorname{\mathsf{RP}}{{\mathscr{A}}}=\operatorname{\mathsf R}'\operatorname{\mathsf P}{{\mathscr{A}}}$ for all function clones ${{\mathscr{A}}}$.
Let us conclude this section with an analogue of Birkhoff’s HSP theorem for closure under [reflections]{} and products.
\[cor:linear-birkhoff\] Let ${\mathcal{K}}$ be a nonempty class of algebras of the same signature $\tau$.
- ${\mathcal{K}}$ is closed under $\operatorname{\mathsf R}$ and $\operatorname{\mathsf P}$ if and only if ${\mathcal{K}}$ is the class of models of some set of $\tau$-identities of height $1$.
- ${\mathcal{K}}$ is closed under $\operatorname{\mathsf R_{ret}}$ and $\operatorname{\mathsf P}$ if and only if ${\mathcal{K}}$ is the class of models of some set of $\tau$-identities of height at most $1$.
The implications from right to left are trivial since $\operatorname{\mathsf P}$ preserves all identities and since $\operatorname{\mathsf R}$ and $\operatorname{\mathsf R_{ret}}$ preserve identities of height $1$ and identities of height at most $1$, respectively.
Suppose that ${\mathcal{K}}$ is closed under $\operatorname{\mathsf R}$ and $\operatorname{\mathsf P}$, and let ${\mathbf{B}}$ be any $\tau$-algebra satisfying the set $\Sigma$ of those $\tau$-identities of height $1$ which hold in all algebras of ${\mathcal{K}}$. Pick for every $\tau$-identity of height $1$ which is not contained in $\Sigma$ an algebra in ${\mathcal{K}}$ which does not satisfy this identity, and let ${\mathbf{A}}\in{\mathcal{K}}$ be the product of all such algebras. Then the mapping which sends every $\tau$-term of ${\mathbf{A}}$ to the corresponding term of ${\mathbf{B}}$ preserves identities of height $1$, and so ${\mathbf{B}}\in \operatorname{\mathsf{RP}}{\mathbf{A}}$ by Proposition \[prop:abstract\]. Hence, ${\mathbf{B}}\in{\mathcal{K}}$.
The proof of (ii) is identical.
Topological Linear Birkhoff {#sec:cont}
===========================
Finite goal structures
----------------------
In Proposition \[prop:abstract\] we obtained a characterization when ${\mathscr{B}} \in \operatorname{\mathsf{ERP}_{fin}}{\mathscr{A}}$ for function clones on finite domains. For function clones on arbitrary sets, even for polymorphism clones of countable $\omega$-categorical structures, we are generally forced to take infinite powers, obstructing the combination with Corollary \[cor:pp\_mut\]. By considering the topological structure of function clones in addition to their algebraic structure, a characterization of when ${\mathscr{B}} \in \operatorname{\mathsf{HSP}_{fin}}{\mathscr{A}}$ has been obtained for polymorphism clones of $\omega$-categorical structures [@Topo-Birk] – cf. Theorem \[thm:old\_infinite\]. We will now obtain a similar characterization for our operators in the case where ${\mathscr{B}}$ has a finite domain. This is, in particular, interesting in the light of Conjecture \[conj:old\] which states that for a certain class of $\omega$-categorical structures, the only reason for hardness of the CSP is reduction of the CSP of a structure on a 2-element domain whose only polymorphisms are projections.
In the following proposition, item (i) is a variant of a statement in [@Topo-Birk] which uses stronger general assumptions, namely, that ${{\mathscr{A}}}$ is dense in the polymorphism clone of a countable $\omega$-categorical structure; on the other hand, it uses a formally weaker statement in one of the sides of the equivalence, namely, continuity rather than uniform continuity. Continuity and uniform continuity, however, turn out to be equivalent for that setting; cf. also Section \[sect:cont-unifcont\]. Our variant presented here, first observed in [@GPin15], follows directly from the right interpretation of the proof in [@Topo-Birk].
We say that the domain of a function clone ${{\mathscr{B}}}$ is *finitely generated by ${{\mathscr{B}}}$* iff the algebra obtained by viewing the elements of ${{\mathscr{B}}}$ as the operations of the algebra is finitely generated; that is, there is a finite subset $B'$ of $B$ such that every element of $B$ is of the form $f(b_1,\ldots,b_n)$, where $b_1,\ldots,b_n\in B'$ and $f\in {{\mathscr{B}}}$. We remark that domains of polymorphism clones of countable $\omega$-categorical structures are finitely generated by those clones.
\[prop:topological\] Let ${\mathscr{A}}, {\mathscr{B}}$ be function clones.
- If the domain of ${{\mathscr{B}}}$ is finitely generated by ${{\mathscr{B}}}$, then ${\mathscr{B}}\in \operatorname{\mathsf{EHSP}_{fin}}({\mathscr{A}})$ iff there exists a uniformly continuous clone homomorphism $\xi\colon {\mathscr{A}}{\rightarrow}{\mathscr{B}}$.
- If the domain of ${{\mathscr{B}}}$ is finite, then ${\mathscr{B}}\in \operatorname{\mathsf{ERP}_{fin}}({\mathscr{A}})$ iff there exists a uniformly continuous h1 clone homomorphism $\xi\colon {\mathscr{A}}{\rightarrow}{\mathscr{B}}$.
As in Proposition \[prop:abstract\], the implications from left to right are trivial.
For the other direction, we follow the proof of that proposition, but use uniform continuity to replace arbitrary powers by finite ones. To do this for item (i), let $b_1,\ldots,b_n$ be generators of $B$. By uniform continuity, there exist $m\geq 1$ and $a^1,\ldots,a^m\in A^n$ such that for all $n$-ary functions $f,g\in{{\mathscr{A}}}$ we have that if $f,g$ agree on all tuples $a^1,\ldots,a^m$, then $\xi(f)(b_1,\ldots,b_n)=\xi(g)(b_1,\ldots,b_n)$. For all $1\leq i\leq n$, let $a_i\in A^m$ consist of the $i$-th components of the tuples $a^j$. Then, $f(a_1,\ldots,a_n)=g(a_1,\ldots,a_n)$, calculated componentwise, implies that $\xi(f)(b_1,\ldots,b_n)=\xi(g)(b_1,\ldots,b_n)$. Let $S\subseteq A^m$ be the set obtained by applying $n$-ary functions in ${{\mathscr{A}}}$ to $a_1,\ldots,a_n$ componentwise. The remainder of the proof is identical with the second proof of item (i) of Proposition \[prop:abstract\].
For (ii), we proceed the same way and then as in the corresponding item of Proposition \[prop:abstract\], assuming that $\{b_1,\ldots,b_n\}$ actually equals $B$. The reason why we need the stronger assumption of finiteness is that we cannot use the generating process since $\xi$ does not necessarily preserve identities (in particular, we would be unable to define the mapping $h'$ as in the proof of Proposition \[prop:abstract\]).
We remark that the condition in item (i) of Proposition \[prop:topological\] that the domain of ${{\mathscr{B}}}$ be finitely generated is reasonable (and, in general, necessary): if ${{\mathscr{A}}}$ is a transformation monoid, then we can let it act on arbitrarily many disjoint copies of its domain simultaneously; this new action ${{\mathscr{B}}}$ will not be in $\operatorname{\mathsf{HSP}_{fin}}({{\mathscr{A}}})$ for reasons of cardinality of the domain if we take enough copies, but ${{\mathscr{A}}}$ and ${{\mathscr{B}}}$ will be isomorphic as monoids via a homeomorphism. We can do the same with a function clone which is a transformation monoid in disguise in the sense that all of its functions depend on at most one variable.
Similarly, the finiteness condition in item (ii) seems to be necessary in general: note, for example, that any mapping between transformation monoids preserves identities of height 1 since that notion of preservation only becomes non-trivial for functions of several variables. We cannot, however, expect endomorphism monoids of completely unrelated structures to be related via the operators $\operatorname{\mathsf R}$ and $\operatorname{\mathsf P_{fin}}$.
Continuity versus uniform continuity {#sect:cont-unifcont}
------------------------------------
We will now observe conditions ensuring that continuity of a mapping between function clones implies uniform continuity, in particular in order to shed light on the question why uniform continuity appears in Proposition \[prop:topological\], whereas it does not in earlier results such as Theorem \[thm:old\_infinite\].
Let ${{\mathscr{A}}}$ be a function clone. An *invertible* of ${{\mathscr{A}}}$ is a unary bijection in ${{\mathscr{A}}}$ whose inverse is also an element of ${{\mathscr{A}}}$.
\[defn:outerinvertibles\] A mapping $\xi\colon {{\mathscr{A}}}{\rightarrow}{{\mathscr{B}}}$ between function clones ${{\mathscr{A}}}, {{\mathscr{B}}}$ *preserves composition with invertibles from the outside* iff $\xi(\alpha f)=\xi(\alpha)\xi(f)$ for all invertibles $\alpha\in{{\mathscr{A}}}$ and all $f\in{{\mathscr{A}}}$.
We briefly mentioned the following fact in the discussion preceding Proposition \[prop:topological\]; it follows from the material in [@Topo-Birk], but we give a compact proof here.
\[prop:uniform\] Let $\mathscr A,\mathscr B$ be function clones, where $\mathscr A$ is the polymorphism clone of a countable $\omega$-categorical structure. Then any continuous mapping $\xi\colon \mathscr A{\rightarrow}\mathscr B$ preserving composition with invertibles from the outside is uniformly continuous.
Let $k, j\geq 1$ and $w_1,\ldots,w_k\in B^{j}$ be given; we have to show that there exist $m\geq 1$ and $q_1,\ldots,q_k\in A^{m}$ such that $f(q_1,\ldots,q_k)=g(q_1,\ldots,q_k)$ implies $\xi(f)(w_1,\ldots,w_k)=\xi(g)(w_1,\ldots,w_k)$ for all $k$-ary $f,g\in\mathscr A$.
For $k,m\geq1$, $q_1,\ldots,q_k\in A^{m}$, and $q\in A^m$ we write $O_{q_1,\ldots,q_k}^q$ for the basic open set of $\mathscr A$ which consists of those $k$-ary functions $f\in \mathscr A$ for which $f(q_1,\ldots,q_k)=q$. We use the same notation for the basic open sets of $\mathscr B$. We further write $U_{q_1,\ldots,q_k}^q$ for the open set of those $k$-ary functions $f$ in $\mathscr A$ for which $f(q_1,\ldots,q_k)$ lies in the orbit of $q$ with respect to the action of the invertible elements of $\mathscr A$ on $k$-tuples.
By continuity, for every $k$-ary $f\in\mathscr A$ there exist $m^f\geq 1$ and $q_1^f,\ldots,q_k^f, q^f \in A^{m^f}$ such that $g\in
O_{q_1^f,\ldots,q_k^f}^{q^f}$ implies $\xi(f)(w_1,\ldots,w_k)=\xi(g)(w_1,\ldots,w_k)$, for all $k$-ary $g\in \mathscr
A$. By a standard compactness argument, the space $\mathscr A \cap {A^{A^k}}$ is covered by finitely many sets of the form $U_{q_1^f,\ldots,q_k^f}^{q^f}$; write $f_1,\ldots,f_n$ for the functions involved in this covering. Set $q_1,\ldots,q_k\in A^{m}$ to be the vectors obtained by gluing everything together, i.e., $q_i$ is obtained by joining the vectors $q_i^{f_1}$, …, $q_i^{f_n}$, and $m$ is the length of the vectors obtained this way. Suppose $f(q_1,\ldots,q_k)=g(q_1,\ldots,q_k)$. There exists $h\in\{f_1,\ldots,f_n\}$ such that $f\in U_{q_1^h,\ldots,q_k^h}^{q^h}$. Thus there exists a unary invertible $\alpha\in\mathscr A$ such that $\alpha(f)\in
O_{q_1^h,\ldots,q_k^h}^{q^h}$. Because $f(q_1,\ldots,q_k)=g(q_1,\ldots,q_k)$, we also have $\alpha(g)\in O_{q_1^h,\ldots,q_k^h}^{q^h}$. By definition, $\xi(\alpha(f))(w_1,\ldots,w_k)=\xi(h)(w_1,\ldots,w_k)=\xi(\alpha(g))(w_1,\ldots,w_k)$. Hence, because $\xi$ preserves composition with invertibles from the outside, $$\begin{aligned}
\xi(f)(w_1,\ldots,w_k)&=\xi(\alpha^{-1}\alpha f)(w_1,\ldots,w_k)\\
&=\xi(\alpha^{-1})\xi(\alpha f)(w_1,\ldots,w_k)\\
&=\xi(\alpha^{-1})\xi(\alpha g)(w_1,\ldots,w_k)\\
&=\xi(g)(w_1,\ldots,w_k)\;. \qedhere\end{aligned}$$
\[defn:outerinvertibles2\] Let ${{\mathscr{A}}}$ be a function clone, and consider it as an algebra with signature ${{\mathscr{A}}}$ and domain $A$. An identity over the signature ${{\mathscr{A}}}$ is *of height 1 modulo outer invertibles* if it is of the form $$\alpha f(x_{1},\ldots,x_{n}) \approx \beta g(y_{1},\ldots,y_{m})$$ where $f,g\in{{\mathscr{A}}}$, $\alpha,\beta\in{{\mathscr{A}}}$ are invertibles, and $x_1,\dots,x_n, y_1,\dots, y_m$ are not necessarily distinct variables.
Clearly, a mapping $\xi\colon{{\mathscr{A}}}{\rightarrow}{{\mathscr{B}}}$ preserves identities of height 1 modulo outer invertibles iff it is an h1 clone homomorphism preserving composition with invertibles from the outside.
\[cor:outer\] Let $\mathscr A,\mathscr B$ be function clones, where $\mathscr A$ is the polymorphism clone of an $\omega$-categorical structure, and ${{\mathscr{B}}}$ has a finite domain. If there exists a continuous mapping $\xi\colon\mathscr A{\rightarrow}\mathscr B$ which preserves height 1 identities modulo outer invertibles, then $\mathscr B\in\operatorname{\mathsf{ERP}_{fin}}(\mathscr A)$.
This follows from Propositions \[prop:uniform\] and \[prop:topological\].
Notice that the operator $\operatorname{\mathsf R}$ preserves height 1 identities, but not necessarily identities which are of height 1 modulo outer invertibles, depriving us in Corollary \[cor:outer\] of an equivalence similar to the one in Proposition \[prop:topological\].
Infinite goal structures
------------------------
In infinite domain constraint satisfaction, structures are generally studied relative to a base structure: one usually starts with a structure ${\mathbb{A}}$ and studies all structures which are first-order definable in ${\mathbb{A}}$, called *reducts* of ${\mathbb{A}}$ [@BodPin-Schaefer-both; @tcsps-journal; @BodDalMarPin; @BodMarMot; @BPT-decidability-of-definability]. When ${\mathbb{A}}$ is countable and $\omega$-categorical, then this amounts to the study of all structures whose polymorphism clone contains the automorphism group of ${\mathbb{A}}$. So in a sense, one considers function clones *up to automorphisms of ${\mathbb{A}}$*. In this section, we see that we can make some of our results work for infinite goal structures when we assume that mappings between function clones are compatible with composition with automorphisms. Definitions \[defn:outerinvertibles\] and \[defn:outerinvertibles2\] pointed in that direction, but as it turns out, we need to compose functions with invertibles from the inside rather than the outside.
\[defn:innerinvertibles\] Let ${{\mathscr{A}}}$ be a function clone, and consider it as an algebra with signature ${{\mathscr{A}}}$ and domain $A$. An identity over the signature ${{\mathscr{A}}}$ is *of height 1 modulo inner invertibles* if it is of the form $$f(\alpha_1(x_{1}),\ldots,\alpha_n(x_{n})) \approx g(\beta_1(y_{1}),\ldots,\beta_m(y_{m}))$$ where $f,g\in{{\mathscr{A}}}$, $\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,b_m\in{{\mathscr{A}}}$ are invertibles, and $x_1,\dots,x_n, y_1,\dots, y_m$ are not necessarily distinct variables.
A mapping $\xi\colon {{\mathscr{A}}}\to {{\mathscr{B}}}$ between function clones *preserves height 1 identities modulo inner invertibles* iff $\xi(f(\alpha_1,\dots,\alpha_n)) = \xi(f)(\xi(\alpha_1),\dots,\xi(\alpha_n))$ for all invertibles $\alpha_1,\dots,\alpha_n \in {{\mathscr{A}}}$ and all $n$-ary $f\in {{\mathscr{A}}}$.
\[prop:inner\] Let $\mathscr A, \mathscr B$ be function clones, and let $\xi\colon \mathscr A{\rightarrow}\mathscr B$
- be uniformly continuous;
- preserve height 1 identities modulo inner invertibles;
- be so that the invertibles of the image $\xi[\mathscr A]$ act with finitely many orbits on $B$.
Then $\mathscr B\in \operatorname{\mathsf{ERP}_{fin}}(\mathscr A)$.
Let $d_1,\ldots,d_k\in B$ be representatives of the orbits of the action of the invertibles of $\xi[\mathscr A]$ on $B$. By uniform continuity, there exist $m\geq 1$ and $c_1,\ldots,c_k\in A^m$ such that for all $k$-ary $f,g\in\mathscr A$ we have that $f(c_1,\ldots,c_k)=g(c_1,\ldots,c_k)$ implies $\xi(f)(d_1,\ldots,d_k)=\xi(g)(d_1,\ldots,d_k)$. Define a mapping $h_1\colon A^m{\rightarrow}B$ by setting $h_1(f(c_1,\ldots,c_k)):=\xi(f)(d_1,\ldots,d_k)$ for all elements of the form $f(c_1,\ldots,c_k)$ for some $f\in \mathscr A$, and by extending it arbitrarily to $A^m$. By the choice of $c_1,\ldots,c_k\in A^m$, this mapping is well-defined. Next define $h_2\colon B{\rightarrow}A^m$ as follows: for each $d\in B$, pick an invertible $\alpha\in \mathscr A$ and $1\leq i\leq k$ such that $d=\xi(\alpha)(d_i)$, and set $h_2(d):=\alpha(c_i)$. We claim that for all $k$-ary $f\in\mathscr A$ we have $\xi(f)=h_1(f(h_2(x_1),\ldots,h_2(x_k)))$. It then follows that $\xi[\mathscr A]$ is a [reflection]{} of the action of $\mathscr A$ on $A^m$, proving the statement.
To see the claim, let $f$ be given, and let $s_1,\ldots,s_k\in B$. Pick invertibles $\alpha_1,\ldots,\alpha_k\in\mathscr A$ and $i_1,\ldots,i_k\in\{1,\ldots,k\}$ such that $\xi(\alpha_j)(d_{i_j})=s_j$ for all $1\leq j\leq k$. Then $$\begin{aligned}
h_1(f(h_2(s_1),\ldots,h_2(s_k)))&=h_1(f(h_2(\xi(\alpha_1)(d_{i_1})),\ldots,h_2(\xi(\alpha_k)(d_{i_k}))))\\
&=h_1(f(\alpha_1(c_{i_1}),\ldots,\alpha_k(c_{i_k})))\\
&=h_1(f(\alpha_1,\ldots,\alpha_k)(c_{i_1},\ldots,c_{i_k}))\\
&=\xi(f(\alpha_1,\ldots,\alpha_k))(d_{i_1},\ldots,d_{i_k}))\\
&=\xi(f)(\xi(\alpha_1)(d_{i_1}),\ldots,\xi(\alpha_k)(d_{i_k})))\\
&=\xi(f)(s_1,\ldots,s_k).\end{aligned}$$ Here we use the definitions of $h_1$ and $h_2$ and that $\xi$ preserves height 1 identities modulo inner invertibles.
Observe that strong h1 clone homomorphisms between function clones which preserve identities of height 1 modulo inner invertibles send invertibles to invertibles. In particular, the image of the group of invertibles of a function clone under such a mapping is a group.
Let ${\mathbb{A}}, {\mathbb{B}}$ at most countable $\omega$-categorical relational structures. Suppose that $\xi\colon \mathscr A{\rightarrow}\mathscr B$
- is uniformly continuous;
- preserves identities of height 1 modulo inner invertibles;
- is so that the invertibles of $\xi[\mathscr A]$ act with finitely many orbits on $B$.
Then ${\mathbb{B}}$ is pp-constructible from ${\mathbb{A}}$.
Coloring of clones by relational structures {#sec:snek}
===========================================
In order to define the central concept, we first introduce some notation, a piece of which has already appeared in the proof of Proposition \[prop:abstract\].
Let ${{\mathscr{A}}}$ be a clone and $B$ a set. For an element $b\in B$, let $\pi^B_b\in A^{A^B}$ be the function which projects every tuple in $A^B$ onto the $b$-th coordinate, let $F_{{{\mathscr{A}}}}(B) \subseteq A^{A^B}$ be the closure of $\{\pi^B_b : b\in B\}$ under the componentwise action of ${{\mathscr{A}}}$, and let ${\mathscr{F}}_{{{\mathscr{A}}}}(B)$ be the corresponding clone acting on $F_{{{\mathscr{A}}}}(B)$. (We mentioned in the proof of Proposition \[prop:abstract\] that $F_{{{\mathscr{A}}}}(B)$, denoted $S$ there, is the universe of the free algebra generated by the algebra $(A; (f)_{f\in {{\mathscr{A}}}})$ with generators $\{\pi^B_b : b\in B\}$.) Note that for a finite $B = \{0,1, \dots, n-1\}$, $F_{{{\mathscr{A}}}}(B)$ is equal to the set of $n$-ary operations in ${{\mathscr{A}}}$.
For a relation $R \subseteq B^k$ we define a relation $R^{{\mathscr{A}}}\subseteq F_{{{\mathscr{A}}}}(B)^k$ as the closure of the set $$\{ (\pi_{b_1}^B, \dots, \pi_{b_k}^B) : (b_1,\dots,b_k) \in R\}$$ under the componentwise action of ${\mathscr{F}}_{{{\mathscr{A}}}}(B)$. In this way, each relational structure ${{\mathbb{B}}}$ with universe $B$ lifts to a relational structure with universe $F_{{{\mathscr{A}}}}(B)$ of the same signature. Colorings are defined as homomorphisms from the lifted structure to ${\mathbb{B}}$:
Let ${{\mathscr{A}}}$ be a function clone and let ${{\mathbb{B}}}$ be a relational structure. We say that a mapping $c\colon F_{{{\mathscr{A}}}}(B) \to B$ is a *coloring of ${{\mathscr{A}}}$ by ${{\mathbb{B}}}$* if for all relations $R$ of ${{\mathbb{B}}}$ and all tuples $ (f_1,\dots,f_k)
\in R^{{\mathscr{A}}}$ we have $ (c(f_1),\dots,c(f_k)) \in R $. A *strong coloring* is a coloring that in addition satisfies $c(\pi_b^B) = b$ for all $b
\in B$. A clone is (strongly) *${{\mathbb{B}}}$-colorable* if there exists a (strong) coloring $c$ of the clone by ${{\mathbb{B}}}$.
Sequeira’s notion of compatibility with projections [@sequeira01] is equivalent to strong colorings by relational structures consisting of equivalence relations.
The proof of the following proposition illustrates how a specific Maltsev condition for $n$-permutability is translated to strong non-colorability.
\[prop:hm\] A variety ${\mathcal{V}}$ is congruence $n$-permutable for some $n$ if and only if ${\mathrm{clo}({\mathcal{V}})}$ is not strongly $(\{0,1\}; \leq)$-colorable.
Hagemann-Mitschke operations are ternary operations $p_1$, …, $p_{n-1}$ such that $
p_1(x,y,y) \approx x $, $
p_{n-1}(x,x,y) \approx y$, and $
p_i(x,x,y) \approx p_{i+1}(x,y,y)
$ for every $i=1,\dots,n-2$. By [@hagemann.mitschke73], a variety ${\mathcal{V}}$ is $n$-permutable if and only if ${{\mathscr{A}}}= {\mathrm{clo}({\mathcal{V}})}$ contains Hagemann-Mitschke operations.
The relation $\leq^{{{\mathscr{A}}}}$ is the closure of $\{(\pi^{\{0,1\}}_0,\pi^{\{0,1\}}_0), (\pi^{\{0,1\}}_0,\pi^{\{0,1\}}_1), (\pi^{\{0,1\}}_1,\pi^{\{0,1\}}_1)\}$ under the componentwise action of ${{\mathscr{A}}}$, therefore it is equal to $$\{(t(\pi^{\{0,1\}}_0,\pi^{\{0,1\}}_0,\pi^{\{0,1\}}_1), t(\pi^{\{0,1\}}_0,\pi^{\{0,1\}}_1,\pi^{\{0,1\}}_1)): t \in {{\mathscr{A}}}, \mbox{ $t$ is ternary }\}.$$ In other words, for two binary operations $f,g$ in ${{\mathscr{A}}}$ we have $f \leq^{{\mathscr{A}}}g$ if and only if there exists a ternary operation $t\in {{\mathscr{A}}}$ satisfying $t(x,x,y) \approx f(x,y)$ and $t(x,y,y) \approx
g(x,y)$. It follows that if a clone contains Hagemann-Mitschke operations then it is not strongly $(\{0,1\};\leq)$-colorable since such operations force $c(\pi_1^{\{0,1\}}) \le c(\pi_0^{\{0,1\}})$, a contradiction with $c(\pi_i^{\{0,1\}}) = i$. For the other implication, we define a strong coloring $c$ by $c(h) = 0$ iff there exists a Hagemann-Mitschke chain connecting $\pi^{\{0,1\}}_0$ and $h$, i.e., there is $n$, and operations $p_1,\dots,p_n \in {{\mathscr{A}}}$ such that $
p_1(x,y,y) \approx x $, $
p_n(x,x,y) \approx h(x,y) $, and $
p_i(x,x,y) \approx p_{i+1}(x,y,y)
$ for every $i=1,\dots,n-1$. Since ${{\mathscr{A}}}$ does not contain Hagemann-Mitschke operations, we get that $c(\pi_1^{\{0,1\}}) = 1$. The rest is an easy exercise.
A similar characterization of congruence modularity follows from [@sequeira01]:
\[prop:day\] A variety ${\mathcal{V}}$ is congruence modular if and only if ${\mathrm{clo}({\mathcal{V}})}$ is not strongly ${\mathbb{D}}$-colorable, where ${\mathbb{D}} = (D; \alpha, \beta, \gamma)$, $D = \{1,2,3,4\}$, and $\alpha$, $\beta$, $\gamma$ are equivalence relations on $D$ defined by partitions $12|34$, $13|24$, $12|3|4$.
The connection between colorings of clones by relational structures and h1 clone homomorphisms is presented in the following proposition.
\[prop:coloring-and-h1\] Let ${{\mathscr{A}}}$ be a function clone and let ${{\mathbb{B}}}$ be a relational structure.
1. ${{\mathscr{A}}}$ is ${{\mathbb{B}}}$-colorable if and only if there is an h1 clone homomorphism $\xi\colon {{\mathscr{A}}}\to {{\mathscr{B}}}$, and
2. ${{\mathscr{A}}}$ is strongly ${{\mathbb{B}}}$-colorable if and only if there is a strong h1 clone homomorphism $\xi\colon {{\mathscr{A}}}\to {{\mathscr{B}}}$.
The proof relies on Proposition \[prop:abstract\] and its proof.
To prove (i), suppose that $c\colon F_{{{\mathscr{A}}}}(B) \to B$ is a coloring of ${{\mathscr{A}}}$ by ${{\mathbb{B}}}$ and consider the [reflection]{} ${\mathscr{C}} = \{f': f\in{{\mathscr{A}}}\}$ of ${\mathscr{F}}_{{{\mathscr{A}}}}(B)$ given by $c$ and the mapping $b \mapsto \pi_b^B$ (thus the clone ${\mathscr{C}}$ acts on $B$). We claim that each $f'$ is a polymorphism of ${\mathbb{B}}$. To verify this, consider a relation $R$ of ${{\mathbb{B}}}$ and tuples $(b_{i1},\dots,b_{ik})\in R$, $i=1,\dots,n$. We have $
\big(
f(\pi_{b_{11}}^B,\dots,\pi_{b_{n1}}^B), \dots,
f(\pi_{b_{1k}}^B,\dots,\pi_{b_{nk}}^B)
\big) \in R^{{\mathscr{A}}}$, and then $$\big(
f'(b_{11},\dots,b_{n1}), \dots, f'(b_{1k},\dots,b_{nk})
\big)
=
\big(
c(f(\pi_{b_{11}}^B,\dots,\pi_{b_{n1}}^B)), \dots,
c(f(\pi_{b_{1k}}^B,\dots,\pi_{b_{nk}}^B))
\big) \in R$$ from the definition of coloring. We have shown that ${\mathscr{C}} \subseteq {{\mathscr{B}}}$, therefore ${{\mathscr{B}}}\in \operatorname{\mathsf{ERP}}{{\mathscr{A}}}$ since ${\mathscr{C}} \in \operatorname{\mathsf R}\operatorname{\mathsf S}\operatorname{\mathsf P}{{\mathscr{A}}}= \operatorname{\mathsf R}\operatorname{\mathsf P}{{\mathscr{A}}}$. From Proposition \[prop:abstract\] it now follows that there exists an h1 clone homomorphism ${{\mathscr{A}}}\to {{\mathscr{B}}}$.
For the other implication suppose that we have an h1 clone homomophism $\xi$ from ${{\mathscr{A}}}$ to ${{\mathscr{B}}}$. Then, from the proof of Proposition \[prop:abstract\], we know that ${{\mathscr{B}}}$ is an expansion of the [reflection]{} of ${\mathscr{F}}_{{{\mathscr{A}}}}(B)$ given by the mappings $h = c \colon f(\pi_{b_1}^B,\dots,\pi_{b_n}^B)
\mapsto \xi(f)(b_1,\dots,b_n)$ and $h'\colon b \mapsto \pi_b^B$. We will show that $c$ is a coloring. Let $R$ be a relation in ${{\mathbb{B}}}$ and $(f_1,\dots,f_k) \in R^{{\mathscr{A}}}$. By the definition of $R^{{{\mathscr{A}}}}$, there exists $f\in {{\mathscr{A}}}$ and tuples $(b_{11},\dots,b_{1k}), \dots, (b_{n1},\dots,b_{nk}) \in R$ such that $f_i =
f(\pi_{b_{1i}}^B,\dots,\pi_{b_{ni}}^B)$ for all $i=1,\dots,n$. Therefore, $$\begin{gathered}
\big(
c(f_1), \dots, c(f_k)
\big) =
\big(
c(f(\pi_{b_{11}}^B,\dots,\pi_{b_{n1}}^B)), \dots,
c(f(\pi_{b_{1k}}^B,\dots,\pi_{b_{nk}}^B))
\big) \\
= \big(
\xi(f)(b_{11},\dots,b_{n1}), \dots, \xi(f)(b_{1k},\dots,b_{nk})
\big) \in R\end{gathered}$$ since $\xi(f)$ is compatible with $R$. This concludes the proof of (i); the proof of (ii) is analogous.
As a corollary of the last three propositions we get that a variety is congruence $n$-permutable for some $n$ (modular, respectively) if and only if its clone does not have a strong h1 clone homomorphism to the polymorphism clone of $(\{0,1\}; \leq)$ (the structure ${\mathbb{D}}$ from Proposition \[prop:day\], respectively).
The application of colorings formulated as Theorem \[thm:perm-and-modular\] is based on the following consequence of Proposition \[prop:abstract\] and Corollary \[cor:linear-birkhoff\].
Let ${\mathcal{V}}$ be a variety and ${{\mathscr{B}}}$ be a function clone.
1. If ${\mathcal{V}}$ is defined by identities of height 1 and there is an h1 clone homomorphism from ${\mathrm{clo}({\mathcal{V}})}$ to ${{\mathscr{B}}}$, then there is also a clone homomorphism from ${\mathrm{clo}({\mathcal{V}})}$ to ${{\mathscr{B}}}$.
2. If ${\mathcal{V}}$ is defined by identities of height at most 1 and there is a strong h1 clone homomorphism from ${\mathrm{clo}({\mathcal{V}})}$ to ${{\mathscr{B}}}$, then there is also a clone homomorphism from ${\mathrm{clo}({\mathcal{V}})}$ to ${{\mathscr{B}}}$.
We prove the first part, the second part is analogous. Since there is an h1 clone homomorphism from ${\mathrm{clo}({\mathcal{V}})}$ to ${{\mathscr{B}}}$, we have ${{\mathscr{B}}}\in \operatorname{\mathsf{ERP}}{\mathrm{clo}({\mathcal{V}})}$, therefore ${{\mathscr{B}}}\in \operatorname{\mathsf{ERP}}{{\mathbf{A}}}$ for a generator ${\mathbf{A}}$ of ${\mathcal{V}}$. But ${\mathcal{V}}$ is defined by identities of height 1, so it is closed under $\operatorname{\mathsf R}$ and $\operatorname{\mathsf P}$, hence ${{\mathscr{B}}}\in \operatorname{\mathsf E}{\mathcal{V}}$, which in turn implies ${{\mathscr{B}}}\in \operatorname{\mathsf{EHSP}}{\mathrm{clo}({\mathcal{V}})}$. It follows that there exists a clone homomorphism from ${\mathrm{clo}({\mathcal{V}})}$ to ${{\mathscr{B}}}$.
A combination of this proposition and Proposition \[prop:coloring-and-h1\] immediately gives the following.
\[cor:join\] Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be varieties and let ${\mathbb{B}}$ be a relational structure.
- If ${\mathcal{V}}$ and ${\mathcal{W}}$ are defined by identities of height $1$, and ${\mathrm{clo}({\mathcal{V}})}$ and ${\mathrm{clo}({\mathcal{W}})}$ are ${\mathbb{B}}$-colorable, then so is ${\mathrm{clo}({\mathcal{V}} \vee {\mathcal{W}})}$.
- If ${\mathcal{V}}$ and ${\mathcal{W}}$ are defined by identities of height at most $1$, and ${\mathrm{clo}({\mathcal{V}})}$ and ${\mathrm{clo}({\mathcal{W}})}$ are strongly ${\mathbb{B}}$-colorable, then so is ${\mathrm{clo}({\mathcal{V}} \vee {\mathcal{W}})}$.
Back to the Introduction {#sec:wrapup}
========================
Our results imply Theorem \[thm:new\_infinite\] as follows; note that Theorem \[thm:new\_finite\] is a special case thereof. The two statements in (i) are equivalent by Corollary \[cor:relational\]. They are equivalent to (ii) by Corollary \[cor:pp\_mut\], and to (iii) by Proposition \[prop:topological\].
Item (i) in Theorem \[thm:equiv\_conditions\] is trivially implied by item (ii). The other direction follows from Taylor’s theorem [@T77] which implies that the non-existence of a clone homomorphism from an idempotent clone ${{\mathscr{B}}}$ to the clone of projections is witnessed by non-trivial height 1 identities satisfied by operations in ${{\mathscr{B}}}$. These identities prevent an h1 clone homomorphism to the projection clone $\mathbf 1$. Items (i) and (ii) are equivalent to (iii) by [@BK12] and to (iv) by [@KMM14] (which is a strengthening of [@Sig10]). The following corollary implies that items (i)–(iv) are equivalent to their primed versions.
\[cor:weak\_homo\_to\_core\] Let ${\mathbb{A}}$ be at most countable $\omega$-categorical structure and let ${\mathbb{B}}$ be the model-complete core of ${\mathbb{A}}$ expanded by finitely many singleton unary relations. Then there exist uniformly continuous h1 clone homomorphisms ${{\mathscr{A}}}\to {{\mathscr{B}}}$ and ${{\mathscr{B}}}\to {{\mathscr{A}}}$.
Since ${\mathbb{B}}$ is pp-constructible from ${\mathbb{A}}$ and ${\mathbb{A}}$ is pp-constructible from ${\mathbb{B}}$, the claim follows from Theorem \[thm:new\_infinite\].
The old Conjecture \[conj:old\] implies the new Conjecture \[conj:new\].
If the first item of Conjecture \[conj:old\] holds for a structure, then so does the first item of Conjecture \[conj:new\]. Indeed, let ${\mathbb{C}}$ be an expansion of the model-complete core ${\mathbb{B}}$ of an $\omega$-categorical structure ${\mathbb{A}}$ such that ${\mathscr{C}}$ has a continuous homomorphism $\xi$ to the projection clone $\mathbf 1$. Then $\xi$ is uniformly continuous by Proposition \[prop:uniform\]. By composing $\xi$ with a uniformly continuous h1 clone homomorphism from ${{\mathscr{A}}}$ into ${{\mathscr{C}}}$, provided by Corollary \[cor:weak\_homo\_to\_core\], we obtain a uniformly continuous h1 clone homomorphism from ${{\mathscr{A}}}$ to $\mathbf 1$.
We do not know whether the converse holds, i.e., whether or not the first item of Conjecture \[conj:new\] implies the first item of Conjecture \[conj:old\]. This problem also provides a possible approach to disproving Conjecture \[conj:old\].
Let ${{\mathbb{A}}}$ be a reduct of a finitely bounded homogeneous structure, and let ${{\mathbb{B}}}$ be its model-complete core. Suppose that ${{\mathscr{A}}}$ maps to $\mathbf 1$ via a uniformly continuous h1 clone homomorphism (and hence, $\operatorname{CSP}({{\mathbb{A}}})$ is NP-hard).\
Do there exist elements $b_1,\ldots,b_n$ in ${{\mathbb{B}}}$ such that the polymorphism clone of the expansion of ${{\mathbb{B}}}$ by those constants maps homomorphically and continuously to $\mathbf 1$?
Let us discuss the results on colorings in Section \[sect:intro\_colorings\]. Theorem \[thm:coloring-and-h1\] is the first part of Proposition \[prop:coloring-and-h1\], and Theorem \[thm:perm-and-modular\] is a consequence of Corollary \[cor:join\], Proposition \[prop:hm\], and Proposition \[prop:day\].
The correspondence between Maltsev conditions and h1 clone homomorphism suggests an approach to Conjecture \[conj:taylor\] and similar problems: for a given clone ${\mathscr{B}}$ (which characterizes the Maltsev condition in question via h1 clone homomorphisms), find an upward directed class of clones such that, for every ${{\mathscr{A}}}$, ${{\mathscr{A}}}$ has an h1 clone homomorphism to ${\mathscr{B}}$ if and only if ${\mathscr{A}}$ has a clone homomorphism to some member of the class. Encouraging results in this direction are [@kearnes.tschantz07] and [@valeriote.willard14], where such a class was found for idempotent ${\mathscr{A}}$s and the ${\mathscr{B}}$s characterizing congruence permutability [@kearnes.tschantz07] and $n$-permutability [@valeriote.willard14]. Is it possible that such a class exists even for every clone ${\mathscr{B}}$?
[^1]: Also known as the *double shrink*.
[^2]: Often denoted by $\mathsf D$, for *double shrink*.
[^3]: Also see the comment below Definition \[def:5.2\].
|
---
abstract: 'The precise data for the total cross section $\sigma(e^+e^-\to\mbox{hadrons})$ from the charm threshold region, when combined with the evaluation of moments with three loop accuracy, lead to a direct determination of the short distance $\overline{\rm MS}$ charm quark mass $m_c(m_c)=1.304(27)$. Applying the same approach to the bottom quark we obtain $m_b(m_b)=4.191(51)$ GeV. A complementary method for the determination of $m_b$ is based of the analysis of the $\Upsilon(1S)$ system which is confronted with a next-to-next-to-next-to-leading order calculation of the corresponding energy level. This leads to $m_b(m_b)=4.346(70)$ GeV.'
address: |
II. Institut für Theoretische Physik, Universität Hamburg,\
Luruper Chaussee, D-22761 Hamburg, Germany
author:
- 'M. Steinhauser [^1]'
title: '$\sigma(e^+e^-\to\mbox{hadrons})$ and the Heavy Quark Masses'
---
Introduction
============
During the past years new and more precise data for $\sigma(e^+e^-\to\mbox{hadrons})$ have become available in the low energy region between 2 and 10 GeV. At the same time increasingly precise calculations have been performed in the framework of perturbative QCD (pQCD), both for the cross section as a function of the center-of-mass energy $\sqrt{s}$, including quark mass effects, and for its moments which allow for a precise determination of the quark mass. A fresh look at the evaluation of the charm quark mass with the help of sum rules is thus an obvious task. We will concentrate on low moments as suggested by the ITEP group long ago [@NSVZ]. This is a natural route to determine directly a short distance mass, say $m_c(m_c)$, in the $\overline{\rm MS}$ scheme [@NSVZ; @ShiVaiZak79].
The same method can also be applied to the bottom system leading directly to $m_b(m_b)$.
A complementary method for the determination of $m_b$ is based of the analysis of the $\Upsilon(1S)$ system. Recently a major improvement on the theoretical side has been achieved by the evaluation of the next-to-next-to-next-to-leading order (N$^3$LO) corrections to the energy level. The disadvantage of this method is due to the large non-perturbative effects.
\[sec::mcmb\]Charm and bottom quark mass from low-order moment sum rules
========================================================================
A very elegant method for the determination of the charm quark mass is based on the direct comparison of theoretical and experimental moments of the charm quark contribution to the photon polarization function. In the limit of small momentum the latter can be cast into the form [@CheKueSte96] $$\begin{aligned}
\Pi_c(q^2) &=& Q_c^2 \frac{3}{16\pi^2} \sum_{n\ge0}
\bar{C}_n z^n
\,,
\label{eq:pimom}\end{aligned}$$ with $z=q^2/(4m_c^2)$ where $m_c=m_c(\mu)$ is the $\overline{\rm MS}$ charm quark mass at the scale $\mu$. The perturbative series for the coefficients $\bar{C}_n$ up to $n=8$ is known analytically [@CheKueSte96] up to order $\alpha_s^2$. We define the moments through $$\begin{aligned}
{\cal M}_n^{\rm exp} &=& \int \frac{{\rm d}s}{s^{n+1}} R_c(s)
\nonumber\\
&=&
{\cal M}_n^{\rm th} \,\,=\,\,
\frac{9}{4}Q_c^2
\left(\frac{1}{4 m_c^2}\right)^n \bar{C}_n
\,,
\label{eq:Mth}\end{aligned}$$ which leads to the following formula for the charm quark mass $$\begin{aligned}
m_c(\mu) &=& \frac{1}{2}
\left(\frac{\bar{C}_n(\ln m_c)}{{\cal M}_n^{\rm exp}}\right)^{1/(2n)}
\,.
\label{eq:mc1}\end{aligned}$$ The analysis of the experimental moments with $n=1,\ldots,4$ leads to the results displayed in Fig. \[fig:mom\] [@KueSte01].
The moment with $n=1$ is evidently least sensitive to non-perturbative contributions from condensates, to the Coulombic higher order effects, the variation of $\mu$ and the parametric $\alpha_s$ dependence. Hence $$\begin{aligned}
m_c(m_c) &=& 1.304(27)~\mbox{GeV}
\,.
\label{eq:mcfinal}\end{aligned}$$ is adopted as the final result [@KueSte01]. In principle the same analysis can be performed using the pole mass sheme for the quarks. However, the final results are quite unstable in contrast to the $\overline{\rm MS}$ scheme.
The same approach is also applicable to the determination of $m_b$. Again a significant improvement of the stability of the prediction after inclusion of the NNLO terms is observed. As final result one finds $$\begin{aligned}
m_b(m_b) &=& 4.191(51)~\mbox{GeV}
\,,
\label{eq:mbmb}\end{aligned}$$
[c]{}
The $\Upsilon(1S)$ system and the bottom quark mass
===================================================
In contrast to the previous section one has to deal with a non-relativistic system of a bound state of a heavy quark-antiquark pair which is governed by a complicated multiscale dynamics. In the nonrelativistic regime, where the heavy-quark velocity $v$ is of the order of the strong-coupling constant $\alpha_s$, the Coulomb effects are crucial and have to be taken into account to all orders in $\alpha_s$. This makes the use of the effective theory mandatory. This approach allows us to separate the scales and to implement the expansion in $v$ at the level of the Lagrangian.
The dynamics of a nonrelativistic quark-antiquark pair is characterized by four different regions, the hard region, the soft region, the potential region, and the ultrasoft region. Nonrelativistic QCD (NRQCD) [@CasLep] is obtained by integrating out the hard modes. Subsequently integrating out the soft modes and the potential gluons results in the effective theory of potential NRQCD (pNRQCD) [@PinSot1], which contains potential heavy quarks and ultrasoft gluons, ghosts, and light quarks as active particles. The effect of the modes that have been integrated out is two-fold: higher-dimensional operators appear in the effective Hamiltonian, corresponding to an expansion in $v$, and the Wilson coefficients of the operators in the effective Hamiltonian acquire corrections, which are series in $\alpha_s$. In [@KPSS1; @KPSS2] the ingredients for the N$^3$LO Hamiltonian have been completed using an efficient combination of the effective theory formalism and the threshold expansion [@BenSmi].
Once the Hamiltonian is available the only task is to solve the corresponding Schrödinger equation up to third order using the usual time-independent perturbation theory. A new feature which appears for the first time at N$^3$LO are the retardation effects which arise from the chromoelectric dipole interaction of the heavy quarkonium with a virtual ultrasoft gluon.
The third order correction to the lowest energy level was calculated in [@KPSS2; @PS1] and will be used in this work in order to determine the bottom quark mass. The application to the top quark system can be found in [@Peninproc]. Recently the N$^3$LO Hamiltonian has been used to compute the corrections of order $\alpha_s^3\ln\alpha_s$ to the wave function [@KPSS3].
The perturbative expansion of the energy level with quantum number $n$ looks as follows $$\begin{aligned}
E_n^{\rm p.t.}=E^C_n+\delta E^{(1)}_n+\delta E^{(2)}_n+\delta
E^{(3)}_n
+\ldots\,,\end{aligned}$$ where the ${\cal O}\left(\alpha_s^3\right)$ correction, $E^{(3)}_n$, arises from the following sources:
- matrix elements of the N$^3$LO operators of the effective Hamiltonian between Coulomb wave functions;
- higher iterations of the NLO and NNLO operators of the effective Hamiltonian in time-independent perturbation theory;
- matrix elements of the N$^3$LO instantaneous operators generated by the emission and absorption of ultrasoft gluons; and
- the retarded ultrasoft contribution.
--------
=7.2cm
(a)
=7.2cm
(b)
--------
The analytical result for $E^{(3)}_n$ can be found in [@PS1]. In numerical form it reads (adopting the choice $\mu_s=C_F\alpha_s(\mu_s) m_q$) for the bottom and top system $$\begin{aligned}
\delta E^{(3)}_1&=& \alpha_s^3(\mu_s)E_1^C\left[
\left(
\begin{array}{c}
70.590|_{\mbox{\tiny bottom}}\\
56.732|_{\mbox{\tiny top}}
\end{array}
\right)
\right.
\nonumber\\&&
\left.
+ 15.297 \ln(\alpha_s(\mu_s)) + 0.001\,a_3
\right.
\nonumber\\&&
\left.
+ \left(
\begin{array}{c}
34.229|_{\mbox{\tiny bottom}}\\
26.654|_{\mbox{\tiny top}}
\end{array}
\right)\bigg|_{\beta_0^3}
\right]\,,
\label{eq:dele3num}\end{aligned}$$ where we have separated the contributions arising from $a_3$ and $\beta_0^3$. The only unknown ingredient in the result for $\delta E^{(3)}_1$ is the three-loop $\overline{\rm MS}$ coefficient $a_3$ of the corrections to the static potential. Up to now there are only estimates based on Padé approximation [@ChiEli] which is used in our analysis. However, the final result only changes marginally even for a rather large deviations of $a_3$ from its Padé estimate.
The starting point for the determination of the bottom quark mass is its relation to the mass of the $\Upsilon(1S)$ resonance $$\begin{aligned}
M_{\Upsilon(1S)} = 2m_b + E_1^{\rm p.t.}+ \delta^{\rm n.p.}E_1\,,
\label{eq:M1SE}\end{aligned}$$ with $M_{\Upsilon(1S)}=9.46030(26)$ GeV. Here $\delta^{\rm n.p.}E_1$ is the nonperturbative correction to the ground state energy. The dominant contribution is associated with the gluon condensate and gives [@PS1] $$\delta^{\rm n.p.}E_1=60\pm 30~{\rm MeV}\,.
\label{nptb}$$
Combining Eq. (\[eq:M1SE\]) with the result for the perturbative perturbative ground state energy up to ${\cal O}(m_q\alpha_s^5)$ one obtains the bottom quark mass as a function the renormalization scale of the strong coupling constant normalization, $\mu$, which is plotted in Fig. \[fig:botpol\](a). For the numerical evaluation we extract $\alpha_s^{(4)}(m_b)$ with $m_b=4.83$ GeV from its value at $M_Z$ using four-loop $\beta$-function accompanied with three-loop matching[^2]. $\alpha_s^{(4)}(m_b)$ is used as starting point in order to evaluate $\alpha_s^{(4)}(\mu)$ at N$^k$LO with the help of the $k$-loop $\beta$ function. From Fig. \[fig:botpol\](a) we see that the dependence on the renormalization scale becomes very strong below $\mu\sim 2$ GeV which indicates that the perturbative corrections are not under control. However, even above this scale the perturbative series for the pole mass shows no sign of convergence. This means that one can assign a numerical value to the pole mass only in a specified order of perturbation theory.
On the contrary, it is widely believed that the $\overline{\rm MS}$ mass ${\overline m}_b(\mu)$ at the scale $\mu={\overline m}_b(\mu)$ is a short-distance object which has much better perturbative properties. Thus, it seems to be reasonable to convert our result for the pole mass into ${\overline m}_b({\overline m}_b)$. The relation between $m_b$ and ${\overline m}_b({\overline m}_b)$ is known up to three-loop approximation [@CheSte99; @MelRit99] and shows sizable perturbative corrections. For this reason we suggest the following procedure to take into account these corrections in a most accurate way. The idea is that from the one-parametric family of ${\overline m}_b(\mu)$ we can choose a representative corresponding to some scale $\mu^\star$ in such a way that $${\overline m}_b(\mu^\star)=m_b\,.
\label{mustar}$$ For a given fixed-order value of the pole mass Eq. (\[mustar\]) can be solved for $\mu^\star$. In particular, for a pole mass to N$^k$LO we use the $k$-loop relation between the $\overline{\rm MS}$ and pole mass in Eq. (\[mustar\]). Afterwards ${\overline m}_b({\overline m}_b)$ can be computed from $\overline{m}_b(\mu^\star)$ solving the renormalization group (RG) equation. The advantage of this approach is obvious: we use the finite order relation between $\overline{\rm MS}$ and pole mass at the scale where they are perturbatively close while the large difference between ${\overline m}_b({\overline m}_b)$ and $m_b$ is completely covered by the RG evolution which can be computed with very high accuracy as the corresponding anomalous dimension is known to four-loop approximation [@Chetyrkin:1997dh; @Vermaseren:1997fq]. The only restriction on the method could be connected to the value of $\mu^\star$. It should be large enough to allow for a reliable use of the RG equation which in practice indeed is the case [@PS1].
In Fig. \[fig:botpol\](b) our result for ${\overline m}_b({\overline m}_b)$ is plotted at NLO, NNLO and N$^3$LO as a function of the normalization scale $\mu$ which is used to obtain the pole mass $m_b$ (cf. Fig. \[fig:botpol\](a)). It is remarkable that close to $\mu= 2.7$ GeV, which is consistent with the physically motivated soft scale $\mu_s\approx 2$ GeV, both the second and the third order corrections vanish. This fact is a rather strong indication of the convergence of the series for ${\overline m}_b({\overline m}_b)$.
The uncertainties in the obtained value of ${\overline m}_b({\overline m}_b)$ have been investigated in detail in [@PS1]. Finally, the prediction for the $\overline{\rm MS}$ bottom quark mass reads $${\overline m}_b({\overline m}_b)=4.346\pm 0.070~{\rm GeV}
\,.
\label{finalmb1}$$
Conclusions
===========
For the comparison of the results discussed in this contribution with the literature we refer to [@KueSte01; @PS1]. It is, however, interesting to compare the value for the charm quark mass with a very recent result obtained in a lattice calculation [@Rolf:2002gu]. Their final result in quenched approximation, $m_c(m_c)=1.301(34)$ GeV, is impressively close to ours (cf. Eq. (\[eq:mcfinal\])) with comparable errors. In order to estimate the uncertainty induced by the quenched approximation we performed a “perturbative quenching” by setting the number of active flavours to zero in the calculation of [@KueSte01]. This leads to a small uncertainty of $\approx30$ MeV, which agrees with the estimate of [@Rolf:2002gu].
Although formally slightly beyond 1$\sigma$ the $\overline{\rm MS}$ quark mass obtained from the low-moment sum rule approach is in very good agreement with the one from the $\Upsilon(1S)$ system. One needs to have in mind that both the experimental data and the theoretical calculations are completely different. Whereas no further improvement in the $\Upsilon(1S)$ method can be expected there is significant improvement possible in the approach discussed in Section \[sec::mcmb\]. In particular, after reducing the experimental error of $R(s)$ in the charm and bottom threshold region and the one of the leptonic widths of the narrow resonances to roughly 2% a reduction of the uncertainty in the charm and bottom quark mass to 15 MeV and 30 MeV, respectively, can be expected. More details can be found in Ref. [@Kuhn:2002zr].
Acknowledgment {#acknowledgment .unnumbered}
==============
I would like to thank the organizers of RADCOR/Loops and Legs 2002 conference for the kind inviation and B.A. Kniehl, J.H. Kühn, A.A. Penin and V.A. Smirnov for a fruitful collaboration.
[99]{}
\#1\#2\#3[ [Nucl. Phys. B ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[ [Phys. Lett. B ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[ [Phys. Rev. Lett. ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[ [Z. Phys. C ]{}[**\#1**]{} (\#2) \#3]{}
V. A. Novikov [*et al.*]{}, Phys. Rep. C [**41**]{} (1978) 1.
M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, ; .
K. G. Chetyrkin, J. H. Kühn, and M. Steinhauser, ; ; .
J.H. Kühn and M. Steinhauser, Nucl. Phys. B [**619**]{} (2001) 588, (E) ibid. B [**640**]{} (2002) 415.
W.E. Caswell and G.P. Lepage, Phys. Lett. B 167 (1986) 437; G.T. Bodwin, E. Braaten, and G.P. Lepage, Phys. Rev. D 51 (1995) 1125; (E) ibid. D 55 (1997) 5853.
A. Pineda and J. Soto, Nucl. Phys. B (Proc. Suppl.) 64 (1998) 428.
B.A. Kniehl, A.A. Penin, V.A. Smirnov, and M. Steinhauser, Phys. Rev. D [**65**]{} (2002) 091503(R).
B.A. Kniehl, A.A. Penin, V.A. Smirnov, and M. Steinhauser, Nucl. Phys. [**B635**]{}, 357 (2002).
M. Beneke and V.A. Smirnov, Nucl. Phys. B 522 (1998) 321.
A.A. Penin and M. Steinhauser, Phys. Lett. b [**538**]{} (2002) 335.
A.A. Penin, these proceedings, hep-ph/0210201.
B.A. Kniehl, A.A. Penin, V.A. Smirnov, and M. Steinhauser, Report No. DESY 02-134, hep-ph/0210161.
F.A. Chishtie and V. Elias, Phys. Lett. B 521 (2001) 434.
K. G. Chetyrkin, J. H. Kühn, and M. Steinhauser, Comput. Phys. Commun. [**133**]{} (2000) 43.
K. G. Chetyrkin and M. Steinhauser, ; .
K. Melnikov and T. v. Ritbergen, Phys. Lett. B [**482**]{} (2000) 99.
K. G. Chetyrkin, Phys. Lett. B [**404**]{} (1997) 161. J. A. Vermaseren, S. A. Larin, and T. van Ritbergen, Phys. Lett. B [**405**]{} (1997) 327. J. Rolf and S. Sint \[ALPHA Collaboration\], Report No.: hep-ph/0209255.
J.H. Kühn and M. Steinhauser, JHEP [**0210**]{} (2002) 018.
[^1]: Talk given at RADCOR/Loops and Legs, September 2002.
[^2]: We use the package [ RunDec]{} [@Chetyrkin:2000yt] to perform the running and matching of $\alpha_s$
|
---
abstract: 'A comparative analysis has been done of the formerly established two self-consistent solutions for the density of quasiparticle states in doped d-wave superconductors, having strikingly different and disputed behavior near the Fermi energy. One of them (1) remains finite in this limit, while the other (2) tends to zero. To resolve this discrepancy, the known Ioffe-Regel criterion for band states, widely used for doped semiconductors, was applied to these solutions. It is shown that both them are valid in limited and different energy regions, where the corresponding quasiparticles are weakly damped. In particular, density of states of nodal quasiparticles near the Fermi level is provided by the (2) solution, while the (1) only applies far enough from this level.'
author:
- 'V.M. Loktev$^{1}$'
- 'Yu.G. Pogorelov$^{2}$'
title: 'Nodal quasiparticles in doped d-wave superconductors: self-consistent T-matrix approach'
---
ł ø § Ø
\#1[(\[\#1\])]{}
The self-consistent T-matrix approximation (SCTMA [@bay], or FLEX method [@bick]) is extensively used for description of quasiparticles and their density of states (DOS) in disordered crystals, in particular, in the doped high-Tc superconducting cuprates. Its advantage consists in relative simplicity and transparency for numeric calculations. However, this simplicity can be sometimes mischievous if the solutions are used without limitations on their validity. In fact, a number of controversies emerge when comparing some SCTMA results [@zieg] with those of other approaches [@ners],[@atk], which suggest the need for a more detailed substantiation of the method. Perhaps the central point in this discussion is now the question about the quasiparticle DOS $\r (\e)$ close to the center of superconducting gap, that is at $\e \ra 0$, where the DOS of clean d-wave superconductor vanishes as $\r_{d}(\e )\propto \mathrm{const}\cdot|\e|$. The theoretical predictions for its behavior in presence of impurity scattering include: *i*) various kinds of tendency to zero [@ners], [@atk], [@sent], [@lp], *ii*) tendency to a constant value [@gor], [@lee], and even *iii*) to infinity [@pep]. Such extreme ambiguity motivates us to reconsider this situation in a more general context of the theory of elementary excitations in disordered systems [@lgp].
There is a consensus on that the single-particle excitations in disordered sytems can be either of band (extended) type or localized type [@mott], both types forming certain continuous regions of spectrum separated by the so-called mobility edges $\e _{c}$ (Fig. \[fig1\]). The extended states can be approximately described by the wavevector $\bk$, through a dispersion law $E_{\bk}$ and a broadening $\G_{\bk}$, as far as the Ioffe-Regel criterion (IRC) [@ir] for the mean free path $\ell $ and the wavelength $\l$ holds: ł, k | E\_/| \_. Then the real and imagnary parts of self-energy $\S_{\bk}$ in the disorder averaged Green function (GF) $G_{\bk}=(\e - \e_{\bk}-\S_{\bk})^{-1}$ permit to define: $E_{\bk}=\e_{\bk}-\mathrm{Re}\S_{\bk}\left( E_{\bk}\right)$, $\G_{\bk}=\mathrm{Im}\S_{\bk}\left( E_{\bk}\right)$ [@bb], and the modified DOS $\r(\e)=\pi^{-1}\mathrm{Im}G \approx \r_{0}(\e)/
\left(\pd E_{\bk}/\pd\e_{\bk} \right)_{E_{\bk}=\e}$ ($\e_{\bk}$ and $\r_{0}(\e)$ being respectively the dispersion law and DOS in pure crystal). The condition $\ell
\sim \l$ is reached when $E_{\bk}$ approaches the mobility edges, then the very notion of self-energy correction to the initial band spectrum $\e_{\bk}$ ceases to make sense and the averaged properties of localized states are only described by their DOS $\r(\e)$.
In the case of disorder due to fixed impurity perturbation $V_{\mathrm{L}}$ at random sites $\bp$ (Lifshitz model) for the normal metal quasiparticles, the modification of band spectrum can involve new specific features like local or resonance levels [@lif], and there are various ways to expand the self-energy in groups of interacting impurity centers [@ilp], analogous to the classical Ursell-Mayer group expansions (GE) for statistical sum [@hill]. For instance, the so-called fully renormalized GE [@iv] reads: §\_&=&\_, where $A_{\bp,\bp^\prime}=V_{\mathrm{L}}G_{\bp,\bp^\prime}/(1-GV_{\mathrm{L}})$ includes $G_{\bp,\bp^\prime}=
N^{-1}\sum_{\bk^\prime \neq \bk} \mathrm{e}^{i\bk^\prime\left( \bp-\bp^\prime \right)} G_{\bk}$ and $G=G_{\bp,\bp}$. Other types of GE can differ from Eq.\[eq2\] either in the structure of next to unity terms and in the degree of renormalization of $G$ and $A$ functions present in them. The relevant expansion parameter is not simply the impurity concentration $c=\sum_{\bp}N^{-1}$ (supposedly small, $c\ll 1$) but the ”gas parameter” $c\sum_{\mathbf{n}\neq 0}
A_{0,\bn}^{2}$ for the ”non-ideal gas” of impurities with effective interaction described by the (energy dependent) functions $A_{\bp,\bp^\prime}$. Hence the convergence of the series (\[eq2\]) turns also energy dependent, reflecting the division between the above referred types of states. It can be shown that this convergence is equivalent to validity of the IRC [@is]. Within the energy domain of convergence, the self-energy can be approximated by Eq.\[eq2\] with only unity term retained in the brackets: $\S_{\bk}\approx \S =cV_{\mathrm{L}}/(1-GV_{\mathrm{L}})$, the momentum-independent SCTMA form. But beyond this domain, the SCTMA does not make sense and a better description of DOS is obtained with GE’s, different from Eq.\[eq2\]. Below we check the fulfillment of IRC for nodal quasiparticle states in a d-wave superconductor with dopants and conclude on the validity of the SCTMA solutions.
Let us start from the most common model Hamiltonian for this problem:
H&=&\_\_\^(\_\_[3]{}+\_\_[1]{}) \_\
&&- \_[,\_,]{}\^[i(-\^)]{} \_\^\_[3]{}\_[\^]{}, Here the Nambu spinors $\psi _{\bk}^{\dagger }=\left( c_{\bk,\ua}^{\dagger},
c_{-\bk,\da}\right) $ include Fermi operators of normal quasiparticles with the simplest 2D dispersion law $\xi_{\bk}=2t\left( 2-\cos ak_{x}-\cos ak_{y}\right) -\m$ in square lattice, approximately constant DOS $\r(\e)\approx \r_{0}=4/\left( \pi W\right) $ where $W=8t$ is the bandwidth, and chemical potential $\m$; $\widehat{\t}_{j}$ are the Pauli matrices. The d-wave gap function is $\D_{\bk}=\D \theta \left( \e_{D}^{2}-\xi _{\bk}^{2}\right)
\cos 2\varphi _{\bk}$, where the ”Debye” energy $\e_{D} \gg \D$, and $\varphi _{\bk}=\arctan k_{y}/k_{x}$ defines the nodal lines $k_{x}=\pm k_{y}$. The Lifshitz perturbation term in Eq.\[eq3\] produces scattering of quasiparticles, modelling the impurity effect of dopants.
The relevant GF is a Nambu matrix $\widehat{G}_{\bk}=\left\langle \left\langle \psi _{\bk}|
\psi _{\bk}^{\dagger }\right\rangle \right\rangle $ with matrix elements being Fourier transformed two-time GF’s: a|b\_=i\_[0]{}\^\^[i(i0)t]{}{ a( t) ,b( 0) } dt where $\left\langle \ldots \right\rangle $ is the quantum statistical average with the Hamiltonian \[eq3\] and $\left\{ . , . \right\} $ is the anticommutator of Heisenberg operators. In analogy with the above scalar GF $G_{\bk}$ for normal quasiparticles, the general solution for this matrix is \_=( --\_\_[3]{} - \_\_[1]{} - \_)\^[-1]{}. All the impurity effects are now accounted for by a GE for the self-energy matrix $\widehat{\S}_{\bk}$ [@po1] (cf. Eq.\[eq2\]): \_ &=&\_ ( 1 - )\^[-1]{} { 1 + .\
&& + \_[\^]{}\
&& . \^[-1]{} + …} . Here the matrices: $\widehat{V}=V_{\mathrm{L}}\widehat{\t}_{3}$, $\widehat{G}=N^{-1}\sum_{\bk}\widehat{G}_{\bk}$, and \_[,\^]{}=N\^[-1]{}\_[\^]{} \^[i\^ ( - \^)]{} \_[\^]{} ( 1- )\^[-1]{}, and some additional restrictions are imposed on summation in momenta in the products like $\widehat{A}_{\bp,\bp^\prime} \widehat{A}_{\bp^\prime,\bp}$, resulting from a specific procedure of consecutive elimination of GF’s in the infinite chain of coupled Dyson equations. There are possible different such procedures and, respectively, different types of GE [@ilp]. Generally, GE’s are only asymptotically convergent and the best choice between them is determined by their convergence range with respect to energy $\e $.
The conditions for convergence of different GE’s were studied in detail for a number of types of elementary excitations in crystals with impurities [@ip],[@lp], and this permitted to establish certain general criteria for the corresponding characteristic regions of spectrum. In particular, the region of band states is best described by the so-called fully renormalized GE which ceases to converge at approaching the mobility edges where IRC, Eq.\[eq1\], fails. For the GE, this is expressed by the tendency of all its terms, next to the unity in curled brackets in Eq.\[eq5\], to become $\sim$1.
Alike the above mentioned scalar case, the SCTMA just corresponds to the fully renormalized GE, restricted to only its first term. Hence it is only justified when IRC holds. Bearing this in mind, let us analyze the SCTMA solutions in the vicinity of the Fermi energy for the system, described by the Hamiltonian \[eq3\].
Then, using the $\bk$-independent SCTMA self-energy $\widehat{\S}=\widehat{V}
\left( 1-\widehat{G}\widehat{V}\right) ^{-1}$ and following the procedure of Refs. [@lp2],[@lp], one can arrive at the explicit average local GF: G==\_[0]{}. In Eq.\[eq6\], the renormalized energy $\widetilde{\e}=\e -\S \left( \widetilde{\e}\right)$ includes the scalar value $\S =\mathrm{Tr}\widehat{\S}/2$, the parameter $\widetilde{\m}=\m
(1-\m \r_{0}/2)$, and $\mathrm{K}$ is the 1st kind full elliptic integral. Having the explicit relation [@lp] $\Sigma =c\widetilde{V}^{2}G/(1-\widetilde{V}^{2}G^{2})$ between the self-energy and GF, where the renormalized perturbation parameter $\widetilde{V} = V_{\mathrm{L}}
/ [1+V_{\mathrm{L}}\r_{0}\ln (1-\widetilde{\m}/\m)]$, one obtains the self-consistent equation for $G$ which can be solved in principle numerically. Since the analytic structure of Eq.\[eq6\] involves singular points in the complex $G$ plane (including essential singularity of the $\mathrm{K}$-function), it possesses multiple solutions. The physical solutions among them should be then selected by IRC, as a necessary condition for SCTMA validity.
The analysis turns most transparent in the important limit $\e \ra 0$ (related to the Fermi energy). There are two characteristic solutions [@foot] of Eq.\[eq6\] in this limit. One of them, $G\left( \varepsilon \right) = G^{\left( 1\right) }\left(\e
\right) $, tending to a constant imaginary value, $G^{\left(1\right)}\left( \e \ra 0\right)
\ra i\cdot \mathrm{const}$, was first obtained by Gor’kov and Kalugin in the Born scattering limit [@gor] and then by P.A. Lee in the unitary scattering limit [@lee]. Later on, it was repeatedly reproduced by various numerical techniques [@atk] and hence believed to be the unique SCTMA solution. Neverthless, it was shown recently by the authors [@lp] that another solution exists, $G\left( \e \right) = G^{\left( 2\right) }\left(\e \right) $, with low energy asymptotics: G\^[( 2) ]{}( ) , which tends to zero with $\e $. The behavior of real and imaginary parts of the two solutions in function of energy for a particular choice of parameters: $c=10\%$, $V_{\mathrm{L}}=0.3$ eV, $W=2$ eV, and $\D =30$ meV, is shown in Fig.\[fig2\]. Notice that at low energies, $\e \ll \D $, the solution $G^{\left( 1\right) }$ is dominated by the above mentioned imaginary constant, presented as $i\pi c\r_{0}g_{0}$ where $g_{0}\lesssim 1$ is a root of a certain transcendental equation [@lp]. In contrary, the tendency of $G^{\left( 2\right) }$ to zero is characterized by the progressive domination of its real part. The renormalized dispersion law $\widetilde{E}_{\bk}$ (as far as the condition \[eq1\] holds) is given by the common equation [@bb] \_-§( \_) =E\_, where $E_{\bk}=\sqrt{\xi _{\bk}^{2}+\D_{\bk}^{2}}$ is the non-perturbed superconducting dispersion law and $\mathrm{\S }$ is specified for particular $G^{\left( j\right) }$, $j=1,2$. Then the IRC is written down as: ( -\_[0]{} ) \_\_ \_, near a nodal point $\bk_{0}$ where a nodal line crosses the Fermi surface.
The low energy asymptotics of Eq.\[eq8\], corresponding to the $G^{\left(2\right) }$ solution, Eq.\[eq7\], is: $\widetilde{E}_{\bk}^{\left(2\right) } \approx (\pi c\widetilde{V}^{2}\r_{0}/\D)
E_{\bk}\ln (2\D/E_{\bk})$, and with the related damping $\G_{\bk}^{\left( 2\right) }=\mathrm{Im}\S^{\left( 2\right) }
\left(\widetilde{E}_{\bk}^{\left( 2\right) }\right) \approx E_{\bk}/\ln (2\D/E_{\bk})$, we arrive at the condition: E\_ ( -) , which defines a narrow enough vicinity of the Fermi energy where this solution makes sense.
Applying the same treatment to the $G^{\left( 1\right) }$ solution, which formally defines the low energy dispersion law $\widetilde{E}_{\bk}^{\left( 1\right) }
\approx E_{\bk}$ and the damping $\G_{\bk}^{\left( 1\right) }=\mathrm{Im}
\S ^{\left( 1\right)}\left( \widetilde{E}_{\bk}^{\left( 1\right) }\right) \approx
\pi c \widetilde{V}^{2}\r_{0}g_{0}$, we obtain the condition E\_ c\^[2]{}\_[0]{}g\_[0]{}, so that this solution is valid far enough from the nodal points, where it provides also a correct limit of pure d-wave DOS. However, this solution is clearly eliminated near the nodal point. Thus we come to the conclusion that the only SCTMA solution, valid in the close vicinity of the Fermi energy, is that given by Eq.\[eq7\]. A physical consequence of vanishing DOS at $\e \to
0$ for this solution is that the much disputed conjecture of universal electric and thermal conductivity [@lee], [@durst] turns impossible. Nevertheless, if the validity range for the $G^{\left(2\right)}$ solution, Eq.\[eq9\], is very narrow, these conductivities, as far as being defined by the $G^{\left(1\right)}$ solution, can display an apparent tendency to those universal values.
Notably, the two estimates, Eqs.\[eq9\],\[eq10\], do not necessarily assure the overlap between the two validity regions, so that for $\pi c
\widetilde{V}^{2}\r_{0} \gg \D $ there can exist some intermediate energy range where neither of SCTMA solutions applies. This range roughly corresponds to the broad linewidth of the known impurity resonance $\e_{res}$ [@lp] where DOS cannot be rigorously obtained even with use of the next terms from GE, Eq.\[eq6\], though some plausible interpolation is possible between the two SCTMA asymptotics.
Finally it is worthwhile to notice that other known non-perturbative solutions for d-wave disordered systems with DOS vanishing at $\e \ra 0$ as a certain power law: $\r
\left( \e \right) \sim \e^{\a}$ [@ners],[@sent], also have to satisfy IRC since they use field theoretic approach, only compatible with band-like states. But it can be easily shown that this criterion can be only fulfilled for such DOS if the power is $\a >1$, while the reported values are $\a =1/7$ [@ners] and $\a =1$ [@sent].
In fact, let the renormalized radial dispersion law (in the low energy limit) behave as $\widetilde{\xi }_{k}\sim (k-k_{\mathrm{F}})^{\n } \propto \xi ^{\n}$ with certain $\n >0$, then the simplest estimate for d-wave DOS is ( ) &&\_[0]{}\^ dd( \^[2]{} - \^[2]{} - \^[2]{})\
&&\_[0]{}\^=\^[3-2]{}, that is $\a =3-2\n$. In the considered field models, DOS defines the quasiparticle broadening $\G_{\bk}=u^{2}\r \left( \widetilde{\xi }_{k}\right) $, with a disorder parameter $u$. Then the criterion, Eq.\[eq1\], is reformulated as u\^[2]{}( \_[k]{}) , leading to the condition $\xi ^{\n }\gg \mathrm{const\cdot }\xi ^{\n \left( 3-2\n \right)}$, and in the limit $\xi \ra 0$ this is only possible if $3-2\n >1$, that is $\a >1$.
So, the above considerations essentially restrict possible candidate solutions for quasiparticle spectrum in the disordered d-wave superconductor and in fact suggest Eq.\[eq7\] as the only known consistent low energy solution for the problem.
We thank Hans Beck, Valery Gusynin, Vladimir Miransky, and Sergei Sharapov for valuable and stimulating discussions. We also acknowledge the Swiss Science Foundation for the partial support of this research (the SCOPES grant 7UKPJ062150.00/1) and Neuchatel University for kind hospitality.
[99]{} G. Baym, Phys. Rev. [**127**]{}, 1391 (1962).
N.E. Bickers, D.J. Scalapino, S.R. White, Phys. Rev. Lett. [**62**]{}, 961 (1989).
K. Ziegler, M.H. Hettler, P.J. Hischfeld, Phys. Rev. Lett. [**77**]{}, 3013 (1996).
A.N. Nersesyan, A.M. Tsvelik, F. Wenger, Nucl. Phys. B [**438**]{}, 561 (1995).
W.A. Atkinson, P.J. Hischfeld, A.H. MacDonald, Phys. Rev. Lett. [**85**]{}, 3922 (2000).
T. Senthil, M.P.A. Fisher, Phys. Rev. B [**60**]{}, 6893 (1999).
V.M. Loktev, Yu.G. Pogorelov, Europhys. Lett. [**58**]{}, 549 (2002).
L.P. Gor’kov, P.A. Kalugin, Sov. Phys. JETP Lett. [**41**]{}, 253 (1985).
P.A. Lee, Phys. Rev. Lett. [**71**]{}, 1887 (1993).
C. Pepin, P.A. Lee, Phys. Rev. Lett. [**81**]{}, 2779 (1998).
I.M. Lifshitz, S.A. Gredescul, L.A. Pastur, Introduction to the Theory of Disordered Systems, Wiley, NY, 1988.
N.F. Mott, Adv. Phys. [**16**]{}, 49 (1967).
A.F. Ioffe, A.R. Regel, Prog. Semicond. [**4**]{}, 237 (1960).
V.L. Bonch-Bruevich, S.V. Tyablikov, The Green Function Method in Statistical Mechanics, North Holland, Amsterdam, 1962.
I.M. Lifshitz, Adv. Phys. [**13**]{}, 483 (1964).
M.A. Ivanov, V.M. Loktev, Yu.G. Pogorelov, Phys. Rep. [**153**]{}, 209 (1987).
T.L. Hill, Statistical Mechanics, Principles and Selected Applications, Dover, NY, 1987.
M.A. Ivanov, Sov. Phys. Sol. State [**12**]{}, 1508 (1971).
M.A. Ivanov, Yu.V. Skrypnyk, Phys. Sol. State [**36**]{}, 51 (1994).
Yu.G. Pogorelov, Solid State Commun. [**95**]{}, 245 (1995).
It can be shown that, besides these two, formally there are also other solutions, but all they fail to satisfy IRC.
M.A. Ivanov, Yu.G. Pogorelov. Zh. Eksp. Teor. Fiz. (in Russian), [**72**]{}, 2198 (1977).
V.M. Loktev, Yu.G. Pogorelov. Physica C [**272**]{}, 151 (1996).
V.M. Loktev, Yu.G. Pogorelov. Low. Temp. Phys. [**27**]{}, 1039 (2001).
A.C. Durst, P.A. Lee. Phys. Rev. B [**62**]{}, 1270 (2000).
|
---
abstract: 'Random band matrices relevant for open chaotic systems are introduced and studied. The scattering model based on such matrices may serve for the description of preequilibrium chaotic scattering. In the limit of a large number of open channels we calculate the average $S$-matrix and $S$-matrix’s pole distribution which are found to reduce to those of the full matrix (GOE) case under proper renormalization of the energy scale and strength of coupling to the continuum.'
address: ' Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia '
author:
- 'D.V. Savin [^1]'
date: ' November 28, 1995 '
title: |
Random band matrix approach to chaotic scattering:\
the average $S$-matrix and its pole distribution.
---
[**1.**]{} It is now generally acknowledged that the Random Matrix Theory (RMT) provides an adequate tool for the description of statistical properties of chaotic quantum systems [@Mehta]–[@Brody]. The most of works in the RMT deals with the Hermitian matrices which are, strictly speaking, appropriate for the description of closed systems. Meanwhile, any exited state of a quantum system has a finite lifetime, decaying eventually into open channels. It is well known [@MW-69]–[@SZ-89] that the openness of a system can be incorporated into consideration by means of the effective nonhermitian Hamiltonian $$\label{ham}
{\cal H}_{nm} = H_{nm} - i \sum_{c\,(open) } V^c_n V^c_m\ .$$ Due to coupling to the continuum, the effective Hamiltonian (\[ham\]) acquires, apart from the intrinsic part $H$, the antihermitian part which consists of the sum of products of the transitions amplitudes $V^c_n$ between $N$ internal $(|n\rangle)$ and $M$ open channel $(|c\rangle)$ states. For the T-invariant theory these amplitudes as well as $H$ can be chosen to be real, the matrix ${\cal H}$ being symmetric. The eigenvalues ${\cal E}_n=E_n-\frac{i}{2}\Gamma_n$ of ${\cal H}$ are the complex energy levels of an unstable system, with $E_n$ and $\Gamma_n$ being, respectively, the energy and width of the $n$-th level.
Assuming the intrinsic dynamics to be chaotic, the hermitian part of ${\cal H}$ is usually supposed to belong to the Gaussian Orthogonal Ensemble (GOE) of random matrices. As to the coupling amplitudes, they are considered to be either fixed [@W-84; @VWZ-85] or random [@SZ-89]. Resulting generalization of the RMT on the unstable systems proved [@W-84]–[@LSSS-95b] to be useful for the description of statistical properties of quantum chaotic scattering. The resonance part of the scattering matrix is represented (see, e.g. [@MW-69]) as $$\label{S}
S_{ab}(E) = \delta_{ab} -
2i \sum_{nm} V^a_n[(E-{\cal H})^{-1}]_{nm}V^b_m\ .$$ in terms of the effective Hamiltonian (\[ham\]) which describes the evolution of the unstable system formed at the intermediate stage of the collision. Therefore, fluctuations in scattering reflect statistical properties of the complex energy levels of this intermediate system.
Recently, the considerable progress has been achieved in the study of dynamical and statistical properties of $S$-matrix’s poles and their connection to those of scattering [@SZ-89] -[@LSSS-95b]. The limit of very large number of equivalent channels has a special interest since it can be related to the semiclassical approach [@LW-91; @LSSS-95a]. The noteworthy property of the pole distribution was found [@HILSS-92; @LSSS-95a] in this limit: there always exists a finite gap between the upper edge of the distribution of complex energies of resonances and the real energy axis. This gap turns out to be the important characteristic of local fluctuations in chaotic scattering [@LSSS-95a; @LSSS-95b].
The Gaussian distribution of the matrix elements of the internal Hamiltonian $H$ implies the invariance with respect to the choice of the intrinsic basis. However, the mean-field basis seems to play an exceptional role [@Z-93] in many-body chaotic systems. The realistic Hamiltonian matrix has a banded structure in such a representation [@FGGK-94; @ZHB-95]. In fact, Wigner was the first who considered band matrices in connection with properties of (stable) complicated systems [@W-55]. He assumed the nonzero matrix elements to be Gaussian random variables. Now, the theory of random band matrices (RBM) attracts a great interest and is claimed to be relevant for studying quantum chaos (for review see [@CC-95]). The most of known analytical results for RBM has been recently obtained by Fyodorov and Mirlin [@FM-91; @FM-95] who studied RBM in the context of the localization, having reduced the RBM problem to the Efetov’s supersymmetric $\sigma$-model [@E-83] with the diffusion constant proportional to the square of the bandwidth.
The purpose of this Letter is to extend the RBM approach to the consideration of open chaotic systems described by the effective Hamiltonian (\[ham\]). The scattering model based on such random matrices can be related to preequilibrium chaotic scattering in a sense intermediate between fully chaotic scattering described by the GOE model and multistep compound reactions [@NVWY-86]. In the “pure” GOE models all internal degrees of freedom being uniformly involved, the intermediate unstable system has already attained its complete thermodynamical equilibrium before a decay takes place. The band structure of $H$ leads to the localization of intrinsic wave functions, an intermediate decaying state concluding only a part of the degrees of freedom. Therefore, in addition to the decay time, the new diffusion time scale appears in the problem which characterizes the internal relaxation time. In this Letter we calculate the average $S$-matrix and $S$-matrix’s pole distribution in the limit of very large number of open equivalent channels.
[**2.**]{} We suppose that the hermitian part of (\[ham\]) belongs to the Gaussian RBM ensemble defined by $$\label{rbm}
\langle H_{nm}H_{nm}\rangle = \lambda^2 J_{nm}(1+\delta_{nm}) \ ,$$ where the function $J_{nm}\equiv J(|n-m|)$ decreases sufficiently fast then $|n-m|>b$, with $b\gg 1$ being the effective bandwidth. In the GOE case, when all matrix elements $J_{nm}$ are equal to $\frac{1}{N}$, $2 \lambda$ determines the radius of Wigner’s semicircle low of the eigenvalue distribution [@Mehta]. The transition amplitudes $V^a_n$ are considered as fixed quantities subject to the condition [@W-84; @VWZ-85] $$\label{ampl}
\sum_n V^a_nV^b_n = \gamma \lambda \delta^{ab} \ ,$$ which turns out to be enough for excluding direct reactions. The dimensionless parameter $\gamma$ characterizes the strength of coupling between the internal motion and channels.
We calculate the average $S$-matrix and distribution of complex energy levels in the limit $M, N\rightarrow\infty$, with $m=M/N<1$ fixed. It turns out that the band structure of the hermitean part $H$ does not lead to the essential complication when evaluating the one-point average characteristics and all calculations can be done in close analogy with those of the GOE case [@HILSS-92; @LSSS-95a]. Below we present rather short description, referring for details to [@HILSS-92; @LSSS-95a].
[**3.**]{} The calculation of the average $S$-matrix is related with that of the average Green function $$\label{green}
\langle {\cal G}(z) \rangle =
\Bigl\langle \left(\frac{1}{z-{\cal H}}\right) \Bigr\rangle \, ,
\ \ z=E+i0 \ ,$$ which governs the evolution of the intermediate unstable chaotic system. It is convenient to use the following representation for $\langle{\cal G}_{nm}(z)\rangle$ [@W-84] $$\begin{aligned}
\label{green-z}
& & \langle {\cal G}_{nm}(z) \rangle =
2\frac{\partial}{\partial I_{nm}}\langle\ln Z(z,I)\rangle\Big|_{I=0} \ , \\
& & Z(z,I)= \det(z - H + i VV^T - I)^{-\frac{1}{2}} \ . \nonumber\end{aligned}$$
To carry out averaging in (\[green-z\]) we use the replica method [@EA-75; @W-84]. The generating function $Z(z,I)$ can be represented as a multivariable Gaussian integral that makes averaging over the Gaussian RBM ensemble (\[rbm\]) trivial. The further integration can be performed by means of the saddle-point approximation. The saddle-point solution turns out to be stable in the replica space and proportional to $\delta_{nn'}$. Therefore, the replicas decouple and, as was pointed in [@HILSS-92], it is enough to calculate $\ln\langle Z(z,I) \rangle$.
One has $$\begin{aligned}
\label{z-aver}
\langle Z(z,I)\rangle \cong \int\!& d [\phi]
& \,\exp\Bigr\{ \sum_{n,m=1}^N \bigr[ - \frac{\lambda^2 }{4}
J_{nm}\phi^2_n \phi^2_m \nonumber \\
& + & \frac{i}{2}\phi_n(z+i VV^T-I)_{nm} \phi_m \bigl] \Bigl\} \ ,\end{aligned}$$ where $d[\phi]$ means the product of differentials and above equality is valid to irrelevant constant. To make the integration over $\phi$ doable, we introduce, following [@HILSS-92; @LSSS-95a], new variables $\sigma_n=\lambda\phi_n^2$ with the help of $\delta$-functions defined by their Fourier representation $$\label{delta}
\delta(\sigma_n-\lambda\phi_n^2) = \frac{1}{\pi}\int\!d\hat\sigma_n \exp
\{ - \frac{i}{2}(\sigma_n\hat\sigma_n-\lambda\phi_n\hat\sigma_n\phi_n) \}$$ After the Gaussian integration over $\phi$ being done, the subsequent integration can be performed in the saddle-point approximation justified by two large parameters $N,b \gg 1$. We find after some algebra that the average Green function is determined in the following way $$\label{green-l}
\langle {\cal G}_{nm}(z) \rangle = \bigl[(z + \lambda\hat\sigma_{s.p.}
+ i VV^T )^{-1}\bigr]_{nm}$$ by the solution of the saddle-point equations (with respect to $\sigma$ and $\hat\sigma$) for the “Lagrangian” ${\cal L}$ $$\begin{aligned}
\label{lagr}
{\cal L} &=& - \frac{1}{4}\sum_{nm}J_{nm}\sigma_n\sigma_m - \sum_n
(\frac{i}{2}\sigma_n\hat\sigma_n + \ln(z+\lambda\hat\sigma_n)) \nonumber \\
&\ & - \frac{1}{2}\mbox{tr}_{c}\ln [1+i V^T(z+\lambda\hat\sigma)^{-1}V] \ ,\end{aligned}$$ where the diagonal matrix $\hat\sigma=\mbox{diag}(\hat\sigma_1,\ldots,\hat\sigma_N)$. Note that the trace in the last term in (\[lagr\]) runs over the channel ($M$-dimensional) space.
Similar to the pure RBM case [@FM-91], the saddle point equations possess the translation invariant (independent of $n$) solution provided that the relation (\[ampl\]) is fulfilled. Going along the same line as in [@HILSS-92; @LSSS-95a], we arrive at $$\label{green-aver}
\langle{\cal G}_{nm}(z)\rangle =
\frac{ \delta_{nm}(z - J_0\lambda^2 g(z) + i\gamma\lambda)
- i \sum_c V^c_nV^c_m }{ (z-J_0\lambda^2 g(z) )
( z-J_0\lambda^2 g(z)+i\gamma\lambda) } \ .$$ Here the notation $J_0\equiv\sum_r J(|r|)$ is introduced. Due to the translation invariance mentioned above the average Green function depends on the only combination of transition amplitudes. This fact makes the average $S$-matrix be diagonal and equal to $$\label{s-aver}
\langle S^{ab}(z)\rangle = \delta^{ab}
\frac{z - J_0\lambda^2 g(z) - i\gamma\lambda}{
z - J_0\lambda^2 g(z) + i\gamma\lambda } \ .$$ The function $g(z)$ denotes the trace of the average Green function which is found to satisfy the cubic equation $$\label{cubic}
g(z)\bigl(J_0\lambda^2 g(z) - z \bigr) + 1
+ \frac{i m\gamma\lambda}{J_0\lambda^2 g(z) - z - i\gamma\lambda} = 0 \ ,$$ where the (unique) solution with a negative imaginary part has to be chosen [@LSSS-95a].
For a closed system ($\gamma=0$), this cubic equation is reduced to the quadratic one which determines the density of the RBM eigenvalues to be given by Wigner’s semicircle law [@KLH-91] with the half-radius $$\label{scale1}
\widetilde \lambda = \lambda\sqrt{J_0} \ ,$$ renormalized by the factor $\sqrt{J_0}$ (this factor reduces to unity in the case of the GOE). For an open system, additional rescaling of the coupling constant $$\label{scale2}
\widetilde \gamma = \frac{\gamma}{\sqrt{J_0}},$$ reduces eqs.(\[green-aver\])–(\[cubic\]) to corresponding ones [@SZ-92; @LSSS-95a] for the full matrix case.
[**4.**]{} The calculation carried out above is valid only for the upper half of the complex plane where the Green function is analytical, all $S$-matrix singularities being located into the lower part (see eq.(\[S\])). The analyticity is broken where the density of complex energies differs from zero. A special regularization procedure of the pole singularities has been proposed in [@HILSS-92] to calculate the average $S$-matrix’s pole distribution in the lower halfplane of the complex variable $z=x+iy$. Basing on a convenient electrostatic analogy [@SZ-89; @HILSS-92], the distribution of complex levels is considered to be the source of the two-dimensional electrostatic field with the potential $$\label{pot}
\Phi(x,y) = \frac{1}{N} \langle \ln\det \bigl\{
(z^*-{\cal H}^{\dagger})(z-{\cal H})+\delta \bigr\}^{-1} \rangle \ .$$ The limit $\delta\rightarrow 0^+$ should be taken at the very end of calculations. Applying the two-dimensional Laplacian to $\Phi$, one gets the average density of complex levels (“charges”) $$\label{dens}
4\pi\rho(x,y) = - \Delta \Phi(x,y) \ .$$
The replica trick can be used again for performing the ensemble averaging. The infinitesimal positive $\delta$ makes the matrix in the rhs of eq.(\[pot\]) positive definite for any $z$. Therefore, the determinant may be represented as the Gaussian integral over a complex $N$-vector $\psi(1)$. To make the ensemble averaging possible, the product $(z^*-{\cal H}^{\dagger})(z-{\cal H})$ is decoupled [@HILSS-92; @LSSS-95a] with the help of the complex Habbard-Stratonovich transformation ($N$-vector $\psi(2)$). One finally arrives at the integral representation $$\label{pot-l}
\exp\{ N\Phi \} \cong \int\!d[\psi]\,\bigl\langle\,
\exp\{i \psi^{\dagger} {\cal M} \psi' \} \bigr\rangle$$ where $2N\times 2N$ matrix ${\cal M}$ is defined as $${\cal M}=\left[ \begin{array}{cc}
z^{\ast}-{\cal H}^{\dagger} & i \delta \\
i & z-{\cal H} \end{array} \right] \ ,$$ introduced $2N$-vectors being $\psi^T=\bigl(\psi(1)^T,\psi(2)^T\bigr)$ and $\psi'^T=\bigl(\psi(2)^T,\psi(1)^T\bigr)$.
Performing the same steps as in [@HILSS-92; @LSSS-95a], we find the potential $N\Phi$ to be determined by the saddle-point value of the “Lagrangian” $$\begin{aligned}
\label{l-dens}
&& {\cal L} =
- \frac{1}{2}\sum_{nm} J_{nm}\mbox{tr}_{\alpha}(\sigma_n\sigma_m)
- i\sum_n \mbox{tr}_{\alpha}(\sigma_n\hat\sigma_n) \\
&& - \sum_n\mbox{tr}_{\alpha}\ln(z_{\delta}+\lambda\hat\sigma_n) \nonumber
- \mbox{tr}_{\alpha c} \ln \bigl[1 -
i V^T(z_{\delta}+\lambda\widehat\Sigma)^{-1}V l\bigr] \ .\end{aligned}$$ Each of $N$ matrices $\sigma_n$ in (\[l-dens\]) has the following structure $$\sigma=\left( \begin{array}{cc} w & u \\ v & w^{\ast} \end{array} \right)$$ with the real positive $u$ and $v$ whereas $\hat\sigma_n$ stands for its Fourier counterpart. We have also introduced $2N\times 2N$ block-diagonal matrices $z_{\delta}$ obtained from ${\cal M}$ by setting there $\cal H$ equal to zero and $\widehat\Sigma_{\alpha\beta\ nm}=(\hat\sigma_n)_{\alpha
\beta} \delta_{nm}$, the $2M\times 2M$ block-diagonal matrix $l$ is unity in the channel subspace and equals to $l=\mbox{diag}(1,-1)$ in the replica subspace, and $V$ is the unit matrix for replica indices and corresponds to $V^{c}_{n}$ for others. In the saddle-point we have the translation invariant saddle-point equations $$\begin{aligned}
\label{saddle}
&& \hat\sigma=iJ_0\sigma \nonumber \\
&& \frac{i\sigma}{\lambda}(z_{\delta} + i\lambda J_0 \sigma ) + 1
- i\frac{m\gamma\lambda l}{ z_{\delta}
+ i\lambda J_0 \sigma - i\gamma\lambda l} = 0\end{aligned}$$ which differ from the corresponding equations for the full matrix case only in appearing the renormalizing factor $J_0$. Therefore, the explicit solution can be found in our case in the same way as it has been done in [@LSSS-95a]. As a result, one concludes that all poles (charges) lie in the finite domain of the lower part ($y<0$) of the complex energy plane defined by the condition $x^2 \leq {\sf x}^2(y)$ with $$\label{bound}
{\sf x}^2(y) =
- \frac{4m\gamma\lambda^3 J_0}{y} - \Bigl[ \frac{m\lambda^2 J_0}{y}
+ \frac{1-m}{\gamma\lambda + y}\lambda^2 J_0 - \gamma\lambda \Bigl]^2\ .$$ Inside this region the density of complex energy levels is equal to $$\label{dens-aver}
4\pi\rho(x,y) = \frac{m}{y^2} + \frac{1-m}{(\gamma\lambda+y)^2}
- \frac{1}{J_0\lambda^2} \ .$$ One can easily see that under rescaling (\[scale1\]),(\[scale2\]) the average complex level distribution reduces again to that [@LSSS-95a] of the full matrix case.
In conclusion, the band structure of the hermitean part $H$ results only in the renormalization of the energy scale $\lambda$ (\[scale1\]) and coupling constant $\gamma$ (\[scale2\]) as compared to the GOE model [@LSSS-95a]. In particular, the condition of the “width collectivization” [@SZ-89; @SZ-92], $\widetilde\gamma \sim 1$, implies again the natural physical condition of the average partial width being comparable with the average level spacing [@SZ-92]. One should expect nontrivial consequences of the band structure to appear only while considering the higher correlation functions.
I am indebted to V.V. Sokolov for bringing my interest to the subject, critical reading of the manuscript, and illuminating discussions. The partial financial support from the International Science Foundation (grants RB7000 and RB7300) and INTAS (grant 94-2058) is acknowledged with thanks.
[99]{}
M.L. Mehta, [*Random Matrices*]{}, (Academic Press, NY, 1991).
C.E. Porter, [*Statistical Theories of Spectra: Fluctuations*]{}, (Academic Press, NY, 1965).
T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Rev. Mod. Phys. [53]{} (1981) 385.
C. Mahaux and H.A. Weidenmüller, [*Shell-model Approach to Nuclear Reactions*]{}, (North-Holland, Amsterdam, 1969).
H.A. Weidenmüller, Ann. of Phys. [158]{} (1984) 120.
J.J.M. Verbaarschot, H.A. Weidenmüller, and M.R. Zirnbauer, Phys. Rep. [129]{} (1985) 367.
V.V. Sokolov and V.G. Zelevinsky, Phys. Lett. [B202]{} (1988) 140; Nucl. Phys. [A504]{} (1989) 562.
F. Haake, F.M. Izrailev, N. Lehmann, D. Saher, and H.-J. Sommers, Z. Phys. [B88]{} (1992) 359.
N. Lehmann, D. Saher, V.V. Sokolov, and H.-J. Sommers, Nucl. Phys. [A582]{} (1995) 223.
N. Lehmann, D.V. Savin, V.V. Sokolov, and H.-J. Sommers, Physica D [86]{} (1995) 572.
C.H. Lewenkopf and H.A. Weidenmüller, Ann. Phys. (N.Y.) [212]{} (1991) 53.
V.G. Zelevinsky, Nucl. Phys. [A555]{} (1993) 109.
V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, and M.G. Kozlov, Phys. Rev. [A50]{} (1994) 267.
V. Zelevinsky, M. Horoi, and B.A. Brown, Phys. Lett. [B350]{} (1995) 141; M. Horoi, V. Zelevinsky, and B.A. Brown, Phys. Rev. Lett. [74]{} (1995) 5194.
E. Wigner, Ann. Math. [62]{} (1955) 548; ibid [65]{} (1957) 203.
G. Casati and B.V. Chirikov, eds., [*Quantum Chaos*]{}, (Cambridge Univ. Press, Cambridge, 1995).
Y.V. Fyodorov and A.M. Mirlin, Phys. Rev. Lett. [67]{} (1991) 2405; ibid [69]{} (1992) 1093; ibid [71]{} (1993) 412;
Y.V. Fyodorov and A.M. Mirlin, Int. J. Mod. Phys. [8]{} (1994) 3795.
K.B. Efetov, Advan. in Phys. [32]{} (1983) 53.
H. Nishioka, J.J.M. Verbaarschot, H.A. Weidenmüller, and S. Yoshida, Ann. Phys. [172]{} (1986) 67.
S.F. Edwards and P.W. Anderson, J. Phys. F[5]{} (1975) 965.
M. Kus, M. Lewenstein, and F. Haake, Phys. Rev. [A44]{} (1991) 2800.
V.V. Sokolov and V.G. Zelevinsky, Ann. Phys. [216]{} (1992) 323.
[^1]: e-mail: savin@inp.nsk.su
|
---
abstract: 'Each acyclic graph, and more generally, each acyclic orientation of the graph associated to a Cartan matrix, allows to define a so-called frise; this is a collection of sequences over $\mathbf N$, one for each vertex of the graph. We prove that if these sequences satisfy a linear recurrence, then the Cartan matrix is of Dynkin type (if the sequences are bounded) or of Euclidean type (if the sequences are unbounded). We prove the converse in all cases, except for the exceptional Euclidean Cartan matrices; we show even that the sequences are rational over the semiring $\mathbf N$. We generalize these results by considering frises with variables; as a byproduct we obtain, for the cases case $A_m$ and $\tilde A_m$, explicit formulas for the cluster variables, over the semiring of Laurent polynomials over $\mathbf N$ generated by the initial variables (which explains simultaneously positivity and the Laurent phenomenon). The general tool are the so-called $SL_2$-tilings of the plane; these are fillings of the whole discrete plane by elements of a ring, in such a way that each $2\times 2$ connected submatrix is of determinant 1.'
author:
- Ibrahim Assem
- Christophe Reutenauer
- David Smith
title: Frises
---
Introduction
============
To each acyclic quiver (that is, to each acyclic directed graph) and more generally, to each Cartan matrix), we associate its so-called [*frise*]{}. This construction is motivated by the theory of cluster algebras of Fomin-Zelevinsky, and was shown to two of the authors by Philippe Caldero some years ago. For each such graph, this construction gives sequences of positive rational numbers, one for each vertex of the graph. Actually, the Laurent phenomenon of Fomin-Zelevinsky implies that these sequences are integer-valued.
The present work was motivated by a simple example, shown in Section 2: the quiver with two vertices and two edges from one to the other. Then the numbers which appears in the sequences of the frise are the Fibonacci numbers of even rank. This lead us to the question of determining those graphs for which the sequences are [*rational*]{} (that is, satisfy some linear recursion, as do the Fibonacci numbers). We conjecture that these graphs are exactly the Dynkin and Euclidean (also called affine) graphs, with some orientation.
We show in Theorem 1 that this conjecture holds in one direction: if a Cartan matrix gives rise to a frise with rational sequences, then this Cartan matrix is of Dynkin or Euclidean type. For this, we apply criteria of Vinberg and Berman-Moody-Wonenburger, which characterize these graphs combinatorially, through the so-called additive and subadditive functions. In order to obtain such functions, we take, very roughly speaking, the logarithm of the sequences of the frise (there are some technicalities, due to the fact that rational sequences do not grow very smoothly in general).
We prove the opposite direction of the conjecture for all but a finite number of cases (Theorem 2): for each Dynkin and for each non-exceptional Euclidean graph, the sequences of the frise are rational. Note that the Dynkin case is immediately settled as a consequence of the finite-type classification of cluster algebras by Fomin and Zelevinsky [@FZ2]. So only remains to prove the case of Euclidean graphs.
The rationality for the Euclidean graphs is proved by using a mathematical object, which seems interesting in itself: the [*$SL_2$-tilings*]{} of the plane. This is a filling of the whole discrete plane by elements of a commutative ring, in such a way that each 2 by 2 connected submatrix is of determinant 1. Note that an analogue mathematical object has been already considered in the literature: the so-called “frieze patterns” in [@Co], [@Coco1], [@Coco2]. In our settings, the latter would be called partial $SL_2$-tilings, since they do not cover the whole plane.
We give a way to construct $SL_2$-tilings, where the whole plane is filled with natural integers. As a corollary, we obtain Theorem 2 for the case $\tilde A_m$. The case $\tilde D_m$ is more involved, since we have to study special tilings containing perfect squares. The other Euclidean non-exceptional cases may be reduced to the two previous ones.
At the end of the article, we generalize all this by considering frises and $SL_2$-tilings with variables. This is directly motivated by the theory of cluster algebras, and particularly by the search for formulas expressing cluster variables.
Recall that cluster algebras were introduced by Fomin and Zelevinsky [@FZ1], [@FZ2] in order to explain the connection between the canonical basis of a quantised enveloping algebra and total positivity for algebraic groups. Since then, they turned out to have important ramifications in several fields of mathematics. Roughly speaking, a cluster algebra is an integral domain with a possibly infinite family of distinguished generators (called cluster variables) grouped into (overlapping) clusters of the same finite cardinality and computed recursively from an initial cluster. By construction, every cluster variable can be uniquely expressed as a rational function of the elements of any given cluster. The Laurent phenomenon, established in [@FZ1], asserts that these rational functions are in fact Laurent polynomials with integral coefficients. Moreover, it was conjectured by Fomin and Zelevinsky that these coefficients are positive. This latter conjecture is known as the positivity conjecture.
Our motivation in the last part of the present article was to develop computational tools to derive direct formulas for the cluster variables without going through the recursive process. We give explicit and simple formulas, involving only matrix products, for all cluster variables (or all but finitely many cluster variables) for cluster algebras without coefficients of Dynkin type $A_m$ (or Euclidean type $\tilde A_m$, respectively), explaining simultaneously the Laurent phenomenon and the positivity for initial acyclic clusters.
It has to be noted that the positivity conjecture, and direct formulas for computing the cluster variables, have been obtained in special cases only (see, for instance, [@FZ2; @CZ; @MS; @S; @S1; @CR; @CK; @MSW; @MuPro]). In particular, in [@CK; @CR], the positivity conjecture was established for cluster algebras having no coefficients and an acyclic initial cluster, by using Euler-Poincaré characteristics of appropriate Grassmannians of quiver representations. Note also that summation formulas (where the sum is over perfect matchings of certain graphs) for cluster variables appear in [@MSW], over the semiring of Laurent polynomials over $\mathbf N$; these formulas are valid for a wide class of cluster algebras.
Our paper provides, through a novel approach leading to much simpler formulas, a new elementary proof of the positivity conjecture for all Dynkin diagrams of type $A$ and all Euclidean diagrams of type $A$. Our approach uses, instead of summation formulas, only products of 2 by 2 matrices over the Laurent polynomial semiring over $\mathbf N$.
For type $A_n$, we use partial $SL_2$-tilings, that are equivalent to the frieze patterns of Coxeter and Conway [@Co; @Coco1; @Coco2]. Their importance for cluster algebras of this type has already been noted by Caldero and Chapoton [@Cacha], see also [@BaMaTho]. Our approach provides matrix product formulas to compute them, which seem to be new (however, see [@Pro], where a similar matrix approach is used to compute frieze patterns giving the Markoff numbers).
For details about cluster algebras and cluster categories, we refer the reader to [@FZ1; @FZ2; @BMRRT], and for information concerning linear recurrences and rational series, we refer to [@EPSW; @BR].
[**Acknowledgments**]{}: the authors thank Anissa Amroun for computer calculations which helped to determine many frises, and Vestislav Apostolov, François Bergeron, Yann Bugeaud, Christophe Hohlweg, Ralf Schiffler and Lauren Williams for discussions and mail exchanges which helped to clarify several points leading to the present article. Special thanks to Gregg Musiker, who indicated to us the notion of frieze patterns in the literature.
An introductory example
=======================
Given a quiver $Q$ (in other words, a directed graph with possibly multiple edges), which we assume to be acyclic, let $V$ be its set of vertices. Define for each $v$ in $V$ a sequence $v(n)$ by the initial condition $v(0)=1$ and the recursion $$v(n+1)=\frac{1}{v(n)} (1+\prod_{v\rightarrow w}w(n) \prod_{w\rightarrow v}w(n+1)).$$ The fact that these equations define uniquely the sequences $v(n)$ follows from the acyclicity of the graph. The previous recursion formula may be represented by defining the [*frise* ]{} associated to the quiver: it is an infinite graph with set of vertices $V\times \mathbb N$ and edges $(v,n)\rightarrow (w,n)$ if $v\rightarrow w$ is in $Q$, and edges $(v,n)\rightarrow (w,n+1)$ if $v\leftarrow w$ in $Q$. Then the sequence $v(n)$ labels the vertex $(v,n)$ of the frise, $l(v,n)=v(n)$ say, and the recursion reads $$l(v,n+1)=\frac{1}{l(v,n)}(1+\prod_{(w,i)\rightarrow (v,n+1)} l(w,i)),$$ with the initial conditions $l(v,0)=1$. Note that only $i=n$ or $i=n+1$ may occur in the product. As an example take the [ Kronecker quiver]{}, with two vertices and two edges from one to the other. The frise is represented in Figure \[Kronecker\], together with the labels.
It turns out that the numbers $1,1,2,5,13,34,...$ are the Fibonacci numbers of even rank $F_{2n}$, if one defines $F_0=F_1=1$ and $F_{n+2}=F_{n+1}+F_n$. This may be proved for instance by using the identity $$\left | \begin{array}{ll} F_{2n+4} & F_{2n+2}\\ F_{2n+2}& F_{2n} \end{array} \right | = 1,$$ which is a consequence of the fact that the Fibonacci numbers of even rank satisfy the recursion $F_{2n+4}=3F_{2n+2}-F_{2n}$, as is well-known.
Actually, as mentioned before, we shall also deal with more general frises: the initial values $v(0)$ will be variables. In the example of the Kronecker quiver, we may take $u_0=a$, $u_1=b$ and the recursion $u_{n+2}=\frac{1+u_{n+1}^2}{u_{n}}$, which shortcuts the frise: $u_0,u_2,u_4,\dots$ label the vertices $(v,0),(v,1),(v,2),\dots$ and $u_1,u_3, u_5\dots$ the vertices $(w,0),(w,1),(w,2)\dots$. Then one has also a linear recursion, generalizing the recursion for Fibonacci numbers of even rank: $$u_{n+2}=\frac{a^2+b^2+1}{ab}u_{n+1}-u_n.$$ Moreover, $u_n=\frac{1}{a^{n-2}b^{n-1}}(1,b)M^{n-2}\left ( \begin{array}{l} 1 \\ b\end{array} \right )$, where $$M=\left ( \begin{array}{lc}a^2+1 & b \\ b&b^2\end{array} \right ).$$ A summation formula for $u_n$ has already been given by Caldero and Zelevinsky [@CZ] Th.4.1.
For quivers of type $A_m$ and $\tilde A_m$, we shall obtain these kinds of formulas, which explain simultaneously the Laurent phenomenon (the denominator is a monomial) of Fomin and Zelevinsky, and the positivity of the formulas.
Frises associated to Cartan matrices
====================================
Recall that a [*Cartan matrix*]{} $C=(C_{ij})_{1\leq i,j\leq d}$ is defined by the following properties:
\(i) $C_{ij}\in \mathbf Z$;
\(ii) $C_{ii}=2$;
\(iii) $C_{ij}\leq 0$, if $i\neq j$;
\(iv) $C_{ij} \ne 0 \Leftrightarrow C_{ji}\ne 0$.
The [*simple graph associated to*]{} $C$ has set of vertices $\{1,\dots,d\}$ and an undirected edge $\{i,j\}$ if $i\neq j$ and $C_{ij}\neq 0$. The Cartan matrix is completely described by its [*diagram*]{}, which is the previous graph, with [*valuations*]{} on it; the couple $(|C_{ij}|,|C_{ji}|)$ is represented on the edge $\{i,j\}$ as follows:
If $|C_{ij}|=1$, it is omitted. In this manner, Cartan matrices are equivalent to diagrams.
Without loss of generality, we consider only [*connected*]{} Cartan matrices; this means that the underlying graph is connected. Consider some fixed acyclic orientation of this graph; if the edge $\{i,j\}$ of this graph is oriented from $i$ to $j$, we write $i\rightarrow j$. For each $j=1,...,d$, we define a sequence $a(j,n)$, $n\in \mathbf N$, by the formula, for all $j=1,...,d$:
$$\label{frises_induction}
\, a(j,n)a(j,n+1)=1+( \prod_{j\rightarrow i} a(i,n)^{|C_{ij}|})(\prod_{i\rightarrow j} a(i,n+1)^{|C_{ij}|} )$$
and the initial conditions $a(j,0)=1$. The data of these $d$ sequences is called the [*frise*]{} associated to the Cartan matrix and the given acyclic orientation of its graph. This generalizes the construction of Section 2. The acyclicity of the orientation ensures that these sequences are well-defined. Moreover, they have clearly coefficients in $\mathbf Q_+^*$. Now, it is a consequence of the Laurent phenomenon of Fomin and Zelevinsky that the coefficients are actually positive integers; see [@FZ]; see also [@K].
Recall that a sequence $(a_n)_{n\in \mathbf N}$ of complex numbers [*satisfies a linear recurrence*]{} if for some $k\geq 1$, some $\alpha_1,...,\alpha_k$ in $\mathbf C$, one has: for all $n$ in $\mathbf N$, $a_{n+k}=\alpha_1 a_{n+k-1}+...+\alpha_k a_n$. Equivalently, the series $\sum_{n\in \mathbf N} a_nx^n$ is [*rational*]{}, that is, it is the quotient of two polynomials in $\mathbf C[x]$; we say also that the sequence $(a_n)$ is rational. We say that a frise is [*rational*]{} if the sequences $a(i,n)$ are all rational for $i=1,...,d$.
**Main conjecture**. \[conjecture\] [*A frise associated to a Cartan matrix with some acyclic orientation is rational if and only if the Cartan matrix is of Dynkin or Euclidean type.*]{}
These Cartan matrices are recalled in Section 9.
One direction of the conjecture is completely solved by the following result.
\[theorem1\] Let $C$ be a connected Cartan matrix with some acyclic orientation. Suppose that the associated frise is rational. If the sequences are all bounded, then $C$ is of Dynkin type. If they are not all bounded, then $C$ is of Euclidean type.
Note that since we assume that the Cartan matrix is connected, the sequences are all simultaneously bounded or unbounded. We prove Theorem 1 in Section 4.
We prove the opposite direction of the conjecture in all cases, except for the exceptional Euclidean cases. Actually, we prove more than rationality of the sequences. For this, recall that a series $S=\sum_{n\in \mathbf N} a_n x^n \in \mathbf N[[x]]$ is called [*$\mathbf N$-rational*]{} if it satisfies one of the two equivalent conditions (this equivalence is a particular case of the Kleene-Schützenberger theorem, see [@BR]):
\(i) $S$ belongs to the smallest subsemiring of $\mathbf N[[x]]$ closed under the operation $T\rightarrow T^*=\sum_{n\in \mathbf N} T^n$ (which is defined if $T$ has zero constant term);
\(ii) for some matrices $\lambda \in \mathbf N^{1\times d}$, $M\in \mathbf N^{d\times d}$, $\gamma \in \mathbf N^{d\times 1}$, one has: for all $n$ in $\mathbf N$, $a_n=\lambda M^n \gamma$.
We then say that the sequence $(a_n)$ is $\mathbf N$-rational.
A result which we need later in this article is that $\mathbf N$-rational sequences are closed under Hadamard product; this is the coefficientwise product $((a_n),(b_n))\mapsto(a_nb_n)$. This follows easily from (ii) by tensoring the matrices. Note that if $K$ is any commutative semiring, we may define $K$-rational series in exactly the same way, by replacing above $\mathbf N$ by $K$. See [@BR].
\[theorem2\] If a Cartan matrix is of Dynkin or Euclidean type, but not one of the exceptional Euclidean types, then the sequences of each associated frise are $\mathbf N$-rational, and in particular, rational.
Theorem 2 will be proved in Section 7. Note that in the Dynkin case, this is an immediate consequence of the finiteness of the set of cluster variables, proved in [@FZ2]. Indeed, the construction of the frise is a particular case of the mutations of Fomin and Zelevinsky, by performing mutations only on sources. See for example [@K].
In order to prove Theorem 2, we introduce objects which we call [*$SL_2$-tiling of the plane*]{}. This is a mapping $t:\mathbf Z^2\mapsto \mathbf N$ such that for any $i,j$ in $\mathbf Z$, $$\left | \begin{array}{ll} t(i,j) & t(i,j+1)\\ t(i+1,j) & t(i+1,j+1) \end{array} \right | = 1.$$ An example is given below; the 1’s are boldfaced, since they will play a special role in the sequel; we have not represented the numbers above them. The perfect squares in the tiling will be explained further in the article: see Lemma 4. $$\begin{array}{lllllllllllllll}
&&&&&&&&&&\bf 1&\bf 1&\bf 1&\bf 1& \\
&&&&&&&&&\bf 1&\bf 1&2&3&4& \\
&&&&&&&&&\bf 1&2&5&8&11& \\
&&&&&&&&&\bf 1&3&8&13&18& \\
&&&&...&&...&&\bf 1&\bf 1&2^2&11&18&25& ...\\
&&&&&&&&\bf 1&2&9&5^2&41&57& \\
&&&&&&&&\bf1&3&14&39&8^2&89& \\
& & &\bf 1&\bf 1&\bf 1&\bf 1&\bf 1&\bf 1&4&19&53&87&11^2& \\
\bf 1&\bf 1&\bf 1&\bf 1&2&3&4&5&6&5^2&119&332&545&758& \\
\bf 1&2&3&4&9&14&19&24&29&121&24^2&1607&2368&3669&... \\
&&&&&...&&...&&&&...&&& \\
\end{array}$$ We call [*frontier*]{} a bi-infinite sequence $$\label{frontier}
\ldots x_{-3}x_{-2}x_{-1}x_0x_1x_2x_3 \ldots$$ with $x_i\in \{x,y\}$. It is called [*admissible*]{} if there are arbitrarily large and arbitrarily small $i$’s such that $x_i=x$, and arbitrarily large and arbitrarily small $j$’s such that $x_j=y$; in other words, none of the two sequences $(x_n)_{n\geq0}$ and $(x_n)_{n\leq0}$ is ultimately constant. Each frontier may be embedded into the plane: the $x$ (resp. $y$) determine the horizontal (resp. vertical) edges of a bi-infinite discrete path: $x$ (resp. $y$) corresponds to a segment of the form $[(a,b),(a+1,b)]$ (resp $[(a,b),(a,b+1)]$). The vertices of the path (that is, the endpoints of the previous segments) get the label 1.
An example is the frontier $$\ldots yxxxyxxxxxyyyxyyyxyxxx\ldots$$ corresponding to the 1’s in the above $SL_2$-tiling.
We prove below that an admissible frontier, embedded into the plane, may be extended to a unique $SL_2$-tiling. For this, we need the following notation. Let $$M(x)=\left(\begin{array}{cc}1&1\\0&1\end{array}\right) \quad \mbox{and} \quad M(y)=\left(\begin{array}{cc}1&0\\1&1\end{array}\right).$$
We extend $M$ into a homomorphism from the free monoid generated by $x$ and $y$ into the group $SL_2(\mathbf Z)$.
Given an admissible frontier, embedded in the plane as explained previously, let $(u,v)\in \mathbf Z^2$. Then we obtain a finite word, which is a factor of the frontier, by projecting the point $(u,v)$ horizontally and vertically onto the frontier. We call this word the [*word*]{} of $(u,v)$. It is illustrated in the figure below, where the word of the point $M$ is $yyyxxyx$: $$\begin{array}{ccccccccccccccccc}
&&&&&&&&&&&&. \\
&&&&&&&&&&&. \\
&&&&&&&&&&. \\
&&&&&&&\bf 1&\bf 1&\bf 1 \\
&&&&&\bf 1&\bf 1&\bf 1&|\\
&&&&&\bf 1&&&|\\
&&&&&\bf 1&&&|\\
&&&&\bf 1&\bf 1&-&-&M\\
&&&. \\
&&. \\
&. \\
\end{array}$$ Note that such a word always begins by $y$ and ends by $x$. We define the word of a point only for points below the frontier; for points above, the situation is symmetric and we omit it.
Given an admissible frontier, there exists a unique $SL_2$-tiling of the plane $t$ extending the embedding of the frontier into the plane. It is defined, for any point $(u,v)$ below the frontier, with associated word $x_1x_2...x_{n+1}$, where $n\geq 1$, $x_i\in\{x,y\}$, by the formula $$\label{tilingformula}
t(u,v)=(1,1)M(x_2)\cdots M(x_n)\left(\begin{array}{c}1\\1\end{array}\right).$$
Theorem 3 will be proved in Section 5. In the formula, note that the first and last letter of the word are omitted. An instance of the formula, for the tiling above, is $$14=(1,1)\left(\begin{array}{cc}1&0\\1&1\end{array}\right) \left(\begin{array}{cc}1&1\\0&1\end{array}\right) \left(\begin{array}{cc}1&0\\1&1\end{array}\right)^3 \left(\begin{array}{c}1\\1\end{array}\right),$$ since the word corresponding to 14 in the figure is $y^2xy^3x$.
Theorem 3 will be further generalized, by considering tilings with variables. Indeed, in Section 8, we consider frontiers with variables, instead of 1’s as in Theorem 3. Then we show that one obtains an $SL_2$-tiling whose values lie in the semiring of Laurent polynomials over $\mathbf N$ in these variables.
Proof of Theorem 1
==================
Given two sequences of positive real numbers $(a_k)$ and $(b_k)$, we shall write $a_k\approx b_k$ to express the fact that for some positive constant $C$, one has $\lim_{k\rightarrow \infty}a_k/b_k=C$.
\[asymptotic\] Let $a(j,n)$, for $j=1,...,d$, be $d$ unbounded sequences of positive integers, each satisfying a linear recurrence. There exist an integer $p\geq 1$, real numbers $\lambda(j,l)\geq 1$ and integers $d(j,l)\geq 0$, for $j=1,...,d$ and $l=0,...,p$, and a strictly increasing sequence $(n_k)_{k\in \mathbf N}$ of nonnegative integers, such that:
\(i) for every $j=1,...,d$ and every $l=0,...,p, a(j,pn_k+l)\approx\lambda(j,l)^{n_k} n_k^{d(j,l)}$;
\(ii) for every $j=1,...,d$, there exists $l=0,...,p$ such that $\lambda(j,l)>1$ or $d(j,l)\geq 1$;
\(iii) for every $j=1,...,d$, $\lambda (j,0)=\lambda(j,p)$ and $d(j,0)=d(j,p)$.
This lemma may be well-known to the specialist of linear recurrent sequences. Since we could not find a precise reference, we give a proof below.
*Proof.*
Step 1. Recall that each sequence $(a_n)_{n\in \mathbf N}$ satisfying a linear recurrence has a unique expression, called the [*exponential polynomial*]{}, of the form $$\label{exponential_polynomial}
a_n=\sum_{i=1}^{k} P_i(n)\lambda_i^n,$$ for $n$ large enough, where $P_i(n)$ is a nonzero polynomial in $n$ and the $\lambda_i$ are distinct nonzero complex numbers; see e.g. [@BR] or [@EPSW]. We call the $\lambda_i$ the [*eigenvalues*]{} of the sequence $a_n$. The [*degree*]{} of $\lambda_i$ is $deg(P_i)$. Note that if the $a_n$ are all positive integers, then at least one of its eigenvalues has modulus $|\lambda_i|\geq 1$. The [*principal part*]{} of the above exponential polynomial is $$\label{principal_part}
n^D\sum_{j}\alpha_j\lambda_j^n,$$ where the sum is restricted to those $j$ with $|\lambda_j|$ maximum, $D=deg(P_j)$ maximum for these $j$, and $\alpha_j$ is the coefficient of $n^D$ in $P_j$. We call these $\lambda_j$ the [*dominating eigenvalues*]{}, [*dominating modulus*]{} their modulus and $D$ the [*maximum degree*]{}. Note that if $\lambda_i$ is not a dominating eigenvalue, then either $\lambda_i$ has modulus strictly smaller than the maximum modulus, or its modulus is the maximum modulus, but its degree is strictly smaller than $D$. For further use, note that if Eq.(\[principal\_part\]) is the principal part of $(a_n)$, then the principal part of the sequence $a_{n+H}$ is $$\label{principal_part+H}
n^D\sum_{j}\alpha_j\lambda_j^H\lambda_j^n.$$
Note also that for any $p\geq 1$ and $l\geq 0$, the eigenvalues of the sequence $(a_{np+l})_{n\in\mathbf N}$ are $p$-th powers of eigenvalues of $(a_n)$: it suffices, to see it, to replace $n$ by $np+l$ in Eq. (\[exponential\_polynomial\]).
Step 2. Consider the subgroup $G$ of $\mathbf C^*$ generated by the eigenvalues of the $d$ sequences $a(j,n)$. It is a finitely generated abelian group and therefore, by the fundamental theorem of finitely generated abelian groups, there exists $p\geq 1$ such that the subgroup $G_1$ generated by the $p$-th powers of any set of elements of $G$ is a free abelian group.
Consider the sequence $(a(j,pn+l))_{n\in\mathbf N}$, with $j=1,...,d$ and $l=0,...,p$. The eigenvalues of the sequence $(a(j,pn+l))_{n\in\mathbf N}$ are $p$-th powers of the eigenvalues of $a_n$ (this follows easily from Eq.(\[exponential\_polynomial\])). Denote by $Z_{j,l}$ the set of its dominating eigenvalues and $$Z=\bigcup _{1\geq j\geq d, 0\geq l\geq p} Z_{j,l}.$$ Then $Z\subset G_1$. In particular, no quotient $z/z'$ with $z,z'\in Z$, is a nontrivial root of unity, since $G_1$ is a torsion-free group. Note also that $Z_{j,l}$ is nonempty, since the sequences $a(j,pn+l)$ are positive.
Step 3. Suppose that we have proved the lemma for the $d$ sequences $a'(j,n)=a(j,n+ph)$ for some nonnegative integer $h$. Denote by $\lambda(j,l), d(j,l), (n_k)$ the corresponding numbers and sequences. Then we have $a'(j,pn_k+l)\approx \lambda(j,l)^{n_k} n_k^{d(j,l)}$. Thus $a(j,p(n_k+h)+l)\approx \lambda(j,l)^{n_k} n_k^{d(j,l)}$. This shows that the lemma then holds for the $d$ sequences $a(j,n)$: we simply replace $(n_k)_{k\in \mathbf N}$ by $(n_k+h)_{k\in \mathbf N}$. We choose $h$ below.
Step 4. The principal part of the sequence $a(j,pn+l)$ is of the form $$n^{d(j,l)}\sum_{z\in Z_{j,l}} \alpha_z z^n. \label{principal_part_a(j,pn+l)}$$ Consider the sequence $(\sum_{z\in Z_{j,l}} \alpha_z z^n)_{n\in\mathbf N}$. Since no quotient of distinct elements of $Z_{j,l}$ is a root of unity, we see by the theorem of Skolem-Mahler-Lech (see [@BR] Th.4.1 or [@EPSW] Th.2.1 ) that the previous sequence has only finitely many zeros. Hence, for some $h$, there is no zero for $n\ge h$. We may choose the same $h$ for each $j=1,...,d$ and $l=0,...,p.$
By Step 3 and Eq.(\[principal\_part+H\]) with $H=ph$, we may therefore assume that the principal part of $a(j,pn+l)$ is Eq. (\[principal\_part\_a(j,pn+l)\]), with $\sum_{z\in Z_{j,l}} \alpha_z \neq 0$.
Step 5. Choose some $z(j,l)$ in $Z_{j,l}$. The complex numbers $z/z(j,l)$, for $z\in Z_{j,l}$, $j=1,...,d$ and $l=0,...,p$ have modulus 1 and generate a subroup of $G_1$, which by Step 2 is a finitely generated free abelian subgroup of $\mathbf C^*$.
Observe that if a finite set $E$ of complex numbers of modulus 1 generates a free abelian subgroup of $\mathbf C^*$, then there is a strictly increasing sequence $(n_k)_{k\in \mathbf N}$ of nonnegative integers such that: for every $e\in E, \lim_{k\rightarrow \infty}e^{n_k}=1$. This follows from Kronecker’s simultaneous approximation theorem applied to a basis of the previous free abelian group (write the basis as $\exp(2i\pi x)$, for a finite set of $\mathbf Q$-linear independent real numbers $x$, see [@HW] Th. 442).
Thus we may assume the existence of $n_k$ with $z^{n_k}\sim_{k\rightarrow\infty} z(j,l)^{n_k}$ for any $j=1,...,d$ and $l=0,...,p$.
Step 6. Going back to the principal part Eq.( \[principal\_part\_a(j,pn+l)\]), we see that $$n_k^{d(j,l)} \sum_{z\in Z_{j,l}} \alpha_z z^{n_k}
\sim_{k\rightarrow \infty} n_k^{d(j,l)} z(j,l)^{n_k} \sum_{z\in Z_{j,l}} \alpha_z$$ since the last sum is nonzero.
Now, since the eigenvalues of $a(j,pn+l)$ which are not in $Z(j,l)$ have modulus strictly smaller than that of $z(j,l)$, or have the same modulus but smaller degree, and since $a(j,n)>0$, we obtain that $$a(j,pn_k+l)\approx\lambda(j,l)^{n_k} n_k^{d(j,l)},$$ with $\lambda(j,l)=| z(j,l)|$.
Step 7. In order to prove (ii), we use the fact that $a(j,n)$ is unbounded; hence, for any $j=1,...,d$, there exists $l=0,...,p-1$ such that $a(pn+l)$ is unbounded. Then at least one of its eigenvalues is of modulus $>1$, and $| z(j,l)|>1$, or otherwise, they have all modulus $\leq 1$ and some $d(j,l)$ must be $>1$.
For (iii), note that $a(j,p(n+1))=a(j,pn+p)$, hence the sequences $a(j,pn)$ and $a(j,pn+p)$ have the same eigenvalues and maximum degrees. Thus $ |z(j,0)|=|z(j,p)|$ and $d(j,0)=d(j,p)$.
$\square$
Before proving Theorem 1, we must recall some facts about additive and subadditive functions of diagrams. Let $C=(C_{ij})_{1\leq i,j\leq d}$ be a Cartan matrix. An [*additive*]{} (resp. [*subadditive*]{}) [*function for*]{} $C$ is a function $f: \{1,...,d\} \rightarrow \mathbf R_+^{ \ast}$ such that for any $j=1,...,d$, one has $2f(j)=\sum _{i\neq j} f(i)|C_{ij}|$ (resp. $2f(j)\geq \sum _{i\neq j} f(i)|C_{ij}|$). Note that, by the properties of a Cartan matrix, this may equivalently be rewritten as $\sum _{i}f(i)C_{ij} =0$ (resp. $\sum _{i}f(i)C_{ij} \geq 0$).
The results we need is the following.
[**Theorem**]{}\[additive\] [*A Cartan matrix $C$ is of Euclidean type if and only if there exists an additive function for $C$; it is of Dynkin type if and only if there exists a subadditive function for $C$ which is not additive.*]{}
The second part of this theorem is due to Vinberg [@V] and the first to Berman, Moody and Wonenburger [@BMW]. Both results were proved by Happel, Preiser and Ringel [@HPR], under the assumption that the function takes integer values, although this assumption was unnecessary. We need the generalization involving real valued functions, which thus holds by the proof of [@HPR], Theorem p.286.
The idea of the proof of Theorem 1 is as follows: we show first that if the sequences of the frise are rational and bounded, then there exists a subadditive function which is not additive for the diagram. Hence by the theorem above, the diagram is of Dynkin type. The subadditive function is obtained by multiplying $p$ consecutive values of each sequence, $p$ being a common period, and then taking their logarithm. The recurrence relations Eq.(\[frises\_induction\]) imply that this logarithm is a subadditive function which is not additive.
In the case where the sequences are unbounded, the proof is similar by replacing each sequence by its principal part. However, the proof is more technical, since the growth of a rational sequence is in general not of exponential type. Lemma \[asymptotic\] allows to bypass this difficulty.
**Proof of Theorem 1**
[**Case 1.**]{} The sequences $a(j,n)$, $j=1,...,d$, are all bounded. Since they are integer-valued, they take only finitely many values. Since they satisfy linear recursions, they are ultimately periodic. Let $p$ be a common period and let $n_0$ be such that each sequence is purely periodic for $n\geq n_0$.
Let $b(j)=\prod_{n_0\leq n <n_0+p} a(j,n)$. Note that $b(j)>1$. Indeed, if $a(j,n)=1$, then $a(j,n+1)>1$ by Eq.(\[frises\_induction\]); moreover, each $a(j,n)$ is a positive integer. We have, since $a(j,n_0)=a(j,n_0+p)$, $$\begin{aligned}
b(j)^2 &=& \left(\prod_{n_0\leq n <n_0+p} a(j,n)\right)\left( \prod_{n_0\leq n <n_0+p} a(j,n+1)\right) \\
&=& \prod_{n_0\leq n <n_0+p} a(j,n)a(j,n+1) \\
&=& \prod_{n_0\leq n <n_0+p} (1+(\prod_{j\rightarrow i} a(i,n)^{|C_{ij}|})(\prod_{i\rightarrow j} a(i,n+1)^{|C_{ij}|}))\end{aligned}$$ by Eq.(\[frises\_induction\]). Thus $$\begin{aligned}
b(j)^2 &>& \prod_{n_0\leq n <n_0+p} ((\prod_{j\rightarrow i} a(i,n)^{|C_{ij}|})(\prod_{i\rightarrow j} a(i,n+1)^{|C_{ij}|})) \\
&=& (\prod_{j\rightarrow i} \prod_{n_0\leq n <n_0+p} a(i,n))^{|C_{ij}|}) (\prod_{i\rightarrow j}\prod_{n_0\leq n <n_0+p} a(i,n+1))^{|C_{ij}|}) \\
&=& (\prod_{j\rightarrow i} b(i)^{|C_{ij}|}) (\prod_{i\rightarrow j} b(i)^{|C_{ij}|}) \\
&=& \prod _{i\neq j} b(i)^{|C_{ij}|} .\end{aligned}$$ Taking logarithms, we obtain $$2 log(b(j)) > \sum _{i\neq j} log(b(i)) |C_{ij}|$$ and we have a subadditive function which is not additive, since $b(j)>1$.
[**Case 2.**]{} We assume now that some sequence $a(j,n)$ is unbounded. Then by Eq. (\[frises\_induction\]) and the connectedness of the underlying graph of the Cartan matrix, they are all unbounded.
We show, by using Lemma \[asymptotic\], that there exists an additive function for the Cartan matrix. We use freely the notations of this lemma. Define $$b(j,n)=a(j,n)a(j,n+1)\cdots a(j,n+p-1).$$ Then
$$\label{bjpnk}
b(j,pn_k)\approx \lambda(j,0)^{n_k} n_k^{d(j,0)}\cdots \lambda(j,p-1)^{n_k} n_k^{d(j,p-1)}\approx \lambda(j)^{n_k} n_k^{d(j)}$$
where $\lambda(j)=\lambda(j,0)\cdots\lambda(j,p-1)$ and $d(j)=d(j,0)+\cdots+d(j,p-1)$. Now $$b(j,pn_k)^2=a(j,pn_k)a(j,pn_k+1)a(j,pn_k+1)a(j,pn_k+2)\cdots$$ $$\cdots a(j,pn_k+p-1)a(j,pn_k).$$ By the lemma, $a(j,pn_k)\approx a(j,pn_k+p)$. Thus $$b(j,pn_k)^2 \approx\prod_{0\leq l<p}a(j,pn_k+l)a(j,pn_k+l+1).$$ Using Eq. (\[frises\_induction\]), we obtain $$b(j,pn_k)^2\approx \prod_{0\leq l<p} (1+(\prod_{j\rightarrow i} a(i,pn_k+l)^{|C_{ij}|})(\prod_{i\rightarrow j} a(i,pn_k+l+1)^{|C_{ij}|})).$$ Let $$u_k=(\prod_{j\rightarrow i} a(i,pn_k+l)^{|C_{ij}|})(\prod_{i\rightarrow j} a(i,pn_k+l+1)^{|C_{ij}|}).$$ If $u_k$ is unbounded when $k\rightarrow \infty$, then by (i) in Lemma \[asymptotic\], there exists $i$ with: either $j\rightarrow i$, and $\lambda(i,l)>1$ or $d(i,l)\geq 1$; or $j\leftarrow i$, and $\lambda(i,l+1)>1$ or $d(i,l+1)\geq 1$. Then $\lim_{k\rightarrow\infty}u_k=\infty$ and $u_k\approx 1+u_k$. Otherwise, $u_k$ is bounded and by Lemma \[asymptotic\], $u_k$ is constant, therefore $u_k\approx 1+u_k$. Thus in both cases, $1+u_k\approx u_k$.
We deduce that $$b(j,pn_k)^2$$ $$\approx\prod_{0\leq l<p} (\prod_{j\rightarrow i} a(i,pn_k+l)^{|C_{ij}|})(\prod_{i\rightarrow j} a(i,pn_k+l+1)^{|C_{ij}|})$$ $$\approx\prod_{0\leq l<p}(\prod_{j\rightarrow i}\lambda(i,l)^{n_k|C{ij}|} n_k^{d(i,l)|C_{ij}|}) (\prod_{i\rightarrow j} \lambda(i,l+1)^{n_k|C{ij}|} n_k^{d(i,l+1)|C_{ij}|})$$ $$\approx(\prod_{j\rightarrow i} \prod_{0\leq l<p} \lambda(i,l)^{n_k|C{ij}|} n_k^{d(i,l)|C_{ij}|}) (\prod_{i\rightarrow j} \prod_{0\leq l<p} \lambda(i,l+1)^{n_k|C{ij}|} n_k^{d(i,l+1)|C_{ij}|}).$$ Since $d(i,0)=d(i,p)$ and $\lambda(i,0)=\lambda(i,p)$, we obtain $$\begin{aligned}
b(j,pn_k)^2 &\approx& (\prod_{j\rightarrow i} \lambda(i)^{n_k|C{ij}|} n_k^{d(i)|C_{ij}|}) (\prod_{i\rightarrow j} \lambda(i)^{n_k|C{ij}|} n_k^{d(i)|C_{ij}|}) \\
&\approx& \prod_{i\neq j}\lambda(i)^{n_k|C{ij}|} n_k^{d(i)|C_{ij}|}.\end{aligned}$$ Thus by Eq.(\[bjpnk\]), $$\lambda(j)^{2n_k}n_k^{2d(j)}\approx\prod_{i\neq j}\lambda(i)^{n_k|C{ij}|} n_k^{d(i)|C_{ij}|}.$$ Therefore, since $n_k$ tends to infinity with $k$, for $j=1,...,d$ $$\lambda(j)^2=\prod_{i\neq j}\lambda(i)^{|C{ij}|}$$ and $$2d(j)=\sum_{i\neq j}d(i)|C_{ij}|.$$ If the $d(j)$ are all positive, we have the additive function $d(j)$. If one of them is 0, then they are all 0, by connectedness of the graph and the above equation. In this case, $d(j,l)=0$ for any $j$ and $l$. Thus (ii) in the lemma ensures that for any $j$, some $\lambda(j,l)>1$ and therefore $\lambda(j)>1$. Taking logarithms, we find $$2log(\lambda(j))=\sum_{i\neq j}log(\lambda(i)))|C_{ij}|$$ and we have therefore an additive function.
$\square$
Proof of Theorem 3
==================
Step 1. We prove first that the function $t$ given by Eq.(\[tilingformula\]) is an $SL_2$-tiling of the plane. It is enough to show that for any $(u,v)\in \mathbf Z^2$, the determinant of the matrix $\left ( \begin{array}{ll} t(u,v) & t(u,v+1)\\ t(u+1,v) & t(u+1,v+1) \end{array} \right )$ is equal to 1.
By inspection of the figure below, where $k,l\ge 0$ and $w=x_1\cdots x_n$, $n\geq 0$ and $x_i\in\{x,y\}$, $$\begin{array}{cccccccccccc}
&&&&&&&&&&\bf 1&\bf 1 \\
&&&&&&&&&&. &| \\
&&&&&&&&&&. &| \\
&&&&&&&&&&. &| \\
&&&&&&&&&&\bf 1 &| \\
&&&&&&&&&\bf 1&\bf 1&| \\
&&&&&&&&.&&|&| \\
&&&&&&&w&&&|&| \\
&&&&&&.&&&&|&| \\
&&&&&\bf 1&&&&&|&| \\
\bf 1&.&.&.&\bf 1&\bf 1&-&-&-&-&(u,v)&(u,v+1) \\
\bf 1&-&-&-&-&-&-&-&-&-&(u+1,v)&(u+1,v+1)
\end{array}$$ it is seen that the words associated to the four points $(u,v)$, $(u,v+1)$, $(u+1,v)$ and $(u+1,v+1)$ are respectively of the form $ywx$, $ywxy^{l}x$, $yx^{k}ywx$ and $yx^{k}ywxy^{l}x$. Let $M=M(x_1\ldots x_n)$. Moreover, denote by $S(A)$ the sum of the coefficients of any matrix $A$. Then $t(u,v)=S(M)$, $t(u,v+1)=S(MM(x)M(y)^l)$, $t(u,v+1)=S(M(x)^kM(y)M)$ and moreover $t(u+1,v+1)=S(M(x)^kM(y)MM(x)M(y)^l)$. A straightforward computation, which uses the fact that $det(M)=1$, then shows that $t(u,v)t(u+1,v+1)-t(u,v+1)t(u+1,v)=1$.
Step 2. Clearly $t(u,v)> 0$ for any $(u,v)\in \mathbf Z^2$. Then it is easily deduced, by induction on the length of the word associated to $(u,v)$, that $t(u,v)$ is uniquely defined by the $SL_2$ condition. This proves that the tiling is unique.
$\square$
Properties of $SL_2$-tilings
============================
Rays and periodic frontiers
----------------------------
Given a mapping $t: \mathbf Z^2\rightarrow R$, a point $M\in \mathbf Z^2$ and a nonzero vector $v\in \mathbf Z^2$, we consider the sequence $a_n=t(M+nv)$. Such a sequence will be called a [*ray associated to*]{} $t$. We call $M$ the [*origin*]{} of the ray and $v$ its [*directing vector*]{}. The ray is [*horizontal*]{} if $v=(1,0)$, [*vertical*]{} if $v=(0,-1)$ and [*diagonal*]{} if $v=(1,-1)$.
We say that the frontier Eq.(\[frontier\]) is [*ultimately periodic*]{} if for some $p\geq 1$, called a [*period*]{}, and some $n_0,n_0'\in \mathbf Z$, one has:
\(i) for $n\geq n_0$, $x_n=x_{n+p}$;
\(ii) for $n\leq n'_0$, $x_n=x_{n-p}$.
\[rational\] If the frontier in Theorem 3 is ultimately periodic, then each ray associated to $t$, and whose directing vector is of the form $(a,b)$ with $ab\leq 0$, is $\mathbf N$-rational.
*Proof.*
Step 1. The points $M+nv$ are, for $n$ large enough, all above or all below the frontier, since the frontier is admissible and by the hypothesis on the directing vector. Since rationality is not affected by changing a finite number of values, we may, by symmetry, suppose that they are all below.
Step 2. Let $w_n$ be the word associated to the point $M+nv$. By ultimate periodicity of the frontier, there exists an integer $q\geq 1$ and words $v_0,...,v_{q-1}, u'_0,...,u'_{q-1},u_0,...,u_{q-1}$ such that for any $i=0,...,q-1$ and for $n$ large enough, $w_{i+nq}={u'}_i^nv_iu_i^n$.
Step 3. It follows from Theorem 3 that for some $2\times 2$ matrices $M'_i,N_i,M_i$ over $\mathbf N$, one has for any $i=0,\ldots,q-1$ and $n$ large enough, $a_{i+nq} = \lambda_i {M'_i}^nN_iM_i^n\gamma_i$, where $\lambda_i\in \mathbf N^{1\times 2},\gamma_i\in \mathbf N^{2\times1}$.
Step 4. Since rational series over $\mathbf N$ are closed under Hadamard product, each series $\sum_{n\in \mathbf N}a_{i+nq}x^n$ is rational over $\mathbf N$; indeed, such a series is by the formula in Step 3 an $\mathbf N$-linear combination of products of Hadamard products of two $\mathbf N$-rational series. Therefore $$\sum_{n\in \mathbf N}a_nx^n=\sum_{i=0,...,q-1} x^i (\sum_{n\in \mathbf N}a_{i+nq}(x^q)^n)$$ is also $\mathbf N$-rational.
$\square$
Symmetric frontiers, perfect squares and quadratic relations
------------------------------------------------------------
Given a finite or infinite word $w$ on the alphabet $\{x,y\}$, we call [*transpose*]{} of $w$, and denote it by $^tw$ the word obtained by reversing it and exchanging $x$ and $y$. For instance, $^t(xyyxy)=xyxxy$. If $w$ is a right infinite word, then its transpose is a left infinite word.
Consider an admissible frontier of the form ${}^tsyx^hys$, embedded in the plane, with $h\in\mathbf N$. Let $I,J,K$ be the points of the plane defined as follows: $I$ corresponds to the point between $x^h$ and $y$ on the frontier; $J$ (resp. $K$) is immediately below $I$ (resp. $J$); see the figure. Let $i_n$ (resp. $j_n$, $k_n$) be the horizontal (resp. diagonal) ray of origin $I$ (resp. $J$, $K$) of the $SL_2$-tiling corresponding to the frontier. Then for any $n\in \mathbf N$, $$j_n=(h+1)i_n^2 \,\,\mbox{and} \,\, k_{n}+1=(h+1)i_ni_{n+1}.$$
$$\begin{array}{cccccccccccccccc}
&&&&&&&&&&&. \\
&&&&&&&&&&s \\
&&&&&&&&&. \\
&&&&&x^h&&&| \\
&&&-&.&.&.&-&I &.&.&i_n&i_{n+1}\\
&&&|&&&&&J \\
&&.&&&&&&K&. \\
&{}^ts&&&&&&&&.&.\\
. &&&&&&&&&&.&j_n\\
&&&&&&&&&&&k_n
\end{array}$$
*Proof.* Step 1. We have $$M(yx^hy)=\left(\begin{array}{cc}1&0\\1&1\end{array}\right) \left(\begin{array}{cc}1&h\\0&1\end{array}\right) \left(\begin{array}{cc}1&0\\1&1\end{array}\right)=\left(\begin{array}{cc}h+1&h\\h+2&h+1\end{array}\right).$$ The quadratic form associated to this matrix (which is not symmetric) is therefore $$(a,b) \left(\begin{array}{cc}h+1&h\\h+2&h+1\end{array}\right) \left(\begin{array}{c}a\\b\end{array}\right)=(h+1)a^2+(2h+2)ab+(h+1)b^2=(h+1)(a+b)^2.$$
Step 2. The words associated to $i_n$, $j_n$, $i_{n+1}$, $k_n$ are respectively of the form $yvx$, $y\,{}^tvyx^hyvx$, $yvxy^kx$, $yx^ky\,{}^tvyx^hyvx$; see the figure.
$$\begin{array}{cccccccccccccccccccc}
&&&&&&&&&&&&&&&&&\bf 1&\bf 1 \\
&&&&&&&&&&&&&&&&&. \\
&&&&&&&&&&&&&&&&k&. \\
&&&&&&&&&&&&&&&&&. \\
&&&&&&&&&&&&&&&&\bf 1&\bf 1 \\
&&&&&&&&&&&&&&&. \\
&&&&&&&&&&&&&&v \\
&&&&&&&&&&&&&. \\
&&&&&&&&&&h&&\bf 1 \\
&&&&&&&&\bf 1 &.&.&.&\bf 1&&&&&i_n&i_{n+1}\\
&&&&&&&&\bf 1 \\
&&&&&&&. \\
&&&&&&{}^tv \\
&&&&&. \\
&&k&& \bf 1\\
\bf 1&.&.&.&\bf 1 &&&&&&&&&&&&&j_n\\
\bf 1 &&&&&&&&&&&&&&&&&k_n
\end{array}$$
Step 3. Thus $i_n=(1,1)M(v) \left(\begin{array}{c}1\\1\end{array}\right)$ and $j_n=(1,1)M({}^tvyx^hyv) \left(\begin{array}{c}1\\1\end{array}\right)$. Let $\left(\begin{array}{c}a\\b\end{array}\right)=M(v)\left(\begin{array}{c}1\\1\end{array}\right)$. Then $i_n=a+b$ and $j_n=(1,1)M({}^tv)M(yx^hy)M(v)\left(\begin{array}{c}1\\1\end{array}\right)=(a,b)M(yx^hy)\left(\begin{array}{c}a\\b\end{array}\right)$, since $M({}^tv)={}^tM(v)$. Thus by Step 1, $j_n=(h+1)(a+b)^2=(h+1)i_n^2$.
Step 4. Let $M(v)=\left(\begin{array}{cc}p&q\\r&s\end{array}\right)$. Then $$\begin{array}{lll}
k_n&=&(1,1)M(x^ky\,{}^tvyx^hyv)\left(\begin{array}{c}1\\1\end{array}\right)\\
&=&\left(\begin{array}{cc}1&k\\0&1\end{array}\right) \left(\begin{array}{cc}1&0\\1&1\end{array}\right) \left(\begin{array}{cc}p&r\\q&s\end{array}\right) \left(\begin{array}{cc}h+1&h\\h+2&h+1\end{array}\right) \left(\begin{array}{cc}p&q\\r&s\end{array}\right).
\end{array}$$ Furthermore, $i_n=p+q+r+s$ and $$\begin{array}{lll}
i_{n+1}&=&(1,1)M(vxy^k)\left(\begin{array}{c}1\\1\end{array}\right)\\
&=&(1,1)\left(\begin{array}{cc}p&q\\r&s\end{array}\right) \left(\begin{array}{cc}1&1\\0&1\end{array}\right) \left(\begin{array}{cc}1&0\\k&1\end{array}\right) \left(\begin{array}{c}1\\1\end{array}\right).
\end{array}$$ A straightforward computation then shows that $(h+1)i_ni_{n+1}-k_n=ps-rq$. Since $det(M(v))=1$, the lemma is proved.
$\square$
[**Remark**]{} Denote by $k'_n$ the value of the tiling in the point immediately to the right of the value $j_n$. It is easily shown that one has also $k'_{n}-1=(h+1)i_ni_{n+1}$. Therefore, the $2\times 2$ matrix $\left(\begin{array}{cc}j_n&k'_n\\k_n&j_{n+1}\end{array}\right)$, which appears as a connected submatrix of the tiling, encodes a pythagorean triple: indeed, $(j_{n+1}+j_n, j_{n+1}-j_n, k_n+k'_n)$ is such a triple, because $(j_{n+1}-j_n)^2+(k_n+k'_n)^2=(h+1)^2(i_{n+1}^2-i_n^2)^2+(h+1)^2(2i_ni_{n+1})^2=(h+1)^2(i_{n+1}^2+i_n^2)^2=(j_{n+1}+j_n)^2$. See for example the tiling given in Section 3 and its submatrices $\left(\begin{array}{cc}1&3\\1&2^2\end{array}\right)$, representing the triple $(5,3,4)$, or $\left(\begin{array}{cc}2^2&11\\9&5^2\end{array}\right)$, representing the triple $(29,21,20)$ (here $h=0$).
\[quadratic\] Let $h,h'\in\mathbf N$. Consider a frontier of the form $f=s'xy^{h'}xwyx^hys$, where $w\in\{x,y\}^*$, such that ${}^ts=s'xy^{h'}xw$ and ${}^ts'=wxyx^hys$. Let $P_0,...,P_k$ be the points corresponding to $w$, with $k$ the length of $w$, and $b(j,n)$ be the diagonal rays of origin $P_j$, for $j=0,...,k$. Let $I,J,K,I',J',K'$ be the points defined as follows: $I$ (resp. $I'$) corresponds to the point on the frontier between $x^h$ and $y$ (resp. $x$ and $y^{h'}$), $J$ (resp. $K$) is immediately below $I$ (resp. $J$), $J'$ (resp. $K'$) is immediately to the right of $I'$ (resp. of $J'$). Let $i_n$ (resp. $i'_n$, resp. $ j_n,k_n,j'_n,k'_n$) denote the horizontal (resp. vertical, resp. diagonal) ray of origin $I$ (resp. $I'$, resp. $J,K,J',K'$). Then $j_n=(h+1)i_n^2$, $k_n+1=(h+1)i_ni_{n+1}$, $j'_n=(h'+1){i'}_n^2, k'_n+1=(h'+1)i'_ni'_{n+1}$. Moreover, for $j=1,\ldots ,k-1$, one has, with $w=x_1\cdots x_k$: for any $n\in \mathbf N$, $b(j,n)b(j,n+1)=1+B$, where $$B= \left\{
\begin{array}{lll} b(j-1,n+1)b(j+1,n)&{if} &x_jx_{j+1}=xx \\
b(j-1,n+1)b(j+1,n+1)&{if} & x_jx_{j+1}=xy \\
b(j-1,n)b(j+1,n)&{if} & x_jx_{j+1}=yx\\
b(j-1,n)b(j+1,n+1)&{if} & x_jx_{j+1}=yy
\end{array}
\right.$$
Note that the points given in the lemma need not be all distinct. The lemma is illustrated by the following figure, where $h=1$, $h'=0$, and $w=xyxx$; in bold are represented the 1’s corresponding to the factors $xy^{h'}x=xx$ and $yx^hy=yxy$ of the frontier.
$$\begin{array}{lllllllllllllllllllllllllllllllllllllllllllllll}
&&&&&&&&&1&1 \\
&&&&&&&&&1&2 \\
&&&\cdots&&&&&&1&3 \\
&&&&&&&&1&1&4 \\
&&&&&&&&1&2&9 \\
&&&&&&&&\bf 1&3&14 \\
&&&&&&&\bf 1&{\bf 1}(I)&4&19 \\
&&&&&1(P_2)&1(P_3)&{\bf 1}(P_4)&2\cdot1^2(J)&9&43 \\
&&\bf 1&{\bf 1}(I')&{\bf 1^2}(J'=P_0)&1(K'=P_1)&2&3&7(K)&2\cdot4^2&153 \\
&1&1&2&3&2^2&9&14&33&151&2\cdot19^2\\
&1&2&5&8&11&5^2&39\\
1&1&3&8\\
&&&&&&&&\cdots
\end{array}$$
*Proof.* This follows from Lemma 1, and its symmetrical statement, together with the inspection of the figure below, which shows the four different possible configurations. $$\begin{array}{lllllllllllllll}
P_{j-1}&P_j&P_{j+1} \\
&.&.&. \\
&&.&.&. \\
&&&&b(j-1,n)&b(j,n)&b(j+1,n) \\
&&&&&b(j-1,n+1)&b(j,n+1)\\
\\
\\
&P_{j+1} \\
P_{j-1}&P_j&. \\
&.&.&. \\
&&.&.&b(j+1,n) \\
&&&b(j-1,n)&b(j,n)&b(j+1,n+1) \\
&&&&b(j-1,n+1)&b(j,n+1)
\\
\\
P_j&P_{j+1} \\
P_{j-1}&.&. \\
&.&.&. \\
&&.&b(j,n)&b(j+1,n) \\
&&&b(j-1,n)&b(j,n+1) \\
\\
\\
P_{j+1} \\
P_j &.\\
P_{j-1}&.&. \\
&.&.&b(j+1,n) \\
&&.&b(j,n)&b(j+1,n+1) \\
&&&b(j-1,n)&b(j,n+1) \\
\end{array}$$
$\square$
\[periodic-exists\] Let $h,h'$ in $\mathbf N$ and $w$ in $\{x,y\}^*$. Then there exists a periodic frontier $f=s'xy^{h'}xwyx^hys$ satisfying the hypothesis of Lemma \[quadratic\].
*Proof.* Let indeed $$\label{periodic_frontier}
f={}^\infty(wxy^hx\,{}^twxy^{h'}x)(wyx^hy\,{}^twyx^{h'}y)^{\infty}.$$ In other words, we take $s=({}^twyx^{h'}ywyx^hy)^\infty$ and $s'={}^\infty(xy^{h'}xwxy^hx\,{}^tw)$. Then ${}^ts={}^\infty(xy^hx\,{}^twxy^{h'}xw)={}^\infty(xy^{h'}xwxy^hx\,{}^tw)xy^{h'}xw=s'xy^{h'}xw$. Similarly ${}^ts'=wyx^hys$.
$\square$
Proof of Theorem 2
==================
We completely omit the case of Dynkin diagrams, since Theorem 2 in this case follows immediately from the finiteness of the set of cluster variables, see [@FZ2].
The case $\tilde A_m$
---------------------
Let $1,...,m+1$ be the vertices of the graph $\tilde A_m$, with edges $\{j,j+1\}$, $j=1,...,m+1$, with $j+1$ taken $mod. m+1$. An acyclic orientation being given, let $x_j=x$ if the orientation is $j\rightarrow j+1$ and $x_j=y$ if it is $j\leftarrow j+1$. Let $a(j,n)$ be the sequences of the frise, $j=1,\ldots,m+1$. We extend the notation $a(j,n)$ to $j\in \mathbf Z$ by taking $j$ $mod. m+1$. Let $w$ be the word $x_1\cdots x_{m+1}$, which encodes the orientation. Then $^\infty w^\infty$ is an admissible frontier; indeed, $x$ and $y$ appear both in $w$, since the orientation is acyclic. Embed this frontier into the plane and denote by $P_j$ , with $j\in \mathbf Z$, the successive points of this embedding, in such a way that $P_j$ corresponds to the point between $x_{j-1}$ and $x_j$, with $j$ taken $mod.\, m+1$. Let $t$ be the tiling given by Theorem 3. Because of the periodicity of the frontier, the diagonal ray $b(j,n)$ of origin $P_j$, $j\in \mathbf Z$, depends only on the class of $j$ $mod. \, m$.
We claim that the ray $b(j,n)$ is equal to $a(j,n)$, for $j=1,..,m+1$. This is true for $n=0$, since both are equal to 1. It is enough to show that $b(j,n)$ satisfies Eq.(1). Fix $j=1,...,m+1$. We have four cases according to the relative positions of $P_{j-1},P_j,P_{j+1}$ (see the figure in the proof of Lemma \[quadratic\]).
They correspond to the four possible values of the couple $(x_{j-1},x_j)$: $$(x,x), (x,y), (y,x),(y,y).$$ By definition of $w$, these four cases correspond to the four possible orientations: $$j-1\rightarrow j\rightarrow j+1, \,j-1\rightarrow j\leftarrow j+1,$$ $$\, j-1\leftarrow j\rightarrow j+1,\, j-1\leftarrow j\leftarrow j+1.$$ Thus, by Eq.(1), they correspond to the four induction formulas $a(j,n+1)=\frac{1+A}{a(j,n)}$, where $A$ takes one of the four possible values: $$a(j-1,n+1)a(j+1,n), \quad a(j-1,n+1)a(j+1,n+1),$$ $$a(j-1,n)a(j+1,n), \quad a(j-1,n)a(j+1,n+1).$$ Regarding the tiling, these four cases correspond to the four possible configurations, shown in the same figure. Hence, by the $SL_2$-condition, they correspond to the four induction formulas for $b(j,n)$: $b(j,n+1)=\frac{1+B}{b(j,n)}$, where $B$ takes one of the four possible values: $$b(j-1,n+1)b(j+1,n), b(j-1,n+1)b(j+1,n+1),$$ $$b(j-1,n)b(j+1,n), b(j-1,n)b(j+1,n+1).$$ This concludes the proof, by using Corollary 1.
$\square$
The proof is illustrated in the tiling below and in Figure \[friseA3tilde\], for a specific orientation of $\tilde A_3$.
$$\begin{array}{lllllllllllllllllllllllllllllll}
&&&&&&&&&&&&1 \\
&&&&&\ldots&&&&1&1&1&1 \\
&&&&&&1&1&1&1&2&3&4 \\
&&&1&1&1&1&2&3&4&9&14&19 \\
1&1&1&1&2&3&4&9&14&19&43&67 &&\ldots \\
1&2&3&4&9&14&19&43&67 \\
&&&&\ldots
\end{array}$$
The case $\tilde D_m$
---------------------
We consider an orientation of $\tilde D_m$, of the form shown in Figure \[Dmtilde\], where the orientations of the edges $\{i,i+1\}$ for $i=2,...,m-3$ are arbitrary. The other cases, which differ from the case considered here by changing the orientation of the forks, are similar.
An example is shown in Figure \[friseD7tilde\]. The reader may recognize that this frise is encoded in the tiling shown in Section 6.2 after the statement of Lemma 3.
Step 1. We define a word $w$, which encodes the orientation, as follows: $w=x_1x_2\cdots x_{m-3}$, with $$x_i=\left\{
\begin{array}{ll}
x & \mbox{if $i\rightarrow i+1$} \\
y & \mbox{if $i\leftarrow i+1$}.
\end{array}
\right.$$
Note that by our choice in Figure \[Dmtilde\], $x_1=x$. In the example, $w=xyxx$. We consider now the frontier given by Lemma \[periodic-exists\], with $h'=0$ and $h=1$; see Eq.(\[periodic\_frontier\]). The associated rays $b(j,n)$ are all rational by Corollary \[rational\].
Step 2. Taking the notations of Lemma \[quadratic\], we have for any $n\in\mathbf N$: $$j'_n=b(0,n), \, k'_n=b(1,n), \, k_n=b(m-3,n+1).$$ For $j'_n$ and $k'_n$ this follows from $h'=0$ and therefore $J'=P_0$, $K'=P_1$, hence $j'_n$ (resp. $k'_n$) and $b(0,n)$ (resp. $b(1,n)$) are diagonal rays with the same origin. Moreover $w$ is of length $k$ in Lemma \[quadratic\] and here of length $m-3$, hence $k=m-3$. Since $h=1$, we have the configuration shown below.
$$\begin{array}{lllll}
&1 \\
1&1(I) \\
1(P_{m-3})&(J) \\
&(K)
\end{array}$$
This implies $k_n=b(m-3,n+1)$.
Step 3. In accordance with Lemma \[quadratic\], we have for $j=1,\ldots, k$, $$x_j=\left\{
\begin{array}{ll}
x & \mbox{if $[P_{j-1},P_j]$ is horizontal} \\
y & \mbox{if $[P_{j-1},P_j]$ is vertical}.
\end{array}
\right.$$
Step 4. We define $m+1$ sequences $a'(j,n)$ for $j=0,\ldots m$. First, for any $n\in\mathbf N$, $$a'(0,n)=a'(0,n)=a'(1,n)=i'_n.$$ Now, for $j=2,...,m-2$, $$a'(j,n)=b(j-1,n).$$ Furthermore, $$a'(m-1,n)=\left\{
\begin{array}{ll}
i_n & \mbox{if $n$ is even} \\
2i_n & \mbox{if $n$ is odd}.
\end{array}
\right.$$ Finally, for $n\ge 1$, $$a'(m,n)=\left\{
\begin{array}{ll}
i_{n-1} & \mbox{if $n$ is even} \\
2i_{n-1} & \mbox{if $n$ is odd},
\end{array}
\right.$$ with $a'(m,0)=1$.
Observe that for any $n\in\mathbf N$, $a'(m-1,n)a'(m,n+1)=2i_n^2$. Moreover, since the sequences $i_n$, $i'_n$ and $b(j,n)$ are rays, they are rational. Hence, so are the sequences $a'(j,n)$ (for $a'(m-1,n)$ and $a'(m,n)$, this follows from standard constructions on rational sequences).
Step 5. In order to end the proof, it is enough to show that $a(j,n)=a'(j,n)$. First note that $a(j,0)=a'(j,0)$, as is easily verified. Thus it suffices to show that $a'(j,n)$ satisfies the same recursion formula as $a(j,n)$, that is, Eq.(\[frises\_induction\]).
Step 6. We have $$\begin{array}{llll}
a(0,n+1)a(0,n)&=& i'_{n+1}i'_n & \mbox{by Step 4} \\
&=& 1+k'_n & \mbox{by Lemma \ref{quadratic}} \\
&=& 1+b(1,n) & \mbox{by Step 2} \\
&=& 1+a'(2,n) & \mbox{by Step 4}.
\end{array}$$ Similarly $$a'(1,n+1)a'(1,n)=1+a'(2,n).$$ This is the good recursion for $a'(0,n)$ and $a'(1,n)$ since, by Figure \[Dmtilde\] and Eq.(1), $$a(0,n+1)a(0,n)=1+a(2,n),$$ $$a(1,n+1)a(1,n)=1+a(2,n).$$
Step 7. Let $$n'=\left\{
\begin{array}{ll}
n & \mbox{if $x_2=x$} \\
n+1& \mbox{if $x_2=y$}.
\end{array}
\right.$$ We have by Step 4, $a'(2,n+1)a'(2,n)=b(1,n+1)b(1,n)$. Looking at the figures below, where we use Step 3:
Case $n'=n,x_2=x$ $$\begin{array}{llllll}
P_0&P_1&P_2&b(1,n)&b(2,n)\\
&&&b(0,n+1)&b(1,n+1)
\end{array}$$ Case $n'=n+1, x_2=y$ $$\begin{array}{lllll}
&P_2&b(1,n)&b(2,n+1)\\
P_0&P_1&b(0,n+1)&b(1,n+1)
\end{array}$$ we see that this is equal to $1+b(0,n+1)b(2,n')$. Using Step 2, Step 4, Lemma \[quadratic\] then Step 4 again, we obtain $$\begin{array}{lll}
b(0,n+1)b(2,n')&=&j'_{n+1}a'(3,n')\\
&=&{i'}_{n+1}^2a'(3,n')\\
&=&a'(0,n+1)a'(1,n+1)a'(3,n').
\end{array}$$ Thus $a'(2,n+1)a'(2,n)=1+a'(0,n+1)a'(1,n+1)a'(3,n')$. This corresponds to Eq.(1) for $(2,n)$, that is, $a(2,n+1)a(2,n)=1+a(0,n+1)a(1,n+1)a(3,n')$.
Step 8. Let $$n''=\left\{
\begin{array}{ll}
n & \mbox{if $x_{m-3}=y$.} \\
n+1& \mbox{if $x_{m-3}=x$}.
\end{array}
\right.$$ We have $$\begin{array}{llll}
a'(m-2,n+1)a'(m-2,n)&=&b(m-3,n+1)b(m-3,n)&\mbox{by Step 4}\\
&=&1+b(m-4,n'')j_n,
\end{array}$$ where the second equality follows from figures \[FriseA\] and \[FriseB\], which use Step 3:
Case $n''=n, x_{m-3}=y$ $$\begin{array}{lllll}
P_{m-3}&J&&b(m-3,n)&j_n\\
P_{m-4}&&&b(m-4,n)&b(m-3,n+1)
\end{array}$$ Case $n''=n+1, x_{m-3}=x$ $$\begin{array}{llllll}
P_{m-4}&P_{m-3}&J&&b(m-3,n)&j_n\\
&&&&b(m-4,n+1)&b(m-3,n+1)
\end{array}$$
By Lemma \[quadratic\], $j_n=2i_n^2$ and by Step 4, $2i_n^2=a'(m-1,n)a'(m,n+1)$. Furthermore, $b(m-4,n'')=a'(m-3,n'')$. Thus $$a'(m-2,n+1)a'(m-2,n)=1+a'(m-3,n'')a'(m-1,n)a'(m,n+1).$$ This equality corresponds to Eq.(1) for $a(m-2,n)$, if we look at the two figures \[FriseA\] and \[FriseB\], representing the frise in the two cases $n''=n, x_{n-3}=y$ and $n''=n+1, x_{n-3}=x$.
Step 9. We have $$\begin{array}{lllll}
a'(m-1,n+1)a,(m-1,n)&=&2i_ni_{n+1}&\mbox{by Step 4}\\
&=&1+k_n&\mbox{by Lemma \ref{quadratic}}\\
&=&1+b(m-3,n+1)&\mbox{by Step 2}\\
&=&1+a'(m-2,n+1)&\mbox{by Step 4}\\
\end{array}$$ in accordance with Eq.(1), which gives $a(m-1,n+1)a(m-1,n)=1+a(m-2,n+1)$. Moreover, for $n\geq 1$, $$\begin{array}{lllll}
a'(m,n+1)a'(m,n)&=&2i_ni_{n-1}&\mbox{by Step 4}\\
&=&1+k_{n-1}&\mbox{by Lemma \ref{quadratic}}\\
&=&1+b(m-3,n)&\mbox{by Step 2}\\
&=&1+a'(m-2,n)&\mbox{by Step 4},\\
\end{array}$$ in accordance with Eq.(1), which gives $a(m,n+1)a(m,n)=1+a(m-2,n)$, noting that for $n=0$: $a'(m,1)a'(m,0)=2i_0$ (by Step 4)$=2$ and $1+a'(m-2,0)=1+1=2$.
Cases $\tilde B_m$, $\tilde C_m$, $\tilde BC_m$, $\tilde BD_m$, $\tilde CD_m$.
------------------------------------------------------------------------------
Each frise in these cases is reduced to a frise of type $\tilde A$ or $\tilde D$. Precisely: $\tilde B_m$ is reduced to $\tilde D_{m+2}$; $\tilde C_m$ is reduced to $\tilde A_{2m}$; $\tilde {BC}_m$ is reduced to $\tilde D_{2m+2}$; $\tilde {BD}_m$ is reduced to $\tilde D_{m+1}$; $\tilde {CD}_m$ is reduced to $\tilde D_{2m}$.
We give no formal proof, but an example; it should convince the reader. In this example, we show how a frise of type $\tilde {BC}_4$ can be simulated by a frise of type $\tilde D_{10}$. See Figure \[BC4tilde\].
Frises and tilings with variables
=================================
As mentioned in the introduction, if we replace the inital values $a(j,0)$ in Section 3 by commuting variables, and keep the recurrence of Eq.(1) unchanged, we obtain frises of variables. The variables therefore obtained are usual cluster variables, in the sense of Fomin and Zelevinsky. It is well-known that all cluster variables, but finitely many of them, can be obtained in this way (those not obtained in this way being the cluster variables corresponding to the exceptional objects, in the corresponding cluster category, lying in tubes or $\mathbb{Z}A_{\infty}$ components).
Likewise, we generalize the $SL_2$-tilings; these are simply fillings of the discrete plane by elements of a ring $R$ such that each $2\times 2$ connected minor is of determinant $1$. We generalize Theorem 3 by putting variables on the frontier.
Case $\tilde A_m$
-----------------
We call ([*generalized*]{}) [*frontier*]{} a bi-infinite sequence $$\label{frontiervariable}
\ldots x_{-2}a_{-2}x_{-1}a_{-1}x_0a_0x_1a_1x_2a_2x_3a_3 \ldots$$ where $x_i\in \{x,y\}$ and $a_i$ are variables, for any $i\in \mathbf Z$. It is called [*admissible*]{} if there are arbitrarily large and arbitrarily small $i$’s such that $x_i=x$, and similarly for $y$; in other words, none of the two sequences $(x_n)_{n\geq0}$ and $(x_n)_{n\leq0}$ is ultimately constant. The $a_i$’s are called the [*variables*]{} of the frontier. Each frontier may be embedded into the plane: the variables label points in the plane, and the $x$ (resp. $y$) determine a bi-infinite discrete path, in such a way that $x$ (resp. $y$) corresponds to a segment of the form $[(a,b),(a+1,b)]$ (resp $[(a,b),(a,b+1)]$). For example, corresponding to the frontier $\ldots a_{-2}xa_{-1}xa_0ya_1ya_2xa_3 \ldots$ is given below:
$$\begin{array}{lllllllllllllll}
&&&&&&&&&. \\
&&&&&&&&. \\
&&&&&&&. \\
&&&&&a_2&a_3 \\
&&&&&a_1 \\
&&&a_{-2}&a_{-1}&a_0 \\
&&. \\
&. \\
. \\
\end{array}$$ Formally we do as follows. We define a partial function$f$ from $\mathbf Z^2$ into the semiring of Laurent polynomials over $\mathbf N$ generated by the variable, defined up to translation, as follows: fix some $(k,l)\in \mathbf Z^2$ and $i \in \mathbf Z$; then $f(k,l)=a_i$; moreover, if $x_{i+1}x_{i+2}...x_p$ labels the discrete path from $(k,l)$ to $(k',l')$, then $f(k',l')=a_p$; furthermore, if $x_px_{p+1}...x_i$ labels the path from $(k',l')$ to $(k,l)$, then $f(k',l')=a_{p-1}$.
We see below that an admissible frontier, embedded into the plane, may be extended to an $SL_2$-tiling. For this, we need the following notation. Let $$M(a,x,b)=\left(\begin{array}{cc}a&1\\0&b\end{array}\right) \,\mbox{and} \, \,M(a,y,b)=\left(\begin{array}{cc}b&0\\1&a\end{array}\right).$$
Note that these matrices reduce to the matrices $M(x)$ and $M(y)$ when the variables $a$ and $b$ are set to 1.
Given an admissible frontier, embedded in the plane as explained previously, let $(u,v)\in \mathbf Z^2$. Then we obtain a finite word, which is a factor of the frontier, by projecting the point $(u,v)$ horizontally and vertically onto the frontier. We call this word the [*word*]{} of $(u,v)$. It is illustrated in the figure below, where the word of the point $M=(k,l)$ is $a_{-3}ya_{-2}ya_{-1}ya_0xa_1xa_2ya_3xa_4$: $$\begin{array}{ccccccccccccccccc}
&&&&&&&&&&&&. \\
&&&&&&&&&&&. \\
&&&&&&&&&&. \\
&&&&&&&a_3&a_4&a_5 \\
&&&&&a_0&a_1&a_2 &|\\
&&&&&a_{-1} &&&|\\
&&&&&a_{-2} &&&|\\
&&&&a_{-4}&a_{-3} &-&-&M\\
&&. \\
&. \\
\end{array}$$ We define the word of a point only for points below the frontier; for points above, things are symmetric and we omit this case. We call [*denominator*]{} of the point $M$ the product of the variables of its word, excluding the two extreme ones. In the example, its denominator is $a_{-2}a_{-1}a_0a_1a_2a_3$.
\[tiling-variables\] Given an admissible frontier, there exists a unique $SL_2$-tiling $t$ of the plane, with values in the semiring of Laurent polynomials over $\mathbf N$ generated by the variables lying on the frontier, extending the embedding of the frontier into the plane. It is defined, for any point $(u,v)$ below the frontier, with associated word $a_0x_1a_1x_2...x_{n+1}a_{n+1}$, where $n\geq 1$ and $x_i\in\{x,y\}$, by the formula $$\label{tilingformulvariable}
t(u,v)=\frac{1}{a_1a_2...a_n} (1,a_0)M(a_1,x_2,a_2)M(a_2,x_3,a_3)\cdots M(a_{n-1},x_n,a_n)\left(\begin{array}{c}1\\a_{n+1}\end{array}\right).$$
In order to prove the theorem, we need two lemmas, where $R$ denotes some commutative ring. We extend the notation $M(a,x,b)$ and $M(a,y,b)$ for $a,b$ in $R$.
\[lemmapqrsa\] (i) Let $A\in R^{2\times 2}$, $\lambda, \lambda'\in R^{1\times 2}$, $\gamma, \gamma'\in R^{2\times 1}$, and define $p=\lambda A\gamma$, $q=\lambda A\gamma'$, $r=\lambda' A\gamma$, $s=\lambda' A\gamma'$. Then $det\left(\begin{array}{cc}p&q\\r&s\end{array}\right) =det(A)det\left(\begin{array}{c}\lambda\\\lambda'\end{array}\right) det(\gamma,\gamma')$.
\(ii) Let $a,b_1,...,b_k,b\in R$, $\lambda'=(1,a)M(b_1,x,b_2)\cdots M(b_{k-1},x,b_k)M(b_k,y,b)$ and $\lambda=(1,b_k)$. Then $det\left(\begin{array}{c}\lambda'\\\lambda\end{array}\right)=b_1\cdots b_kb$.
*Proof.*
\(i) This follows since $\left(\begin{array}{cc}p&q\\r&s\end{array}\right) =\left(\begin{array}{c}\lambda\\\lambda'\end{array}\right) A (\gamma,\gamma')$.
\(ii) Let $N=M(b_1,x,b_2)\cdots M(b_{k-1},x,b_k)$. Then $N=\left(\begin{array}{cc}b_1\cdots b_{k-1}&u\\0&b_2\cdots b_k\end{array}\right)$ and $(1,a)N=(b_1\cdots b_{k-1},u+ab_2\cdots b_k)$. Thus $$\lambda'=(1,a)N\left(\begin{array}{cc}b&0\\1&b_k\end{array}\right)=(bb_1\cdots b_{k-1}+u+ab_2\cdots b_k,ub_k+ab_2\cdots b_{k-1}b_k^2).$$ It follows that $$det\left(\begin{array}{c}\lambda'\\ \lambda\end{array}\right)=(bb_1\cdots b_{k-1}+u+ab_2\cdots b_k)b_k-(ub_k+ab_2\cdots b_{k-1}b_k^2)=bb_1\cdots b_k.$$
$\square$
\[lemmapqrsb\] Let $A\in R^{2\times 2}$ and $a,b_1,...,b_k,b,c,c_1,...,c_l,d\in R$, $k,l\geq 1$. Define $$\begin{array}{lll}
p&=&(1,b_k)A\left(\begin{array}{c}1\\ c_1\end{array}\right),\\
q&=&(1,b_k)A M(c,x,c_1)M(c_1,y,c_2)\cdots M(c_{l-1},y,c_l) \left(\begin{array}{c}1\\ d\end{array}\right), \\
r&=&(1,a)M(b_1,x,b_2)\cdots M(b_{k-1},x,b_k)M(b_k,y,b) A \left(\begin{array}{c}1\\ c_1\end{array}\right),
\\
s&=&(1,a)M(b_1,x,b_2)\cdots M(b_{k-1},x,b_k)M(b_k,y,b) AM(c,x,c_1)M(c_1,y,c_2)\cdots \\
&&M(c_{l-1},y,c_l) \left(\begin{array}{c}1\\ d\end{array}\right).
\end{array}$$ Then $det\left(\begin{array}{cc}p&q\\r&s\end{array}\right) =b_1\cdots b_k b c c_1\cdots c_l det(A)$.
*Proof.* Let $\lambda=(1,b_k)$, $\gamma=\left(\begin{array}{c}1\\ c_1\end{array}\right)$, $\gamma'=M(c,x,c_1)M(c_1,y,c_2)\cdots M(c_{l-1},y,c_l) \left(\begin{array}{c}1\\ d\end{array}\right)$, $\lambda'=(1,a)M(b_1,x,b_2)\cdots M(b_{k-1},x,b_k)M(b_k,y,b)$. By Lemma \[lemmapqrsa\], $det\left(\begin{array}{cc}p&q\\r&s\end{array}\right) =det(A)det\left(\begin{array}{c}\lambda\\\lambda'\end{array}\right) det(\gamma,\gamma')$. By Lemma \[lemmapqrsa\] again, $det\left(\begin{array}{c}\lambda\\\lambda'\end{array}\right)=-b_1\cdots b_kb$ and symmetrically, $det(\gamma,\gamma')=-cc_1\cdots c_l$, which ends the proof.
$\square$
**Proof of Theorem \[tiling-variables\].** We prove that $t$ given by Eq. (\[tilingformula\]) is an $SL_2$-tiling of the plane. It is enough to show that for any $(u,v)\in \mathbf Z^2$, the determinant of the matrix $\left ( \begin{array}{ll} t(u,v) & t(u,v+1)\\ t(u+1,v) & t(u+1,v+1) \end{array} \right )$ is equal to 1.
By inspection of the figure below, where $k,l\ge 1$ and $a,b_1,...,b_k,b,c,c_1,..,c_l,d$ are in $R$ and $w=x_1e_1\cdots e_{n-1}x_n$, $n\geq 0$, with $e_i\in R$ and $x_i\in\{x,y\}$, $$\begin{array}{cccccccccccc}
&&&&&&&&&&c_l&d \\
&&&&&&&&&&. &| \\
&&&&&&&&&&. &| \\
&&&&&&&&&&. &| \\
&&&&&&&&&&c_2 &| \\
&&&&&&&&&c&c_1&| \\
&&&&&&&&.&&|&| \\
&&&&&&&w&&&|&| \\
&&&&&&.&&&&|&| \\
&&&&&b&&&&&|&| \\
b_1&.&.&.&b_{k-1}&b_k&-&-&-&-&(u,v)&(u,v+1) \\
a&-&-&-&-&-&-&-&-&-&(u+1,v)&(u+1,v+1)
\end{array}$$ it is seen that the words associated to the four points $(u,v)$, $(u,v+1)$, $(u+1,v)$ and $(u+1,v+1)$ are respectively of the forms $b_kybwcxc_1$, $b_kybwcxc_1yc_2\cdots yc_lxd$, $ayb_1x\cdots b_{k-1}xb_kybwcxc_1$ and $ayb_1x\cdots b_{k-1}xb_kybwcxc_1yc_2\cdots yc_lxd$.
Let $A=M(b,x_1,e_1)M(e_1,x_2,e_2)\cdots M(e_{n-1},x_n,c)$ and $D=be_1e_2\cdots e_{n-1}c$. Then define $p,q,r,s$ as in Lemma \[lemmapqrsb\]. Thus we have $t(u,v)=\frac{p}{D}$, $t(u,v+1)=\frac{q}{Dc_1\cdots c_l}$, $t(u+1,v)=\frac{r}{b_1\cdots b_kD}$ and $t(+1,v+1)=\frac{s}{b_1\cdots b_kDc_1\cdots c_l}$. Therefore by Lemma \[lemmapqrsb\], $\Delta=t(u,v)t(u+1,v+1)-t(u,v+1)t(u+1,v)$ $=\frac{1}{b_1\cdots b_kD^2c_1\cdots c_l}(pq-rs)$ $=\frac{b_1\cdots b_kbcc_1\cdots c_l}{b_1\cdots b_kD^2c_1\cdots c_l} det(A)$. Now $det(A)=be_1^2e_2^2\cdots e_{n-1}^2c$ and $D^2=b^2e_1^2e_2^2\cdots e_{n-1}^2c^2$. Therefore $\Delta=1$.
$\square$
The sequences of a frise of type $\tilde A_m$, whose initial values are variables, are rational over the semiring of Laurent polynomials generated by these variables.
The proof is quite analogue to the proof in Section 7.1. Observe that if one follows the lines of the proof, one may recover the formula for the sequence associated to frise of the Kronecker quiver, as given in Section 2. In particular, one verifies that $$M(a,x,b)M(b,y,a)=M,$$ with the notations in Section 2.
Partial tilings and explicit formulas in case $A_n$
---------------------------------------------------
We call [*partial $SL_2$-tiling of the plane*]{} a filling of a subset of $\mathbf Z^{2}$ such that each connected $2\times 2$ submatrix is of determinant 1. We construct some of these partial tilings. Note that our construction below is somewhat equivalent to the construction of frieze patterns of Coxeter [@Co] and [@Cacha].
We begin with a construction which we call the [*cross construction*]{}: consider a word $w$, for example $w=aybycxdxexfxgyhyiyj$ (with $a,b,c,d,e,f,g,h,i,j$ either variables or set equal to 1) and its transpose $^tw=jxixhxgyfyeydycxbxa$. These two words determine two discrete paths in the plane; we represent each path by the sequence of variables of the word; we put the second one at the south-east of the first, separated by a cross, as shown below by asterisks, and then fill diagonally by 1’s, italicized below. $$\begin{array}{cccccccccccc}
&&&&j&\it 1 \\
&&&&i&*&\it 1 \\
&&&&h&*&&\it 1 \\
c&d&e&f&g&*&&&\it 1 \\
b&&&&&*&&P&&\it 1 \\
a&&&&&*&&&&&\it 1 \\
\it 1&*&*&*&*&*&*&*&*&*&*&\it 1 \\
&\it 1&&&&*&&&&c&b&a \\
&&\it 1&&&*&&&&d \\
&&&\it 1&&*&&&&e \\
&&&&\it 1&*&&&&f \\
&&&&&\it 1&j&i&h&g
\end{array}$$
Then there exists a unique partial $SL_2$-tiling which fits into this figure. The proof of this result is a straightforward generalization of the proofs given so far. The only new thing is the definition of the word associated to each point in the region of the plane constructed above. Note that this region has naturally four components, separated by the cross (although they are not disjoint).
For the north-west component, the word is defined by projection on the frontier, exactly as it has been done in Section 8.1.
For the north-east component, one projects horizontally the point on the north-west frontier, and vertically on the south-east frontier: this give two words and the actual word is given by intersecting them. We do it on an example, which should be explicit enough. Consider the point denoted by $P$ above. Then the horizontal projection gives the point corresponding to the variable $b$ on the north-west frontier, and the vertical projection gives the point corrresponding to the variable $i$ on the south-east frontier. Then, the word associated to $P$ is by definition the word $bycxdxexfxgyhyi$; its denominator is, similarly to Section 8.1, the product of all the variables of the word, except the extreme ones. Thus, for $P$, it is $cdefgh$.
Then the value of the tiling at $P$ is $$\frac{1}{cdefgh}(1,b)M(c,x,d)M(d,x,e)M(e,x,f)M(f,x,g)M(g,y,h) \left( \begin{array}{l}i\\1\end{array} \right),$$ similarly to Eq. (\[tilingformulvariable\]), except that the column matrix in the product is changed:
For the two other components, things are symmetric.
For the example, we show this tiling below in the case where all variables are set equal to 1. $$\begin{array}{cccccccccccc}
&&&&1&\it 1 \\
&&&&1&2&\it 1 \\
&&&&1&3&2&\it 1 \\
1&1&1&1&1&4&3&2&\it 1 \\
1&2&3&4&5&21&16&11&6&\it 1 \\
1&3&5&7&9&38&29&20&11&2&\it 1 \\
\it 1&4&7&10&13&55&42&29&16&3&2&\it 1 \\
&\it 1&2&3&4&17&13&9&5&1&1&1 \\
&&\it 1&2&3&13&10&7&4&1 \\
&&&\it 1&2&9&7&5&3&1 \\
&&&&\it 1&5&4&3&2&1 \\
&&&&&\it 1&1&1&1&1
\end{array}$$
The corresponding frise is then easily constructed; indeed, the word coding the south-east frontier is the transpose of the word coding the frontier; thus one continues the cross-construction with this new word. The tiling obtained has only to be repeated indefinitely; it is then seen to be periodic and its period is 2 + the length of the frontier $w$ (as follows already from the work of Fomin and Zelevinsky). If the frontier is an anti-palindrome, that is, $^tw=w$, then the period is the half of this number.
As a consequence of our results, we get the following well-known result (see [@CK] for instance).
The Laurent phenomenon and the positivity conjecture hold for cluster algebras of type $\tilde{A}_n$ or $A_m$ with acyclic initial clusters.
Appendix: Cartan matrices of Dynkin and Euclidean types
=======================================================
We represent Cartan matrices by their diagram. The Dynkin diagrams are in four infinite series, shown in Figure \[Dynkinseries\], where $m$ represents the number of vertices of each diagram, or one of the five exceptional ones, shown in Figure \[Dynkinexcept\].
Furthermore, the Euclidean diagrams are in one of the seven infinites series shown in Figure \[Euclideanseries\], where $m$ is one less than the number of vertices of the diagram, or one of the nine exceptional ones, shown in Figure \[Euclideanexcept\].
[EvdPSW03]{}
A.B. Buan, R. Marsh, I. Reiten, M. Reineke, and G. Todorov. Tilting theory and cluster combinatorics. , 204:572–618, 2006.
K. Baur, R.J. Marsh, and H. Thomas. Frieze patterns for punctured discs. to appear.
S. Berman, R. Moody, and M. Wonenburger. artan matrices with null roots and finite [C]{}artan matrices. , 21:1091–1099, 1972.
J. Berstel and C. Reutenauer. . Cambridge University Press, to appear.
J.H. Conway and H.S.M. Coxeter. Triangulated polygons and frieze patterns. , 57:87–94, 1973.
J.H. Conway and H.S.M. Coxeter. Triangulated polygons and frieze patterns (continued). , 57:175–183, 1973.
P. Caldero and F. Chapoton. Cluster algebras as [H]{}all algebras of quiver representations. , 81:595–616, 2006.
P. Caldero and B. Keller. From triangulated categories to cluster algebras. , to appear.
H.S.M. Coxeter. Frieze patterns. , 18:298–310, 1971.
P. Caldero and M. Reineke. On the quiver grassmannian in the acyclic case. , 212:2369–2380, 2008.
P. Caldero and A. Zelevinsky. Laurent expansions in cluster algebras. , 6:411–429, 2006.
G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward. . Mathematical surveys and monographs AMS, 2003.
S. Fomin and A. Zelevinsky. Cluster algebras [I]{}: Foundations. , 15:497–529, 2002.
S. Fomin and A. Zelevinsky. The [L]{}aurent phenomenon. , 28:119–144, 2002.
S. Fomin and A. Zelevinsky. Cluster algebras [II]{}: finite type classification. , 154:63–121, 2003.
D. Happel, U. Preiser, and C.M. Ringel. Vinberg’s characterization of [D]{}ynkin diagrams using subadditive functions with applications to [DT]{}r-periodic modules. , 832:280–294, 1980.
G.H. Hardy and E.M. Wright. . Clarendon Press, Oxford, 1978.
B. Keller. Cluster algebras, quiver representations and triangulated categories. , 2006.
G. Musiker and J. Propp. Combinatorial interpretation for rank-two cluster algebras of affine type. , 14, 2007.
G. Musiker and R. Schiffler. Cluster expansion formulas and perfect matchings. , 2008.
G. Musiker, R. Schiffler, and L. Williams. Cluster algebras and positivity. to appear.
J. Propp. The combinatorics of frieze patterns and [M]{}arkoff numbers. , 2008.
R. Schiffler. A cluster expansion formula (${A}\sb n$ case). , 15, 2008.
R. Schiffler. On cluster algebras arising from unpunctured surfaces ii. to appear.
E.B. Vinberg. Discrete linear groups that are generated by reflections. , 35:1072–1112, 1971.
|
[**M. R. Setare[^1]\
V. Kamali[^2]**]{}\
[ Department of Science, University of Kurdistan, Sanandaj, IRAN.]{}\
[**[Abstract]{}**]{}\
In the present work, we study warm tachyon inflation model in the context of “logamediate inflation” where the cosmological scale factor expands as $a=a_0\exp(A[\ln t]^{\lambda})$. The characteristics of this model in slow-roll approximation are presented in two cases: 1- Dissipative parameter $\Gamma$ is a function of tachyon field $\phi$. 2- $\Gamma$ is a constant parameter. The level of non-Gaussianity of this model is found for these two cases. Scalar and tensor perturbations for this scenario are presented. The parameters appearing in our model are constrained by recent observational data (WMAP7). Harrison-Zeldovich spectrum, i.e. $n_s=1,$ is approximately obtained for large values of $\lambda$ (i.e. $\lambda\simeq 50$)and normal value of number of e-folds, i.e. $N\simeq 60$. On the other hand, the scale invariant spectrum (Harrison-Zeldovich spectrum) is given by using two parameters in our model. For intermediate inflation, where the cosmological scale factor expands as $a=a_0\exp(A t^{f}),$ this spectrum has been exactly obtained using one parameter, i.e. $f=\frac{2}{3}$ [@2-n].
PACS numbers: 98.80.Cq \
Keywords: Warm inflation, Tachyon field, logamediate inflation, slow-roll approximation, WMAP data
Introduction
============
Big Bang model has many long-standing problems (horizon, flatness,...). These problems are solved in a framework of inflationary universe model [@1-i]. Scalar field as a source of inflation provides the causal interpretation of the origin of the distribution of large scale structure and also observed anisotropy of cosmological microwave background (CMB) [@2-i]. Standard models for inflationary universe are divided into two regimes, slow-roll and reheating epochs. In slow-roll period kinetic energy remains small compared to the potential terms. In this period, all interactions between scalar fields (inflatons) and other fields are neglected and as a result the universe inflates. Subsequently, in reheating period, the kinetic energy is comparable to the potential energy that causes the inflaton starts an oscillation around minimum of the potential losing their energy to other fields present in the theory. After this period, the universe is filled with radiation.\
In warm inflation scenario radiation production occurs during inflationary period and reheating is avoided [@4]. Thermal fluctuations may be generated during warm inflationary era. These fluctuations could play a dominant role to produce initial fluctuations which are necessary for Large-Scale Structure (LSS) formation. In this model, density fluctuation arises from thermal rather than quantum fluctuation [@3-i]. Warm inflationary period ends when the universe stops inflating. After this period the universe enters in the radiation phase smoothly [@4]. Finally, remaining inflatons or dominant radiation fields create matter components of the universe. Some extensions of this model are found in Ref.[@new].\
Friedmann-Robertson-Walker (FRW) cosmological models in the context of string/M-theory have related to brane-antibrane configurations [@4-i]. Tachyon fields associated with unstable D-branes may be responsible for inflation in early time [@5-i]. If the tachyon field start to roll down the potential, then universe dominated by a new form of matter will smoothly evolve from inflationary universe to an era which is dominated by a non-relativistic fluid [@1]. So, we could explain the phase of acceleration expansion (inflation) in term of tachyon field.\
It has been shown that, there are eight possible asymptotic solutions for cosmological dynamics [@v1]. Three of these solutions have non-inflationary scale factor and another three one’s of solutions give de Sitter (with scale factor $a(t)=a_0\exp(H_0 t)$), intermediate (with scale factor $a(t)=a_0\exp(At^f), 0<f<1$) and power-low (with scale factor $a(t)=t^p, p>1$), inflationary expansions. Finally, two cases of these solutions have asymptotic expansion with scale factor($a=a_0\exp(A(\ln t)^{\lambda})$. This version of inflation is named “logamediate inflation”. This model is found in a number of scalar-tensor theories [@1-n]. We have considered Warm-tachyon inflationary model in the scenario of “intermedaite inflation” in Ref. [@2-n]. The expansion scale factor of FRW universe evolves as: $a(t)=a_0\exp(At^f)$ where $A>0$ and $ 0<f<1$. The expansion of the universe is faster than power-low inflation model and slower than de Sitter which arises as $f=1$. It was shown, to first order, the Harrizon-Zeldovich spectrum [@3-n] of density perturbation (i.e. $n_s=1$ [@4-n]) for warm-tachyon inflation arises when $f=\frac{2}{3}$.\
In this work we would like to consider the tachyonic warm inflationary universe in the particular scenario “logamediate inflation” which is denoted by scale factor $a(t)=a_0\exp(A[\ln t]^{\lambda}),\lambda>1, A>0$ [@v1], [@6-i]. For $\lambda=1$ case this model is converted to power-law inflation ($a=t^A; A>1$). The study of logamediate inflationary model is motivated by imposing weak general conditions on the cosmological model which has indefinite expansion [@v1]. The effective potential of this model which arises with the above scale factor has been studied in dark energy models [@v2]. This potential also is found in supergravity, super-string models and Kaluza-Klein theories [@v3]. For this inflationary model the power spectrum may be either blue or red tilted in term of the value of the parameters of the model [@6-i]. Curvaton reheating in logamediate inflationary models have been studied in Ref.[@5-n].\
The tachyon inflation is a k-inflation model [@n-1] for scalar field $\phi$ with a positive potential $V(\phi)$. Tachyon potentials have two special properties: 1- A maximum of these potentials is obtained where $\phi\rightarrow 0.$ 2- A minimum of these potentials is obtained where $\phi\rightarrow \infty$. In our logamediate model we find an exact solution for tachyon field potential in slow-roll approximation which has the form $V(\phi)\sim(\ln [\phi]^{-\beta}/\phi^4)$. This form of potential has the above properties of tachyon field potential. This special form of potential is everlasting, so we consider our model in the context of the warm inflation to bring an end for inflation period. In the warm inflationary models, dissipative effect arises from a friction term. This effect could describe the mechanisms that scalar fields decay into a thermal bath via its interaction with other fields that causes the warm inflation ends when the universe heats up to become radiation dominant. After the inflation period the universe smoothly connected with the radiation Big Bang phase. Logamediate inflation in standard gravity with the presence of tachyon field on the brane has been considered in Ref.[@8], and non-tachyon warm-logamediate inflationary model has been considered in Ref.[@6-n]. Also, the warm tachyon inflation in standard gravity has been studied in Ref.[@3]. To the best of our knowledge, warm tachyon inflation model in the context of logamediate scenario has not been yet considered.\
The paper is organized as: In section II, we give a brief review about the tachyonic warm inflationary universe and its perturbation parameters in high dissipative regime. In section III, we consider high dissipative warm-logamediate inflationary phase in two cases: 1- Dissipative parameter $\Gamma$ as a function of tachyon field $\phi$. 2- A constant dissipative parameter $\Gamma$. In this section we also, investigate the cosmological perturbations and non-Gaussianity. Finally in section IV, we close by concluding remarks.
Tachyon warm inflationary universe
==================================
Tachyonic inflation model in a spatially flat Friedmann Robertson Walker (FRW) is described by an effective fluid which is recognized by energy-momentum tensor $T^{\mu}_{\nu}=diag(-\rho_{\phi},p_{\phi},p_{\phi},p_{\phi})$ [@1], where energy density $\rho_{\phi}$, and pressure for the tachyon field are defined by $$\begin{aligned}
\label{1}
\rho_{\phi}=\frac{V(\phi)}{\sqrt{1-\dot{\phi}^2}}~~~~~~~~\\
\nonumber p_{\phi}=-V(\phi)\sqrt{1-\dot{\phi}^2},\end{aligned}$$ respectively, where $\phi$ denotes the tachyon scalar field and $V(\phi)$ is the effective scalar potential associated with the tachyon field. Characteristics of any tachyon field potential are $\frac{dV}{d\phi}<0$ and $V(\phi\rightarrow 0)\rightarrow V_{max}$ [@2]. Friedmann equation for spatially flat universe and conservation equation, in the warm tachyon inflationary scenario are given by [@3] $$\begin{aligned}
\label{2}
H^2=\frac{8\pi}{3m_p^2}[\rho_{\phi}+\rho_{\gamma}],\end{aligned}$$
$$\begin{aligned}
\label{3}
\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})=-\Gamma
\dot{\phi}^2\Rightarrow~~~~\frac{\ddot{\phi}}{1-\dot{\phi}^2}+3H\dot{\phi}+\frac{V'}{V}=-\frac{\Gamma}{V}\sqrt{1-\dot{\phi}^2}\dot{\phi},\end{aligned}$$
and $$\begin{aligned}
\label{4}
\dot{\rho}_{\gamma}+4H\rho_{\gamma}=\Gamma \dot{\phi}^2,\end{aligned}$$ where $H=\frac{\dot{a}}{a},$ is Hubble parameter, $a$ is scale factor, $m_p$ represents the Planck mass and $\rho_{\gamma}$ is energy density of the radiation. Dissipative coefficient $\Gamma$ has the dimension $m_p^5$. In the above equations dots $"^{.}"$ mean derivative with respect to time and prime $"'"$ is derivative with respect to $\phi$. During inflation epoch the energy density (\[1\]) is the order of the potential, i.e. $\rho_{\phi}\sim V$. Tachyon energy density $\rho_{\phi}$ in this era dominates over the energy density of radiation, i.e. $\rho_{\phi}>\rho_{\gamma}$. In slow-roll regime, i.e. $\dot{\phi}\ll
1$ and $\ddot{\phi}\ll(3H+\Gamma/V)\dot{\phi}$ [@4] and when the radiation production in warm inflation era is quasi-stable, i.e. $\dot{\rho}_{\gamma}\ll 4H\rho_{\gamma},
\dot{\rho}_{\gamma}\ll\Gamma\dot{\phi}^2,$ the equations (\[2\]), (\[3\]) and (\[4\]) are reduced to $$\begin{aligned}
\label{5}
H^2=\frac{8\pi}{3m_p^2}V,\end{aligned}$$
$$\begin{aligned}
\label{6}
3H(1+r)\dot{\phi}=-\frac{V'}{V},\end{aligned}$$
$$\begin{aligned}
\label{7}
\rho_{\gamma}=\frac{\Gamma\dot{\phi}^2}{4H},
\end{aligned}$$
where $r=\frac{\Gamma}{3HV}$. The main problem of inflation theory is how to attach the universe to the end of the inflation period. One of the solutions of this problem is the study of inflation in the context of warm inflation [@m1]. In this model radiation is produced during inflation period where its energy density is kept nearly constant. This is phenomenologically fulfilled by introducing the dissipation term $\Gamma$, in the equation of motion as we have seen in Eq.(\[3\]). This term grants a continuous energy transfer from scalar field energy into a thermal bath. In this article we consider high dissipation regimen, i.e. $r\gg 1,$ where the dissipation coefficient $\Gamma$ is much greater than the $3HV$. High dissipative and weak dissipative regimes for non-tachyonic warm inflation have been studied in Refs.[@m1] and [@m2] respectively. The main attention was given to the high dissipative regime. This regime is the more difficult of the two, and when high dissipative regime is understood, the weak dissipative regime could follow. Also, in Refs.[@m2] and [@m3], the study of warm inflation in the context of quantum field theory has been fulfilled in high dissipative regime. Dissipative parameter $\Gamma$ could be a constant or a positive function of $\phi$ by the second law of thermodynamics. In some works $\Gamma$ and potential of inflation have the same form[@3], [@m4]. In Ref.[@3], perturbation parameters for warm tachyon inflation have been obtained where $\Gamma=\Gamma_0=const$ and $\Gamma=\Gamma(\phi)=V(\phi)$. So, in this work we will study the logamediate tachyon warm inflation in high dissipative regime for these two cases.\
From Eqs. (\[6\]) and (\[7\]) in high dissipation regime we get $$\begin{aligned}
\label{8}
\rho_{\gamma}=\frac{m_p^2}{32\pi r}[\frac{V'}{V}]^2.\end{aligned}$$ We introduce the slow-roll parameter $$\begin{aligned}
\label{9}
\epsilon=-\frac{\dot{H}}{H^2}=\frac{m_p^2}{16\pi
r}[\frac{V'}{V}]^2\frac{1}{V}.\end{aligned}$$ Using Eqs. (\[8\]) and (\[9\]) in slow-roll regime ($\rho_{\phi}\sim V$) we find a relation between the energy densities $\rho_{\phi}$ and $\rho_{\gamma}$ as $$\begin{aligned}
\label{10}
\rho_{\gamma}=\frac{\epsilon}{2}\rho_{\phi}.\end{aligned}$$ The second slow-roll parameter $\eta$ is given by $$\begin{aligned}
\label{11}
\eta=-\frac{\ddot{H}}{H\dot{H}}\simeq\frac{m_p^2}{8r\pi V}[\frac{V''}{V}-\frac{1}{2}(\frac{V'}{V})^2].\end{aligned}$$ The warm inflationary condition, i.e. $\ddot{a}>0,$ may be obtained by parameter $\epsilon$, satisfying the relation $$\begin{aligned}
\label{12}
\epsilon <1.\end{aligned}$$ From above equation and Eq.(\[10\]) the tachyon warm inflation epoch could take place when $$\label{13}
\rho_{\phi}>2\rho_{\gamma}.$$ For the tachyon field in warm inflationary universe (in slow-roll and high dissipative regime) the power spectrum of the curvature perturbation and amplitude of tensor perturbation (which would produce gravitational waves during inflation) are given by [@3] $$\begin{aligned}
\label{14}
P_R\simeq\frac{\sqrt{3}}{30\pi^2}\exp(-2\Im(\phi))[(\frac{1}{\epsilon})^3\frac{9m_p^4}{128\pi^2r^2\sigma
V}]^{\frac{1}{4}}\end{aligned}$$ $$\begin{aligned}
\label{15}
P_T=\frac{16\pi}{m_p^2}(\frac{H}{2\pi})^2\coth[\frac{k}{2T}]\simeq\frac{32V}{3m_p^4}\coth[\frac{k}{2T}],\end{aligned}$$ respectively. Temperature $T$ in extra factor $\coth[\frac{k}{2T}]$ denotes the temperature of the thermal background of gravitational wave [@6] and $$\begin{aligned}
\label{16}
\Im(\phi)=-\int[\frac{1}{3Hr}(\frac{\Gamma}{V})'+\frac{9}{8}\frac{V'}{V}[1-\frac{(\ln
\Gamma)'(\ln V)'}{36H^2r}]]d\phi.\end{aligned}$$ In high dissipative regime ($r\gg 1$), from Eqs.(\[14\]) and (\[15\]), tensor-scalar ratio is obtained $$\begin{aligned}
\label{17}
R(k_0)\approx
\frac{240\sqrt{3}}{25m_p^2}[\frac{r^{\frac{1}{2}}\epsilon
H^3}{T_r}\exp[2\Im(\phi)]\coth(\frac{k}{2T})]|_{k=k_0}.\end{aligned}$$ $R$ is important parameter. We can use the seven-year Wilkinson Microwave Anisotropy Probe (WMAP7) data to find an upper bound for $R$, from these results we have $P_R\simeq 2.28\times
10^{-9}, R=0.21<0.36$ [@2-i]. Spectral indices $n_g$ and $n_s$ were presented in [@3] $$\begin{aligned}
\label{18}
n_g=-2\epsilon~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,\\
\nonumber
n_s=1-[\frac{3}{2}\eta+\epsilon(\frac{2V}{V'}[2\Im'(\phi)-\frac{r'}{4r}]-\frac{5}{2})].\end{aligned}$$ From above equations we could find interesting inflationary parameters in slow-roll approximation in the context of logamediate inflation. The non-Gaussainty level for warm tachyon inflation in high dissipative regime may be obtained by $f_{NL}$ parameter ―cite[2-n]{} $$\begin{aligned}
\label{Ga}
f_{NL}=-\frac{5}{3}(\frac{\dot{\phi}}{H})[\frac{1}{H}\ln(\frac{k_F}{H})(\frac{V'''(\phi_0(t_F))+2k_F^2V'(\phi_0(t_F))}{\Gamma})]\end{aligned}$$ where $t_F$ is the time when the last three wave-vector thermalize and $k_{F}=\sqrt{\frac{\Gamma H}{V}}$ is freeze-out momentum.
Logamediate inflation
=====================
In this section we consider high dissipative warm tachyon logamediate inflation. In logamediate inflation the scale factor follows the law $$\begin{aligned}
\label{19}
a(t)=a_0\exp(A[\ln t]^{\lambda}) ~~~~~~\lambda>1,\end{aligned}$$ where $A$ is a positive constant. From above scale factor, the number of e-folds may be found $$\begin{aligned}
\label{20}
N=\int_{t1}^{t2} H dt=A[(\ln t_2)^{\lambda}-(\ln t_1)^{\lambda}]\end{aligned}$$ We would like to consider our model in two important cases [@3]: 1- $\Gamma$ is a function of $\phi$ ($\Gamma=f(\phi)=\Gamma_1V(\phi)$). 2- $\Gamma$ is a constant parameter.
$\Gamma=\Gamma{(\phi)}=\Gamma_1V(\phi)$
---------------------------------------
At the late time and by using Eqs. (\[5\]), (\[6\]) and (\[19\]) with $\Gamma=\Gamma_1V(\phi)$ where $\Gamma_1>0$, we get tachyon scalar field $\phi$ $$\begin{aligned}
\label{21}
\Gamma_1\dot{\phi}^2=-\frac{2\dot{H}}{H}=\frac{2}{t}\Rightarrow \sqrt{\Gamma_1}(\phi-\phi_0)=2\sqrt{2t}\end{aligned}$$ where $$\begin{aligned}
\label{22}
\dot{H}=\frac{\lambda A(\ln t)^{\lambda-1}}{t^2}[\frac{\lambda-1}{\ln t}-1]\end{aligned}$$ At the late time the first term in the above equation is omitted. Effective potential in terms of tachyon field is given by $$\begin{aligned}
\label{23}
V(\phi)=\frac{24 m_P^2(\lambda A)^2}{\Gamma_1^2\pi}\frac{[\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{2\lambda-2}}{(\phi-\phi_0)^4}\end{aligned}$$ The Hubble parameter in term of tachyon field $\phi$ becomes $$\begin{aligned}
\label{24}
H(\phi)=\frac{\lambda A [\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{\lambda-1}}{\Gamma_1(\phi-\phi_0)^2}\end{aligned}$$ From equation (\[23\]), $V(\phi)$ has the characteristic of tachyon field potential ($\frac{dV}{d\phi}<0$ and $V(\phi\rightarrow 0)\rightarrow V_{max}$), also these potentials at the late time do not have a minimum [@5]. Slow-roll parameters $\epsilon$ and $\eta$ in term of tachyon field $\phi$ are obtained from Eqs. (\[9\]) and (\[11\]). $$\begin{aligned}
\label{25}
\epsilon =―frac{[\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{1-\lambda}}{\lambda A}\end{aligned}$$
$$\begin{aligned}
\label{26}
\eta =\frac{2[\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{1-\lambda}}{\lambda A}\end{aligned}$$
respectively. From Eq.(\[20\]), the number of e-folds between two fields $\phi_1=\phi(t_1)$ and $\phi_2=\phi(t_2)$ is given by $$\begin{aligned}
\label{27}
N(\phi)=A([\ln(\Gamma_1\frac{(\phi_2-\phi_0)^2}{8})]^{\lambda}-[\ln(\Gamma_1\frac{(\phi_1-\phi_0)^2}{8})]^{\lambda})\end{aligned}$$ $\phi_1$ is obtained at the begining of inflation ($\epsilon=1$), $\phi_1=\frac{2\sqrt{2}}{\sqrt{\Gamma_1}}\exp(\frac{(\lambda A)^{\frac{1}{1-\lambda}}}{2})+\phi_0$. The value of $\phi_2$ could be determined in terms of $N$, $A$ and $\lambda$ parameters $$\begin{aligned}
\label{28}
\phi_2=\frac{2\sqrt{2}}{\sqrt{\Gamma_1}}\exp([\frac{1}{2}(\frac{N}{A}+(\lambda A)^{\frac{\lambda}{1-\lambda}})^{\frac{1}{\lambda}}])+\phi_0\end{aligned}$$
In logamediate inflation, we obtain perturbation parameters in term of tachyon field. Firstly, from Eq.(\[16\]) in this case ($\Gamma=\Gamma_1V$) we find $$\begin{aligned}
\label{29}
\Im(\phi)_1=-―frac{9}{8}\ln[V(\phi)]-\frac{3―Gamma_1}{8\lambda A}\gamma[2-\lambda,\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]\end{aligned}$$ where $\gamma[a,t],$ is incomplete gamma function [@gamma]. From above equation and Eq.(\[14\]) we obtain spectrum of curvature perturbation in slow-roll limit $$\begin{aligned}
\label{}
P_R=\alpha [\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{\frac{3}{4}(\lambda-1)}\end{aligned}$$ where $\alpha=\frac{\exp(-2\Im(\phi)_1)}{10\Gamma_1\pi^2}(\frac{3m_p^2\Gamma_1^2(\lambda A)^3}{1024\pi\sigma})^{\frac{1}{4}}$. In high dissipative regime ($r\gg 1$) we have $$\begin{aligned}
\label{30}
P_R=\beta\frac{[\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{\frac{21}{4}(\lambda-1)}}{(\phi-\phi_0)^4}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\nonumber
=\frac{\Gamma_1^2\beta}{64}\exp(-2(\frac{N}{A}+(\lambda A)^{\frac{\lambda}{1-\lambda}})^{\frac{1}{\lambda}})(\frac{N}{A}+(\lambda A)^{\frac{\lambda}{1-\lambda}})^{\frac{21(\lambda-1)}{4\lambda}}\end{aligned}$$ where $\beta=\frac{1}{10\Gamma_1\pi^2}(\frac{3m_p^2\Gamma_1^2(\lambda A)^3}{1024\pi\sigma})^{\frac{1}{4}}(\frac{24m_p^2(\lambda A)^2}{\Gamma_1^2\pi})^{\frac{9}{4}}$. This parameter may be constrained by WMAP7 data [@2-i]. The amplitude of tensor perturbation in this case, from Eq.(\[15\]) becomes $$\begin{aligned}
\label{31}
P_T=\frac{256 (\lambda A)^2}{\Gamma_1^2\pi}\frac{[\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{2\lambda-2}}{(\phi-\phi_0)^4}\coth[\frac{k}{2T}]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\nonumber
=\frac{4(\lambda A)^2}{\pi}\exp(-2(\frac{N}{A}+(\lambda A)^{\frac{\lambda}{1-\lambda}})^{\frac{1}{\lambda}})(\frac{N}{A}+(\lambda A)^{\frac{\lambda}{1-\lambda}})^{\frac{2\lambda-1}{\lambda}}\coth[\frac{k}{2T}]\end{aligned}$$ From Eq.(\[18\]) the spectral indices $n_g$ and $n_s$ are given by $$\begin{aligned}
\label{32}
n_g=-2\frac{[\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{1-\lambda}}{\lambda A}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\nonumber
n_s=1-\frac{3}{2}\eta+\frac{3}{4}\epsilon(1+\epsilon)\simeq 1-\frac{9[\ln(\Gamma_1\frac{(\phi-\phi_0)^2}{8})]^{1-\lambda}}{4\lambda A}\\
\nonumber
=1-\frac{9}{4\lambda A}(\frac{N}{A}+(\lambda A)^{\frac{\lambda}{1-\lambda}})^{\frac{1-\lambda}{\lambda}}\end{aligned}$$ In Fig.(1), the dependence of spectral index on the number of e-folds of inflation is shown (for $\lambda=5$ and $\lambda=50$ cases). It is observed that small values of number of e-folds are assured for large values of $\lambda$ parameter.
From Eq.(\[17\]), we could find the tensor-scalar ratio as $$\begin{aligned}
\label{33}
R=\frac{256(\lambda A)^2}{\Gamma_1^2\beta}(\frac{N}{A}+(\lambda A)^{\frac{\lambda}{1-\lambda}})^{\frac{-13\lambda+17}{4\lambda}}\coth[\frac{k}{2T}]\end{aligned}$$ The above parameter is found from WMAP7 observational data. Non-Gaussianity in term of tachyon field in this case is obtained from Eq.(\[Ga\]) $$\begin{aligned}
\label{34}
f_{NL}=\frac{5}{3}\Gamma_1\ln(\Gamma_1)[\ln(\frac{\Gamma_1(\phi-\phi_0)^2}{8})]^{1-\lambda}=\frac{5}{3}\Gamma_1\ln(\Gamma_1)[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1-\lambda}{\lambda}}\end{aligned}$$ Where $\lambda>1$ this parameter have small amount at the late time. FIG.(1) shows the Harrison-Zeldovich spectrum, i.e. $n_s=1,$ could approximately obtained for $(\lambda,N)=(50,60)$. In this case, we have small level of non-Gaussianity. From Eq.(\[32\]) and (\[33\]) we can find the tensor-scalar ratio $R$ versus spectral index $n_s$. $$\begin{aligned}
\label{}
R=\frac{256(\lambda A)^2}{\Gamma_1^2\beta}(\frac{4\lambda A}{9}(1-n_s))^{\frac{13\lambda-17}{4(\lambda-1)}}\coth[\frac{k}{2T}]\end{aligned}$$ In Fig.(2), two trajectories in the $n_s-R$ plane are shown. There is a range of values of $R$ and $n_s$ which is compatible with the WMAP7 data. The scale-invariant spectrum (Harrison-Zeldovich spectrum, i.e. $n_s=1$) may be obtained for $(\lambda,\Gamma_1)=(50,37.5)$.
$\Gamma=\Gamma_0=const$
------------------------
By using Eqs. (\[5\]), (\[6\]) and (\[19\]) with $\Gamma=\Gamma_0$ at the late time , we get tachyon scalar field $\phi$ $$\begin{aligned}
\label{34}
\phi-\phi_0=\frac{\Upsilon_{\lambda}[t]}{\varpi}\end{aligned}$$ where $\varpi=\frac{\sqrt{4\pi\Gamma_0}}{m_p\sqrt{3}\lambda A 2^{\lambda}}$ and $\Upsilon=\gamma[\lambda,\frac{\ln t}{2}]$ ($\gamma[a,t]$ is incomplete gamma function [@gamma]). The Hubble parameter and effective potential are obtained as $$\begin{aligned}
\label{35}
H(\phi)=\frac{\lambda A[\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)]]^{\lambda-1}}{\Upsilon^{-1}[\varpi(\phi-\phi_0)]}\end{aligned}$$ and $$\begin{aligned}
\label{36}
V(\phi)=\frac{3m_p^2(\lambda A)^2[\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)]]^{2\lambda-2}}{8\pi(\Upsilon^{-1}[\varpi(\phi-\phi_0)])^2}\end{aligned}$$ respectively. $\Upsilon^{-1}$ is inverse function of the $\Upsilon$. From Eqs.(\[9\]) and (\[11\]) we could find slow-roll parameters $\epsilon$ and $\eta$ in term of tachyon field (\[34\]), $$\begin{aligned}
\label{37}
\epsilon=\frac{(\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)])^{1-\lambda}}{\lambda A}\end{aligned}$$ and $$\begin{aligned}
\label{38}
\eta=\frac{2(\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)])^{1-\lambda}}{\lambda A}\end{aligned}$$ respectively. Using Eq.(\[20\]) the number of e-folds is obtained as $$\begin{aligned}
\label{39}
N=A[(\ln[\Upsilon^{-1}_{\lambda}(\varpi(\phi_2-\phi_0)])^{\lambda}-(\ln[\Upsilon^{-1}_{\lambda}(\varpi(\phi_1-\phi_0))])^{\lambda}]\end{aligned}$$ At the begining of inflation ($\epsilon=1$), $\phi_1=\frac{\Upsilon[\exp([A\lambda]^{\frac{1}{1-\lambda}})]}{\varpi}$, so tachyon field $\phi_2$ at the end of inflation in terms of number of e-folds becomes $$\begin{aligned}
\label{40}
\phi_2-\phi_0=\frac{\Upsilon[\exp([\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1}{\lambda}})]}{\varpi}\end{aligned}$$ In this case ($\Gamma=\Gamma_0$), perturbation parameters may be obtained in terms of tachyon field. Therefore from Eq.(\[16\]) we have $$\begin{aligned}
\label{41}
\Im(\phi)_2=-\frac{\ln V}{8}\end{aligned}$$ We obtain spectrum of curvature perturbation in slow-roll limit, from above equation and Eq.(\[14\]) $$\begin{aligned}
\label{42}
P_R=\alpha' \frac{(\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)])^{\frac{9\lambda-9}{4}}}{(\Upsilon^{-1}[\varpi(\phi-\phi_0)])^{\frac{3}{2}}}=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\nonumber
\alpha'[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{9\lambda-9}{4\lambda}}\exp(-\frac{3}{2}[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1}{\lambda}})\end{aligned}$$ where $\alpha'=\frac{\sqrt{3}}{30\pi^2}(\frac{3m_p(\lambda A)^3}{8\pi})^{\frac{3}{4}}(\frac{9}{16\pi\sigma\Gamma_0})^{\frac{1}{4}}$. This parameter may be constrained by WMAP7 observational data [@2-i]. From Eqs.(\[15\]) and (\[36\]), the amplitude of tensor perturbation in this case, becomes $$\begin{aligned}
\label{43}
P_T=\frac{4(\lambda A)^2}{\pi m_p^2}\frac{(\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)])^{2\lambda-2}}{(\Upsilon^{-1}[\varpi(\phi-\phi_0)])^2}=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\
\nonumber
\frac{4(\lambda A)^2}{\pi m_p^2}[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{2\lambda-2}{\lambda}}\exp(-2[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1}{\lambda}})\coth[\frac{k}{2T}]\end{aligned}$$ From Eq.(\[18\]) the spectral indices $n_s$ and $n_g$ are given by $$\begin{aligned}
\label{44}
n_g=-\frac{2(\ln \Upsilon^{-1}[\varpi(\phi-\phi_0)])^{1-\lambda}}{\lambda A}~~~~~~~~~~~~~~~~~~~~~~~~~\\
\nonumber
n_s=1-\frac{3}{4}\eta-\frac{9}{4}\epsilon=1-\frac{13(\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)])^{1-\lambda}}{4\lambda A}\\
\nonumber
=1-\frac{13}{4\lambda A}[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1-\lambda}{\lambda}}~~~~~~~~~~~~~~~~~\end{aligned}$$ In Fig.(3), the dependence of spectral index on the number of e-folds of inflation is shown (for $\lambda=5$ and $\lambda=50$ cases). It is observed that small values of number of e-folds are assured for large values of $\lambda$ parameter.
We could find the tensor-scalar ratio as (from Eq.(\[17\])) $$\begin{aligned}
\label{45}
R=\frac{9(\lambda A)^2}{\pi m_p^2\alpha'}[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1-\lambda}{4\lambda}}\exp(-\frac{1}{2}[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1}{\lambda}})\coth[\frac{k}{2T}]\end{aligned}$$ this parameter is found from WMAP7 observational data [@2-i]. From Eq.(\[Ga\]) the non-Gaussianity in this case is obtained as $$\begin{aligned}
\label{}
f_{NL}=\frac{5}{3}(\ln\Upsilon^{-1}[\varpi(\phi-\phi_0)])^{1-\lambda}=\frac{5}{3}[\frac{N}{A}+(\lambda A)^{\frac{1}{1-\lambda}}]^{\frac{1-\lambda}{\lambda}}\end{aligned}$$ Where $\lambda>1$ the non-Gaussianity has small level at the late time. FIG.(3) shows the scale invariant spectrum, (Harrison-Zeldovich spectrum, i.e. $n_s=1$) could be approximately obtained for ($\lambda,$$N$)=(50,60). In this case, the small level of non-Gaussianity is found. From Eq.(\[44\]) and (\[45\]) we can find the tensor-scalar ratio $R$ versus spectral index $n_s$. $$\begin{aligned}
\label{}
R=\frac{9(\lambda A)^2}{\pi m_p^2\alpha'}[\frac{4\lambda A}{3}(1-n_s)]^{\frac{1}{4}}\exp(-\frac{1}{2}[\frac{4\lambda A}{3}(1-n_s)]^{\frac{1}{1-\lambda}})\coth[\frac{k}{2T}]\end{aligned}$$ In Fig.(4), two trajectories in the $n_s-R$ plane are shown. There is a range of values of $R$ and $n_s$ which is compatible with the WMAP7 data. The scale-invariant spectrum (Harrison-Zeldovich spectrum, i.e. $n_s=1$) may be obtained for $(\lambda,\Gamma_0)=(50,37.5)$.
Using WMAP7 data, $P_R(k_0)\simeq 2.28\times 10^{-9}$, $R(k_0)\simeq 0.21$ and the characteristic of warm inflation $T>H$ [@3], we may restrict the values of temperature to $T_r>5.47\times 10^{-5}$ using Eqs.(\[14\]),(\[17\]) or equivalently Eqs.(\[42\]), (\[45\]), (see Fig.(5)). We have chosen $k_0=0.002 Mpc^{-1}$ and $T\simeq T_r$.
![In this graph we plot the Hubble parameter $H$ in term of the temperature $T_r$. We can find the minimum amount of temperature $T_r=5.47\times 10^{-5}$ in order to have the necessary condition for warm inflation model ($T_r>H$). []{data-label="fig:F3"}](a.eps){width="10cm"}
Conclusion
==========
In this article we have investigated the warm-tachyon-logamediate inflationary model. In this model we have found an everlasting form of potential. This form of potential agrees with tachyon potential properties. Warm inflation is an important model as a mechanism which gives an end for inflation epoch. Therefore we have considered the tachyon warm-logamediate inflationary model. This model has been developed for $\Gamma$ as a function of tachyon field $\phi$ and for $\Gamma=\Gamma_0$. For these two cases we have extracted the form of potential and Hubble parameters as a function of tachyon field $\phi$. Explicit expressions for tensor-scalar ratio $R$, and spectrum indices $n_g$ and $n_s$ in slow-roll were obtained. We also have constrained these parameters by WMAP7 results.\
In $\Gamma=\Gamma(\phi)=V(\phi)$ case we have obtained the form of tachyon field as $\phi\propto\sqrt{t}$ which agrees with intermediate model [@2-n], where $\phi\propto\sqrt{(1-f)t}$. The form of potential in logamediate model is $V(\phi)\propto\frac{(\ln\phi)^{\lambda-1}}{\phi^2}$ but in intermediate model tachyon potential has the form $V(\phi)\propto\phi^{-4\frac{f-1}{2f-1}}$. These two forms of potential have the special properties of the tachyon potentials. On the other hand the Harrison-Zeldovich spectrum of density perturbation (i.e. $n_s=1$) was obtained in warm-tachyon-intermediate inflation model [@2-n], for fixed value of parameter $f$ where $f=\frac{2}{3}$, but in warm-tachyon-logamediate inflation model the scale invariant spectrum is not obtained for one fixed value of parameter $\lambda$. In logamediate scenario, we could approximately obtain the Harrison-Zeldovich spectrum for large values of parameter $\lambda$ ($\lambda\geq50$). FIGs.(1) and (3) show the scale invariant spectrum may be presented for ($\lambda, N)=(50,60)$. FIG.(2) shows the Harrison-Zeldovich spectrum could be approximately obtained for $(\lambda,\Gamma_1)=(50,37.5)$. FIG.(4) shows the Harrison-Zeldovich spectrum could be approximately obtained for $(\lambda,\Gamma_0)=(50,37.5)$. Scale-invariant spectrum in logamediate scenario is given by two parameters ($\lambda,N$) or ($\lambda,\Gamma_i$), but in intermediate inflation the scale-invariant spectrum was presented by one parameter $f$. [@2-n] We also have found the small level of non-Gaussianity for large value of parameter $\lambda$ at the late time.
[99]{} M. R. Setare and V. Kamali, JCAP 08,034, (2012). A. Guth,Phys. Rev. D 23, 347, (1981); A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220, (1982); A complete description of inflationary scenarios can be found in the book by A. Linde, “Particle physics and inflationary cosmology,” (Gordon and Breach, New York, 1990). E. Komatsu et al. arXiv:1001.4538 \[astro-ph.CO\]; B. Gold et al. arXiv:1001.4555 \[astro-ph.GA\]; D. Larson et al.arXiv:1001.4635 \[astro-ph.CO\]. A. Berera, Phys. Rev. Lett. 75, 3218, (1995); Phys. Rev. D 55, 3346, (1997). L. M. H. Hall, I. G. Moss and A. Berera, Phys.Rev.D 69, 083525 (2004); I.G. Moss, Phys.Lett.B 154, 120 (1985); A. Berera, Nucl.Phys B 585, 666 (2000). Y. -F. Cai, J. B. Dent and D. A. Easson, Phys. Rev. D [**83**]{}, 101301 (2011) \[arXiv:1011.4074 \[hep-th\]\]; R. Cerezo and J. G. Rosa, JHEP [**1301**]{}, 024 (2013) \[arXiv:1210.7975 \[hep-ph\]\]; S. Bartrum, A. Berera and J. G. Rosa, Phys. Rev. D [**86**]{}, 123525 (2012) \[arXiv:1208.4276 \[hep-ph\]\]; M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, JCAP [**1301**]{}, 016 (2013) \[arXiv:1207.0445 \[hep-ph\]\]; M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, Phys. Lett. B [**712**]{}, 425 (2012) \[arXiv:1110.3971 \[hep-ph\]\]; M. Bastero-Gil, A. Berera and J. G. Rosa, Phys. Rev. D [**84**]{}, 103503 (2011) \[arXiv:1103.5623 \[hep-th\]\]. A. Sen, JHEP 0204, 048, (2002). A. Sen, Mod. Phys. Lett. A 17, 1797, (2002); M. Sami, P. Chingangbam and T. Qureshi, Phys. Rev. D 66, 043530, (2002). G. W. Gibbons, (2002) Phys. Lett. B 537, 1. J. D. Barrow, Class. Quantum Grav. 13, 2965 (1996). J. D. Barrow, Phys. Rev. D 51, 2729 (1995). E. R. Harrison, Phys. Rev. D1, 2726 (1970), Y. B. Zeldovich, Mon. Not. Roy. Astron. Soc. 160, 1P (1972). J. D Barrow and A. R. Liddle, Phys. Rev. D 47, R5219 (1993); A. Vallinotto, E. J. Copeland, E. W. Kolb, A. R. Liddle and D. A. Steer, Phys. Rev. D 69, 103519 (2004); A. A. Starobinsky JETP Lett. 82, 169 (2005). J. D. Barrow and N. J. Nunes, Phys. Rev. D 76, 043501 (2007); J. Yokoyama and K. Maeda, Phys.Lett. B 207, 31 (1988). A. K. Sanyal, Phys. Lett. B. 645, 1 (2007). P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003). P. G. Ferreira and M. Joyce, Phys. Rev. D 58, 023503 (1998); J. D. Barrow, Phys. Rev. D 51, 2729 (1995). C. Campuzano, S. del. Campo, R. Herrera, E. Rojas and J. Saavedra, Phys.Rev.D 80, 123531, (2009). C. Armendariz-Picon, T. Damour and V. Mukhanov, Phys. Lett. B. 458, 209 (1999). R. Herrera, and M. Olivares, Mod. Phys. Lett. A 1250101 (2012). R. Herrera, and M. Olivares, arXiv:1205.2365. R. Herrera, S. del Campo, C. Campuzano, JCAP 0610, 009, (2006). A. Sen, JHEP 9808, 012 (1998). A. Berera and L.Z. Fang, Phys. Rev. Lett. 74 1912 (1995); H. P. de Oliveria and R. O. Ramos, Phys. Rev. D57 741 (1998); H. P. de Oliveira and S. E. Jors, Phys. Rev. D64 063513 (2001). A. Berera, M. Gleiser and R. O. Ramos, Phys. Rev. D58 123508 (1998); Phys. Rev. Lett. 83 264 (1999). J. Yokoyama, A. Linde, Phys. Rev. D60, 083509 (1999); A. Berera and R. O. Ramos, Phys. Rev. D63 103509 (2001); A. Berera and T. W. Kephart, Phys.Lett. B456 (1999); Phys. Rev. Lett. 83 1084 (1999). M. A. Cid, S. del Campo and R. Herrera, JCAP [**0710**]{} (2007) 005 \[arXiv:0710.3148 \[astro-ph\]\]. K. Bhattacharya, S. Mohanty and A. Nautiyal, Phys. Rev. Lett.97, 251301, (2006). S. del Campo and R. Herrera, Phys. Rev. D 76, 103503 (2007). Eds. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972), 9th printing; G. Arfken, The incomplete gamma function and related functions, Mathematical Methods for Physicists, 3rd edn. (Academic Press, 1985).
[^1]: rezakord@ipm.ir
[^2]: vkamali1362@gmail.com
|
---
abstract: 'Let $L$ be a nef line bundle on a projective scheme $X$ in positive characteristic. We prove that the augmented base locus of $L$ is equal to the union of the irreducible closed subsets $V$ of $X$ such that $L\vert_V$ is not big. For a smooth variety in characteristic zero, this was proved by Nakamaye using vanishing theorems.'
address:
- 'Department of Mathematics, Imperial College London, London SW7 2AZ, UK'
- 'Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA'
- 'Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA'
author:
- Paolo Cascini
- James M^c^Kernan
- Mircea Mustaţă
title: The augmented base locus in positive characteristic
---
[^1]
Introduction
============
Let $X$ be a projective scheme over an algebraically closed field $k$, and $L$ a line bundle on $X$. The base locus $\operatorname{Bs}(L)$ of $L$ is the closed subset of $X$ consisting of those $x\in X$ such that every section of $L$ vanishes at $x$. It is easy to see that if $m_1$ and $m_2$ are positive integers such that $m_1$ divides $m_2$, then $\operatorname{Bs}(L^{m_2})\subseteq \operatorname{Bs}(L^{m_1})$. It follows from the Noetherian property that $\operatorname{Bs}(L^m)$ is independent of $m$ if $m$ is divisible enough; this is the *stable base locus* $\operatorname{\mathbf{SB}}(L)$ of $L$.
The stable base locus is a very interesting geometric invariant of $L$, but it is quite subtle: for example, there are numerically equivalent Cartier divisors whose stable base loci are different. Nakamaye introduced in [@Nakamaye] the following upper approximation of $\operatorname{\mathbf{SB}}(L)$, the *augmented base locus* $\operatorname{\mathbf{B}_{+}}(L)$. If $L\in\operatorname{Pic}(X)$ and $A\in \operatorname{Pic}(X)$ is ample, then $$\operatorname{\mathbf{B}_{+}}(L):=\operatorname{\mathbf{SB}}(L^m\otimes A^{-1}),$$ for $m\gg 0$. It is easy to check that this is well-defined, it is independent of $A$, and only depends on the numerical equivalence class of $L$. The following is our main result.
\[thm\_main\] Let $X$ be a projective scheme over an algebraically closed field of positive characteristic. If $L$ is a nef line bundle on $X$, then $\operatorname{\mathbf{B}_{+}}(L)$ is equal to $L^{\perp}$, the union of all irreducible closed subsets $V$ of $X$ such that $L\vert_V$ is not big.
We note that since $L$ is nef, for an irreducible closed subset $V$ of $X$, the restriction $L\vert_V$ is not big if and only if $V$ has positive dimension and $(L\vert_V^{\dim(V)})=0$. When $X$ is a smooth projective variety in characteristic zero, the above theorem was proved in [@Nakamaye], making use of the Kawamata-Viehweg vanishing theorem. It is an interesting question whether the result holds in characteristic zero when the variety is singular.
The proof of Theorem \[thm\_main\] makes use in an essential way of the Frobenius morphism. The following is a key ingredient in the proof.
\[thm\_main2\] Let $X$ be a projective scheme over an algebraically closed field of positive characteristic. If $L$ is a nef line bundle on $X$ and $D$ is an effective Cartier divisor such that $L(-D)$ is ample, then $\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{B}_{+}}(L\vert_D)$.
In the proofs of the above results we make use of techniques introduced by Keel in [@Keel]. In fact, if we replace in Theorem \[thm\_main2\] the two augmented base loci by the corresponding stable base loci, we recover one of the main results in [@Keel]. We give a somewhat simplified proof of this result (see Corollary \[cor2\_thm\_Keel\] below), and this proof extends to give also Theorem \[thm\_main2\].
In the next section we recall some basic facts about augmented base loci. The proofs of Theorems \[thm\_main2\] and \[thm\_main\] are then given in §3.
Acknowledgment
--------------
We are indebted to Rob Lazarsfeld for discussions that led to some of the results in this paper. We would also like to thank Seán Keel for several very useful discussions and the referee for some useful comments.
Augmented base loci and big line bundles
========================================
In this section we review some basic facts about the augmented base locus. This notion is usually defined for integral schemes. However, even if one is only interested in this restrictive setting, for the proof of Theorem \[thm\_main\] we need to also consider possibly reducible, or even non-reduced schemes. We therefore define the augmented base locus in the more general setting that we will need. Its general properties follow as in the case of integral schemes, for which we refer to [@ELMNP].
Let $X$ be a projective scheme over an algebraically closed field $k$. If $L$ is a line bundle on $X$ and $s\in H^0(X,L)$, then we denote by $Z(s)$ the zero-locus of $s$ (with the obvious scheme structure). Note that $Z(s)$ is defined by a locally principal ideal, but in general it is not an effective Cartier divisor (if $X$ is reduced, then $Z(s)$ is an effective Cartier divisor if and only if no irreducible component of $X$ is contained in $Z(s)$). The base locus of $L$ is by definition the closed subset of $X$ given by $$\operatorname{Bs}(L):=\bigcap_{s\in H^0(X,L)}Z(s)_{\rm red}.$$
If $m$ is a positive integer and $s\in H^0(X,L)$, then it is clear that $Z(s)_{\rm red}=Z(s^{\otimes m})_{\rm red}$, hence $\operatorname{Bs}(L^m)\subseteq \operatorname{Bs}(L)$. More generally, we have $\operatorname{Bs}(L^{mr})\subseteq \operatorname{Bs}(L^r)$ for every $m,r\geq 1$, hence by the Noetherian property there is $m_0\geq 1$ such that $$\operatorname{\mathbf{SB}}(L):=\bigcap_{r\geq 1}\operatorname{Bs}(L^r)$$ is equal to $\operatorname{Bs}(L^m)$ whenever $m$ is divisible by $m_0$. The closed subset $\operatorname{\mathbf{SB}}(L)$ of $X$ is the *stable base locus* of $L$. It follows by definition that $\operatorname{\mathbf{SB}}(L)=\operatorname{\mathbf{SB}}(L^r)$ for every $r\geq 1$.
Since $X$ is projective, every line bundle is of the form $\cO_X(D)$, for some Cartier divisor $D$ (see [@Nakai]). We will sometimes find it convenient to work with Cartier divisors, rather than line bundles. Let $\operatorname{Cart}(X)_{\QQ}:=\operatorname{Cart}(X)\otimes_{\ZZ}\QQ$ denote the group of Cartier $\QQ$-divisors and $\operatorname{Pic}(X)_{\QQ}:=\operatorname{Pic}(X)\otimes_{\ZZ}\QQ$. For a Cartier divisor $D$ , we put $\operatorname{\mathbf{SB}}(D)=\operatorname{\mathbf{SB}}(\cO_X(D))$. Since $\operatorname{\mathbf{SB}}(D)=\operatorname{\mathbf{SB}}(rD)$ for every $r\geq 1$, the definition extends in the obvious way to $\operatorname{Cart}(X)_{\QQ}$.
Given a Cartier $\QQ$-divisor $D$, the augmented base locus of $D$ is $$\operatorname{\mathbf{B}_{+}}(D):=\bigcap_A\operatorname{\mathbf{SB}}(D-A),$$ where the intersection is over all ample Cartier $\QQ$-divisors on $X$. It is easy to see that if $A_1$ and $A_2$ are ample Cartier $\QQ$-divisors such that $A_1-A_2$ is ample, then $\operatorname{\mathbf{SB}}(D-A_2)\subseteq\operatorname{\mathbf{SB}}(D-A_1)$. It follows from the Noetherian property that there is an ample Cartier $\QQ$-divisor $A$ such that $\operatorname{\mathbf{B}_{+}}(D)=\operatorname{\mathbf{SB}}(D-A)$. Furthermore, in this case if $A'$ is ample and $A-A'$ is ample, too, then $\operatorname{\mathbf{B}_{+}}(D)=\operatorname{\mathbf{SB}}(D-A')$. It is then clear that if $H$ is any ample Cartier divisor on $X$, then for $m\gg 0$ we have $$\operatorname{\mathbf{B}_{+}}(D)=\operatorname{\mathbf{SB}}\left(D-\frac{1}{m}H\right)=\operatorname{\mathbf{SB}}(mD-H).$$
The following properties of the augmented base locus are direct consequences of the definition (see [@ELMNP §1]).
1. For every Cartier $\QQ$-divisor $D$, we have $\operatorname{\mathbf{SB}}(D)\subseteq\operatorname{\mathbf{B}_{+}}(D)$.
2. If $D_1$ and $D_2$ are numerically equivalent Cartier $\QQ$-divisors, then $\operatorname{\mathbf{B}_{+}}(D_1)=\operatorname{\mathbf{B}_{+}}(D_2)$.
If $D$ is a Cartier divisor and $L=\cO_X(D)$, we also write $\operatorname{\mathbf{B}_{+}}(L)$ for $\operatorname{\mathbf{B}_{+}}(D)$.
\[lem1\] If $L$ is a line bundle on the projective scheme $X$, and $Y$ is a closed subscheme of $X$, then
1. $\operatorname{\mathbf{SB}}(L\vert_Y)\subseteq\operatorname{\mathbf{SB}}(L)$.
2. $\operatorname{\mathbf{B}_{+}}(L\vert_Y)\subseteq \operatorname{\mathbf{B}_{+}}(L)$.
The first assertion follows from the fact that if $s\in H^0(X,L)$, then $Z(s\vert_Y)\subseteq Z(s)$, hence $\operatorname{Bs}(L^m\vert_Y)\subseteq \operatorname{Bs}(L^m)$ for every $m\geq 1$. For the second assertion, fix an ample line bundle $A$ on $X$, and let $m\gg 0$ be such that $\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{SB}}(L^m\otimes A^{-1})$. Since $A\vert_Y$ is ample on $Y$, using i) and the definition of the augmented base locus of $L\vert_Y$, we obtain $$\operatorname{\mathbf{B}_{+}}(L\vert_Y)\subseteq \operatorname{\mathbf{SB}}(L^m\vert_Y\otimes A^{-1}\vert_Y)\subseteq\operatorname{\mathbf{SB}}(L^m\otimes A^{-1})
=\operatorname{\mathbf{B}_{+}}(L).$$
Recall that a line bundle $L$ on an integral $n$-dimensional scheme $X$ is *big* if there is $C>0$ such that $h^0(X,L^m)\geq Cm^{n}$ for $m\gg 0$. Equivalently, this is the case if and only if there are Cartier divisors $A$ and $E$, with $A$ ample and $E$ effective, such that $L^m\simeq
\cO_X(A+E)$ for some $m\geq 1$. We refer to [@positivity §2.2] for basic facts about big line bundles on integral schemes. The following lemma is well-known, but we include a proof for completeness.
\[lem2\] Let $X$ be an $n$-dimensional projective scheme and $L$ a line bundle on $X$. For every coherent sheaf $\cF$ on $X$, there is $C>0$ such that $h^0(X,\cF\otimes L^m)\leq Cm^n$ for every $m\geq 1$.
Let us write $L\simeq A\otimes B^{-1}$ for suitable very ample line bundles $A$ and $B$. For every $m\geq 1$, the line bundle $B^m$ is very ample. By choosing a section $s_m\in H^0(X,B^m)$ such that $Z(s_m)$ does not contain any of the associated subvarieties of $\cF$, we obtain an inclusion $H^0(X,\cF\otimes L^m)\hookrightarrow
H^0(X,\cF\otimes A^m)$. Since $h^0(X,\cF\otimes A^m)=P(m)$ for $m\gg 0$, where $P$ is a polynomial of degree $\leq n$, we obtain the assertion in the lemma.
If $X$ is reduced, and $A$, $D$ are Cartier divisors on $X$ with $A$ ample and $D$ effective, then the restriction of $\cO_X(A+D)$ to every irreducible component $Y$ of $X$ is big (note that the restriction $D\vert_{Y}$ is well-defined and gives an effective divisor on $Y$). As a consequence of the next lemma, we will obtain a converse to this statement.
\[lem3\] Let $X$ be a reduced projective scheme. Given line bundles $L$ and $A$ on $X$, with $A$ ample, if $m\gg 0$ and $s\in H^0(X, L^m\otimes A^{-1})$ is general, then for every irreducible component $Y$ of $X$ such that $L\vert_Y$ is big, we have $Y\not\subseteq Z(s)$.
Note that since $A$ is ample, if $s\in H^0(X,L^m\otimes A^{-1})$ and $Y'$ is an irreducible component of $X$ (considered with the reduced scheme structure) such that $L\vert_{Y'}$ is not big, then $Y'\subseteq Z(s)$.
Suppose that $Y$ is an irreducible component of $X$ (considered with the reduced structure) such that $L\vert_Y$ is big, but such that for infinitely many $m$ we have $Y\subseteq Z(s)$ for every $s\in H^0(X,L^m\otimes A^{-1})$. If $W$ is the union of the other irreducible components of $X$, also considered with the reduced scheme structure, then we have an exact sequence $$0{\xrightarrow{\ \ }}\cO_X{\xrightarrow{\ \ }}\cO_Y\oplus\cO_W{\xrightarrow{\ \ }}\cO_{Y\cap W}{\xrightarrow{\ \ }}0,$$ where $Y\cap W$ denotes the (possibly non-reduced) scheme-theoretic intersection of $Y$ and $W$. After tensoring with $L^m\otimes A^{-1}$ and taking global sections, this induces the exact sequence $$0{\xrightarrow{\ \ }}H^0(X, L^m\otimes A^{-1}){\xrightarrow{\ \ }}H^0(Y,L^m\otimes A^{-1}\vert_Y)\oplus
H^0(W,L^m\otimes A^{-1}\vert_W)$$ $${\xrightarrow{\ \ }}H^0(Y\cap W,L^m\otimes A^{-1}\vert_{Y\cap W}).$$ By assumption, the map $H^0(X, L^m\otimes
A^{-1}){\xrightarrow{\ \ }}H^0(Y,L^m\otimes A^{-1}\vert_Y)$ is zero for infinitely many $m$, in which case the above exact sequence implies $$\label{eq_lem3}
h^0(Y,L^m\otimes A^{-1}\vert_Y)\leq h^0(T, L^m\otimes A^{-1}\vert_{T}).$$ Let $n=\dim(Y)$. Since $\dim(T)\leq n-1$, it follows from Lemma \[lem2\] that we can find $C>0$ such that $$h^0(T, L^m\otimes A^{-1}\vert_{T})\leq C m^{n-1}$$ for all $m$. On the other hand, since $L\vert_Y$ is big, it is easy to see that there is $C'>0$ such that $h^0(Y,L^m\otimes A^{-1}\vert_Y)\geq C'm^n$ for all $m\gg 0$. These two estimates contradict (\[eq\_lem3\]) when $m\gg 0$.
Let $L$ be a line bundle on the reduced projective scheme $X$. If the restriction of $L$ to every irreducible component of $X$ is big, then for every ample line bundle $A$ and every $m\gg 0$, the zero locus of a general section in $H^0(X,L^m\otimes A^{-1})$ defines an effective Cartier divisor on $X$.
Main results
============
In this section we assume that all our schemes are of finite type over an algebraically closed field $k$ of characteristic $p>0$. For such a scheme $X$ we denote by $F=F_X$ the absolute Frobenius morphism of $X$. This is the identity on the topological space, and it takes a section $f$ of $\cO_X$ to $f^p$. Note that $F$ is a finite morphism of schemes (not preserving the structure of schemes over $k$). We will also consider the iterates $F^e$ of $F$, for $e\geq 1$.
Let us recall some basic facts concerning pull-back of line bundles, sections, and subschemes. Suppose that $L$ is a line bundle on $X$ and $Z$ is a closed subscheme of $X$.
1. There is a canonical isomorphism of line bundles $(F^e)^*(L)\simeq L^{p^e}$.
2. The scheme-theoretic inverse image $Z^{[e]}:=(F^e)^{-1}(Z)$ is a closed subscheme of $X$ defined by the ideal $I_Z^{[p^e]}$, such that if $I_Z$ is locally generated by $(f_i)_i$, then $I_Z^{[p^e]}$ is defined by $(f_i^{p^e})_i$. In particular, if $Y$ is another closed subscheme of $X$, having the same support as $Z$, there is some $e$ such that $Y$ is a subscheme of $Z^{[e]}$.
3. If $s\in H^0(Z,L\vert_Z)$, then $(F^e)^*(s)$ is a section in $H^0(Z^{[e]},(F^e)^*(L)\vert_{Z^{[e]}})$, whose restriction to $Z$ gets identified with $s^{\otimes p^e}\in H^0(Z,L^{p^e}\vert_Z)$.
\[lem5\] If $X$ is a projective scheme over $k$ and $L$ is a line bundle on $X$, then
$\operatorname{\mathbf{SB}}(L)=\operatorname{\mathbf{SB}}(L\vert_{X_{\rm red}})$.
$\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{B}_{+}}(L\vert_{X_{\rm red}})$.
The inclusions “$\supseteq$" in both i) and ii) follow from Lemma \[lem1\]. Let us prove the reverse implication in i). Let $m$ be such that $\operatorname{\mathbf{SB}}(L\vert_{X_{\rm red}})=
\operatorname{Bs}(L^m\vert_{X_{\rm red}})$. Given $x\in X$, suppose that $x\not\in \operatorname{Bs}(L^m\vert_{X_{\rm red}})$. Consider $s\in H^0(X_{\rm red},L^m\vert_{X_{\rm red}})$ such that $x\not\in Z(s)$. Let $J$ denote the ideal defining $X_{\rm red}$, and let $e\gg 0$ be such that $J^{[p^e]}=0$. In this case $(F^e)^*(s)$ gives a section in $H^0(X,L^{mp^e})$ whose restriction to $X_{\rm red}$ is equal to $s^{\otimes p^e}$. In particular, $x\not\in Z((F^e)^*(s))$. We conclude that $x\not\in \operatorname{Bs}(L^{mp^e})$, hence $x\not\in\operatorname{\mathbf{SB}}(L)$. This completes the proof of i).
Let $A$ be an ample line bundle on $X$, and let $m\gg 0$ be such that $\operatorname{\mathbf{B}_{+}}(L\vert_{X_{\rm red}})=\operatorname{\mathbf{SB}}(L^m\otimes A^{-1}\vert_{X_{\rm red}})$ and $\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{SB}}(L^m\otimes A^{-1})$. The assertion in ii) now follows by applying i) to $L^m\otimes A^{-1}$.
The following is a key result from [@Keel]. We give a different proof, that has the advantage that it will apply also when replacing the stable base loci by the augmented base loci.
\[thm\_Keel\] If $L$ is a nef line bundle on a projective scheme $X$, and $D$ is an effective Cartier divisor on $X$ such that $L(-D)$ is ample, then $$\operatorname{\mathbf{SB}}(L)=\operatorname{\mathbf{SB}}(L\vert_D).$$
We isolate the key point in the argument in a lemma that we will use several times.
\[key\_lemma\] Let $A$ be an ample line bundle on a projective scheme $X$, and $D$ an effective Cartier divisor on $X$. If $L:=A\otimes\cO_X(D)$ is nef, then for every $m\geq 1$ and every section $s\in H^0(D,L^m\vert_D)$, there is $e\geq 1$ such that $s^{\otimes p^e}\in H^0(D,L^{mp^e}\vert_D)$ is the restriction of a section in $H^0(X,L^{mp^e})$.
Consider the short exact sequence $$0{\xrightarrow{\ \ }}L^m(-D){\xrightarrow{\ \ }}L^m{\xrightarrow{\ \ }}L^m\vert_D{\xrightarrow{\ \ }}0.$$ Pulling-back by $F^e$ gives the exact sequence $$0{\xrightarrow{\ \ }}L^{mp^e}(-p^eD){\xrightarrow{\ \ }}L^{mp^e}{\xrightarrow{\ \ }}L^{mp^e}\vert_{p^eD}{\xrightarrow{\ \ }}0.$$ Note that $L^m(-D)=L^{m-1}\otimes L(-D)$ is ample, since $L$ is nef and $L(-D)$ is ample. By asymptotic Serre vanishing, we conclude that for $e\gg 0$ we have $H^1(X, L^{mp^e}(-p^eD))=0$, and therefore the restriction map $$H^0(X,L^{mp^e}){\xrightarrow{\ \ }}H^0(X,L^{mp^e}\vert_{p^eD})$$ is surjective. Therefore there is $t\in H^0(X,L^{mp^e})$ such that $t\vert_{p^eD}=
(F^e)^*(s)$. In this case the restriction of $t$ to $D$ is equal to $s^{\otimes p^e}$.
It follows from Lemma \[lem1\] that it is enough to show that if $P$ is a point on $X$ that does not lie in $\operatorname{\mathbf{SB}}(L\vert_D)$, then $P$ does not lie in $\operatorname{\mathbf{SB}}(L)$. If $P$ does not lie on $D$, then it is clear that $P\not\in \operatorname{\mathbf{SB}}(L)$, since $A:=L\otimes\cO_X(-D)$ is ample. On the other hand, if $P\in D$, let $m\geq 1$ be such that there is a section $s\in H^0(D,L^m\vert_D)$, with $P\not\in Z(s)$. Since $Z(s^{\otimes p^e})_{\rm red}=Z(s)_{\rm red}$, in order to show that $P\not\in\operatorname{\mathbf{SB}}(L)$ it is enough to show that for some $e$, the section $s^{\otimes p^e}$ lifts to a section in $H^0(X, L^{mp^e})$. This is a consequence of Lemma \[key\_lemma\].
\[cor1\_thm\_Keel\] Let $X$ be a reduced projective scheme. If $L$ and $A$ are line bundles on $X$, with $A$ ample and $L$ nef, and $Z=Z(s)$ for some $s\in
H^0(X,L\otimes A^{-1})$, then $\operatorname{\mathbf{SB}}(L)=\operatorname{\mathbf{SB}}(L\vert_Z)$.
Let $X'$ be the union of the irreducible components of $X$ that are contained in $Z$, and let $X''$ be the union of the other components (we consider on both $X'$ and $X''$ the reduced scheme structures). If $X'=X$, then $Z=X$ and there is nothing to prove, while if $X'=\emptyset$, then $Z$ is an effective Cartier divisor and the assertion follows from Theorem \[thm\_Keel\]. Therefore we may and will assume that both $X'$ and $X''$ are non-empty.
Using the fact that $A$ is ample and the definition of the stable base locus, we obtain $\operatorname{\mathbf{SB}}(L)\subseteq\operatorname{\mathbf{SB}}(L\otimes A^{-1})\subseteq Z$. As in the proof of Theorem \[thm\_Keel\], we see that it is enough to show that if $t\in H^0(Z,L^m\vert_Z)$ for some $m$, then there is $e\geq 1$ such that $t^{\otimes p^e}$ can be lifted to a section in $H^0(X,L^{mp^e})$. By applying Lemma \[key\_lemma\] to $X''$, $D=Z\cap X''$ and the ample line bundle $L\vert_{X''}\otimes\cO_{X''}(-D)$, we see that for some $e$ we can lift $t^{\otimes p^e}\vert_{X''\cap Z}$ to a section $t''\in H^0(X'',
L^{mp^e}\vert_{X''})$. Since $X'\subseteq Z$, the restriction of $t''$ to $X''\cap X'$ is equal to $t^{\otimes p^e}\vert_{X'\cap X''}$. Therefore we can glue $t^{\otimes
p^e}\vert_{X'}$ with $t''$ to get a section in $H^0(X,L^{mp^e})$ lifting $t^{\otimes
p^e}$.
Recall that if $L$ is a nef line bundle on the projective scheme $X$, then the *exceptional locus* $L^{\perp}$ is the union of all closed irreducible subsets $V\subseteq X$ such that $L\vert_V$ is not big. Since $L$ is nef, this condition is equivalent to the fact that $\dim(V)>0$ and $(L\vert_V^{\dim(V)})=0$.
\[rem1\] It is easy to see by induction on $\dim(X)$ that $L^{\perp}$ is a closed subset of $X$. Note first that if $X_1,\ldots,X_r$ are the irreducible components of $X$ (with the reduced scheme structures), then clearly $L^{\perp}=(L\vert_{X_1})^{\perp}\cup\ldots\cup (L\vert_{X_r})^{\perp}$. Therefore we may assume that $X$ is integral. In this case, if $L$ is not big, then $L^{\perp}=X$. Otherwise, we can find an effective Cartier divisor $D$ and a positive integer $m$ such that $L^m(-D)$ is ample. It is clear that if $L\vert_V$ is not big, then $V\subseteq D$. Therefore $L^{\perp}=(L\vert_D)^{\perp}$, hence it is closed by induction.
The following result is one of the main results from [@Keel]. As we will see, this is an easy consequence of Corollary \[cor1\_thm\_Keel\].
\[cor2\_thm\_Keel\] If $L$ is a nef line bundle on the projective scheme $X$, then $\operatorname{\mathbf{SB}}(L)=\operatorname{\mathbf{SB}}(L\vert_{L^{\perp}})$.
Arguing by Noetherian induction, we may assume that the result holds for every proper closed subscheme of $X$. Since $L^{\perp}=(L\vert_{X_{\rm red}})^{\perp}$, it follows from Lemma \[lem5\] that we may assume that $X$ is reduced. If the restriction of $L$ to every irreducible component of $X$ is not big, then $L^{\perp}=X$, and there is nothing to prove. From now on we assume that this is not the case, and let $X'$ and $X''$ be the union of those irreducible components of $X$ on which the restriction of $L$ is not (respectively, is) big. On both $X'$ and $X''$ we consider the reduced scheme structures. Note that by assumption $X''$ is nonempty.
Consider an ample line bundle $A$ on $X$. It follows from Lemma \[lem3\] that if $m\gg 0$, there is a section $s\in H^0(X,L^m\otimes A^{-1})$ such that no irreducible component of $X''$ is contained in $Z=Z(s)$ (but such that $X'\subseteq Z$). It is clear that if $V$ is an irreducible closed subset of $X$ such that $L\vert_V$ is not big, then $V\subseteq Z$. Therefore $L^{\perp}=
(L\vert_Z)^{\perp}$. Since $X''$ is nonempty, it follows that $Z\neq X$, hence the inductive assumption gives $\operatorname{\mathbf{SB}}(L\vert_Z)=\operatorname{\mathbf{SB}}(L\vert_{L^{\perp}})$. On the other hand, Corollary \[cor1\_thm\_Keel\] gives $$\operatorname{\mathbf{SB}}(L)=\operatorname{\mathbf{SB}}(L^m)=\operatorname{\mathbf{SB}}(L^m\vert_Z)=\operatorname{\mathbf{SB}}(L\vert_Z),$$ which completes the proof.
We can now prove the second theorem stated in the Introduction.
We suitably modify the argument in the proof of Theorem \[thm\_Keel\]. By Lemma \[lem1\], it is enough to prove the inclusion $\operatorname{\mathbf{B}_{+}}(L)\subseteq\operatorname{\mathbf{B}_{+}}(L\vert_D)$. Furthermore, Lemma \[lem5\] implies $\operatorname{\mathbf{B}_{+}}(L\vert_D)=\operatorname{\mathbf{B}_{+}}(L\vert_{2D})=\operatorname{\mathbf{B}_{+}}(L^2\vert_{2D})$ and we have $\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{B}_{+}}(L^2)$, hence we may replace $L$ by $L^2$ and $D$ by $2D$ to assume that $L(-D)\simeq A^2$, for some ample line bundle $A$.
Suppose that $P$ is a point that does not lie on $\operatorname{\mathbf{B}_{+}}(L\vert_D)$. If $P\not\in D$, since $L(-D)$ is ample, it follows that $P\not\in\operatorname{\mathbf{B}_{+}}(L)$. Hence from now on we may assume that $P\in D$. By assumption, for $m\gg 0$ we have $P\not\in\operatorname{\mathbf{SB}}(L^m\otimes A^{-1}\vert_D)$. Let us choose $r\geq 1$ such that there is $t\in H^0(D, L^{rm}\otimes A^{-r}\vert_D)$ with $P\not\in Z(t)$. Furthermore, since we may take $r$ large enough, we may assume that $A^{r-1}\vert_{D}$ is globally generated. Let $t'\in H^0(D, A^{r-1}\vert_D)$ be such that $P\not\in Z(t')$. Therefore $t\otimes t'\in
H^0(D, L^{rm}\otimes A^{-1})$ is such that $P\not\in Z(t\otimes t')$. Note that $L^{rm}\otimes A^{-1}(-D)\simeq L^{rm-1}\otimes A$ is ample, since $L$ is nef and $A$ is ample. Therefore Lemma \[key\_lemma\] implies that for some $e\geq 1$, the section $t^{\otimes p^e}\otimes {t'}^{\otimes p^e}$ can be lifted to a section in $H^0(X, L^{rmp^e}\otimes A^{-p^e})$, and this section clearly does not vanish at $P$. This shows that $P\not\in\operatorname{\mathbf{B}_{+}}(L)$, and completes the proof of the theorem.
\[cor\_thm\_main2\] Let $X$ be a reduced projective scheme. If $L$ and $A$ are line bundles on $X$, with $L$ nef and $A$ ample, and $Z=Z(s)$ for some $s\in H^0(X,L\otimes A^{-1})$, then $\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{B}_{+}}(L\vert_Z)$.
We modify slightly the argument in the proof of Theorem \[thm\_main2\], along the lines in the proof of Corollary \[cor1\_thm\_Keel\]. By Lemma \[lem1\], it is enough to show that if $P\not\in\operatorname{\mathbf{B}_{+}}(L\vert_Z)$, then $P\not\in\operatorname{\mathbf{B}_{+}}(L)$. Let $X'$ be the union of the irreducible components of $X$ that are contained in $Z$, and $X''$ the union of the other components, both considered with the reduced scheme structures. If $X'=X$, then $Z=X$ and there is nothing to prove, while if $X'=\emptyset$, then $Z$ is an effective Cartier divisor, and the assertion follows from Theorem \[thm\_main2\]. From now on, we assume that both $X'$ and $X''$ are nonempty.
After replacing $L$ and $A$ by $L^2$ and $A^2$, respectively, and $s$ by $s^{\otimes 2}$, we may assume that $A=B^2$, for some ample line bundle $B$ (note that $\operatorname{\mathbf{B}_{+}}(L\vert_{Z(s)})=\operatorname{\mathbf{B}_{+}}(L\vert_{Z(s^{\otimes 2})})$ by Lemma \[lem5\]). Suppose that $P\not\in\operatorname{\mathbf{B}_{+}}(L\vert_Z)$. If $P\not\in Z$, then $P\not\in\operatorname{\mathbf{SB}}(L\otimes
A^{-1})$; since $A$ is ample, we have $\operatorname{\mathbf{B}_{+}}(L)\subseteq \operatorname{\mathbf{SB}}(L\otimes A^{-1})$, hence $P\not\in \operatorname{\mathbf{B}_{+}}(L)$. From now on we assume that $P$ lies in $Z$.
Arguing as in the proof of Theorem \[thm\_main2\], we find a section $$t\otimes t'\in H^0(Z, L^{rm}\otimes A^{-1}\vert_Z)$$ such that $P\not\in Z(t\otimes t')$, and we use Lemma \[key\_lemma\] to deduce that for some $e\geq 1$, we can lift $t^{\otimes p^e}\otimes {t'}^{p^e}\vert_{Z\cap X''}$ to a section $t''\in H^0(X'',L^{rmp^e}\otimes A^{-p^e}\vert_{X''})$. Recall that $X'\subseteq Z$, hence $X'\cap X''\subseteq Z\cap X''$, and therefore $t''\vert_{X'\cap X''}=t^{\otimes p^e}\otimes {t'}^{\otimes p^e}\vert_{X'\cap X''}$. Since $X$ is reduced, it follows that we can glue $t''$ and $t^{\otimes p^e}\otimes {t'}^{\otimes p^e}\vert_{X'}$ to a section in $H^0(X,L^{rmp^e}\otimes A^{-p^e})$ that does not vanish at $P$. Therefore $P\not\in\operatorname{\mathbf{B}_{+}}(L)$, which concludes the proof.
We now give the proof of the characteristic $p$ version of Nakamaye’s theorem.
We argue as in the proof of Corollary \[cor2\_thm\_Keel\]. By Noetherian induction, we may assume that the theorem holds for every proper closed subscheme of $X$. Lemma \[lem5\] implies $\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{B}_{+}}(L\vert_{X_{\rm red}})$, and since $L^{\perp}=
(L\vert_{X_{\rm red}})^{\perp}$, we may assume that $X$ is reduced.
Note that the inclusion $L^{\perp}\subseteq\operatorname{\mathbf{B}_{+}}(L)$ is clear: if $V$ is a closed irreducible subset of $X$ that is not contained in $\operatorname{\mathbf{B}_{+}}(L)$, then we can find an ample line bundle $A$, a positive integer $m$, and $s\in H^0(X,L^m\otimes A^{-1})$ such that $V\not\subseteq Z(s)$. Therefore $s\vert_V$ gives a nonzero section of $L^m\otimes A^{-1}\vert_V$, hence $L\vert_V$ is big. This shows that it is enough to prove the inclusion $\operatorname{\mathbf{B}_{+}}(L)\subseteq L^{\perp}$.
If the restriction of $L$ to all the irreducible components of $X$ is not big, then $L^{\perp}=X$, and the assertion is clear. Otherwise, let $X'$ denote the union of the irreducible components of $X$ on which the restriction of $L$ is not big, and $X''$ the union of the other components, both with the reduced scheme structures. It follows from Lemma \[lem3\] that given any ample line bundle $A$, we can find $m\geq 1$ and a section $s\in H^0(X,L^m\otimes A^{-1})$ whose restriction to every component of $X''$ is nonzero (and whose restriction to $X'$ is zero). Let $Z=Z(s)$. By assumption $X''$ is nonempty, and therefore $Z$ is a proper closed subscheme of $X$, hence by the inductive assumption we have $\operatorname{\mathbf{B}_{+}}(L\vert_Z)=(L\vert_Z)^{\perp}$. If $V\subseteq X$ is an irreducible closed subset such that $L\vert_V$ is not big, then $V\subseteq Z$, hence $L^{\perp}=(L\vert_Z)^{\perp}$. On the other hand, Corollary \[cor\_thm\_main2\] gives $\operatorname{\mathbf{B}_{+}}(L)=\operatorname{\mathbf{B}_{+}}(L\vert_Z)$, and we conclude that $\operatorname{\mathbf{B}_{+}}(L)=L^{\perp}$.
[Nakamaye]{}
L. Ein, R. Lazarsfeld, M. Mustaţǎ, M. Nakamaye and M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) **56** (2006), 1701–1734.
S. Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) **149** (1999), 253–286.
R. Lazarsfeld, *Positivity in algebraic geometry* II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Vol. 49, Springer-Verlag, Berlin, 2004.
Y. Nakai, Some fundamental lemmas on projective schemes, Trans. Amer. Math. Soc. **85** (1963), 296–302.
M. Nakamaye, Stable base loci of linear series, Math. Ann. **318** (2000), 837–847.
[^1]: 2010*Mathematics Subject Classification*. Primary 14A15; Secondary 14E99. The first author was partially supported by an EPSRC grant, the second author was partially supported by NSF research grant no: 0701101, and the third author was partially supported by NSF research grant no: 1068190 and by a Packard Fellowship.
|
---
abstract: 'Ground states of the Edwards-Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on ${\mathbb{Z}}^d$ for any $d$. This problem can be seen as the spin glass version of determining the number of infinite geodesics in first-passage percolation or the number of ground states in the disordered ferromagnet. It was recently shown by Newman, Stein and the two authors that, on the half-plane ${\mathbb{Z}}\times{\mathbb{N}}$, there is a unique ground state (up to global flip) arising from the weak limit of finite-volume ground states for a particular choice of boundary conditions. In this paper, we study the entire set of ground states on the infinite graph, proving that the number of ground states on the half-plane must be two (related by a global flip) or infinity. This is the first result on the entire set of ground states in a non-trivial dimension. In the first part of the paper, we develop tools of interest to prove the analogous result on ${\mathbb{Z}}^d$.'
author:
- |
Louis-Pierre Arguin [^1]\
Université de Montréal\
Montréal, QC H3T 1J4 Canada
- |
Michael Damron[^2]\
Princeton University\
Princeton, NJ 08544, USA
date: 'October 30, 2011'
title: 'On the Number of Ground States of the Edwards-Anderson Spin Glass Model'
---
Introduction
============
The model and the main result
-----------------------------
We study the Edwards-Anderson (EA) spin glass model on an infinite graph $G=(V,E)$ of finite degree. We mostly take $G={\mathbb{Z}}^d$ (and further, $d=2$), and $G={\mathbb{Z}}\times{\mathbb{N}}$, a half-plane of ${\mathbb{Z}}^2$.
For a finite set $A \subseteq V$, consider the set of spin configurations $\Sigma_A = \{-1,+1\}^A$ and for $\sigma \in \Sigma_A$, the Hamiltonian (with free boundary conditions) $$\label{eqn: H}
H_{J,A}(\sigma) = -\sum_{\substack{\{x,y\}\in E\\ x,y \in A}} J_{xy}\sigma_x\sigma_y\ ,$$ where the $J_{xy}$’s (the [*couplings*]{}) are taken from an i.i.d. product measure $\nu$. We assume that the distribution of each $J_{xy}$ is continuous with support equal to $\mathbb{R}$. For inverse temperature $\beta>0$ the Gibbs measure for $A$ is $$\text{{\bf G}}_{J,A}(\sigma) = \frac{1}{Z_{J,A}} \exp(-\beta H_{J,A}(\sigma)) \ , \ Z_{J,A}=\sum_{\sigma\in\Sigma_A} \exp(-\beta H_{J,A}(\sigma))\ .$$ As temperature approaches 0 ($\beta \to \infty$) the Gibbs measure converges weakly to a sum of two delta masses, supported on the spin configurations with minimal value of $H_{J,A}$. These spin configurations (related by global flip) can be seen to be characterized by the following local flip property: for each $B \subseteq A$, we have $$\sum_{\substack{\{x,y\} \in \partial B \\ x,y \in A}} J_{xy}\sigma_x\sigma_y \geq 0\ .$$ Here the set $\partial B\subset E$ is defined as all edges $\{x,y\}$ such that $x \in B$ and $y \notin B$. The advantage is that this definition makes sense for infinite sets $A$. For this reason, we define the [*set of ground states on the infinite graph $G$ at couplings $J$*]{} by $$\label{eq:GSproperty}
\mathcal{G}(J)=\{\sigma\in \{-1,+1\}^{V}: \forall A\subset V \text{ finite, }\sum_{\{x,y\}\in\partial A}J_{xy}\sigma_x\sigma_y\geq 0\}\ .$$ In other words, elements of $\mathcal{G}(J)$ are the spin configurations minimizing the Hamiltonian locally for the coupling realization $J$. Clearly, $\sigma\in \mathcal{G}(J)$ if and only if $-\sigma\in \mathcal{G}(J)$. The goal of this paper is not to determine precisely the cardinality of $\mathcal{G}(J)$ but rather to rule out possibilities other than two or infinity. Our main result is to prove such a claim in the case of the EA model on the two-dimensional half-plane.
\[thm: mainthm\] For the EA model on the half-plane $G={\mathbb{Z}}\times {\mathbb{N}}$, the number of ground states $|\mathcal{G}(J)|$ is either $2$ with $\nu$-probability one or $\infty$ with $\nu$-probability one.
Previous results
----------------
A main question in the theory of short-range spin glasses is to understand the structure of the set $\mathcal{G}(J)$, and in particular its cardinality. This problem is the zero-temperature equivalent of understanding the structure and the cardinality of the set of pure states, the set of infinite-volume Gibbs measures of the EA model that are extremal. It is easy to check that for $G={\mathbb{Z}}$, $\mathcal{G}(J)$ has only two elements: the flip-related configurations $\sigma$ defined by the identity $\sigma_x\sigma_y=\text{sgn} J_{xy}$. However, it is not known how many elements are in $\mathcal{G}(J)$ for $G={\mathbb{Z}}^d$ when $d>1$. (We will see in the next section that the cardinality of $\mathcal{G}(J)$ must be a constant number $\nu$-almost surely.) It is expected that $|\mathcal{G}(J)|=2$ for $d=2$ [@Middleton99; @PY99] (see also [@Loebl04] for a possible counterargument to this). There are competing predictions for higher dimensions. The Replica Symmetry Breaking (RSB) scenario would predict $|\mathcal{G}(J)|=\infty$ for $d$ high enough, and the droplet/scaling proposal would be consistent with $|\mathcal{G}(J)|=2$ in every dimension. We refer to [@N97; @NS03] for a detailed discussion on ground states of disordered systems or pure states at positive temperature.
There have been several works on ground states of the EA model in the physics and mathematics literature; a partial list includes [@ADNS10; @FH86; @Loebl04; @Middleton99; @N97; @NS01; @NS03; @PY99]. The present work appears to give the first rigorous result about the entire set of ground states $\mathcal{G}(J)$. Previous rigorous results have focused on the so-called [*metastates on ground states*]{}. A metastate is a $J$-dependent probability measure on $\{-1,+1\}^V$ supported on ground states. It is constructed using a sequence of finite graphs $G_n$ converging to $G$. For a given realization $J$ and $n$, the ground state on $G_n$ is unique up to a global flip. We identify the flip-related configurations and write $\sigma_n^*(J)$ for them. A metastate is obtained by considering a converging subsequence of the measures $\big(\nu(dJ) ~ \delta_{\sigma_n^*(J)}\big)_n$, where $\delta_{\sigma_n^*(J)}$ is the delta measure on the ground state of $G_n$ for the coupling realization $J$. If $\kappa$ denotes a subsequential limiting measure, then sampling from $\kappa$ gives a pair $(J,\sigma)\in {\mathbb{R}}^E \times \{-1,+1\}^V$. A metastate is the conditional measure $\kappa$ given $J$ and is denoted $\kappa_J$. It is not hard to verify that $\kappa_J$ is supported on $\mathcal{G}(J)$.
It was proved in [@ADNS10] that the ground state of the EA model on the half-plane with horizontal periodic boundary conditions and free boundary condition at the bottom is unique in the metastate sense. Precisely, for a sequence of boxes $G_n$ that converges to the half-plane, the limit $\kappa_J$ produced by the metastate construction is unique and is given by a delta measure on two flip-related ground states. Though the metastate construction is very natural, it is important to stress that the measure thus obtained is not necessarily supported on the whole set $\mathcal{G}(J)$. It may be that some elements of $\mathcal{G}(J)$ do not appear in the support of the metastate, due to the choice of boundary conditions on $G_n$ or to the fact that the subsequence in the metastate construction is chosen independently of $J$. Therefore uniqueness in the metastate sense does not answer the more general question of the number of ground states.
It is natural from a statistical physics perspective to study the set $\mathcal{G}(J)$ by looking at probability measures on it. One challenge is to construct probability measures on $\mathcal{G}(J)$ that have a nice dependence on $J$, namely measurability and translation covariance. The metastate (with suitably chosen boundary conditions) briefly described above is one such measure. The main idea of the present paper is to consider another measure, the [*uniform measure on $\mathcal{G}(J)$*]{} $$\mu_J:= \frac{1}{|\mathcal{G}(J)|}\sum_{\sigma\in \mathcal{G}(J)} \delta_{\sigma}\ .$$ For $\mu_J$ to be well-defined it is necessary to assume that $|\mathcal{G}(J)|$ is finite. Like the metastate, the uniform measure on ground states depends nicely on $J$: see Proposition \[prop: mu mble\] and Lemma \[lem: covariance\]. The strategy to prove a “two-or-infinity" result is to assume that $|\mathcal{G}(J)|<\infty$ and to conclude that it implies that $\mu_J$ is supported on two spin configurations related by a global flip (that is, $|\mathcal{G}(J)| = 2$). The approach is similar to the proof of uniqueness in [@ADNS10] using the interface between ground states, though new tools need to be developed. For spin configurations $\sigma$ and $\sigma'$, define the [*interface*]{} $\sigma \Delta \sigma'$ as $$\sigma \Delta \sigma'=\{\{x,y\}\in E:\sigma_x\sigma_y\neq \sigma'_x\sigma'_y \}\ .$$ It will be shown for the half-plane that $$\int \nu(dJ) ~ \mu_J\times\mu_J\{(\sigma,\sigma'):\sigma\Delta \sigma'=\emptyset\}=1 \ .$$ This implies that $\mu_J$ is supported on two flip-related configurations for $\nu$-almost all $J$ since $\sigma\Delta\sigma'=\emptyset$ if and only if $\sigma=\sigma'$ or $\sigma=-\sigma'$.
Before going into the details of the proofs, we remark that the problem of determining the number of ground states for the EA model can be seen as a spin-glass version of a first-passage percolation problem. Indeed, one question in two-dimensional first-passage percolation is to determine whether there exist infinite geodesics. These are doubly-infinite curves that locally minimize the sum of the random weights between vertices of the graph. This problem is equivalent to determining whether there exist more than two flip-related ground states in the (ferromagnetic) Ising model with random couplings. The Hamiltonian of the ferromagnetic model is the same as in , but the distribution of $J$ is restricted to the positive half-line. The reader is referred to [@Wehr97] for the details of the correspondence. It was proved by Wehr in [@Wehr97] that the number of ground states for this model is either two or infinity in dimensions greater or equal to two. On the half-plane, it was shown by Wehr and Woo [@WehrWoo98] that the number of ground states is two. Contrary to the ferromagnetic case, the study of ground states of the EA spin glass model presents technical difficulties that stem from the presence of positive and negative couplings. This feature rules out monotonicity of the partial sums of couplings along an interface.
The paper is organized into two main parts as follows. The first part develops general tools to study ground states of the EA model. Precisely, in Section 2, elementary properties of the set $\mathcal{G}(J)$ are derived for general graphs. In particular, the dependence of $\mathcal{G}(J)$ on a single coupling is studied. Properties of probability measures on $\mathcal{G}(J)$ are investigated in Section 3 with an emphasis on the uniform measure on $\mathcal{G}(J)$. The second part of the paper consists of the proof of Theorem \[thm: mainthm\] and is contained in Section 4.\
[**Acknowledgements**]{} Both authors are indebted to Charles Newman and Daniel Stein for having introduced them to the subject of short-range spin glasses and for numerous discussions on related problems. L.-P. Arguin thanks also Janek Wehr for discussions on the problem of the number of ground states in spin glasses and in disordered ferromagnets.
Elementary properties of the set of ground states {#sec:gs}
=================================================
In this section, unless otherwise stated, we consider the EA model on a connected graph $G=(E,V)$ of finite degree. We assume that there exists a sequence of subgraphs $(G_n)$ that converges locally to $G$. Throughout the paper, we will use the following notation: $\Omega_1 = {\mathbb{R}}^{E}$, and ${\cal F}_1$ is the Borel sigma-algebra generated by its product topology; $\Omega_2 = \{-1,+1\}^{V}$ and ${\cal F}_2$ is the corresponding product sigma-algebra.
Measurability
-------------
We first note that the set of ground states is compact.
$\mathcal{G}(J)$ is a non-empty compact subset of $\Omega_2$ (in the product topology) for all $J$. In particular, the set of probability measures on $\mathcal{G}(J)$ is compact in the weak-\* topology on the set of probability measures on $\Omega_2$.
The fact that $\mathcal{G}(J)$ is non-empty follows by a standard compactness argument, taking a subsequence of ground states for the Hamiltonian with $A=G_n$. The function $\sigma\mapsto \sum_{\{x,y\}\in\partial A}J_{xy}\sigma_x\sigma_y$ is continuous in the product topology for a given finite $A$ and $J$. Therefore, the set $\{\sigma\in \{-1,+1\}^{V}: \sum_{\{x,y\}\in\partial A}J_{xy}\sigma_x\sigma_y\geq 0\}$ is closed. Since $\mathcal{G}(J)$ is the intersection of these sets over all finite $A$ by , it is closed. Being a closed subset of the compact space $\Omega_2$, it is also compact. The second statement of the lemma follows from the first.
The next result is necessary for the uniform measure to be well-behaved and to later apply the ergodic theorem to $|\mathcal{G}(J)|$.
\[prop:mble\] The random variable $J\mapsto |\mathcal{G}(J)|$ is $\mathcal{F}_1$-measurable.
Consider a sequence of finite graphs $\Lambda_n\subset G$, a configuration $\sigma_n$ on $\Lambda_n$ and a configuration $\bar \sigma_n$ on the external boundary of $\Lambda_n$ (that is, all vertices that are not in $\Lambda_n$ but are adjacent to vertices in it). The condition that $\sigma_n$ is a ground state in $\Lambda_n$ with boundary conditions $\bar \sigma_n$ is a finite list of conditions of the form $$\label{eq:loopcondition}
\sum_{\{ x,y \} \in S} J_{xy} (\sigma_n)_x (\sigma_n)_y \geq 0 \mbox{ or } \sum_{\{ x,y \} \in S} J_{xy} (\sigma_n)_x (\bar \sigma_n)_y \geq 0$$ for specific finite sets $S$ of edges. For any given $S$, the set of $J \in \Omega_1$ such that condition holds for fixed $\sigma_n$ and $\bar \sigma_n$ is then measurable (that is, it is in $\mathcal{F}_1$). Intersecting over all relevant sets $S$, we see that the following set is measurable: $${\cal J}(\sigma_n,\bar \sigma_n) := \{ J ~:~ \sigma_n \mbox{ is a ground state in } \Lambda_n \mbox{ for the boundary condition } \bar \sigma_n\} \ .$$
Next take $m < n$ and fixed configurations $\sigma_m$ on $\Lambda_m$, $\sigma_{m,n}$ on $\Lambda_n \setminus \Lambda_m$ and $\bar \sigma_n$ on the boundary of $\Lambda_n$. By a similar argument to the one given above, the set ${\cal J}(\sigma_m,\sigma_{m,n},\bar \sigma_n)$ of $J$ such that the concatenation of $\sigma_m$ and $\sigma_{m,n}$ is a ground state on $\Lambda_n$ with boundary condition $\bar \sigma_n$ is measurable. Taking the union over all $\sigma_{m,n}$ and $\bar \sigma_n$ for a fixed $\sigma_m$, we get that for $m<n$ and $\sigma_m$ fixed, the following set is measurable: $${\cal J}(\sigma_m,n) := \{J ~:~ \mbox{ there is a ground state in } \Lambda_n \mbox{ (for some $\bar \sigma_n$) that equals } \sigma_m \mbox{ on } \Lambda_m\}\ .$$
If there exists a sequence of (possibly $J$-dependent) configurations $(\bar \sigma_n)$ such that there are ground states $(\sigma_n)$ on $\Lambda_n$ with boundary condition $\bar \sigma_n$ that converge to $\sigma$, then $\sigma$ is in $\mathcal{G}(J)$. Conversely, if $\sigma\in \mathcal{G}(J)$, such a sequence $(\sigma_n)$ exists by taking $\bar{\sigma}_n$ to be the restriction of $\sigma$ to the boundary. It follows that $\cap_{n\geq m} {\cal J}(\sigma_m,n)$ is the event that there is an infinite-volume ground state $\sigma$ for couplings $J$ that equals $\sigma_m$ on $\Lambda_m$. This event is thus measurable.
For fixed $m$ and a configuration $\sigma_m$ on $\Lambda_m$, let $F_{\sigma_m}(J)$ be the indicator of the event that there is an infinite-volume ground state $\sigma$ for couplings $J$ equal to $\sigma_m$ on $\Lambda_m$. By the above, it is ${\mathcal{F}}_1$-measurable. The proposition will then be proved once we show: $$\label{eq:numbergs}
|\mathcal{G}(J)|=\sup_m \sum_{\sigma_m} F_{\sigma_m}(J)\ .$$ Here the sum is over all $\sigma_m$ on $\Lambda_m$. For any $m$, the sum $\sum_{\sigma_m} F_{\sigma_m}(J)$ equals the number of different $\sigma_m$’s that are equal to restrictions on $\Lambda_m$ of elements of $\mathcal{G}(J)$ . So for each $m$, $$\sum_{\sigma_m} F_{\sigma_m}(J) \leq |\mathcal{G}(J)|$$ and the right side of is at most $|\mathcal{G}(J)|$. To show equality in , suppose first that $|\mathcal{G}(J)|$ is finite. We can choose $n$ so that the restriction to $\Lambda_n$ of each element of $\mathcal{G}(J)$ is different. For this $n$, $\sum_{\sigma_n} F_{\sigma_n}(J) = |\mathcal{G}(J)|$ and is established. If $|\mathcal{G}(J)|=\infty$, then for any $k\in{\mathbb{N}}$, we can find $n_k$ such that $\sum_{\sigma_{n_k}} F_{\sigma_{n_k}}(J)\geq k$. This is because we can take $\Lambda_n$ large enough so that there are at least $k$ elements of $\mathcal{G}(J)$ that are distinct on $\Lambda_n$. Taking the supremum over $k$ completes the proof of .
In the case $G={\mathbb{Z}}^d$, it is easy to see that for any translation $T_a$ by a vector $a \in {\mathbb{Z}}^d$, $|\mathcal{G}(J)|=|\mathcal{G}(T_a J)|$ where $(T_a J)_{xy} = J_{T_a(x)T_a(y)}$. The ergodic theorem then implies that the random variable $|\mathcal{G}(J)|$ is constant $\nu$-almost surely. The same holds when $G$ is the half-plane by considering only horizontal translations.
\[cor: constant\] For $G={\mathbb{Z}}^d$ or $G={\mathbb{Z}}\times {\mathbb{N}}$, the number of ground states $|\mathcal{G}(J)|$ is a constant $\nu$-almost surely.
The next result shows that if $|\mathcal{G}(J)|<\infty$ then the uniform measure $\mu_J$ is a random variable over $\mathcal{F}_1$.
\[prop: mu mble\] Let $B \in {\cal F}_2$ and assume that $|\mathcal{G}(J)|<\infty$. The map $$J \mapsto \mu_J(B)$$ is $\mathcal{F}_1$-measurable. Similarly, if $B'$ is a Borel set in $\Omega_2\times \Omega_2$, then the map $J \mapsto \mu_J\times \mu_J(B')$ is $\mathcal{F}_1$-measurable.
By a standard approximation, it is sufficient to prove the statement for $B$ of the form $$B = \{ \sigma~:~ \sigma = s_A \mbox{ on } A\}$$ for some finite set $A$ and fixed configuration $s_A$ on $A$.
Take a sequence of finite graphs $\Lambda_n$ converging to $G$. We define $$F_{s_A}(J) = \mbox{ number of } \sigma\in \mathcal{G}(J) \mbox{'s that equal } s_A \mbox{ on } A\ .$$ Note that $\mu_J(B)$ is simply $F_{s_A}(J)$ divided by $|\mathcal{G}(J)|$. The variable $\mathcal{G}(J)$ is ${\mathcal{F}}_1$-measurable by Proposition \[prop:mble\]. Thus it remains to show that $F_{s_A}(J)$ is also.
Exactly as in the last proof, if $n$ is so large that $\Lambda_n$ contains $A$ and if $s_{A,n}$ is any fixed spin configuration on $\Lambda_n \setminus A$, then the set ${\cal J}(s_A,s_{A,n})$ of all $J$ such that there is an element of $\mathcal{G}(J)$ that (a) equals $s_A$ on $A$ and (b) equals $s_{A,n}$ on $\Lambda_n \setminus A$ is measurable. Let $F_{s_A,s_{A,n}}(J)$ be the indicator of the event ${\cal J}(s_A,s_{A,n})$ and consider the random variable $$\sup_n \sum_{s_{A,n}} F_{s_A,s_{A,n}}(J)\ .$$ Here the supremum is over all $n$ such that $A \subseteq \Lambda_n$. The same reasoning to prove shows that $F_{s_A}(J)$ is equal to the above and is thus measurable. This completes the proof of the first claim. The second assertion is implied by the first one since by a standard approximation, any measurable function on $\Omega_2\times\Omega_2$ can be approximated by linear combinations of indicator functions of sets of the form $$B_A:=\{(\sigma,\sigma'): \sigma=s_A \text{ on $A$ }, \sigma'=s_{A'} \text{ on $A'$ }\}$$ for two finite sets $A$ and $A'$ of $G$. Since $\mu_J\times \mu_J(B_A)$ is equal to the product of the $\mu_J$-probability of each coordinate, measurability follows from the first part of the proposition.
Properties of the set of ground states
--------------------------------------
In this section, we establish some elementary properties of the dependence of the set of ground states $\mathcal{G}(J)$ on a finite number of couplings.
Fix an edge $e=\{x,y\}$. We will sometimes abuse notation and write for simplicity $$J_e:=J_{xy} ~~\text{and}~~ \sigma_e:=\sigma_x\sigma_y \ .$$ We are interested in studying how $\mathcal{G}(J)$ varies when $J_e$ is modified. For simplicity, we will fix all other couplings and write $\mathcal{G}(J_e)$ for the set of ground states to stress the dependence on $J_e$. From the definition , it is easy to see that if $\sigma\in \mathcal{G}(J_e)$ and $\sigma_e=+1$, then $\sigma$ remains a ground state for coupling values greater than $J_e$. More generally:
\[lem: monotone\] Fix an edge $e=\{x,y\}$. If $J_e\leq J_e'$ then $$\begin{aligned}
&\mathcal{G}(J_e)\cap \{\sigma: \sigma_e=+1\}\subseteq \mathcal{G}(J'_e)\cap \{\sigma: \sigma_e=+1\}\\
&\mathcal{G}(J_e)\cap \{\sigma: \sigma_e=-1\}\supseteq \mathcal{G}(J'_e)\cap \{\sigma: \sigma_e=-1\}
\end{aligned}$$
In view of the above monotonicity of the set of ground states, it is natural to introduce the [*critical value*]{} of $\sigma\in \mathcal{G}(J_e)$ at $e$. Namely, we define the critical value $C_e$ as $$\begin{aligned}
C_e(J,\sigma):=\begin{cases}
\inf\{J_e: \sigma\in \mathcal{G}(J_e)\} \text{ if $\sigma_e=+1$;}\\
\sup\{J_e: \sigma\in \mathcal{G}(J_e)\}\text{ if $\sigma_e=-1$.}
\end{cases}
\end{aligned}$$ For future reference, we remark that from the definition, $$\label{eqn: equiv}
\begin{aligned}
&\sigma\in \mathcal{G}(J_e)\text{ and }\sigma_e=+1 \Longrightarrow J_e\geq C_e(J,\sigma)\\
&\sigma\in \mathcal{G}(J_e)\text{ and }\sigma_e=-1 \Longrightarrow J_e\leq C_e(J,\sigma)\ .
\end{aligned}$$
An elementary correspondence exists between the critical values and the energy required to flip finite sets of spins.
\[lem: critical formula\] Let $\sigma\in \mathcal{G}(J_e)$. Then $$\label{eqn: c_e}
\sigma_eC_e(J,\sigma)=-\inf_{A~:~e \in \partial A}\sum_{\substack{\{z,w\}\in \partial A\\ \{z,w\}\neq e}} J_{zw}\sigma_z\sigma_w\ ,$$
In particular, for a given $\sigma$, $C_e(J,\sigma)$ does not depend on $J_e$.
In this section, we will often omit the dependence on $J$ in the notation and write $C_e(\sigma)$ for simplicity. From the above result, we see that this notation is consistent with the fact that all couplings other than $J_e$ are fixed in this section.
The independence assertion is straightforward from the expression. We prove the equation in the case of $\sigma_e=+1$. The other case is similar. Let $-\widetilde{C}_e(\sigma)$ be the right side of . If $C_e(\sigma)+\widetilde C_e(\sigma)>0$, there exists $\delta>0$ such that $$C_e(\sigma)-\delta+ \inf_{A: e\in \partial A}\sum_{\substack{\{z,w\}\in \partial A\\ \{z,w\}\neq e}} J_{zw}\sigma_z\sigma_w>0\ .$$ In particular, $\sigma\in \mathcal{G}(J'_e)$ for $J'_e=C_e(\sigma)-\delta$, contradicting $C_e(\sigma)$ as the infimum of such values. On the other hand if $C_e(\sigma)+\widetilde C_e(\sigma)<0$, there must exist a finite set $A$ such that $$C_e(\sigma)+\sum_{\substack{\{z,w\}\in \partial A\\ \{z,w\}\neq e}} J_{zw}\sigma_z\sigma_w<0\ .$$ In particular this would hold for $C_e(\sigma)$ replaced by some $J_e>C_e(\sigma)$, contradicting the definition of $C_e(\sigma)$, because we should have $\sigma\in \mathcal{G}(J_e)$ for all $J_e>C_e(\sigma)$.
The distance $|J_e-C_e(\sigma)|$ from $J_e$ to the critical value is called the [*flexibility of $e$*]{} and is denoted $F_e(\sigma)$. (This quantity was first introduced in [@NS01].) From above, it has a useful representation: $$\label{eqn: flex}
F_e(\sigma):=|J_e-C_e(\sigma)|=\inf_{A:e \in \partial A}\sum_{\{z,w\}\in \partial A} J_{zw}\sigma_z\sigma_w\ .$$
In the same spirit as the critical values, for any edge $e$ and $\sigma \in \mathcal{G}(J_e)$, we define the set of [*critical droplets*]{} for $e$ in $\sigma$. These are the limit sets of the infimizing sequences of finite sets in the expression of the critical value $C_e(\sigma)$. Precisely, if $(\Lambda_n)$ is a sequence of vertex sets, we say that $\Lambda_n \to \Lambda$ if each vertex $v \in V$ is in only finitely many of the sets $\Lambda_n \Delta \Lambda$ (here $\Delta$ denotes the symmetric difference of sets). We will say that $\Lambda$ is a critical droplet for $e$ in $\sigma$ if there exists a sequence of finite vertex sets $(\Lambda_n)$ such that $\Lambda_n \to \Lambda$, $e \in \partial \Lambda_n$ for all $n$ and $$-\sum_{\substack{\{x,y\} \in \partial \Lambda_n\\ \{x,y\} \neq e}} J_{xy} \sigma_x\sigma_y \to \sigma_eC_e(\sigma) \text{ as } n \to \infty\ .$$ Write $CD_e(\sigma)$ for the set of critical droplets of $e$ in $\sigma$. By compactness, this set is nonempty.
Since the critical values are values of $J_e$ where there is a change in the set $\mathcal{G}(J_e)$, it will be useful to get bounds on them that are functions of the couplings only (not of $\sigma\in \mathcal{G}(J_e)$). In this spirit, similarly to [@NS01], we define the [*super-satisfied value*]{} for an edge $e=\{x,y\}$ as $$\label{eqn: supersat}
\mathcal{S}_e:=\min \left\{\sum_{\substack{z \neq y\\ \{x,z\}\in E}}|J_{xz}|, \sum_{\substack{z \neq x\\ \{y,z\}\in E}} |J_{yz}| \right\}\ .$$ We will say that an edge $e$ is [*super-satisfied*]{} if $|J_e|> \mathcal{S}_e$. The terminology is explained by the following fact: by taking $A=\{x\}$ and $A=\{y\}$ in , one must have $$\label{eq: supersat sign}
\begin{aligned}
J_e>\mathcal{S}_e &\Longrightarrow \sigma_e=+1 \text{ for all $\sigma\in \mathcal{G}(J_e)$}\\
J_e<-\mathcal{S}_e &\Longrightarrow \sigma_e=-1 \text{ for all $\sigma\in \mathcal{G}(J_e)$}\ .
\end{aligned}$$ Moreover, for the same choice of $A$, we get from Lemma \[lem: critical formula\] $$\label{eq: critical first bound}
C_e(\sigma)\geq-\mathcal{S}_e \text{ if $\sigma_e=+1$} \text{ and } C_e(\sigma)\leq\mathcal{S}_e \text{ if $\sigma_e=-1$.}$$ Our next goal is to prove that in fact $|C_e(\sigma)|\leq \mathcal{S}_e$ (cf. Corollary \[cor: bound\]). This is done by establishing a correspondence between the two following sets: $$\begin{aligned}
\mathcal{G}_{+_e}&=\{\sigma\in \Omega_2: \sigma_e=+1, \forall A\subset V \text{ finite with $e\notin\partial A$, }\sum_{\{x,y\}\in\partial A}J_{xy}\sigma_x\sigma_y\geq 0\}\ ;\\
\mathcal{G}_{-_e}&=\{\sigma\in \Omega_2: \sigma_e=-1, \forall A\subset V \text{ finite with $e\notin\partial A$, }\sum_{\{x,y\}\in\partial A}J_{xy}\sigma_x\sigma_y\geq 0\}\ .
\end{aligned}$$ In other words, $\mathcal{G}_{\pm_e}$ are the sets of ground states on the graph $G$ minus the edge $e$, where the spins of the vertices of $e$ are restricted to have the same/opposite sign. Note that these sets depend on the couplings but not on $J_e$. Clearly, if $\sigma\in \mathcal{G}(J_e)$ then either $\sigma\in \mathcal{G}_{+_e}$ or $\sigma\in \mathcal{G}_{-_e}$ depending on its sign at $e$. Moreover by , if $J_e>\mathcal{S}_e$, then $\mathcal{G}(J_e)\subseteq \mathcal{G}_{+e}$ and if $J_e<-\mathcal{S}_e$, then $\mathcal{G}(J_e)\subseteq \mathcal{G}_{-e}$. Equality is derived in Corollary \[cor: supersat3\] from the following correspondence.
\[prop: pair\] For $\sigma\in \mathcal{G}_{+_e}$ and $\Lambda \in CD_e(\sigma)$, consider $\widetilde \sigma$ where $\sigma\Delta\widetilde \sigma=\partial \Lambda$, that is $$\label{eq:mapping}
\widetilde \sigma_v = \begin{cases}
\sigma_v & v \notin \Lambda \\
-\sigma_v & v \in \Lambda
\end{cases}\ .$$ Then $\widetilde \sigma\in \mathcal{G}_{-_e}$ and $C_e(\widetilde \sigma)\geq C_e(\sigma)$. A similar statement holds for $\sigma \in \mathcal{G}_{-_e}$ with $\widetilde \sigma \in \mathcal{G}_{+_e}$ and $C_e(\widetilde \sigma) \leq C_e(\sigma)$.
Write ${\cal D}$ for the collection of sets of edges $S$ such that $S=\partial A$ for some finite set of vertices $A$. We will use the following fact, which is verified by elementary arguments, and which was also noticed in [@Fink10]: if $S_1, S_2 \in {\cal D}$, then $S_1 \Delta S_2 \in {\cal D}$.
We will prove the proposition in the case $\sigma \in \mathcal{G}_{+_e}$. The other case is similar. Choose a sequence of finite vertex sets $(\Lambda_n)$ such that $e \in \partial \Lambda_n$ for all $n$, $\Lambda_n \to \Lambda$, and $$-\sum_{\substack{\{x,y\} \in \partial \Lambda_n\\ \{x,y\} \neq e}} J_{xy} \sigma_x\sigma_y \to C_e(\sigma) \text{ as } n \to \infty\ .$$
Write $S_n = \partial \Lambda_n$, $S = \partial \Lambda$, let $T \in {\cal D}$ and take $n$ so large that $T\cap S_n = T \cap S$ and $T \setminus S_n = T \setminus S$. Let $\widetilde J$ be the coupling configuration with value $\widetilde J_f=J_f$ for $f \neq e$ and $\widetilde J_e=C_e(\sigma)$ at $e$. $$\begin{aligned}
\sum_{\{x,y\} \in T} \widetilde J_{xy} \widetilde \sigma_x \widetilde \sigma_y &=& \sum_{\{x,y\} \in T \cap S_n} \widetilde J_{xy} \widetilde \sigma_x \widetilde \sigma_y + \sum_{\{x,y\} \in T \setminus S_n} \widetilde J_{xy} \widetilde \sigma_x \widetilde \sigma_y \\
&\overset{\mbox{\eqref{eq:mapping}}}{=}& -\sum_{\{x,y\} \in T \cap S_n} \widetilde J_{xy} \sigma_x\sigma_y + \sum_{\{x,y\} \in T \setminus S_n} \widetilde J_{xy} \sigma_x\sigma_y \\
&=& \sum_{\{x,y\} \in T \Delta S_n} \widetilde J_{xy} \sigma_x\sigma_y - \sum_{\{x,y\} \in S_n} \widetilde J_{xy} \sigma_x\sigma_y\ .\end{aligned}$$ Since $T \Delta S_n \in \mathcal{D}$ and $\sigma \in \mathcal{G}(\widetilde J)$, we have $\sum_{\{x,y\} \in T \Delta S_n} \widetilde J_{xy} \sigma_x\sigma_y \geq 0$. Therefore, $$\sum_{\{x,y\} \in T} \widetilde J_{xy} \widetilde \sigma_x \widetilde \sigma_y \geq -\sum_{\{x,y\} \in S_n} \widetilde J_{xy} \sigma_x\sigma_y\ .$$ The right side tends to 0 as $n \to \infty$ by the definition of $S$ and $\widetilde{J}$, so $\sum_{\{x,y\} \in T} \widetilde J_{xy} \widetilde \sigma_x \widetilde \sigma_y \geq 0$ and $\widetilde \sigma \in \mathcal{G}(\tilde J)$. Clearly, $\widetilde \sigma \in \mathcal{G}_{-_e}$, and by , $C_e(\widetilde \sigma) \geq \widetilde J_e=C_e(\sigma)$.
We prove three corollaries of the proposition. The first is the claimed bounds on $C_e(\sigma)$.
\[cor: bound\] Let $e$ be an edge. If $\sigma\in \mathcal{G}(J_e)$, then $$|C_{e}(\sigma)|\leq \mathcal{S}_e\ .$$
We prove the bound when $\sigma\in \mathcal{G}_{+_e}$. The case $\sigma\in \mathcal{G}_{-_e}$ is similar. The lower bound was noticed in . As for the upper bound, by Lemma \[prop: pair\], there exists $\widetilde \sigma\in \mathcal{G}_{-_e}$ such that $ C_e(\sigma)\leq C_e(\widetilde\sigma)$. The claim then follows from $C_e(\widetilde\sigma)\leq \mathcal{S}_e$ again by .
A useful fact about Corollary \[cor: bound\] is that it replaces the critical value that a priori depends on an infinite number of couplings by a quantity that depends on finitely many. Another corollary is that for $J_e$ low enough or large enough, the set $\mathcal{G}(J)$ is independent of $J_e$:
\[cor: supersat3\] If $J_e> \mathcal{S}_e$, then $\mathcal{G}(J_e)=\mathcal{G}_{+_e}$. If $J_e< -\mathcal{S}_e$, then $\mathcal{G}(J_e)=\mathcal{G}_{-_e}$.
Suppose first that $J_e > \mathcal{S}_e$. Then, from and Corollary \[cor: bound\], one has $\mathcal{G}(J_e)\subseteq \mathcal{G}_{+_e}$. Conversely, if $\sigma\in \mathcal{G}_{+_e}$, it suffices to show that for any finite set of vertices $A$ with $e\in \partial A$ $$J_e+ \sum_{\substack{\{z,w\}\in \partial A\\\{z,w\}\neq e}}J_{zw}\sigma_z\sigma_w\geq 0\ .$$ By Corollary \[cor: bound\], we have $\mathcal{S}_e- C_e(\sigma)\geq 0$ and, using formula , we see that the above holds for $J_e> \mathcal{S}_e$. The proof for $\mathcal{G}_{-_e}$ is similar.
Finally, we show that an infimizing sequence of sets for the critical values of an edge can never contain certain super-satisfied edges. For this we need to introduce for $e=\{x,y\}$ $$\label{eq: vertexSSvalue}
\mathcal{S}_e^x = \sum_{\substack{\{x,z\}\in E,z \neq y}} |J_{xz}|\ .$$ Note that by definition, $ \mathcal{S}_e=\min\{\mathcal{S}_e^x,\mathcal{S}_e^y\}$. If $d$ and $e$ are two different edges, there exists a vertex $x$ which is an endpoint of $d$, but not of $e$. Having $|J_d|>\mathcal{S}_d^x$ guarantees that the edge $d$ is super-satisfied independently of the value of $J_e$.
\[cor: supersat2\] Let $d =\{x,y\}$ and $e$ be edges such that $x$ is not an endpoint of $e$ and $|J_d|>\mathcal{S}_d^x$. If $\sigma \in \mathcal{G}(J)$ then no element $\Lambda$ of $CD_e(\sigma)$ has $d \in \partial \Lambda$.
Let $\sigma\in{\mathcal{G}}(J)$ for some fixed $J$ such that $|J_d|>\mathcal{S}_d^x$. Suppose $d \in \partial \Lambda$ for some $\Lambda \in CD_e(\sigma)$. Define $\widetilde \sigma$ as in Proposition \[prop: pair\], so that $\sigma \Delta \widetilde \sigma = \partial \Lambda$. For $y \in \mathbb{R}$, let $J(e,y)$ be the coupling configuration that equals $J_f$ at $f \neq e$ and $y$ at $e$. On one hand, note that, by Proposition \[prop: pair\], $\sigma_d=-\widetilde \sigma_d$ and that $\widetilde \sigma\in \mathcal{G}(J(e,y))$ for either small or large values of $y$. On the other hand, if $|J_d|>\mathcal{S}_d^x$ for $J$, then $|J_d| > \mathcal{S}_d^x$ in $J(e,y)$ for all $y\in{\mathbb{R}}$, because $x$ is not shared by $d$ and $e$. In particular, this implies by Corollary \[cor: supersat3\] that the sign at the edge $d$ of the elements of $\mathcal{G}(J(e,y))$ must be the same for all $y\in{\mathbb{R}}$. This contradicts $\sigma_d=-\widetilde \sigma_d$.
The uniform measure on the set of ground states {#sec: properties}
===============================================
In this section we assume that $$\text{$|\mathcal{G}(J)|$ is constant $\nu$-a.s. and $|\mathcal{G}(J)| < \infty$}\ .$$ The first assertion holds for graphs with translation symmetry by the ergodic theorem as noted in Corollary \[cor: constant\]. We consider the family $(\mu_J)$ consisting of the uniform measures on $\mathcal{G}(J)$ indexed by $J\in\Omega_1$. Recall from Proposition \[prop: mu mble\] that this family has a measurable dependence on $J$. For concision, the following notation will be used throughout the paper for the product measures on $J$ and on one or two replicas of the spin configurations: $$\label{eqn: M}
M=\nu(dJ) ~ \mu_J \qquad \text{ or } \qquad M=\nu(dJ) ~ \mu_J\times\mu_J\ ,$$ where the appropriate case will be clear from the context. In the first part, we use the monotonicity of the measure (defined below) to prove several facts, for example that the critical droplet of any edge is unique. Second, we focus on the properties of the interface sampled from $M$ and prove that, if it exists, any given edge lies in it with positive probability.
Properties of the measure {#sect: monotonicity}
-------------------------
We first introduce the [*monotonicity property*]{} of the family $(\mu_J)$. It is the analogue of the monotonicity of $\mathcal{G}(J)$ in Lemma \[lem: monotone\] at the level of measures. To define it, we give the following notation. For any coupling configuration $J=(J_f)_{f\in E}$, fixed edge $e$ and real number $y$, let $J(e,y)$ be the coupling configuration given by $$\label{eqn: J(e,s)}
(J(e,y))_f = \begin{cases}
y & \text{ if } f=e \\
J_f & \text{ if } f \neq e
\end{cases}\ .$$ Consider any event $A\subseteq\Omega_1\times\{\sigma:\sigma_e=+1\}$. A simple consequence of Lemma \[lem: monotone\], since $|{\mathcal{G}}(J)|$ is a.s. constant, is that for almost all $J$ and for almost all $y\geq J_e$: $$\label{eqn: increasing}
\begin{aligned}
\mu_J\{\sigma: (J,\sigma)\in A\}&\leq \mu_{J(e,y)}\{\sigma: (J,\sigma)\in A\}\ ;
\end{aligned}$$ on the other hand, if $A\subseteq\Omega_1\times\{\sigma:\sigma_e=-1\}$, then for almost all $J$ and almost all $y\leq J_e$: $$\label{eqn: decreasing}
\begin{aligned}
\mu_J\{\sigma: (J,\sigma)\in A\}&\geq \mu_{J(e,y)}\{\sigma: (J,\sigma)\in A\}.
\end{aligned}$$ Similar statements hold for the product $\mu_J\times \mu_J$. For example, the mixed case $A\subseteq \Omega_1\times\{\sigma:\sigma_e=+1\}\times\{\sigma':\sigma'_e=-1\}$ yields for almost all $J$ and almost all $y\geq J_e$ and $y'\leq J_e$: $$\label{eqn: mixed}
\begin{aligned}
\mu_J\times \mu_J \{(\sigma,\sigma'): (J,\sigma,\sigma')\in A\}&\leq \mu_{J(e,y)}\times \mu_{J(e,y')}\{(\sigma,\sigma'): (J,\sigma,\sigma')\in A\}.
\end{aligned}$$
We refer to , and as the [*monotonicity*]{} of the family $(\mu_J)$. It is a natural property to expect from a family of measures on ground states. The results of this section, with the exception of Lemma \[lem: backmodify\], are derived solely from it and no other finer properties of the uniform measure. The main use of the monotonicity property is to decouple the dependence on $J_e$ in $\mu_J$ from the dependence on $J_e$ in the considered event. This trick will appear frequently. The results of this section are stated for the measure $M$ in with one replica of $\sigma$ for concision. They also hold for the measure $M$ on two replicas.
A useful consequence of , , , and the continuity of $\nu$ is that $\nu$-almost surely no coupling value is equal to its critical value. This is a special case of the next proposition, taking $B=\{e\}$ and $h_{B^c}=C_e$.
\[prop: decoupling\] Let $B\subset E$ be a finite set of edges and $h_{B^c}:{\mathbb{R}}^E\times \{-1,+1\}^V\to{\mathbb{R}}$ be a function that does not depend on couplings of edges in $B$. Then for any given linear combination $\sum_{b\in B}J_bs_b$, provided that the coefficients $s_b\in{\mathbb{R}}$ are not all zero, $$M\{(J,\sigma): h_{B^c}(J,\sigma)=\sum_{b\in B}J_bs_b\}=0\ .$$ The same statement holds if $h_{B^c}$ is a function of the couplings and two replicas $(J,\sigma,\sigma')\mapsto h_{B^c}(J,\sigma,\sigma')$ that does not depend on the couplings of edges in $B$.
The event $\{\sigma: h_{B^c}(J,\sigma)=\sum_{b\in B}J_bs_b\}$ can be decomposed by taking the intersection with all possible spin configurations on $B$. Suppose first that $\sigma_b=+1$ for all $b\in B$ and define, for a given $J\in{\mathbb{R}}^E$, $J(B, y)$ for $y\in{\mathbb{R}}^B$ similarly to $$(J(B, y))_e=\begin{cases}
y_e, \text{ $e\in B$}\\
J_e, \text{ $e\notin B$}\ .
\end{cases}$$ By , $\mu_J\{\sigma: h_{B^c}(J,\sigma)=\sum_{b\in B}J_b s_b, ~\sigma_b=+1~ \forall b\in B\}$ is smaller than the probability of the same event under the measure averaged over larger $J_b$’s. Writing $\{J_B^\geq\}$ for the event that $y_b\geq J_b$ for all $b\in B$, $$\begin{aligned}
&&\int \nu(dJ_B) \mu_J\{\sigma: h_{B^c}(J,\sigma)=\sum_{b\in B}J_bs_b, ~\sigma_b=+1~ \forall b\in B\} \\
&\leq&\int \nu(dJ_B) \frac{1}{\nu\{J_B^\geq\}}\int_{\{J_B^\geq\}}\nu(dy) ~ \mu_{J(B,y)}\{\sigma: h_{B^c}(J,\sigma)=\sum_{b\in B}J_b s_b, ~\sigma_b=+1~ \forall b\in B\}\ .\end{aligned}$$
Integrating $y$ over all of ${\mathbb{R}}^B$ and dropping $\{\sigma_b=+1~ \forall b\in B\}$ gives the upper bound: $$\int \nu(dJ_B) \frac{1}{\nu\{J_B^\geq\}} \int \nu(dy) \mu_{J(B,y)}\{\sigma:h_{B^c}(J(B,y),\sigma) = \sum_{b \in B} J_b s_b\} \ .$$ Note $h_{B^c}(J(B,y),\sigma)=h_{B^c}(J,\sigma)$ as $h_{B_c}$ does not depend on couplings in $B$. Now use Fubini: $$\int \nu(dy) \int d\mu_{J(B,y)}(\sigma) \left[ \int \nu(dJ_B) \nu\{J_B^\geq\}^{-1} 1_{\{\sum_{b \in B} J_b s_b = h_{B^c}(J(B,y),\sigma)\}}(J_B) \right]\ ,$$ where $1_A(J_B)$ denotes the indicator function of the event $A$. Because the linear combination of $J_b$’s is non-trivial and $h_{B^c}(J(B,y),\sigma)$ does not depend on $J_B$, the indicator function is equal to 1 on a set of $J_B$’s that is a hyperplane of dimension at most $|B|-1$. Therefore it is $\nu$-almost surely zero, and the inner integral equals zero. This completes the proof in the case that $\sigma_b=+1$ for all $b \in B$. To prove the other cases where $\sigma_b=-1$ for some $b \in B$, it suffices to average over $\{J_b^\leq\}$ (where this event is defined in the obvious way) for $b$ and use . The proof of the second claim when $h_{B^c}$ is a function of the couplings and two replicas $(J,\sigma,\sigma')\mapsto h_{B^c}(J,\sigma,\sigma')$ is done the same way. In the case that $\sigma_b=+1$ and $\sigma'_b=-1$, one uses and bounds $\mu_J\times\mu_J$ by the average of $\mu_{J(b,y)}\times\mu_{J(b,y')}$ over $\{J_b^\geq\}\times \{J_b^\leq\}$ .
One consequence of the above proposition is that the critical droplet $CD_e(\sigma)$ set cannot contain two non-flip-related elements. In other words, infimizing sequences of finite sets of edges entering in the definition of the critical value converge to a unique set. This implies in particular that the mapping of Lemma \[prop: pair\] is well-defined.
\[cor: droplet\] For any edge $e\in E$, $M\{ (J,\sigma): \exists ~T_1 \neq T_2 \in CD_e(\sigma) \text{ with } T_1 \neq G \setminus T_2\} = 0$.
Suppose that $CD_e(\sigma)$ contains at least two critical droplets, $T_1$ and $T_2$, not related by $T_1 = G \setminus T_2$, with positive probability. Let $S_1$ be the set of edges connecting $T_1$ to $T_1^c$ (similarly for $S_2$). Either $S_1 \setminus S_2$ or $S_2 \setminus S_1$ is non-empty. We may assume that $S_1 \setminus S_2$ is non-empty. So there exists $b$ such that $$\label{eq: endeq1}
M\{ (J,\sigma): \exists~ T_1, T_2 \in CD_e(\sigma) \text{ with } b \in S_1 \setminus S_2\}>0\ .$$ Assume that $\sigma_e=+1$ and $\sigma_b=+1$; the other cases are similar. Define $$C_{b,e}(J,\sigma) = - \inf_{\substack{A:b,e \in \partial A \\ A \text{ finite}}} \sum_{\substack{\{x,y\} \in \partial A \\ \{x,y\} \neq b,e}} J_{xy} \sigma_x \sigma_y \text{ and }
C_{e}^b(J,\sigma) = - \inf_{\substack{A:e \in \partial A \\ b \notin \partial A \\ A \text{ finite}}} \sum_{\substack{\{x,y\} \in \partial A \\ \{x,y\} \neq e}} J_{xy} \sigma_x \sigma_y\ .$$ On the event in , we have $C_{b,e}(J,\sigma)-J_b = C_e(J,\sigma)=C_e^b(J,\sigma)$ because $T_1$ and $T_2$ are in $CD_e(\sigma)$. Thus implies that $$M\{ (J,\sigma): \sigma_e=\sigma_b=+1, C_{b,e}(J,\sigma) - C_e^b(J,\sigma) = J_b\} > 0\ .$$ This contradicts Proposition \[prop: decoupling\] using $B=\{e\}$ and $h_{B^c}(J,\sigma)= C_{b,e}(J,\sigma)-C_e^b(J,\sigma)$.
We now state a lemma that will be used in Section \[sec: I=0\]. By Corollary \[cor: supersat2\], the critical droplet cannot go through certain super-satisfied edges. Therefore if there are such super-satisfied edges forcing the critical droplet of an edge $f$ to go through some fixed edges $e_1$ or $e_2$, then the flexibility of $f$, by definition, cannot be smaller than both of those of $e_1$ and $e_2$. The situation is depicted in Figure \[fig: magic\_rung\] where the super-satisfied edges appear in grey. As in Corollary \[cor: supersat2\], the edges need to be super-satisfied independently of the value of $J_f$. For this reason, we work with the value $\mathcal{S}^x_e$ defined in .
\[lem: cylinder\] Let $e_1,e_2, f$ be edges. Let $U$ be a set of edges with the property that all finite sets $A$ with $f\in\partial A$ and $\partial A \cap U = \varnothing$ must have either $e_1$ or $e_2$ in $\partial A$. For each $e \in U$ pick $x(e)$ to be an endpoint of $e$ that is not an endpoint of $f$. Then $$M \{(J,\sigma):F_f(J,\sigma) \geq \min \{ F_{e_1}(J,\sigma),F_{e_2}(J,\sigma)\},~ \forall e\in U~ |J_e| > \mathcal{S}_e^{x(e)}\} = 1\ .$$
We will now prove two lemmas about the measure $M$ that will be useful later. They require an extra assumption on the type of events under consideration; see for example and . The results show that an event of positive probability remains of positive probability after a certain coupling modification. They in fact provide explicit lower bounds which will be needed when dealing with weak limits of the measure $M$ in Section 4.
\[lem: SStypemod\] Let $A \subseteq \Omega_1\times \{\sigma:\sigma_e=+1\}$ be such that $$\label{eqn: event monotone}
\text{If }(J,\sigma) \in A \text{ then }(J(e,s), \sigma) \in A \text{ for all }s \geq J_e.$$ Then for each $\lambda \in \mathbb{R}$, $$\label{eq:monotoneestimate}
M(A,~J_e \geq \lambda) \geq (1/2) \nu([\lambda,\infty))~M(A)\ .$$ If instead, we have $A \subseteq \Omega_1\times \{\sigma:\sigma_e=-1\}$ and $(J(e,s),\sigma) \in A$ for all $s \leq J_e$ then $$M(A,~J_e \leq \lambda) \geq (1/2) \nu((-\infty, \lambda])~M(A)\ .$$
We will prove the first statement; the second is similar. The left side of equals $$\int \nu(dJ_{\{e\}^c}) \left[ \int_\lambda^\infty \nu(dJ_e) \mu_J\{\sigma: (J,\sigma) \in A\}\right] \ ,$$ where the first integral is over all couplings $J_b$ for $b \neq e$, and the second is over $J_e$. This is $$\begin{aligned}
&&\int \nu(dJ_{\{e\}^c}) \left[ \int_\lambda^\infty \nu(dJ_e) \frac{1}{\nu((-\infty,\lambda))} \int_{-\infty}^\lambda \mu_J\{\sigma:(J,\sigma) \in A\} \nu(dy) \right] \\
&{\overset{\mbox{\eqref{eqn: increasing}}}{\geq}}& \int \nu(dJ_{\{e\}^c}) \left[ \int_\lambda^\infty \nu(dJ_e) \frac{1}{\nu((-\infty,\lambda))} \int_{-\infty}^\lambda \mu_{J(e,y)}\{\sigma:(J,\sigma) \in A\} \nu(dy) \right] \\
&\overset{\mbox{\eqref{eqn: event monotone}}}{\geq}& \int \nu(dJ_{\{e\}^c}) \left[ \int_\lambda^\infty \nu(dJ_e) \frac{1}{\nu((-\infty,\lambda))} \int_{-\infty}^\lambda \mu_{J(e,y)}\{\sigma:(J(e,y),\sigma) \in A\} \nu(dy) \right] \\
&\geq& \nu([\lambda,\infty))~M(A,~J_e<\lambda)\ ,\end{aligned}$$ where the third inequality comes from dropping $ \nu((-\infty,\lambda))^{-1}$. From this computation, $$\begin{aligned}
M(A,~J_e \geq \lambda) &\geq& (1/2)\left\{\nu([\lambda,\infty)) ~M(A,~J_e < \lambda) + M(A,~J_e \geq \lambda)\right\} \\
&\geq& (1/2)\nu([\lambda,\infty))~M(A)\ .\end{aligned}$$
The next lemma does not use the monotonicity property, but its proof is similar in spirit to the previous one. Instead of considering coupling values that are far from the critical value, we now consider values that are close. To show that an event of positive probability remains of positive probability after bringing the coupling closer to the critical value, we need to use the fact that by definition, a ground state remains in the support of the uniform measure for all values of $J_e$ up to the critical value.
\[lem: backmodify\] Let $c <d \in \mathbb{R}$ and $A \subseteq \{(J,\sigma):\sigma \in \mathcal{G}(J),~\sigma_e=+1\} \subseteq \Omega_1 \times \Omega_2$ be such that $$\label{eqn: event monotone 2}
\text{If $(J,\sigma) \in A$ and $J_e \geq c$ then $(J(e,y),\sigma) \in A$ for all $y\geq c$.}$$ Then for all $d >c$, $$M(A,~J_e \in [c,d]) \geq \nu([c,d])~M(A,~J_e \geq c)\ .$$
From the second condition, for a fixed $J$ with $J_e \geq c$, $$\sharp \{\sigma:(J,\sigma) \in A\} \leq \sharp \{ \sigma:(J(e,y),\sigma) \in A \} \text{ for all } y \geq c\ .$$ Since $\mu_J$ is the uniform measure and $A \subseteq \{(J,\sigma):\sigma \in \mathcal{G}(J)\}$, this implies $\nu$-almost surely $$\mu_J\{\sigma:(J,\sigma) \in A\} \leq \mu_{J(e,y)}\{\sigma:(J(e,y),\sigma) \in A\} \text{ for all } y \geq c\ .$$ Therefore $M(A,~J_e \geq c)$ equals $$\begin{aligned}
& &\int \nu(dJ_{\{e\}^c}) \int_c^\infty \nu(dJ_{e}) \frac{1}{\nu([c,d])} \left[ \int_c^{d} \mu_J\{\sigma:(J,\sigma) \in A\} \nu(dy) \right] \\
&\overset{\mbox{\eqref{eqn: event monotone 2}}}{\leq}& \int \nu(dJ_{\{e\}^c}) \int_c^\infty \nu(dJ_{e}) \frac{1}{\nu([c,d])} \left[ \int_c^{d} \mu_{J(e,y)}\{(\sigma,\sigma'):(J(e,y),\sigma,\sigma') \in A\} \nu(dy) \right] \\
&=& \frac{\nu([c,\infty))}{\nu([c,d])} \int \nu(dJ_{\{e\}^c}) \int_c^{d} \mu_{J(e,y)}\{(\sigma,\sigma'):(J(e,y),\sigma,\sigma')\in A\}\nu(dy) \ ,\end{aligned}$$ which is smaller than $\frac{M(A,~J_e \in [c,d])}{\nu([c,d])}$. This implies the lemma.
Properties of the interface
---------------------------
We now turn to properties of the interface $\sigma\Delta\sigma'$ under the measure $$M=\nu(dJ) ~ \mu_J\times \mu_J\ .$$
The main result of this section is that if $\sigma\Delta \sigma'$ is not empty, then it can be made to contain any fixed edge of the graph with positive probability. A similar statement has been proved in [@ADNS10 Corollary 2.9] for the metastate measure on ground states. The conclusion is straightforward by translation invariance in the case $G={\mathbb{Z}}^2$. A different approach is needed for the half-plane $G={\mathbb{Z}}\times {\mathbb{N}}$. For the sake of simplicity, we prove the statement in the case that the graph is planar and each face has four edges. The general statement for a graph $G=(V,E)$ with finite degree can be proved the same way.
\[prop: touch\] If there exists an edge $e\in E$ such that $M\{ (J,\sigma,\sigma'): e\in \sigma\Delta \sigma'\} >0$, then for any edge $b\in E$, $M\{ (J,\sigma,\sigma'): b\in \sigma\Delta \sigma'\} >0$.
Before turning to the proof, we record a fact: if $\sigma$ and $\sigma'$ are spin configurations then a cycle (in particular, a face) of the graph cannot have an odd number of edges in $\sigma \Delta \sigma'$. This is a direct consequence of the following elementary lemma; see for example Theorem 1 in [@Bieche].
\[lem: parity\] For any finite cycle $\mathcal{C}$ in the graph $G$, the parity of $\#\{e\in \mathcal{C}: J_e<0\}$ equals the parity of $\#\{e\in \mathcal{C}: \sigma_e\neq sgn J_e\}$.
The following lemma interprets the event that an edge is in the interface in terms of the critical values of $e$ in the two ground states.
\[lem: critical-dw\] For any edge $e$, $M\{ (J,\sigma,\sigma'): e\in \sigma\Delta \sigma'\} >0$ if and only if $M\{ (J,\sigma,\sigma'): C_e(J,\sigma)\neq C_e(J,\sigma')\}> 0$.
$\Longrightarrow$. By assumption, $$M\{(J,\sigma,\sigma'): \sigma_e=+1, \sigma'_e=-1\}>0\ .$$ By , $\sigma\in \mathcal{G}(J)$ and $\sigma_e=+1$ together imply that $J_e\geq C_e(J,\sigma)$. Similarly, $\sigma'\in \mathcal{G}(J)$ and $\sigma'_e=-1$ together imply that $J_e\leq C_e(J,\sigma')$. Therefore $$\label{eqn: equality}
M\{(J,\sigma,\sigma'): \sigma_e=+1, \sigma'_e=-1,C_e(J,\sigma)\leq J_e\leq C_e(J,\sigma')\}>0\ .$$
To complete the proof, observe that Proposition \[prop: decoupling\] implies $$M\{ (J,\sigma,\sigma'): C_e(J,\sigma)= J_e \text{ or } C_e(J,\sigma') = J_e\} =0\ .$$
$\Longleftarrow$. We may assume that with positive probability, on the event $\{C_e(J,\sigma)=C_e(J,\sigma')\}$, $\sigma$ and $\sigma'$ have the same sign at $e$. Without loss of generality, taking $\sigma_e=\sigma'_e=+1$, $$M\{ (J,\sigma,\sigma'): \sigma_e=\sigma'_e=+1, C_e(J,\sigma)\neq C_e(J,\sigma')\}>0\ .$$ In particular, there exists a deterministic $\delta>0$ such that $$M\{ (J,\sigma,\sigma'): \sigma_e=\sigma'_e=+1, C_e(J,\sigma') >C_e(J,\sigma)+\delta \}>0\ .$$ Hence there is a subset of the couplings of positive $\nu$-probability such that on this set $$\mu_J\times\mu_J\{ (\sigma,\sigma'): \sigma_e=\sigma'_e=+1, C_e(J,\sigma') >C_e(J,\sigma)+\delta \}>0$$ Fix the couplings other than $J_e$ and take $(\sigma,\sigma')$ in the above event. By , we must have $J_e\geq C_e(J,\sigma)$ and $J_e\geq C_e(J,\sigma')$. From Proposition \[prop: pair\], there exists $\sigma''\in \mathcal{G}_{-_e}$ such that $C_e(J,\sigma'')\geq C_e(J,\sigma')$. In particular, by Corollary \[cor: supersat3\], $\sigma''\in \mathcal{G}(J)$ for $J_e$ in the non-empty interval $(C_e(J,\sigma), C_e(J,\sigma''))$. Since $\mu_J$ is supported on a finite number of spin configurations, this implies that on a subset of positive $\nu$-probability $$\mu_J\times\mu_J\{ (\sigma,\sigma''): \sigma_e=+1,\sigma''_e=-1 \}>0\ .$$ Integrating over $J$ completes the proof.
By Lemma \[lem: critical-dw\], it suffices to show that $$\label{eqn: to prove edge}
M\{(J,\sigma,\sigma'): C_b(J,\sigma)\neq C_b(J,\sigma')\} >0\ .$$ Assume that $$\label{eqn: touch assumption}
M\{(J,\sigma,\sigma'):\sigma_e\neq\sigma'_e\} >0\ .$$ Without loss of generality, we can assume that $b$ and $e$ are edges of the same face. Otherwise, we simply apply the same argument successively on a path of neighboring faces from $b$ to $e$. Let us denote the other edges of the square face by $\tilde{b}$ and $\tilde{e}$.
$\sigma\Delta\sigma'$ contains $e$ with positive probability. By the paragraph preceding the statement of the proposition, if it contains $e$ it must also contain another edge of the face. If it contains $b$ with positive probability we are done, so suppose it contains $\tilde{e}$ with positive probability. Suppose also that with positive probability $\widetilde{b}$ is not in the interface. The other case is proved the same way and is simpler. We will indicate how to deal with it at the end of the proof.
In our notation, $e,\tilde e \in \sigma \Delta \sigma'$ and $b,\tilde b \notin \sigma \Delta \sigma'$. Therefore $\sigma_e\neq \sigma'_e$, $\sigma_{\tilde e} \neq \sigma'_{\tilde e}$, $\sigma_b = \sigma'_b$ and $\sigma_{\tilde{b}} = \sigma'_{\tilde b}$ on this event. The hypothesis now reduces to $M(B) >0$ for the event $$B=\{(\sigma,\sigma'): ~\sigma_{b}=\sigma'_{b},~\sigma_{\tilde b}=\sigma'_{\tilde b}, ~\sigma_{e}\neq \sigma'_{e}, \sigma_{\tilde e}\neq \sigma'_{\tilde e}\}\ .$$ By , for any $J$ such that $\mu_J\times\mu_J(B\cap\{\sigma:\sigma_{\tilde b}=+1\})>0$, if $J'$ is a configuration with $J'_{\tilde b}>J_{\tilde b}$, and $J'_a=J_a$ for $a \neq \tilde b$, then $\mu_{J'}\times\mu_{J'}(B)>0$. Similarly, for any $J$ such that $\mu_J\times\mu_J(B\cap\{\sigma:\sigma_{\tilde b}=-1\})>0$, if $J'$ is a configuration with $J'_{\tilde b}<J_{\tilde b}$, and $J'_a=J_a$ for $a \neq \tilde b$, then $\mu_{J'}\times\mu_{J'}(B)>0$. In particular, this implies that if $x$ is one of the endpoints of $\tilde b$ that is not also an endpoint of $b$, $$\int_{\{J:|J_{\tilde b}|>\mathcal{S}_{\tilde b}^x\}} \nu(dJ) ~\mu_J\times\mu_J(B) ~ >0\ .$$ We show that $$\label{eqn: to show touch}
\int_{\{J:|J_{\tilde b}|>\mathcal{S}_{\tilde b}^x\}} \nu(dJ) ~\mu_J\times\mu_J(B\cap\{(\sigma,\sigma'): C_b(J,\sigma)\neq C_b(J,\sigma')\}) ~ >0\ ,$$ thereby proving and the proposition.
The expression for the critical value $C_b(J,\sigma)$ can be written as follows. Let $F=\{b,\tilde{b},e,\tilde e\}$. For $I$ a non-empty subset of $\{\tilde b,\tilde e, e\}$, write $\mathcal{I}_{b,I}$ for the collection of finite sets of vertices $A$ whose boundary $\partial A$ intersected with $F$ equals the union of $\{b\}$ with $I$. This collection might be empty for some choice of $I$. We restrict only to sets $I$ for which $\mathcal{I}_{b,I}$ is not empty. Let $$C_{b,I}(J,\sigma)=\sup_{A \in \mathcal{I}_{b,I}}\left\{-\sum_{\substack{\{x,y\}\in\partial A\\ \{x,y\}\notin F }} J_{xy}\sigma_x\sigma_y\right\}\ .$$ In this notation, the expression becomes $$C_b(J,\sigma)=\max_{I\subseteq F\setminus \{b\} } \left\{\sum_{c\in I} -J_c\sigma_c + C_{b,I}(J,\sigma)\right\}\ .$$ Let $\Lambda \in CD_b(\sigma)$, $\Lambda' \in CD_b(\sigma')$ and note that both $\partial \Lambda$ and $\partial \Lambda'$ must contain at least one edge of the face other than $b$. When $|J_{\tilde b}| > \mathcal{S}_{\tilde b}^x$, Corollary \[cor: supersat2\] gives that neither can contain $\tilde{b}$, so they must both contain $b$ and other edges in $\{e,\tilde{e}\}$. Therefore on this event, the above definition of the critical values reduces to $$C_b(J,\sigma)=\max_{I\subseteq\{e,\tilde e\}} \left\{\sum_{c\in I}-J_c\sigma_c + C_{b,I}(J,\sigma)\right\}\ .$$ Since the max is attained, it holds on the event $\{J: |J_{\tilde b}|>\mathcal S_{\tilde b}^x\}$ that $$\begin{aligned}
&\mu_J\times\mu_J\Big(B\cap \{C_b(J,\sigma)=C_b(J,\sigma')\}\Big)\leq \\
&\qquad \sum_{I\subseteq\{e,\tilde e\}, I'\subseteq\{e,\tilde e\} }\mu_J\times\mu_J
\left \{\sum_{c\in I} -J_c\sigma_c + C_{b,I}(J,\sigma)=\sum_{c'\in I'} -J_{c'}\sigma'_{c'}
+ C_{b,I'}(J,\sigma')\right\}\ .
\end{aligned}$$ The right-hand side is the same as $$\sum_{I\subseteq\{e,\tilde e\}, I'\subseteq\{e,\tilde e\} }\mu_J\times\mu_J
\Big\{ C_{b,I}(J,\sigma)- C_{b,I'}(J,\sigma')=\sum_{c\in I} J_c\sigma_c -\sum_{c'\in I'} J_{c'}\sigma'_{c'}\Big\}\ .$$ The right-hand side of the equality in the event is a linear combination of the $J_c$’s, $c\in I\cup I'$, where the coefficients, which we call $s_c$, can only take the values $0,\pm1,\pm2$. Most importantly, for each choice of $I,I'$, the $s_c$’s cannot all be zero since $I$ and $I'$ are not empty, and $\sigma_c=-\sigma'_c$ for $c\in \{e,\tilde{e}\}$. Letting $\mathcal{J}_{I,I'}$ be the set of non-zero $\{0,\pm 1, \pm 2\}$-valued vectors $s$, with each entry corresponding to an element in $I \cup I'$, we see that the above is smaller than $$\sum_{I\subseteq\{e,\tilde e\}, I'\subseteq\{e,\tilde e\} } \sum_{ s \in \mathcal{J}_{I,I'}} \mu_J\times\mu_J
\Big\{ C_{b,I}(J,\sigma)- C_{b,I'}(J,\sigma')=\sum_{c\in I\cup I'} J_cs_c\Big\}\ .$$ To show , integrate over $\nu$ and use Proposition \[prop: decoupling\] with $B=\{e,\tilde{e}\}$ and $h_{B^c}=C_{b,I}(J,\sigma)- C_{b,I'}(J,\sigma')$.
This completes the proof in the case that $\tilde b$ is not in the interface. If the probability of this is zero (that is, if does not hold), then the proof is easier. We do not need to supersatisfy $J_{\tilde b}$; we simply take $I,I'$ to be subsets of $\{\tilde b, e, \tilde e\}$ and complete the proof from after equation .
Before turning to the proof of the main result, we mention that in the case that the graph is invariant under a set of transformations (for example, translations), the uniform measure inherits a covariance property. Translation-covariant measures on ground states are typically not easy to construct. The only other example known to the authors is the metastate on ground states constructed from suitable boundary conditions. An advantage of a translation-covariant measure is that the corresponding $\nu$-averaged measure is preserved under translations.
\[lem: covariance\] Let $G={\mathbb{Z}}^d$ or $G={\mathbb{Z}}\times{\mathbb{N}}$ and suppose $|\mathcal{G}(J)|<\infty$. The uniform measure $\mu_J$ is translation-covariant. That is, if $T$ is a translation of ${\mathbb{Z}}^d$ or a horizontal translation of ${\mathbb{Z}}\times{\mathbb{N}}$, then for any $B \in \mathcal{F}_2$, $$\mu_{TJ}(B)=\mu_{J}\{\sigma: T \sigma\in B\} \text{ for $\nu$-almost all $J$.}$$ In particular, the measure $M$ on $\Omega_1\times \Omega_2$ (or on $\Omega_1\times\Omega_2\times\Omega_2$) is translation-invariant.
Using the fact that $|\mathcal{G}(J)|$ is constant $\nu$-almost surely, one gets $$\begin{aligned}
\mu_{TJ}(B)=\frac{\#\{\sigma \in \mathcal{G}(TJ): \sigma\in B\}}{|\mathcal{G}(TJ)|}=\frac{\#\{\sigma \in \mathcal{G}(J): T \sigma\in B\}}{|\mathcal{G}(J)|}
=\mu_{J}\{\sigma: T \sigma\in B\}\ .
\end{aligned}$$
For the second assertion, let $B'\subset {\mathbb{R}}^E\times \{-1,+1\}^V$. Define $T^{-1}B'=\{(J,\sigma): (TJ,T\sigma)\in B' \}$. Then the first claim implies that the probability of $T^{-1}B'$ is $$M(T^{-1}B')
= \int \nu(dJ) \mu_J \{\sigma: (TJ,T\sigma)\in B'\} = \int \nu(dJ) \mu_{TJ} \{\sigma:(TJ,\sigma)\in B'\}\ .$$ As $\nu$ is translation-invariant, we may replace $\nu(dJ)$ by $\nu(dTJ)$ on the right side. The right side then equals $\int \nu(dJ) \mu_{J}(\sigma:(J,\sigma)\in B')=M(B')$ as claimed.
The main result on the half-plane
=================================
Preliminaries
-------------
In this section, we consider the EA model on the half-plane $H={\mathbb{Z}}\times {\mathbb{N}}$ with free boundary conditions at the bottom. Recall from Corollary \[cor: constant\] that the number of ground states $|\mathcal{G}(J)|$ is non-random. We continue to assume that $|\mathcal{G}(J)| < \infty$. Write $$M = \nu(dJ) \times \left( \mu_J \times \mu_J \right)\ ,$$ where $\mu_J$ is the uniform measure on $\mathcal{G}(J)$. We will use the notation that sampling from $M$ amounts to obtaining a triple $(J,\sigma,\sigma')$ from the space $$\Omega := \mathbb{R}^{E_H} \times \{-1,+1\}^{V_H} \times \{-1,+1\}^{V_H}\ ,$$ where $E_H$ and $V_H$ denote the edges and vertices of the half-plane respectively. To show Theorem \[thm: mainthm\], it is sufficient to prove that $M\{(J,\sigma,\sigma'): \sigma\Delta\sigma'\neq\emptyset\}=0$. This implies that if $|\mathcal{G}(J)|<\infty$, then $|\mathcal{G}(J)|=2$. We will derive a contradiction from the following: $$\label{eqn: assumption}
\text{ assume that } M\{(J,\sigma,\sigma'): \sigma\Delta\sigma'\neq\emptyset\}>0\ .$$
For this purpose, a representation of the interface $\sigma\Delta\sigma'$ in the dual lattice will be used. Instead of thinking of an edge $e$ as being in the interface, we think of the dual edge crossing $e$ as being in it. We denote this dual edge by $e^*$. The interface represented this way is a collection of paths in the dual lattice. The reader is referred to Figure \[fig: dw\] for an illustration of this representation. Note that these dual paths cannot contain loops; otherwise, $\sigma$ or $\sigma'$ would violate the ground state property . Moreover, it is elementary to see that the interface cannot have dangling ends – dual vertices with degree one in the interface (for example, using Lemma \[lem: parity\]). A [*domain wall*]{} refers to a connected component of $\sigma \Delta \sigma'$, viewed as edges in the dual lattice. In the case of the half-plane $G={\mathbb{Z}}\times{\mathbb{N}}$, we call any domain wall that crosses the $x$-axis a [*tethered domain wall*]{}.
![An example of an interface between ground states on the half-plane. The edges in $\sigma\Delta\sigma'$ are the thick ones. The representation of the interface as dual paths is depicted by the dotted lines. In this example, there are two domain walls and they are both tethered.[]{data-label="fig: dw"}](interface.eps){height="6cm"}
The method used to derive a contradiction is similar in spirit to the one in [@ADNS10]. From $M$ we construct a measure on ground states in ${\mathbb{Z}}^2$ (denoted by $\widetilde{M}$) with two contradicting properties: on the one hand any interface sampled from $\widetilde M$ must be disconnected; on the other hand it must be connected. The construction of $\widetilde{M}$ is outlined below and some properties are proved. The proof of non-connectivity is given in Section \[sec: disconnected\]. The proof of connectivity follows the method of Newman & Stein [@NS01] and is in Section \[sec: connected\].
The first step is to extend the measure $M$ to include the critical values. This extension is needed because the critical values are not continuous functions of $(J,\sigma)$ in the product topology; they depend on the couplings in a non-local manner, as can be seen from the formula . Therefore their distribution is not automatically preserved under weak limits. Enlarging the probability space to include them will bypass this obstacle. For illustration, consider the event that a fixed edge $e$ has $C_e(J,\sigma) \in I$, for some fixed open interval $I$. The probability of this event is not necessarily preserved under weak limits. However, after we include the variables $C_e(J,\sigma)$ in our space, this event becomes a cylinder event and therefore its probability will behave nicely after taking limits.
We remark that a different type of extension (but with the same spirit) was done in [@ADNS10]. Namely, a measure called [*the excitation metastate*]{} (introduced first in [@NS01]) was defined to include the critical values but also all information about local changes of the couplings. Implementing this type of construction turns out to be more delicate in the case of the uniform measure. We therefore abandon it and turn to a simpler framework. The monotonicity property defined in Section \[sect: monotonicity\] is the key tool for this approach.
For a fixed $J$, edge $e$, and $\sigma \in \mathcal{G}(J)$, recall the definition of the critical value $C_e(J,\sigma)$ from Lemma \[lem: critical formula\]. Define the map $$\label{eq: phidef}
\Phi \text{ by } (J,\sigma,\sigma') \mapsto (J,\sigma,\sigma',\{C_e(J,\sigma)\}_e,\{C_e(J,\sigma')\}_e)\ ,$$ where the last two coordinates are the collections of critical values of all edges. (This map is only defined for $\sigma,\sigma' \in \mathcal{G}(J)$ but this does not create a problem because the support of $\mu_J \times \mu_J$ is equal to $\mathcal{G}(J) \times \mathcal{G}(J)$.) Let $M^*$ be the push-forward of $M$ by $\Phi$ on the space $$\label{eqn: Omega*}
\Omega^* := {\mathbb{R}}^{E_H} \times \{-1,+1\}^{V_H} \times \{-1,+1\}^{V_H} \times \mathbb{R}^{E_H} \times \mathbb{R}^{E_H}\ .$$ Sampling from $M^*$ amounts to obtaining a configuration $$\omega = (J,\sigma,\sigma',\{C_e\}_e,\{C_e'\}_e) \in \Omega^*\ .$$ We have not indicated the dependence of $C_e$ on $\sigma$ and $J$, for example, because on $\Omega^*$, it is no longer a function of the other variables. Note that the marginal of $M^*$ on $(J,\sigma,\sigma')$ is $M$.
We now construct a translation-invariant measure $\widetilde{M}$ on $$\widetilde \Omega = {\mathbb{R}}^{E_{{\mathbb{Z}}^2}} \times \{-1,+1\}^{{\mathbb{Z}}^2} \times \{-1,+1\}^{{\mathbb{Z}}^2} \times {\mathbb{R}}^{E_{{\mathbb{Z}}^2}} \times {\mathbb{R}}^{E_{Z^2}}\ .$$ from the measure $M^*$ using a standard procedure. An event in $\widetilde \Omega$ that only involves, in a measurable way, a finite number of vertices of ${\mathbb{Z}}^2$ in $\sigma$ and $\sigma'$, and a finite number of edges through the couplings $J_e$ and the critical values $C_e$ and $C_e'$ will be called a [*cylinder event*]{}. Let $T$ be the translation of $\mathbb{Z}^2$ that maps the origin to the point $(0,-1)$ and for each $n \geq 0$ define $$\label{eqn: M*}
M^*_n = \frac{1}{n+1} \sum_{k=0}^n T^k M^*\ .$$ Note that the translated measure $T^k M^*$ is well-defined on cylinder events for $k$ large enough. (If it is not defined, we can take it to be zero without affecting the limit below.) Moreover, the sequence of measures $M^*_n$ is tight. This is obvious for the marginal on $(J,\sigma,\sigma')$. The fact that it holds also when including the critical values is a direct consequence of Corollary \[cor: bound\]. Therefore there exists a sequence $(n_k)$ such that $M^*_{n_k}$ converges as $k \to \infty$, in the sense of finite-dimensional distributions, to a translation invariant measure on $\widetilde{\Omega}$. Call this limiting measure $\widetilde M$. The weak convergence of the measures $M^*_n$ to $\widetilde M$ implies that for any event $B$ in $\widetilde{\Omega}$ $$\label{eqn: convergence sets}
\begin{aligned}
\liminf_{n\to\infty} M^*_n(B)&\geq \widetilde M(B) \text{ if $B$ is open;}\\
\limsup_{n\to\infty} M^*_n(B)&\leq \widetilde M(B) \text{ if $B$ is closed;}\\
\lim_{n \to \infty} M^*_n(B)&=\widetilde M(B) \text{ if } \widetilde M(\partial B)=0\ .
\end{aligned}$$ (See, for example, Theorem 4.25 of [@Kallenberg].) Here we are using the fact that $\widetilde \Omega$ is metrizable, as these statements are true in general for probability measures on metric spaces. The boundary $\partial B$ is the closure of $B$ minus its interior in $\widetilde \Omega$ (not to be confused with $\partial A$ for $A$ a finite set of vertices in the graph). Examples of open (resp. closed) cylinder sets are $\{h(J,\{C_e\}_e,\{C_e'\}_e)\in O\}$ where $h$ is a continuous function depending only on a finite number of edges, and $O$ is an open (resp. closed) set of ${\mathbb{R}}$.
\[rem: rabbit\] Note that if $B$ only depends on the spins of a finite number of vertices and not on the couplings and critical values, actual convergence of the probability holds, since $B$ is open and closed thus $\partial B=\varnothing$. This same conclusion is true if $B$ is an event of the form $\{(\sigma,\sigma') \in D, J \in I\}$ for events $D$ that depend on finitely many spins and sets $I$ in some finite dimensional Euclidean space with boundary of zero Lebesgue measure. Indeed, it is a general fact that for any two events $B$ and $B'$, $\partial (B\cap B')\subseteq \partial B \cup \partial B'$; therefore $\partial B\subseteq \partial \{(\sigma,\sigma') \in D\}\cup \partial\{J \in I\}$. It follows that the set $\partial B$ has $\widetilde M$-measure zero, since $\widetilde M(\partial\{J \in I\})=\nu(\partial\{J \in I\})=0$ (by the continuity of $\nu$) and $\widetilde M( \partial\{(\sigma,\sigma') \in D\})=\widetilde M(\varnothing)=0$.
Since $\widetilde M$ will be our object of study for the remainder of the paper, we will spend some time explaining its basic properties. Suppose $\omega=(J,\sigma,\sigma',\{C_e\}_e,\{C_e'\}_e)$ is sampled from $\widetilde M$. First, it follows directly from the construction that $\sigma$ and $\sigma'$ are almost-surely ground states on $\mathbb{Z}^2$. Also if we define $F_e=|J_e-C_e|$ and $F_e'=|J_e-C_e'|$ to be the [*flexibility of the edge $e$*]{} in $\sigma$ and in $\sigma'$, then for any finite set $A$ with $e\in \partial A$, $$\label{eq: tacos}
F_e\leq \sum_{\{x,y\} \in \partial A} J_{xy} \sigma_x \sigma_y\ ~\widetilde M\text{-a.s.}$$ and similarly for $F_e'$. This is true because this relation holds with $M^*$-probability one on the space $\Omega^*$ and for its translates by $k$ (for $k$ large enough that $A \subseteq T^kV_H$) by . Moreover, both sides are continuous functions of $\omega$. Thus the $\omega$’s satisfying the relation form a closed set. Equation then follows from . It remains to take the infimum over all (countably many) finite sets $A$ to conclude the following lemma.
\[lem: flex ineq\] Let $I_e(J,\sigma):=\inf_{\substack{A:e \in \partial A \\ A \text{ finite}}}\sum_{\{x,y\} \in \partial A} J_{xy} \sigma_x \sigma_y$. For any edge $e$, $$\widetilde M\{F_e \leq I_e(J,\sigma)\}= 1\ .$$ The corresponding statement holds for $\sigma'$.
In other words, flexibilities produced by the weak limit procedure from half-planes are no bigger than the ones computed directly from in the full plane. This is to be expected since the former also take into account sets $A$ that touch the boundaries of some translated half-planes. The last basic property we need is a result analogous to Proposition \[prop: decoupling\] (specifically the consequence of that proposition that $M(C_e=J_e)=0$) for the weak limit $\widetilde{M}$.
\[lem: flexnonzero\] For any edge $e$, $$\widetilde M\{F_e = 0\} = 0\ .$$ The corresponding statement holds for $F_e'$.
It suffices to prove the statement for $F_e$. Because $\{F_e=0\}$ is not an open set, we cannot simply take limits in Proposition \[prop: decoupling\] to obtain the result. Consider the cylinder event $\{|J_e - C_e| < {\varepsilon}, |J_e|<N\}$ for ${\varepsilon}>0$ and $N>0$. Note that this set is open. (The cutoff in $J_e$ seems superfluous first but is useful in the estimate below.) The conclusion will follow from once we show that for each fixed $N$, $$\label{eq: limuniform}
T^k M\{|J_e - C_e(J,\sigma)|<{\varepsilon},~|J_e|< N\}$$ can be made arbitrarily small uniformly in $k$ (for $k$ such that $e \in T^k E_H$) by taking ${\varepsilon}$ small.
We prove the estimate for $k=0$ only. It will be clear that the same proof holds for any $k$. Using the monotonicity and the notation $J(e,s)$ of , we have $$\begin{aligned}
&&M(|J_e - C_e(J,\sigma)|<{\varepsilon},~ J_e< N,~ \sigma_e=+1) \\
&=&\int \nu(dJ_{\{e\}^c}) \int_{-\infty}^{N} \nu(dJ_e) \frac{1}{\nu(J_e,\infty)} \int_{J_e}^\infty \nu(ds) ~\mu_J \{\sigma: |J_e-C_e(J,\sigma)| < {\varepsilon},~ \sigma_e = +1\} \\
&\leq& \frac{1}{\nu(N,\infty)} \int \nu(dJ_{\{e\}^c}) \int \nu(dJ_e) \int \nu(ds) ~\mu_{J(e,s)} \{\sigma: |J_e-C_e(J,\sigma)| < {\varepsilon},~ \sigma_e = +1\} \\\end{aligned}$$ We now exchange integrals using Fubini and integrate over $J_e$ first to get the upper bound $$\frac{1}{\nu(N,\infty)} \int \nu(dJ_{\{e\}^c}) \int \nu(ds)\int \mu_{J(e,s)}(d\sigma) \nu\{J_e: |J_e - C_e(J,\sigma)| < {\varepsilon}\}\ .$$ Recall that $C_e(J,\sigma)$ does not depend on $J_e$. The interval $\{J_e: |J_e - C_e(J,\sigma)| < {\varepsilon}\}$ has length $2{\varepsilon}$, hence given $\delta>0$, its $\nu$-probability can be made smaller than $\delta$, independently of $C_e(J,\sigma)$, by the continuity of $\nu$. We have thus shown $$M(|J_e - C_e(J,\sigma)|<{\varepsilon},~ J_e< N,~ \sigma_e=+1)\leq \frac{\delta}{\nu(N,\infty)}$$ Repeating the same proof, but using monotonicity in the other direction and taking $J_e > -N$, $$M(|J_e-C_e(J,\sigma)|< {\varepsilon},~ J_e > -N,~\sigma_e=-1) \leq \frac{\delta}{\nu(N,\infty)}\ .$$ This estimate holds for any $k$ and can be made uniformly small by taking ${\varepsilon}$ small.
Non-connectivity of the interface {#sec: disconnected}
---------------------------------
In this section we show
\[prop: not connected\] If holds, then $
\widetilde M\{\sigma \Delta \sigma' \text{ is not connected}\} > 0\ .
$
The first key ingredient is to show that with positive $M$-probability, there are infinitely many tethered domain walls in the interface on the half-plane.
If holds, then with positive $M$-probability, $\sigma\Delta\sigma'$ crosses the $x$-axis. Moreover, with positive $M$-probability, $\sigma\Delta\sigma'$ has infinitely many domain walls.
The first claim is a direct application of Proposition \[prop: touch\]. For the second, note that a connected component of $\sigma\Delta\sigma'$ cannot cross the $x$-axis twice. If it did, it would contain a dual path whose union with the $x$-axis encloses a finite set of vertices $S$. We must have $\sum_{\{x,y\}\in \partial S}J_{xy}\sigma_x\sigma_y\geq 0$ and similarly in $\sigma'$ by . Since $\sigma_x\sigma_y=-\sigma'_x\sigma'_y$ on $\partial S$, we conclude $\sum_{\{x,y\}\in \partial S}J_{xy}\sigma_x\sigma_y=0$, and this has probability zero by the continuity of $\nu$. Therefore to each dual edge crossing the $x$-axis contained in $\sigma\Delta\sigma'$, there corresponds a unique connected component of $\sigma\Delta\sigma'$. By horizontal translation-invariance of $M$ (Lemma \[lem: covariance\]), if $\sigma\Delta\sigma'$ contains one such dual edge, it must contain infinitely many. This gives the second claim.
The next step is to prove that distinct connected components sampled from $M$ do not disappear after constructing $\widetilde M$. This is done by showing that the expected number of components intersecting a fixed box is uniformly bounded below in $k$. This is the content of the next lemma. We omit the proof; it is exactly the same as that of [@ADNS10 Proposition 3.4]. For any $k \geq 0$ and $n \geq 1$, let $$I_{n,k} = [-n,n] \times \{k\}$$ and let $N_{n,k}$ be the number of distinct tethered domain walls that cross the line segment $I_{n,k}$. Write ${\mathbb{E}}_M$ for the expectation with respect to $M$.
\[lem: tethered\] For fixed $k \geq 0$, the sequence $({\mathbb{E}}_{M} N_{n,k})_n$ is sub-additive. Therefore $$\lim_{n \to \infty} (1/n) {\mathbb{E}}_{ M} N_{n,k} \text{ exists }.$$ Furthermore if holds then there exists $c>0$ such that for all $k \geq 0$ and $n \geq 1$, $${\mathbb{E}}_{M} N_{n,k} \geq cn\ .$$
This lemma yields Proposition \[prop: not connected\]. We omit the proof as it is identical to [@ADNS10 Proposition 3.5]. The proof there only deals with cylinder events involving only spins, and therefore limits go through using Remark \[rem: rabbit\].
The Newman-Stein technique {#sec: connected}
--------------------------
In this section, we show
\[prop: connected\] $
\widetilde M(\sigma \Delta \sigma' \text{ is not connected}) =0 \ .
$
This contradicts Proposition \[prop: not connected\] and finishes the proof of Theorem \[thm: mainthm\]. We will apply the Newman-Stein technique from [@NS01]. The idea is to construct a random variable $I$ (see below) that is defined on the event $\{\sigma \Delta \sigma' \text{ is not connected}\}$. Proposition \[prop: connected\] will follow from both $$\label{eqn: 1}
\widetilde{M}\{I\leq 0, \sigma \Delta \sigma' \text{ is not connected}\}=0$$ and $$\label{eqn: 2}
\widetilde{M}\{I> 0, \sigma \Delta \sigma' \text{ is not connected}\}=0\ .$$
### The definition of $I$
We first need information about the topology of interfaces $\sigma \Delta \sigma'$ sampled from $\widetilde M$. This is the content of the following proposition, which is analogous to Theorem 1 in [@NS01]. The proof of part 1 relies on translation invariance and part 2 is a consequence of Lemma \[lem: parity\]. The proof of part 3 uses ideas of Burton & Keane [@BK89].
\[prop: dwproperties\] With $\widetilde M$ probability one, the following statements hold.
1. If $\sigma \Delta \sigma'$ is nonempty, then it has positive density.
2. If $\sigma \Delta \sigma'$ is nonempty, then it does not contain any dangling ends or three-branching points.
3. If $\sigma \Delta \sigma'$ is nonempty, then it contains no four-branching points. In particular, each dual vertex in the domain wall has degree two; thus each domain wall is a doubly infinite dual path. Moreover, each component of the complement (in $\mathbb{R}^2$) of $\sigma \Delta \sigma'$ is unbounded and has no more than two topological ends in the following sense. If $C$ is such a component then for all bounded subsets $B$ of $\mathbb{R}^2$, the set $C \setminus B$ does not have more than two unbounded components.
Parts 2 and 3 of the proposition tell us that the regions between domain walls are topologically either strips or half-spaces. This implies that there is a natural ordering on domain walls: each domain wall has 0, 1 or 2 well-defined neighboring domain walls. In particular, dual paths from one domain wall to a neighboring one are well-defined:
\[def: rung\] A [**rung**]{} is a non-self intersecting finite dual path that starts at a dual vertex in a domain wall and ends at a dual vertex in a different domain wall. No other dual vertices on the path are in a domain wall.
Let $h=h(\omega)$ be the first horizontal edge in the interface starting from the origin to the right. For almost every configuration $\omega$ such that the interface is nonempty, such an $h$ exists because of translation and rotation invariance of $\widetilde M$. So we can define $$I = \inf_R E(R)\ ,$$ where the infimum is over all rungs $R$ touching the domain wall of $h^*$ and $E(R)$ is the energy: $$E(R) = \sum_{\{ x,y \}^* \in R} J_{xy} \sigma_x \sigma_y\ .$$ See Figure \[fig: rung\] for a depiction of $h$ and a rung under consideration.
![An example of rung from the domain wall of $h$ to another domain wall.[]{data-label="fig: rung"}](magic_rung.eps){height="8cm"}
Note that since no edge of a rung is in the interface $\sigma \Delta \sigma'$, we must have $\sigma_x\sigma_y = \sigma_x'\sigma_y'$ for all edges $\{ x,y \} \in R$. Therefore in the definition of $E(R)$ it does not matter if we choose $\sigma$ or $\sigma'$ to perform the computation.
### $I\leq 0$ has zero probability
We will now $$\label{eq: assumption3}
\text{assume that }\widetilde{M}\{I\leq 0, \sigma \Delta \sigma' \text{ is not connected}\}>0\ ,$$ and derive a contradiction. For a dual edge $e^*$, ${\varepsilon}>0$ and a positive integer $K$, let $A_e({\varepsilon},K)$ be the event that (a) $e^*$ is in a rung (between any two domain walls) with energy less than ${\varepsilon}$ and (b) this rung has length (number of dual edges) at most $K$. Whenever $I\leq0$ and ${\varepsilon}>0$ there must exist a rung starting from the domain wall containing $h^*$ with energy less than ${\varepsilon}$. So $A_e({\varepsilon},K)$ occurs for some $e$ and $K$, and under , there exists ${\varepsilon}>0$ and $K$ such that $$\sum_{e \in \mathbb{E}^2} \widetilde M(A_e({\varepsilon},K)) \geq \widetilde M \left(\cup_{e \in \mathbb{E}^2} A_e({\varepsilon},K)\right) > 0\ .$$ By translation invariance, $\widetilde M(A_e({\varepsilon},K))>0$ for all $e$.
Let us say that dual edges $e_1^*$ and $e_2^*$ are [*on the same side*]{} of a domain wall $D$ if they both have a dual endpoint in the same connected component of the complement of $D$. The following lemma is the same as Lemma 1 in [@NS01].
\[lem: infinitelymanyedges\] With $\widetilde M$-probability one, the following holds. If $\sigma \Delta \sigma'$ is not connected, then for each domain wall $D$, either there are infinitely many dual edges $e^*$ touching $D$ such that $A_e({\varepsilon},K)$ occurs (in [*both*]{} directions along $D$ and on each side of $D$) or there are zero.
For an edge $e$, let $B_e({\varepsilon},K)$ be the event that (a) $A_e({\varepsilon},K)$ occurs and (b) there exists a domain wall $D$ such that $e^*$ touches $D$ and in at least one direction on $D$, there are no endpoints of dual edges $h^*$ for which $A_h({\varepsilon},K)$ occurs for the same domain wall $D$ on the same side. For each $e$ such that $B_e({\varepsilon},K)$ occurs we may associate $e$ to a domain wall $D$. Note that in each realization $\omega$ in the support of $\widetilde M$, there are at most 4 edges associated with each domain wall (counting two directions and two sides of the domain wall).
Let $B(n)$ be the box of side length $n$ centered at the origin, and let $N_n$ be the number of domain walls which have a dual vertex in $B(n)$. Last, let us use the notation that $e \in B(n)$ if both of $e$’s endpoints are in $B(n)$. The above arguments imply that $$\begin{aligned}
\sum_{e \in B(n)} \widetilde M (B_e({\varepsilon},K)) &=& {\mathbb{E}}_{\widetilde M} \left( \sum_{e \in B(n)} 1(B_e({\varepsilon},K)) \right)
\leq 4 {\mathbb{E}}_{\widetilde M} N_n\ .\end{aligned}$$ Here $\mathbb{E}_{\widetilde M}$ stands for expectation with respect to $\widetilde M$. Distinct domain walls do not intersect so we can associate to each dual edge of the outer edge boundary $\partial_e B(n)$ (that is, having one endpoint in $B(n)$ and one in $B(n)^c$) at most one domain wall that contains it. Therefore for some suitable constants $C_1,C_2>0$ $$\frac{1}{|B(n)|} \sum_{e \in B(n)} \widetilde M(B_e({\varepsilon},K)) \leq C_1 \frac{1}{|B(n)|} |\partial_e B(n)| \leq C_2 |B(n)|^{-1/2} \to 0$$ as $n\to\infty$. By translation invariance, $\widetilde M(B_e({\varepsilon},K))$ is the same for all $e$ and thus equals $0$, completing the proof.
\[rem: rem1\] Although the previous lemma was stated for the events $A_e({\varepsilon},K)$, the same proof can be used for a number of different events like $A_e({\varepsilon},K)$. In [@NS01], these events were called “geometrically defined.” Examples of such events are (a) the event that $e^*$ is in a domain wall and is adjacent to a rung with a specified energy and (b) the event that $e^*$ is in a domain wall and has a specified flexibility in $\sigma$ or $\sigma'$. We will use these facts later in Section \[sec: I=0\]. Note that it is not enough to use only translation-invariance in the proof, as we would need to use (random) translations along a domain wall.
For an edge $e$, ${\varepsilon}>0$ and a positive integer $K$, let $A_e^0({\varepsilon},K)$ be the event that $A_e({\varepsilon},K)$ occurs and one of the endpoints of $e^*$ is in the domain wall of $h^*$. If $I \leq 0$, then for each ${\varepsilon}$ there exists $K$ such that $A_e^0({\varepsilon},K)$ occurs. By Lemma \[lem: infinitelymanyedges\], we may find infinitely many dual edges $e_n^*$ and $f_n^*$ (in both directions along the domain wall of $h^*$ but on the same side as $e^*$) such that $A_{e_n}({\varepsilon},K)$ and $A_{f_n}({\varepsilon},K)$ occur. The $e_n$’s are chosen in one direction and the $f_n$’s in the other. Let $R_n$ be a rung corresponding to $e_n$ and let $S_n$ corresponding to $f_n$. By relabeling the sequences $(e_n)$ and $(f_n)$ we may ensure that $R_n$ does not intersect $S_n$ for any $n$. (Here we are using the fact that the rungs have length at most $K$ and so for a fixed $n_0$, there are finitely many $n$’s such that $R_{n_0}$ intersects $S_n$.) Calling $D_0$ the domain wall containing $h^*$, both rungs $S_n$ and $R_n$ connect $D_0$ to the same domain wall, say, $D_1$.
Since $R_n$ and $S_n$ are disjoint, the dual path $P$ consisting of $R_n$, $S_n$, the piece of $D_0$ between $e_n^*$ and $f_n^*$ (call it $P_0$) and the corresponding piece of $D_1$ between the intersection points of $R_n$ and $S_n$ with $D_1$ (call it $P_1$) is a circuit in the dual lattice. See Figure \[fig: circuit\] for a depiction.
\[fig: circuit\]
{height="8cm"}
The spin configurations $\sigma$ and $\sigma'$ sampled from $\widetilde M$ are ground states, hence $$\sum_{\{x,y\}^* \in P} J_{xy} \sigma_x \sigma_y \geq 0 \text{ and } \sum_{\{x,y\}^* \in P} J_{xy} \sigma'_x \sigma'_y \geq 0\ .$$ For each edge $\{x,y\}$ whose dual edge is in either $R_n$ or $S_n$, we have $\sigma_x \sigma_y = \sigma'_x \sigma'_y$. For each edge $\{x,y\}$ whose dual edge is on either $P_1$ or $P_0$ we have $\sigma_x \sigma_y = - \sigma'_x \sigma'_y$. Using the fact that the energies of the rungs $R_n$ and $S_n$ are below ${\varepsilon}$, the above two inequalities reduce to $$\left| \sum_{\{x,y\}^* \in P_0} J_{xy}\sigma_x \sigma_y + \sum_{\{x,y\}^* \in P_1} J_{xy} \sigma_x \sigma_y \right| < 2 {\varepsilon}\ ,$$ and so $\sum_{\{x,y\}^* \in P} J_{xy} \sigma_x \sigma_y < 4{\varepsilon}\ .$ As ${\varepsilon}$ is arbitrary, the edge $h$ has flexibility zero by Lemma \[lem: flex ineq\] (since $I_h=0$). By Lemma \[lem: flexnonzero\], this has zero probability, proving .
### $I>0$ has zero probability {#sec: I=0}
We now show by assuming $$\label{eqn: assum I>0}
\widetilde{M}\{I> 0, \sigma \Delta \sigma' \text{ is not connected}\}>0\ .$$ and deriving a contradiction. The idea is that if $I > 0$ then we can find one rung near the origin whose energy we can lower by making a local modification to the couplings. The contradiction follows because the first edge in this rung will be the only one touching its domain wall with a certain energy property. This violates a variation of Lemma \[lem: infinitelymanyedges\]. In this section we will write $I=I(\omega)$ to emphasize the dependence of $I$ on the configuration $\omega \in \widetilde \Omega$.
For each edge $e$ let $$\widetilde F_e:= \min\{ F_e, F'_e \}\ .$$ By Lemma \[lem: flexnonzero\], $$\label{eq: doubleflexnonzero}
\widetilde M(\widetilde F_e > 0 \text{ for all } e) = 1\ .$$
The rest of this subsection will serve to prove the following proposition. Fix ${\varepsilon}>0$ and let $f$ be the edge connecting $(1,0)$ and $(1,1)$. Also define $g$ to be the edge connecting the origin to $(1,0)$. Let $X_{\varepsilon}$ be the intersection of the following events:
1. $\sigma\Delta\sigma'$ is disconnected and $I>0$;
2. $g^* \in \sigma \Delta \sigma'$;
3. $f^*$ is in a rung $R$ that satisfies $E(R) < I(\omega)+{\varepsilon}/2$.
Note that on $X_{\varepsilon}$, the edge $h$ (used in the definition of $I$ in the previous section) equals $g$.
\[prop: xe\] If holds, there exists ${\varepsilon}_0$ such that for all but countably many $0<{\varepsilon}<{\varepsilon}_0$, $\widetilde M\left(X_{\varepsilon},~ \widetilde F_f>{\varepsilon}\right)>0$.
We begin by finding deterministic replacements for many local quantities. Let $E_1$ be the event that $g^*\in\sigma\Delta\sigma'$, $f^*\notin\sigma\Delta\sigma'$, $\sigma\Delta\sigma'$ is disconnected and $I>0$. By translation invariance and by the assumption , we have $\widetilde{M}(E_1)>0$. We denote the domain wall of $g^*$ by $D_0(\omega)$ for $\omega\in E_1$. By , we may choose ${\varepsilon}_0>0$ such that whenever $0<{\varepsilon}<{\varepsilon}_0$, $$\widetilde M(E_1,~\text{there exists } e^* \in D_0(\omega) \text{ such that } \widetilde F_e>{\varepsilon}) > 0\ .$$ Furthermore, note that the distribution of $\widetilde F_e$ (for any edge $e$) under the measure $\widetilde M$ can only have countably many atoms. We fix any such $0<{\varepsilon}< {\varepsilon}_0$ in the complement of this set for the rest of the proof, so that $$\label{eq: rigging2}
\widetilde M(\widetilde F_e = {\varepsilon}\text{ for some } e) = 0\ .$$ Let $E_2=E_1\cap\{\exists e^* \in D_0(\omega) \text{ such that } \widetilde F_e>{\varepsilon}\}$.
If $\omega\in E_2$, we may find a rung $R(\omega)$ touching $D_0(\omega)$ such that $$\label{eq: Rdef}
E(R(\omega))<I(\omega)+{\varepsilon}/2\ .$$ This is by the definition of $I(\omega)$. Let $f^*(\omega)$ be the dual edge in $R(\omega)$ that touches $D_0(\omega)$. There are countably many choices for $f^*(\omega)$, so we may find a deterministic $\widetilde f^*$ such that $$\widetilde M(E_2,~f^*(\omega)=\widetilde f^*)>0\ .$$ In fact, by rotation and translation invariance we can take $\widetilde f^*$ to be the fixed dual edge $f^*$: $$\widetilde M(E_2,~f^*(\omega)=f^*)>0\ .$$
By an argument identical to that given in Lemma \[lem: infinitelymanyedges\], for $\widetilde M$-almost all $\omega \in E_2$, there are infinitely many dual edges $e^* \in D_0(\omega)$ (in both directions along $D_0(\omega)$) for which $\widetilde F_e>{\varepsilon}$. (See Remark \[rem: rem1\].) Therefore, for $\widetilde M$-almost every $\omega\in E_2$, we may find dual edges $e_1^*(\omega)$ and $e_2^*(\omega)$ on $D_0(\omega)$ such that the piece of $D_0(\omega)$ from $e_1^*(\omega)$ to $e_2^*(\omega)$ contains $g^*$ and such that $\widetilde F_{e_1}$ and $\widetilde F_{e_2}$ are bigger than ${\varepsilon}$. For any $N$, let $B(0;N)$ be the box of side length $N$ centered at the origin and for a spin configuration $\sigma$, let $\sigma_N$ be the restriction to $B(0;N)$. There are only countably many choices, so we may find deterministic values of $e_1,e_2,N,\sigma_N,\sigma_N'$ and $R$ (whose first dual edge is $f^*$) such that with positive $\widetilde M$-probability on $E_2$:
1. $ B(0;N/2)$ contains $R$, $e_1^*$, $e_2^*$ and the piece of $D_0(\omega)$ between $e_1^*$ and $e_2^*$;
2. $\sigma(\omega)\Big|_{B(0;N)}=\sigma_N$, $\sigma'(\omega)\Big|_{B(0;N)}=\sigma'_N$, $\widetilde F_{e_1} > {\varepsilon}, \widetilde F_{e_2}>{\varepsilon}$ ;
3. $R$ is a rung with $E(R) < I(\omega) + {\varepsilon}/2$.
Call $E_3$ the set of configurations satisfying the three above conditions. By construction, $\widetilde M(E_3\cap \{I>0\})>0$. Note that by the choice of $\sigma_N$ and $\sigma_N'$, their interface contains $g^*$, $e_1^*$ and $e_2^*$ (and they are all connected through a single domain wall in $B(0;N)$), but the interface does not contain $f^*$. In addition, if $E_3$ occurs then $R$ is a rung, and $\sigma \Delta \sigma'$ must be disconnected. Therefore $E_2$ contains $E_3 \cap \{I>0\}$. The same arguments also show that $$\label{eq: E3eq}
X_\varepsilon \supseteq E_3 \cap \{I>0\}\ .$$
Now, write $D$ for the (deterministic) set of edges in $B(0;N)$ that are in $\sigma_N \Delta \sigma'_N$ and can be connected to $g^*$ by a path of dual edges in $\sigma_N \Delta \sigma'_N$ that stay in $B(0;N)$. This is just the connected “piece” of $D_0(\omega)$ in $B(0;N)$ for configurations $\omega \in E_3$. Let $f_1^*, \ldots, f_n^*$ be the dual edges with both endpoints in $B(0;N)$ that are (a) incident to $D$, (b) not in $\sigma_N \Delta \sigma_N'$ and (c) not equal to $f$. A depiction of these definitions is given in Figure \[fig: magic\_rung\].
![Depiction of definitions on the event $E_3$. The two domain walls are the dual dotted lines. The rung $R$ is the thick path between the two domain walls. The edges $f_1, f_2,\dots,f_n$ along the domain wall are the grey edges.[]{data-label="fig: magic_rung"}](magic_rung2.eps){height="8cm"}
We claim that we can order the $f_i^*$’s so that for each $i=1, \ldots, n-1$, $f_i$ has an endpoint $x_i$ that does not touch any edge from the set $\{f_{i+1}, \ldots, f_n, f\}$ (note here we are considering edges, not dual edges). To explain why this is true, we consider the graph whose edge set is equal to the union of the $f_i$’s (in the original lattice). Note that if $C_1, \ldots , C_p$ are the components of this graph then it suffices to give an ordering of each component and then concatenate these orderings together. So we may consider just one component, say, $C_1$. We will choose the edges $g_1, \ldots, g_k$ of $C_1$ in reverse order, so that our final ordering of $C_1$ will be $g_k, \ldots, g_1$. The desired condition on the $f_i$’s becomes the following for the $g_i$’s: for each $i=1, \ldots, k$, $g_i$ has an endpoint that does not touch any edge from the set $\{f, g_1, \ldots, g_{i-1}\}$.
We now note that the graph whose edges are $f,f_1, \ldots, f_n$ does not contain any cycles. If there were a cycle then it would force the interface $\sigma \Delta \sigma'$ in the dual graph to have one too, which is impossible. Therefore the component $C_1$ above can have at most one edge that touches $f$. If there is such an edge, we let $g_1$ be it; otherwise, we choose $g_1$ arbitrarily in $C_1$. We now add edges in steps: at each step $j \geq 2$ we let $G_j$ be the current connected subgraph of $C_1$ (that is, the graph whose edges are $\{g_1, \ldots, g_{j-1}\}$) and add $g_j$ to our collection of edges so that it connects $G_j$ to its complement. This is always possible because $C_1$ does not contain a cycle. We finish at step $k$ with the desired ordering of the $g_j$’s, which, when reversed, gives the desired ordering of $C_1$.
We claim $$\label{eq: SSNS}
\widetilde M(E_3,~\cap_{i=1}^n\{|J_{f_i}| > \mathcal{S}_{f_i}^{x_i}\})>0\ ,$$ where $\mathcal{S}_{f_i}^{x_i}$ is the super-satisfied value of the edge $f_i$ defined in . Essentially, the claim means that the event $E_3$ is somewhat stable under modifications of couplings. Equation will be proved in the lemma below. We first show how this implies the claim of the proposition using Lemma \[lem: cylinder\]. Let $U$ be the set of all $f_i$’s. Note that by construction, for any finite set $A$ such that $f\in\partial A$ and $\partial A\cap U=\varnothing$, we must have $e_1$ or $e_2$ in $\partial A$. Let $\widetilde G$ be the event that $F_f \geq \min\{F_{e_1},F_{e_2}\}$ and $|J_{f_i}|\geq\mathcal{S}_{f_i}^{x_i}$ for all $f_i \in U$. The probability of $\widetilde G$ under any translates of $M$ is equal to that of $\widetilde G$, which is $1$ by Lemma \[lem: cylinder\]. On the other hand, $\widetilde G$ is a closed event so $\widetilde M(\widetilde G)$ is no smaller than $\limsup_k M_k^*(\widetilde G) = 1$. This implies from $$\widetilde M(E_3,~ \widetilde F_{f}>{\varepsilon}) \geq \widetilde M(E_3, \widetilde F_f >{\varepsilon}, ~\cap_{i=1}^n\{|J_{f_i}| \geq \mathcal{S}_{f_i}^{x_i}\} ) = \widetilde M(E_3,~\cap_{i=1}^n\{|J_{f_i}| \geq \mathcal{S}_{f_i}^{x_i}\})>0\ .$$ Since $X_{\varepsilon}\supseteq E_3 \cap \{I>0\}$ (and $\widetilde M(I=0)=0$ by ), this concludes the proof of Proposition \[prop: xe\].
\[lem: almostSS\] Let $E_3$ be the event defined above . Define $f_i$ and $x_i$, $i=1,...,n$ as above . If $\widetilde M(E_3)>0$, then $\widetilde M(E_3,~\cap_{i=1}^n\{|J_{f_i}| > \mathcal{S}_{f_i}^{x_i}\})>0$.
Write $S$ for the set of dual edges in $B(0;N)$ that are not equal to any of the $f_i^*$’s or to $f^*$. Since $\widetilde M(E_3)>0$, we can choose $\lambda>0$ such that $$\widetilde M(E_3,~|J_e|\leq \lambda \text{ for all } e^* \in S)>0\ .$$ Write $E_4$ for this event. We will show that $\widetilde M(E_4,~\cap_{i=1}^n\{|J_{f_i}| > \mathcal{S}_{f_i}^{x_i}\})>0$. This will follow if we find positive numbers $a_1, \ldots, a_n,$ $b_1, \ldots, b_n$ such that the following hold:
1. $a_i<b_i$ for all $i$;
2. $a_{i+1}> 4b_i$ for $i=0,1, \ldots, n-1$ and $b_0:= \lambda$.
3. $\widetilde M(E_4,~|J_{f_i}|\in[a_i,b_i] \text{ for all $i$}\})>0$;
These conditions imply that $|J_{f_i}|>\mathcal{S}_{f_i}^{x_i}$ for all $i$, as $|J_{f_i}|\geq a_i> 4 b_{i-1}$ and $4b_{i-1}>\mathcal{S}_{f_i}^{x_i}$. Here we are using the fact that $x_i$ does not touch the set $\{f,f_{i+1}, \ldots, f_n\}$. For $q>1$, define $$E_4^q:= E_4 \cap \{|J_i| \in [a_i,b_i] \text{ for all }i=1, \ldots, q-1\}$$ and for $q=1$ define $E_4^q := E_4$. We will proceed by induction to show that if $\widetilde M(E_4^{q})>0$ then $\widetilde M(E_4^{q+1})>0$ with appropriately chosen $a_q,b_q$, for $q=1,...,n-1$. Note that $\widetilde M(E_4)>0$. The case $q=n-1$ gives the desired conclusion. For the rest of the proof, we assume that the spins at the endpoints of $f_q$ are the same. The subsequent argument is similar in the other case. The idea is to use Lemma \[lem: SStypemod\], which shows that the probability mass is somewhat conserved when one value of the coupling is increased for events satisfying . Two obstacles have to be overcome. First, the properties of $M$ (in particular, the monotonicity property) needed in Lemma \[lem: SStypemod\] do not directly carry over under weak limits to $\widetilde M$. Therefore, we need to go back to $M$ to apply the lemma. Second, weak convergence of the measures applies to cylinder events. Note that, from its definition, $E_4^q$ is an intersection of a finite number of cylinder events except for $\{R \text{ is a rung with } E(R) < I(\omega) + {\varepsilon}/2\}$. To apply Lemma \[lem: SStypemod\], we thus need to find a cylinder approximation for this condition.
Let $\widetilde B^R \subseteq \widetilde \Omega$ be the event $\{R \text{ is a rung with } E(R) < I(\omega) + {\varepsilon}/2\}$ intersected with the event $\{g^* \in \sigma \Delta \sigma'\}$. We will first define a double sequence of cylinder events $(\widetilde B^R_{j,l})$ in $\widetilde \Omega$ with $$\label{eq: symmetricdiff}
\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M(\widetilde B^R \Delta \widetilde B^R_{j,l}) = 0\ ,$$ where $\Delta$ represents the symmetric difference of events.
Let $B(0;j)$ be the box of side-length $j$ centered at $0$ and let $l \geq j$. For arbitrary spin configurations $\sigma$ and $\sigma'$, the interface $\sigma \Delta \sigma'$ splits into different connected components in the following way. Two dual edges in $B(0;j) \cap \sigma \Delta \sigma'$ (that is, they have both endpoints in $B(0;j)$) are said to be [*$l$-connected*]{} if they are connected by a path of dual edges in $\sigma \Delta \sigma'$, all of which remain in $B(0;l)$. Let $D_0(j,l), D_1(j,l), \ldots, D_t(j,l)$ be the $l$-connected components of such edges in $B(0;j)$, where $D_0(j,l)$ is the connected component containing $g^*$ (if one exists). Call these the [*$(j,l)$-domain walls*]{} (see Figure \[fig: (j,l)\]). We define a [*$(j,l)$-rung*]{} as a finite path of dual edges in $B(0;j)$ which starts in a $(j,l)$-domain wall and ends in a different one, and no dual vertices on the path except for the starting and ending points are on a $(j,l)$-domain wall.
![In this figure, when we restrict the interface to $B(0;j)$, there are three components (connected inside this box). However, two of them are $l$-connected. Therefore, there are two $(j,l)$-domain walls in $B(0;j)$. []{data-label="fig: (j,l)"}](jlrung.eps){height="7cm"}
On the event $\widetilde B^R$, $R$ is a $(j,l)$-rung for all $l \geq j\geq N$. Let $\widetilde B^R_{j,l}\subseteq \widetilde \Omega$ be the event that
1. $g^* \in \sigma \Delta \sigma'$;
2. $R$ is a $(j,l)$-rung;
3. no other $(j,l)$-rung between $D_0(j,l)$ and another $(j,l)$-domain wall has energy less than the energy of $R$ minus ${\varepsilon}/2$.
We start by showing that $$\label{eq: stapler1}
\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M(\widetilde B^R \setminus \widetilde B^R_{j,l}) = 0\ .$$ Consider $\omega \in \widetilde B^R$. It suffices to prove that there exists $J(\omega)$ and for each $j\geq J(\omega)$ there is a $L(j,\omega)$ such that $$\label{eq: decondition}
j\geq J(\omega) \text{ and } l \geq L(j,\omega) \text{ implies } \omega \in \widetilde B^R_{j,l}\ .$$ This implies because if $\widetilde B^R \setminus \widetilde B^R_{j,l}$ occurs then either $j \leq J(\omega)$ or both $j \geq J(\omega)$ and $l \leq L(j,\omega)$. Therefore the limit in is bounded above by $$\lim_{j \to \infty} \lim_{l \to \infty} \left[ \widetilde M(j \leq J(\omega)) + \widetilde M(l \leq L(j, \omega)) \right] = 0\ .$$ Take $j\geq N$. Note that there are at most $|B(0;j)|$ number of $(j,l)$-domain walls in $B(0;j)$. We claim that there exists $L(j)$ such that all $(j,l)$-rungs are rungs for $l\geq L(j)$. Indeed, if $S$ is a rung then it is plainly a $(j,l)$-rung. On the other hand, if $S$ is a $(j,l)$-rung, then either it connects distinct domain walls in $\sigma\Delta\sigma'$ or simply two pieces of the same domain wall of $\sigma\Delta\sigma'$ that are $l$-connected for $l$ large enough. Now, since $\omega \in \widetilde B^R$, we must have $g^* \in \sigma \Delta \sigma'$. Moreover, $R$ is a rung and so it is also a $(j,l)$-rung for any $l$. By definition of $\widetilde B^R$, no rung touching $D_0(\omega)$ can have energy less than the energy of $R$ minus ${\varepsilon}/2$. Therefore for $l \geq L(j)$, no $(j,l)$-rung can either, and we see that $\omega \in\widetilde B^R_{j,l}$ for $J(\omega) = N$ and $L(j,\omega) = L(j)$ in .
To show the other half of , it remains to prove that $$\label{eq: B^R_ij minus B^R}
\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M(\widetilde B^R_{j,l} \setminus \widetilde B^R) = 0\ .$$ We claim that if $\omega \notin \widetilde B^R$, there exists $J(\omega)$ such that for each $l \geq j \geq J(\omega)$, $\omega \notin \widetilde B^R_{j,l}$ as well. This implies by the same argument as before. Since $\omega \notin \widetilde B^R$, at least one of three defining conditions of $\widetilde B^R$ must fail. In each case, we will show that $\omega$ cannot be in $\widetilde B^R_{j,l}$ for all large $j$ and $l$. First if $g^* \notin \sigma \Delta \sigma'$ then we will never have $\omega \in B^R_{j,l}$, so we may assume the contrary. If $R$ is a $(j,l)$-rung for some $l \geq j \geq N$ then it connects two $(j,l)$-domain walls. As in the previous paragraph, either these $(j,l)$-domain walls are in fact distinct domain walls or they are part of the same domain wall for $j$ and $l$ large enough. This argument shows that if $R$ is not a rung, there exists $J(\omega)$ such that it will also not be a $(j,l)$-rung for $l \geq j \geq J(\omega)$. Finally, if $g^* \in \sigma \Delta \sigma'$ and $R$ is a rung, suppose that there is another rung $S$ touching $D_0(\omega)$ with energy less than the energy of $R$ minus ${\varepsilon}/2$. Then the same argument as above shows there exists $J'(\omega)$ such that for $l \geq j \geq J'(\omega)$, $S$ will be a $(j,l)$-rung with energy less than $E(R)-{\varepsilon}/2$ and therefore $\omega \notin \widetilde B^R_{j,l}$. This proves and thus .
Recall that the event $E_4^q$ is the intersection of the following:
1. the three events that comprise $E_3$ defined above (the last one of which we can replace by $\widetilde B^R$);
2. $|J_e| \leq \lambda$ for all $e^* \in S$;
3. $|J_{f_i}| \in [a_i,b_i]$ for all $i=1, \ldots, q-1$.
Let $E_{j,l}^q$ be the cylinder approximation of $E_4^q$ that is, the event $E_4^q$ where $\widetilde B^R$ is replaced by the cylinder event $\widetilde B_{j,l}^R$. Note that $E_{j,l}^q$ can be seen as an event in the translated space $T^k\Omega$ for $k$ large enough such that the box $B(0;l)$ is contained in $T^k V_H$. Recall that in $\Omega$ as well as in $T^k\Omega$, the flexibilities $F_e$ and $F_e'$ are functions of $J$ and $\sigma,\sigma'$ given by the formula . Note also by directly applying , we find $$\label{eq: E_4^q minus E_jl^q}
\lim_{l \to \infty} \limsup_{j \to \infty} \widetilde M(E_4^q \Delta E_{j,l}^q) = 0\ .$$
We claim that $E^q_{j,l}$ (and $T^k E^q_{j,l}$) has the property : $$\label{eq: Ckproperty}
\text{If }(J,\sigma,\sigma') \in E^q_{j,l} \text{ then }(J(f_q,s),\sigma,\sigma') \in E^q_{j,l} \text{ whenever }s \geq J_{f_q}\ .$$ To check this, we first remark that if $|J_{f_i}| \in [a_i,b_i]$ for all $i=1, \ldots, q-1$ and $|J_e|\leq \lambda$ for all $e \in S$ for $(J,\sigma,\sigma')$ then this is plainly true for $(J(f_q,s),\sigma,\sigma')$ for any $s$. This handles conditions (b) and (c) of $E_{j,l}^q$. To address condition (a), we first note that the event that $g^* \in \sigma \Delta \sigma'$ (part of $\widetilde B_{j,l}^R$ in the third part of (a)) is unaffected by $J_{f_q}$, so it will continue to hold. In the other two parts of (a), no conditions involve the couplings except for $F_e>\varepsilon$, $F_e'>\varepsilon$. But since the spins at the endpoint of $f_q$ are the same, increasing $J_{f_q}$ can only possibly increase $F_e$ and $F_e'$ as seen from . (Note here that $F_e$ and $F_e'$ are simply images under $\Phi$ of $F_e(J,\sigma)$ and $F_e'(J,\sigma)$ on $\Omega$ or $T^k \Omega$, so since this argument is valid on these spaces, it holds as stated on $\Omega^*$ or $T^k \Omega^*$.) Finally, to establish , it remains to show that if $(J,\sigma,\sigma')\in \widetilde B^R_{j,l}$, then $(J(f_q,s),\sigma,\sigma')\in \widetilde B^R_{j,l}$ for $s \geq J_{f_q}$. Note that because $l \geq j \geq N,$ the set $D$ (defined before the statement of the present proposition) is contained in the $(j,l)$-domain wall of $g^*$ and since $f_q^*$ is adjacent to $D$, no $(j,l)$-rung containing $f^*$ can contain $f_q^*$. So increasing the value of $J_{f_q}$ to $s$ can only increase the energies of $(j,l)$-rungs that do not contain $f^*$. This means that if no $(j,l)$-rungs have energy less than the energy of $R$ minus ${\varepsilon}/2$ in $(J,\sigma,\sigma')$ then the same will be true in $(J(f_q,s),\sigma,\sigma')$ for $s \geq J_{f_q}$. We have thus proved .
We are now in a position to use Lemma \[lem: SStypemod\]. Since $T^kM$ is just a translate of $M$, the lemma holds for the measure $T^k M$ as well, so we conclude that for all $a\in {\mathbb{R}}$ and $k \geq l \geq j \geq N$, $$\label{eq: floss}
T^k M(E_{j,l}^q,~ J_{f_q} \geq a) \geq (1/2)\nu([a,\infty)) ~T^kM(E_{j,l}^q)\ .$$ This holds trivially for $M$ replaced by $M^*$, on the space $\Omega^*$ in , where the flexibilities are added to the coordinates. We would like to take limits in this inequality. For this purpose, the reader may trace through the definition of $E_{j,l}^q$ and see that this event is an intersection of a cylinder event $Y$ involving only spins and couplings and another event $Z$ equal to $\{\widetilde F_{e_1} >{\varepsilon}\text{ and } \widetilde F_{e_2} > {\varepsilon}\}$. The boundary $\partial Y$ is included in the union of $\partial\{|J_e| \leq \lambda ~ \forall e^* \in S\}$, $\partial\{|J_{f_i}| \in [a_i,b_i]: \forall i=1, \ldots, q-1\}$, $\partial\{g^* \in \sigma \Delta \sigma'\}$, $\partial\{\text{$R$ is a $(j,l)$-rung}\}$, and the boundary of the event [*$\{$no other $(j,l)$-rung between $D_0(j,l)$ and another $(j,l)$-domain wall has energy less than the energy of $R$ minus ${\varepsilon}/2\}$*]{}. It is straightforward to see that the first four have $\widetilde M$-probability zero. As for the fifth one, notice that the energy of a $(j,l)$-rung is a linear function of the couplings in the box $B(0;j)$ with coefficients $+1$ or $-1$. There are only a finite number of such linear combinations. Therefore, the probability that the difference of energy between any two rungs is exactly ${\varepsilon}/2$ is $0$. By condition , we also have $\partial Z$ of $\widetilde M$-probability zero. Therefore by the discussion preceding Remark \[rem: rabbit\], we have $$\lim_{k \to \infty} M_k^*(E_{j,l}^q) = \widetilde M(E_{j,l}^q)\ .$$ A similar argument holds for the left side of . Averaging over $k$ and taking limits in this inequality, we find $$\widetilde M(E_{j,l}^q,~ J_{f_q} \geq a) \geq (1/2)\nu([a,\infty)) ~\widetilde M(E_{j,l}^q)\ ,$$ Now we take $l \to \infty$ and $j \to \infty$, using to obtain $$\widetilde M(E_4^q,~ J_{f_q} \geq a) \geq (1/2)\nu([a,\infty)) ~\widetilde M(E_4^q)\ .$$ By the induction hypothesis, $\widetilde M(E_4^q) >0$. To finish the proof of the lemma, it thus suffices to take $a=a_q = 4b_{q-1} +1$ and choose any $b_q>a_q$.
### Finishing the proof
In this subsection we use Proposition \[prop: xe\] to prove a final proposition about rung energies. This will allow us to reach a contradiction and establish .
Recall that $f$ refers to the fixed edge connecting $(0,1)$ to $(1,1)$ and $g$ is the edge connecting the origin to $(1,0)$, see Figure \[fig: magic\_rung\]. Our goal in this section is to show that $J_f$ can be modified so that the energy of some rung that contains $f^*$ decreases below the energies of all rungs that do not contain $f^*$. To do this, we introduce two variants of $I(\omega)$, dealing with rungs that contain $f^*$ and rungs that do not.
On the event $X_{\varepsilon}$, we define the variable $I'(\omega)$ to be the infimum of energies of all rungs that touch $D_0(\omega)$ (the domain wall that contains $h^*=g^*$) and that do not contain $f^*$. Also we define $\widetilde I(\omega)$ to be the infimum of energies of all rungs that contain $f^*$. Later in the proof we will use a small technical fact: the distribution of $\widetilde I(\omega) - I'(\omega)$ (under $\widetilde M$) can have only countably many point masses. Therefore we may choose ${\varepsilon}$ small enough so that the conclusion of Proposition \[prop: xe\] holds and so that $$\label{eq: technical}
\widetilde M(\omega: \widetilde I(\omega) - I'(\omega) = {\varepsilon}/2 \text{ or } \widetilde I(\omega) - I'(\omega) = -{\varepsilon}/4) = 0\ .$$ This ${\varepsilon}$ will be fixed for the rest of the paper.
Let $Y_{\varepsilon}$ be the event that:
1. $\sigma\Delta\sigma'$ is disconnected and $I>0$;
2. $g^* \in \sigma \Delta \sigma'$;
3. $f^*$ is in a rung $R$ that satisfies $E(R) <I'(\omega)-{\varepsilon}/4$.
The next two propositions establish the desired contradiction. The idea is that, on the one hand (cf. Proposition \[prop: ye=0\]), $Y_{\varepsilon}$ must have zero probability since by Lemma \[lem: infinitelymanyedges\] and Remark \[rem: rem1\] an event along the domain wall occurs infinitely often, whereas $f^*$ must be unique along the domain wall by the definition of $I'$. On the other hand, we will use Proposition \[prop: xe\] in Proposition \[prop: ye\] to show that the event $Y_\varepsilon$ must have positive probability.
\[prop: ye=0\] The following statement holds. $$\widetilde M(Y_{\varepsilon})=0\ .$$
\[prop: ye\] If $\widetilde M(X_{\varepsilon}\cap \{\widetilde F_f>{\varepsilon}\})>0$, then $$\widetilde M(Y_{\varepsilon})>0\ .$$
For a dual vertex $b^*$, let $Y_{\varepsilon}(b) \subseteq \widetilde \Omega$ be the event that
1. $\sigma\Delta\sigma'$ is disconnected and $I>0$;
2. $b^* \in \sigma \Delta \sigma'$;
3. there is a dual edge $e^*$, sharing a dual endpoint with $b^*$, that is the first edge of a rung $R$ with $E(R) <I'_{b,e}(\omega)-{\varepsilon}/4$. Here $I'_{b,e}$ is the infimum of energies of the rungs not containing $e^*$ and touching the domain wall of $b^*$.
In this notation, the $Y_{\varepsilon}$ corresponds to the case $b=g$ and $e=f$. By definition of $I'_{b,e}$, for each domain wall $D$, there are at most two dual edges $b^*$ such that $Y_{\varepsilon}(b)$ occurs (one for each side of $D$). By the same argument as in Lemma \[lem: infinitelymanyedges\] (with $B_e({\varepsilon},K)$ replaced by $Y_{\varepsilon}(b)$), it follows that $\widetilde M(Y_{\varepsilon}(b))=0$ for all dual edges $b$ (see Remark \[rem: rem1\]), so $\widetilde M(Y_{\varepsilon})=0$.
On the event $X_{\varepsilon}\cap \{\widetilde F_f>{\varepsilon}\}$, either the spins at the endpoints of $f$ are the same or they are different (in both $\sigma$ and $\sigma'$). Let us suppose that: $$\widetilde M(X_{\varepsilon},~\widetilde F_f>{\varepsilon},~\sigma_f =\sigma_f' =+1)>0\ .$$ The subsequent argument can easily be modified in the case $\sigma_f=\sigma'_f=-1$ (using an obvious analogue of Lemma \[lem: backmodify\].) Define $\widetilde C_f = \max\{C_f, C_f'\}$. We may choose $a \in \mathbb{R}$ such that $$\label{eq: toaster1}
\widetilde M(X_{\varepsilon},~\widetilde F_f>{\varepsilon},~\sigma_f=\sigma'_f=+1,~\widetilde C_f \in (a,a+{\varepsilon}/8))>0$$ and because the distribution of $\widetilde C_f$ can have countably many point masses, we may further restrict our choice of $a$ so that $$\label{eq: rigging}
\widetilde M(\widetilde C_f = a \text{ or } a+{\varepsilon}/8) = 0\ .$$ By property , for each $k$, $$T^k M((J,\sigma,\sigma')~:~\sigma_f= \sigma'_f =+1,~J_f < \max\{C_f(J,\sigma),C_f(J,\sigma')\}) = 0\ .$$ This is an open cylinder event in $\Omega^*$, thus after averaging and taking liminf, $$\label{eq: toaster2}
\widetilde M(\sigma_f=\sigma'_f=+1,~J_f < \widetilde C_f) \leq \liminf_{k \to \infty} M_k^*(\sigma_f=\sigma'_f=+1,~J_f < \widetilde C_f) = 0\ .$$ If $J_f\geq\widetilde C_f$ and $\widetilde F_f=\max\{|J_f-C_f|,|J_f-C_f'|\}>{\varepsilon}$, then $J_f>C_f+{\varepsilon}$ and $J_f'>C_f'+{\varepsilon}$. By combining and , we thus find $$\widetilde M(X_{\varepsilon},~\widetilde C_f \in (a,a+{\varepsilon}/8), J_f \geq a+{\varepsilon})>0\ .$$
Recall that $\widetilde I(\omega)$ is the infimum of energies of all rungs that contain $f^*$. On the event $X_{\varepsilon}$, we have $\widetilde I(\omega)< I'(\omega)+{\varepsilon}/2$. Therefore if $\widetilde B$ is the event that
1. $g^* \in \sigma \Delta \sigma'$ but $f^* \notin \sigma \Delta \sigma'$;
2. $\widetilde I(\omega)<I'(\omega)+{\varepsilon}/2$,
then $$\label{eq: tildeB4}
\widetilde M(\widetilde B,~ \widetilde C_f \in (a,a+{\varepsilon}/8),~J_f \geq a+{\varepsilon})>0\ .$$ Note that condition 2 of $\widetilde B$ only makes sense if $f^*$ is actually in a rung; however, in the support of $\widetilde M$, $\sigma$ and $\sigma'$ are ground states, so their interface does not contain loops. Thus when condition 1 of $\widetilde B$ holds and $\omega$ is in the support of $\widetilde M$, $f^*$ is in a rung.
From this point on, the strategy is similar to the proof of Lemma \[lem: almostSS\]. The idea is to use Lemma \[lem: backmodify\] to lower $\widetilde I(\omega)$ below $I'(\omega)-{\varepsilon}/4$. Let $\widetilde P$ be the event $\widetilde B$ with the condition $\widetilde I(\omega)<I'(\omega)+{\varepsilon}/2$ replaced by $\widetilde I(\omega)<I'(\omega)-{\varepsilon}/4$. We will show that $$\widetilde M(\widetilde P)>0\ .$$ A quick look at can convince us that this is possible since $J_f$ could be lowered by $3{\varepsilon}/4$ and still not reach the critical value. Since $\widetilde I(\omega)$ depends linearly on $J_f$ by definition, it will be itself lowered by $3{\varepsilon}/4$ and become lower than $I'(\omega)$ by ${\varepsilon}/4$. To make this reasoning rigorous, as in the proof of Lemma \[lem: almostSS\], we must bring the problem back to the half-plane measure $M$ and find a cylinder approximation for both $\widetilde B$ and $\widetilde P$.
Let $B(0;j)$ be the box of side-length $j$ centered at $0$ and let $l \geq j$. Recall the definitions of $(j,l)$-domain walls and $(j,l)$-rungs below . Let $D_0(j,l), D_1(j,l), \ldots, D_t(j,l)$ be the $(j,l)$-domain walls in $B(0;j)$ and $D_0(j,l)$ be the one containing $g^*$ (if it exists). For $l \geq j$ and $\omega \in \widetilde \Omega$ such that $g^* \in \sigma \Delta \sigma'$, write $I_{j,l}'(\omega)$ (the cylinder approximation of $I'(\omega)$) as the infimum of all energies of $(j,l)$-rungs which touch $D_0(j,l)$ but do not contain the dual edge $f^*$. Write $\widetilde I_{j,l}(\omega)$ (the cylinder approximation of $\widetilde I(\omega)$) for the infimum of all energies of $(j,l)$-rungs which contain $f^*$. Let $\widetilde B_{j,l}\subseteq \widetilde \Omega$ be the cylinder approximation of $\widetilde B$:
1. $g^* \in \sigma \Delta \sigma'$ but $f^* \notin \sigma \Delta \sigma'$.
2. $\widetilde I_{j,l}(\omega)<I'_{j,l}(\omega)+{\varepsilon}/2$.
We define the cylinder approximation $\widetilde P_{j,l}$ of $\widetilde P$ similarly with ${\varepsilon}/2$ replaced by $-{\varepsilon}/4$. There may be no $(j,l)$ rungs, but their existence is implicit in condition 2 (in other words, it is implied in condition 2 that the variables $\widetilde I_{j,l}(\omega)$ and $I'_{j,l}(\omega)$ are defined). We claim that $$\label{eq: tildeB3}
\begin{aligned}
&\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M(\widetilde B_{j,l} \Delta \widetilde B) = 0\\
&\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M( P_{j,l} \Delta P_{\varepsilon}) = 0\ .
\end{aligned}$$ We give the proof for $\widetilde{B}$. The proof for $\widetilde P$ is identical with ${\varepsilon}/2$ replaced by $-{\varepsilon}/4$.
To begin with, let $\omega$ be a configuration such that $g^* \in \sigma \Delta \sigma'$ and $f \notin \sigma \Delta \sigma'$ (this is true for all configurations in $\widetilde B$ or in $\widetilde B_{j,l}$). Note that for fixed $j$, $$\widetilde I_j(\omega):= \lim_{l \to \infty} \widetilde I_{j,l}(\omega) \text{ exists }$$ and equals the infimum of energies of all rungs that stay in $B(0;j)$ and contain $f^*$. Clearly, $$\lim_{j \to \infty} \widetilde I_j(\omega) = \widetilde I(\omega)\ .$$ The analogous statements are true for $I'(\omega)$ (defining $I'_j(\omega)$ similarly). Therefore given $\delta>0$ we may choose $J(\omega)$ such that $j \geq J(\omega)$ implies that $$|\widetilde I_j(\omega) - \widetilde I(\omega)|<\delta/2 \text{ and } |I'_j(\omega) - I'(\omega)|<\delta/2\ .$$ For any such $j$ we can find $L(j,\omega)$ such that for $l \geq L(j,\omega)$, $$|\widetilde I_{j,l}(\omega) - \widetilde I_j(\omega)| < \delta/2 \text{ and } |I'_{j,l}(\omega)-I'_j(\omega)|<\delta/2\ .$$ Therefore for $j \geq J(\omega)$ and $l \geq L(j,\omega)$, $$\label{eq: hamster}
|\widetilde I_{j,l}(\omega) - \widetilde I(\omega)|<\delta \text{ and } |I'_{j,l}(\omega)-I'(\omega)|<\delta\ .$$ We first show that $$\label{eqn: B minus B_ij}
\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M(\widetilde B \setminus \widetilde B_{j,l}) = 0\ .$$ Suppose that $\omega \in \widetilde B$. Then $\widetilde I(\omega)< I'(\omega)+{\varepsilon}/2$ and, combining this with , we may choose $\delta = \delta(\omega)$ so small that for $j \geq J(\omega)$ and $l \geq L(j,\omega)$, $$\widetilde I_{j,l}(\omega)<I'_{j,l}(\omega) + {\varepsilon}/2\ .$$ Because $\omega \in \widetilde B$, the first condition of $\widetilde B_{j,l}$ holds directly. Equation follows from this using the same reasoning as for .
We now prove that $$\label{eq: hamster2}
\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M(\widetilde B_{j,l} \setminus \widetilde B) = 0\ .$$ As before, we need to show that if $\widetilde I(\omega)\geq I'(\omega)+{\varepsilon}/2$ then there is $J(\omega)$ such that for each $j \geq J(\omega)$, there is an $L(j,\omega)$ such that if $l \geq L(j,\omega)$ then $\widetilde I_{j,l}(\omega) \geq I_{j,l}'(\omega)$. If $\widetilde I(\omega)>I'(\omega)+{\varepsilon}/2$ then the arguments leading up to prove this immediately. In the other case, let $\widetilde U$ be the event that $\widetilde I(\omega) = I'(\omega)+{\varepsilon}/2$. This event has $\widetilde M$-probability zero by (for the approximation of $\widetilde P$, one has $\widetilde I(\omega) = I'(\omega)-{\varepsilon}/4$ instead). So $$\lim_{j \to \infty} \limsup_{l \to \infty} \widetilde M( (\widetilde B_{j,l} \cap \widetilde U^c) \setminus (\widetilde B \cap \widetilde U^c)) = 0\ .$$ However, $\widetilde U$ has $\widetilde M$-probability zero, so this proves .
Notice that $$\widetilde B_{j,l} \cap \{\widetilde C_f \in (a,a+{\varepsilon}/8) \} \cap \{J_f \geq a+{\varepsilon}\}$$ is a cylinder event in $\widetilde \Omega$. This event also makes sense under the measure $T^kM$ on the half-plane for $\widetilde C_f= \max \{C_f(J,\sigma),C_f(J,\sigma')\}$ (where the critical values are functions as defined in ) and for $k\geq l\geq j$ so that the boxes are contained in $T^k V_H$. We now analyze the probability $T^kM(\widetilde B_{j,l},~C_f\in (a,a+{\varepsilon}/8),~J_f \geq a+{\varepsilon})$. Let $\widetilde K_{j,l}(\omega):= \widetilde I_{j,l}(\omega)-J_f$ be the infimum of the energies of $(j,l)$-rungs where the contribution from the edge $f$ is removed. If $\widetilde I_{j,l}(\omega)<I'_{j,l}(\omega)+{\varepsilon}/2$ and $J_f\geq a+{\varepsilon}$, then $$\label{eqn: K}
\widetilde K_{j,l}(\omega)= \widetilde I_{j,l}(\omega)-J_f < I'_{j,l}(\omega)-a - {\varepsilon}/2\ .$$ Define $A_{j,l} \subseteq T^k\Omega$ as the intersection of the following events.
1. $f^* \notin \sigma \Delta \sigma'$ and $\sigma_f=\sigma'_f=+1$.
2. $g^* \in \sigma \Delta \sigma'$ and $\widetilde K_{j,l}(\omega) < I'_{j,l}(\omega)-a - {\varepsilon}/2$.
3. $\widetilde C_f \in (a,a+{\varepsilon}/8)$.
4. $\sigma, \sigma'$ are in ${\mathcal{G}}(J)$, the ground states in ${\mathbb{Z}}\times{\mathbb{N}}$.
Implicit in the second condition is that the variables $\widetilde K_{j,l}(\omega)$ and $I'_{j,l}(\omega)$ are actually defined; in particular, $f^*$ must be in some $(j,l)$-rung. Although the last condition does not give a cylinder event, it will be used to apply Lemma \[lem: backmodify\]. $A_{j,l}$ is an intermediary event between $\widetilde B_{j,l}$ and $P_{j,l}$. On the set $\{\sigma,\sigma' \in {\mathcal{G}}(J)\}$, $\widetilde B_{j,l} \cap \{\widetilde C_f \in (a,a+{\varepsilon}/8) \} \cap \{J_f \geq b\}$ implies $A_{j,l}$ by , so $$\label{eq: Btilde2}
T^kM(\widetilde B_{j,l},~\widetilde C_f \in (a,a+{\varepsilon}/8),~J_f \geq a+{\varepsilon}) \leq T^kM(A_{j,l})\ .$$
We claim that $A_{j,l}$ (and $T^kA_{j,l}$) has the property of Lemma \[lem: backmodify\]: $$\text{If $(J,\sigma,\sigma') \in A_{j,l}$ and $J_f \geq a+{\varepsilon}/8$, then $(J(f,s),\sigma,\sigma') \in A_{j,l}$ for all $s \geq a+{\varepsilon}/8$}\ .$$ To verify this, note that the defining condition 1 of $A_{j,l}$ does not depend on $J_f$, so if $(J,\sigma,\sigma')$ satisfies it, so will $(J(f,s),\sigma,\sigma')$ for all $s$. Next we argue that $\sigma,\sigma' \in {\mathcal{G}}(J(f,s))$ for all $s \geq a+{\varepsilon}/8$. This holds because $\sigma_f=\sigma'_f=+1$, $J_f \geq a+{\varepsilon}/8 >\widetilde C_f=\max\{C_f(J,\sigma),C_f(J,\sigma')\}$. Clearly condition 3 holds for $(J(f,s),\sigma,\sigma')$ as the critical values do not depend on the coupling at $f$. Last, because $\sigma_f=\sigma'_f=+1$, we see that $\widetilde K_{j,l}(J(f,s),\sigma,\sigma')$ does not depend on $s$ since the contribution of $J_f$ to $\widetilde I_{j,l}$ is removed. Also the variable $I_{j,l}'$ does not depend on $J_f$ by construction. Therefore condition 4 holds for $(J(f,s),\sigma,\sigma')$.
We are now in the position to apply Lemma \[lem: backmodify\]. Because $A_{j,l}$ satisfies the hypotheses of the lemma for $c=a+{\varepsilon}/8$, we select $d=a+{\varepsilon}/4$ and find $$T^kM(A_{j,l},~J_f \in [a+{\varepsilon}/8,a+{\varepsilon}/4]) \geq \nu([a+{\varepsilon}/8,a+{\varepsilon}/4]) ~T^kM(A_{j,l})\ .$$ When $A_{j,l}$ occurs and $J_f \leq a+{\varepsilon}/4$, $$\begin{aligned}
\widetilde I_{j,l}(\omega) = \widetilde K_{j,l}(\omega) + J_f &<& I'_{j,l}(\omega) -a -{\varepsilon}/2 + a +{\varepsilon}/4 \\
&=& I'_{j,l}(\omega)-{\varepsilon}/4\ .\end{aligned}$$ Therefore, writing $r = \nu([a+{\varepsilon}/8,a+{\varepsilon}/4])$, $$T^k M(A_{j,l},~\widetilde I_{j,l}(\omega) \leq I'_{j,l}(\omega)-{\varepsilon}/4) \geq r~T^kM(A_{j,l})\ ,$$ and by , $$\label{eq: Btilde3}
T^k M(A_{j,l},~\widetilde I_{j,l}(\omega) \leq I'_{j,l}(\omega)-{\varepsilon}/4) \geq r~T^kM(\widetilde B_{j,l}, \widetilde C_f \in (a,a+{\varepsilon}/8), J_f \geq a+{\varepsilon})\ .$$ Now $A_{j,l}\cap \{\widetilde I_{j,l}(\omega) \leq I'_{j,l}(\omega)-{\varepsilon}/4\}$ is contained in $\widetilde P_{j,l}$. By , $$\label{eq: lastreally1}
T^kM( \widetilde P_{j,l}) \geq r~T^kM(\widetilde B_{j,l}, \widetilde C_f \in (a,a+{\varepsilon}/8),J_f \geq a+{\varepsilon})\ .$$
We now want to average over $k$ and take the limit in . First note that $\widetilde P_{j,l}$ is an event that only involves spins and couplings. Furthermore, the only non-trivial contribution to the boundary $\partial \widetilde P_{j,l}$ is $\partial \{\widetilde I_{j,l}(\omega) \leq I'_{j,l}(\omega)-{\varepsilon}/4\}$. This event is contained in the event that there are two distinct $(j,l)$-rungs in the box $B(0;N)$ whose energies differ by exactly ${\varepsilon}/4$. Since the energy is a linear function of the couplings and of the spins, and since there are only a finite number of possible rungs in $B(0;N)$, this event has $\widetilde M$-probability zero by the continuity of $\nu$. Thus by the discussion preceding Remark \[rem: rabbit\] we may take the limit on the left to get $$\lim_{k \to \infty} M_k^*(P_{j,l}) = \widetilde M(P_{j,l})\ .$$ By , and reasoning similar to above, the boundary of the event on the right side of also has $\widetilde M$-probability zero. Therefore we can average over $k$ in and take the limit to finally get $$\label{eq: finaleq11}
\widetilde M( P_{j,l}) \geq r~\widetilde M(\widetilde B_{j,l},\widetilde C_f \in (a,a+{\varepsilon}/8),J_f \geq a+{\varepsilon})\ .$$ Finally, it suffices to take $l \to \infty$ and $j \to \infty$. By , the right side converges to $$r\widetilde M(\widetilde B, \widetilde C_f \in (a,a+{\varepsilon}/8),J_f \geq a+{\varepsilon}) >0\ .$$ The probability is positive by . The left side of converges to $\widetilde M (\widetilde P)$ again by . Thus $\widetilde M(P_{\varepsilon}) >0$. As $P_{\varepsilon}\cap \{I>0\} \subset Y_{\varepsilon}$ and $\widetilde M(I=0)=0$, this completes the proof.
[1]{}
<span style="font-variant:small-caps;">Arguin, L.-P.</span>, <span style="font-variant:small-caps;">Damron, M.</span>, <span style="font-variant:small-caps;">Newman, C. M.</span> and <span style="font-variant:small-caps;">Stein, D. L.</span> (2010) Uniqueness of ground states for short-range spin glasses in the half-plane. [*Commun. Math. Phys.*]{} [**300**]{}(3), 641–675.
<span style="font-variant:small-caps;">Bieche I.</span>, <span style="font-variant:small-caps;">Maynard R.</span>, <span style="font-variant:small-caps;">Rammal R.</span>, <span style="font-variant:small-caps;">Uhryt J.P.</span> (1980) On the ground states of the frustration model of a spinglass by a matching method of graph theory. [*J. Phys. A*]{} [**13**]{}, 2553–2576.
<span style="font-variant:small-caps;">K. Binder</span> and <span style="font-variant:small-caps;">A. P. Young</span> (1986) Spin glasses: experimental facts, theoretical concepts, and open questions. [*Rev. Mod. Phys.*]{} [**58**]{}, 801–976.
<span style="font-variant:small-caps;">R.M. Burton</span> and <span style="font-variant:small-caps;">M. Keane</span> (1989) Density and uniqueness in percolation. [*Comm. Math. Phys.*]{} [**121**]{}, 501–505.
<span style="font-variant:small-caps;">S. Edwards</span> and <span style="font-variant:small-caps;">P.W. Anderson</span> (1975) Theory of spin glasses. [*J. Phys. F*]{} [**5**]{}, 965–974.
<span style="font-variant:small-caps;">D.S. Fisher</span> and <span style="font-variant:small-caps;">D.A. Huse</span> (1986) Ordered Phase of Short-Range Ising Spin-Glasses. [*Phys. Rev. Lett.*]{} [**56**]{} No. 15, 1601–1604
<span style="font-variant:small-caps;">J. Fink</span> (2010) Towards a theory of ground state uniqueness. [*Excerpt from Ph.D. thesis.*]{}
<span style="font-variant:small-caps;">O. Kallenberg</span> (2002) [*Foundations of Modern Probability*]{} (Springer, Berlin) 2nd Edition
<span style="font-variant:small-caps;">M. Loebl</span> (2004) Ground State Incongruence in 2D Spin Glasses Revisited [*Electr. J. Comb*]{} [**11**]{} R40.
<span style="font-variant:small-caps;">M. Mézard</span>, <span style="font-variant:small-caps;">G. Parisi</span> and <span style="font-variant:small-caps;">M.A. Virasoro</span> (1987) [*Spin Glass Theory and Beyond*]{} (World Scientific, Singapore).
<span style="font-variant:small-caps;">A.A. Middleton</span> (1999) Numerical investigation of the thermodynamic limit for ground states in models with quenched disorder. [*Phys. Rev. Lett.*]{} [**83**]{}, 1672–1675.
<span style="font-variant:small-caps;">C. Newman</span> (1997) [*Topics in Disordered Systems*]{} (Birkhaüser, Basel).
<span style="font-variant:small-caps;">C.M. Newman</span> and <span style="font-variant:small-caps;">D.L. Stein</span> (2001) Are there incongruent ground states in 2D Edwards-Anderson spin glasses? [*Comm. Math. Phys.*]{} [**224**]{}(1), 205-218.
<span style="font-variant:small-caps;">C.M. Newman</span> and <span style="font-variant:small-caps;">D.L. Stein</span> (2003) Topical Review: Ordering and Broken Symmetry in Short-Ranged Spin Glasses. [*J. Phys.: Cond. Mat.*]{} [**15**]{}, R1319 – R1364.
<span style="font-variant:small-caps;">M. Palassini</span> and <span style="font-variant:small-caps;">A.P. Young</span> (1999) Evidence for a trivial ground-state structure in the two-dimensional Ising spin glass [*Phys. Rev. B*]{} [**60**]{}, R9919–R9922.
<span style="font-variant:small-caps;">D. Sherrington</span> and <span style="font-variant:small-caps;">S. Kirkpatrick</span> (1975) Solvable model of a spin glass. [*Phys. Rev. Lett.*]{} [**35**]{}, 1792-1796.
<span style="font-variant:small-caps;">J. Wehr</span> On the Number of Infinite Geodesics and Ground States in Disordered Systems. [*J. Stat. Phys.*]{} [**87**]{}, 439-447
<span style="font-variant:small-caps;">J. Wehr</span> and <span style="font-variant:small-caps;">J. Woo</span> (1998) Absence of Geodesics in First-Passage Percolation on a Half-Plane. [*Ann. Prob.*]{} [**26**]{}, 358-367
[^1]: L.-P. Arguin was supported by the NSF grant DMS-0604869 during part of this work.
[^2]: M. Damron is supported by an NSF postdoctoral fellowship.
|
---
abstract: 'We give a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, which is due to P. Deligne as a by-product of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the configuration spaces of points on curves are also studied.'
address: |
Institut de Mathématiques de Jussieu (IMJ)\
4 place Jussieu\
75005 Paris, France
author:
- Clément Dupont
bibliography:
- 'biblio.bib'
title: 'Hypersurface arrangements and a global Brieskorn-Orlik-Solomon theorem'
---
Introduction
============
Let $X$ be a complex manifold of dimension $n$. A hypersurface arrangement in $X$ is a union $$L=L_1\cup\cdots\cup L_l$$ of smooth hypersurfaces $L_i\subset X$, $i=1,\cdots,l$, that locally looks like a union of hyperplanes in ${\mathbb{C}}^n$: around each point of $X$ we can find a system of local coordinates in which each $L_i$ is defined by a linear equation.\
This generalizes the notion of a (simple) normal crossing divisor: a hypersurface arrangement is a normal crossing divisor if the local linear equations defining the $L_i$’s are everywhere linearly independent; in other words, if we can always choose local coordinates $(z_1,\cdots,z_n)$ such that $L$ is locally defined by the equation $z_1\cdots z_r=0$ for some $r$.\
Besides normal crossing divisors, examples of hypersurface arrangements include unions of hyperplanes in a projective space ${\mathbb{P}}^n({\mathbb{C}})$, or unions of diagonals $\Delta_{i,j}=\{y_i=y_j\}\subset Y^n$ inside the $n$-fold cartesian product of a Riemann surface $Y$. The class of hypersurface arrangements is also closed under certain blow-ups (see §\[secblowups\]).\
The aim of this paper is to define and study a model $M^\bullet(X,L)$ for the cohomology algebra over ${\mathbb{Q}}$ of the complement $X\setminus L$ of a hypersurface arrangement, when $X$ is a smooth projective variety over ${\mathbb{C}}$.\
Our model, which we call the Gysin model, has *combinatorial* inputs coming from the theory of hyperplane arrangements (the local setting) and *geometric* inputs coming from the cohomology of smooth hypersurface complements in a smooth projective variety (the global setting). Roughly speaking, it is the *direct product* of two classical tools related to these two situations, that we first recall.
at (-6.6,0) (loc) [The Orlik-Solomon algebra of a central hyperplane arrangement.]{}; at (0,1.6) (globncd) [The Gysin long exact sequence for a smooth hypersurface in a smooth projective variety.]{}; at (0,0) (globhyp) [The Gysin model $M^\bullet(X,L)$ (Theorem \[intromaintheorem\]).]{}; at (-6.6,0.9) [Combinatorics]{}; at (0,2.6) [Geometry]{}; (loc) to (globhyp); (globncd) to (globhyp);
- *Combinatorics: the Orlik-Solomon algebra.* Let $L$ be a union of hyperplanes in ${\mathbb{C}}^n$ that contain the origin, and call any intersection of hyperplanes of $L$ a stratum of $L$. The strata of $L$ form a poset which is graded by the codimension of the strata, and denoted by ${\mathscr{S}}_\bullet(L)$. In [@orliksolomon], Orlik and Solomon introduced ${\mathbb{Q}}$-vector spaces $A_S(L)$ for every stratum $L$, and gave the direct sum $$\label{introdirectsumS}
A_\bullet(L)=\bigoplus_{S\in {\mathscr{S}}_\bullet(L)}A_S(L)$$ the structure of a graded algebra, via product maps $$\label{introprodS}
A_S(L)\otimes A_{S'}(L)\rightarrow A_{S\cap S'}(L).$$ Furthermore, there are natural morphisms $$\label{introdiffS}
A_S(L)\rightarrow A_{S'}(L).$$ for any inclusion $S\subset S'$ of strata of $L$ such that $\mathrm{codim}(S')=\mathrm{codim}(S)-1$. The crucial fact is that the Orlik-Solomon algebra is a combinatorial object, which means that it only depends on the poset of the strata of $L$.\
We now recall the classical Brieskorn-Orlik-Solomon theorem (see Theorem \[bos\] for a more precise statement). Here $H^\bullet({\mathbb{C}}^n\setminus L)$ denotes the cohomology of the complement ${\mathbb{C}}^n\setminus L$ with rational coefficients.
We have an isomorphism of graded algebras $$H^\bullet({\mathbb{C}}^n\setminus L)\cong A_\bullet(L).$$
One may define an Orlik-Solomon algebra $A_\bullet(L)$ for $L$ any hypersurface arrangement inside a complex manifold $X$. We still have a direct sum decomposition (\[introdirectsumS\]), with ${\mathscr{S}}_\bullet(L)$ the graded poset of the strata of $L$, as well as product maps (\[introprodS\]) and natural morphisms (\[introdiffS\]). As in the local case, the Orlik-Solomon algebra $A_\bullet(L)$ only depends on the poset of the strata of $L$. It is functorial with respect to $(X,L)$ in the sense that any holomorphic map $\varphi:X\rightarrow X'$ such that $\varphi^{-1}(L')\subset L$ induces a map of graded algebras $A_\bullet(\varphi):A_\bullet(L')\rightarrow A_\bullet(L)$.\
- *Geometry: the Gysin long exact sequence.* For a smooth hypersurface $V$ inside a smooth projective variety $X$ over ${\mathbb{C}}$, the Gysin morphisms of the inclusion $V\subset X$ are the morphisms $H^{k-2}(V)(-1)\rightarrow H^k(X)$ , where $(-1)$ denotes a Tate twist, obtained as the Poincaré duals of the natural morphisms $H^{2n-k}(X)\rightarrow H^{2n-k}(V)$ where $n=\mathrm{dim}_{\mathbb{C}}(X)$. They fit into a long exact sequence, called the Gysin long exact sequence: $$\label{introgysinles}
\cdots \rightarrow H^{k-2}(V)(-1)\rightarrow H^k(X) \rightarrow H^k(X\setminus V)\rightarrow H^{k-1}(V)(-1) \rightarrow \cdots$$ It is worth nothing that the connecting homomorphisms $H^k(X\setminus V)\rightarrow H^{k-1}(V)(-1) $ are residue morphisms, which are easily described using logarithmic forms.
We can now state our main theorem (see Theorem \[maintheorem\] for more precise statements).
\[intromaintheorem\] Let $X$ be a smooth projective variety over ${\mathbb{C}}$ and $L$ a hypersurface arrangement in $X$.
1. For integers $q$ and $n$ let us consider $$M_q^n(X,L)=\bigoplus_{S\in{\mathscr{S}}_{q-n}(L)}H^{2n-q}(S)(n-q)\otimes A_S(L)$$ where $(n-q)$ denotes a Tate twist, viewed as a pure Hodge structure of weight $q$. Then the direct sum $$M^\bullet(X,L)=\bigoplus_{q}M^\bullet_q(X,L)$$ has the structure of a dga in the (semi-simple) category of split mixed Hodge structures over ${\mathbb{Q}}$. The product in $M^\bullet(X,L)$ is induced by the product maps (\[introprodS\]) of the Orlik-Solomon algebra and the cup-product on the cohomology of the strata. The differential in $M^\bullet(X,L)$ is induced by the natural morphisms (\[introdiffS\]) and the Gysin morphisms $$H^{2n-q}(S)(n-q)\rightarrow H^{2n-q+2}(S')(n+1-q)$$ of the inclusions of strata $S\subset S'$. The dga $M^\bullet(X,L)$ is functorial with respect to $(X,L)$ in the sense explained above.
2. The dga $M^\bullet(X,L)$ is a model for the cohomology of $X\setminus L$ in the following sense: we have isomorphisms of pure Hodge structures over ${\mathbb{Q}}$ $${\mathrm{gr}}^W_q H^n(X\setminus L)\cong H^n(M_q^\bullet(X,L))$$ which are compatible with the algebra structures. These isomorphisms are functorial with respect to $(X,L)$.
The precise definition of the Gysin model $M^\bullet(X,L)$ is given in §\[defM\]. Theorem \[intromaintheorem\] generalizes the case of normal crossing divisors, which is due to P. Deligne ([@delignehodge2], see also [@voisin], 8.35) as a by-product of the definition of the mixed Hodge structure on the cohomology of smooth varieties over ${\mathbb{C}}$.\
We should say a word on the usefulness of our generalization from normal crossing divisors to hypersurface arrangements. Indeed, Deligne’s approach relies on the fact that any smooth variety over ${\mathbb{C}}$ can be viewed as the complement of a normal crossing divisor inside a smooth projective variety, using Nagata’s compactification theorem and Hironaka’s resolution of singularities. Thus the case of normal crossing divisors is (in principle) sufficient to give a model for the cohomology of *any* smooth variety over ${\mathbb{C}}$.\
In the framework of Theorem \[intromaintheorem\], we may even produce, following [@deconciniprocesi], [@fultonmcpherson], [@hu], [@li], an explicit sequence of blow-ups (see Theorem \[seqblowups\]) $$\pi:{\widetilde{X}}\rightarrow X$$ sometimes called a “wonderful compactification”, that transforms $L$ into a normal crossing divisor ${\widetilde{L}}=\pi^{-1}(L)$ inside ${\widetilde{X}}$ and induces an isomorphism $$\pi:{\widetilde{X}}\setminus{\widetilde{L}}{\overset{\simeq}{\rightarrow}}X\setminus L.$$ Thus Deligne’s special case of Theorem \[intromaintheorem\] applied to $({\widetilde{X}},{\widetilde{L}})$ gives a model $M^\bullet({\widetilde{X}},{\widetilde{L}})$ for the cohomology of $X\setminus L$. The functoriality of our construction gives a quasi-isomorphism of dga’s $$M^\bullet(\pi):M^\bullet(X,L)\rightarrow M^\bullet({\widetilde{X}},{\widetilde{L}})$$ that we may compute explicitly (see Theorem \[formulaMpi\] and the example in §\[basicexample\]).\
The model $M^\bullet(X,L)$ has three advantages over $M^\bullet({\widetilde{X}},{\widetilde{L}})$. Firstly, it is in general smaller (see the example in §\[basicexample\]). Secondly, its definition only uses geometric and combinatorial information from the pair $(X,L)$ without having to look at the blown-up situation $({\widetilde{X}},{\widetilde{L}})$. Thirdly, it is functorial with respect to the pair $(X,L)$.\
The reader may want to skip directly to §\[basicexample\] to look at an example of a computation using the model $M^\bullet(X,L)$. In the next paragraph, we first give some details on the proof of Theorem \[intromaintheorem\].
Logarithmic forms and mixed Hodge theory
----------------------------------------
Our approach to proving Theorem \[intromaintheorem\] follows Deligne’s proof of the case of normal crossing divisors, hence makes extensive use of logarithmic forms and the formalism of mixed Hodge structures.\
Let $X$ be a smooth projective variety and $L=L_1\cup\cdots\cup L_l$ a hypersurface arrangement in $X$. The first task is to define a complex of sheaves on $X$, denoted by $\Omega^\bullet_{\langle X,L\rangle}$, of meromorphic forms on $X$ with logarithmic poles along $L$. In local coordinates where each $L_i$ is defined by a linear equation $f_i=0$, a section of $\Omega^\bullet_{\langle X,L\rangle}$ is a meromorphic differential form on $X$ which is a linear combination over ${\mathbb{C}}$ of forms of the type $$\label{eqlogformsncd1}\eta\wedge\dfrac{df_{i_1}}{f_{i_1}}\wedge\cdots\wedge\dfrac{df_{i_s}}{f_{i_s}}$$ with $\eta$ a holomorphic form and $1\leq i_1< \cdots< i_s\leq l$. It has to be noted that the complex $\Omega^\bullet_{\langle X,L\rangle}$ is in general a strict subcomplex of the complex $\Omega^\bullet_X(\log L)$ introduced by Saito ([@saitologarithmic]), even though the two complexes coincide in the case of a normal crossing divisor.\
The main point of the complex $\Omega^\bullet_{\langle X,L\rangle}$ is that it computes the cohomology of the complement $X\setminus L$. More precisely, if we denote by $j:X\setminus L\hookrightarrow X$ the open immersion of the complement of $L$ inside $X$, we prove the following theorem (Theorem \[qisglobal\]):
\[introqis\] The inclusion $\Omega^\bullet_{\langle X,L\rangle}\hookrightarrow j_*\Omega^\bullet_{X\setminus L}$ is a quasi-isomorphism, and hence induces isomorphisms $$\label{isoiso}\mathbb{H}^n(\Omega^\bullet_{\langle X,L\rangle})\cong H^n(X\setminus L,{\mathbb{C}}).$$
It has to be noted (Remark \[conjterao\]) that according to this theorem, a conjecture of H. Terao ([@teraologarithmic]) is equivalent to the fact that the inclusion $\Omega^\bullet_{\langle X,L\rangle}\subset\Omega^\bullet_X(\log L)$ is a quasi-isomorphism.\
The proof of Theorem \[introqis\] is local and relies on the Brieskorn-Orlik-Solomon theorem. Another central technical tool is the weight filtration $W$ on $\Omega^\bullet_{\langle X,L\rangle}$: we define $W_k\Omega^\bullet_{\langle X,L\rangle}\subset \Omega^\bullet_{\langle X,L\rangle}$ to be the subcomplex spanned by the forms (\[eqlogformsncd1\]) with $s\leq k$. In view of the isomorphism (\[isoiso\]), we get a filtration on the cohomology of $X\setminus L$ which is proved to be defined over ${\mathbb{Q}}$. Together with the Hodge filtration $F^p\Omega^\bullet_{\langle X,L\rangle}=\Omega^{\geq p}_{\langle X,L\rangle}$, it defines a mixed Hodge structure on $H^\bullet(X\setminus L)$. The functoriality of our construction then implies that this is the same as the mixed Hodge structure defined by Deligne.\
According to the general theory of mixed Hodge structures, the hypercohomology spectral sequence associated to the weight filtration degenerates at the $E_2$-term, hence the $E_1$-term gives a model for the cohomology of $X\setminus L$. We then prove that this model is the Gysin model $M^\bullet(X,L)$ described in the previous paragraph. This concludes the proof of Theorem \[intromaintheorem\].\
After a first preprint version of this article was released, E. Looijenga kindly informed us that this spectral sequence had already appeared in [@looijenga], §2. Our approach is more down-to-earth in that we prove that the Gysin spectral sequence is compatible with Hodge structures using only mixed Hodge theory *à la* Deligne. With Looijenga’s formalism, one would have to use Saito’s theory of mixed Hodge modules (see also [@getzlerresolvingmhm]). Our use of logarithmic forms allows us to manipulate explicit resolutions of the (complexes of) sheaves that appear in [@looijenga].
Configuration spaces of points on curves
----------------------------------------
Let $Y$ be a compact Riemann surface and $n$ an integer. For all $1\leq i<j\leq n$ we have a diagonal $$\Delta_{i,j}=\{y_i=y_j\}\subset Y^n$$ inside the $n$-fold cartesian product of $Y$. Any union of $\Delta_{i,j}$’s then defines a hypersurface arrangement in $Y^n$. For example, if we consider the union of all diagonals, the complement is the configuration space of $n$ ordered points in $Y$: $$C(Y,n)=\{(y_1,\cdots,y_n)\in Y^n\,\,|\,\,y_i\neq y_j\ \,\, \textnormal{for}\,\, i\neq j\}.$$ Theorem \[intromaintheorem\] hence gives a Gysin model for the cohomology of $C(Y,n)$. This model is isomorphic to the one independently found by I. Kriz ([@kriz]) and B. Totaro ([@totaroconf]), as we prove in Theorem \[compbloch\].\
In the one hand, the functoriality of our constructions imposes the existence of a quasi-isomorphism $M^\bullet(\pi)$ associated to any wonderful compactification $\pi$, as the one considered by Kriz (though we have not checked whether $M^\bullet(\pi)$ is the quasi-isomorphism $\varphi$ considered in [@kriz]). On the other hand, our method is close to Totaro’s method, since the Gysin spectral sequence that we are considering in §\[gysinss\] is indeed the Leray spectral sequence of the inclusion $j:X\setminus L\hookrightarrow X$.\
As a natural generalization, we consider the union of only certain diagonals $\Delta_{i,j}$. Such a generalization has been recently studied by S. Bloch ([@blochtreeterated]), who gives a model in the spirit of Kriz and Totaro’s model. We prove that this model is also isomorphic to our Gysin model.
A basic example {#basicexample}
---------------
Let $X$ be a smooth projective surface ($\mathrm{dim}_{\mathbb{C}}(X)=2$) and $L=L_1\cup L_2\cup L_3$ a hypersurface arrangement in $X$ such that $L_{12}=L_{13}=L_{23}=P$ a point (one can think of $3$ lines in ${\mathbb{P}}^2({\mathbb{C}})$ that meet at a point).\
Let us consider the blow-up $\pi:{\widetilde{X}}\rightarrow X$ of $X$ along $P$. We get a normal crossing divisor ${\widetilde{L}}=E\cup {\widetilde{L}}_1\cup {\widetilde{L}}_2\cup {\widetilde{L}}_3$ in ${\widetilde{X}}$, where $E=\pi^{-1}(P)$ is the exceptional divisor and ${\widetilde{L}}_i$ is the strict transform of $L_i$, and $\pi$ induces an isomorphism $\pi:{\widetilde{X}}\setminus{\widetilde{L}}{\overset{\simeq}{\rightarrow}}X\setminus L$.
(0,.7) – (0,3); (0.5,0.5) to (2,2); (-0.5,0.5) to (-2,2); (1,0) arc \[radius=1,start angle=0, end angle=180\]; at (0,1) ; at (45:1) ; at (135:1) ; at (0,3.3) [$\widetilde{L}_2$]{}; at (2.1,2.3) [$\widetilde{L}_3$]{}; at (-2.1,2.3) [$\widetilde{L}_1$]{}; at (-1.3,0.2)[$E$]{}; at (3.5,1.05) [${\widetilde{X}}$]{};
at (0,0) (1); at (2,0) (2); (1) to node\[above\][$\pi$]{} (2);
(0,-.3) – (0,3); (-0.2,-0.2) to (2,2); (0.2,-0.2) to (-2,2); at (0,0) ; at (0,3.3) [$L_2$]{}; at (2.1,2.3) [$L_3$]{}; at (-2.1,2.3) [$L_1$]{}; at (-0.5,0) [$P$]{}; at (-3.5,1) [$X$]{};
We explicitly describe, for $q=2$ and $q=4$, the complexes $M_q^\bullet(X,L)$ and $M_q^\bullet({\widetilde{X}},{\widetilde{L}})$, and the quasi-isomorphism $M_q^\bullet(\pi):M_q^\bullet(X,L)\overset{\sim}{\rightarrow}M_q^\bullet({\widetilde{X}},{\widetilde{L}})$, described in general by the formulas in Theorem \[formulaMpi\]. For the sake of simplicity, we do not write the Tate twists.\
- $q=2$. The quasi-isomorphism $M_2^\bullet(\pi):M_2^\bullet(X,L)\overset{\sim}{\rightarrow}M_2^\bullet({\widetilde{X}},{\widetilde{L}})$ is described by the following diagram: $$\xymatrix{
M_2^\bullet(X,L):\ar[d]^{M_2^\bullet(\pi)}&0\ar[r] & H^0(L_1)\oplus H^0(L_2)\oplus H^0(L_3)\ar[r]\ar[d]_{M_2^0(\pi)}& H^2(X)\ar[r]\ar[d]_{M_2^1(\pi)}& 0 \\
M_2^\bullet({\widetilde{X}},{\widetilde{L}}):&0\ar[r] & H^0(E)\oplus H^0({\widetilde{L}}_1)\oplus H^0({\widetilde{L}}_2)\oplus H^0({\widetilde{L}}_3)\ar[r] & H^2({\widetilde{X}})\ar[r]& 0\\
}$$
Here the Gysin models are concentrated in degrees $0$ and $1$. We do not mention the factors $A_S(L)$ and $A_S({\widetilde{L}})$ since they are all $1$-dimensional. The differentials are given by the obvious Gysin morphisms. In degree $1$, $M^1_2(\pi)$ is just the pull-back $\pi^*:H^2(X)\rightarrow H^2({\widetilde{X}})$. In degree $0$, $M^0_2(\pi)$ is defined by $$H^0(L_i)\rightarrow H^0(E)\oplus H^0({\widetilde{L}}_i)\,,\,1\mapsto (1,1).$$ The fact that $M_2^\bullet(\pi)$ is a morphism of complexes is equivalent to the following equality in $H^2({\widetilde{X}})$: $$\pi^*([L_i])=[E]+[{\widetilde{L}}_i].$$ It is easy to check by hand that $M_2^\bullet(\pi)$ is a quasi-isomorphism. Roughly speaking, the cohomology classes that are added in the blown-up situation “do not contribute” to the cohomology of the complex.\
- $q=4$. The quasi-isomorphism $M_4^\bullet(\pi):M_4^\bullet(X,L)\overset{\sim}{\rightarrow}M_4^\bullet({\widetilde{X}},{\widetilde{L}})$ is described by the following diagram: $$\xymatrix@C-6mm{
0\ar[r]& H^0(P)\otimes A_P(L) \ar[r]\ar[d]_{M_4^0(\pi)} & H^2(L_1)\oplus H^2(L_2)\oplus H^2(L_3)\ar[r]\ar[d]_{M_4^1(\pi)}& H^4(X)\ar[r]\ar[d]_{M_4^2(\pi)}& 0 \\
0\ar[r]& H^0(E\cap{\widetilde{L}}_1)\oplus H^0(E\cap{\widetilde{L}}_2)\oplus H^0(E\cap{\widetilde{L}}_3) \ar[r] & H^2(E)\oplus H^2({\widetilde{L}}_1)\oplus H^2({\widetilde{L}}_2)\oplus H^2({\widetilde{L}}_3)\ar[r] & H^4({\widetilde{X}})\ar[r]& 0\\
}$$
Here all the factors $A_S(L)$ and $A_S({\widetilde{L}})$ are all $1$-dimensional except $A_P(L)$, which is the quotient of the $3$-dimensional space ${\mathbb{Q}}e_{12}\oplus{\mathbb{Q}}e_{13}\oplus {\mathbb{Q}}e_{23}$ by the relation $e_{12}-e_{13}+e_{23}=0$. Thus $M_4^0(X,L)=H^0(P)\otimes A_P(L)$ has dimension $2$ and we may view it as the quotient of $H^0(L_{12})\oplus H^0(L_{13})\oplus H^0(L_{23})$ by the sub-vector space spanned by the elements $(x,-x,x)$.\
The differentials are still given by the Gysin morphisms (with the usual signs that ensure that the differentials square to zero).\
In degrees $1$ and $2$, $M^\bullet_4(\pi)$ is given by the pull-backs $\pi^*:H^2(L_i)\rightarrow H^2({\widetilde{L}}_i)$ and $\pi^*:H^4(X)\rightarrow H^4({\widetilde{X}})$. In degree $0$, $M^0_4(\pi)$ is defined by $$H^0(L_{ij})\rightarrow H^0(E\cap{\widetilde{L}}_i)\oplus H^0(E\cap{\widetilde{L}}_j)\,,\,1\mapsto(-1,1).$$ It is easy to check by hand that this passes to the quotient and defines a quasi-isomorphism. In fact, in this particular case, the complexes $M_4^\bullet(X,L)$ and $M_4^\bullet({\widetilde{X}},{\widetilde{L}})$ are acyclic.
Outline of the paper
--------------------
In §$2$ we recall some classical facts about the Orlik-Solomon algebra and the Brieskorn-Orlik-Solomon theorem in the framework of central hyperplane arrangements, and introduce the Orlik-Solomon algebra of a hypersurface arrangement.\
In §$3$, we introduce the complex of logarithmic forms along a central hyperplane arrangement and its weight filtration, and prove the local form (Theorem \[qis\]) of the comparison theorem \[introqis\]. Then we globalize our results to the framework of hypersurface arrangements (Theorem \[qisglobal\]).\
In §$4$, we use the formalism of mixed Hodge complexes to give an alternative definition of the mixed Hodge structure on the cohomology of $X\setminus L$. This allows us to prove Theorem \[intromaintheorem\] (Theorem \[maintheorem\]).\
In §$5$, we study the functoriality of the Gysin model with respect to blow-ups, giving explicit formulas (Theorem \[formulaMpi\]).\
In §$6$, we apply our results to configuration spaces of points on curves and prove (Theorem \[compbloch\]) the isomorphism between the Gysin model and the model proposed by Kriz and Totaro and generalized by Bloch.
Conventions and notations
-------------------------
1. *(Coefficients)* Unless otherwise stated, all vector spaces, algebras, and Hopf algebras are implicitly defined over ${\mathbb{Q}}$, as well as tensor products of such objects. All (mixed) Hodge structures are implicitly defined over ${\mathbb{Q}}$.\
2. *(Cohomology)* If $Y$ is a complex manifold, we will simply write $H^p(Y)$ for the $p$-th singular cohomology group of $Y$ with rational coefficients. We will write $H^p(Y,{\mathbb{C}})=H^p(Y)\otimes{\mathbb{C}}$ for the $p$-th singular cohomology group of $Y$ with complex coefficients. This group is naturally isomorphic, via the de Rham isomorphism, to the $p$-th de Rham cohomology group of $Y$ tensored with ${\mathbb{C}}$, hence we allow ourselves to use smooth differential forms as representatives for cohomology classes.\
3. *(Holomorphic forms)* If $Y$ is a complex manifold, we write $\Omega^p_Y$ for the sheaf of holomorphic $p$-forms on $Y$ and $\Omega^p(Y)=\Gamma(Y,\Omega^p_Y)$ for the vector space of global holomorphic $p$-forms on $Y$.\
4. *(Filtrations and spectral sequences)* Our convention for spectral sequences uses decreasing filtrations. One passes from an increasing filtration $\{W_p\}_{p\in\mathbb{Z}}$ to a decreasing filtration $\{W^p\}_{p\in\mathbb{Z}}$ by putting $W^p=W_{-p}$.\
5. *(Signs)* If $I$ and $J$ are disjoint subsets of a linearly ordered set $\{1,\cdots,n\}$, we define a sign ${\mathrm{sgn}}(I,J)\in\{\pm1\}$ as follows. In the exterior algebra on $n$ independent generators $x_1,\cdots,x_n$, we write $x_I=x_{i_1}\wedge\cdots\wedge x_{i_k}$ for a set $I=\{i_1<\cdots<i_k\}\subset\{1,\cdots,n\}$. Then ${\mathrm{sgn}}(I,J)$ is defined by the equation $x_{I\cup J}={\mathrm{sgn}}(I,J)x_I\wedge x_J$. For example we get ${\mathrm{sgn}}(\{i_r\},I\setminus\{i_r\})=(-1)^{r-1}$.
Acknowledgements
----------------
The author thanks Francis Brown for many corrections and comments on this article, Spencer Bloch for helpful discussions and for giving him a preliminary version of [@blochtreeterated], and Eduard Looijenga for pointing out to him the reference [@looijenga] which helped simplify the presentation. This work was partially supported by ERC grant 257638 “Periods in algebraic geometry and physics”.
The Orlik-Solomon algebra of a hypersurface arrangement
=======================================================
In §\[OScentral\], \[delres\], \[secBOS\], we recall some classical facts about central hyperplane arrangements: the Orlik-Solomon algebra, the deletion-restriction exact sequence and the Brieskorn-Orlik-Solomon theorem. The interested reader will find more details in the expository book [@orlikterao], or surveys such as [@yuzvinskiorliksolomon]. In §\[OSglobal\], we introduce hypersurface arrangements and define their Orlik-Solomon algebras. In §\[secfunctoriality\] we discuss the functoriality of the Orlik-Solomon algebra.
The Orlik-Solomon algebra of a central hyperplane arrangement {#OScentral}
-------------------------------------------------------------
A **central hyperplane arrangement** in ${\mathbb{C}}^n$ is a finite set $L$ of hyperplanes of ${\mathbb{C}}^n$, all containing the origin. For a matter of notation, we will implicitly fix a linear ordering on the hyperplanes and write $L=\{L_1,\cdots,L_l\}$. Nevertheless, the objects that we will define out of a central hyperplane arrangement will be independent of such an ordering.\
We will use the same letter $L$ to denote the union of the hyperplanes: $$L=L_1\cup\cdots\cup L_l.$$
For a subset $I\subset\{1,\cdots, l\}$, the [**stratum**]{} of the arrangement $L$ indexed by $I$ is the vector space $$L_I=\bigcap_{i\in I}L_i$$ with the convention $L_{\varnothing}={\mathbb{C}}^n$. We write ${\mathscr{S}}_\bullet(L)$ for the set of strata of $L$, graded by the codimension, so that ${\mathscr{S}}_0(L)=\{{\mathbb{C}}^n\}$ and ${\mathscr{S}}_1(L)=\{L_1,\cdots,L_l\}$. With the order given by reverse inclusion, ${\mathscr{S}}(L)$ is given the structure of a graded poset, called the **poset** of the central hyperplane arrangement $L$.\
We set $\Lambda_\bullet(L)=\Lambda^\bullet({\mathbb{Q}}e_1\oplus\cdots\oplus {\mathbb{Q}}e_l)$, the exterior algebra over ${\mathbb{Q}}$ with a generator $e_i$ in degree $1$ for each $L_i$. Let $\delta:\Lambda_\bullet(L)\rightarrow \Lambda_{\bullet-1}(L)$ be the unique derivation of $\Lambda_\bullet(L)$ such that $\delta(e_i)=1$ for $i=1,\cdots,l$.\
For $I=\{i_1<\cdots<i_k\}\subset\{1,\cdots,l\}$ we set $e_I=e_{i_1}\wedge \cdots\wedge e_{i_k}\in \Lambda_k(L)$ with the convention $e_\varnothing=1$. The derivation $\delta$ is then given by the formula $$\delta(e_I)=\sum_{s=1}^{k}(-1)^{s-1}e_{i_1}\wedge\cdots\wedge\widehat{e_{i_s}}\wedge\cdots\wedge e_{i_k}=\sum_{i\in I}{\mathrm{sgn}}(\{i\},I\setminus\{i\})e_{I\setminus\{i\}}.$$
A subset $I\subset\{1,\cdots,l\}$ is said to be [**dependent**]{} (resp. [**independent**]{}) if $\mathrm{codim}(L_I)<|I|$ (resp. $\mathrm{codim}(L_I)=|I|$), which is equivalent to saying that the linear forms defining the $L_i$’s, for $i\in I$, are linearly dependent (resp. independent).
Let $J_\bullet(L)$ be the homogeneous ideal of $\Lambda_\bullet(L)$ generated by the elements $\delta(e_I)$ for $I\subset\{1,\cdots,l\}$ dependent. The quotient $$A_\bullet(L)=\Lambda_\bullet(L)/J_\bullet(L)$$ is a graded algebra over ${\mathbb{Q}}$ called the [**Orlik-Solomon algebra**]{} of the central hyperplane arrangement $L$.
\[remposet\] It is important to notice that the Orlik-Solomon algebra $A_\bullet(L)$ only depends on the poset of $L$.
\[linearpresos\] As a ${\mathbb{Q}}$-vector space, $A_\bullet(L)$ is spanned by the monomials $e_I$ for $I$ independent. The only linear relations between these elements are given, for $I'$ dependent, by $$\sum_{\substack{i\in I'\\ I'\setminus\{i\} \textnormal{ indep.}}}{\mathrm{sgn}}(\{i\},I'\setminus\{i\})e_{I'\setminus\{i\}}=0.$$
For $I$ dependent and any $i\in I$, $e_I=\pm e_i\wedge\delta e_I \in J(L)$ and the first assertion follows. For the second assertion, it is enough to prove that the ideal $J(L)$ is spanned as a ${\mathbb{Q}}$-vector space by the elements $e_I$ and $\delta e_I$ for $I$ dependent. Now $J(L)$ is spanned by elements $e_T\wedge\delta e_I$ for $I$ dependent and $T$ any subset of $\{1,\cdots,l\}$. Using the Leibniz rule we see that $$\pm e_T\wedge \delta e_I=\delta(e_T\wedge e_I) - \sum_{t\in T}\pm e_{T\setminus\{t\}}\wedge e_I.$$ Since $I$ is dependent, $e_T\wedge e_I$ (resp. $e_{T\setminus\{t\}}\wedge e_I$) is either $0$ or an $e_{I'}$ for $I'$ dependent. This completes the proof of the lemma.
If $I'\subset\{1,\cdots,l\}$ is dependent and $I'\setminus\{i\}$ is independent, then for dimension reasons we have $L_{I'\setminus\{i\}}=L_{I'}$. Thus all the sets $I'\setminus\{i\}$ appearing in the relation in the above lemma define the same stratum. Thus, if we define, for a stratum $S$, $A_S(L)$ to be the sub-vector space of $A_\bullet(L)$ spanned by the monomials $e_I$ for $I$ such that $L_I=S$, we have a direct sum decomposition $$\label{eqdirectsumS}
A_\bullet(L)=\bigoplus_{S\in{\mathscr{S}}_\bullet(L)}A_S(L).$$
\[remposetLS\] One may note that as a ${\mathbb{Q}}$-vector space, $A_S(L)$ has the same presentation as in the above lemma, restricting to monomials $e_I$ such that $L_I=S$. Thus, it only depends on the central hyperplane arrangement consisting of the hyperplanes in $L$ that contain $S$ (and more precisely on its poset, see Remark \[remposet\]).
The product in $A_\bullet(L)$ splits with respect to the direct sum decomposition (\[eqdirectsumS\]), with components $$\label{eqprodS}
A_S(L)\otimes A_{S'}(L)\rightarrow A_{S\cap S'}(L)$$ which are zero if $\mathrm{codim}(S\cap S')<\mathrm{codim}(S)+\mathrm{codim}{S'}$.\
The derivation $\delta$ induces a derivation $\delta:A_{\bullet}(L)\rightarrow A_{\bullet-1}(L)$ which splits with respect to the direct sum decomposition (\[eqdirectsumS\]), with components $$\label{eqderivS}
A_S(L)\rightarrow A_{S'}(L)$$ for $S\subset S'$, $\mathrm{codim}(S')=\mathrm{codim}(S)-1$.
Deletion and restriction {#delres}
------------------------
Let $L=\{L_1,\cdots,L_l\}$ be a central hyperplane arrangement in ${\mathbb{C}}^n$ such that $l\geq 1$. In this article we will only be concerned about deletion and restriction with respect to the last hyperplane $L_l$. The **deletion** of $L$ (with respect to $L_l$) is the arrangement $L'=\{L_1,\cdots,L_{l-1}\}$ in ${\mathbb{C}}^n$. The **restriction** of $L$ (with respect to $L_l$) is the arrangement $L''$ on $L_l\cong {\mathbb{C}}^{n-1}$ consisting of all the intersections of $L_l$ with the $L_i$’s, $i=1,\cdots,l-1$. If the hyperplanes $L_i$ are not in general position, it may happen that the cardinality $l''$ of $L''$ is less than $l-1$. For an index $i=1,\cdots,l-1$ of a hyperplane in $L$, we will write $\lambda(i)$ for the corresponding index of $L_l\cap L_i$ in $L''$: $L''_{\lambda(i)}=L_l\cap L_i$.\
For all $k$, we have a short exact sequence of ${\mathbb{Q}}$-vector spaces, called the **deletion-restriction short exact sequence**: $$\label{eqdelres}
0\rightarrow A_k(L')\stackrel{i}{\rightarrow} A_k(L) \stackrel{j}{\rightarrow} A_{k-1}(L'')\rightarrow 0.$$ One may find a detailed proof in [@orlikterao], Theorem 3.65 or [@yuzvinskiorliksolomon], Corollary 2.17. Here $i$ is the natural map defined by $i(e_I)=e_I$ for $I\subset\{1,\cdots,l-1\}$, and $j$ is defined by $j(e_I)=0$ if $I$ does not contain $l$, and $j(e_{i_1}\wedge\cdots\wedge e_{i_{k-1}}\wedge e_l)=e_{\lambda(i_1)}\wedge\cdots\wedge e_{\lambda(i_{k-1})}$ for $i_1<\cdots<i_{k-1}<l$.\
This deletion-restriction exact sequence splits with respect to the direct sum decomposition (\[eqdirectsumS\]). Let $S$ be a stratum of $L$, then there are three cases:
- $S$ is not contained in $L_l$, then it is not a stratum of $L''$ but is a stratum of $L'$, and we just get an isomorphism $$0\rightarrow A_S(L')\rightarrow A_S(L)\rightarrow 0\rightarrow 0.$$
- $S$ is contained in $L_l$ but is not a stratum of $L'$, and we just get an isomorphism $$0\rightarrow 0\rightarrow A_S(L)\rightarrow A_S(L'')\rightarrow 0.$$
- $S$ is contained in $L_l$ and is a stratum of $L'$, and we get a short exact sequence $$0\rightarrow A_S(L')\rightarrow A_S(L)\rightarrow A_S(L'')\rightarrow 0.$$
In any case, the maps in the short exact sequence are given by the same formulas as (\[eqdelres\]).
The Brieskorn-Orlik-Solomon theorem {#secBOS}
-----------------------------------
Let $L=\{L_1,\cdots,L_l\}$ be a central hyperplane arrangement in ${\mathbb{C}}^n$. For $i=1,\cdots,l$ we fix a linear form $f_i$ on ${\mathbb{C}}^n$ that defines $L_i$: $L_i=\{f_i=0\}$. Such a form is unique up to a non-zero multiplicative constant.\
We set $$\omega_i=\dfrac{df_i}{f_i}$$ viewed as a holomorphic $1$-form on ${\mathbb{C}}^n\setminus L$. For a subset $I=\{i_1<\cdots<i_k\}\subset\{1,\cdots,l\}$ we set $$\omega_I=\omega_{i_1}\wedge\cdots\wedge\omega_{i_k}.$$
Let $\Omega^\bullet({\mathbb{C}}^n\setminus L)$ be the algebra of global holomorphic forms on ${\mathbb{C}}^n\setminus L$ and $R^\bullet(L)\subset \Omega^\bullet({\mathbb{C}}^n\setminus L)$ be the subalgebra over ${\mathbb{Q}}$ generated by $1$ and the forms $\frac{1}{2i\pi}\omega_i$ for $i=1,\cdots,l$. We define a morphism of graded algebras $u:\Lambda_\bullet(L)\rightarrow R^\bullet(L)$ by the formula $$u(e_i)=\dfrac{1}{2i\pi}\omega_i.$$
A simple computation shows:
\[osforms\] For $I\subset\{1,\cdots,l\}$ a dependent subset, we have the relation in $\Omega^\bullet({\mathbb{C}}^n\setminus L)$: $$\sum_{i\in I}{\mathrm{sgn}}(\{i\},I\setminus\{i\})\omega_{I\setminus\{i\}}=0.$$
Thus $u$ passes to the quotient and defines a map of graded algebras $$u:A_\bullet(L)\rightarrow R^\bullet(L)$$
Each form $\frac{1}{2i\pi}\omega_i$ is closed and its class is in the cohomology of ${\mathbb{C}}^n\setminus L$ with *rational* (and even integer) coefficients. Indeed, let $f_i^*:H^1({\mathbb{C}}^*)\rightarrow H^1({\mathbb{C}}^n\setminus L)$ be the pullback map induced by $f_i$ in cohomology. Then the class of $\frac{1}{2i\pi}\omega_i$ is the pullback of the class of $\frac{1}{2i\pi}\frac{dz}{z}$ which is known to be in the rational (and even integer) cohomology of ${\mathbb{C}}^*$, dual to the class of the loop $t\mapsto e^{2i\pi t}$ around $0$.\
All the differential forms in $R^\bullet(L)$ being closed, there is a well-defined map of graded algebras $$v:R^\bullet(L)\rightarrow H^\bullet({\mathbb{C}}^n\setminus L).$$
\[bos\] The maps $u$ and $v$ are isomorphisms of graded algebras: $$A_\bullet(L) \overset{\substack{u\\\simeq}}{\longrightarrow} R^\bullet(L) \overset{\substack{v\\\simeq}}{\longrightarrow} H^\bullet({\mathbb{C}}^n\setminus L).$$
The fact that $v$ is an isomorphism was conjectured by Arnol’d ([@arnold]) and proved by Brieskorn ([@brieskorn]). The fact that $u$ is an isomorphism was proved by Orlik and Solomon ([@orliksolomon]). A proof may be found in [@orlikterao], Theorems 3.126 and 5.89.
The Orlik-Solomon algebra of a hypersurface arrangement {#OSglobal}
-------------------------------------------------------
We write $\Delta=\{|z|<1\}\subset{\mathbb{C}}$ for the open unit disk and $\Delta^n\subset{\mathbb{C}}^n$ for the unit $n$-dimensional polydisk. Let $X$ be a complex manifold. The following terminology is borrowed from P. Aluffi ([@aluffi]).
A finite set $L=\{L_1,\cdots,L_l\}$ of smooth hypersurfaces of $X$ is a **hypersurface arrangement** if around each point of $X$ we may find a system of local coordinates in which each $L_i$ is defined by a linear equation. In other words, $X$ is covered by charts $V\cong \Delta^n$ such that for all $i$, $L_i\cap V$ is the intersection of $\Delta^n$ with a linear hyperplane in ${\mathbb{C}}^n$.
As for central hyperplane arrangements, the objects that we will define out of a hypersurface arrangement will be independent of the linear ordering on the hypersurfaces $L_i$. We use the same letter $L$ to denote the union of the hypersurfaces: $$L=L_1\cup\cdots\cup L_l.$$
The notion of hypersurface arrangement generalizes that of (simple) normal crossing divisor: a hypersurface arrangement is a normal crossing divisor if the local linear equations defining the $L_i$’s are everywhere linearly independent. In other words, we can always choose local coordinates such that the irreducible components $L_i$ are coordinate hyperplanes.\
For a subset $I\subset\{1,\cdots,l\}$, we still write $L_I=\bigcap_{i\in I}L_i$, which is a disjoint union of complex submanifolds of $X$. Throughout this article and for simplicity,
*we will always assume that for all $I$, $L_I$ is connected*
(that includes the case $L_I=\varnothing$) and leave it to the interested reader to extend our results to the general case.\
A **stratum** of $L$ is a non-empty $L_I$, it is a complex submanifold of $X$. We write ${\mathscr{S}}_\bullet(L)$ for the set of strata of $L$, graded by the codimension. We give ${\mathscr{S}}_\bullet(L)$ the structure of a graded poset using reverse inclusion, and call it the **poset** of the hypersurface arrangement $L$.\
Let $p$ be a point in ${\mathbb{C}}^n$ and $V$ a neighbourhood of $p$. Then any chart $V\cong\Delta^n$ as in the above definition defines a central hyperplane arrangement denoted $L^{(p)}$ in ${\mathbb{C}}^n$. It is an abuse of notation since choosing another chart gives a different central hyperplane arrangement, but it will not matter since we will only be interested in the poset of $L^{(p)}$, which is well-defined. More intrinsically, $L^{(p)}$ may be read off the tangent space of $X$ at $p$.
Let $S$ be a strata of $L$. For all $p\in S$, we consider the poset of the strata of $L^{(p)}$ that contain $S$. This poset is independent of the point $p\in S$.
It is obviously true locally, and the claim follows from the fact that $S$ is connected.
According to the above lemma and Remark \[remposetLS\], we may define $$A_S(L)=A_S(L^{(p)})$$ for any point $p\in S$. Let us then define $$A_\bullet(L)=\bigoplus_{S\in{\mathscr{S}}_\bullet(L)}A_S(L).$$ We want to give $A_\bullet(L)$ the structure of a graded algebra. Let $S$ and $S'$ be two strata of $L$. If $S\cap S'=\varnothing$ or $\mathrm{codim}(S\cap S')<\mathrm{codim}(S)+\mathrm{codim}(S')$ then we define the product $A_S(L)\otimes A_{S'}(L)\rightarrow A_\bullet(L)$ to be zero. Else we have a product $$\label{eqprodglobal}
A_S(L)\otimes A_{S'}(L)\rightarrow A_{S\cap S'}(L)$$ given by (\[eqprodS\]) by choosing any point $p\in S\cap S'$.
The graded algebra $A_\bullet(L)$ is called the **Orlik-Solomon algebra** of the hypersurface arrangement $L$.
For $S\subset S'$ an inclusion of strata of $L$ such that $\mathrm{codim}(S')=\mathrm{codim}(S)-1$, we define $$\label{eqderivglobal}
A_S(L)\rightarrow A_{S'}(L)$$ as in the local case (\[eqderivS\]) by choosing any point $p\in S$. One should note that in general the map $A_\bullet(L)\rightarrow A_{\bullet-1}(L)$ induced by (\[eqderivglobal\]) is not a derivation of the Orlik-Solomon algebra.
The Orlik-Solomon algebra of a hypersurface arrangement $L=\{L_1,\cdots,L_l\}$ has a presentation similar to that of a central hyperplane arrangement. A subset $I\subset\{1,\cdots,l\}$ is said to be **null** if $L_I=\varnothing$ and **dependent** (resp. **independent**) if $L_I\neq \varnothing$ and $\mathrm{codim}(L_I)<|I|$ (resp. $\mathrm{codim}(L_I)=|I|$. Then $A_\bullet(L)$ is the quotient of $\Lambda^\bullet(e_1,\cdots,e_l)$ by the homogeneous ideal generated by the monomials $e_I$ for $I$ null and $\delta(e_I)$ for $I$ dependent. In the case of a general hyperplane arrangement (the hyperplanes do not necessarily contain the origin), we recover the classical definition ([@orlikterao], Definition 3.45).\
The presentation of $A_\bullet(L)$ as a vector space (Lemma \[linearpresos\]) also holds for all hypersurface arrangements.\
One may note that this remark only holds under the assumption that for all $I$, $L_I$ is connected. Without this assumption, the Orlik-Solomon algebra may not even be generated in degree $1$.
Functoriality of the Orlik-Solomon algebra {#secfunctoriality}
------------------------------------------
Let $L=\{L_1,\cdots,L_l\}$ and $L'=\{L'_1,\cdots,L'_{l'}\}$ be central hyperplane arrangements respectively in ${\mathbb{C}}^n$ and ${\mathbb{C}}^{n'}$. Let $\varphi:\Delta^n\rightarrow\Delta^{n'}$ be a holomorphic map such that $\varphi^{-1}(L')\subset L$, i.e. $\varphi(\Delta^n\setminus L)\subset\Delta^{n'}\setminus L'$.\
Then $\varphi$ induces a map $\varphi^*:H^\bullet(\Delta^{n'}\setminus L')\rightarrow H^\bullet(\Delta^n\setminus L)$ in cohomology. The inclusions $\Delta^n\setminus L\subset {\mathbb{C}}^n\setminus L$ and $\Delta^{n'}\setminus L'\subset {\mathbb{C}}^{n'}\setminus L'$ are retractions and hence induce isomorphisms in cohomology. Thus the Brieskorn-Orlik-Solomon theorem \[bos\] implies that there is a unique map of graded algebras $$A_\bullet(\varphi):A_\bullet(L')\rightarrow A_\bullet(L)$$ that fits into the following commutative square.
$$\xymatrix{
A_\bullet(L') \ar[r]^{A_\bullet(\varphi)} \ar[d]^{\cong}& A_\bullet(L) \ar[d]^{\cong}\\
H^\bullet(\Delta^{n'}\setminus L') \ar[r]^{\varphi^*} & H^\bullet(\Delta^n\setminus L)
}$$
\[lemlocalfunctoriality\] For $j=1,\cdots,l'$, there is an equality $$f'_j\circ\varphi=u_j\prod_i f_i^{m_{ij}}$$ between germs at $0$ of holomorphic functions on $\Delta^n$, with $u_j$ a holomorphic function such that $u_j(0)\neq 0$ and $m_{ij}\geq 0$.
Let us denote by $\mathcal{O}_n$ the ring of germs at $0$ of holomorphic functions on $\Delta^n$. We have $\varphi^{-1}(L'_j)\subset L$, hence the analytic Nullstellensatz (see [@kaupkaup]) implies that we have an integer $N\geq 0$ and an element $a\in\mathcal{O}_n$ such that $$a.f'_j\circ\varphi=(\prod_if_i)^N.$$ The claim then follows from the fact that $\mathcal{O}_n$ is a factorial ring (see [@kaupkaup]).
With the notations of Lemma \[lemlocalfunctoriality\], $A_\bullet(\varphi):A_\bullet(L')\rightarrow A_\bullet(L)$ is the unique map of graded algebras such that for $j=1,\cdots,l'$, $$A_1(\varphi)(e'_j)=\sum_i m_{ij}e_i.$$
We may assume that we have an equality $$f'_j\circ\varphi=u_j\prod_i f_i^{m_{ij}}$$ defined on $\Delta^n$. We then have $$\varphi^*(\omega'_j)=\dfrac{du_j}{u_j}+\sum_{i}m_{ij}\omega_i$$ and the claim follows from the fact that $\dfrac{du_i}{u_i}$ is $0$ in $H^1({\mathbb{C}}^n\setminus L)$.
More generally, if we write $m_{IJ}=\det\left(m_{ij}\right)_{i\in I,j\in J}$ for $|I|=|J|$, then the formula reads $$A_\bullet(\varphi)(e'_J)=\sum_I m_{IJ}e_I.$$ We will write $A_{S,S'}(\varphi):A_{S'}(L')\rightarrow A_S(L)$ for the components of $A_\bullet(\varphi)$; they are zero if $\mathrm{codim}(S)\neq \mathrm{codim}(S')$.\
We may globalize this construction; if $L$ (resp. $L'$) is a hypersurface arrangement in a complex manifold $X$ (resp. $X'$), and $\varphi:X\rightarrow X'$ a holomorphic map such that $\varphi^{-1}(L')\subset L$, then we define $$\label{functorialityS}
A_{S,S'}(\varphi):A_{S'}(L')\rightarrow A_S(L)$$ for strata $S\in{\mathscr{S}}_\bullet(L)$ and $S'\in{\mathscr{S}}_\bullet(L')$ by looking at $\varphi$ in local charts and applying the above definition. It is clear that this defines a map of graded algebras $A_\bullet(\varphi):A_\bullet(L)\rightarrow A_\bullet(L')$ that is functorial in the sense that we have $A_\bullet(\psi\circ\varphi)=A_\bullet(\varphi)\circ A_\bullet(\psi)$ whenever this is meaningful.
Logarithmic forms and the weight filtration
===========================================
We define and study the forms with logarithmic poles along a central hyperplane arrangement. In §\[seclog\], \[secres\], \[weight\], \[seccomp\], we focus on central hyperplane arrangements (the local case). The main results are Theorem \[gr\] which computes its graded pieces, and Theorem \[qis\] which states that the logarithmic complex computes the cohomology of the complement of the hyperplane arrangement. Then in §\[secglobalforms\] we extend our constructions and results to the case of hypersurface arrangements (the global case).
The logarithmic complex {#seclog}
-----------------------
Let $L=\{L_1,\cdots,L_l\}$ be a central hyperplane arrangement in ${\mathbb{C}}^n$. We recall that we defined some differential forms $\omega_i=\frac{df_i}{f_i}$ for $i=1,\cdots,l$, and $\omega_I=\omega_{i_1}\wedge\cdots\wedge\omega_{i_k}$ for $I=\{i_1<\cdots<i_k\}$, which is zero if $I$ is dependent.
A meromorphic form on ${\mathbb{C}}^n$ is said to have **logarithmic poles along $L$** if it is a linear combination over ${\mathbb{C}}$ of forms of the type $\eta\wedge\omega_I$ for some $I\subset\{1,\cdots,l\}$, where $\eta$ is a holomorphic form on ${\mathbb{C}}^n$.
We define $\Omega^p\langle L\rangle$ to be the ${\mathbb{C}}$-vector space of meromorphic $p$-forms on ${\mathbb{C}}^n$ with logarithmic poles along $L$. These forms are stable under the exterior differential, hence we get a complex $\Omega^\bullet\langle L\rangle$ that embeds into the complex of holomorphic forms on ${\mathbb{C}}^n\setminus L$: $$\Omega^\bullet\langle L\rangle \hookrightarrow \Omega^\bullet({\mathbb{C}}^n\setminus L)$$ which we call the **complex of logarithmic forms** of $L$.
\[logcomplex\] This definition is not standard in the theory of hyperplane arrangements. In [@orlikterao], following Saito ([@saitologarithmic]), one defines a complex $\Omega^\bullet(\log L)$ in the following way. Let $Q=f_1\cdots f_l$ be a defining polynomial for the arrangement. Then $\Omega^p(\log L)$ is the set of meromorphic $p$-forms $\omega$ on ${\mathbb{C}}^n$ such that $Q\omega$ and $Qd\omega$ are holomorphic.\
We have an inclusion $$\Omega^\bullet\langle L\rangle\subset\Omega^\bullet(\log L)$$ which is an equality if and only if $L=\{L_1,\cdots,L_l\}$ is independent.
In ${\mathbb{C}}^2$ with coordinates $x$ and $y$, let $L_1=\{x=0\}$, $L_2=\{y=0\}$, $L_3=\{x=y\}$. Then $Q=xy(x-y)$ and the closed form $$\omega=\dfrac{dx\wedge dy}{xy(x-y)}$$ is in $\Omega^2(\log L)$. One easily checks that $\omega$ is not in $\Omega^2\langle L\rangle$.
Residues {#secres}
--------
We briefly recall the notion of residue of a form with logarithmic poles along a central hyperplane arrangement. In the case of dimension $n=1$, this is the usual Cauchy residue in complex analysis; the general notion of residue is due to Poincaré and Leray ([@leray]).\
We fix a central hyperplane arrangement $L=\{L_1,\cdots,L_l\}$ in ${\mathbb{C}}^n$. The proof of the following easy lemma is left to the reader.
\[lemmadiv\] Let $L'=\{L_1,\cdots,L_{l-1}\}$ the deletion of $L$ with respect to $L_l=\{f_l=0\}$. Let $\omega$ be a meromorphic $p$-form on ${\mathbb{C}}^n$ with poles along $L'$. If $df_l\wedge \omega=0$ then there exists a meromorphic $(p-1)$-form $\theta$ with poles along $L'$ such that $\omega=df_l\wedge\theta$.
\[existenceres\] Let $L'$ (resp. $L''$) the deletion (resp. the restriction) of $L$ with respect to $L_l=\{f_l=0\}$. Let $\omega$ be a $p$-form on ${\mathbb{C}}^n$ with logarithmic poles along $L$. Then there exists a $(p-1)$-form $\alpha$ and a $p$-form $\beta$, both of which have logarithmic poles along $L'$, such that $$\omega=\alpha\wedge\omega_l+\beta$$ The restriction $\alpha_{|L_l}$ is independent of the choices. It is a $(p-1)$-form on $L_l$ with logarithmic poles along $L''$.
Let us write $$\omega=\sum_{I}\eta_I\wedge\omega_I$$ where the $\eta_I$’s are holomorphic forms on ${\mathbb{C}}^n$. Then we can define $$\alpha=\sum_{I'\subset\{1,\cdots,l-1\}}\eta_{I'\cup\{l\}}\wedge\omega_{I'} \hspace{.5cm} \textnormal{and} \hspace{.5cm} \beta=\sum_{l\notin I}\eta_I\wedge\omega_I$$ which are logarithmic forms with poles along $L'$. We then have $$\alpha_{|L_l}=\sum_{I'\subset\{1,\cdots,l-1\}}\eta_{I'\cup\{l\}|L_l}\wedge\omega_{\lambda(I')}$$ which is a logarithmic form on $L_l$ with poles along $L''$. For the uniqueness statement it is enough to prove that if $$\alpha\wedge\omega_l+\beta=0$$ then $\alpha_{|L_l}=0$.\
Since $df_l\wedge\beta=0$, by Lemma \[lemmadiv\] there is a meromorphic $(p-1)$-form $\beta'$ with poles along $L'$ such that $\beta=df_l\wedge\beta'$. Now $$0=f_l\left(\alpha\wedge\omega_l+df_l\wedge\beta'\right)=df_l\wedge\left((-1)^{p-1}\alpha+f_l\beta'\right)$$ so again by Lemma \[lemmadiv\], there exists a meromorphic $(p-2)$-form $\beta''$ with poles along $L'$ such that $$(-1)^{p-1}\alpha+f_l\beta'=df_l\wedge\beta''.$$ Hence we have $$\alpha=(-1)^{p-1}(df_l\wedge\beta''-f_l\beta')$$ which is zero when restricted to $L_l=\{f_l=0\}$.
With the notations of the above proposition, we set $$\mathrm{Res}_{L_l}(\omega)=2i\pi\,\alpha_{|L_l}.$$ It is a meromorphic $(p-1)$-form on $L_l$ with logarithmic poles along $L''$, called the **residue** of $\omega$ along $L_l$.\
For $\eta$ a holomorphic form on ${\mathbb{C}}^n$, we have $\mathrm{Res}_{L_l}(\eta\wedge\omega_I)=0$ if $I$ does not contain $l$, and $\mathrm{Res}_{L_l}(\eta\wedge\omega_{i_1}\wedge\cdots\wedge\omega_{i_{k-1}}\wedge\omega_l)=2i\pi \,\eta_{|L_l}\wedge\omega_{\lambda(i_1)}\wedge\cdots\wedge\omega_{\lambda(i_{k-1})}$ for $i_1<\cdots<i_{k-1}<l$.\
It is easy to see that the residue gives a morphism of complexes $$\mathrm{Res}_{L_l}:\Omega^\bullet\langle L \rangle \rightarrow \Omega^{\bullet-1}\langle L''\rangle$$ where $L''$ is the restriction of $L$ with respect to $L_l$.\
We then have a sequence of morphisms of complexes $$(\mathcal{R}): \hspace{.2cm} 0\rightarrow \Omega^\bullet\langle L'\rangle\stackrel{i}{\rightarrow} \Omega^\bullet\langle L\rangle \overset{\mathrm{Res}_{L_l}}{\longrightarrow} \Omega^{\bullet-1}\langle L''\rangle\rightarrow 0$$ where $i$ is the natural inclusion. It is obvious from the definitions that $\mathrm{Res}_{L_l}\circ i=0$, that $i$ is injective and $\mathrm{Res}_{L_l}$ is surjective. We will prove in the next paragraph that $\mathrm{ker}(\mathrm{Res}_{L_l})\subset \mathrm{Im}(i)$, so that the above sequence is a short exact sequence.
\[iteratedresidues\] When taking iterated residues, one should note that they “do not commute” in general, even when this has a clear meaning. For example, if $L_1=\{x=0\}$, $L_2=\{y=0\}$, $L_3=\{x=y\}$ in ${\mathbb{C}}^2$ and $\omega=\dfrac{dx}{x}\wedge\dfrac{dy}{y}\in \Omega^2\langle L\rangle$, we have $\mathrm{Res}_{L_2\cap L_3}\mathrm{Res}_{L_2}(\omega)=(2i\pi)^2$ and $\mathrm{Res}_{L_3\cap L_2}\mathrm{Res}_{L_3}(\omega)=0$ because $\mathrm{Res}_{L_3}(\omega)=0$.
The weight filtration {#weight}
---------------------
The following terminology is borrowed from P. Deligne ([@delignehodge2], 3.1.5). We fix a central hyperplane arrangement $L=\{L_1,\cdots,L_l\}$ in ${\mathbb{C}}^n$.
For $k\geq 0$, we define $W_k\Omega^\bullet\langle L\rangle \subset \Omega^\bullet\langle L\rangle$ to be the subcomplex spanned by the forms that are of the type $\eta\wedge\omega_I$ with $|I|\leq k$, where $\eta$ is a holomorphic form on ${\mathbb{C}}^n$. These subcomplexes define an ascending filtration $$W_0\Omega^\bullet\langle L\rangle \subset W_1\Omega^\bullet\langle L\rangle \subset \cdots$$ on $\Omega^\bullet\langle L\rangle$ called the **weight filtration**.
We have $W_0\Omega^\bullet\langle L\rangle=\Omega^\bullet({\mathbb{C}}^n)$ and $W_p\Omega^p\langle L\rangle=\Omega^p\langle L\rangle$.\
By definition, the residue morphisms induce morphisms $\mathrm{Res}_{L_l}:W_k\Omega^\bullet\langle L\rangle\rightarrow W_{k-1}\Omega^{\bullet-1}\langle L''\rangle$ which are easily seen to be surjective. Thus the sequence $(\mathcal{R})$ induces sequences $$\label{WR}(W_k\mathcal{R}): \hspace{.2cm} 0\rightarrow W_k\Omega^\bullet\langle L'\rangle\stackrel{i}{\rightarrow} W_k\Omega^\bullet\langle L\rangle \overset{\mathrm{Res}_{L_l}}{\longrightarrow} W_{k-1}\Omega^{\bullet-1}\langle L''\rangle\rightarrow 0$$ and $$\label{grR}({\mathrm{gr}}_k^W\mathcal{R}): \hspace{.2cm}0\rightarrow {\mathrm{gr}}_k^W\Omega^\bullet\langle L'\rangle\stackrel{i}{\rightarrow} {\mathrm{gr}}_k^W\Omega^\bullet\langle L\rangle \overset{\mathrm{Res}_{L_l}}{\longrightarrow} {\mathrm{gr}}_{k-1}^W\Omega^{\bullet-1}\langle L''\rangle\rightarrow 0.$$ We will prove that they are short exact sequences. For now, the only easy facts are that $(W_k\mathcal{R})$ is exact on the left and on the right, and that $({\mathrm{gr}}_k^W\mathcal{R})$ is exact on the right.\
The following lemma is easily proved by choosing appropriate coordinates on ${\mathbb{C}}^n$.
\[weightres\] Let $I\subset\{1,\cdots,l\}$, $|I|=k$, be an independent subset and $\eta$ a holomorphic form on ${\mathbb{C}}^n$. If $\eta_{|L_I}=0$ then $\eta\wedge\omega_I\in W_{k-1}\Omega^\bullet\langle L\rangle$.
For all $k$, we define $$G_k^\bullet(L)=\bigoplus_{S\in{\mathscr{S}}_k(L)}\Omega^{\bullet-k}(S)\otimes A_S(L).$$
This is a complex of ${\mathbb{C}}$-vector spaces. We define a morphism of complexes $$\Phi:G_k^\bullet(L)\rightarrow {\mathrm{gr}}_k^W\Omega^\bullet\langle L\rangle$$ in the following way. For $I$ independent of cardinality $k$, for $\eta\in\Omega^{\bullet-k}(L_I)$, we set $$\Phi(\eta\otimes e_I)=(2i\pi)^{-k}{\widetilde{\eta}}\wedge\omega_I$$ where ${\widetilde{\eta}}\in\Omega^{\bullet-k}({\mathbb{C}}^n)$ is any form such that ${\widetilde{\eta}}_{|L_l}=\eta$. Lemma \[weightres\] implies that this does not depend on the choice of ${\widetilde{\eta}}$ and Lemma \[osforms\] shows that it passes to the quotient that defines the groups $A_S(L)$. It is then easy to check that $\Phi$ is a morphism of complexes.
\[gr\] The morphism $$\Phi:G_k^\bullet(L)\rightarrow {\mathrm{gr}}_k^W\Omega^\bullet\langle L\rangle$$ is an isomorphism of complexes.
The surjectivity is trivial and we prove the injectivity by induction on the cardinal $l$ of the arrangement.\
For $l=0$, the only non-trivial case is $k=0$ and $\Phi$ is just the identity of $\Omega^\bullet({\mathbb{C}}^n)$.\
Suppose that the statement is proved for arrangements of cardinality $\leq l-1$ and take an arrangement $L$ of cardinality $l$. Tensoring the deletion-restriction short exact sequence from §\[delres\] with the complexes $\Omega^{\bullet-k}(S)$ we get a short exact sequence of complexes of ${\mathbb{C}}$-vector spaces $$0\rightarrow G_k^\bullet(L')\rightarrow G_k^\bullet(L)\rightarrow G_{k-1}^{\bullet-1}(L'')\rightarrow 0.$$ We then have a diagram $$\xymatrix{
0 \ar[r]& G_k^\bullet(L') \ar[r]\ar[d]^{\Phi} & G_k^\bullet(L) \ar[r]\ar[d]^{\Phi} & G_{k-1}^{\bullet-1}(L'') \ar[r]\ar[d]^{\Phi} & 0 \\
0 \ar[r]& {\mathrm{gr}}_k^W\Omega^\bullet\langle L'\rangle \ar[r] & {\mathrm{gr}}_k^W\Omega^\bullet\langle L\rangle \ar[r] & {\mathrm{gr}}_{k-1}^W\Omega^{\bullet-1}\langle L''\rangle \ar[r]& 0 \\
}$$ where the bottom row is the sequence (\[grR\]). This diagram is easily seen to be commutative.\
By the inductive hypothesis, the vertical arrows on the right and on the left are isomorphisms. Thus a diagram chase shows that the bottom row is exact in the middle.\
Now the complexes (\[WR\]) and (\[grR\]) give rise to a short exact sequence of complexes $$0\rightarrow (W_{k-1}\mathcal{R})\rightarrow (W_k\mathcal{R}) \rightarrow ({\mathrm{gr}}_{k}^W\mathcal{R})\rightarrow 0.$$ The long exact sequence in cohomology tells us that if $(W_{k-1}\mathcal{R})$ is exact in the middle then it is also the case for $(W_k\mathcal{R})$. Since $(W_0\mathcal{R})$ is just the sequence $$0\rightarrow \Omega^\bullet({\mathbb{C}}^n)\overset{\mathrm{id}}{\rightarrow} \Omega^\bullet({\mathbb{C}}^n)\rightarrow 0\rightarrow 0$$ an induction on $k$ shows that $(W_k\mathcal{R})$ is exact in the middle, hence a short exact sequence, for all $k$. Again, the long exact sequence in cohomology shows that $({\mathrm{gr}}_k^W\mathcal{R})$ is also a short exact sequence for all $k$.\
Thus, in the above commutative diagram, both rows are exact and a diagram chase (the $5$-lemma) shows that the middle $\Phi$ is injective. This completes the induction and the proof of the theorem.
\[rempsi\] The inverse morphism $\Psi:{\mathrm{gr}}_k^W\Omega^\bullet\langle L\rangle \rightarrow G_k^\bullet(L)$ is given, for $\eta$ holomorphic and $I$ independent of cardinality $k$, by $$\Psi(\eta\wedge\omega_I)=(2i\pi)^k\,\eta_{|L_I}\in \Omega^{\bullet-k}(L_I)$$ For $k=1$ this is exactly the definition of a residue, but for $k>1$ one should note that this has nothing to do with an “iterated residue” (see Remark \[iteratedresidues\]).
As a corollary of the proof of Theorem \[gr\] we get the following:
\[resexact\] The sequences $(\mathcal{R})$, $(W_k\mathcal{R})$ and $({\mathrm{gr}}_k^W\mathcal{R})$ are short exact sequences of complexes.
The only thing that has to be noticed is that $(\mathcal{R})=(W_k\mathcal{R})$ for $k$ large enough.
The comparison theorem {#seccomp}
----------------------
\[qis\] The inclusion $\Omega^\bullet\langle L\rangle \hookrightarrow \Omega^\bullet({\mathbb{C}}^n\setminus L)$ is a quasi-isomorphism.
Since ${\mathbb{C}}^n\setminus L$ is a smooth affine algebraic variety over ${\mathbb{C}}$, the cohomology of $\Omega^\bullet({\mathbb{C}}^n\setminus L)$ is the cohomology of ${\mathbb{C}}^n\setminus L$ with complex coefficients. Thus we have to prove that the natural map $$H^p(\Omega^\bullet\langle L\rangle)\rightarrow H^p({\mathbb{C}}^n\setminus L,{\mathbb{C}})$$ is an isomorphism for all $p$.\
Let us consider the spectral sequence associated to the filtered complex $(\Omega^\bullet\langle L\rangle, W)$. In view of Theorem \[gr\] we have $$E_0^{-p,q}=\mathrm{gr}_{p}^W \Omega^{-p+q}\langle L\rangle\cong G_{p}^{-p+q}(L)=\bigoplus_{S\in {\mathscr{S}}_p(L)}\Omega^{-2p+q}(S) \otimes A_S(L)$$ the differential being the exterior differential on forms. At the next page of the spectral sequence we get $$E_1^{-p,q}=\bigoplus_{S\in {\mathscr{S}}_p(L)}H^{-2p+q}(S,{\mathbb{C}})\otimes A_S(L).$$ Since the strata $S$ are vector spaces, hence contractible, the only non-zero terms are $$E_1^{-p,2p}= \bigoplus_{S\in{\mathscr{S}}_p(L)} {\mathbb{C}}\otimes A_S(L)=A_p(L)\otimes{\mathbb{C}}.$$ For degree reasons, all the differentials in the pages $E_1$, $E_2$, etc are zero, hence the spectral sequence degenerates at $E_1$: $E_\infty=E_1$. Thus the only non-zero graded pieces of the cohomology of $\Omega^\bullet\langle L\rangle$ are $$H^p(\Omega^\bullet\langle L\rangle)={\mathrm{gr}}^W_pH^p(\Omega^\bullet\langle L\rangle)\cong E_\infty^{-p,2p}= E_1^{-p,2p} \cong A_p(L)\otimes{\mathbb{C}}$$ hence the filtration $W$ on $H^p(\Omega^\bullet\langle L\rangle)$ is trivial and we get an isomorphism $H^p(\Omega^\bullet\langle L\rangle)\cong A_p(L)\otimes{\mathbb{C}}$ which by definition sends $\omega_I$ to $(2i\pi)^pe_I$ for $|I|=p$. Composing with the Brieskorn-Orlik-Solomon isomorphism from Theorem \[bos\] we get an isomorphism $$H^p(\Omega^\bullet\langle L\rangle)\cong A_p(L)\otimes{\mathbb{C}}\cong H^p({\mathbb{C}}^n\setminus L,{\mathbb{C}})$$ which sends $\omega_I$ to its cohomology class, and the Theorem is proved.
Since $(\Omega^\bullet\langle L\rangle, W)$ is a filtered differential graded algebra, there is an algebra structure on the spectral sequence considered in the proof of Theorem \[qis\] and it is easy to check that the product $$E_1^{-p,2p}\otimes E_1^{-p',2p'}\rightarrow E_1^{-(p+p'),2(p+p')}$$ is indeed the product in the Orlik-Solomon algebra (tensored with ${\mathbb{C}}$).
Using the residue exact sequence (Theorem \[resexact\]), the deletion-restriction short exact sequence, and the $5$-lemma, one gets another proof of the fact by $H^p(\Omega^\bullet\langle L\rangle)\cong A_p(L)\otimes {\mathbb{C}}$ by induction on the cardinality $l$ of the arrangement. We have chosen to present the above proof because it shows in a more transparent way the relationship between the weight filtration and the Orlik-Solomon algebra.
\[conjterao\] We have the inclusions of complexes $$\Omega^\bullet\langle L\rangle \overset{i_1}{\hookrightarrow} \Omega^\bullet(\log L) \overset{i_2}{\hookrightarrow} \Omega^\bullet({\mathbb{C}}^n\setminus L)$$ where $\Omega^\bullet(\log L)$ has been defined in Remark \[logcomplex\].\
A conjecture by H. Terao ([@teraologarithmic]) states that $i_2$ is a quasi-isomorphism. According to Theorem \[qis\], the composite $i_2\circ i_1$ is a quasi-isomorphism, hence Terao’s conjecture is equivalent to the fact that $i_1$ is a quasi-isomorphism. This is equivalent to the acyclicity of the quotient complex $\Omega^\bullet(\log L)/\Omega^\bullet\langle L\rangle$.
Logarithmic forms along hypersurface arrangements {#secglobalforms}
-------------------------------------------------
In this paragraph we globalize the definitions of the logarithmic complex and the weight filtration. As in the local case, we determine the graded parts of the logarithmic complex and prove a comparison theorem. This generalizes the case of normal crossing divisors, studied by Deligne in [@delignehodge2], 3.1.\
Let $X$ be a complex manifold and $L$ a hypersurface arrangement in $X$. A meromorphic form on $X$ is said to have **logarithmic poles along $L$** if it is locally a linear combination over ${\mathbb{C}}$ of forms of the type $$\label{localform}\eta\wedge\frac{df_{i_1}}{f_{i_1}}\wedge\cdots\wedge\frac{df_{i_r}}{f_{i_r}}$$ with $\eta$ holomorphic and the $f_i$’s local defining (linear) equations for the $L_i$’s.\
The meromorphic forms on $X$ with logarithmic poles along $L$ form a complex of sheaves of ${\mathbb{C}}$-vector spaces on $X$, that we denote by $\Omega^\bullet_{\langle X,L\rangle}$. If $L=D$ is a normal crossing divisor, then $\Omega^\bullet_{\langle X,D\rangle}=\Omega^\bullet_X(\log D)$ as defined in [@delignehodge2], 3.1.
As already noticed in Remark \[logcomplex\], the definition of $\Omega^\bullet_{\langle X,L\rangle}$ does not agree with the definition of $\Omega^\bullet_X(\log L)$ given in [@saitologarithmic] or other references. In general, our definition gives a smaller complex, although the two definitions are equivalent in the case of normal crossing divisors (and only in this case).
The complex of sheaves $\Omega^\bullet_{\langle X,L\rangle}$ is functorial in $(X,L)$ in the following sense. If $L'$ is another hypersurface arrangement in a complex manifold $X'$, and if we have a holomorphic map $\varphi:X\rightarrow X'$ such that $\varphi^{-1}(L')\subset L$, then there is a pull-back map $$\varphi^*:\Omega^\bullet_{\langle X',L' \rangle}\rightarrow \varphi_*\Omega^\bullet_{\langle X,L \rangle}$$ that is compatible with composition in the usual sense. This follows from the discussion in §\[secfunctoriality\].\
Now we define the **weight filtration** on $\Omega^\bullet_{\langle X,L\rangle}$ exactly in the same fashion as in the local case: $W_k\Omega^\bullet_{\langle X,L\rangle}\subset\Omega^\bullet_{\langle X,L\rangle}$ is the subcomplex of sheaves spanned by the forms that are locally of type (\[localform\]) with $r\leq k$.\
We have $W_0\Omega^\bullet_{\langle X,L\rangle}=\Omega^\bullet_X$ and $W_p\Omega^p_{\langle X,L\rangle}=\Omega^p_{\langle X,L\rangle}$. The weight filtration is also functorial.\
For a stratum $S$ we denote by $i_S:S\hookrightarrow X$ the closed immersion of $S$ inside $X$.\
We globalize the definition of $G_k^\bullet(L)$ from §\[weight\], defining $$\mathcal{G}^\bullet_k(X,L)=\bigoplus_{S\in{\mathscr{S}}_k(L)} (i_S)_*\Omega^{\bullet-k}_{S} \otimes A_S(L).$$ This is a complex of sheaves of ${\mathbb{C}}$-vector spaces on $X$.\
As in the local case, we may define a morphism of complexes of sheaves $$\Phi:\mathcal{G}^\bullet_k(X,L)\rightarrow {\mathrm{gr}}_k^W\Omega^\bullet_{\langle X,L\rangle}$$ by putting $$\Phi(\eta\otimes e_I)=(2i\pi)^{-k}\,{\widetilde{\eta}}\wedge\frac{df_{i_1}}{f_{i_1}}\wedge\cdots\wedge\frac{df_{i_k}}{f_{i_k}}$$ for $I=\{i_1<\cdots<i_k\}$, $\eta\in\Omega^{\bullet-k}_S$ a local section, ${\widetilde{\eta}}\in\Omega^{\bullet-k}_X$ a local extension of $\eta$, and the $f_i$’s local defining equations for the $L_i$’s.\
It has to be noted that this definition is independent from the choice of local equations for the $L_i$’s. Indeed if we write $f_i'=f_iu_i$ with $u_i$ a holomorphic function that is non-zero near the origin, then $$\frac{df_i'}{f_i'}=\frac{df_i}{f_i}+\frac{du_i}{u_i}$$ with $\dfrac{du_i}{u_i}$ being a holomorphic $1$-form near the origin, hence we have $$\frac{df'_{i_1}}{f'_{i_1}}\wedge\cdots\wedge\frac{df'_{i_k}}{f'_{i_k}}=\frac{df_{i_1}}{f_{i_1}}\wedge\cdots\wedge\frac{df_{i_k}}{f_{i_k}} \hspace{3mm}\mathrm{mod}\,\, W_{k-1}\Omega^\bullet_{\langle X,L\rangle}.$$
The following theorem is a global version of Theorem \[gr\].
The morphism $$\Phi:\mathcal{G}^\bullet_k(X,L)\rightarrow {\mathrm{gr}}_k^W\Omega^\bullet_{\langle X,L\rangle}$$ is an isomorphism of complexes of sheaves of ${\mathbb{C}}$-vector spaces.
It is enough to prove that for every chart $V\cong\Delta^n$ on which $L$ is a central hyperplane arrangement, the morphism $$\Gamma(V,\mathcal{G}^\bullet_k(X,L))\rightarrow \Gamma(V,{\mathrm{gr}}_k^W\Omega^\bullet_{\langle X,L\rangle})$$ is an isomorphism. This is exactly Theorem \[gr\] with the ambient space ${\mathbb{C}}^n$ replaced by the polydisk $\Delta^n$. One can check that the proof of Theorem \[gr\] can be copied word for word in that local setting.
The inverse morphism $\Psi: {\mathrm{gr}}_k^W\Omega^\bullet_{\langle X,L\rangle}\rightarrow\mathcal{G}^\bullet_k(X,L)$ is given locally by the same formula as in Remark \[rempsi\]. As already noted, this should not be mistaken with an iterated residue, unless $L$ is a normal crossing divisor (in this case, Deligne calls $\Psi$ the Poincaré residue, see [@delignehodge2], 3.1.5.2).
Let $j:X\setminus L\hookrightarrow X$ be the open immersion of the complement of $L$ inside $X$. The following theorem is a global version of Theorem \[qis\]:
\[qisglobal\] The inclusion $\Omega^\bullet_{\langle X,L\rangle}\hookrightarrow j_*\Omega^\bullet_{X\setminus L}$ is a quasi-isomorphism.
It is enough to prove that for every chart $V\cong\Delta^n$ on which $L$ is a central hyperplane arrangement, the morphism $$\Gamma(V,\Omega^\bullet_{\langle X,L\rangle}) \rightarrow \Gamma(V,j_*\Omega^\bullet_{X\setminus L})=\Omega^\bullet(V\setminus L)$$ is a quasi-isomorphism. This is exactly Theorem \[qis\] with the ambient space ${\mathbb{C}}^n$ replaced by the polydisk $\Delta^n$. One can check that the proof of Theorem \[qis\] can be copied word for word in the local setting. The argument that the strata $L_I$ are contractible has to be replaced by the fact that the local strata $\Delta^n\cap L_I$ are contractible (because they are polydisks). The Brieskorn-Orlik-Solomon theorem remains true in the local setting because the inclusion $\Delta^n\setminus L\subset{\mathbb{C}}^n\setminus L$ is a retraction and hence induces an isomorphism in cohomology.
A functorial mixed Hodge structure and the Gysin model
======================================================
If $X$ is a smooth projective variety and $L$ is a hypersurface arrangement in $X$, we put a functorial mixed Hodge structure on the cohomology of the complement $X\setminus L$. Our construction mimicks Deligne’s ([@delignehodge2]) in the case of normal crossing divisors.
Reminders on mixed Hodge complexes
----------------------------------
In this paragraph we recall some facts on mixed Hodge complexes, defined by P. Deligne. The interested reader will find all the details in [@delignehodge3], 7.1, 8.1. Although the formalism of mixed Hodge complexes, notably the use of (filtered) derived categories, can seem cumbersome at first, it is a useful tool to put mixed Hodge structures on (hyper)cohomology groups of sheaves.\
Let us recall that if $\mathfrak{C}$ is an abelian category, the **filtered derived category** $\mathrm{D^+F}(\mathfrak{C})$ is obtained from the category of filtered (bounded from above) complexes $(K^\bullet,F)$ in $\mathfrak{C}$ after inverting the filtered quasi-isomorphisms. We also have the **bifiltered derived category** $\mathrm{D^+F}_2(\mathfrak{C})$ where we start with bifiltered complexes $(K^\bullet,W,F)$. The letter $W$ denotes an increasing filtration, and $F$ denotes a decreasing filtration. If $Y$ is a complex manifold and $\mathbb{K}$ is ${\mathbb{Q}}$ or ${\mathbb{C}}$, then we write $\mathrm{D^+}(Y,\mathbb{K})$ ($\mathrm{D^+F}(Y,\mathbb{K})$, $\mathrm{D^+F}_2(Y,\mathbb{K})$) for the (filtered, bifiltered) derived category of sheaves of $\mathbb{K}$-vector spaces on $Y$.\
Let $Y$ be a complex manifold and $w$ an integer. A **cohomological Hodge complex** of weight $w$ on $Y$ is a triple $${\mathcal{K}}=({\mathcal{K}}_{\mathbb{Q}},({\mathcal{K}}_{\mathbb{C}},F),\alpha)$$ with ${\mathcal{K}}_{\mathbb{Q}}\in \mathrm{D^+}(Y,{\mathbb{Q}})$, $({\mathcal{K}}_{\mathbb{C}},F)\in \mathrm{D^+F}(Y,{\mathbb{C}})$ and $\alpha:{\mathcal{K}}_{\mathbb{Q}}\otimes{\mathbb{C}}\cong {\mathcal{K}}_{\mathbb{C}}$ an isomorphism in $\mathrm{D^+}(Y,{\mathbb{C}})$, satisfying some conditions that basically mean that for all $p$ the filtration $F$ defines on $\mathbb{H}^p({\mathcal{K}}_{\mathbb{Q}})$ a Hodge structure of weight $p+w$.\
The following fact is a reformulation of classical Hodge theory: if $Y$ is a smooth projective variety over ${\mathbb{C}}$ (or more generally a compact Kähler manifold) then we have a cohomological complex $${\mathcal{K}}(Y)=({\mathcal{K}}_{\mathbb{Q}}(Y),({\mathcal{K}}_{\mathbb{C}}(Y),F),\alpha)$$ of weight $w=0$ on $Y$ consisting of
1. ${\mathcal{K}}_{\mathbb{Q}}(Y)={\mathbb{Q}}_Y$, the constant sheaf.
2. ${\mathcal{K}}_{\mathbb{C}}(Y)=\Omega^\bullet_Y$ with the Hodge filtration $F^p\Omega^\bullet_Y=\Omega^{\geq p}_Y$.
3. the holomorphic Poincaré lemma $\alpha:{\mathbb{Q}}_Y\otimes {\mathbb{C}}\cong{\mathbb{C}}_Y\cong\Omega^\bullet_Y$.
If ${\mathcal{K}}=({\mathcal{K}}_{\mathbb{Q}},({\mathcal{K}}_{\mathbb{C}},F),\alpha)$ is a Hodge complex of weight $w$ on $Y$ and $k$ is an integer, then we can define the translation $${\mathcal{K}}[k]=({\mathcal{K}}_{\mathbb{Q}}[k],({\mathcal{K}}_{\mathbb{C}}[k],F),\alpha[k])$$ which is a Hodge complex of weight $w+k$, and the tate Twist $${\mathcal{K}}(k)=({\mathcal{K}}_{\mathbb{Q}}\otimes{\mathbb{Q}}(2i\pi)^k,({\mathcal{K}}_{\mathbb{C}},F[k]),\alpha)$$ which is a Hodge complex of weight $w-2k$.\
Now if $Y$ is any complex manifold, a **cohomological mixed Hodge complex** on $Y$ is a triple $${\mathcal{K}}=(({\mathcal{K}}_{\mathbb{Q}},W),({\mathcal{K}}_{\mathbb{C}},W,F),\alpha)$$ with $({\mathcal{K}}_{\mathbb{Q}},W)\in \mathrm{D^+F}(Y,{\mathbb{Q}})$, $({\mathcal{K}}_{\mathbb{C}},W,F)\in \mathrm{D^+F}_2(Y,{\mathbb{C}})$ and $\alpha:({\mathcal{K}}_{\mathbb{Q}},W)\otimes{\mathbb{C}}\cong({\mathcal{K}}_{\mathbb{C}},W)$ an isomorphism in $\mathrm{D^+F}(Y,{\mathbb{C}})$, such that for all $k$, the triple $${\mathrm{gr}}_k^W{\mathcal{K}}=({\mathrm{gr}}_k^W{\mathcal{K}}_{\mathbb{Q}},({\mathrm{gr}}_k^W{\mathcal{K}}_{\mathbb{C}},F),{\mathrm{gr}}_k^W\alpha)$$ is a Hodge complex of weight $k$ on $Y$.
The following theorem ([@delignehodge3], 8.1.9) is the fundamental theorem of mixed Hodge complexes.
\[mhc\] Let $Y$ be a complex manifold and ${\mathcal{K}}=(({\mathcal{K}}_{\mathbb{Q}},W),({\mathcal{K}}_{\mathbb{C}},W,F),\alpha)$ a cohomological mixed Hodge complex on $Y$.
1. For all $n$, the filtration $W[-n]$ and the filtration $F$ define a mixed Hodge structure on $\mathbb{H}^n({\mathcal{K}}_{\mathbb{Q}})$.
2. Let ${{}_{\mathrm{w}}\!E}$ be the cohomological spectral sequence defined by $({\mathcal{K}}_{\mathbb{Q}},W)$. Then for all $(p,q)$, the filtration $F$ induces on ${{}_{\mathrm{w}}\!E}_1^{-p,q}=\mathbb{H}^{-p+q}({\mathrm{gr}}_p^W{\mathcal{K}}_{\mathbb{Q}})$ a Hodge structure of weight $q$ and the differentials $d_1^{-p,q}$ are morphisms of Hodge structures.
3. The spectral sequence ${{}_{\mathrm{w}}\!E}$ degenerates at $E_2$: ${{}_{\mathrm{w}}\!E}_2^{-p,q}={{}_{\mathrm{w}}\!E}_\infty^{-p,q}={\mathrm{gr}}_{p}^W\mathbb{H}^n({\mathcal{K}}_{\mathbb{Q}})={\mathrm{gr}}_q^{W[-n]}\mathbb{H}^n({\mathcal{K}}_{\mathbb{Q}})$ for $n=-p+q$.
A functorial mixed Hodge structure
----------------------------------
Let $X$ be a smooth projective variety over ${\mathbb{C}}$ and $L$ a hypersurface arrangement in $X$. We use the previous constructions to put a functorial mixed Hodge structure on the cohomology $H^\bullet(X\setminus L)$ of the complement, using the formalism of mixed Hodge complexes. This generalizes the case of normal crossing divisors, studied by Deligne in [@delignehodge2], 3.2, and summarized in terms of mixed Hodge complexes in [@delignehodge3], 8.1.8. We recall the notation $j:X\setminus L\hookrightarrow X$.\
We define a triple $${\mathcal{K}}(X,L)=(({\mathcal{K}}_{\mathbb{Q}}(X,L),W),({\mathcal{K}}_{\mathbb{C}}(X,L),W,F),\alpha)$$ in the following way:
1. ${\mathcal{K}}_{\mathbb{Q}}(X,L)=Rj_*{\mathbb{Q}}_{X\setminus L}$ with the filtration $W=\tau$, the canonical filtration ([@delignehodge2], 1.4.6).
2. ${\mathcal{K}}_{\mathbb{C}}(X,L)=\Omega^\bullet_{\langle X,L\rangle}$ with the weight filtration $W$ defined in §\[secglobalforms\], and the Hodge filtration $F$ defined by $$F^p\Omega^\bullet_{\langle X,L\rangle}=\Omega^{\geq p}_{\langle X,L\rangle}.$$
3. We have isomorphisms in $\mathrm{D^+}(X,{\mathbb{C}})$: $$Rj_*{\mathbb{Q}}_{X\setminus L}\otimes{\mathbb{C}}\cong Rj_*{\mathbb{C}}_{X\setminus L}\cong j_*\Omega^\bullet_{X\setminus L}\cong\Omega^\bullet_{\langle X,L\rangle}$$ the last one being the quasi-isomorphism of the comparison theorem \[qisglobal\].\
Hence we have an isomorphism $(Rj_*{\mathbb{Q}}_{X\setminus L}\otimes{\mathbb{C}},\tau)\cong(\Omega^\bullet_{\langle X,L\rangle},\tau)$ in $\mathrm{D^+F}(X,{\mathbb{C}})$. Finally the identity gives a filtered quasi-isomorphism $(\Omega^\bullet_{\langle X,L\rangle},\tau)\cong(\Omega^\bullet_{\langle X,L\rangle},W)$, as follows from the same proof as in [@delignehodge2], 3.1.8, in view of the comparison theorem \[qisglobal\]. This gives the isomorphism $$\alpha:(Rj_*{\mathbb{Q}}_{X\setminus L},\tau)\otimes{\mathbb{C}}\cong(\Omega^\bullet_{\langle X,L\rangle},W)$$ in $\mathrm{D^+F}(X,{\mathbb{C}})$.
\[mhs\] The triple ${\mathcal{K}}(X,L)$ is a cohomological mixed Hodge complex on $X$, which is functorial with respect to the pair $(X,L)$. It thus defines a functorial mixed Hodge structure on $\mathbb{H}^n(Rj_*{\mathbb{Q}}_{X\setminus L})\cong H^n(X\setminus L)$ for all $n$.
Here, functoriality has to be understood in the sense of §\[secfunctoriality\].
Theorem \[gr\] gives an isomorphism $${\mathrm{gr}}_k^W\Omega^\bullet_{\langle X,L\rangle}\cong\bigoplus_{S\in{\mathscr{S}}_k(L)}(i_S)_*\Omega^{\bullet-k}_S \otimes A_S(L).$$ A local computation as in [@peterssteenbrink], Lemma 4.9, shows that this isomorphism is defined over ${\mathbb{Q}}$ if we take care of the Tate twists. In other words we have a commutative diagram: $$\xymatrix{
{\mathrm{gr}}_k^W\Omega^\bullet_{\langle X,L\rangle} \ar[r]^-{\cong} & \bigoplus_{S\in{\mathscr{S}}_k(L)}(i_S)_*\Omega^{\bullet}_S [-k]\otimes A_S(L) \\
{\mathrm{gr}}_k^\tau Rj_*{\mathbb{C}}_U \ar[r]^-{\cong} \ar[u]^{\cong} & \bigoplus_{S\in{\mathscr{S}}_k(L)}(i_S)_*{\mathbb{C}}_S [-k]\otimes A_S(L) \ar[u]^{\cong}\\
{\mathrm{gr}}_k^\tau Rj_*{\mathbb{Q}}_U \ar[r]^-{\cong} \ar[u] & \bigoplus_{S\in{\mathscr{S}}_k(L)}(i_S)_*{\mathbb{Q}}_S [-k](-k)\otimes A_S(L) \ar[u]\\
}$$ To complete the proof it is enough to notice that the top row of this diagram is compatible with the Hodge filtrations. Hence we get $${\mathrm{gr}}_k^W{\mathcal{K}}(X,L)=\bigoplus_{S\in{\mathscr{S}}_k(L)} (i_S)_*{\mathcal{K}}(S)[-k](-k)\otimes A_S(L)$$ which is a cohomological Hodge complex of weight $k$.\
The functoriality statement follows from the functoriality of the sheaves of logarithmic forms.
The following theorem shows that the Hodge structures that we have just defined are indeed the functorial Hodge structures defined by Deligne.
\[functoriality\] Let $U$ be a smooth quasi-projective variety over ${\mathbb{C}}$.
1. There exists a smooth projective variety $X$ and an open immersion $U\hookrightarrow X$ such that the complement $L=X\setminus U$ is a hypersurface arrangement in $X$.
2. Given two such compactifications $(X_1,L_1)$ and $(X_2,L_2)$, the mixed Hodge structures on $H^\bullet(U)$ defined via $(X_1,L_1)$ and $(X_2,L_2)$ are the same.
3. The mixed Hodge structure on $H^\bullet(U)$ defined in Theorem \[mhs\] is the same as the mixed Hodge structure defined by Deligne in [@delignehodge2].
<!-- -->
1. This follows from Nagata’s compactification theorem and Hironaka’s resolution of singularities. In fact, we can assume that $L$ is a normal crossing divisor.
2. Using resolution of singularities, we can always embed $U$ in a smooth projective variety $X$ such that $X\setminus U=L$ is a simple normal crossing divisor (and hence a hypersurface arrangement), and such that there exists morphisms $$(X_1,X_1\setminus L_1)\leftarrow (X,X\setminus L)\rightarrow (X_2,X_2\setminus L_2)$$ that are the identity on $U$. Hence by functoriality the two mixed Hodge structures are isomorphic to the mixed Hodge structure defined via $(X,L)$.
3. The claim follows from (2) and the fact that for a given $U$, one can always choose $(X,L)$ such that $L$ is a normal crossing divisor (using resolution of singularities).
The Gysin spectral sequence {#gysinss}
---------------------------
Let $X$ be a smooth projective variety and $L$ a hypersurface arrangement in $X$. In the previous paragraph we defined a cohomological mixed Hodge complex on $X$ that defines a mixed Hodge structure on the cohomology of $X\setminus L$. The general formalism of mixed Hodge complexes (Theorem \[mhc\]) tells us that the [**Gysin spectral sequence**]{} ${{}_{\mathrm{w}}\!E}_r^{p,q}$ associated to the weight filtration degenerates at $E_2$. In this section we make the $E_1$ term explicit. We will write ${{}_{\mathrm{w}}\!E}_r^{p,q}={{}_{\mathrm{w}}\!E}_r^{p,q}(X,L)$ when confusion might occur.\
By definition we have $${{}_{\mathrm{w}}\!E}_1^{-p,q}=\mathbb{H}^{-p+q}({\mathrm{gr}}_p^W{\mathcal{K}}_{\mathbb{Q}}(X,L)).$$ From the proof of Theorem \[mhs\] we get an isomorphism $${\mathrm{gr}}_p^W{\mathcal{K}}(X,L)\cong \bigoplus_{S\in{\mathscr{S}}_p(L)} (i_S)_*{\mathcal{K}}(S)[-p](-p)\otimes A_S(L)$$ and thus $${{}_{\mathrm{w}}\!E}_1^{-p,q}= \bigoplus_{S\in{\mathscr{S}}_p(L)} H^{-2p+q}(S)(-p)\otimes A_S(L).$$
The next proposition unravels the algebra structure on the $E_1$ term.
\[product\] Let $S$ and $S'$ be two strata of $L$ such that $S\cap S'\neq\varnothing$, $\mathrm{codim}(S)=p$, $\mathrm{codim}(S')=p$ and $\mathrm{codim}(S\cap S')=p+p'$. Then the component of the product $$\label{prodE}
{{}_{\mathrm{w}}\!E}_1^{-p,q}\otimes{{}_{\mathrm{w}}\!E}_1^{-p',q'}\rightarrow{{}_{\mathrm{w}}\!E}_1^{-(p+p'),q+q'}$$ indexed by $(S,S')$ is induced by the product morphism (\[eqprodglobal\]) $$A_S(L)\otimes A_{S'}(L)\rightarrow A_{S\cap S'}(L)$$ and by the morphism $$H^{-2p+q}(S)\otimes H^{-2p'+q'}(S')\rightarrow
H^{-2p+q}(S\cap S')\otimes H^{-2p'+q'}(S\cap S')\overset{\cup}{\rightarrow}
H^{-2(p+p')+(q+q')}(S\cap S')$$ multiplied by the sign $(-1)^{pq'}$. The above morphism is the composition of the restriction morphisms for the inclusion of $S\cap S'$ inside $S$ and $S'$, followed by the cup-product on $S\cap S'$.\
The other components of the product (\[prodE\]) are zero.
*First step*: we do the proof in the case where $L=D=\{D_1,\cdots,D_l\}$ is a normal crossing divisor. In this case we have $A_{D_I}(D)={\mathbb{Q}}e_I$ for every $I\subset\{1,\cdots,l\}$ such that $D_I\neq\varnothing$, and the product (\[eqprodglobal\]) is given by $e_I\otimes e_{I'}\mapsto e_I\wedge e_{I'}= {\mathrm{sgn}}(I,I')e_{I\cup I'}$ if $I\cap I'=\varnothing$ and $I\cup I'$ independent. We have $${{}_{\mathrm{w}}\!E}_1^{-p,q}=\bigoplus_{|I|=p}H^{-2p+q}(D_I)$$ and we want to prove that the product is given by $$H^{-2p+q}(D_I)\otimes H^{-2p'+q'}(D_{I'})\rightarrow
H^{-2p+q}(D_{I\cup I'})\otimes H^{-2p'+q'}(D_{I\cup I'})\overset{\cup}{\rightarrow}
H^{-2(p+p')+(q+q')}(D_{I\cup I'})$$ multiplied by the sign $(-1)^{pq'}{\mathrm{sgn}}(I,I')$.\
In the normal crossing case we have $\Omega^\bullet_{\langle X,D\rangle}=\Omega^\bullet_X(\log D)$ as defined in [@delignehodge2]. It is enough to do the proof for the cohomology with complex coefficients and hence leave the Tate twists aside. To work with explicit representatives we have to work with smooth forms. We sketch the argument, which can be found in a more complete form in [@voisin], Proposition 8.34.\
If $Y$ is any complex manifold, let ${\mathcal{A}}^\bullet_Y$ be the complex of sheaves of smooth forms (with complex coefficients) on $Y$, and let $A^\bullet(Y)=\Gamma(Y,{\mathcal{A}}^\bullet_Y)$ be the complex of global smooth forms on $Y$. The inclusion $\Omega^\bullet_Y\hookrightarrow {\mathcal{A}}^\bullet_Y$ is a quasi-isomorphism.\
We may define a complex of sheaves ${\mathcal{A}}^\bullet_X(\log D)$ on $X$ whose sections are smooth forms which are locally linear combinations over ${\mathbb{C}}$ of forms $\eta\wedge\omega_I$ with $\eta$ smooth. This complex is endowed with a weight filtration defined in the same way as for the holomorphic case. The inclusion $\Omega^\bullet_X(\log D)\hookrightarrow {\mathcal{A}}^\bullet_X(\log D)$ is then a filtered quasi-isomorphism. Besides, we have a commutative diagram of quasi-isomorphisms (the top horizontal arrow being an isomorphism): $$\xymatrix{
{\mathrm{gr}}_k^W\Omega^\bullet_X(\log D) \ar[r]^-{\cong} \ar[d]^-{\cong}& \bigoplus_{|I|=k} (i_{D_I})_*\Omega^{\bullet-k}_{D_I} \ar[d]^-{\cong} \\
{\mathrm{gr}}_k^W{\mathcal{A}}^\bullet_X(\log D) \ar[r]^-{\cong} & \bigoplus_{|I|=k} (i_{D_I})_*{\mathcal{A}}^{\bullet-k}_{D_I}\\
}$$ Since the complexes ${\mathcal{A}}^\bullet_X(\log D)$ and ${\mathcal{A}}^\bullet_{D_I}$ are made of fine sheaves, we can take them as resolutions to compute the spectral sequence and thus work with closed smooth global forms. Let us write $A^\bullet_X(\log D)$ for the complex of global sections of ${\mathcal{A}}^\bullet_X(\log D)$.\
For $i=1,\cdots,l$ let $\eta_i\in A^1_X(\log D)$ be a $(1,0)$-form such that locally around $D_i$ we have $$\eta_i=\dfrac{1}{2i\pi}\dfrac{df_i}{f_i}\hspace{3mm}\mathrm{mod}\,A^1(X)$$ where $f_i$ is a local equation for $D_i$.\
Let us fix a subset $I\subset\{1,\cdots,l\}$ of cardinality $p$ and let $\alpha$ be a closed smooth $(-2p+q)$-form on $D_I$. We fix $I'$ of cardinality $p'$ and $\alpha'$ in the same way. In order to compute the product of the classes of $\alpha$ and $\alpha'$ in the spectral sequence we have to find representatives for $\alpha$ and $\alpha'$ in $A^{-p+q}_X(\log D)$ and $A^{-p'+q'}_X(\log D)$ respectively. Let ${\widetilde{\alpha}}$ be a smooth closed $(-2p+q)$-form on $X$ such that ${\widetilde{\alpha}}_{|D_I}=\alpha$, and ${\widetilde{\alpha}}'$ the same for $\alpha'$, then representatives are given by $$\omega={\widetilde{\alpha}}\wedge\eta_I \hspace{5mm}\textnormal{and}\hspace{5mm}\omega'={\widetilde{\alpha}}'\wedge\eta_{I'}$$ respectively. We then look at the form $$\omega\wedge\omega'=({\widetilde{\alpha}}\wedge\eta_I)\wedge({\widetilde{\alpha}}'\wedge\eta_{I'})=(-1)^{pq'}{\widetilde{\alpha}}\wedge{\widetilde{\alpha}}'\wedge\eta_I\wedge\eta_{I'}.$$ If $I\cap I'\neq\varnothing$, then $\eta_I\wedge\eta_{I'}=0$ in ${\mathrm{gr}}_p^WA^{-(p+p')+(q+q')}_X(\log D)$. Else, $\eta_I\wedge\eta_{I'}={\mathrm{sgn}}(I,I')\eta_{I\cup I'}$ and the product we are looking for is $$(-1)^{pq'}{\mathrm{sgn}}(I,I')({\widetilde{\alpha}}\wedge{\widetilde{\alpha}}')_{|D_{I\cup I'}}=(-1)^{pq'}{\mathrm{sgn}}(I,I')\alpha_{|D_{I\cup I'}}\wedge\alpha'_{|D_{I\cup I'}}$$ hence the result.\
*Second step*: We deduce the general case from the functoriality of the Gysin spectral sequence. According to Lemma \[linearpresos\], the ${\mathbb{Q}}$-vector space $A_S(L)$ is spanned by monomials $e_I$ with $I$ independent. We fix monomials $e_I\in A_S(L)$ and $e_{I'}\in A_{S'}(L)$ with $I$ and $I'$ independent. Let us write $J=I\cup I'$ and let $L(J)=\bigcup_{j\in J}L_j$, which is a hypersurface arrangement in $X$. From the functoriality of the spectral sequence, there is a map of spectral sequences $${{}_{\mathrm{w}}\!E}_1^{-p,q}(X,L(J))\rightarrow {{}_{\mathrm{w}}\!E}_1^{-p,q}(X,L)$$ which is easily seen to be injective (this follows from the injectivity in the deletion-restriction short exact sequence). Thus the product of elements in $H^{-2p+q}(S)\otimes{\mathbb{Q}}e_I$ and $H^{-2p'+q'}(S')\otimes{\mathbb{Q}}e_{I'}$ can be read off ${{}_{\mathrm{w}}\!E}_1^{p,q}(X,L(J))$. If $L(J)$ is not a normal crossing divisor, then there is no independent subset of $J$ of cardinal $p+p'$, hence ${{}_{\mathrm{w}}\!E}_1^{p+p',q+q'}(X,L(J))=0$ and the product is zero. We are then reduced to the first step.
\[differential\] Let $S\subset S'$ be an inclusion of strata of $L$ with $\mathrm{codim}(S)=p$ and $\mathrm{codim}(S')=p-1$. Then the component of the differential $$d_1:{{}_{\mathrm{w}}\!E}_1^{-p,q}\rightarrow{{}_{\mathrm{w}}\!E}_1^{-p+1,q}$$ indexed by $S$ and $S'$ is induced by the natural morphism (\[eqderivglobal\]) $$A_S(L)\rightarrow A_{S'}(L)$$ and by the Gysin morphism $$H^{-2p+q}(S)(-p)\rightarrow H^{-2p+q+2}(S')(-p+1)$$ multiplied by the sign $(-1)^{q-1}$. The other components of $d_1$ are zero.
*First step*: If $L=D=\{D_1,\cdots,D_l\}$ is a normal crossing divisor, this is Proposition 8.34 in [@voisin] (see also [@peterssteenbrink], Proposition 4.7). Indeed in this case we have for every subset $I\subset\{1,\cdots,l\}$, $A_{D_I}(D)={\mathbb{Q}}e_I$ , and the morphisms $A_{D_I}\rightarrow A_{D_{I\setminus\{i\}}}$ send $e_I$ to ${\mathrm{sgn}}(\{i\},I\setminus\{i\})e_{I\setminus\{i\}}$.\
*Second step*: We deduce the general case from the functoriality of the Gysin spectral sequence. According to Lemma \[linearpresos\], the ${\mathbb{Q}}$-vector space $A_S(L)$ is spanned by monomials $e_I$ with $I$ independent. Let $e_I$ be such a monomial and let us write $L(I)=\bigcup_{i\in I} L_i$, which is a normal crossing divisor in $X$. From the functoriality of the spectral sequence, there is a map of spectral sequences $${{}_{\mathrm{w}}\!E}_1^{-p,q}(X,L(I))\rightarrow {{}_{\mathrm{w}}\!E}_1^{-p,q}(X,L)$$ which is easily seen to be injective (this follows from the injectivity in the deletion-restriction short exact sequence). Thus the differential of an element in $H^{-2p+q}(S)\otimes {\mathbb{Q}}e_I$ can be read off ${{}_{\mathrm{w}}\!E}_1^{p,q}(X,L(I))$. We are then reduced to the first step.
We now turn to the functoriality of the Gysin spectral sequence.
\[propfunctoriality\] Let $L$ (resp. $L'$) be a hypersurface arrangement in a smooth projective variety $X$ (resp. $X'$), and $\varphi:X\rightarrow X'$ a holomorphic map such that $\varphi^{-1}(L')\subset L$. Let $S$ and $S'$ be strata of codimension $p$ respectively of $L$ and $L'$ such that $\varphi(S)\subset S'$ and let us denote by $\varphi_{S,S'}:S\rightarrow S'$ the restriction of $\varphi$. Then the component of the morphism $${{}_{\mathrm{w}}\!E}_1^{-p,q}(\varphi):{{}_{\mathrm{w}}\!E}_1^{-p,q}(X',L')\rightarrow {{}_{\mathrm{w}}\!E}_1^{-p,q}(X,L)$$ indexed by $S$ and $S'$ is induced by the morphism (\[functorialityS\]) $$A_{S,S'}(\varphi):A_{S'}(L')\rightarrow A_S(L)$$ and the pull-back morphism $$\varphi_{S,S'}^*:H^{-2p+q}(S')\rightarrow H^{-2p+q}(S).$$ The other components of ${{}_{\mathrm{w}}\!E}_1^{p,q}(\varphi)$ are zero.
It is enough to do the proof over ${\mathbb{C}}$. We work with smooth forms and use the notations of the proof of Proposition \[product\]. Let us fix a stratum $S'$ of $L'$ of codimension $p$, and let $\alpha\in A^{-2p+q}(S')$ be a closed smooth form. We look at an element $[\alpha]\otimes e'_J\in {{}_{\mathrm{w}}\!E}_1^{-p,q}$, with $|J'|=p$ and $L'_J=S'$. It corresponds to the logarithmic form $$\omega={\widetilde{\alpha}}\wedge\eta'_J$$ where ${\widetilde{\alpha}}\in A^{-2p+q}(X)$ is such that ${\widetilde{\alpha}}_{|S'}=\alpha$. Then we look at $$\varphi^*(\omega)=\varphi^*({\widetilde{\alpha}})\wedge\varphi^*(\eta'_J)$$ Using the notations of §\[secfunctoriality\], we have a local expression $$\varphi^*(\eta'_J)=\sum_{I}m_{IJ}\eta_I$$ in ${\mathrm{gr}}_p^WA^{p}_W(\log L)$ and hence $$\varphi^*(\omega)=\sum_Im_{IJ}\varphi^*({\widetilde{\alpha}})\wedge\eta_I$$ in ${\mathrm{gr}}_p^WA^{q-p}_W(\log L)$. If $m_{ij}\neq 0$ then $\varphi(L_i)\subset L'_j$, hence we easily see that if $m_{IJ}\neq 0$ then $\varphi(L_I)\subset L'_J=S'$. Let us then fix a stratum $S$ such that $\varphi(S)\subset S'$. The corresponding component in ${{}_{\mathrm{w}}\!E}_1^{-p,q}(X,L)$ is then $$[\varphi^*({\widetilde{\alpha}})_{|S}]\otimes\sum_{I\,,\,L_I=S}m_{IJ} e_I=[\varphi_{S,S'}^*(\alpha)]\otimes A_{S,S'}(\varphi)(e'_J).$$
If $X$ is just a compact complex manifold, then we can also consider the Gysin spectral sequence converging to the cohomology of $X\setminus L$, and the formulas for the $E_1$ term remain valid. The only thing that we gain when assuming that $X$ is a projective variety is the degeneracy of this spectral sequence at the $E_2$ term, by Theorem \[mhc\].
The Gysin model and the main theorem {#defM}
------------------------------------
We restate the results of the previous paragraph. Let $X$ be a smooth projective variety and $L$ a hypersurface arrangement in $X$. Let us define $$M_q^n(X,L)=\bigoplus_{S\in{\mathscr{S}}_{q-n}(L)} H^{2n-q}(S)(n-q) \otimes A_S(L)$$ viewed as a Hodge structure of weight $q$.\
1. We define a *product* $$\label{prodM}
M_q^n(X,L)\otimes M_{q'}^{n'}(X,L)\rightarrow M_{q+q'}^{n+n'}(X,L).$$ Let $S$ and $S'$ be two strata of $L$ such that $S\cap S'\neq\varnothing$, $\mathrm{codim}(S)=q-n$, $\mathrm{codim}(S')=q'-n'$ and $\mathrm{codim}(S\cap S')=(q+q')-(n+n')$. Then the component of the product (\[prodM\]) indexed by $(S,S')$ is induced by the product morphism (\[eqprodglobal\]) $$A_S(L)\otimes A_{S'}(L)\rightarrow A_{S\cap S'}(L)$$ and by the morphism $$H^{2n-q}(S)\otimes H^{2n'-q'}(S')\rightarrow
H^{2n-q}(S\cap S')\otimes H^{2n'-q'}(S\cap S')\overset{\cup}{\rightarrow}
H^{2(n+n')-(q+q')}(S\cap S')$$ multiplied by the sign $(-1)^{(q-n)q'}$. The above morphism is the composition of the restriction morphisms for the inclusion of $S\cap S'$ inside $S$ and $S'$, followed by the cup-product on $S\cap S'$.\
The other components of the product (\[prodM\]) are zero.
2. We define a *differential* $$\label{diffM}
d:M_q^n(X,L)\rightarrow M_q^{n+1}(X,L).$$ Let $S\subset S'$ be an inclusion of strata of $L$ with $\mathrm{codim}(S)=q-n$ and $\mathrm{codim}(S')=q-(n+1)$. Then the component of the differential (\[diffM\]) indexed by $S$ and $S'$ is induced by the natural morphism (\[eqderivglobal\]) $$A_S(L)\rightarrow A_{S'}(L)$$ and by the Gysin morphism $$H^{2n-q}(S)(n-q)\rightarrow H^{2n-q+2}(S')(n-q+1)$$ multiplied by the sign $(-1)^q$. The other components of the differential (\[diffM\]) are zero.
3. Let $X'$ be another smooth projectiver variety, $L'$ be a hypersurface arrangement in $X'$ and $\varphi:X\rightarrow X'$ a holomorphic map such that $\varphi^{-1}(L')\subset L$. Then we define a map $$\label{functorialityM}
M^\bullet(\varphi):M^\bullet(X',L')\rightarrow M^\bullet(X,L).$$ Let $S$ and $S'$ be strata of codimension $q-n$ respectively of $L$ and $L'$ such that $\varphi(S)\subset S'$, and let $\varphi_{S,S'}:S\rightarrow S'$ be the restriction of $\varphi$. Then the component of $M^n_q(\varphi)$ indexed by $S$ and $S'$ is induced by the morphism (\[functorialityS\]) $$A_{S,S'}(\varphi):A_{S'}(L')\rightarrow A_S(L)$$ and by the pull-back morphism $$\varphi_{S,S'}^*:H^{2n-q}(S')\rightarrow H^{2n-q}(S).$$ The other components of $M^\bullet(\varphi)$ are zero.
In the next theorem, a **split mixed Hodge structure** is a mixed Hodge structure that is a direct sum of pure Hodge structures.\
Recall that a graded algebra $B=\oplus_{n\geq 0}B_n$ is said to be graded-commutative if for homogeneous elements $x$ and $x'$ in $B$ we have $xx'=(-1)^{|x||x'|}x'x$.
\[maintheorem\] Let $X$ be a smooth projective variety over ${\mathbb{C}}$ and $L$ a hypersurface arrangement in $X$.
1. The direct sum $M^\bullet(X,L)=\bigoplus_{q}M^\bullet_q(X,L)$ is a graded-commutative differential graded algebra in the category of split mixed Hodge structures. It is functorial with respect to $(X,L)$, using (\[functorialityM\]).
2. We have isomorphisms of algebras in the category of split mixed Hodge structures: $${\mathrm{gr}}^W H^{\bullet}(X\setminus L)\cong H^\bullet(M^\bullet(X,L)) .$$ They are functorial with respect to $(X,L)$.
Since the differential is given by Gysin morphisms, we call $M^\bullet(X,L)$ the **Gysin model** of the pair $(X,L)$.
1. Note that we have multiplied the differential by $-1$ for more comfort; this gives an isomorphic differential graded algebra. The assertion is a consequence of the previous paragraph (Propositions \[product\], \[differential\] and \[propfunctoriality\]).
2. The isomorphism is just, after the change of variables $n=-p+q$, the fact that the spectral sequence ${{}_{\mathrm{w}}\!E}_r^{p,q}$ degenerates at $E_2$ and converges to the cohomology of $X\setminus L$: $$H^{p}({{}_{\mathrm{w}}\!E}_1^{-\bullet,q})\cong{\mathrm{gr}}^W_q H^{-p+q}(X\setminus L).$$
\[remGysinquotient\] Using the presentation of the groups $A_S(L)$ given in Lemma \[linearpresos\], we may give a presentation of the Gysin model that is more suitable in certain situations. For $S$ a stratum of $L$ and $I\subset\{1,\cdots,l\}$ an independent subset such that $L_I=S$, we have a monomial $e_I\in A_S(L)$. If we identify $H^{2n-q}(S)\otimes {\mathbb{Q}}e_I = H^{2n-q}(L_I)$, then we see that $M_q^n(X,L)$ is the quotient of $$\bigoplus_{\substack{|I|=q-n\\I\textnormal{ indep.}}}H^{2n-q}(L_I)(n-q)$$ by the sub-vector space spanned by the images of the morphisms $$H^{2n-q}(L_{I'})\rightarrow\bigoplus_{\substack{i\in I'\\ I'\setminus\{i\}\textnormal{ indep.}}} H^{2n-q}(L_{I'\setminus\{i\}})$$ for $I'$ dependent. The above morphism is ${\mathrm{sgn}}(\{i\},I'\setminus\{i\})$ times the identity on the component indexed by $i$ (if $I'$ is dependent and $I'\setminus\{i\}$ is independent, then $L_{I'\setminus\{i\}}=L_{I'}$ for dimension reasons).
Wonderful compactifications and the Gysin model
===============================================
Hypersurface arrangements and wonderful compactifications {#secblowups}
---------------------------------------------------------
Let $L=\{L_1,\cdots,L_l\}$ be a central hyperplane arrangement in ${\mathbb{C}}^n$, $S$ a stratum of $L$. We say that $S$ is a **good stratum** if there exists a direct sum decomposition ${\mathbb{C}}^n=S\oplus U$ such that the hyperplanes $L_i$ that do not contain $S$ contain $U$. We say that $S$ is a **very good stratum** if furthermore the hyperplanes $L_i$ that do not contain $S$ are independent.
In other words, $S$ is a good stratum if there exists coordinates $(z_1,\cdots,z_n)$ on ${\mathbb{C}}^n$ such that $S=\{z_1=\cdots=z_r=0\}$ for some $r$, and for each $i=1,\cdots,l$, $L_i$ is either of the type $\{a_1z_1+\cdots+a_rz_r=0\}$ or of the type $\{a_{r+1}z_{r+1}+\cdots+a_nz_n=0\}$. It is a very good stratum if furthermore we can choose the coordinates so that the hyperplanes of the second type are among $\{z_{r+1}=0\},\cdots,\{z_n=0\}$.
1. The stratum $\{0\}$ is very good.
2. In ${\mathbb{C}}^3$, let $L_1=\{x=0\}$, $L_2=\{y=0\}$, $L_3=\{z=0\}$, $L_4=\{x=y\}$. Then the stratum $\{x=y=0\}$ is very good. The stratum $\{z=0\}$ is good, but not very good. The stratum $\{x=z=0\}$ is not good.
Let $L=\{L_1,\cdots,L_l\}$ be a hypersurface arrangement in a complex manifold $X$, $S$ a stratum of $L$. We say that $S$ is a **good stratum** (resp. a **very good stratum**) if in every local chart where the $L_i$’s are hyperplanes, it is a good stratum (resp. a very good stratum) in the sense of the above definition. A stratum of dimension $0$ (a point) is always very good. In the case of a normal crossing divisor, all non-empty strata are very good.
\[blowuphyparr\] Let $L=\{L_1,\cdots,L_l\}$ be a hypersurface arrangement in a complex manifold $X$, $S$ a good stratum of $L$, and $$\pi:{\widetilde{X}}\rightarrow X$$ the blow-up of $X$ along $S$. Let $E=\pi^{-1}(S)$ be the exceptional divisor, and for all $i$, let ${\widetilde{L}}_i$ be the strict transform of $L_i$. Then ${\widetilde{L}}=\{E,{\widetilde{L}}_1,\cdots,{\widetilde{L}}_l\}$ is a hypersurface arrangement in ${\widetilde{X}}$.
It is enough to do the proof for $X=\Delta^n$ and the $L_i$’s hyperplanes. We choose coordinates $(z_1,\cdots,z_n)$ such that $S=\{z_1=\cdots=z_r=0\}$ and for each $i=1,\cdots,l$, $L_i$ is either of the type $\{a_1z_1+\cdots+a_rz_r=0\}$ or $\{a_{r+1}z_{r+1}+\cdots+a_nz_n=0\}$.\
We have $r$ natural local charts ${\widetilde{X}}_k\cong\Delta^n$ on ${\widetilde{X}}$, $k=1,\cdots,r$. On the chart ${\widetilde{X}}_k$, the blow-up morphism is given by $$\pi(z_1,\cdots,z_n)=(z_1z_k,\cdots,z_{k-1}z_k,z_k,z_{k+1}z_k,\cdots,z_rz_k,z_{r+1},\cdots,z_n)$$ In this chart, $E$ is defined by the equation $z_k=0$. The strict transform of a hyperplane of the type $\{a_1z_1+\cdots+a_rz_r=0\}$ is given by the equation $a_1z_1+\cdots+a_{k-1}z_{k-1}+a_k+a_{k+1}z_{k+1}+\cdots+a_rz_r=0$. The strict transform of a hyperplane of the type $\{a_{r+1}z_{r+1}+\cdots+a_nz_n=0\}$ is defined by the same equation.\
To sum up, in the chart ${\widetilde{X}}_k$, all the hypersurfaces of ${\widetilde{L}}$ are given by affine equations. Up to some translations, we can then find smaller charts where all the equations are linear. This completes the proof.
With the notations of the above lemma, we will simply write that $$\pi:({\widetilde{X}},{\widetilde{L}})\rightarrow (X,L)$$ is the blow-up of the pair $(X,L)$ along the good stratum $S$. We stress the fact that ${\widetilde{L}}$ is the hypersurface arrangement consisting of the exceptional divisor $E$ and all the proper transforms ${\widetilde{L}}_i$ of the hypersurfaces $L_i$.\
The blow-ups along good strata (and in fact, of very good strata) are enough to resolve the singularities of a hypersurface arrangement, as the following theorem shows.
\[seqblowups\] Let $L$ be a hypersurface arrangement in a complex manifold $X$. There exists a sequence $$({\widetilde{X}},{\widetilde{L}})= (X^{(N)},L^{(N)}) \overset{\pi_N}{\longrightarrow} (X^{(N-1)},L^{(N-1)}) \overset{\pi_{N-1}}{\longrightarrow} \cdots \overset{\pi_1}{\longrightarrow} (X^{(0)},L^{(0)})=(X,L)$$ where
1. for all $k$, $X^{(k)}$ is a complex manifold and $L^{(k)}$ a hypersurface arrangement in $X^{(k)}$
2. for all $k$, $\pi_k:(X^{(k)},L^{(k)})\rightarrow (X^{(k-1)},L^{(k-1)})$ is the blow-up of $(X^{(k-1)},L^{(k-1)})$ along a very good stratum of $L^{(k-1)}$
3. ${\widetilde{L}}$ is a normal crossing divisor in ${\widetilde{X}}$.
We use the blow-up procedure described by Y. Hu in [@hu]. Such a procedure has become standard, so we do not give too many details. One may notice that Hu works with algebraic varieties, even though his results are valid in the framework of complex manifolds. The reader may assume that $X$ is a smooth algebraic variety over ${\mathbb{C}}$, and the $L_i$’s are algebraic. The sequence of blow-ups is the following:\
$(0)$ We first blow up all the strata of dimension $0$ (the points).\
$(1)$ We then blow up all the strict transforms of the strata of dimension $1$ (the lines).\
$\cdots$\
$(d)$ We blow up all the strict transforms of the strata of dimension $d$.\
$\cdots$\
It has to be noted that at step $(d)$, all the centers are pairwise disjoint, so that we may blow them up in any order. We want to prove that all the centers that appear in this blow-up procedure are very good strata.\
At step $(0)$, it is clear since the points are always very good strata. Suppose now that we are at step $(d)$ with $d\geq 1$. For every stratum $C$ of $L$ of dimension $\leq d-1$, we have added an exceptional divisor ${\widetilde{C}}$. Let $S$ be a stratum of $L$ of dimension $d$, we want to prove that its strict transform ${\widetilde{S}}$ is a good stratum.\
For $i=1,\cdots,l$, if ${\widetilde{S}}\cap L_i \neq \varnothing$, then it is easy to see that $S\subset L_i$, so that ${\widetilde{S}}\subset{\widetilde{L}}_i$.\
For $C$ a stratum of $L$ of dimension $\leq d-1$, if ${\widetilde{S}}\cap{\widetilde{C}}\neq\varnothing$, then $C\subset S$, hence we only have to consider the strict transforms of the strata of $L$ contained in $S$. According to [@hu], Theorem 1.1, at each point of $S$ we only have to consider those fitting into a certain flag $C_1\subset C_2\subset\cdots\subset C_k\subset S$. The result then follows from a local computation on the iterated blow-up of a flag of subspaces in ${\mathbb{C}}^n$.
In the above proof, we have described a particular blow-up procedure. As has been known since the works of De Concini-Procesi ([@deconciniprocesi]) and Fulton-MacPherson ([@fultonmcpherson]), recently generalized by L. Li ([@li]), there are lots of other blow-up strategies that can be applied. The beautiful combinatorial notion of a building set is very useful to understand the multiplicity of such strategies. Those sequences of blow-ups are sometimes referred to as “wonderful compactifications”.
Functoriality of the Gysin model with respect to blow-ups
---------------------------------------------------------
Let us consider a sequence of blow-ups along (very) good strata as in Theorem \[seqblowups\]: $$({\widetilde{X}},{\widetilde{L}})= (X^{(N)},L^{(N)}) \overset{\pi_N}{\longrightarrow} (X^{(N-1)},L^{(N-1)}) \overset{\pi_{N-1}}{\longrightarrow} \cdots \overset{\pi_1}{\longrightarrow} (X^{(0)},L^{(0)})=(X,L).$$ Then by the functoriality of the Gysin model we get a sequence of morphisms of dga’s (in the category of split mixed Hodge structures): $$M^\bullet(X,L)=M^\bullet(X^{(0)},L^{(0)}) \overset{\substack{M^\bullet(\pi_1)\\\sim}}{\longrightarrow} \cdots \overset{\substack{M^\bullet(\pi_{N-1})\\\sim}}{\longrightarrow} M^\bullet(X^{(N-1)},L^{(N-1)}) \overset{\substack{M^\bullet(\pi_{N})\\\sim}}{\longrightarrow}M ^\bullet(X^{(N)},L^{(N)})=M^\bullet({\widetilde{X}},{\widetilde{L}}).$$ For each $k$, $M^\bullet(\pi_k)$ is a quasi-isomorphism since $\pi_k$ induces an isomorphism $X^{(k)}\setminus L^{(k)}{\overset{\simeq}{\rightarrow}}X^{(k-1)}\setminus L^{(k-1)}$. Thus we get a natural quasi-isomorphism between the Gysin model of $(X,L)$ and that of $({\widetilde{X}},{\widetilde{L}})$.\
In the following theorem, we investigate the case of a single blow-up and give explicit formulas. We use the presentation of the Gysin model given in Remark \[remGysinquotient\].
\[formulaMpi\] Let $X$ be a smooth projective variety over ${\mathbb{C}}$ and $L$ a hypersurface arrangement in $X$. Let $S$ be a good stratum of $L$ and $$\pi:({\widetilde{X}},{\widetilde{L}})\rightarrow(X,L)$$ the blow-up of $(X,L)$ along $S$. Let $$M^\bullet(\pi):M^\bullet(X,L)\rightarrow M^\bullet({\widetilde{X}},{\widetilde{L}})$$ be the morphism induced by $\pi$ on the Gysin models. Then
1. $M^\bullet(\pi)$ is a quasi-isomorphism.
2. the components of $M^n_q(\pi)$ are given, for $I$ independent of cardinality $q-n$, by
1. the pull-back morphism $H^{2n-q}(L_I)\overset{\pi^*}{\rightarrow} H^{2n-q}({\widetilde{L}}_I)$.
2. for all $i\in I$ such that $S\subset L_i$, the morphism $H^{2n-q}(L_I)\rightarrow H^{2n-q}(E\cap{\widetilde{L}}_{I\setminus\{i\}})$ which is the pull-back morphism corresponding to $E\cap{\widetilde{L}}_{I\setminus\{i\}}\stackrel{\pi}{\rightarrow} S\cap L_{I\setminus\{i\}}=S\cap L_I\hookrightarrow L_I$, multiplied by the sign ${\mathrm{sgn}}(\{i\},I\setminus\{i\})$.
<!-- -->
1. This is obvious by Theorem \[maintheorem\], since $\pi$ induces an isomorphism ${\widetilde{X}}\setminus{\widetilde{L}}{\overset{\simeq}{\rightarrow}}X\setminus L$.
2. It is a consequence of the general formula for functoriality given in §\[defM\]. Using the notation $E={\widetilde{L}}_0$ and Lemma \[lemlocalfunctoriality\], a local computation shows that we have the following formula for $A_\bullet(\pi):A_\bullet(L)\rightarrow A_\bullet({\widetilde{L}})$. $$A_1(\pi)(e_i)=\begin{cases} e_i & \textnormal{if }L_i\textnormal{ does not contain } S\\ e_0+e_i & \textnormal{if }L_i\textnormal{ contains } S\end{cases}$$
Thus we get $$A_\bullet(\pi)(e_I)=e_I+\sum_{\substack{i\in I\\S\subset L_i}}{\mathrm{sgn}}(\{i\},I\setminus\{i\})e_0\wedge e_{I\setminus\{i\}}$$ and the claim follows.
Configuration spaces of points on curves {#confcurves}
========================================
Configuration spaces associated to graphs
-----------------------------------------
Let $Y$ be a complex manifold, and $n\geq 1$ an integer. The configuration space of $n$ ordered points on $Y$ is by definition $$C(Y,n)=\{(y_1,\cdots,y_n)\in Y^n\,\,|\,\,y_i\neq y_j\ \,\, \textnormal{for}\,\, i\neq j\}=Y^n\setminus\bigcup_{i<j}\Delta_{i,j}$$ where $\Delta_{i,j}=\{y_i=y_j\}$ is a diagonal.\
As a natural generalization, we can consider the space obtained from $Y^n$ after deleting only certain diagonals $\Delta_{i,j}$. We use graphs to parametrize such spaces.\
Let $\Gamma$ be a finite unoriented graph with no multiple edges and no self-loops, with $V$ its set of vertices and $E$ its set of edges. Let $Y^V$ be the cartesian power of $Y$ indexed by $V$, with coordinates $y_v$. For $v\in V$, we have a projection $$p_v:Y^V\rightarrow Y.$$ Every edge $e\in E$ with endpoints $v$ and $v'$ defines a diagonal $$\Delta_e=\{y_v=y_{v'}\}\subset Y^V$$ which is the locus where the coordinates corresponding to the two endpoints of $e$ are equal. We define $$\Delta_\Gamma=\bigcup_{e\in E}\Delta_e$$ and then the configuration space of points on $Y$ associated to $\Gamma$: $$C(Y,\Gamma)=Y^V\setminus\Delta_\Gamma.$$ In the case where $\Gamma$ is the complete graph on $n$ vertices, we recover the configuration space $C(Y,n)$.\
In the rest of §\[confcurves\], we focus on the case where $Y$ is a compact Riemann surface, i.e. a smooth projective curve.
A model for the cohomology
--------------------------
In [@kriz] and [@totaroconf], I. Kriz and B. Totaro independently found a model for the cohomology of $C(Y,n)$. Their result has been recently generalized to $C(Y,\Gamma)$ by S. Bloch in [@blochtreeterated] (even though Bloch’s framework is slightly more general, with external edges in $\Gamma$ labeled by points of $Y$). We recall the definition of this model. Here $Y$ has dimension $1$, but the general definition is similar.\
If $B=\oplus_{n\geq 0}B_n$ is a graded-commutative graded algebra and $\{x_\alpha\}$ are indeterminates with prescribed degrees $\{d_\alpha\}$, then there is a well-defined notion of graded-commutative algebra generated by the $x_\alpha$’s over $B$. This is a graded-commutative graded algebra which is the quotient of $B[\{x_\alpha\}]$ by the relations $bx_\alpha=(-1)^{|b|d_\alpha}x_\alpha b$ for $b$ homogeneous, and $x_\beta x_\alpha=(-1)^{d_\alpha d_\beta}x_\alpha x_\beta$ for all $\alpha$ and $\beta$. For example, if $B$ is a field concentrated in degree $0$ then we recover the exterior algebra generated by the $x_\alpha$’s. We use the wedge notation $x_\alpha\wedge x_\beta$ to remember the graded-commutativity property.\
Let us define, following [@blochtreeterated], a graded-commutative dga $N^\bullet(Y,\Gamma)$ in the following way. It is generated (as a graded-commutative algebra) by the cohomology $H^\bullet(Y^V)$ and elements $G_e$ in degree $1$ for every edge $e\in E$, modulo the relations:
1. $p_{v}^*(c)G_e=p_{v'}^*(c)G_e$ for every class $c\in H^\bullet(Y)$, where $v$ and $v'$ are the endpoints of $e$ in $\Gamma$.
2. $\sum_{i=1}^r(-1)^{i-1}G_{e_1}\wedge\cdots\wedge\widehat{G_{e_i}}\wedge\cdots\wedge G_{e_r}=0$ if $\{e_1,\cdots,e_r\}\subset E$ contains a loop.
In order to define a differential on $N^\bullet(Y,\Gamma)$ we need a lemma, which generalizes the above relation $(\mathrm{R1})$.
\[kunneth\] For a finite set $T$ and a map $u:T\rightarrow V$, we write $P_u:Y^V\rightarrow Y^{T}$ for the map induced by $u$, whose $t$-th component is $p_{u(t)}$ for $t\in T$. Let $I\subset E$ and let $u,u':T\rightarrow V$ such that for all $t\in T$, the vertices $u(t)$ and $u'(t)$ are linked by a path in $\Gamma$ made of edges in $I$. Then for all $c\in H^\bullet(Y^{T})$ we have the relation $$P_u^*(c)G_I=P_{u'}^*(c)G_I$$ in $N^\bullet(Y,\Gamma)$.
In the case where $I=\{e\}$ is made of a single edge, $T=\{1\}$, $u(1)$ and $u'(1)$ are the endpoints of $e$, then the above Lemma is nothing but the relation $(\mathrm{R1})$ in the definition of $N^\bullet(Y,\Gamma)$.
We write $T=\{1,\cdots,k\}$. Thanks to the Künneth formula we may write the cohomology class $c$ as $$c=\sum c_{i_1}\cdots c_{i_k}$$ with $c_{i_r}\in H^\bullet(Y)$, so that $$P_u^*(c)=\sum_{r=1}^kp_{u(1)}^*(c_{i_1})\cdots p_{u(k)}^*(c_{i_k})$$ and the same for $P_{u'}^*(c)$. Then it is enough to show that for $v$ and $v'$ two vertices that are linked by a path in $\Gamma$ made of edges in $I$, and for any $c\in H^\bullet(Y)$, we have $$p_v^*(c)G_I=p_{v'}^*(c)G_I.$$ This follows from an induction on the length of the path, the length one case being implied by the relation $(\mathrm{R1})$ in the definition of $N^\bullet(Y,\Gamma)$.
Defining a differential on $N^\bullet(Y,\Gamma)$ is the same as defining a differential on $H^\bullet(Y^V)$ and defining the images of the elements $G_e$. We define the differential to be zero on $H^\bullet(Y^V)$ and given on the elements $G_e$ by the formula $$d(G_e)=[\Delta_e]\in H^2(Y^V)$$
The differential $d$ is well-defined and makes $N^\bullet(Y,\Gamma)$ into a graded-commutative dga.
The fact that $d^2=0$ follows from $d^2(G_e)=d([\Delta_e])=0$ and the Leibniz rule.\
The compatibility of $d$ with the relation $(\mathrm{R1})$ follows from the relation $$p_v^*(c)[\Delta_e]=p_{v'}^*(c)[\Delta_e]$$ if $v$ and $v'$ are the endpoints of $e$, for all $c\in H^\bullet(Y)$. Let now $R$ be the expression in the relation $(\mathrm{R2})$. As in the proof of Lemma \[linearpresos\], it is enough to show that $d(R)=0$ when $\{e_1,\cdots,e_r\}$ is a simple loop. We compute $$\begin{aligned}
d(R)&= & \sum_{i=1}^r\sum_{j<i}(-1)^{i+j}[\Delta_{e_j}]G_{e_1}\wedge\cdots\wedge\widehat{G_{e_j}}\wedge\cdots\wedge\widehat{G_{e_i}}\wedge\cdots\wedge G_{e_r}\\
&+&\sum_{i=1}^r\sum_{i<j}(-1)^{i+j-1}[\Delta_{e_j}]G_{e_1}\wedge\cdots\wedge\widehat{G_{e_i}}\wedge\cdots\wedge\widehat{G_{e_j}}\wedge\cdots\wedge G_{e_r}
\end{aligned}$$ and it is enough to show that for $i<j$ we have $$[\Delta_{e_i}]G_{e_1}\wedge\cdots\wedge\widehat{G_{e_i}}\wedge\cdots\wedge\widehat{G_{e_j}}\wedge\cdots\wedge G_{e_r}=
[\Delta_{e_j}]G_{e_1}\wedge\cdots\wedge\widehat{G_{e_i}}\wedge\cdots\wedge\widehat{G_{e_j}}\wedge\cdots\wedge G_{e_r}.$$ Let $v_i$, $v_i'$ be the endpoints of $e_i$ and $v_j$, $v_j'$ the endpoints of $e_j$. Since $\{e_1,\cdots,e_r\}$ is a simple loop, we can assume that there is a path in $\Gamma$ between $v_i$ and $v_j$ made of edges in $\{e_1,\cdots,\widehat{e_i},\cdots,\widehat{e_j},\cdots,e_r\}$, and the same thing for $v_i'$ and $v_j'$. Let $P_i:Y^V\rightarrow Y^2$ be the projection $(p_{v_i},p_{v_i'})$ and $P_j:Y^V\rightarrow Y^2$ be the projection $(p_{v_j},p_{v_j'})$. We then have $[\Delta_{e_i}]=P_{i}^*([\Delta])$ and $[\Delta_{e_j}]=P_{j}^*([\Delta])$. The desired equality thus follows from Lemma \[kunneth\] with $I=\{e_1,\cdots,\widehat{e_i},\cdots,\widehat{e_j},\cdots,e_r\}$.
The isomorphism with the Gysin model
------------------------------------
By choosing charts on $Y$, one easily sees that $L=\Delta_\Gamma$ is a hypersurface arrangement in $X=Y^V$. Thus theorem \[maintheorem\] can be applied to the pair $(Y^V,\Delta_\Gamma)$ and gives a model for the cohomology of $C(Y,\Gamma)=Y^V\setminus\Delta_\Gamma$. We fix an linear order on the set $E$ of edges of $\Gamma$, hence on the irreducible components $\Delta_e$ of $\Delta_\Gamma$. This allows us to consider the Gysin model $M^\bullet(Y^V,\Delta_\Gamma)$, with its presentation given by Remark \[remGysinquotient\]. Thus $M_q^n(Y^V,\Delta_\Gamma)$ is a quotient of $$\bigoplus_{\substack{I\subset E\\ |I|=q-n\\I\textnormal{ indep.}}} H^{2n-q}(\Delta_I)(n-q).$$ We note that a subset $I\subset E$ is dependent if and only if it contains a loop, and is a circuit if and only if it is a simple loop.\
We define a morphism of dga’s $$\alpha:N^\bullet(Y,\Gamma)\rightarrow M^\bullet(Y^V,\Delta_\Gamma)$$ in the following way.\
First we note that for all $n$ we have $M^n_n(Y^V,\Delta_\Gamma)=H^n(Y^V)$, and we easily see that the resulting (injective) map $H^\bullet(Y^V)\rightarrow M^\bullet(Y^V,\Delta_\Gamma)$ is a map of graded algebras. Then we define $\alpha(G_e)$ to be a generator $g_e$ of $H^0(\Delta_e)(-1)\subset M^1_2(Y^V,\Delta_\Gamma)$.
The morphism $\alpha$ is well-defined and compatible with the differentials. It is thus a map of dga’s.
First we show that $\alpha$ respects the relations $(\mathrm{R1})$ and $(\mathrm{R2})$. For the relation $(\mathrm{R1})$ we see that by definition $$\alpha(p_v^*(c)G_e)=p_v^*(c)g_e=p_v^*(c)_{|\Delta_e}\in H^\bullet(\Delta_e).$$ This equals $i_e^*(p_v^*(c))=(p_v\circ i_e)^*(c)$ where $i_e:\Delta_e\hookrightarrow Y^V$ is the inclusion of $\Delta_e$. The relation then follows from the equality $p_v\circ i_e=p_{v'}\circ i_e$.\
For the relation $(\mathrm{R2})$ we can assume that we have $e_1<\cdots<e_r$. Then if $R$ is the expression in the relation $(\mathrm{R2})$ we have $$\alpha(R)=\sum_{i=1}^r(-1)^{i-1}g_{e_1}\cdots\widehat{g_{e_i}}\cdots g_{e_r}$$ and $g_{e_1}\cdots\widehat{g_{e_i}}\cdots g_{e_r}$ is a generator of $H^0(\Delta_{e_1}\cap\cdots\cap\widehat{\Delta_{e_i}}\cap\cdots\cap\Delta_{e_r})(-r+1)$. Since $\{\Delta_{e_1},\cdots,\Delta_{e_r}\}$ is dependent, $\alpha(R)$ is thus killed by the quotient that defines $M^\bullet(Y^V,\Delta_\Gamma)$.\
We then show that $\alpha$ is compatible with the differentials. By definition, the differential is zero on $H^\bullet(Y^V)\subset M^\bullet(Y^V,\Delta_\Gamma)$. Furthermore, $d\alpha(G_e)=d(g_e)$ is, by definition of the Gysin morphism, the class of $\Delta_e$ in $H^2(Y^V)$. This completes the proof.
\[compbloch\] The morphism $\alpha:N^\bullet(Y,\Gamma)\rightarrow M^\bullet(Y^V,\Delta_\Gamma)$ is an isomorphism of differential graded algebras.
We sketch the proof and leave the details to the reader. We define the inverse morphism $\beta$ in the following way. Let $I\subset E$ be an independent set of edges of $\Gamma$ of cardinality $|I|=q-n$, let $i_I:\Delta_I\hookrightarrow Y^V $ be the inclusion of the corresponding stratum. Let $f_I:Y^V\rightarrow \Delta_I$ be any natural splitting of $i_I$ defined out of projections $p_v$’s. Then we define the component of $\beta$: $$\beta^n_q:H^{2n-q}(\Delta_I)\rightarrow H^{2n-q}(Y^V)G_I$$ to be the pull-back $f_I^*$. The degrees match since $H^{2n-q}(Y^V)G_I$ is in degree $2n-q+|I|=n$. It remains to prove that $\beta$ passes to the quotient that defines $M^\bullet(Y^V,\Delta_\Gamma)$, and defines an inverse to $\alpha$.
It is striking that Kriz and Totaro’s model works for configuration spaces of points on any smooth projective variety $Y$, where the diagonals can have any codimension. It is then tempting to ask for a generalization of the Gysin model to the cohomology of $X\setminus L$ where $L\subset X$ locally looks like a union of sub-vector spaces of any codimension inside ${\mathbb{C}}^n$. In [@totaroconf], B. Totaro suggests a particular case of the previous question, focusing on vector spaces $V_i$ of a fixed codimension $c$ such that all intersections $V_{i_1}\cap\cdots\cap V_{i_r}$ have codimension a multiple of $c$ (this paper handles the case $c=1$).
|
---
abstract: 'Isospin relaxation times characterizing isospin transport processes between the projectile and the target with different $N/Z$ ratios and that between the neck and the spectator with different isospin asymmetries and densities in intermediate-energy heavy-ion collisions are studied within an isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model using the lattice Hamiltonian approach. The respective roles and time scales of the isospin diffusion and drift as the major mechanisms of isospin transport in intermediate-energy heavy-ion collisions are discussed. Effects of nuclear symmetry energy and neutron-proton effective mass splitting on the isospin relaxation times are examined.'
author:
- 'Han-Sheng Wang'
- 'Jun Xu[^1]'
- 'Bao-An Li'
- 'Wen-Qing Shen'
title: 'Revisiting the isospin relaxation time in intermediate-energy heavy-ion collisions'
---
Introduction
============
Understanding properties of isovector nuclear interactions as well as the related nuclear symmetry energy and the neutron-proton effective mass splitting in neutron-rich matter is a major thrust of nuclear science. In particular, the density dependence of nuclear symmetry energy $E_{sym}(\rho)$ has important ramifications in not only nuclear structures and nuclear reactions but also several areas of astrophysics and cosmology. Despite of the great efforts made over the last few decades, $E_{sym}(\rho)$ at both subsaturation and suprasaturation densities are still uncertain, see, e.g., Refs. [@ireview98; @Bar05; @Ste05; @Li08; @Tra12; @EPJA; @Hor14; @Bal16; @Oer17; @Li17; @Lyn18] for reviews. The nucleon effective mass is a fundamental quantity characterizing the nucleon’s propagation in nuclear medium [@Jeuk76; @Jam89; @Sjo76], and it is related to the momentum/energy dependence of the nucleon potential in the non-relativistic approach. In recent years, whether the neutron-proton effective mass splitting $m_{n-p}^{\ast}(m_{n-p}^{\ast}\equiv m_{n}^{\ast}-m_{p}^{\ast})$ is negative, zero, or positive in neutron-rich matter becomes a hotly debated topic. It affects the isospin dynamics in nuclear reactions [@Riz05; @Gio10; @Fen11; @Zha14; @Xie14; @kong15; @Cou16], thermodynamic and transport properties of neutron-rich matter [@Ou11; @Beh11; @xu15; @Zha10; @xu15b], and isovector giant dipole resonances in neutron-rich nuclei [@zhangzhen16; @kong17]. Moreover, based on the Hugenholtz-Van Hove theorem, the isospin splitting of the nucleon effective mass is closely related to the nuclear symmetry energy [@XuC10; @BAL13]. For a very recent review on the nucleon effective mass in neutron-rich medium, we refer the reader to Ref. [@Li18].
Heavy-ion reactions at intermediate energies provide a means to probe the nuclear symmetry energy and the neutron-proton effective mass splitting in neutron-rich matter. In particular, both the degree and time scale for isospin transport in heavy-ion reactions are known to be affected by nuclear isovector interactions [@Li95; @Shi03; @Riz08]. There are two driving mechanisms for isospin transport, i.e., the isospin diffusion and the isospin drift. The isospin diffusion is the dominating effect when the projectile and the target nuclei have different $N/Z$ ratios [@Li98]. The degree of isospin mixing as a result of isospin transport between the two nuclei is quantitatively described by the so-called isospin transport ratio. The latter was proposed to be a useful probe of the nuclear symmetry energy [@Tsa04; @Tsa09]. It was later realized that the isospin transport ratio is affected by the momentum dependence of the nucleon potential [@Che05], the in-medium nucleon-nucleon scattering cross section [@Li05], and the neutron-proton effective mass splitting [@Zha14]. On the other hand, since different density regions can be reached in intermediate-energy heavy-ion collisions, they generally have different isospin asymmetries due to the isospin fractionation effect depending on the density dependence of the nuclear symmetry energy, i.e., the low-density neck region in non-central heavy-ion collisions is more neutron-rich compared to the normal-density spectator. The isospin transport between the neck and the spectator is driven by both the isospin diffusion and the isospin drift. While various observables have been proposed to measure the degree of isospin transport in heavy-ion reactions, it has been rather challenging to obtain experimental information about the time scale of isospin transport. Very interestingly, the isospin relaxation time for the neck and the spectator in the projectile-like fragment (PLF) or target-like fragment (TLF) to reach isospin equilibrium was recently extracted by a group at Texas A&M University (TAMU) [@Je17]. It is thus physically useful and timely to know how sensitive the isospin relaxation time in PLF or TLF is to $E_{sym}(\rho)$ and/or $m_{n-p}^{\ast}$, and whether the new data is precise enough for constraining the properties of isovector nuclear interactions within the model considered. For these purposes, we carry out a study within an isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model using an improved isospin- and momentum-dependent interaction (ImMDI) [@xu15]. In order to improve the stability for the momentum-dependent mean-field potential at lower beam energies, the lattice Hamiltonian (LH) method [@Lenk89] was employed for calculating the mean-field potential. Appreciable effects of $E_{sym}(\rho)$ and $m_{n-p}^{\ast}$ on the isospin relaxation time are observed. However, they are much smaller than the current uncertainty range of the isospin relaxation time extracted from the experiment by the TAMU group.
The rest part of the manuscript is organized as follows. Section \[framework\] briefly introduces the ImMDI interaction as well as the LH method in calculating the mean-field potential for the IBUU transport simulation. We discuss the isospin transport process between projectile and target in central $^{40}\textrm{Ca}+^{124}\textrm{Sn}$ collisions, and study the isospin transport process between neck and spectator within the PLF in non-central $^{70}$Zn+$^{70}$Zn collisions in Sec. \[results\]. A summary is made in Sec. \[summary\].
Theoretical framework {#framework}
=====================
An improved isospin- and momentum-dependent interaction
-------------------------------------------------------
The potential energy density of the ImMDI interaction can be obtained from an effective two-body interaction with a zero-range density-dependent term and a finite-range Yukawa-type term based on the Hartree-Fock calculation [@Das03; @xu10]. In the asymmetric nuclear matter with isospin asymmetry $\delta$ and nucleon number density $\rho$, it has the following form [@Das03; @xu15] $$\begin{aligned}
V(\rho ,\delta ) &=&\frac{A_{u}\rho _{n}\rho _{p}}{\rho _{0}}+\frac{A_{l}}{%
2\rho _{0}}(\rho _{n}^{2}+\rho _{p}^{2})+\frac{B}{\sigma+1}\frac{\rho^{\sigma +1}}{\rho _{0}^{\sigma }} \notag \\
& &\times (1-x\delta ^{2})+\frac{1}{\rho _{0}}\sum_{\tau ,\tau^{\prime}}C_{\tau ,\tau ^{\prime }} \notag \\
& &\times \int \int d^{3}pd^{3}p^{\prime }\frac{f_{\tau }(\vec{r}, \vec{p}%
)f_{\tau ^{\prime }}(\vec{r}, \vec{p}^{\prime })}{1+(\vec{p}-\vec{p}^{\prime})^{2}/\Lambda ^{2}}. \label{MDIV}\end{aligned}$$In the above, $\rho_n$ and $\rho_p$ are number densities of neutrons and protons, respectively, $\rho _{0}$ is the saturation density, $\delta =(\rho _{n}-\rho _{p})/\rho$ is the isospin asymmetry, and $f_{\tau }(\vec{r}, \vec{p})$ is the phase-space distribution function, with $\tau=1(-1)$ for neutrons (protons) being the isospin index. The single-particle mean-field potential for a nucleon with momentum $\vec{p}$ and isospin $\tau$ in the asymmetric nuclear matter with isospin asymmetry $\delta$ and nucleon number density $\rho$ can be obtained from Eq. (\[MDIV\]) through the variational principle as $$\begin{aligned}
U_\tau(\rho ,\delta ,\vec{p}) &=&A_{u}\frac{\rho _{-\tau }}{\rho _{0}}%
+A_{l}\frac{\rho _{\tau }}{\rho _{0}} \notag \\
& & +B\left(\frac{\rho }{\rho _{0}}\right)^{\sigma }(1-x\delta ^{2})-4\tau x\frac{B}{%
\sigma +1}\frac{\rho ^{\sigma -1}}{\rho _{0}^{\sigma }}\delta \rho
_{-\tau }
\notag \\
& & +\frac{2C_{\tau,\tau}}{\rho _{0}}\int d^{3}p^{\prime }\frac{f_{\tau }(%
\vec{r}, \vec{p}^{\prime })}{1+(\vec{p}-\vec{p}^{\prime })^{2}/\Lambda ^{2}}
\notag \\
& & +\frac{2C_{\tau,-\tau}}{\rho _{0}}\int d^{3}p^{\prime }\frac{f_{-\tau }(%
\vec{r}, \vec{p}^{\prime })}{1+(\vec{p}-\vec{p}^{\prime })^{2}/\Lambda ^{2}},
\label{MDIU}
\end{aligned}$$where the four parameters $A_{u}$, $A _{l}$, $C_{\tau,\tau}$, and $C_{\tau,-\tau}$ can be expressed as [@xu15] $$\begin{aligned}
A_{l}(x,y)&=&A_{0} + y + x\frac{2B}{\sigma +1}, \label{AlImMDI}\\
A_{u}(x,y)&=&A_{0} - y - x\frac{2B}{\sigma +1}, \label{AuImMDI}\\
C_{\tau,\tau}(y)&=&C_{l0} - \frac{2yp^2_{f0}}{\Lambda^2\ln [(4 p^2_{f0} + \Lambda^2)/\Lambda^2]}, \label{ClImMDI}\\
C_{\tau,-\tau}(y)&=&C_{u0} + \frac{2yp^2_{f0}}{\Lambda^2\ln[(4 p^2_{f0} + \Lambda^2)/\Lambda^2]}. \label{CuImMDI}\end{aligned}$$ In the above, $p_{f0}=\hbar(3\pi^{2}\rho_0/2)^{1/3}$ is the nucleon Fermi momentum in symmetric nuclear matter at saturation density. The isovector parameters $x$ and $y$ are introduced to mimic the density dependence of the symmetry energy, i.e., the slope parameter $L=3\rho _{0}(d E_{sym} /d \rho) _{\rho =\rho _{0}}$, and the momentum dependence of the symmetry potential or the neutron-proton effective mass splitting. The values of the parameters are $A_{0}=-66.6973$ MeV, $C_{u0}=-99.67$ MeV, $C_{l0}=-60.36$ MeV, $B=141.697$ MeV, $\sigma=1.2658$, and $\Lambda=2.423p_{f0}$, in order to obtain the empirical nuclear matter properties: the saturation density $\rho_{0}=0.16$ fm$^{-3}$, the binding energy $E_{0}(\rho_{0})=-16$ MeV, the incompressibility $K_{0}=230$ MeV, the symmetry energy $E_{sym}(\rho_{0})=32.5$ MeV, the isoscalar potential at infinitely large momentum $U_{0,\infty}=75$ MeV, and the isoscalar effective mass at saturation density $m^{*}_{s}=0.7 m$, with $m$ being the nucleon mass in vacuum. The non-relativistic k-mass in the present study is defined as $$\begin{aligned}
\frac{m_{n(p)}^{*}}{m}=\left( 1+\frac{m}{p}\frac{\partial U_{n\left ( p \right )}}{\partial p}\right) ^{-1}.\end{aligned}$$
Lattice Hamiltonian approach within the IBUU transport model
------------------------------------------------------------
The IBUU transport model [@ireview98] has incorporated properly the isospin degree of freedom into the BUU transport model [@Bertsch88], with the later basically solving numerically the BUU equation $$\begin{aligned}
& &\frac{\partial f}{\partial t}+\nabla _{\vec{p}}U \cdot \nabla _{\vec{r}}f-\nabla _{\vec{r}}U \cdot \nabla _{\vec{p}}f \notag \\ &=&-\frac{1}{\left (2\pi \right )^{6}}\int d^3\vec{p}_2d^3\vec{p}_{2^{\prime }}d\Omega \frac{d\sigma}{d\Omega}v_{12} \notag \\
& &\times \left [ ff_2(1-f_{1^{\prime }})(1-f_{2^{\prime }})-f_{1^{\prime }}f_{2^{\prime }}(1-f)(1-f_{2})\right ] \notag \\
& &\times (2\pi)^3\delta^{(3)}(\vec{p}+\vec{p}_2-\vec{p}_{1^{\prime }}-\vec{p}_{2^{\prime }}),\end{aligned}$$where $\frac{d\sigma}{d\Omega}$ and $v_{12}$ are respectively the nucleon-nucleon differential cross section and relative velocity. The left-hand side of the above BUU equation describes the time evolution of the phase-space distribution function $f(\vec{r}, \vec{p})$ in the mean-field potential, and this can be approximately realized by solving the canonical equations of motion for test particles [@Won82; @Bertsch88]. In this approach, the phase-space distribution $f(\vec{r}, \vec{p})$ as well as the local density can be obtained by averaging $N$ parallel collision events: $$\begin{aligned}
f(\vec{r}, \vec{p})&=&\frac{1}{N}\sum_{i}^{AN} h(\vec{r}-\vec{r}_{i})\delta(\vec{p}-\vec{p}_{i}), \\
\rho (\vec{r})&=&\frac{1}{N}\sum_{i}^{AN}h(\vec{r}-\vec{r}_{i}),\end{aligned}$$ where $h$ is a smooth function in coordinate space, and $A$ is the number of real particles, with each represented by $N$ test particles.
In order to improve the stability for the momentum-dependent mean-field potential especially at lower collision energies, we improve the calculation by using a better function $h$ based on the lattice Hamiltonian framework as in Ref. [@Lenk89]. The average density $\rho_L$ at the sites of a three-dimensional cubic lattice is defined as $$\begin{aligned}
\rho_L(\vec{r}_{\alpha})=\sum_{i}^{AN}S(\vec{r}_{\alpha}-\vec{r}_i),\end{aligned}$$ where $\alpha$ is a site index and $\vec{r}_{\alpha}$ is the position of site $\alpha$. $S$ is the shape function describing the contribution of a test particle at $\vec{r}_i$ to the value of the average density $\rho_L(\vec{r}_{\alpha})$ at $\vec{r}_{\alpha}$, i.e., $$\begin{aligned}
S(\vec{r})=\frac{1}{N(nl)^6}g(x)g(y)g(z)\end{aligned}$$ with $$\begin{aligned}
g(q)=(nl-|q|)\Theta(nl-|q|).\end{aligned}$$ In the above, $l$ is the lattice spacing, $n$ determines the range of $S$, and $\Theta$ is the Heaviside function. In the following study, we adopt the values of $l=1$ fm and $n=2$.
After using the above smooth function $\rho_L(\vec{r}_{\alpha})$, the Hamiltonian of the system can be expressed as $$H=\sum_{i}^{AN}\frac{\vec{p}_{i}^{2}}{2m}+N\widetilde{V},$$ with the total potential energy expressed as $$\begin{aligned}
\widetilde{V}&=&l^3\sum_{\alpha}V_{\alpha} \notag \\
&=& l^3\sum_{\alpha} \Bigg \{ \frac{A_{u}\rho _{L,n}(\vec{r}_{\alpha})\rho _{L,p}(\vec{r}_{\alpha})}{\rho _{0}}+\frac{A_{l}}{2\rho _{0}}[\rho _{L,n}^{2}(\vec{r}_{\alpha}) \notag \\
& & +\rho _{L,p}^{2}(\vec{r}_{\alpha})]+\frac{B}{\sigma+1}\frac{\rho_{L}^{\sigma +1}(\vec{r}_{\alpha})}{\rho _{0}^{\sigma }}(1-x\delta ^{2})+\frac{1}{\rho _{0}} \notag \\
& & \times \sum_{i,j} \sum_{\tau_{i} ,\tau_{j}}C_{\tau_{i} ,\tau_{j}}
\frac{S(\vec{r}_{\alpha}-\vec{r}_i)S(\vec{r}_{\alpha}-\vec{r}_j)}{1+(\vec{p}_{i}
-\vec{p}_{j})^{2}/\Lambda ^{2}} \Bigg \}, \label{MDIVT}\end{aligned}$$ where $\rho _{L,n}(\vec{r}_{\alpha})$ and $\rho _{L,p}(\vec{r}_{\alpha})$ are respectively the number density of neutrons and protons at $\vec{r}_{\alpha}$. The canonical equations of motion for the $i$th test particle of isospin $\tau_i$ from the above Hamiltonian can thus be written as $$\begin{aligned}
\frac{d\vec{r}_{i}}{dt}&=&\frac{\partial H}{\partial\vec{p}_{i}}
= \frac{\vec{p}_i}{m} + N\frac{\partial\widetilde{V}}{\partial\vec{p}_{i}} \notag \\
&=& \frac{\vec{p}_i}{m} - Nl^3\sum_{\alpha}\frac{4}{\rho _{0}}\sum_{j}\sum_{\tau_{j}}C_{\tau_{i} ,\tau_{j}} S(\vec{r}_{\alpha}-\vec{r}_i) \notag \\
& &\times \frac{S(\vec{r}_{\alpha}-\vec{r}_j)(\vec{p}_{i}-\vec{p}_{j})}{[1+(\vec{p}_{i}-\vec{p}_{j})^{2}/\Lambda ^{2}]^{2}/\Lambda ^{2}}, \label{rt}\\
\frac{d\vec{p}_{i}}{dt} &=&-\frac{\partial H}{\partial\vec{r}_{i}}
= -N\frac{\partial\widetilde{V}}{\partial\vec{r}_{i}} \notag \\
&=& -Nl^3\sum_{\alpha}\frac{\partial S(\vec{r}_{\alpha}-\vec{r}_i)}{\partial\vec{r}_{i}} \Bigg \{ A_{u}\frac{\rho _{L,-\tau_{i} }(\vec{r}_{\alpha})}{\rho _{0}} \notag \\
& & +A_{l}\frac{\rho _{L,\tau_{i} }(\vec{r}_{\alpha})}{\rho _{0}}+B\left[\frac{\rho_L(\vec{r}_{\alpha}) }{\rho _{0}}\right]^{\sigma }(1-x\delta ^{2})
\notag \\
& &-4\tau_{i} x\frac{B}{\sigma +1}\frac{\rho_L^{\sigma -1}(\vec{r}_{\alpha})}{\rho _{0}^{\sigma }}\delta \rho _{L,-\tau_{i} }(\vec{r}_{\alpha})+\frac{2}{\rho _{0}} \notag \\
& &\times \sum_{j}\sum_{\tau_{j}}C_{\tau_{i} ,\tau_{j}} \frac{S(\vec{r}_{\alpha}-\vec{r}_j)}{1+(\vec{p}_{i}-\vec{p}_{j})^{2}/\Lambda ^{2}} \Bigg \}. \label{pt}\end{aligned}$$
Results and discussions {#results}
=======================
In the following study, we employ the improved IBUU transport model using the LH approach for the mean-field potential from the ImMDI interaction to investigate the isospin transport in heavy-ion collisions at intermediate energies. Generally speaking, effects of the isospin transport in intermediate-energy heavy-ion collisions may manifest itself in both the single-nucleon momentum spectra and fragment distributions in the final state [@Zhang99; @Toro01; @Liu03; @Bar05prc; @Nap10; @Col14; @Koh14; @Fil14; @Hud14; @Hag14]. While the IBUU transport model does not have the dynamical cluster formation mechanism, it is a useful tool for investigating the isospin transport dynamics by tracing the evolution of the isospin asymmetry during the reaction. Our following study is divided into two parts. In the first part, we study effects of the symmetry energy $E_{sym}(\rho)$ and the neutron-proton effective mass splitting $m_{n-p}^{\ast}$ on the isospin transport process between the projectile and the target with different $N/Z$ ratios. The degree and time scale of the isospin transport are investigated by using a method similar to that used in Ref. [@Li98]. In the second part, we investigate the isospin transport process between the low-density neutron-rich neck and the normal-density but less neutron-rich spectator in the projectile-like fragment as in the recent experiment done at TAMU [@Je17]. By varying values of the $x$ and $y$ parameters in the ImMDI interaction, heavy-ion collisions are simulated with different slope parameters $L$ of the symmetry energy and the neutron-proton effective mass splittings $m_{n-p}^{\ast}$. Typical isospin splittings of the nucleon effective mass used in the following studies are $m_{n-p}^{\ast}/m=0.426 \delta$ by setting $y=-115$ MeV as an example of $m_{n-p}^{\ast}>0$, and $m_{n-p}^{\ast}/m=-0.251 \delta$ by setting $y=115$ MeV as an example of $m_{n-p}^{\ast}<0$. We note that the parameter sets ($x=0$, $y=-115$ MeV) and ($x=1$, $y=115$ MeV) give the same symmetry energy with $L=60$ MeV but different $m_{n-p}^{\ast}$ [@xu15]. The initial density distribution of the projectile and the target nucleus is sampled according to that generated from the Skyrme-Hartree-Fock calculation with the same nuclear matter properties as in the ImMDI interaction, so the neutron skin effect is properly taken into account. The initial nucleon momentum distribution is sampled using the local Thomas-Fermi approximation with the isospin-dependent nucleon Fermi momentum determined by the local neutron or proton density.
Isospin transport between projectile and target with different N/Z ratios {#In central collisions}
-------------------------------------------------------------------------
As an example for studying the isospin transport process between the projectile and the target with different $N/Z$ ratios, $^{40}\textrm{Ca}+^{124}\textrm{Sn}$ collisions at an impact parameter of 1 fm and beam energies from 25 to 300 AMeV are simulated with the improved IBUU transport model, with each case 10 runs and each run 100 test particles. Similar to Ref. [@Li98], the relative neutron/proton ratios in the bounded residue (defined as regions where $\rho>\rho_0/8$) at forward and backward rapidities in the center-of-mass frame of the projectile-target system $$\lambda(t) \equiv \frac{(n/p)_{y>0}}{(n/p)_{y<0}}$$ is used to measure the degree of isospin equilibrium.
![(Color online) Time evolution of the ratio of the particle number in the resides ($\textrm{A}_\textrm{res}$) to the total particle number ($\textrm{A}_\textrm{targ}+\textrm{A}_\textrm{proj}$) in $^{40}\textrm{Ca}+^{124}\textrm{Sn}$ collisions at different beam energies using the parametrization ($x=0$, $y=-115$ MeV) for the ImMDI interaction.[]{data-label="A-res"}](A-res.eps)
The fractions of particles in the resides in $^{40}\textrm{Ca}+^{124}\textrm{Sn}$ collisions at beam energies from 25 to 300 AMeV using the parameter set ($x=0$, $y=-115$ MeV) for the ImMDI interaction are shown in Fig. \[A-res\]. The time evolutions of these fractions mainly reflect the time scales of particle emissions. Reaching a flat fraction of bounded particles indicates that the particle emission is over. It is seen that this time scale drops quickly with increasing beam energy. As the beam energy changes from 25 to 300 AMeV, the particle emission time scale changes approximately from 150 fm/c to about 75 fm/c. Such time scales set a useful reference for discussing the isospin relaxation times.
![(Color online) Time evolution of the isospin equilibration meter $[\lambda (t)-1]/[\lambda (0)-1]$ in $^{40}\textrm{Ca}+^{124}\textrm{Sn}$ collisions and beam energies of 25 (a), 100 (b), 200 (c), and 300 AMeV (d) from calculations using different symmetry energies and neutron-proton effective mass splittings.[]{data-label="F1"}](lambda.eps)
In order to reveal the symmetry energy effect on the isospin relaxation, we have done the same calculations with the parametrization ($x=1$, $y=-115$ MeV), which leads to the same neutron-proton effective mass splitting as ($x=0$, $y=-115$ MeV) but a softer $E_{sym}(\rho)$ with a slope parameter $L=10$ MeV. With different slope parameters $L$ of the symmetry energy and the neutron-proton effective mass splittings, the time evolutions of the isospin equilibration meter $[\lambda (t)-1]/[\lambda (0)-1]$ are displayed in Fig. \[F1\] as functions of time at various beam energies. As in Ref. [@Li98], the isospin relaxation time $\tau$ is defined as the time when $[\lambda (t)-1]/[\lambda (0)-1]$ approaches 0 for the first time. It is an approximate measure of how fast the isospin transport happens. Obviously, the complete isospin equilibrium does not occur even at the lowest energy considered as indicated by the oscillating $[\lambda (t)-1]/[\lambda (0)-1]$ values. Moreover, as indicated in Fig. \[A-res\], the fraction of masses in the resides are still decreasing as the isospin oscillations continue. More quantitatively, with the parameter set of ($x=0$, $y=-115$ MeV), the fractions of masses in the residues at 25 AMeV are about 84% and 71%, respectively, when $[\lambda (t)-1]/[\lambda (0)-1]$ reaches zero for the first and the second time, respectively. For the reaction at 300 AMeV, they are about 89% and 13%, respectively.
![(Color online) Beam energy dependence of the isospin relaxation time in $^{40}\textrm{Ca}+^{124}\textrm{Sn}$ collisions from calculations using different symmetry energies and neutron-proton effective mass splittings.[]{data-label="F2"}](Graph8.eps)
The isospin relaxation times from simulations using different $L$ and $m_{n-p}^{\ast}$ at beam energies from 25 to 300 AMeV are compared in Fig. \[F2\]. The decreasing trend of the isospin relaxation time with the increasing collision energy, as already observed in Fig. \[F1\], is due to stronger dissipations as a result of more successful nucleon-nucleon collisions at higher beam energies. Generally, a softer symmetry energy with $L=10$ MeV leads to a shorter isospin relaxation time. This is understandable since the symmetry energy at the dominating low-density phase acts as a restoring force for the system to reach isospin equilibrium, and the time for reaching isospin equilibrium becomes shorter if this force is stronger. The case with $m_{n-p}^{\ast}<0$ generally leads to a longer isospin relaxation time, especially at higher collision energies. This is due to the weaker symmetry potential at lower momenta for $m_{n-p}^{\ast}<0$ than that from $m_{n-p}^{\ast}>0$, especially when the density increases, as can be seen from Fig. 8 of Ref. [@xu15]. The above calculations were done with the isospin-dependent in-medium nucleon-nucleon scattering cross sections scaled by the nucleon effective mass [@Li05]. We have also tried free-space nucleon-nucleon scattering cross sections in the calculations, and found that the difference is much smaller compared to those caused by the nuclear symmetry energy and the neutron-proton effective mass splitting. Generally speaking, smaller in-medium cross sections reveal more about the mean-field potential effects on the isospin transport.
Isospin transport between neck and spectator in non-central $^{70}$Zn+$^{70}$Zn collisions {#In peripheral collisions}
------------------------------------------------------------------------------------------
Because the symmetry energy generally increases with increasing density, a more neutron-rich neck compared to the less neutron-rich spectator is expected to be formed in non-central heavy-ion reactions as a result of the isospin fractionation effect. Such effect has been studied extensively in the literature and is well understood, see, e.g., Refs. [@Bar05; @Li08] for reviews. However, it is not so clear how fast the neutron-rich neck exchanges its isospin asymmetry with the spectator and how this process may depend on the properties of isovector nuclear interactions. Interestingly, an experimental investigation on the isospin transport process between the neck and the spectator in non-central $^{70}$Zn+$^{70}$Zn collisions at a beam energy of 35 AMeV was recently carried out by the TAMU group [@Je17]. It was assumed that the PLF will rotate in a constant angular frequency after the breakup of the neck while the more neutron-rich light fragment (LF) from the neck and the less neutron-rich heavy fragment (HF) from the spectator evolve towards an isospin equilibrium state. The alignment angle serves as a clock once the angular momentum of the PLF is known, and the difference in isospin asymmetry between the LF and the HF was found to decrease with the increasing alignment angle.
![(Color online) Contours of the isospin asymmetry $\delta =(\rho _{n}-\rho _{p})/\rho$ in the reaction plane ($x-o-z$) at different times in $^{70}$Zn+$^{70}$Zn collisions at the beam energy of 35 AMeV and the impact parameter of 4 fm with ($L=60$ MeV, $m_n^*>m_p^*$) (first row), ($L=60$ MeV, $m_n^*<m_p^*$) (second row), and ($L=90$ MeV, $m_n^*>m_p^*$) (third row). The fourth row is for testing purpose by setting ($L=90$ MeV, $m_n^*>m_p^*$) at $t=0-170$ fm/c and ($L=60$ MeV, $m_n^*>m_p^*$) at $t=170-300$ fm/c.[]{data-label="F3"}](delta4.eps)
The neck formation and fragmentation were previously investigated using the constrained molecular dynamics model [@Sti14]. Although the fragmentation process is not properly described in the IBUU transport model, some useful information can still be obtained by tracing the isospin asymmetry in heavy-ion collisions. Plotted in Fig. \[F3\] are the isospin asymmetry contours from calculations using different symmetry energies and neutron-proton effective mass splittings, from averaging 200 runs for each case and 200 test particles for each run. The rotation of the whole system can be clearly observed. Moreover, the time evolutions of the less neutron-rich normal-density phase and the more neutron-rich low-density phase are vividly shown. A stiffer symmetry energy with a larger slope parameter $L$ generally leads to a more neutron-rich neck, while the neutron-proton effective mass splitting seems to have only small effects on the evolution of the isospin asymmetry. To further examine effects of the symmetry energy and isospin splittings of the nucleon effective mass on the isospin fractionation, the correlation between the isospin asymmetry $\delta$ and the reduced nucleon number density $\rho/\rho_0$ is shown in Fig. \[delta-rho\]. It is more clearly seen that a stiffer symmetry energy leads to a more neutron-rich low-density phase, while the isospin asymmetry of the low-density phase is insensitive to the isospin splitting of the nucleon effective mass. The case with $L=90$ MeV for the first half of the reaction but $L=60$ MeV for the latter half in the bottom row of Fig. \[F3\] is to study the isospin transport between the neck and the spectator with different symmetry energies but starting from the same initial isospin asymmetry difference. This will be further discussed later.
![(Color online) Correlation between the isospin asymmetry $\delta$ and the reduced nucleon number density $\rho/\rho_0$ at $t=170$ fm/c in non-central $^{70}$Zn+$^{70}$Zn collisions at the beam energy of 35 AMeV from calculations using different symmetry energies and neutron-proton effective mass splittings, corresponding to the reactions in Fig. \[F3\].[]{data-label="delta-rho"}](delta-rho.eps)
As seen from Fig. \[F3\], the neutron-rich neck is gradually assimilated by the spectator in the later stage, and the PLF will eventually reach an isospin equilibrium. In our IBUU calculations, the PLF, defined as bounded nucleons ($\rho>\rho_0/8$) at $z>0$, doesn’t break up into a neutron-rich LF and a less neutron-rich HF. In order to describe quantitatively the isospin relaxation within the PLF, we examine the isovector dipole moment $$\vec{D}(t) \equiv \vec{R}_Z(t)-\vec{R}_N(t),$$ where $\vec{R}_Z(t)$ and $\vec{R}_N(t)$ are the centers of mass of neutrons and protons in the PLF, respectively. This quantity is similar to the operator for isovector giant dipole resonances (IVGDR) [@kong17]. The full isospin equilibrium in the PLF is reached when $|\vec{D}(t)|$ is 0. Figure \[F4\] displays the time evolution of $|\vec{D}(t)|$ in the later stage of non-central $^{70}\textrm{Zn}+^{70}\textrm{Zn}$ reactions from simulations using different symmetry energies and neutron-proton effective mass splittings, corresponding to the four scenarios in Fig. \[F3\]. The instant $t=170$ fm/c is taken as the initial time when the norm of the dipole moment is the largest. The different initial $|\vec{D}(t)|$ values correspond to different isospin asymmetries of the neck from using different symmetry energies. The $|\vec{D}(t)|$ shows not only an exponential decay but also a damped oscillation, with the later similar to that of an IVGDR. Based on this observation, we fit the time evolution of $|\vec{D}(t)|$ using $$\begin{aligned}
\label{fit}
| \vec{D}(t)|&=& a \exp[-(t-170)/\tau_1] \notag \\
&+&b \cos[\omega \cdot (t-t_0)]\exp[-(t-170)/\tau_2].\end{aligned}$$ The second term in the above expression is also used in our previous study of IVGDR [@kong17]. The simulation results of $|\vec{D}(t)|$ are fitted reasonably well with Eq. (\[fit\]) as shown by the solid black lines in Fig. \[F4\]. The fitting parameters in the four scenarios are given in Table \[T0.7\]. Comparing results from using the same $L$ but different $m_{n-p}^{\ast}$, it is seen that the difference is mainly in the oscillation part, i.e., the second term in Eq. (\[fit\]). A slower decay of the oscillation magnitude and a lower frequency are observed for the case of $m_n^*<m_p^*$ compared with the $m_n^*>m_p^*$ case. This is qualitatively consistent with that observed in Ref. [@kong17], as a result of the weaker symmetry potential in the case of $m_n^*<m_p^*$ at lower nucleon momenta [@xu15]. From Fig. \[F4\], $|\vec{D}(t)|$ is seen to decrease more slowly for $m_n^*<m_p^*$ than for $m_n^*>m_p^*$, due to the difference in the second term of Eq. (\[fit\]) as discussed above. This means that in the presence of oscillations the measure of the isospin relaxation time $\tau$ should consider the second term. Here, we define the isospin relaxation time $\tau$ as the time needed for the upper envelope of $|\vec{D}(t)|$, i.e., $a \exp[-(t-170)/\tau_1] + b \exp[-(t-170)/\tau_2]$, to decrease to $1/e$ of its initial value, i.e., $(a+b)/e$. The values of $\tau$ are shown in the final column of Table \[T0.7\]. It is worth noting that the isospin relaxation time $\tau$ is quite long for $m_n^*<m_p^*$, qualitatively consistent with our findings in Sec. \[In central collisions\].
![(Color online) Time evolution of the magnitude of the isovector dipole moment for the projectile-like fragment in the later stage of non-central $^{70}\textrm{Zn}+^{70}\textrm{Zn}$ collisions from simulations using different symmetry energies and neutron-proton effective mass splittings corresponding to the four scenarios in Fig. \[F3\]. The scatters are results from simulations, while the solid lines are from the fit according to Eq. (\[fit\]).[]{data-label="F4"}](dipole4.eps)
$L$ (MeV) $m_{n-p}^{\ast}$ ($m$) $a$ (fm) $b$ (fm) $\tau_1$ (fm/c) $\tau_2$ (fm/c) $\omega$ \[rad(fm/c)$^{-1}$\] $\tau$ (fm/c)
------------- ------------------------ ------------------ ------------------ ----------------- ------------------- ------------------------------- -----------------
$60$ 0.426 $\delta$ $0.064\pm 0.001$ $0.012\pm 0.001$ $67.43\pm 0.62$ $71.11\pm 4.53$ $0.089\pm 0.001$ $68.00\pm 1.22$
$60$ -0.251 $\delta$ $0.071\pm 0.002$ $0.011\pm 0.001$ $66.03\pm 1.64$ $155.14\pm 25.62$ $0.062\pm 0.002$ $73.52\pm 3.45$
$90$ 0.426 $\delta$ $0.102\pm 0.002$ $0.036\pm 0.001$ $59.47\pm 0.86$ $61.61\pm 1.59$ $0.068\pm 0.001$ $60.02\pm 1.05$
$90$ and 60 0.426 $\delta$ $0.079\pm 0.001$ $0.056\pm 0.001$ $60.34\pm 0.10$ $51.05\pm 1.02$ $0.074\pm 0.001$ $56.30\pm 0.51$
\[T0.7\]
It is interesting to note that for the same $m_{n-p}^{\ast}$, the calculation with $L=90$ MeV leads to a shorter isospin relaxation time $\tau$ than that with $L=60$ MeV. However, this seems to be opposite to what we found in Sec. \[In central collisions\]. This discrepancy is mainly due to different initial $|\vec{D}(t)|$ values from different $L$. Neglecting the effective mass difference between neutrons and protons, the isovector current can be expressed as [@Shi03; @Bar05prc; @Riz08; @Li08] $$\vec{j}_n-\vec{j}_p= (D_n^{\rho }-D_p^{\rho })\nabla \rho - (D_n^I-D_p^I)\nabla \delta, \label{Current}$$ where the difference of the drift coefficient $D_N^{\rho}$ and the diffusion coefficient $D_N^I$ between neutrons and protons is related to the nuclear symmetry energy via $$\begin{aligned}
D_n^{\rho }-D_p^{\rho } \propto 4\delta \frac{\partial E_{sym}}{\partial \rho}, \notag \\
D_n^I-D_p^I \propto 4\rho E_{sym}. \label{Coefficients}\end{aligned}$$ In the analysis in Sec. \[In central collisions\], it is understood that the isovector current is dominated by the isospin diffusion, i.e., mainly due to the gradient of the isospin asymmetry $\nabla\delta$ as a result of different $N/Z$ ratios between the projectile and the target. A smaller $L$ corresponding to a larger symmetry energy at the dominating low-density phase leads to a larger isovector diffusion coefficient $D_n^I-D_p^I$, and thus a stronger isovector current $\vec{j}_n-\vec{j}_p$. In the analysis of isospin transport between the neck and the spectator, the isovector current is driven by both the isospin diffusion and the isospin drift, i.e., due to the gradients of both the isospin asymmetry $\nabla\delta$ and the density $\nabla\rho$. For different $L$ values, the dynamics leads to similar $\nabla\rho$ but different $\nabla\delta$ values. The longer isospin relaxation time from $L=90$ MeV is likely due to the larger $\nabla\delta$ and the larger isovector drift coefficient $D_n^{\rho }-D_p^{\rho }$, although the isovector diffusion coefficient $D_n^I-D_p^I$ is smaller, compared to the $L=60$ MeV case. To further understand the difference, we perform a simulation with $L=90$ MeV from 0 to 170 fm/c, and $L=60$ MeV for the rest of the reactions. As shown in Fig. \[F3\], the evolution of the isospin asymmetry becomes different for $t>170$ fm/c as expected. In Fig. \[F4\], it is seen that $|\vec{D}(t)|$ is the same in the initial stage, but drops more quickly and oscillates more strongly in the later stage, compared to the scenario with a fixed $L=90$ MeV throughout the simulation. After considering the oscillation, the overall isospin relaxation time $\tau$ is shorter as shown Table \[T0.7\], due to the same initial $\nabla\delta$ from $L=90$ MeV at $t=170$ fm/c but a stronger restoring force from $L=60$ MeV at $t>170$ fm/c.
The above analyses were done at the impact parameter of 4 fm, which is larger than the average value of mini-bias $^{70}$Zn+$^{70}$Zn collisions. This is similar to the experimental situation where more peripheral collision events were chosen [^2]. With a smaller impact parameter, there will be more participating nucleons, a higher-density and less neutron-rich neck, and thus a weaker isovector current due to the smaller gradients of the density and isospin asymmetry according to Eq. (\[Current\]). From our simulations with similar analysis method, we found that the isospin relaxation time generally increases with a smaller impact parameter.
Summary
=======
Within an improved isospin-dependent Boltzmann-Uehling-Uhlenbeck transport model using the lattice Hamiltonian method to calculate the mean-field potential, we have studied the effects of the nuclear symmetry energy and the neutron-proton effective mass splitting on the isospin relaxation time in two different isospin transport processes in intermediate-energy heavy-ion collisions. In the isospin transport process dominated by the isospin diffusion between the projectile and the target with different $N/Z$ ratios, the isospin relaxation time is generally shorted for a softer symmetry energy compared with a stiffer one, and longer for $m_n^*<m_p^*$ compared with $m_n^*>m_p^*$. The situation is different in the isospin transport process between the low-density neutron-rich neck and the normal-density but less neutron-rich spectator driven by both the isospin diffusion and the isospin drift mechanisms in non-central heavy-ion collisions. In this case, the isospin relaxation time is shorter for a stiffer symmetry energy because the isospin asymmetry of the neck is also affected by the symmetry energy, while the effect from the isospin splitting of the nucleon effective mass is qualitatively similar. Although the extracted isospin relaxation time in $^{70}$Zn+$^{70}$Zn collisions from the present study is within the experimental uncertainty range, i.e, $0.3\pm _{0.2}^{0.7}$ zs ($100\pm _{67}^{233}$ fm/c) from Ref. [@Je17], significant improvement of the accuracy for measuring experimentally the isospin relaxation time and additional information about the collision centrality are necessary to extract useful information about the symmetry energy and the neutron-proton effective mass splitting from comparing quantitatively the model calculations with the experimental result. Meanwhile, our study may help better understand the isospin diffusion and the isospin drift mechanisms for the isospin transport in intermediate-energy heavy-ion collisions.
We thank Chen Zhong for maintaining the high-quality performance of the computer facility. This work was supported by the Major State Basic Research Development Program (973 Program) of China under Contract No. 2015CB856904, the National Natural Science Foundation of China under Grant Nos. 11475243, 11320101004, and 11421505, the Shanghai Key Laboratory of Particle Physics and Cosmology under Grant No. 15DZ2272100, the U.S. Department of Energy, Office of Science, under Award Number de-sc0013702, and the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei) under the US Department of Energy Grant No. DEFG02-13ER42025.
[99]{} B. A. Li, C. M. Ko, and W. Bauer, Int. J. Mod. Phys. E **7**, 147 (1998). V. Baran, M. Colonna, V. Greco, and M. Di Toro, Phys. Rep. **410**, 335 (2005). A. W. Steiner, M. Prakash, J. M. Lattimer, and P. J. Ellis, Phys. Rep. **411**, 325 (2005). B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. **464**, 113 (2008).
W. Trautmann and H. H. Wolter, Int. J. Mod. Phys. E **21**, 1230003 (2012).
B. A. Li, À. Ramos, G. Verde, and I. Vidaňa, *Topical issue on nuclear symmetry energy*, Eur. Phys. J. A **50**, No.2 (2014).
C. J. Horowitz, E. F. Brown, Y. Kim, W. G. Lynch, R. Michaels, A. Ono, J. Piekarewicz, M. B. Tsang, and H. H. Wolter, J. Phys. G **41**, 093001 (2014).
M. Baldo and G. F. Burgio, Prog. Part. Nucl. Phys. **91**, 203 (2016).
M. Oertel, M. Hempel, T. Klähn, and S. Typel, Rev. Mod. Phys. **89**, 015007 (2017).
B. A. Li, Nuclear Physics News, Vol. **27**, No. 4, 7-11 (2017).
W. G. Lynch and M. B. Tsang, arXiv:1805.10757 \[nucl-ex\].
J. P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rep. **25**, 83 (1976). M. Jaminon and C. Mahaux, Phys. Rev. C **40**, 354 (1989). O. Sjöberg, Nucl. Phys. A **265**, 511 (1976).
J. Rizzo, M. Colonna, and M. Di Toro, Phys. Rev. C **72**, 064609 (2005). V. Giordano [*et al.*]{}, Phys. Rev. C **81**, 044611 (2010). Z. Q. Feng, Phys. Rev. C **84**, 024610 (2011). Y. X. Zhang, M. B. Tsang, Z. X. Li, and H. Liu, Phys. Lett. B **732**, 186 (2014). W. J. Xie and F. S. Zhang, Phys. Lett. B **735**, 250 (2014). H. Y. Kong, Y. Xia, J. Xu, L. W. Chen, B. A. Li, and Y. G. Ma, Phys. Rev. C [**91**]{}, 047601 (2015). D. D. S. Coupland [*et al.*]{}, Phys. Rev. C **94**, 011601(R) (2016).
L. Ou, Z. Li, Y. Zhang, and M. Liu, Phys. Lett. B **697**, 246 (2011). B. Behera, T. R. Routray, and S. K. Tripathy, J. Phys. G **38**, 115104 (2011). J. Xu, L. W. Chen, and B. A. Li, Phys. Rev. C [**91**]{}, 014611 (2015). H. F. Zhang, U. Lombardo, and W. Zuo, Phys. Rev. C **82**, 015805 (2010). J. Xu, Phys. Rev. C **91**, 037601 (2015).
Z. Zhang and L. W. Chen, Phys. Rev. C [**93**]{}, 034335 (2016). H. Y. Kong, J. Xu, L. W. Chen, B. A. Li, and Y. G. Ma, **95**, 034324 (2017).
C. Xu, B. A. Li, and L. W. Chen, Phys. Rev. C **82**, 054607 (2010). B. A. Li and X. Han, Phys. Lett. B **727**, 276 (2013).
B. A. Li, B. J. Cai, L. W. Chen, and J. Xu, Prog. Part. Nucl. Phys. **99**, 29 (2018).
B. A. Li and S. J. Yennello, Phys. Rev. C **52**, 1746(R) (1995). L. Shi and P. Danielewicz, Phys. Rev. C **68**, 064604 (2003). J. Rizzo, M. Colonna, V. Baran, M. Di Toro, H. H. Wolter, and M. Zielinska-Pfabe, Nucl. Phys. A **806**, 79 (2008).
B. A. Li and C. M. Ko, Phys. Rev. C **57**, 2065 (1998).
M. B. Tsang [*et al.*]{}, Phys. Rev. Lett. **92**, 062701 (2004). M. B. Tsang [*et al.*]{}, Phys. Rev. Lett. **102**, 122701 (2009). L. W. Chen, C. M. Ko, and B. A. Li, Phys. Rev. Lett. **94**, 032701 (2005). B. A. Li and L. W. Chen, Phys. Rev. C **72**, 064611 (2005).
A. Jedele, A. B. McIntosh, K. Hagel, M. Huang, L. Heilborn, Z. Kohley, L. W. May, E. McCleskey, M. Youngs, A. Zarrella, and S. J. Yennello, Phys. Rev. Lett. **118**, 062501 (2017).
R. J. Lenk and V. R. Pandharipande, Phys. Rev. C **39**, 2242 (1989).
C. B. Das, S. Das Gupta, C. Gale, and B. A. Li, Phys. Rev. C [**67**]{}, 034611 (2003). J. Xu and C. M. Ko, Phys. Rev. C **82**, 044311 (2010).
G. F. Bertsch and S. Das Gupta, Phys. Rep. **160**, 189 (1988). C. Y. Wong, Phys. Rev. C **25**, 1460 (1982).
F. S. Zhang, L. W. Chen, Z. Y. Ming, and Z. Y. Zhu, Phys. Rev. C **60**, 064604 (1999). M. Di Toro, V. Baran, M. Colonna, S. Maccarone, M. Zielinska-Pfabe, and H. H. Wolter, Nucl. Phys. A **681**, 426c (2001). J. Y. Liu, W. J. Guo, Y. Z. Xing, and H. Liu, Nucl. Phys. A **726**, 123 (2003). V. Baran, M. Colonna, M. Di Toro, M. Zielinska-Pfabe, and H. H. Wolter, Phys. Rev. C **72**, 064620 (2005). P. Napolitani, M. Colonna, F. Gulminelli, E. Galichet, S. Piantelli, G. Verde, and E. Vient, Phys. Rev. C **81**, 044619 (2010). M. Colonna, V. Baran, and M. Di Toro, Eur. Phys. J. A **50**, 30 (2014). Z. Kohley and S. J. Yennello, Eur. Phys. J. A **50**, 31 (2014). E. De Filippo and A. Pagano, Eur. Phys. J. A **50**, 32 (2014). S. Hudan and R. T. de Souza, Eur. Phys. J. A **50**, 36 (2014). K. Hagel, J. B. Batowitz, and G. Ropke, Eur. Phys. J. A **50**, 39 (2014).
K. Stiefel, Z. Kohley, R. T. deSouza, S. Hudan, and K. Hammerton, Phys. Rev. C **90**, 061605(R) (2014).
[^1]: corresponding author: xujun@sinap.ac.cn
[^2]: S. J. Yennello, private communication.
|
---
author:
- 'Ittai Abraham[^1]'
- 'David Durfee[^2]'
- 'Ioannis Koutis[^3]'
- 'Sebastian Krinninger[^4]'
- 'Richard Peng[^5]'
bibliography:
- 'references.bib'
title: On Fully Dynamic Graph Sparsifiers
---
[^1]: VMware Research
[^2]: Georgia Institute of Technology
[^3]: University of Puerto Rico, Rio Piedras
[^4]: Max Planck Institute for Informatics, Saarland Informatics Campus, Germany
[^5]: Georgia Institute of Technology
|
---
abstract: 'In a public-key infrastructure (PKI), clients must have an efficient and secure way to determine whether a certificate was revoked (by an entity considered as legitimate to do so), while preserving user privacy. A few certification authorities (CAs) are currently responsible for the issuance of the large majority of TLS certificates. These certificates are considered valid only if the certificate of the issuing CA is also valid. The certificates of these important CAs are effectively *too big to be revoked*, as revoking them would result in massive collateral damage. To solve this problem, we redesign the current revocation system with a novel approach that we call PKI Safety Net (PKISN), which uses publicly accessible logs to store certificates (in the spirit of Certificate Transparency) and revocations. The proposed system extends existing mechanisms, which enables simple deployment. Moreover, we present a complete implementation and evaluation of our scheme.'
author:
-
bibliography:
- 'references.bib'
title: |
PKI Safety Net (PKISN):\
Addressing the Too-Big-to-Be-Revoked Problem of the TLS Ecosystem
---
Introduction {#sec:intro}
============
Background {#sec:pre}
==========
[PKISN]{}Overview {#sec:overview}
=================
[PKISN]{}Details {#sec:details}
================
Deployment {#sec:deployment}
==========
Security Analysis {#sec:analysis}
=================
Realization in Practice {#sec:implementation}
=======================
Evaluation {#sec:eval}
==========
Discussion {#sec:discussion}
==========
Conclusion {#sec:conclusions}
==========
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank our shepherd Bart Preneel, the anonymous reviewers, and Franz Saller for their valuable feedback. We gratefully acknowledge support from ETH Zurich and from the Zurich Information Security and Privacy Center (ZISC).
|
---
address: |
Whipple Observatory, Harvard-Smithsonian Center for Astrophysics,\
P.O. Box 97, Amado, AZ 85645-0097 U.S.A.\
e-mail: tweekes@cfa.harvard.edu
author:
- 'Trevor C. Weekes'
title: 'Status of VHE Astronomy c.2000'
---
Retrospective
=============
As we enter a new century of high energy astrophysics, it behooves us to consider briefly how far we have come and far we have yet to go. It is some 47 years since Galbraith and Jelley mounted a ten inch mirror in a garbage can with a photomultiplier at its focus and under the impossibly damp skies of the Berkshires detected the first Cherenkov light pulses from air showers . Within a decade the broad outlines of the atmospheric Cherenkov technique, as it applied to the search for very high energy gamma rays, had been defined and significant experiments built to pursue these searches. Reading again these early papers [@chudakov65] one cannot but marvel at the prescience of the authors and the courage of the exponents who were taking on what must have seemed an impossible task. It is amazing how much was anticipated in these classic papers and how much is “rediscovered” by later exponents of the technique with large teams of experimenters, sophisticated telescopes, and vast simulations. Using simple, but elegant, telescopes, analytical models, and considerable foresight, much of what was later to become the basis of the multi-million dollar observatories now in operation was anticipated. The early pioneers of the field, W. Galbraith and J.V. Jelley of the Atomic Energy Research Establishment, Harwell, England, N.A. Porter of University College, Dublin, Ireland, A.E. Chudakov and V.I. Zatsepin of the Lebedev Institute, Moscow, U.S.S.R., were the giants on whose work all future progress depended; their names should be inscribed large within the High Energy Astrophysics Hall of Fame! It is but an accident of nature that these early experiments failed to detect significant signals from supernova remnants and quasars since their sensitivities were remarkably close to subsequent detection levels.
To consider the state of high energy astrophysics in 1960 is to appreciate their courage in undertaking very high energy gamma-ray experiments at that time. In fact “high energy astrophysics” per se did not exist as a discipline at that time. Although the seminal papers containing estimates of fluxes of 100 MeV and TeV gamma-rays to be expected from cosmic sources were already in print [@morrison58] [@cocconi60], they were, with hindsight, hopelessly optimistic. One can only hope that contemporary estimates of TeV fluxes of neutrinos from cosmic sources are on a more solid base! Gamma-ray astronomy at any energy was in its infancy with no sources detected. X-ray astronomy did not exist and was not even seriously considered. Since cosmic rays were both the prime motivator to look for gamma-ray sources and the chief source of information as to what properties such sources might have, the conventional wisdom was that the dominant energy spectrum would be that of the cosmic radiation, i.e. a differential exponent of -2.7. Such an exponent did not bode well for the chances of detecting very high energy gamma-rays. Since space gamma-ray telescopes could be shielded with active detectors to exclude the charged cosmic radiation, the chances of detecting a signal in a non-shielded ground-based detector seemed vanishingly small. Optical detection methods which depended on an unstable atmosphere and on the detection of a weak optical signal against a background of natural and man-made light sources did not seem promising. Only a fool or someone with great courage would choose to pursue such techniques in the face of such uncertainty.
The eventual detection of a mixed population of very high energy (VHE) gamma-ray sources is a vindication of these early efforts and a tribute to the pioneers.
Gamma-ray Astronomy’s Great Failure!
====================================
Although GeV-TeV gamma-ray astronomy has had a number of outstanding successes (the detection of blazars, the GEV component in solar flares, the GeV-TeV component of GRBs, pulsars, shell and plerionic supernova remnants, the galactic plane, to mention but a few), the single great motivator, the conclusive solution to the problem of the origin of the cosmic radiation, is still elusive. It was this problem more than any other that led to the development of high energy gamma-ray astronomy, both in space and on the ground, but in some ways, at least observationaly, we are no closer to identifying the source than we were 40 years ago. While we can celebrate the contributions of gamma-ray studies to pulsar phenomenology, to limiting the infrared background, to the study of jets in extragalactic sources, we cannot yet proclaim with any confidence “Gamma-ray astronomers find the Origin of the Cosmic Radiation”.
As pointed out previously [@weekes99] every gamma-ray source detected to date can be explained as a source of cosmic electron acceleration and interaction; there is no source that can be conclusively attributed to hadron acceleration. The $\pi$$^o$ bump is only observed in the galactic plane and there we observe the propagation, not the source, of the cosmic rays.
The title to this section is deliberately provocative and hopefully will be obsolete before this volume is off the press. The number of sensitive ground-based telescopes now on-line, or shortly to come on-line, ensures that a vast number of new candidate sources are under ever increasingly sensitive scrutiny and may soon supply the vital data necessary to solve this problem.
What the Imaging A.C.T. has done for Astrophysics:
==================================================
Existence of Sources
--------------------
The outstanding contribution of the imaging atmospheric Cherenkov technique (IACT) must surely be that it has elevated ground-based gamma-ray astronomy techniques above that critical threshold where sources are detected with some credibility. In short the IACT has confirmed that there is a gamma-ray sky at energies above 200 GeV (Figure \[skymap\]), that the upper energy limit of space gamma-ray telescopes is an instrumental limit, not an astrophysical one, that there are a variety of different kinds of source populations, that there are both galactic and extragalactic sources, both steady and variable sources, and both point-like and extended. Furthermore the energy spectra extend to 50 GeV where the flux sensitivity of the IACT is not large so that the construction of new telescopes that are sensitive above this energy is now justified.
Half of the reported sources are galactic. These include plerions, shell SNRs in which the progenitor particles are probably electrons and an X-ray binary. The extragalactic sources are all blazars. All but one are X-ray-selected BL Lacs; this population is quite different from the population of EGRET-detected blazars.
Although a large fraction of the sky has not been surveyed with high sensitivity, there are a number of objects for which significant upper limits have been established. These have the effect of severely limiting some classes of models and thus contribute to high energy astrophysics in a negative sense. These non-sources include shell-type SNRs, in which the progenitor particles are hadrons, the galactic plane and various pulsars.
The energy range covered by atmospheric Cherenkov telescopes extends from 50 GeV to 50 TeV. Thus far, the IACT has only been used down to 200 GeV but it is possible to reach as low as 50 GeV using Solar Arrays. By observing at low elevations the IACT can reach energies as high as 50 TeV.
Source Catalog
--------------
It is a matter of some disappointment (given the number of new observatories now in operation) that the source catalog (Table \[catalog\]) shows no changes since the one assembled at the time of the Snowbird Workshop [@weekes99]. In particular the credibility rating of sources has not changed.
Source Type z Discovery EGRET Grade
------------------------------- ---------- ------- ----------- ------- -------
[**Galactic Sources**]{}
Crab Nebula Plerion 1989 yes A
PSR 1706-44 Plerion? 1995 no A
Vela Plerion? 1997 no B
SN1006 Shell 1997 no B$-$
RXJ1713.7-3946 Shell 1999 no B
Cassiopeia A Shell 1999 no C
Centaurus X-3 Binary 1999 yes C
[**Extragalactic Sources**]{}
Markarian 421 XBL 0.031 1992 yes A
Markarian 501 XBL 0.034 1995 yes A
1ES2344+514 XBL 0.044 1997 no C
PKS2155-304 XBL 0.116 1999 yes B
1ES1959+650 XBL 0.048 1999 no B$-$
3C66A RBL 0.44 1998 yes C
: Source Catalog c.2000 (Heidelberg)[]{data-label="catalog"}
Time Variability
----------------
One of the extraordinary contributions of VHE gamma-ray observations to high energy astrophysics has been the revelation of the variability of blazars on a wide range of time-scales. This has truly opened a new window to the study of these objects and has been a major incentive to improve and extend the observations.
[**Short-term Variations:**]{} The dramatic flare from Markarian 421 observed by the Whipple group on May 15, 1996 [@gaidos96] is still the best instance of a rapid time variation (doubling time $<$ 15 minutes) from an AGN (Figure \[shortflare\]). The observation of such short flares from more distant AGN may have some important consequences for cosmology and quantum gravity [@biller99].
[**Longterm Monitoring:**]{} The ability of VHE telescopes to provide longterm monitoring of variable sources is a little appreciated property of ground-based observatories. Although the optical techniques have a limited duty-cycle, their ability to monitor sources over days, weeks, months and years is a unique feature in gamma-ray astronomy above 1 MeV. This is dramatically illustrated by the HEGRA observations of Markarian 501 in 1997 (Figure \[hegra1997\]). In this instance the observations were significantly extended by the pioneering work of the HEGRA group in observing in moonlight [@kranich99]. Such long-term monitoring can be enhanced by organized observing campaigns involving VHE observatories at different longitudes [@takahashi00]; unfortunately the number of really sensitive VHE telescopes is limited so that it is difficult to get continuous coverage. Such campaigns are further limited by weather patterns. Near continuous coverage will come with the next generation of space telescopes which will use wide field telescopes.
Multiwavelength Observations
----------------------------
For an understanding of the mechanisms at work in AGN jets, multiwavelength campaigns are required. These are notoriously difficult to organize since many traditional observatories require advanced notification and approval and do not readily respond to Targets of Opportunity. VHE observatories have the advantage that they are usually controlled by the principal investigators, do not have rigid observing schedules, and have only a small number of known sources (regrettably!). The correlation of X-ray and VHE observations has been reported in a number of instances (cf. [@cataneseweekes99] and references therein). One such instance is shown in Figure \[maraschi-fig\].
X-ray observations of sources with non-thermal spectra have proved particularly useful in identifying candidate VHE supernova remnants. Hard X-ray observations hold the promise of identifying a new class of extreme blazars which will have detectable and variable TeV emission. The launch of EXIST, the first hard X-ray survey instrument, in the next decade will extend this symbiotic relationship.
Energy Spectrum
---------------
The ability of the IACT to measure energy spectrum in the TeV region has steadily improved and is comparable or better than that achieved in space gamma-ray telescopes. Energy resolutions of single telescope systems are quoted as 35% and for the HEGRA array as low as 10-15%. It is reassuring that observations made by different experiments using different methods of analysis are in good agreement (Figure \[aharonian-fig\]).
Status of VHE Observatories
===========================
The IACT continues to be the favored method of detecting gamma-rays in the 100-10,000 GeV energy range although there are several overlapping techniques at the low and high energy ends of this range. The principal observatories who have reported results using this technique at recent workshops are listed (Table \[IACT\]). Sadly at this meeting we learnt that both the Durham and 7TA observatories, after the successful detection of several sources, have ceased operations. Also listed (Table \[NIACT\]) are several atmospheric Cherenkov observatories which do not use imaging; the Potchefstroom experiment has also recently ceased operation.
In the interval between the death of EGRET and the launch of GLAST it is interesting to explore the lowest energies that can be achieved from the ground. Although it is unlikely that ground-based observations can be really competitive below 50 GeV, there is a window of opportunity between 10 and 100 GeV before the next generation of space telescopes come into operation. Even relatively simple telescopes could elucidate such problems as pulsar spectral cut-offs, AGN absorption in intergalactic space, gamma-ray bursts, etc. The simplest approach is to use large mirror collection areas and these are provided by the relatively crude optics of large solar collectors using heliostats. The four experiments listed in Table \[SOLAR\] use existing solar farms (large fields of heliostats pointing, and roughly focussing, light to a central tower). STACEE uses a facility that is still in operation for experimental solar energy work whereas the other three use facilities that are no longer in use.
The energy threshold of particle air shower detectors (Table \[EAS\]) has gradually been reduced to the point where there is now overlap with IACT experiments. The Tibet experiment has been in operation for some years whereas MILAGRO has just come on-line.
[llcccl]{}
Group/ & Location & Telescope(s) & Camera & Threshold & Ref.\
Countries & & & & &\
& & & Num.$\times$Apert.& Pixels & (TeV)\
\
Whipple & Arizona & 10m & 490 & 0.25 & [@WHIPPLE]\
USA-Ireland-UK & & & & &\
Crimea & Crimea & 6$\times$2.4m & 6$\times$37 & 1 & [@CRIMEA]\
Ukraine & & & & &\
SHALON & Tien Shen & 4m & 244 & 1.0 & [@SHALON]\
Russia & & & & &\
CANG.-II & Woomera & 10m & 256 & 0.5 & [@CANGAROO]\
Japan-Aust. & & & & &\
HEGRA & La Palma. & 6$\times$3m & 5$\times$271 & 0.5 & [@HEGRA]\
German.-Sp.Arm. & & & & &\
CAT & Pyrenées & 4.5m & 600 & 0.25 & [@CAT]\
France & & & & &\
TACTIC & Mt.Abu & 10m & 349 & 0.3 & [@TACTIC]\
India & & & & &\
Durham & Narrabri & 3$\times$7m & 1$\times$109 & 0.25 & [@DURHAM99]\
Uk & & & & &\
7TA & Utah & 7$\times$2m & 7$\times$256 & 0.5 & [@7TEL99]\
Japan & & & & &\
[lllcl]{}
Group & Countries & Type & Telescopes & Ref.\
Potchefstroom & South Africa & Lateral Array & 4 & [@POTCH]\
\
Pachmarhi & India & Lateral Array & 25 & [@PACHMARI99]\
\
Beijing & China & Double & 2 & [@BEIJING]\
\
[cccccl]{}
Group & Countries & Location & Heliostats & Threshold & Ref.\
& & & Now (future) & GeV &\
STACEE & Canada-USA & Albuquerque, USA & 32 (48) & 180 & [@STACEE99]\
\
CELESTE & France & Themis, France & 40 (54) & 50$\pm10$ & [@CELESTE99]\
\
Solar-2 & USA & Barstow, USA & 32 (64) & 20 & [@SOLAR-299]\
\
GRAAL & Germany-Spain & Almeria, Spain & (13-18)x4 & 200 GeV & [@GRAAL99]\
\
[lllllll]{}
Group & Countries & Location & Telescope & Altitude & Threshold & Ref.\
& & & & km & TeV &\
Milagro & USA & Fenton Hill, NM & Water Cher. & 2.6 & 0.5 & [@MILAGRO]\
\
Tibet HD & China-Japan & Tibet & Scintillators & 4.5 & 3 & [@TIBET99]\
\
Next Generation Telescopes
==========================
The next few years will see the completion of several new “next generation” IACTs which will significantly improve the scientific potential of the discipline. These major observatories will probably dominate the field for the next decade and they represent a major transition from the traditional “small science” which characterized the early years of the IACT to multi-national facilities which will serve a larger community as guest investigators. Three of the facilities: CANGAROO-III [@mori99], HESS [@HESS99], and VERITAS [@VERITAS99], build on the IACT array concept that has been demonstrated by HEGRA and are very similar in concept; the first two will be in the southern hemisphere, the third in the northern hemisphere. MAGIC is a single large imaging telescope which will use several new technological approaches [@MAGIC99]. Some of the most important parameters of these telescopes are listed in Table \[telescopes\]. A new addition to the list of new IACTs is MACE, an Indian look-alike of MAGIC; its parameters were described at this meeting.
[**Parameter**]{} MAGIC HESS CANGAROO-III VERITAS
----------------------- -------------- --------------- -------------- -------------------
[**Base** ]{} Munich Heidelberg Tokyo Arizona
[**Country**]{} Germany Germany Japan U.S.A.
[**Partners**]{} Spain, Italy France Australia UK, Ireland
[**Science**]{} AGNs, Bursts SNR Gal. Sources AGNs, SNR, Bursts
[**Location**]{} La Palma Namibia Woomera Arizona
[**Elevation**]{} 2.3 km 1.8 km S.L. 1.4 km
[**\# of tel.**]{} 1 4 (16) 4 7
[**Pattern**]{} - Square Square Hexagon
[**Spacing**]{} - 120m 100m 80m
[**Design**]{} Parabola Davies-Cotton Parabola Davies-Cotton
[**Aperture**]{} 17m 12m 10m 10m
[**Focal length**]{} 20m 15m 8m 12m
[**OSS**]{} Carbon fiber Steel Steel Steel
[**Facets** ]{} 60cm square 60cm circ. 80cm circ. 60cm hex.
[**Material**]{} Al-milled Ground-glass Composite Glass
[**Supplier**]{} Italy Czech/Arm. Japan USA
[**PMTs**]{} EMI Phillips Hama. Hama.?
[**Cabling**]{} Fiber Coax Coax Coax
[**Electronics**]{} FADC - - FADC
[**\# of pixels** ]{} $>$800 800 x 4 512 x 4 499 x 7
[**First light**]{} 2001 2002 2003 2005
: MAGIC, HESS, CANGAROO-III and VERITAS[]{data-label="telescopes"}
[**Sensitivity:**]{} New experiments are often characterized by their energy thresholds and flux sensitivities. The definitions of these quantities are not trivial and often cause confusion. Energy threshold, usually defined as the maximum in the differential response curve for a Crab-like spectrum, is particularly misleading since the event selection can be biassed to give a very low energy threshold but with very small collection area.
Figure \[sensitivity\] gives the integral flux sensitivity for various existing experiments as well as the predicted sensitivity for GLAST and VERITAS [@vassiliev99]. This figure has been widely reproduced. However integral flux sensitivities presuppose knowledge of the source spectrum and are not useful for sources with steep spectra which fall off near the energy threshold. The flux sensitivity for VERITAS is in close agreement with that derived for HESS [@hofmann99] as would be expected. However the sensitivity quoted for CANGAROO-III [@mori99] is much better, particularly at lower energies; this prediction needs to be reconciled with those of VERITAS and HESS since the array parameters are not that different.
[**Schedule:**]{} Of major concern to all those interested in VHE astronomy is the roadmap of HE and VHE experiments in the next decade. Although all proposed launch and construction completion dates are inevitably optimistic, it is hoped that the future depicted in [@weekes99] is not grossly inaccurate. The solar telescopes are already reporting results. The MAGIC web page proudly, if somewhat optimistically, gives a countdown to first light in 2001. Concrete has already been poured for the HESS foundations and one CANGAROO telescope is in place and taking data. Although the first of the next generation projects to be announced, VERITAS, has yet to break ground, first light in 2005 is still feasible. The space telescopes, reported elsewhere in these proceedings, seem to be well on schedule. The scheduled completion date for MACE, the Indian MAGIC look-alike, is 2003.
Although AMS [@AMS99] and AGILE [@AGILE99] will partially fill the HE gap left by the demise of EGRET, they will not provide a significant boost in sensitivity. In the next few years the most exciting new results may well come from the solar telescopes as they explore a new energy domain. Prior to the launch of GLAST [@GLAST99] there will be a spate of new results forthcoming from the new generation of IACT arrays.
[**Acknowledgements:**]{} Research in very high energy gamma-ray astronomy at the Smithsonian Astrophysical Observatory is supported by the U.S. Department of Energy. I am grateful to Tony Hall, Deirdre Horan, Jim Gaidos and Rod Lessard for helpful comments on the manuscript.
Galbraith, W., Jelley, J.V., Nature, [**171**]{}, 349 (1953)
Jelley, J.V., Galbraith, W,., J. Atmos. Phys. [**6**]{}, 304 (1955)
Chudakov, A.E et al., Trans. Consult. Bureau, P.N.Lebedev Inst., [**26**]{}, 99 (1965)
Jelley, J.V., Porter, N.A., M.N.R.A.S. [**4**]{}, 275 (1963)
Morrison, P., Nuovo Cimento, [**7**]{}, 858 (1958)
Cocconi, G., 6th ICRC, (Moscow), [**2**]{}, 309 (1959)
Weekes, T.C. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. August, 1999. AIP Proc. Conf. [**515**]{}, 3
Buckley, J.H. 26th ICRC, (Salt Lake City), AIP Conference Proceedings[**516**]{}, Eds. B.Dingus, D.Kieda, M.Salamon 195 (1999)
Gaidos, J.A. et al., Nature, [**383**]{}, 319 (1996)
Biller, S. et al., Phys. Rev. Lettr. [**83**]{}, 2108 (1999)
Kranich, D. et al., 26th ICRC, (Salt Lake City), [**3**]{}, 358 (1999)
Takahashi, T. et al., Ap. J. (in press) (2000)
Catanese, M., Weekes, T.C., P.A.S.P. [**111**]{}, 1193 (1999)
Maraschi, L. et al., Astropart. Phys. [**11**]{}, 189 (1999)
Aharonian, F.A. et al., A. & A., [**349**]{}, 11 (1999)
Cawley, M.F. et al., Exp. Ast. [**1**]{}, 185 (1990)
Vladimirsky, B.M. et al., Proc. Workshop on VHE Gamma Ray Astronomy, Crimea (April, 1989), 21 (1989)
Nikolsky S.I, Sinitsyna V.G., Proc. Workshop on VHE Gamma Ray Astronony, Crimea (April, 1989), 11 (1989)
Hara, T. et al., Nucl.Inst. Meth. A. [**332**]{}, 300 (1993)
Daum, A. et al., Astropart. Phys. [**8**]{}, 1 (1997)
Goret, P. et al., 25th ICRC, (Durban), [**3**]{}, 173 (1997)
Bhat, C.L. et al., Proc. Workshop on VHE Gamma Ray Astronomy, (Kruger Park), (August, 1997) 196 (1997)
Chadwick, P.M. et al., Proc. Workshop on VHE Gamma Ray Astronomy, (Snowbird), (August, 1999), 223 (1999)
T.Yamamoto et al., 26th ICRC, (Salt Lake City), [**5**]{}, 275 (1999)
DeJager, H.I. et al., South African of J. Phys., [**9**]{} 107 (1986)
Bhat, P.N. et al., 26th ICRC, (Salt Lake City), [**5**]{}, 191 (1999)
Yinlin, J. et al., 21st ICRC, (Adelaide), [**4**]{}, 220 (1990)
Ong, R.A. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. August, 1999. AIP Proc. Conf. [**515**]{}, 401.
de Naurois, M., 26th ICRC, (Salt Lake City), [**5**]{}, 211 (1999)
Zweerink, J.A. et al., 26th ICRC, (Salt Lake City), [bf 5]{}, 223 (1999)
Arqueros, F. et al., 26th ICRC, (Salt Lake City), [**5**]{}, 215 (1999).
Yodh, G.B., Space. Sci. Rev. [**75**]{}, 199 (1996)
Amenomori, M. et al. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. August, 1999. AIP Proc. Conf. [**515**]{}, 459 (1999)
Mori, M. et al. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. August, 1999. AIP Proc. Conf. [**515**]{}, 485
Kohnle, A. et al., 26th ICRC, [Salt Lake City]{}, [**5**]{}, 239 (1999)
Bradbury, S.M. et al., 26th ICRC, [**5**]{}, 280 (1999)
Martinez, M., 26th ICRC, (Salt Lake City), [**5**]{}, 219 (1999).
Hofmann, W. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. August, 1999. AIP Proc. Conf. [**515**]{}, 492 (1999)
Vassiliev, V.V. et al., 26th ICRC, (Salt Lake City), [**5**]{}, 299 (1999)
Mereghetti, S. et al. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. August, 1999. AIP Proc. Conf. [**515**]{}, 467 (1999)
Battiston, R. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. August, 1999. AIP Proc. Conf. [**515**]{}, 474 (1999)
Kniffen, D.A., Bertsch, D.L. and Gehrels, N. “GeV-TeV Gamma Ray Astrophysics Workshop”, Snowbird, UT. AIP Proc. Conf. [**515**]{}, 492 (1999)
|
---
abstract: 'We demonstrate strong coupling of single photons emitted by individual molecules at cryogenic and ambient conditions to individual nanoparticles. We provide images obtained both in transmission and reflection, where an efficiency greater than 55% was achieved in converting incident narrow-band photons to plasmon-polaritons (plasmons) of a silver nanoparticle. Our work paves the way to spectroscopy and microscopy of nano-objects with sub-shot noise beams of light and to triggered generation of single plasmons and electrons in a well-controlled manner.'
author:
- 'M. Celebrano, R. Lettow, P. Kukura, M. Agio, A. Renn, S. Götzinger, V. Sandoghdar'
title: 'Single-Photon Imaging and Efficient Coupling to Single Plasmons'
---
Coupling of light to dipolar radiators lies at the heart of light-matter interaction. Theoretical studies have predicted more than 80% extinction of a focused classical Gaussian beam by a single dipolar radiator [@Zumofen:08; @Mojarad:09], and recent experimental investigations have reported up to 12% extinction of classical light by single quantum emitters [@Vamivakas:07; @Wrigge:08; @Tey:08]. Laboratory realizations of spectroscopy and microscopy on single nano-objects with single-photon illumination, however, have been confronted by obstacles. In particular, excitation of a quantum emitter by individual photons has only been possible in confined geometries [@Eschner:01; @Rempe:03], and quantum optical imaging [@Kolobov:99; @Treps:02; @Lugiato:02] of subwavelength structures has not been explored at the single-photon level. One bottle-neck in spectroscopy with single photons is access to bright, tunable, and narrow-band single-photon sources [@Lounis:05]. Another challenge stems from the fundamental wave character of propagating photons which leads to a weak coupling with matter. In this Letter, we demonstrate more than 55% coupling between a diffraction-limited beam of single photons and single silver nanoparticles, which act as classical dipolar antennae. This strong photon-dipole coupling allows efficient excitation of single plasmon-polaritons (plasmons) [@Akimov:07; @Tame:08] and imaging of nano-objects with nonclassical light.
The source of single photons in our experiments is a single dye molecule embedded in an organic crystalline matrix. Figure \[SPS\]a shows the energy level scheme of such a molecule as well as its excitation and fluorescence channels. We begin with experiments performed at $T\simeq1.4$ K, where we used a tunable narrow-band dye laser to excite dibenzanthanthrene (DBATT) embedded in n-tetradecane on the $S_{0, \rm v=0}\rightarrow S_{1, \rm v=1}$ transition at the wavelength of $\lambda\simeq581$ nm [@Lettow:07]. This state rapidly decays to the $S_{1, \rm v=0}$ state that has a lifetime of 9.5 ns determined by fluorescence to the $S_{0, \rm v}$ states. By filtering the broad Stokes-shifted fluorescence to the $\rm v\neq0$ manifold and collecting the emission on the $S_{1, \rm v=0}\rightarrow S_{0, \rm
v=0}$ zero-phonon line (ZPL), we obtained a source of single photons at $\lambda\simeq589$ nm with a lifetime-limited linewidth of 17 MHz [@Lettow:07]. Figure \[SPS\]b displays a recorded second-order autocorrelation function that proves the strongly photon-antibunched nature of this light. More details of the cryogenic setup and characterization of the single-photon source can be found in Ref. [@Lettow:07]. Here, it suffices to point out that this narrow-band single-photon source delivers a high power of up to 500 fW, corresponding to about $10^6$ detected photons per second.
\[b\]
![ (a) The energy level scheme of a dye molecule. See text for details. (b) An example of the second-order correlation function of a single DBATT molecule under continuous-wave excitation. \[SPS\]](SPS.eps){width="7cm"}
The collimated beam of single photons was coupled into a single-mode fiber and directed to a home-built microscope at room temperature as shown in Fig. \[setup\]a. An oil-immersion objective with a numerical aperture (NA) of 1.4 focused this light onto single silver nanoparticles with nominal diameter of 60 nm (British Biocell), which were spin coated on a glass cover slide and index-matched by immersion in oil (refractive index=1.49). Another microscope objective (NA=1.4) collected the transmitted light and sent it onto an avalanche photodiode (APD). A second APD was used to record the signal in reflection. In addition, we used flip mirrors to couple the light from the dye laser or a white-light source directly to the room-temperature microscope for characterization and diagnostics of the nanoparticles on the sample. By inserting a pinhole in the image plane, we could select each single particle and record its plasmon spectrum using a grating spectrometer.
The red trace in Fig. \[setup\]b plots the plasmon spectrum of a nanoparticle that matched our single photon source at $\lambda=589$ nm indicated by the black curve. This spectrum corresponds to a prolate silver ellipsoid with a short axis of 46 nm and a long axis of 94 nm that is parallel to the substrate. We find good agreement with the results of calculations that considered a dipolar scatterer [@EPAPS] and illumination parallel to the long axis of the particle (see blue curve of Fig. \[setup\]b). Indeed, electron microscopy revealed that the colloidal particles were mostly elongated (see Fig. \[setup\]c) with a notable variation in shape and size. We, thus, selected nanoparticles that matched the wavelength of our single-photon source (see Fig. \[setup\]b) and maximized the signal.
![(a) Single dye molecules embedded in a thin organic matrix at T=1.4 K produce a beam of single photons. This beam is collected and collimated by a solid-immersion lens and an aspherical lens inside the cryostat and then coupled into a single-mode fiber (SMF). The output of this fiber is sent to the sample in a room-temperature microscope. Two avalanche photodiodes rAPD and tAPD register the signal in reflection and transmission, respectively. A spectrometer records the plasmon spectrum of a particle upon illumination by a white-light source. DM: dichroic mirror, BS: beam splitter. (b) The red curve shows the experimentally measured plasmon spectrum of the particle studied in the first experiment. The blue curve displays a theoretical spectrum corresponding to an ellipsoidal silver particle with long and short axes of 94 and 46 nm, respectively. The black curve shows the spectrum of the narrow-band single-photon source. (c) Electron microscope image of a typical silver spheroid. \[setup\]](setup.eps){width="8cm"}
Figures \[SP-zooms\]a and b display images of a nanoparticle recorded simultaneously in transmission and reflection as the sample was scanned by a piezo-electric stage across the focus of the laser beam. The origin of these signals is scattering of the incident light from the nanoparticle [@Bohren-83book], and the details of the contrast mechanism and the detection scheme are discussed in the literature [@Mikhailovsky:03; @Lindfors:04]. Here, we briefly highlight the interferometric character of the signal $I_{\rm d}$ recorded on the detector, which is described as $$I_{\rm d}=\left\vert E_{\rm ref}+E_{\rm sca}\right\vert
^{2}=\left\vert E_{\rm ref}\right\vert ^{2}+ \left\vert E_{\rm
sca}\right\vert^{2}-2\left\vert E_{\rm ref}\right\vert\left\vert
E_{\rm sca}\right\vert \sin \varphi. \label{signal}$$ where $E_{\rm sca}$ is the electric field of the scattered light at the detector position, and $E_{\rm ref}$ denotes the electric field of a “reference" beam. In case of the transmission signal, the reference is the incident light, whereas for the reflection signal, it is produced by a residual reflection of the illumination within the optical setup. Depending on the relation between $E_{\rm ref}$ and $E_{\rm sca}$, the second or the third term of Eq. (1) might dominate and determine the signal contrast [@Lindfors:04]. Furthermore, this is influenced by the scattering phase angle $\varphi$, which depends on the dielectric function of the nanoparticle at the illumination wavelength as well as its size and shape [@Lindfors:04].
Next, in Figs. \[SP-zooms\]d and e we present raster-scan images of the same single nanoparticle recorded under illumination by single photons. As shown by the two cross sections $\delta$ and $\epsilon$ in Fig. \[SP-zooms\]f, we find full width at half-maximum (FWHM) values of 300 nm and 260 nm in transmission and reflection images, respectively. The solid curves display the outcome of rigorous vectorial three-dimensional calculations [@EPAPS] considering a focused Gaussian beam [@Mojarad:09]. Here, the silver nanoparticle was modeled as a dipolar emitter, taking into account radiative and dynamic depolarization corrections [@Meier:83]. We found a good agreement between the theoretical predictions and the experimental results if we considered the polarizability corresponding to the scattering cross section reported in Fig. \[setup\]b [@EPAPS]. A correction to the observed FWHM, $\Delta X_{np}$, for the finite size of the nanoparticle could also be made according to $\Delta
X_{np}\approx\sqrt{(\Delta X_{pd})^2 + D^2_{np}}$ where $D_{np}=46$ nm is the particle size along the scan direction, and $\Delta X_{pd}$ is the FWHM calculated for a point dipole. However, this amounts to an adjustment of only 3 nm, which we have chosen to neglect here. We mention in passing that the appearance of the elliptical images in Figs. \[SP-zooms\]a,b,d,e is a well-known effect for tightly focused linearly-polarized light [@Richards1959]. Another noteworthy point is that because of the nearly index-matched sample, the large cross section of the silver particle, and a tight focusing, the reflection signal is dominated by the second term of equation (1) and, therefore, maps the intensity of the incident beam in the focus spot. The transmission signal, on the other hand, has a substantial contribution from the interference term that depends on the electric field of the excitation light, which has a larger spatial extent than the intensity [@EPAPS]. Indeed a comparison of the data in Figs. \[SP-zooms\]c and \[SP-zooms\]f reveals that the FWHMs in transmission are systematically wider than those in reflection images.
![Transmission (a) and reflection (b) images obtained when the sample was scanned laterally across the focus of a laser beam at a speed of 10 ms per pixel. (c) Cross sections from (a) and (b). Average of 12 transmission (d) and reflection (e) images obtained when the sample was laterally scanned in the focus of the single-photon beam at 40 ms per pixel. (f) Cross sections from (d) and (e). Light beams were polarized along the vertical directions of the images in (a), (b), (d), and (e). Scale bars correspond to 500 nm. \[SP-zooms\]](SP-zooms.eps){width="8cm"}
Tight focusing is key to achieving a strong coupling between a light field and a dipolar emitter [@Zumofen:08; @Mojarad:09; @Wrigge:08]. The cross sections in Fig. \[SP-zooms\]f reveal large extinction and reflection contrasts of 55% in transmission and 22% in reflection. This is in very good agreement with the rigorous vectorial calculations shown by the solid curves, which take into account the modal character of a Gaussian beam as well as the illumination and collection solid angles [@Zumofen:08].
The data presented above clearly show the large effect of a single dipolar oscillator on a propagating light beam. Given that the interaction of the incoming photons with the nanoparticle has been mediated by the excitation of its plasmon-polaritons, these results indicate a high probability that an incident photon excites a single plasmon [@Tame:08]. To define an efficiency for the conversion of a photon to a plasmon, we add the probabilities that it is absorbed and scattered by the particle. In general, computation of this quantity from the reflection and transmission measurements requires a careful consideration of the incident focusing angle $\alpha$ and the collection angle $\beta$ [@Zumofen:08; @Mojarad:09]. However, if $\alpha=\beta$, a simple argument based on energy conservation lets us conclude that the sum of the scattered and absorbed powers equals the power removed from the incident beam, which is directly read from the transmission dip. If $\alpha<\beta$, some of the light that is scattered at larger angles is also collected, which reduces the transmission dip. In our case, the illumination and collection microscope objectives were identical, but the former was not completely filled in order to minimize the loss of photons. Thus, the data in Fig. \[SP-zooms\]f yield a lower bound of 55% for the photon-plasmon conversion efficiency.
The two series of images in Fig. \[SP-zooms\] acquired with laser light and a single photon source appear nearly identical. However, in the first case the signal can be described by the interference of classical fields, while the contrast mechanism of the images recorded by the latter can only be understood as the result of a Young-double-slit type of experiment for single photons [@Taylor:09; @Davis:88]. Here, the two interfering paths for each photon correspond to scattering by the nanoparticle and transmission without any interaction. After averaging the signal accumulated from a large number of single photons at each pixel, one retrieves the results familiar from classical optics.
We have demonstrated that a focused beam of single photons can be used to detect and image nanoparticles. This beam can also be produced in a triggered fashion by using a pulsed excitation of the molecule on the $S_{0, \rm v=0}\rightarrow S_{1, \rm v=1}$ transition [@Ahtee:09]. If the excitation beam is strong enough, one can ensure the production of a photon after each pulse, yielding an intensity-squeezed light source with a well-defined number of photons per unit time. Such a light source would allow the detection of objects with arbitrarily small optical contrast because it eliminates noise on the first term $\left\vert E_{\rm
ref}\right\vert ^{2}$ of Eq. (1) so that the second and third terms can be deciphered regardless of their magnitudes [@Lindfors:04; @Lounis:05; @Wrigge:08b]. One should bear in mind, however, that any loss in the optical system reduces the degree of squeezing [@Kolobov:99]. In our experiment, the central source of loss has been the limited collection angle of the lens used behind the solid-immersion lens in the cryostat (see Fig. \[setup\]a). This can be substantially improved by employing different choices of lenses [@Koyama:99].
![a) A raster-scan image of a silver nanoparticle illuminated by single photons from a terrylene molecule at room temperature. The black region at the bottom indicates loss of signal caused by the photobleaching of the molecule. b) A cross section from part (a). The inset displays the emission spectrum of a single terrylene molecule. \[room-temp\]](room-temp.eps){width="7cm"}
A second major cause for losses in the setup of Fig. \[setup\]a is the coupling into an optical fiber. To verify that use of a single mode and spatial mode filtering is not a strict requirement for the ability to focus single photons to the diffraction limit, we also performed free-beam measurements. Here, we collimated the single photon emission of a room-temperature terrylene molecule embedded in a thin para-terphenyl film [@pfab:04; @Lounis:00], and sent it directly to the second microscope as described earlier in Fig. \[setup\]a. Figure \[room-temp\]a displays an example of an image of a silver nanoparticle recorded in this fashion, while Fig. \[room-temp\]b shows a cross section from it. We find that the full width at half-maximum is as small as 370 nm [@note], demonstrating that freely propagating single photons can be indeed focused tightly. Here, the transmission dip amounts to only 15% because in this experiment we did not search for a nanoparticle with a plasmon resonance that matched the emission spectrum of terrylene. Moreover, the photon-plasmon coupling efficiency in this arrangement is less efficient than the narrow-band single photon source discussed earlier because the broad room-temperature emission of terrylene (see inset in Fig. \[room-temp\]b) does not fully overlap with the particle plasmon resonance.
Strong focusing of single photons demonstrated in this work opens doors to many interesting experiments where photons are to be managed with high efficiency and funneled to other quantum systems in the condensed phase. For example, one can use a silver nanoparticle as a nanoantenna [@Knight:07] to convert single propagating photons to single plasmons in nano-circuits. As opposed to the near-field coupling of photons from pre-positioned emitters to nanowires [@Akimov:07], coupling via propagating beams has the great advantage of being versatile with potential for broad-band communication because a large number of narrow-band single photon beams can be coupled simultaneously or sequentially via the same nano-antenna port. Plasmons can in turn generate electrons in a photovoltaic process where a quantum of excitation at optical frequencies gives birth to an electron [@Falk:09]. These processes would offer interesting possibilities for quantum state engineering of hybrid systems. Another important promise of our work is in the detection and imaging of very small nanoparticles and single molecules [@Lindfors:04; @Lounis:05; @Kukura:09]. To achieve this, we plan to optimize the use of solid-immersion lenses to reach collection efficiencies in excess of 90% [@Koyama:99], and thus produce an intensity-squeezed train of single photons from a single molecule.
We thank J. Hwang; N. Mojarad, and G. Zumofen for fruitful discussions. This work was funded by the Swiss National Science Foundation (SNF).
[28]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , , ****, ().
, , , , ****, ().
, , , , , ****, ().
, *et al.,* ****, ().
, *et al.* , p. ().
, , , , ****, ().
, *et al.*, in **, edited by , , , (, ), pp. .
, ****, ().
, *et al.* , ****, ().
, , , ****, ().
, ****, ().
, *et al.* , ****, ().
, *et al.* , ****, ().
, *et al.* , ****, ().
, ** (, ).
, , , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, *et al.* , ****, ().
, , , , , ****, ().
, , , , , ****, ().
, , , , , , ****, ().
, ****, ().
, , , , , , ****, ().
, *et al.* , ****, ().
, , , , ****, ().
|
---
abstract: 'Let $\kk$ be an algebraically closed field of characteristic 0, and $A=\bigoplus_{i \in \NN} A_i$ a Cohen–Macaulay graded domain with $A_0=\kk$. If $A$ is semi-standard graded (i.e., $A$ is finitely generated as a $\kk[A_1]$-module), it has the [*$h$-vector*]{} $(h_0, h_1, \ldots, h_s)$, which encodes the Hilbert function of $A$. From now on, assume that $s=2$. It is known that if $A$ is standard graded (i.e., $A=\kk[A_1]$), then $A$ is level. We will show that, in the semi-standard case, if $A$ is not level, then $h_1+1$ divides $h_2$. Conversely, for any positive integers $h$ and $n$, there is a non-level $A$ with the $h$-vector $(1, h, (h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).'
address:
- 'Department of Mathematics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, Japan, 603-8555'
- 'Department of Mathematics, Kansai University, Suita, Osaka 564-8680, Japan'
author:
- Akihiro Higashitani
- Kohji Yanagawa
title: 'Non-level semi-standard graded Cohen–Macaulay domain with $h$-vector $(h_0,h_1,h_2)$'
---
[^1]
Introduction
============
Let $\kk$ be a field, and $A=\bigoplus_{i \in \NN} A_i$ a graded noetherian commutative ring with $A_0=\kk$. If $A =\kk[A_1]$, that is, $A$ is generated by $A_1$ as a $\kk$-algebra, we say $A$ is [*standard graded*]{}. If $A$ is finitely generated as a $\kk[A_1]$-module, we say $A$ is [*semi-standard graded*]{}. The [*Ehrhart rings*]{} of lattice polytopes (see §4 below) and the face rings of simplicial posets (see [@St2]) are typical examples of semi-standard graded rings. In this sense, the notion of semi-standard graded rings is natural in combinatorial commutative algebra.
If $A$ is a semi-standard graded ring of dimension $d$, its Hilbert series is of the form $$\sum_{i \in \NN} (\dim_\kk A_i) t^i =\frac{h_0+h_1 t + \cdots +h_st^s}{(1-t)^d}$$ for some integers $h_0, h_1, \ldots, h_s$ with $\sum_{i=1}^s h_i \ne 0$ and $h_s \ne 0$. We call the vector $(h_0,h_1,\ldots,h_s)$ the [*$h$-vector*]{} of $A$. We always have $h_0=1$ and $\deg A=\sum_{i=0}^s h_i$.
If a semi-standard graded ring $A$ is Cohen–Macaulay, its $h$-vector satisfies $h_i \ge 0$ for all $i$. If further $A$ is standard graded, we have $h_i > 0$ for all $i$. For further information on the $h$-vectors of Cohen–Macaulay semi-standard (resp. standard) graded [*domains*]{}, see [@St] (resp. [@Y]).
If a semi-standard graded ring $A$ is Cohen–Macaulay and of dimension $d$, it admits the (graded) canonical module $\omega_A$, which is a $d$-dimensional Cohen–Macaulay $A$-module with the Hilbert series $$\sum_{i \in \ZZ} \dim_\kk (\omega_A)_i t^i =\frac{t^{d-s}(h_s+h_{s-1} t + \cdots +h_0t^s)}{(1-t)^d},$$ where $(h_0, h_1, \ldots, h_s)$ is the $h$-vector of $A$ (see [@BH §3.6 and Theorem 4.4.6]).
In the above situation, if $\omega_A$ is generated by $(\omega_A)_{d-s}$ as an $A$-module (equivalently, $\omega_A$ is generated by elements all of the same degree), we say $A$ is [*level*]{}.
Clearly, the notion of level rings generalizes that of Gorenstein rings. It is easy to see that a Cohen–Macaulay semi-standard graded ring with the $h$-vector $(h_0,h_1)$ or $(h_0, 0, h_2)$ is always level. The following fact, whose assumption that $A$ is a domain is really necessary, is a special case of [@Y Theorem 3.5]. This essentially follows from the theory of algebraic curves, and might be an old result.
Let $\kk$ be an algebraically closed field of characteristic 0, and $A$ a Cohen–Macaulay standard graded $\kk$-algebra. If the $h$-vector of $A$ is of the form $(h_0, h_1, h_2)$ and $A$ is an integral domain, then $A$ is level.
In this paper, we weaken the assumption on $A$ in the above result to be semi-standard graded. The following is the first main result.
\[main0\] Let $\kk$ be an algebraically closed field of characteristic 0, and $A$ a Cohen–Macaulay semi-standard graded domain with the $h$-vector $(h_0,h_1,h_2)$. If $A$ is not level, then ($h_1 \ne 0$ and) $h_1+1$ divides $h_2$.
The outline of the proof is the following. By Bertini’s theorem, we may assume that $\dim A=2$. Under the assumption of the theorem, if $A$ is not level, then the subring $B:=\kk[A_1]$ is isomorphic to the Veronese subring $\kk[x^n, x^{n-1}y, \ldots, xy^{n-1}, y^n]$ of $\kk[x,y]$. Next we regard $A$ as a $B$-module. Then it is a maximal Cohen–Macaulay module, and we consider its direct sum decomposition. However, the classification of indecomposable maximal Cohen–Macaulay modules over $\kk[x^n, x^{n-1}y, \ldots, xy^{n-1}, y^n] \ (\cong B)$ is well-known, and we can determine the $B$-module structure of $A$.
The next result states that the “converse" of the above theorem holds.
For any positive integers $h$ and $n$, there is a Cohen–Macaulay semi-standard graded domain which is not level and has the $h$-vector $(1, h, (h+1)n)$. Moreover, these rings can be constructed as Ehrhart rings (equivalently, as normal affine semigroup rings).
Hence, even if we restrict our attention to Ehrhart rings, Theorem \[main0\] has much sense. However we have no combinatorial proof of this theorem in the Ehrhart ring case.
Preliminaries
=============
In this section, we collect basic facts we will use in the next section. See [@BH] for undefined terminology and basic properties of Cohen–Macaulay rings. Throughout this section, let $A$ be a semi-standard graded ring with $d= \dim A$, and $(h_0, h_1, \ldots, h_s)$ its $h$-vector. Note that $A$ is a graded local ring with the graded maximal ideal $\mm= \bigoplus_{i > 0} A_i$. Since $\dim A = \dim \kk[A_1]$, the ideal of $A$ generated by $A_1$ is $\mm$-primary. Hence, if $|\kk|=\infty$, we can take a system of parameter $\theta_1, \ldots, \theta_d$ of $A$ from $A_1$.
Let $A^\sat$ denote the [*saturation*]{} $A/H_\mm^0(A)$ of $A$. It is clear that $A$ and $A^\sat$ define the same projective scheme, that is, we have $\Proj A = \Proj (A^\sat)$. In this paper, the [*homogeneous coordinate ring*]{} of a projective scheme $X \subset \PP^n$ means the [*standard*]{} graded ring $R$ with $X=\Proj R$ and $R=R^\sat$. Of course, there is a standard graded polynomial ring $S=\kk[x_0, \ldots, x_n]$ with the graded surjection $f: S \to R$ which induces the inclusion map $\Proj R=X \hookrightarrow \PP^n =\Proj S$. We say $X \subset \PP^n$ is [*non-degenerate*]{}, if no hyperplane of $\PP^n$ contains $X$, equivalently, $\dim_\kk R_1= n+1$.
In the rest of this section, we use the following convention.
- $B =\kk[A_1]$ is the subalgebra of $A$ generated by $A_1$ over $\kk$.
- $S=\kk[x_1, \ldots, x_m]$ is a standard graded polynomial ring, where $m=\dim_\kk A_1=h_1+d$. Note that $B$ can be seen as a quotient ring of $S$.
Clearly, $A$ is a finitely generated graded $S$-module. For a finitely generated graded $S$-module $M$, $\beta_{i,j}^S(M)$ (or just $\beta_{i,j}(M)$) denotes the graded Betti number $\dim_\kk [\Tor_i^S(\kk,M)]_j$ of $M$. We also set $\beta_i(M) :=\sum_{j \in \ZZ} \beta_{i,j}(M)$. Since $A_1 =S_1$, we have $\beta_{i,i}(A)=0$ for all $i >1$. If $A$ is Cohen–Macaulay, we have $$\label{Betti dual}
\beta_{i,j}(\omega_A)=\beta_{m-d-i, m-j}(A),$$ where $\omega_A$ is the canonical module of $A$. Let $r(A)$ denote the number of minimal generators of $\omega_A$ as a graded $A$-module, and call it the [*Cohen–Macaulay type*]{} of $A$. Clearly, $A$ is level if and only if $h_s =r(A)$.
For a finitely generated graded $S$-module $M$, $$\reg_S(M) :=\max \{ j-i \mid \beta_{i,j}(M) \ne 0 \}$$ is called the [*Catelnuovo-Mumford regularity*]{} of $M$ ([@EG]). While the theory of Catelnuovo-Mumford regularities is very deep, we only use elementary properties. For example, if $A$ is Cohen–Macaulay, then we have $\reg A=s$.
The following easy result might be well-known to the experts, but we give the proof for the reader’s convenience.
\[h1=0\] If a Cohen–Macaulay semi-standard graded ring $A$ has the $h$-vector of the form $(h_0, 0, h_2)$, then $A$ is level.
We may assume that $|\kk| = \infty$. So we can take a system of parameter $\{\theta_1, \ldots, \theta_d\} \subset A_1$. Then $A/(\theta_1, \ldots, \theta_d)$ has the same $h$-vector as $A$, and $A$ is level if and only if so is $A/(\theta_1, \ldots, \theta_d)$. Hence we may assume that $\dim A=0$. In this case, we have $$\dim_\kk (\omega_A)_i = \begin{cases}
h_s & \text{if $i=-2$,}\\
1 \,(=h_0) & \text{if $i=0$,}\\
0 & \text{otherwise.}
\end{cases}$$ If $A$ is not level, then $\mm \cdot (\omega_A)_{-2}=0$ and $\omega_A = (\omega_A)_{-2} \oplus (\omega_A)_0$ as an $A$-module. This is a contradiction, since $\omega_A$ is indecomposable in general.
\[level by Betti\] Let $A$ be a Cohen–Macaulay semi-standard graded ring with the $h$-vector $(h_0, h_1, h_2)$. Assume that $c:=h_1(=m-d)>0$. Then $A$ is level if and only if $\beta_{c, c+1}(A)=0$.
Since $\reg A =2$, we have $\beta_{c,j}(A)=0$ for all $j \ne c+1, c+2$. Hence if $\beta_{c, c+1}(A)=0$, then $\beta_0(\omega_A) = \beta_{0,d-2}(\omega_A)$ by . It means that $\omega_A$ is generated by $(\omega_A)_{d-2}$ as an $S$-module, but it clearly implies that $\omega_A$ is generated by $(\omega_A)_{d-2}$ as an $A$-module. Hence $A$ is level.
Next we assume that $\beta_{c,c+1}(A) \ne 0$. Then $\beta_{0, d-1}(\omega_A) \ne 0$, and hence $S_1 \cdot (\omega_A)_{d-2} \subsetneq (\omega_A)_{d-1}$. Since $S_1 =A_1$, we have $A_1 \cdot (\omega_A)_{d-2} \subsetneq (\omega_A)_{d-1}$, and $A$ is not level.
\[CM type\] If a Cohen–Macaulay semi-standard graded ring $A$ has the $h$-vector $(h_0, h_1, h_2)$ with $c:=h_1>0$, then we have $r(A)=\beta_c(A)$.
As we have seen in the proof of Lemma \[level by Betti\], we have $\beta_c(A)=\beta_0(\omega_A)= \beta_{0,d-2}(\omega_A)+ \beta_{0,d-1}(\omega_A)$. Since $A_1=S_1$, the number of minimal generators of $\omega_A$ as a graded $A$-module is equal to that as a graded $S$-module.
Assume that $\kk$ is an algebraically closed field. It is a classical result that if $B$ is a domain (but not necessarily Cohen–Macaulay) then we have $$\deg B \ge \codim B+1,$$ where $\codim B:= \dim_\kk B_1-\dim B = m-d=h_1$. If the equality holds, then $B$ is Cohen–Macaulay. Moreover, Del Pezzo–Bertini’s theorem gives a classification of standard graded domains $B$ with $\deg B = \codim B+1$ (see, for example, [@EG Theorem 4.3]). In particular, if $\dim B=2$, then $B$ is the homogeneous coordinate ring of a rational normal curve.
$h_1+1$ divides $h_2$
=====================
In this section, we always assume that the base field $\kk$ is an algebraically closed field of characteristic 0. Let $X$ be a finite set of points in the projective space $\PP^n=\Proj
(\kk[x_0, \ldots, x_n])$, and $R$ its homogeneous coordinate ring. Then $R$ is a Cohen–Macaulay standard graded ring with $\dim R=1$ and $\deg R = \# X$. We define the function $H_X:\ZZ \to \NN$ by $H_X(i)=\dim_\kk R_i$. If $(h_0, h_1, \ldots, h_s)$ is the $h$-vector of $R$, we have $s=\min \{ i \mid H_X(i) = \# X\}$ and $h_i= H_X(i) -H_X(i-1)$ for all $0 \le i \le s$.
Let $X \subset \PP^n$ be a finite set of points. We say that $X$ is in [*uniform position*]{}, if $H_X(1)=n+1$ (i.e., $X$ is non-degenerate) and every subset $Y \subset X$ satisfies $H_Y(i) = \min \{ H_X(i), \# Y \}$ for all $i$.
The usual definition of the uniform position property does not assume that $H_X(1)=n+1$, while many important examples satisfy it. Here we use the above definition for a quick exposition.
Note that if $X \subset \PP^n$ is a finite set of points in uniform position, and $Y \subset X$ is a subset with $\# Y \ge n+1$, then $Y \subset \PP^n$ is in uniform position again.
The following fundamental result is due to J. Harris. An application of this result to commutative algebra is found in the paper [@Y] of the second author.
\[UPL\] If $C \subset \PP^n$ is a reduced, irreducible and non-degenerate curve, then a general hyperplane section $C \cap H$ is a set of points in uniform position in $H \cong \PP^{n-1}$.
The following lemma must be well-known to the specialists, and there are several proofs. We will give one of them for the reader’s convenience.
\[uniform level\] Let $X \subset \PP^n$ be a finite set of points in uniform position, and $S=\kk[x_0, \ldots, x_n]$ (resp. $R$) the homogeneous coordinate rings of $\PP^n$ (resp. $X$). If $\deg R =\#X > n+1$, then $\beta_{n,n+1}^S(R)=0$.
Let $(h_0, h_1, \ldots, h_s)$ be the $h$-vector of $R$. Since $h_0=1$, $h_1 =n$ and $\# X = \deg R= \sum_{i=0}^s h_i$, we have $s \ge 2$. We prove the assertion by induction on $\deg R$. If $\# X=n+2$, then it is easy to see that $$H_X(i)=\begin{cases}
1 & \text{if $i=0$}, \\
n+1 & \text{if $i=1$}, \\
n+2 & \text{if $i \ge 2$}.\\
\end{cases}$$ So [@Kr Corollary 2.5] implies that $R$ is a Gorenstein (note that the Cayley-Bacharach property is weaker than the uniform position property). Hence we have $\beta_n(R)=\beta_{n,n+2}(R)=1$ and $\beta_{n,n+1}(R)=0$.
Next assume that $\# X > n+2$. Set $X':= X \setminus \{ p\}$ for a point $p \in X$, and let $R'$ be the homogeneous coordinate ring of $X'$. Consider the exact sequence $$\label{IRR'}
0 \to I \to R \to R' \to 0.$$ Since $X$ is in uniform position, we have $\min\{ \, i \mid I_i \ne 0 \, \} = s \ge 2$. Applying $\Tor_\bullet^S(\kk,-)$ to the sequence , we have $\beta_{n,n+1}(I)=0$, and hence $\beta_{n,n+1}(R) \le \beta_{n,n+1}(R')=0$. Here the last equality is the induction hypothesis, since $X' \subset \PP^n$ is in uniform position again.
\[deg B > c+1\] Let $A$ be a Cohen–Macaulay semi-standard graded ring with the $h$-vector $(h_0,h_1,h_2)$. Assume that $B=\kk[A_1]$ is a domain (then $\deg B \ge \codim B +1 = h_1 +1$). If $\deg B > h_1 +1$, then $A$ is level.
First, we remark that if $A$ is not level then $\dim B\ge 2$. In fact, if $\dim B=1$, then $B$ is a polynomial ring (since $B$ is a standard graded domain now) and $\deg B =1$. This contradicts the assumption that $\deg B > h_1 +1$.
We use the same notation as the previous section. So $S$ is the standard graded polynomial ring with $S_1 = A_1 (=B_1)$. By Bertini’s theorem, if $\dim B \ge 3$, then there is some $x \in B_1 =S_1$ such that $\tilde{B}:=(B/xB)^\sat$ is a domain with $\dim \tilde{B}=\dim B-1$. Then $x$ is a non-zero divisor of $A$, and $A':=A/xA$ is a Cohen–Macaulay semi-standard graded ring with the $h$-vector $(h_0, h_1, h_2)$. Moreover, $B':=B/(xA \cap B)$ is the subalgebra of $A'$ generated by its degree 1 part. Since $H_\mm^0(B') \subset H_\mm^0(A') =0$, $B'$ is a quotient ring of $\tilde{B}$. However, $\tilde{B}$ is a domain with $\dim B'=\dim A'=\dim \tilde{B}$, hence we have $B'=\tilde{B}$. In particular, $\deg \tilde{B}=\deg B$. Since $A'$ is level if and only if so is $A$, we can reduce the statement on $A$ and $B$ to that on $A'$ and $B'$.
Repeating the above argument, we may assume that $\dim A=2$ (i.e, $\Proj B$ is a curve). In this case, there is some $x \in B_1 =S_1$ such that $\tilde{B}:=(B/xB)^\sat$ is the homogeneous coordinate ring of a finite set of points in uniform position by Theorem \[UPL\]. Let $B':= B/(xA \cap B)$ be the subalgebra of $A':=A/xA$ generated by its degree 1 part. Since $H_\mm^0(B') \subset H_\mm^0(A') =0$, $B'$ is a quotient ring of $\tilde{B}$. Moreover, since $A_1 =B_1$, we have $[(xA \cap B)/xB]_i =0$ for all $i \le 2$, and hence $\tilde{B}_i = (B/xB)_i = B'_i$ for all $i \le 2$.
Set $S':=S/xS$ and $c:= \dim S'-1 =h_1$. For $X:=\Proj \tilde{B}$ and $Y:= \Proj B'$, we have $Y \subset X \subset \PP^c$. Since $\tilde{B}_i =B'_i$ for $i \le 2$, we have $\# Y \ge H_Y(2) = H_X(2)> c+1$. Since $Y$ is in uniform position in $\PP^c$, we have $\beta_{c,c+1}^{S'}(B')=0$ by Lemma \[uniform level\].
Now let us prove that $A'$ is level. By Lemma \[level by Betti\], it suffices to show that $\beta_{c,c+1}^{S'}(A') = 0$. Consider the exact sequence $$\label{A'B'C'}
0 \to B' \to A' \to C' \to 0$$ of $S'$-modules. Since $C'_i=0$ for $i \le 1$, we have $\beta_{i,j}^{S'}(C')=0$ for all $i,j$ with $j \le i+1$. Applying $\Tor_\bullet^{S'}(\kk,-)$ to , it follows that $\beta_{i,i+1}^{S'}(A')=\beta_{i,i+1}^{S'}(B')$ for all $i$. Since $\beta_{c,c+1}^{S'}(B')=0$ as we showed above, we have $\beta_{c,c+1}^{S'}(A')=0$. It means that $A'$ is level, and so is $A$ itself.
\[main1\] Let $A$ be a Cohen–Macaulay semi-standard graded ring with the $h$-vector $(h_0,h_1,h_2)$. If $A$ is not level and $B=\kk[A_1]$ is a domain, then $h_1+1$ divides $h_2$.
If $\dim B=1$, then $B$ is a polynomial ring and $h_1=0$. This contradicts the assumption that $A$ is not level by Lemma \[h1=0\]. Hence we have $\dim B \ge 2$. By the argument using Bertini’s theorem, we can reduce to the case $\dim B = 2$ as in the proof of Proposition \[deg B > c+1\]. By the proposition, we have $\deg B = h_1 +1$, and hence $B$ is the homogeneous coordinate ring of a rational normal curve as we have remarked in the last of the previous section. In other words, $B \cong \kk[x^n, x^{n-1}y, \ldots, xy^{n-1}, y^n]$, where $n+1=h_1 +2=\dim_\kk B_1$, that is, $B$ is isomorphic to the $n$th Veronese subring $T^{(n)}=\bigoplus_{i \in \NN} T_{ni}$ of the 2-dimensional polynomial ring $T= \kk[x,y]$. Let $S$ be the polynomial ring of $n+1$ variables, and regard $B$ as a quotient ring of $S$ as before. Then we have $\reg_S B=2$.
Note that $A$ is a 2-dimensional Cohen–Macaulay graded $B$-module. Next we consider the direct sum decomposition of $A$ as a $B$-module. Identifying $B$ as the $n$th Veronese subring $T^{(n)}$ of $T= \kk[x,y]$, an indecomposable Cohen–Macaulay graded $B$-module of dimension 2 is isomorphic to $$V(m) := \bigoplus_{i \in \NN} T_{m + ni}$$ for some $0 \le m < n$ up to degree shift. This is a classical result, and can be proved by a similar way to its “local version" ([@Yo Proposition 10.5]), since $T^{(n)}$ is an invariant subring of $T$ by a cyclic group of the order $n$.
By the argument using Hilbert functions, we see that $A \cong B \oplus C$ as $B$-modules, where $C$ is a 2-dimensional Cohen–Macaulay module. We have $C_i=0$ for all $i \le 1$ and $\reg C=2$. In other words, $C$ has a 2-linear resolution.
For simplicity, we set the degree 2 part of $V(m)$ as a graded $B$-module (or $S$-module) to be $T_m$. In other words, we use the convention that $V(m)$ is generated by its degree 2 part. The Hilbert series of $V(m)$ is $$\sum_{i \in \NN} (\dim_\kk [V(m)]_i) \cdot t^i = \frac{(m+1)t^2+ (n-1-m)t^3}{(1-t)^2},$$ so $V(m)$ has a 2-linear resolution as a graded $S$-module if and inly if $m = n-1$. Hence we have $C \cong (V(n-1))^{\oplus l}$ for some $l \in \NN$, and the Hilbert series of $C$ is $$\dfrac{nl\cdot t^2}{(1-t)^2} =\dfrac{(h_1+1)l\cdot t^2}{(1-t)^2}.$$ On the other hand, the Hilbert series of $B$ is $(1+h_1 t)/(1-t)^2$, hence the Hilbert series of $A \, (\cong B \oplus C)$ is $$\dfrac{1+h_1 t+ (h_1+1)l\cdot t^2}{(1-t)^2},$$ and the $h$-vector of $A$ is $(1, h_1, (h_1 +1)l)$.
Let $A$ be a Cohen–Macaulay semi-standard graded ring with the $h$-vector $(h_0,h_1,h_2)$. If $A$ is not level and $B$ is a domain, then the Cohen-Macaulay type $r(A)$ of $A$ is equal to $h_1+h_2 \, (=\deg A -1)$.
Recall that $\deg A-1$ is the largest possible value of the Cohen-Macaulay type of $A$ in general.
Since $A$ is not level, we have $c:=h_1 >0$ and we can use Lemma \[CM type\]. With the same notation as in the proof of Theorem \[main1\], we have $$r(A)=\beta_c(A)=\beta_c(B)+\beta_c(C).$$ However, easy calculation shows that $\beta_c(B)=h_1$ and $\beta_c(C)=h_2$. So we are done.
Semi-standard graded normal affine semigroup rings and non-level Ehrhart rings
==============================================================================
In this section, we recall some notions for affine semigroup rings (toric rings) and we discuss semi-standard graded normal affine semigroup rings. We also introduce Ehrhart rings which are some kind of normal affine semigroup rings arising from lattice polytopes. It will be proved that every semi-standard graded normal affine semigroup ring can be always viewed as the Ehrhart ring of some lattice polytope. Finally, we will show the examples of non-level Ehrhart rings whose $h$-vectors are of the form $(1,h,n(h+1))$ (Theorem \[thm:Ehr\]).
For $A \subset \RR^d$, let $\RR_{\geq 0}A$ denote the cone generated by $A$, i.e., $$\RR_{\geq 0}A=\left\{\sum_{v \in A} r_v v \in \RR^d : r_v \in \RR_{\geq 0}\right\}.$$ For $B \subset \ZZ^d$, let $\gp(B)$ denote the group (the lattice) generated by $B$, i.e., $$\gp(B)=\left\{\sum_{v \in B} z_v v \in \ZZ^d : z_v \in \ZZ\right\}.$$
Let $C \subset \ZZ^d$ be an affine semigroup.
- We say that $C$ is [*pointed*]{} if $C$ contains no vector subspace of positive dimension.
- Let $\overline{C}=\RR_{\geq 0}C \cap \gp(C)$. We say that $C$ is [*normal*]{} if $C=\overline{C}$.
- Let $\kk[C]$ be the affine semigroup ring of $C$, i.e., $$\kk[C]:=\kk[{\bf X}^\alpha : \alpha=(\alpha_1,\ldots,\alpha_d) \in C],$$ where ${\bf X}^\alpha=\prod_{i=1}^d X_i^{\alpha_i}$ denotes a Laurent monomial. Note that $\kk[C]$ is positively graded if and only if $C$ is pointed, and $\kk[C]$ is normal if and only if $C$ is normal.
When $\kk[C]$ is $\ZZ$-graded, by abuse of notation, we write $\deg(\alpha)=n$ if $\deg({\bf X}^\alpha)=n$ for $\alpha \in C$.
We also recall what the Ehrhart ring of a lattice polytope is. Let $P \subset \RR^d$ be a lattice polytope, which is a convex polytope all of whose vertices belong to the standard lattice $\ZZ^d$, of dimension $d$. We define the $\kk$-algebra $\kk[P]$ as follows: $$\begin{aligned}
\kk[P]=\kk[ {\bf X}^\alpha Z^n : \alpha \in nP \cap \ZZ^d, \; n \in \NN], \end{aligned}$$ where for $\alpha=(\alpha_1,\ldots,\alpha_d) \in \ZZ^d$, ${\bf X}^\alpha Z^n=X_1^{\alpha_1} \cdots X_d^{\alpha_d} Z^n$ denotes a Laurent monomial in $\kk[X_1^\pm, \ldots,X_d^\pm, Z]$ and $nP=\{nv : v \in P\}$. It is known that $\kk[P]$ is a semi-standard graded normal Cohen–Macaulay domain of dimension $d+1$, where the grading is defined by $\deg ({\bf X}^\alpha Z^n) =n$ for $\alpha \in nP \cap \ZZ^d$. The graded $\kk$-algebra $\kk[P]$ is called the [*Ehrhart ring*]{} of $P$.
\[chuui\] Let $C \subset \ZZ^d$ be a pointed normal affine semigroup and assume that $\kk[C]$ is $\ZZ$-graded. Let $C_1$ be the set of all degree one elements of $C$. If $C$ satisfies $C=\RR_{\geq 0}C_1 \cap \gp(C)$, then there exists a lattice polytope $P$ such that $\kk[C]$ is isomorphic to the Ehrhart ring of $P$ as $\kk$-algebras. In fact, let $Q$ be the convex hull of $C_1$. Then we can easily see that $\kk[C] \cong \kk[Q]$.
Actually, semi-standard graded normal affine semigroup rings are isomorphic to Ehrhart rings of lattice polytopes.
\[key\] Let $C$ be a normal affine semigroup. Assume that $\kk[C]$ is semi-standard graded. Then there exists a lattice polytope $P$ such that $\kk[C]$ is isomorphic to the Ehrhart ring of $P$ as $\kk$-algebras.
We see that $C$ is pointed since $\kk[C]$ is positively graded. Let $C_1$ be the set of all degree one elements of $C$. The normality of $C$ implies that $C$ contains all lattice points contained in $\RR_{\geq 0}C_1$, where “lattice points” stand for the points in $\gp(C)$. Thus, $\RR_{\geq 0}C_1 \cap \gp(C) \subset \overline{C} = C$. It suffices to show the equality $C=\RR_{\geq 0}C_1 \cap \gp(C)$ (see Remark \[chuui\]), in particular, $\RR_{\geq 0}C \subset \RR_{\geq 0}C_1$.
To prove this, it is enough to prove that all $1$-dimensional cones in $\RR_{\geq 0}C$ lie in $\RR_{\geq 0}C_1$. If this were wrong, then there is a lattice point $\alpha \in C \cap \gp(C)$ such that $\RR_{\geq 0}\{\alpha\} \subset \RR_{\geq 0}C \setminus \RR_{\geq 0}C_1$. Then $m\alpha \in (\RR_{\geq 0}C \setminus \RR_{\geq 0}C_1) \cap \gp(C)$ for any $m \in \ZZ_{>0}$. This implies that for any positive integer $m$, ${\bf X}^{m\alpha}$ will be a generator of $\kk[C]$ as a $\kk[C_1]$-module, where $\kk[C_1]$ denotes a subalgebra of $\kk[C]$ generated by its degree one elements. This contradicts the hypothesis that $\kk[C]$ is finitely generated as $\kk[C_1]$-module, i.e., $\kk[C]$ is semi-standard.
Therefore, $C=\RR_{\geq 0}C_1 \cap \gp(C)$. This says that $\kk[C]$ is isomorphic to the Ehrhart ring of some lattice polytope.
In the context of enumerative combinatorics on lattice polytopes, the $h$-vector of the Ehrhart ring $\kk[P]$ of $P$ is often called the [*$h^*$-vector*]{} (or the [*$\delta$-vector*]{}) of $P$. It is known that the $a$-invariant of the Ehrhart ring of $P$ can be computed as follows: $$\begin{aligned}
\label{socle}
a(\kk[P])=-\min\{ \ell \in \ZZ_{> 0} : \ell P^\circ \cap \ZZ^d \not= \emptyset\}, \end{aligned}$$ where $P^\circ$ denotes the interior of $P$. Note that $s=d+1+a(\kk[P])$ holds when the $h^*$-vector of $P$ is $(h_0^*,h_1^*, \ldots,h_s^*)$.
We can discuss whether $\kk[P]$ is level in terms of $P$ as follows.
\[prop:criterion\] Let $P \subset \RR^d$ be a lattice polytope of dimension $d$. Then $\kk[P]$ is level if and only if for each $n \geq -a(\kk[P])$ and for each $\alpha \in nP^\circ \cap \ZZ^d$, there exist $\alpha_1,\ldots,\alpha_{n+a(\kk[P])} \in P \cap \ZZ^d$ and $\beta \in (-a(\kk[P]))P^\circ \cap \ZZ^d$ such that $\alpha=\alpha_1+\cdots+\alpha_{n+a(\kk[P])}+\beta$.
We recall the well-known combinatorial technique how to compute the $h^*$-vector of a lattice [*simplex*]{}. Given a lattice simplex $\Delta \subset \RR^d$ of dimension $d$ with the vertices $v_0, v_1, \ldots, v_d \in \ZZ^d$, we set $$S_\Delta=\left\{ \sum_{i=0}^d r_iv_i \in \ZZ^d : \sum_{i=0}^d r_i \in \NN, \; 0 \leq r_i < 1 \right\}.$$ We define $\height(\alpha)=\sum_{i=0}^d r_i$ for each $\alpha=\sum_{i=0}^dr_iv_i \in S_\Delta$.
\[compute\] Let $(h_0^*,h_1^*,\ldots,h_s^*)$ be the $h^*$-vector of $\Delta$. Then one has $s=\max\{ \height(\alpha) : \alpha \in S_\Delta\}$ and $$h_i^*=|\{ \alpha \in S_\Delta : \height(\alpha)=i\}|$$ for each $i=0,1,\ldots,s$. Moreover, $\sum_{i=0}^d h_i^* = |S_\Delta|=\text{(the volume of $\Delta$)} \cdot d!$.
The following is the main result of this section.
\[thm:Ehr\] Given positive integers $h$ and $n$, there exists a lattice polytope $P_{h,n}$ such that its Ehrhart ring $\kk[P_{h,n}]$ satisfies the following:
- the $h$-vector of $\kk[P_{h,n}]$ (the $h^*$-vector of $P_{h,n}$) is $(1,h,n(h+1))$;
- $\kk[P_{h,n}]$ is non-level.
Let $v_0=(1,1,n), v_1=(0,1,0),v_2=(0,0,1)$ and $v_3=(1,-h,-nh)$ and let $P_{h,n}$ be the convex hull of them. Then $P_{h,n}$ is a lattice simplex of dimension $3$ with its vertices $v_0,v_1,v_2,v_3$. We will prove that $\kk[P_{h,n}]$ satisfies the required properties.
Let $\Delta=P_{h,n}$. First of all, we see that $\vol(\Delta) \cdot 3! =(n+1)(h+1)$ by calculating the determinant of the matrix $(v_1-v_0, v_2-v_0, v_3-v_0)$.
Next, let us compute $S_\Delta$. Let $$v:=\frac{h}{h+1}v_0+\frac{1}{h+1}v_3=(1,0,0) \in \Delta \cap \ZZ^3.$$ Then $v \in S_\Delta$ with $\height(v)=1$.
- For each $i=1,2,\ldots,h-1$, let $$w_i:=\frac{i}{h}v_3+\frac{h-i}{h}v=(1,-i,-ih).$$ Then $w_i \in S_\Delta$ with $\height(w_i)=1$.
- For each $j=1,2,\ldots,n$, let $$w_j':=\frac{j}{n+1}v_0+\frac{n+1-j}{n+1}v_1+\frac{j}{n+1}v_2+\frac{n+1-j}{n+1}v=(1,1,j).$$ Then $w_j' \in S_\Delta$ with $\height(w_j')=2$.
- For each $q=0,1,\ldots,h-1$ and $r=1,2,\ldots,n$, let $$\begin{aligned}
u_{q,r}:&=\frac{rh}{(n+1)h}v_1+\frac{n+1-r}{n+1}v_2+\frac{(n+1)q+r}{(n+1)h}v_3+\frac{(n+1)(h-q)-r}{(n+1)h}v \\
&=(1,-q,1-nq-r). \end{aligned}$$ Then $u_{q,r} \in S_\Delta$ with $\height(u_{q,r})=2$.
Hence, we obtain that $$\begin{aligned}
S_\Delta&=\{(0,0,0)\}\cup\{v\} \cup \{w_i : 1 \leq i \leq h-1\} \\
&\cup \{w_j' : 1 \leq j \leq n\} \cup \{u_{q,r} : 0 \leq q \leq h-1, \; 1 \leq r \leq n\}.\end{aligned}$$ Thus, the $h^*$-vector of $\Delta$ coincides with $(1,h,n(h+1))$ by Lemma \[compute\].
In addition, we also see that $$\begin{aligned}
\Delta \cap \ZZ^3 &= \{v_0,v_1,v_2,v_3\} \cup \{v\} \cup \{w_i : 1 \leq i \leq h-1\}, \text{ and }\\
2\Delta^\circ \cap \ZZ^3 &= \{w_j':1 \leq j \leq n\} \cup \{u_{q,r}: 0 \leq q \leq h-1, 1 \leq r \leq n\}.\end{aligned}$$ Consider $$w:=v_1+v_2+v=(1,1,1) \in 3\Delta^\circ \cap \ZZ^3.$$ Then we can see that there are no $\alpha \in \Delta \cap \ZZ^3$ and $\beta \in 2\Delta^\circ \cap \ZZ^3$ such that $w=\alpha+\beta$. This implies that $\kk[\Delta]$ is non-level by Proposition \[prop:criterion\].
\[degree 2 Ehrhart\] It is known by [@HT] and [@Treut] that for integers $a \ge 0$ and $b >0$, $(1,a,b)$ is the $h^*$-vector of some lattice polytope, i.e., the $h$-vector of some semi-standard graded normal affine semigroup ring if and only if $a \leq 3b+3$ or $(a,b)=(7,1)$ holds. The “If” part was proved in [@HT] and the “Only if” part was proved in [@Treut]. In [@HT], for $a \ge 0$ and $b >0$ satisfying $a \leq 3b+3$ or $(a,b)=(7,1)$, the lattice polytope whose $h^*$-vector coincides with $(1,a,b)$ is given. We can see that the associated Ehrhart rings of their examples are all [*level*]{}. Namely, there exists a level Ehrhart ring with the $h$-vector $(1,h,n(h+1))$ for any positive integers $h$ and $n$.
Let $C$ be a smooth projective curve of genus $g \, (\ge 1)$, and $\cL$ an invertible sheaf on $C$ with $\deg \cL = 2g+c$ for some $c \ge 1$. Then, by [@Mum Corollary to Theorem 6], the ring $$A = \bigoplus_{ n \in \NN} H^0(C, n \cL)$$ is a normal Cohen–Macaulay standard graded domain with the $h$-vector $(1, g+c-1,g)$. (If $c$ is non-positive but not so small, then $A$ has the same property in many cases. This is a classical topic of the curve theory, but we do not argue this direction here.)
Anyway, combining this observation with Remark \[degree 2 Ehrhart\], we see that, for any sequence $(1, a,b)$ for integers $a \ge 0$ and $b >0$, there is a normal Cohen–Macaulay semi-standard graded domain with the $h$-vector $(1, a, b)$. Moreover, we can take such rings from level rings.
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to Professors Chikashi Miyazaki and Kazuma Shimomoto for stimulating discussion. We also thank Professors Takesi Kawasaki and Kazunori Matsuda for valuable comments which helped to improve the exposition of the paper.
[10]{} E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, “Geometry of Algebraic Curves I", Grundlehren Math. Wiss., vol. 167, Springer-Verlag, 1985.
W. Bruns and J. Herzog, “Cohen–Macaulay rings", rev. ed., Cambridge Stud. Adv. Math. **39**, Cambridge Univ. Press, Cambridge, 1998
D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, [*J. Algebra*]{} [**88**]{} (1984), 89–133.
M. Henk and M. Tagami, Lower bounds on the coefficients of Ehrhart polynomials, [*Eur. J. Comb.*]{}, [**30**]{}(1):70–83, 2009.
T. Hibi, “Algebraic Combinatorics on Convex Polytopes,” Carslaw Publications, Glebe NSW, Australia, 1992.
M. Kreuzer, On the canonical module of a O-dimensional scheme, [*Canad. J. Math.*]{} [**46**]{} (1994) 357-379
R. Stanley, $f$-vectors and $h$-vectors of simplicial posets, [*J. Pure Appl. Algebra*]{} [**71**]{} (1991), 319–331.
R.P. Stanley, On the Hilbert function of a graded Cohen–Macaulay domain, [*J. Pure Appl. Algebra*]{} [**73**]{} (1991) 307–314.
J. Treutlein, Lattice polytopes of degree 2. [*J. Comb. Theory, Ser. A*]{} [**117**]{}(3):354–360, 2010.
D. Mumford, Varieties defined by quadratic equations, C.I.M.E. Conference on Questions on Algebraic Varieties, 1969, 31–100.
K. Yanagawa, Castelnuovo’s Lemma and $h$-vectors of Cohen–Macaulay homogeneous domains, [*J. Pure Appl. Algebra*]{} [**105**]{} (1995) 107–116.
Y. Yoshino, “Cohen–Macaulay modules over Cohen–Macaulay rings", London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990.
[^1]: The authors are partially supported by JSPS Grant-in-Aid for Young Scientists (B) 26800015, and Grant-in-Aid for Scientific Research (C) 16K05114, respectively.
|
---
abstract: 'We present an algorithm for calculating the complete data on an event horizon which constitute the necessary input for characteristic evolution of the exterior spacetime. We apply this algorithm to study the intrinsic and extrinsic geometry of a binary black hole event horizon, constructing a sequence of binary black hole event horizons which approaches a single Schwarzschild black hole horizon as a limiting case. The linear perturbation of the Schwarzschild horizon provides global insight into the close limit for binary black holes, in which the individual holes have joined in the infinite past. In general there is a division of the horizon into interior and exterior regions, analogous to the division of the Schwarzschild horizon by the $r=2M$ bifurcation sphere. In passing from the perturbative to the strongly nonlinear regime there is a transition in which the individual black holes persist in the exterior portion of the horizon. The algorithm is intended to provide the data sets for production of a catalog of nonlinear post-merger wave forms using the PITT null code.'
address:
- 'Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260'
- |
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260\
and Max-Planck-Institut f" ur Gravitationsphysik, Albert-Einstein-Institut, 14476 Golm, Germany
author:
- Roberto Gómez
- Sascha Husa and Jeffrey Winicour
title: Complete null data for a black hole collision
---
Introduction
============
In previous work, we developed a model which generates the intrinsic null geometry of an event horizon with the “pair-of-pants” structure characteristic of a binary black hole merger [@ndata; @asym]. In this paper, we extend this approach to determine all extrinsic curvature properties of such horizons, thus providing a complete stand-alone description of the event horizon of binary black holes. We apply this work to study the event horizon of a head-on collision of black holes using a sequence of models which embraces not only the perturbative regime of the close approximation [@pp], where the merger takes place in the distant past, but also includes the highly nonlinear regime. In the perturbative regime the individual black holes merge in an interior region of the horizon, corresponding to the region of the Schwarzschild event horizon lying inside the $r=2M$ bifurcation sphere. But we show how dramatically nonlinear effects can push the merger into the exterior portion of the horizon.
Beyond the investigation of the horizon geometry of a black hole collision, the major motivation for this work is to provide the null data necessary to compute the emitted gravitational wave by means of a characteristic evolution of the exterior spacetime. In the Cauchy problem, the necessary data on a spacelike hypersurface are the intrinsic metric and extrinsic curvature, subject to constraints. On a null hypersurface, such as an event horizon, the situation is quite different. The necessary null data consist of the conformal part of the intrinsic (degenerate) metric, which can be given freely as a function of the affine parameter. Then the surface area of the horizon is determined, via an ordinary differential equation along the null rays (the Raychaudhuri equation), in terms of an integration constant supplied by the mass of the final black hole. Similarly, all extrinsic curvature components of the horizon are determined by ordinary differential equations in terms of integration constants supplied by the final black hole. Whereas in principle the intrinsic conformal geometry is the only data on the horizon necessary for the characteristic initial value problem, in practice the surface area and extrinsic curvature are essential to supply the start-up data for the implementation of a characteristic evolution code, such as the PITT null code [@high; @wobb]. The main results of this paper concern the understanding of the nonlinear nature of the underlying ordinary differential equations from geometrical, physical and numerical points of view.
Given the intrinsic geometry and extrinsic curvature of the horizon, the strategy behind the characteristic approach to the computation of the emitted wave has been outlined elsewhere [@kyoto]. It consists of two evolution stages based upon the double null initial value problem. Referring to Fig. \[fig:strategy\], the necessity of two stages results from the disconnected nature of the two null hypersurfaces on which boundary conditions must be satisfied: (i) the event horizon ${\cal H}^+$, where binary black hole data is prescribed, and (ii) past null infinity ${\cal I}^-$, where ingoing radiation must be absent, at least in the late stage to the future of the hypersurface $\Gamma$ which is decoupled from the formation of the individual black holes. Rather than directly attempting to solve this mixed version of a characteristic initial value problem, the stage I evolution is based upon two intersecting null hypersurfaces consisting of the event horizon, where binary black hole data is prescribed, and an ingoing null hypersurface ${\cal J}^+$ approximating future null infinity ${\cal I}^+$, where data corresponding to no [*outgoing*]{} radiation is prescribed. The characteristic evolution then proceeds [*backward in time*]{} along ingoing null hypersurfaces extending to ${\cal
I}^-$ to determine the spacetime exterior to an ingoing null hypersurface $\Gamma$. As indicated in the figure, the evolution terminates at $\Gamma$ because the horizon splits into two individual black holes. This stage I solution is the [*advanced*]{} solution to the problem in the sense that radiation from ${\cal I}^-$ is absorbed by the black holes but no outgoing wave is radiated to ${\cal I}^+$. Stage II provides the [*retarded*]{} solution, where outgoing but no ingoing radiation is present outside $\Gamma$, by running forward in time a double null evolution based upon the intersecting null hypersurfaces $\Gamma$, where the stage I data is prescribed, and an outgoing null hypersurface ${\cal J}^-$ approximating ${\cal I}^-$, where data corresponding to no ingoing radiation is prescribed to the future of $\Gamma$. The stage II evolution produces the retarded solution for the spacetime outside the world tube $\Gamma$. The approach is a nonlinear version of the standard method of determining the retarded solution $\Phi_{RET}$ of the linear wave equation $\Box \Phi =S$, with source $S$, by first finding the advanced solution $\Phi_{ADV}$ and then superimposing the source-free solution $\Phi_{RET} - \Phi_{ADV}$. In the nonlinear case, where standard Green function techniques cannot be used to define retarded and advanced solutions, they can be defined by requiring the absence of radiation at ${\cal I}^+$ (retarded) or ${\cal I}^-$ (advanced). In the present case, the data on the world tube $\Gamma$ represents the interior source which has led to the formation of the individual black holes. (In a normal physical context, the source consists of two stars undergoing gravitational collapse but in a purely vacuum scenario imploding gravitational waves can play the role of the matter.) The justification of this two stage approach is that it reduces to the standard linear method in the close approximation where the geometry can be regarded as a perturbation of a Schwarzschild background. A purely perturbative characteristic treatment of the close approximation using the characteristic approach has been carried out in a separate study and the advanced solution has been successfully computed [@close1].
We intend to carry out the “backward in time” evolution using the PITT null code [@high; @wobb], which is based upon the Bondi-Sachs version of the characteristic initial value problem [@bondi; @sachs]. The present code is designed to evolve forward in time along a foliation of spacetime by outgoing null hypersurfaces. In this paper, in order to apply it to the backward in time evolution of a black hole horizon, we describe the merger of two black holes in the time reversed scenario of a white hole horizon, in which the black hole merger is now represented by the fission of a white hole. The post-merger era of the black hole horizon then corresponds to the pre-fission era of the white hole; and the proposed backward in time evolution of the black hole horizon to determine the exterior spacetime corresponds to a forward in time evolution of the white hole horizon. Our algorithm provides the complete white hole data necessary to carry out this evolution.
The specific application in this paper is to the axisymmetric head-on collision of two equal mass black holes. However, the algorithm is capable of generating intrinsic geometry and extrinsic curvature of an arbitrary event horizon, including the case of inspiraling binary black holes of non-equal mass. In previous studies of the intrinsic geometry of non-axisymmetric horizons, the approach has revealed new features of the generic collision of two black holes, such as an intermediate toroidal phase which precedes the merger [@asym]. Here we apply the algorithm to gain new insight into the global behavior of the extrinsic curvature properties of the head-on collision.
The material in the paper divides into two types: (i) general features of a binary horizon and (ii) the calculation of the null data on the horizon necessary to initiate the PITT code. Material of the first type, which describes how a binary horizon deviates from the close approximation in the nonlinear regime, appears in Secs. \[sec:horstruct\] and \[sec:confmod\] - \[sec:results\] and can be read independently. This material is described most conveniently in terms of Sachs coordinates, introduced in Sec. \[sec:horstruct\]. Material of the second type involves the transformation from Sachs to Bondi-Sachs coordinates, introduced in Sec. \[sec:Bondisachs\]. This material supplies all data necessary to compute, using the PITT code, the fully nonlinear merger to ringdown wave form, which will be the subject of future work. The details of implementing the horizon data algorithm as a finite difference code are given in the Appendixes. An independent code designed for an axisymmetric horizon is also described and has been used as an independent test of the full code.
We retain the conventions of our previous papers [@ndata; @asym; @high; @wobb], with only minor changes in notation where noted in the text. For brevity, we frequently use the notation $f_{,x}=\partial_x f$ to denote partial derivatives and $\dot f =\partial_u f$ to denote retarded time derivatives.
Characteristic data on a horizon
================================
Horizon structure {#sec:horstruct}
-----------------
The evolution of the exterior spacetime by the PITT code proceeds along a family of outgoing null hypersurfaces. The characteristic initial value problem for the evolution requires an inner boundary condition which can be set either on a timelike world tube or, as a limiting case, on a null world tube. Here we choose the inner boundary to be the null world tube representing a white hole horizon ${\cal H}$. The white hole horizon pinches off in the future where its generators either caustic or cross each other (such as at the vertex of a null cone). As illustrated in Fig. \[fig:whole\], we introduce (i) an affine null coordinate $u$ along the generators of ${\cal H}$, which foliates ${\cal H}$ into cross sections ${\cal S}_u$ and labels the corresponding outgoing null hypersurfaces ${\cal J}_u$ emanating from the foliation; (ii) angular coordinates $x^A$ which are constant both along the generators of ${\cal H}$ and along the outgoing rays and (iii) an affine parameter $\lambda$ along the outgoing rays normalized by $\nabla^{\alpha}u \nabla_{\alpha}\lambda =-1$, with $\lambda =0$ on ${\cal H}$. In the resulting $x^{\alpha}=(u,\lambda ,x^A)$ coordinates, the metric takes the form $$ds^2 = -(W -g_{AB}W^A W^B)du^2
-2dud\lambda -2g_{AB}W^Bdudx^A + g_{AB}dx^Adx^B.
\label{eq:amet}$$ The contravariant components are given by $g^{\lambda u}=-1$, $g^{\lambda
A}=-W^A$, $g^{\lambda\lambda}=W$ and $g^{AB}g_{BC}=\delta^A_C$. In addition, we set $g_{AB}=r^2h_{AB}$, where $\det(h_{AB})=\det(q_{AB})=q(x^A)$, where $q_{AB}$ is some standard choice of unit sphere metric.
These coordinates were first introduced by Sachs to formulate the double-null characteristic initial value problem [@sachsdn]. The Bondi-Sachs coordinate system [@bondi; @sachs] used in the PITT code differs by the use of a surface area coordinate $r$ along the outgoing cones rather than the affine parameter $\lambda$. However, because the horizon is not a surface of constant $r$, except in the special case of an “isolated horizon” [@ih5083], it is advantageous to first determine the necessary data in terms of an affine parameter and later transform to the $r$-coordinate.
The requirement that ${\cal H}$ be null implies that $W=0$ on ${\cal H}$. There is gauge freedom on ${\cal H}$ that we fix by choosing the shift so that $\partial_u$ is tangent to the generators, implying that $W^A=0$ on ${\cal H}$; and by choosing the lapse so that $u$ is an affine parameter, implying that $\partial_\lambda W=0$ on ${\cal H}$. We adopt these choices throughout the paper, and our results generally hold only on $\cal H$ ($\lambda = 0$) and when these conditions are satisfied. Later, in Sec. \[sec:headon\], we also fix the affine freedom in $u$ by specifying it on an initial slice ${\cal S}^-$ of ${\cal H}$, which is located at an early time approximating the asymptotic equilibrium of the white hole at past time infinity $I^-$. The outgoing null hypersurface ${\cal J}^-$ emanating from ${\cal S}^-$ approximates past null infinity ${\cal I}^-$. In Sec. \[sec:sminus\], we discuss the nature of that approximation.
On ${\cal H}$, the affine tangent to the generators $n^a\partial_a=\partial_u$ (see Fig. \[fig:whole\]) satisfies the geodesic equation $n^b\nabla_b n^a=0$ and the hypersurface orthogonality condition $n^{[a}\nabla^b n^{c]}=0$. Following the approach of Refs. [@ndata; @asym], we project 4-dimensional tensor fields into ${\cal H}$ using the operator $$P_a^b = \delta_a^b + n_a l^b,$$ where $l_a = -\nabla_a u$, and we use the shorthand notation $\perp T_a^b$ for the projection (to the tangent space of ${\cal H}$) of the tensor field $T_a^b$. The projection $\perp$ has gauge freedom corresponding to the choice of affine parameter $u$. However, the action of $\perp$ on covariant indices is independent of this freedom and equals the action of the pullback operator to ${\cal H}$.
In addition to the intrinsic geometry of ${\cal H}$, the necessary characteristic data consist of the extrinsic curvature quantities given by $\perp \nabla_a l_b$ (with gauge freedom corresponding to the affine choice of $u$). Since $\nabla_a l_b -\nabla_b l_a =0$, the independent components are determined by $\perp \nabla_{(a} l_{b)}$, which has the decomposition $$\perp \nabla_{(a} l_{b)} = -l_a \omega_b-\omega_a l_b +\tau_{ab}.
\label{eq:gradl}$$ Here $\tau_{ab}$ describes the shear and expansion of the outgoing rays and satisfies $n^a\tau_{ab}=0$ on ${\cal H}$. Following Hayward [@haywsn], we call $\omega_a =\perp n^b \nabla_b l_a$ the twist of the affine foliation of ${\cal H}$. The twist also satisfies $n^a \omega_a =0$. Unlike $\tau_{ab}$, the twist is an invariantly defined extrinsic curvature property of the $u=const$ cross sections of ${\cal H}$, independent of the boost freedom in the extensions of $n_a$ and $l_a$ subject to the normalization $n^a l_a=-1$.
Note that it is natural geometrically to associate extrinsic curvature properties of ${\cal H}$ with $(\perp \nabla_a)n^b$, in analogy with the Weingarten map for a non-degenerate hypersurface [@ih5083]. The normalization $n^a l_a =-1$ then leads, via Eq. (\[eq:gradl\]), to $$l_b(\perp \nabla_a )n^b =-\omega_a.$$ Thus the twist describes an extrinsic curvature property associated with $n^a$ as well as $l_a$. The other components of $(\perp \nabla_a )n^b$ are determined by the shear and expansion of ${\cal H}$ (which vanish in the special case of an isolated horizon).
We consider the double null initial value problem based upon data on the horizon ${\cal H}$ and the outgoing null hypersurface ${\cal J}^-$ emanating from ${\cal S}^-$, with the evolution proceeding along the outgoing null hypersurfaces ${\cal J}_u$ emanating from the $u=const$ foliation ${\cal S}_u$. In this problem, the complete (and unconstrained) characteristic data on ${\cal
H}$ are its affine parametrization $u$ and the conformal part $h_{AB}$ of its degenerate intrinsic metric. Similarly, the characteristic data on ${\cal J}^-$ are its affine parametrization $\lambda$ and its intrinsic conformal metric $h_{AB}$. The remaining data consist of the intrinsic metric and extrinsic curvature of ${\cal S}^-$ (subject to consistency with the characteristic data) [@sachsdn; @haywdn]. In Sachs coordinates, this data on ${\cal S}^-$ consists of $r$, $\dot r$, $\omega_a$ and $r_{,\lambda}$ (which determines the expansion of the outgoing null rays) on ${\cal S}^-$. That completes the data necessary to evolve the exterior spacetime. In carrying out such an evolution computationally, the first step is to propagate the data given on ${\cal S}^-$ along the generators of ${\cal H}$ so that it can be supplied as boundary data to the exterior evolution code. This first step is accomplished by means of certain components of Einstein’s equations.
Einstein’s equation decompose into (i) hypersurface equations intrinsic to the null hypersurfaces ${\cal J}_u$, which determine auxiliary metric quantities in terms of the conformal metric $h_{AB}$; (ii) evolution equations determining the rate of change $\partial_u h_{AB}$ of the conformal metric of ${\cal J}_u$; and (iii) propagation equations which are constraints that need only be satisfied on ${\cal H}$. One of the propagation equations is the ingoing Raychaudhuri equation $R_{uu}=0$ which determines the surface area variable $r$ along the generators of ${\cal H}$ in terms of initial conditions on ${\cal
S}^-$ according to $$\ddot r =\frac{r}{8} \dot h^{AB} \dot h_{AB}.
\label{eq:ruu}$$ The value of $\dot r$ on ${\cal S}^-$ measures the convergence of its ingoing null rays and Eq. (\[eq:ruu\]) implies $\ddot r\le 0$. The remaining propagation equations $R_{Au} =0$ propagate the twist $\omega_a$ along the generators of ${\cal H}$. Our coordinate conditions imply that $\omega_u=\omega_{\lambda}=0$ and $$\omega_A=-\frac{1}{2}\partial_{\lambda}(g_{AB} W^B ). \label{eq:wA}$$ The non-vanishing components propagate according to $$(r^2\omega_A)\dot {}=r^2 D_A(\frac{\dot r}{r})
-\frac{1}{2}h^{BC}D_B(r^2 \dot h_{AC} ),
\label{eq:omegadot}$$ where $D_A$ is the covariant derivative associated with $h_{AB}$. Here $h^{AB}h_{BC}=\delta^A_C$ and $h^{AB}\dot h_{AB} =0$. Having determined $r$, this equation can easily be integrated to determine $\omega_A$ on ${\cal H}$ in terms of initial conditions on ${\cal S}^-$. Once the propagation equations are solved on ${\cal H}$ to determine $r$ and $\omega_a$, the Bianchi identities ensure that they will be satisfied in the exterior spacetime as a result of the ${\cal J}_u$ hypersurface equations and the evolution equations.
The propagation of $\tau_{ab}$ (the outward shear expansion and shear) along ${\cal H}$ requires the $R_{AB}=0$ components of Einstein equations, given by $$\begin{aligned}
R_{AB} &=& h_{AB} \bigg (\frac{ {\cal R}}{2}-D^AD_A \,\log r\bigg )
- D_A \omega_B - D_B \omega_A -2\omega_A\omega_B
\nonumber \\
&+& \frac{2}{r}(\omega_A \partial_B r +\omega_B \partial_A r
-h_{AB}h^{CD}\omega_C \partial_D r)
\nonumber \\
&+& r^2\partial_\lambda \dot h_{AB}
-\frac{r^2}{2} h^{CD}(\dot h_{AC}\partial_\lambda h_{BD}
+\dot h_{BD}\partial_\lambda h_{AC} )
\nonumber \\
&+& 2(\dot r\partial_\lambda r +r\partial_\lambda \dot r)h_{AB}
+r(\partial_\lambda r) \dot h_{AB} + r\dot r \partial_\lambda h_{AB}.
\label{eq:rab}\end{aligned}$$ where ${\cal R}$ represents the curvature scalar of the metric $h_{AB}$. On ${\cal H}$, the equation $R_{AB}=0$ decomposes into the trace $$\partial_u\partial_\lambda (r^2) = +D^AD_A \,\log r
-\frac{1}{2}{\cal R} +D^A\omega_A +h^{AB}\omega_A\omega_B
\label{eq:rlam}$$ and, by introducing the dyad decomposition $h_{AB}=m_{(A}\bar
m_{B)}$, the trace-free part $$m^A m^B \bigg (r\partial_\lambda \partial_u(r h_{AB})-2D_A\omega_B
-2\omega_A\omega_B +\frac{4}{r}\omega_A D_B r \bigg ) =0.
\label{eq:jlamdot}$$ The trace equation propagates the outgoing expansion of the foliation ${\cal
S}_u$, as determined by $r_{,\lambda}$. The trace-free part is the evolution equation for the data $h_{AB}$ on the foliation ${\cal J}_u$, applied at ${\cal
H}$ to evolve $h_{AB,\lambda}$, which describes the shear of the outgoing rays. The outward shear constitutes part of the extrinsic curvature of ${\cal S}^-$. Although its value is implicitly determined by the null data $h_{AB}$ on ${\cal
J}^-$, we view it as part of the initial data that must be specified on ${\cal
S}^-$.
In summary, the data for the double null problem includes the conformal metric on ${\cal H}$ and the quantities $r$, $\dot r$, $\omega_a$, $r_{,\lambda}$ and $h_{AB,\lambda}$ on ${\cal S}^-$. Equations (\[eq:ruu\]), (\[eq:omegadot\]), (\[eq:rlam\]) and (\[eq:jlamdot\]) are then used to propagate the data on ${\cal S}^-$ to all of ${\cal H}$. The choice of foliation of the horizon is an important but complicated aspect of this problem. As in the case of Cauchy evolution, gauge freedom in the foliation introduces arbitrariness into the dynamical description of any black hole process and, in particular, the pair-of-pants structure underlying a black hole merger. In the case of the horizon, the natural choice of an affine foliation $u$ removes any time dependence from this gauge freedom but there remains the affine freedom $u\rightarrow Au+B$, where $A$ and $B$ are ray dependent. In Sec. \[sec:headon\], we use the asymptotic equilibrium of the white hole as $u\rightarrow -\infty$ to fix this freedom. We review in Sec. \[sec:confmod\] how the data $h_{AB}$ and $r$ on ${\cal H}$ describing a white hole fission (binary black hole merger) are provided by a conformal horizon model[@ndata; @asym]. The additional data required on ${\cal S}^-$ can be inferred from the asymptotic properties of the white hole equilibrium at $I^-$, as discussed in Sec. \[sec:headon\]. The evolution of the exterior spacetime by the PITT code requires a transformation of the data on ${\cal H}$ to a Bondi-Sachs coordinate system, as described next in Secs. \[sec:Bondisachs\] - \[sec:bondioff\].
Bondi-Sachs coordinates {#sec:Bondisachs}
-----------------------
The transformation to the Bondi-Sachs coordinates $x^{\alpha}=(u,r,x^A)$ used in the PITT code consists in substituting the surface area coordinate $r$ for the affine parameter $\lambda$. Since the horizon ${\cal H}$ does not in general have constant $r$ it does not lie precisely on radial grid points. Consequently, the assignment of horizon boundary values must be done on the grid points nearest to the boundary. Thus an accurate prescription of boundary conditions in the $r$-grid requires a Taylor expansion of the horizon data.
In Bondi-Sachs variables, the resulting metric takes the form $$ds^2=-\left(e^{2\beta}{V \over r} -r^2h_{AB}U^AU^B\right)du^2
-2e^{2\beta}dudr -2r^2 h_{AB}U^Bdudx^A +r^2h_{AB}dx^Adx^B.
\label{eq:umet}$$ In relating this to the Sachs metric Eq. (\[eq:amet\]), it is simplest to consider the contravariant form in which only the $g^{r\alpha}$ components differ. In terms of the corresponding metric variables, $$\begin{aligned}
g^{rr} &= e^{-2\beta} \frac{V}{r} &= (r_{,\lambda})^2 W
- 2\, r_{,\lambda} r_{,A} W^{A} - 2\, r_{,\lambda} r_{,u}
+ \frac{r_{,A} r_{,B}}{r^2} h^{AB}
\label{eq:VH} \\
g^{rA} &= -e^{-2\beta} U^A &=- r_{,\lambda} W^A + \frac{r_{,B}}{r^2} h^{AB}\\
g^{ru} &= -e^{-2\beta} &= - r_{,\lambda} .
\label{eq:rl}\end{aligned}$$ Restricted to ${\cal H}$, our choice of lapse and shift imply $$\begin{aligned}
\beta &=& -\frac{1}{2}\ln{r_{,\lambda}}
\label{eq:b}
\\
U^A &=& -\frac{e^{2\beta}}{r^2} r_{,B} h^{AB}
\label{eq:U}
\\
V &=& -2\, r r_{,u} + \frac{e^{2\beta}}{r} r_{,A} r_{,B} h^{AB}.
\label{eq:V}\end{aligned}$$
The structure of the ${\cal J}_u$-hypersurface equations and the evolution equations [@newt; @nullinf] reveals the horizon boundary data necessary for characteristic evolution. The hypersurface equations are $$\begin{aligned}
\beta_{,r} &=& \frac{1}{16}rh^{AC}h^{BD}h_{AB,r}h_{CD,r},
\label{eq:beta}
\\
(r^4e^{-2\beta}h_{AB}U^B_{,r})_{,r} &=&
2r^4 \left(r^{-2}\beta_{,A}\right)_{,r}
-r^2 h^{BC}D_{C}h_{AB,r}
\label{eq:u}
\\
2e^{-2\beta}V_{,r} &=& {\cal R} - 2 D^{A} D_{A} \beta
-2 D^{A}\beta D_{A}\beta + r^{-2} e^{-2\beta} D_{A}(r^4U^A)_{,r} \nonumber \\
&-&\frac{1}{2}r^4e^{-4\beta}h_{AB}U^A_{,r}U^B_{,r} \, ,
\label{eq:v}\end{aligned}$$ where here $D_A$ is the covariant derivative and ${\cal R}$ the curvature scalar of the 2-metric $h_{AB}$ of the $r=const$ surfaces (which differ from the corresponding quantities on the $\lambda=const$ surfaces). These equations form a hierarchy which can be integrated radially in order to determine $\beta$, $U^A$ and $V$ on a hypersurface ${\cal J}_u$ in terms of integration constants on ${\cal S}_u$, once the null data $h_{AB}$ has been evolved to ${\cal J}_u$.
The evolution variable $h_{AB}$ can be recast as a single complex function, since $\det (h_{AB})= \det (q_{AB})=q(x^A)$ is independent of $u$ and $r$. The code treats such functions on the sphere in terms of stereographic angular coordinates based upon the auxiliary unit sphere metric $q_{AB}$. Tensor fields are represented by spin-weighted functions using a computational version of the $\eth$-formalism [@competh] based upon a complex dyad $q_A$, satisfying $q_{AB}=q_{(A}\bar q_{B)}$. (Note that this departs from other conventions [@penrin] in order to avoid unnecessary factors of $\sqrt{2}$ which would be awkward in numerical work.) For example, the vector field $v_A$ is represented by the spin-weight 1 function $v=q^Av_A$. Derivatives of tensor fields are represented by $\eth$ operators on spin-weighted functions by taking dyad components of covariant derivatives with respect to the unit sphere metric. Our conventions are fixed in the case of a scalar field $\Psi$ by $\eth \Psi =q^A \nabla_A \Psi$. In these conventions, $$(\bar \eth \eth -\eth \bar \eth)\eta=2s\eta
\label{eq:commut}$$ for a spin-weight $s$ field $\eta$.
The conformal metric $h_{AB}$ is represented by the dyad components $J=h_{AB}q^A q^B /2$ and $K=h_{AB}q^A \bar q^B /2$, with the determinant condition implying $K^2 =1+J\bar J$. As discussed in [@competh], the curvature scalar corresponding to $h_{AB}$ is $${\cal R} = 2K -\eth \bar \eth K
+ {1\over 2} [\bar\eth^2 J + \eth^2 \bar J]
+ {1\over 4 K}[\bar \eth \bar J \eth J-\bar \eth J \eth \bar J ] .
\label{eq:rscalar}$$
In terms of $J$, the evolution equation takes the form $$2 \left(rJ\right)_{,ur}
- \left(r^{-1}V\left(rJ\right)_{,r}\right)_{,r} = S_J
\label{eq:wev}$$ where $S_J$ is explicitly given in term of spin-weighted fields in Ref. [@high]. $S_J$ contains only first derivatives of $J$ and predetermined hypersurface quantities so that it does not play a major role in the integration of Eq. (\[eq:wev\]) on a given null hypersurface ${\cal
J}_u$.
Similarly, in integrating Eq. (\[eq:u\]) for $U^A$, the code uses the spin-weight 1 fields $U=q_A U^A$ and $Q=q^A Q_A$, where $$Q_A = r^2 e^{-2\,\beta} h_{AB} U^B_{,r},$$ so that Eq. (\[eq:u\]) reduces to the first order radial equations $$\begin{aligned}
\left(r^2 Q \right)_{,r} &=& q^A[2r^4 \left(r^{-2} \beta_{,A} \right)_{,r}
-r^2 h^{BC} D_{C} h_{AB,r} ],
\label{eq:Qa}
\\
U_{,r} &=& r^{-2} e^{2\beta} q_A h^{AB}Q_B,
\label{eq:ua}\end{aligned}$$ with the right-hand sides then rewritten in the $\eth$-formalism in terms of spin-weighted fields.
The integration constants necessary to evolve Eqs. (\[eq:beta\]) - (\[eq:wev\]) are the boundary values of $r$, $J$, $\beta$, $Q$, $U$ and $V$ on ${\cal H}$. Part of these boundary conditions are determined by the characteristic data on ${\cal H}$ and the remainder are determined by gauge conditions and the solution of the propagation equations. When the horizon ${\cal H}$ has constant $r$ (as in the Schwarzschild case), this is precisely the data necessary to initiate the radial integrations in the PITT code. In the generic case, ${\cal H}$ does [*not*]{} lie on grid points and the initialization of the $r$-grid boundary also requires $\partial_r J$, $\partial_r \beta$ and $\partial_r V$ on ${\cal H}$ to provide a Taylor expansion consistent with a second order accurate code..
From horizon variables to Bondi metric functions {#sec:fromto}
------------------------------------------------
We now describe the calculation of the Bondi-Sachs metric variables in a neighborhood of the horizon as necessary for a characteristic evolution of the exterior space-time. The essential piece of free horizon data is the conformal metric $h_{AB}$ or equivalently the metric function $J$. In Sec. \[sec:confmod\], this data is supplied for the case of a binary horizon by the conformal model, in which case the value of $r$ on the horizon is also supplied. In a more general situation, where only $J$ is prescribed, $r$ can be determined from its value and time derivative $\dot r$ on $S^{-}$ using the ingoing Raychaudhuri equation (\[eq:ruu\]), which in spin-weighted form is $$\ddot r =-\frac{r}{4} (\dot J \dot {\bar J}-\dot K^2).
\label{eq:ruus}$$ Here, we have used the dyad expansion of the conformal metric $h^{AB}$ which can be written in terms of $J$ and $K$ as $$2\, h^{AB} = - \bar{J} q^A q^B - J \bar{q}^A \bar{q}^B
+ K \left(q^A \bar{q}^B + \bar{q}^A q^B \right).
\label{eq:hAB}$$
Once both the intrinsic geometry $J$ and the radial coordinate $r$ are known, the metric functions $\beta$, $U$ and $W$ (as well as their radial derivatives) can be calculated using a hierarchy of horizon propagation equations \[similar to the hierarchy of hypersurface and evolution equations (\[eq:b\])-(\[eq:wev\]) used to propagate fields along a ${\cal J}_u$ null hypersurface\].
Next in this hierarchy is the propagation Eq. (\[eq:omegadot\]) for the twist $\omega=q^A \omega_A$ , which has spin-weighted form $$\begin{aligned}
(r^2\omega)\dot {}&=& r^2 \eth(\frac{\dot r}{r})
+\frac{1}{2}[ J\bar \eth(r^2 \dot K)+\bar J \eth(r^2\dot J)
-K\bar \eth(r^2\dot J)-K\eth(r^2\dot K)] \nonumber \\
&+& \frac{r^2}{4} ( \dot {\bar J}\eth J -4 \dot K\eth K
+2\dot K\bar \eth J +3\dot J \eth \bar J
-2\dot J \bar \eth K )
\label{eq:omegadots}\end{aligned}$$ and determines $\omega$ given its initial value on the slice $S^{-}$ and the conformal horizon geometry encoded in $J$.
Next, the value of $\beta$ follows from Eq. (\[eq:b\]) with $r_{,\lambda}$ determined from Eq. (\[eq:rlam\]) in the spin-weighted form $$\begin{aligned}
\partial_u\partial_\lambda (r^2) &= &
\frac{1}{2}\bar \eth [K(\eth \log r +\omega)]
+\frac{1}{2} \eth [K(\bar \eth \log r +\bar \omega)]
-\frac{1}{2}\bar \eth [J(\bar \eth \log r +\bar \omega)]
-\frac{1}{2}\eth [\bar J( \eth \log r + \omega)] \nonumber \\
&-&\frac{1}{2}{\cal R}-\frac{1}{2}J\bar \omega^2
-\frac{1}{2}\bar J \omega^2+K \omega\bar \omega ,
\label{eq:rlams}\end{aligned}$$ where Eq. (\[eq:rscalar\]) supplies the spin-weighted form of the curvature scalar ${\cal R}$ of the metric $h_{AB}$. This determines $r_{,\lambda}$ on the entire horizon given its value on the initial slice $S^{-}$.
Next, the Bondi metric functions $U$ and $V$ are evaluated on the horizon using Eqs. (\[eq:U\]) and (\[eq:V\]) in the spin-weighted forms $$\begin{aligned}
U &=& \frac{e^{2\beta}}{r^2}
\left(J \bar\eth r - K \eth r\right)
\label{eq:Uh}
\\
V &=& - 2\, r r_{,u} + \frac{e^{2\beta}}{2r}
\left(- \bar{J} \left(\eth r\right)^2
- J \left(\bar\eth r\right)^2
+ 2\, K \left(\eth r\right) \left(\bar\eth r\right)\right).
\label{eq:Vh}\end{aligned}$$
In summary, the Bondi metric functions $r$, $J$, $\beta$, $U$ and $V$ are determined by the hierarchy of Eqs. (\[eq:ruus\]) and (\[eq:omegadots\]) - (\[eq:Vh\]). When the value of $r$ is given, as in the conformal model, Eq. (\[eq:ruus\]) is not needed.
Extending the Bondi metric off the horizon {#sec:bondioff}
------------------------------------------
The evaluation of the Bondi metric functions on radial grid points in the neighborhood of the horizon requires the $r$-derivatives of $J$, $\beta$, $U$ and $V$. For this purpose we note that, on any given ray, the quantity $\partial_r F$ is known once both $\partial_\lambda F$ and $r_{,\lambda}$ are determined, e.g. $$\partial_r J = \partial_\lambda J / r_{,\lambda}
= e^{2\beta} \partial_\lambda J .
\label{eq:Jr}$$ We assume below that $r_{,\lambda}$ has already been determined on the horizon by integrating Eq. (\[eq:rlams\]).
We obtain $J_{,r}$ on the horizon (and hence $K_{,r}$) in terms of $\partial_\lambda J$, which is determined from its initial value on $S^-$ by integrating Eq. (\[eq:jlamdot\]) in the spin-weighted form $$\begin{aligned}
0&=&2r^2\partial_u \partial_\lambda J +2r\dot r \partial_\lambda J
+2r\dot J \partial_\lambda r
-Jr^2(\dot {\bar J} \partial_\lambda J
+\dot J \partial_\lambda \bar J
-2\dot K\partial_\lambda K) \nonumber \\
&-& (1+K^2)(\eth\omega +\omega^2 -2\omega \eth\log r )
+\omega (J\bar \eth K-K\bar \eth J)
+ \bar\omega(K\eth J-J \eth K)\nonumber \\
&-&J \bigg ( J\bar\eth \bar \omega
-K(\eth\bar\omega +\bar \eth \omega)
+J\bar\omega^2-2K\omega\bar\omega
+2K(\omega\bar\eth \log r+\bar\omega\eth \log r)
-2J\bar\omega\bar\eth \log r \bigg ) .
\label{eq:jlamdots}\end{aligned}$$
We calculate $\beta_{,r}$ from the Raychaudhuri equation for the outgoing null geodesics, $R_{\lambda\lambda}=0$, which in Sachs coordinates takes the form $$r_{,\lambda\lambda} = -\frac{r}{8} h^{AC} h^{BD}
h_{AB,\lambda} h_{CD,\lambda} \, ,$$ with spin-weighted version $$r_{,\lambda\lambda} = \frac{r}{4} \left(
\left(K_{,\lambda} \right)^2 - J_{,\lambda}\bar{J}_{,\lambda}
\right) ,$$ which, together with Eq. (\[eq:b\]), gives $$\beta_{,r} = \frac{r}{8} (J_{,r} \bar J_{,r} - K_{,r}^2) .$$
We obtain $U_{,r}$ (or $Q$) from the twist, which in Bondi coordinates takes the form $$\omega_A = \partial_A \beta +\frac{1}{r}\partial_A r
+\frac{1}{2}\partial_r ( h_{AB})h^{BC}\partial_C r
-\frac{r^2}{2}e^{-2\beta}h_{AB}\partial_r U^B,
\label{eq:Ucr}$$ with spin-weighted version $$\omega = \eth \beta +\frac{1}{r}\eth r
+\frac{1}{2} (\partial_r J)(K\bar \eth r-\bar J \eth r)
+\frac{1}{2}(\partial_r K)(K\eth r-J \bar \eth r)
-\frac{Q}{2}.
\label{eq:twist}$$ This determines the boundary data for $Q$ in terms of quantities that are already known on the horizon. Given $Q$, the value of $\partial_r U$ follows from Eq. (\[eq:ua\]), which has spin-weighted form $$U_{,r} = \frac{e^{2\beta}}{r^2}\left(K Q - J \bar{Q} \right) .$$
Finally, we compute $\partial_r V$ by obtaining $ V_{,\lambda}$ from the $\lambda$-derivative of Eq. (\[eq:VH\]), $$\begin{aligned}
\frac{e^{-2\beta}}{r}
\left[ V_{,\lambda}
- V \left( 2\,\beta_{,\lambda} + \frac{r_{,\lambda}}{r} \right)
\right]
&=& (r_{,\lambda})^2_{,\lambda} W
+ (r_{,\lambda})^2 W_{,\lambda}
- 2\, \left(r_{,\lambda} r_{,A} \right)_{,\lambda} W^{A}
- 2\, r_{,\lambda} r_{,A} W^{A}_{,\lambda} \nonumber \\
&-& 2\, r_{,\lambda\lambda} r_{,u}
- 2\, r_{,\lambda} r_{,\lambda u}
+ 2\, \frac{r_{,\lambda A} r_{,B}}{r^2} h^{AB}
- 2\, \frac{r_{,\lambda} r_{,A} r_{,B}}{r^3} h^{AB} \nonumber \\
&+& \frac{r_{,A} r_{,B}}{r^2} h^{AB}_{,\lambda} .
\label{eq:vlambda}\end{aligned}$$ The first three terms in the right-hand side vanish on ${\cal H}$ due to the gauge conditions. We express the others in spin-weighted form using Eq. (\[eq:wA\]) to obtain $$-2\,r_{,A} W^{A}_{,\lambda} = \frac{2}{r^2} \left(
-\bar{J} \omega \eth r - J \bar{\omega} \bar\eth r
+ K \left( \bar\omega \eth r + \omega \bar\eth r \right)
\right)$$ and Eq. (\[eq:hAB\]) to obtain $$\begin{aligned}
2\, r_{,\lambda A} r_{,B} h^{AB} &=&
- \bar{J} (\eth r_{,\lambda}) (\eth r)
- J (\bar\eth r_{,\lambda}) (\bar\eth r)
+ K \left( (\eth r_{,\lambda}) (\bar \eth r)
+ (\bar\eth r_{,\lambda}) (\eth r) \right)
\\
2\, r_{,A} r_{,B} h^{AB}_{,\lambda} &=&
- \bar{J}_{,\lambda} (\eth r)^2
- J_{,\lambda} (\bar\eth r)^2
+ 2\, K_{,\lambda} (\eth r) (\bar \eth r) .\end{aligned}$$ Then Eq. (\[eq:vlambda\]) gives $$\begin{aligned}
V_{,\lambda} &=& V\left( 2\,\beta_{,\lambda} + \frac{e^{-2\beta}}{r}\right)
+ 4\, r\, r_{,u} \beta_{,\lambda} - 2\, r\, r_{,\lambda u}
+ \frac{2}{r} \left( -\bar{J} \omega \eth r - J \bar{\omega} \bar\eth r
+ K \left( \bar\omega \eth r + \omega \bar\eth r \right) \right)
\nonumber \\
&+& \frac{e^{2\beta}}{r} \left(
- \bar{J} \left(\eth r_{,\lambda}\right) \left(\eth r\right)
- J \left(\bar\eth r_{,\lambda}\right) \left(\bar\eth r\right)
+ K \left(\bar\eth r_{,\lambda}\right) \left(\eth r\right)
+ K \left(\eth r_{,\lambda}\right) \left(\bar\eth r\right)
\right)
\nonumber \\
&+& \frac{1}{r^2} \left(
\bar{J} \left(\eth r\right)^2
+ J \left(\bar\eth r\right)^2
- 2\, K \left(\eth r\right) \left(\bar\eth r\right)
\right)
\nonumber \\
&+& \frac{e^{2\beta}}{2r} \left(
- \bar{J}_{,\lambda} \left(\eth r\right)^2
- J_{,\lambda} \left(\bar\eth r\right)^2
+ 2\, K_{,\lambda} \left(\eth r\right) \left(\bar\eth r\right)
\right)\end{aligned}$$ in terms of previously determined quantities on the right-hand side.
The values of each metric function $(J,\beta,U,V)$ and its first radial derivative can then be used to consistently and accurately initialize $r$-grid points near the horizon. This in turn allows the evolution code to determine the entire region extending from the horizon to ${\cal I}^+$, as long as the coordinate system remains well behaved.
Conformal model for the axisymmetric head-on collision {#sec:confmod}
======================================================
The conformal horizon model [@ndata; @asym] supplies the conformal metric $h_{AB}$ constituting the null data for a binary black hole. The conformal model is based upon the flat space null hypersurface ${\cal H}$ emanating normal from a convex, topological sphere ${\cal S}_0$ embedded at a constant inertial time $\hat t=0$ in Minkowski space. Traced back into the past, ${\cal
H}$ expands to an asymptotically spherical shape. Traced into the future, ${\cal H}$ pinches off where its null rays cross, at points ${\cal X}$, or where neighboring null rays focus, at caustic points ${\cal C}$. Figure \[fig:spheroid\] schematically illustrates the embedding of ${\cal H}$ in Minkowski space for the case when ${\cal S}_0$ is a prolate spheroid. The spheoridal case generates the horizon for the axisymmetric head-on collision of two black holes, or here in the corresponding time reversed scenario, the horizon for an initially Schwarzschild white hole which undergoes an axisymmetric fission into two white hole components. The white hole horizon shares the same submanifold ${\cal H}$ and the same (degenerate) confomal metric as its Minkowski space counterpart but its surface area and affine parametrization differ. As a white hole horizon, ${\cal H}$ extends infinitely far to the past of ${\cal S}_0$ to an asymptotic equilibrium with finite surface area.
The intrinsic white hole geometry of ${\cal H}$ is obtained by the following construction based upon the flat space null hypersurface. A $\hat
t$ foliation of ${\cal H}$ is induced by the $\hat t$ foliation of Minkowski space, with $\hat t$ and the Euclidean coordinates $(x,y,z)$ determining a Minkowski coordinate system. The prolate spheroid ${\cal S}_0$ is given by $$x^2+y^2+\frac{z^2}{1+\epsilon}=a^2 \, ,$$ with $\epsilon>0$, or alternatively in angular coordinates $y^A = (\eta, \phi)$ by $$\begin{aligned}
x&=& a \sin\eta \cos\phi \, \\
y&=& a \sin\eta \sin\phi \, \\
z&=& \sqrt{1+\epsilon} a \cos\eta \, .\end{aligned}$$ In these coordinates, the Minkowski metric induces on ${\cal S}_0$ the intrinsic metric $${ \hat g}_{AB} dy^A dy^B =a^2 \left( \left(1 + \epsilon \sin^2 \eta \right)
d\eta^2 + \sin^2 \eta \, d\phi^2 \right)\, .$$
The principal curvature directions on ${\cal S}_0$ are the polar and azimuthal directions, with corresponding radii of curvature $$r_{\eta} = a \frac{(1 + \epsilon \sin^2 \eta )^{3/2}}{ \sqrt{1+\epsilon}}$$ and $$r_{\phi} = a \frac{(1 + \epsilon \sin^2 \eta )^{1/2}}{\sqrt{1+\epsilon} }
\, .$$ Each generator of ${\cal H}$ encounters two caustics at $\hat t=
r_{\eta}$ and $\hat t= r_{\phi}$ (or one degenerate caustic) if continued into the future but, along a typical generator, ${\cal H}$ first pinches off at a cross over with another generator before a caustic is reached.
The time dependent metric of the $\hat t$ foliation is given by $${\hat g}_{AB} dy^A dy^B = a^2 \left(
\left(1-\frac{\hat t}{r_\eta}\right)^2 (1 + \epsilon \sin^2 \eta) d\eta^2
+ \left(1-\frac{\hat t}{r_\phi}\right)^2 \sin^2 \eta d\phi^2
\right) \, .
\label{eq:ghat}$$ As $\hat t\rightarrow -\infty$, $g_{\phi\phi}/g_{\eta\eta}\rightarrow \sin^2\eta
[1 + \epsilon\sin^2\eta]$ so that the conformal metric in these coordinates does not approach the standard form of the unit sphere conformal metric. For this purpose, it is convenient to introduce new angular coordinates $x^A=(\theta,\phi)$ in which the conformal metric asymptotes to the unit sphere metric $q_{AB}dx^A dx^B=d\theta^2+\sin^2\theta
d\phi^2$. This requires that $$\frac{d\theta}{\sin\theta}= \frac{d\eta}
{\sin\eta \sqrt {1 +\epsilon \sin^2 \eta }}
\label{eq:angles}$$ with the solution $$\tan\theta =\sqrt{1+\epsilon}\tan\eta.$$ (Here the boost freedom corresponding to the unit sphere conformal group has been fixed by requiring the transformation to have reflection symmetry about the equator.)
In these $x^A$ coordinates, $\hat g_{AB}dx^A dx^B= \hat r^2
\hat h_{AB}dx^A dx^B$, where $\det(\hat h_{AB})=\det (q_{AB})=q$, $$\hat r^2 = a^2 \frac{1+\epsilon}{(1 + \epsilon \cos^2 \theta)^2}
\left(1-\frac{\hat t}{r_\theta}\right)\left(1-\frac{\hat t}{r_\phi}\right)
\label{eq:def_rhat}$$ $$r_{\theta} = r_{\eta}= a \frac{1+\epsilon}{(1 + \epsilon\cos^2 \theta)^{3/2}}$$ and $$r_{\phi} = \frac{a}{ (1 + \epsilon \cos^2 \theta)^{1/2}} \, .
\label{eq:rphi}$$ In the prolate case ($\epsilon > 0$), $ r_{\theta} > r_{\phi}$.
We apply the conformal horizon model to endow ${\cal H}$ with the intrinsic metric $g_{AB}=\Omega^2 \hat g_{AB}$ of a white hole horizon. The conformal factor $\Omega$ is designed to stop the expansion of the white hole in the past so that the surface area asymptotically hovers at a fixed radius. As a result, $h_{AB}(\hat t, x^A)=\hat h_{AB}(\hat t, x^A)$ is the intrinsic conformal metric of the horizon (as well as of the flat space null hypersurface). With the dyad choice $q^A=(1,i/\sin\theta)$, Eq. (\[eq:ghat\]) implies that the spin-weight-2 field $$J={1\over 2}q^A q^B h_{AB}(\hat t,x^C)
=\frac{1}{2}\bigg ( \frac { \hat t -r_\theta}
{\hat t -r_\phi} \bigg )
-\frac{1}{2}\bigg ( \frac {\hat t -r_\phi}
{ \hat t -r_\theta} \bigg )
\label{eq:headonj}$$ is the conformal null data for the white hole horizon in the $\hat t$ foliation.
The surface area of the white hole is related to the corresponding surface area of the flat space null hypersurface by $r=\Omega \hat r$. A conformal factor with all required behavior to produce a non-singular white hole is given by [@ndata; @asym] $$\Omega=-R_{\infty}\big( \hat u
+\frac{\sigma^2}{12(p-\hat u )}\big)^{-1},
\label{eq:ansatz}$$ where $R_{\infty}$ is the initial equilibrium radius, $p$ is a model parameter (designated by $\rho$ in Refs. [@ndata; @asym]), $\sigma$ is the difference between the principal curvature radii, $$\sigma = |r_{\theta} - r_{\phi}| =
\frac {a |\epsilon| \sin^2 \theta}
{(1 + \epsilon \cos^2 \theta )^{3/2}} ,
\label{eq:sigma}$$ and $\hat u$ is an affine parameter along the generators of ${\cal H}$ with the same scale as $\hat t$ but with origin $\hat u =0$ chosen to lie midway between the caustics, i.e. $\hat u =\hat t -r_0$ where $$r_0 =\frac {(r_{\theta} + r_{\phi})}{2} =
a \frac{2 + 2\epsilon - \epsilon\sin^2 \theta}
{2(1 + \epsilon \cos^2 \theta )^{3/2}}.
\label{eq:r0}$$ is the mean curvature of ${\cal S}_0$. For an initially Schwarzschild white hole, $R_{\infty}=2M$.
As a 3-manifold with boundary, ${\cal H}$ represents both the white hole horizon and the flat space null hypersurface. Both extend infinitely into the past and continue into the future to the boundary where ${\cal H}$ pinches off at caustics and cross over points. Smoothness of the white hole requires that the parameter $p \ge \sigma_M/\sqrt{13}$, where $\sigma_M$ is the maximum value of $\sigma$ attained on ${\cal S}_0$. For a prolate spheroid, the maximum occurs at the equator and $\sigma_M = a |\epsilon|$.
${\cal H}$ must obey the Raychaudhuri equation (\[eq:ruu\]) both as a flat space null hypersurface and as a white hole horizon. Both geometries have the same intrinsic conformal metric so that they must have the same rate of focusing in terms of their respective affine parameters, in accord with the Sachs optical equations [@sachseq]. As a result of the different behavior of their surface areas due the conformal factor $\Omega$, the affine parameter $t$ on the white hole horizon is related to its flat space counterpart $\hat t$ according to [@ndata] $$\frac {dt}{d \hat t}= \Lambda^{-1}
=\frac{9}{(12 \hat u (\hat u-p) -
\sigma^2)^2} \frac{( 5 p + \mu-2 \hat u)^{2 \,
(2p/\mu +1) \,}}{( 5 p- \mu-2 \hat u)^{2 \,
(2p/\mu -1) \,} } \, ,
\label{eq:lampr}$$ where $$\mu = \sqrt{13p^2 -\sigma^2}.
\label{eq:def_mu}$$ Here the affine scale of $t$ is fixed by the condition $dt/ d\hat t
\rightarrow 1$ as $\hat t \rightarrow -\infty $. Equation (\[eq:lampr\]) determines the deviation of a slicing adapted to an affine parameter of the white hole horizon from the original slicing given by the Minkowski embedding. The angular dependence of the crossover time $t_{\cal X}$ at which the horizon pinches off leads to the change in topology of the white hole associated with a pair-of-pants shaped horizon. From Eq. (\[eq:rphi\]), the pinch-off occurs at $\hat t_{\cal X}
=r_\phi=a/\sqrt{1+\epsilon \cos^2 \theta}$, or $$\hat u_{\cal X} = -\frac {a\epsilon \sin^2 \theta}
{2(1+\epsilon \cos^2 \theta)^{3/2}}.$$ Consequently, $\hat u_{\cal X}$ as well as $\sigma$ vanish at both poles so that integration of Eq. (\[eq:lampr\]) implies that $t_{\cal X} \rightarrow
\infty$ at both poles. This creates the two legs of the pair-of-pants since $t_{\cal X}$ remains finite at all other angles.
Note that $\Lambda = \Lambda(\hat u/a, p/a, \epsilon)$ within our conformal model, so that $t = t(\hat t/a, p/a, \epsilon)$. The same scale dependence on $a$ also holds for the horizon variables $\hat r$ and $J$, as is evident from Eqs. (\[eq:def\_rhat\]) and (\[eq:headonj\]). In addition Eq. (\[eq:ansatz\]) implies $r=r(R_\infty /a,\hat u/a, p/a, \epsilon)$. Thus we can scale $a=1$ without any loss of generality of the model.
The close approximation and data on ${\cal S}^{-}$ {#sec:headon}
==================================================
In addition to the conformal metric on ${\cal H}$ (discussed in the last section), specification of the radius and extrinsic curvature of the initial slice ${\cal S}^{-}$ completes the necessary data on ${\cal H}$. Our strategy is to locate ${\cal S}^{-}$ at an early quasi-stationary era and approximate these data by their equilibrium values as a Schwarzschild white hole. In the linearized approximation, the conformal metric of ${\cal H}$ corresponds to a perturbation of the Schwarzschild background. However, a comparison of how the fully nonlinear data for the head-on collision deviate from their linearized counterparts reveals features which can not be described perturbatively. The chief geometrical issues to be discussed here are the asymptotic properties of the horizon at $I^-$, the behavior where it pinches off and the analogue of the bifurcation sphere occurring in a Schwarzschild horizon. As indicated in Fig. \[fig:spheroid\], we choose ${\cal S}^-$ to correspond to an early Minkowski time $\hat t =\hat t_-$ so that it is initially quasi-spherical for all eccentricities $\epsilon$ of the ellipsoid ${\cal S}_0$ located at $\hat t =0$. The criterion that the initialization be at an early time is $|\hat
t_-|>>r_0$. In that approximation $dt/d\hat t\approx 1$ on ${\cal S}^-$ and Eq. (\[eq:headonj\]) gives $$J_-\approx \frac{ r_\phi -r_\theta}{\hat t_- }.$$
Perturbing a Schwarzschild horizon and the close approximation {#sec:perturbations}
--------------------------------------------------------------
In the Cauchy treatment of the close approximation to the axisymmetric head-on collision of black holes [@pp], the background spacetime is the exterior Kruskal quadrant of the extended Schwarzschild space-time. The trousers shape of the binary horizon is beyond the scope of a perturbative treatment. Here we present a fully nonlinear characteristic treatment of the (time reversed) head-on collision as a 1-parameter sequence of binary collisions with the Schwarzschild case as a limit. A trousers-shaped horizon exists for each non-Schwarzschild member of the sequence so that it is possible to investigate its behavior in the characteristic version of the close approximation.
The conformal horizon model of a Schwarzschild horizon is the highly degenerate case in which ${\cal S}_0$ is geometrically a sphere. Then $\hat h_{ab}=q_{ab}$, $J=0$ and $r=R_{\infty}=2M$, so that the horizon is stationary. In the $(u,\lambda,x^A)$ coordinates of the Sachs metric, the Schwarzschild geometry is described by Eq. (\[eq:amet\]) with $W^A =0$, $r=2M
-\lambda u/(4M)$ and $$W=\frac{2\lambda^2}{\lambda u -8M^2}.$$ These coordinates cover the entire Kruskal manifold with remarkably simple analytic behavior, as observed by Israel [@israel]. On the white hole horizon, given by $\lambda =0$, $u$ is an affine parameter with its origin fixed so that $u=0$ on the $r=2M$ bifurcation sphere where $\partial_\lambda r
=0$.
The spin-weighted versions (\[eq:ruus\]) - (\[eq:jlamdots\]) of the Einstein Eqs. (\[eq:ruu\]) - (\[eq:jlamdot\]) lead to the following linearized equations governing the perturbation of a Schwarzschild horizon. The ingoing Raychaudhuri Eq. (\[eq:ruus\]) simplifies to $$\ddot r = 0,$$ so that we can set $r=2M$ on $\cal H$, where $M$ is the background Schwarzschild mass. Equation (\[eq:omegadots\]) then reduces to $$(r^2\omega)\dot {}= -\frac{1}{2}r^2\bar \eth \dot J,$$ so that $$\omega=-\bar \eth J/2,
\label{eq:pertomega}$$ where we fix the constant of integration by requiring that the perturbed black hole come to equilibrium as a Schwarzschild black hole with $J=\omega=0$ as $u\rightarrow-\infty$. Next Eqs. (\[eq:rlams\]) and (\[eq:jlamdots\]) reduce to $$r\partial_u\partial_\lambda r = -\frac{1}{2}
-\frac{1}{4}(\bar\eth^2 J +\eth^2\bar J)
\label{eq:rlamdp}$$ and $$0=2r\partial_u \partial_\lambda(rJ)+\eth\bar\eth J + J.
\label{eq:jlamdp}$$
It is convenient to set $r=\rho \, r_M$, where $$r_M = 2M -\frac {\lambda u}{4M}
\label{eq:rm}$$ is the background Schwarzschild value and where, restricted to ${\cal H}$, $\rho
=1+O(J^2)$. Then, to first order in the perturbation, Eq. (\[eq:rlamdp\]) reduces to $$r^2\partial_u \partial_\lambda \rho
=-\frac{1}{4}(\bar\eth^2 J +\eth^2\bar J)
\label{eq:rhodot}$$ and Eq. (\[eq:jlamdp\]) reduces to $$0=2r^2\partial_u \partial_\lambda J+\eth\bar\eth J
-u\dot J.
\label{eq:jlamdp2}$$ Note that in the perturbative limit the right-hand sides of of the equations for the extrinsic quantities $\omega$, $r_{,\lambda}$ and $J_{,\lambda}$ only depend on the intrinsic geometry.
As the parameter $\epsilon$ describing the eccentricity of the spheroid approaches zero, the binary black hole horizon approaches the Schwarzschild horizon of a single spherically symmetric black hole. Thus the close approximation for a binary black hole can be described by a perturbation expansion in $\epsilon$. In order to integrate the perturbation Eqs. (\[eq:rhodot\]) and (\[eq:jlamdp2\]), with $J$ supplied by the conformal model, we must relate the affine parameter $t$ of the conformal model to the restriction to ${\cal H}$ of the coordinate $u$ of the Sachs metric. We set $t=Au$, where $$A = \frac {1} {(1 + \epsilon\cos^2\theta)^{3/2}} =1+O(\epsilon)
\label{eq:capa}$$ fixes the relative scale freedom in the affine parameters so that the perturbation has the early time behavior $$J\sim -\frac{a \epsilon\sin^2\theta}{u}$$ in accord with the quadrupole nature of the close approximation.
Data on ${\cal S}^{-}$ {#sec:sminus}
----------------------
The conformal model described in Sec. \[sec:confmod\] supplies the values of $J$ and $r$ on the entire horizon. Other quantities have to be initiated at an early cross section ${\cal S}^-$, near which the horizon behaves as a perturbed Schwarzschild horizon. As indicated in Fig. \[fig:spheroid\], we locate ${\cal S}^-$ at a constant Minkowski time $\hat t=\hat t_-$, as well as at a constant white hole affine time $u=u_-$. The relationship $\hat t(u)$ is then determined by integrating $du/d\hat t = 1/(A \Lambda)$ with initial conditions determined on ${\cal S}^-$ and $\Lambda$ and $A$ given in Eqs. (\[eq:lampr\]) and (\[eq:capa\]). This determines the horizon data $J(u,x^A)$ from $J(\hat t,x^A)$ given in Eq. (\[eq:headonj\]). Similarly, $r(u,x^A)$ is determined from $r(\hat t,x^A)=\Omega (\hat t,x^A) \hat r (\hat
t,x^A)$, with $\hat r$ and $\Omega$ given by Eqs. (\[eq:def\_rhat\]) and (\[eq:ansatz\]), respectively. The requirement that ${\cal S}^-$ be located in the quasi-equilibrium era implies that $r_- \approx 2M$ to an excellent approximation.
The way in which the values of $u_-$ and $\hat t_-$ determine the location of ${\cal S}_0$ relative to ${\cal S}^-$ depends upon the parameters entering the conformal model. Even in the limit $\epsilon \rightarrow0$, where the conformal model yields a Schwarzschild horizon, this relation depends upon the model parameter $p$. The simplest limiting case is when $p$ and $\epsilon$ both vanish. Then $A=\Lambda =1$ so that $\hat t- \hat t_-=u-u_-$ and the Minkowski spheroid ${\cal S}_0$ (where $\hat t=0$) is located at $u=u_- -\hat
t_-$. Thus a negative value of $\hat t_-$ locates ${\cal S}_0$ to the future of ${\cal S^-}$.
In the Schwarzschild limit, $\partial_\lambda r \rightarrow \partial_\lambda
r_M = -u/4M$, where $u=0$ on the $r=2M$ bifurcation sphere ${\cal B}$. This allows us to use the initial value of $\partial_\lambda r$ to determine the location of ${\cal S}^-$ on the horizon by setting $$\partial_\lambda r_- = - \frac{u_-}{4M}
\label{eq:uinit}$$ on ${\cal S}^-$. Thus specification of the initial outward expansion of ${\cal
S}^-$ fixes the translation freedom in the affine parameter $u$. The requirement $|u_-| >>M$ ensures that the initialization be at an early time.
A 1-parameter family of horizon data for a head-on collision results from choosing ${\cal S}_0$ to be an $\epsilon$-family of spheroids (with $\epsilon\ge 0$). The close approximation, corresponding to the behavior linear in $\epsilon$, provides insight into the asymptotic structure of the horizon at early times. In this linear approximation, Eq. (\[eq:headonj\]) implies $$J\approx -\frac {\epsilon a \sin^2\theta}{\hat t -a}$$ on the horizon; and Eqs. (\[eq:sigma\]) and (\[eq:r0\]) imply $$\sigma \approx \epsilon a\sin^2\theta$$ and $$r_0 \approx \frac{r_\theta +r_\phi}{2}= a\left(1+\frac{\epsilon}{2}
-\epsilon \cos^2\theta \right).$$ In addition, $r=2M+O(\epsilon^2)$. (We use $\approx$ to denote approximations valid for small $\epsilon$ and $\sim$ to denote asymptotic approximations at early times.)
The early time asymptotic behavior of all the horizon variables can be explicitly evaluated in this approximation. Equation (\[eq:pertomega\]) determines the asymptotic dependence $$\omega\sim -\bar \eth J/2,
\label{eq:asymomega}$$ on the assumption of an initially non-spinning Schwarzschild white hole. Using the identities $\eth^2\cos^2 \theta=2\sin^2\theta$, $(\bar\eth^2\eth^2+\eth^2\bar\eth^2)\cos^2\theta =
48(\cos^2\theta-\frac{1}{3})$, the commutation relation (\[eq:commut\]) and the property $\bar\eth\eth Y_{\ell m}=-\ell(\ell+1)Y_{\ell m}$, Eq. (\[eq:rhodot\]) reduces to $$r^2\partial_{\hat t} \partial_\lambda \rho \approx
\frac{6\epsilon a}{\Lambda (\hat t-a)} \left(\cos^2\theta-\frac{1}{3} \right).
\label{eq:rhoearly}$$ At early times, where $\Lambda\sim 1$, this integrates to give $$r^2 \partial_\lambda \rho \sim
6\epsilon a \left(\cos^2\theta-\frac{1}{3} \right)
\log\bigg(\frac{\hat t - a}{\hat t_ - a} \bigg),$$ where we set the integration constant so that $\partial_\lambda\rho=0$ on ${\cal S}^-$ in accord with Eq. (\[eq:uinit\]).
Similarly, at early times, the integral of Eq. (\[eq:jlamdp2\]) gives $$\partial_\lambda J \sim -\frac{3\epsilon
a\sin^2\theta}{2r^2} \log\bigg (\frac{\hat t -a}{\hat t_- -a} \bigg ).
\label{eq:jlamint}$$
Note that $\partial_\lambda \rho$ and $\partial_\lambda J$ have logarithmic asymptotic behavior at $I^-$. In the exterior evolution code this singular behavior is renormalized by dealing instead with the quantities $\partial_r
\rho =\partial_\lambda \rho /\partial_\lambda r$ and $\partial_r
J=\partial_\lambda J/\partial_\lambda r$. Since $\partial_\lambda r \sim
-u/4M$ at $I^-$, these quantities both go to zero as $\log u /u$ as $u\rightarrow -\infty$. This justifies initializing these quantities to zero on ${\cal S^-}$. The initialization error is $O(\log u_-/u_-)$ and converges to $0$ as $u_-\rightarrow-\infty$. In summary, given $J$ and $r$ on the horizon via the conformal model, the remaining data necessary on ${\cal S^-}$ are initialized according to $$\begin{aligned}
\omega_- & = & -\frac{1}{2}\bar \eth J_-, \\
\label{eq:omegaminus}
\partial_\lambda \rho_- & = & 0, \\
\label{eq:rhominus}
\partial_\lambda J_- & = & 0.
\label{eq:jminus}\end{aligned}$$ With this initialization, all Bondi-Sachs start-up variables for the exterior evolution are asymptotically well defined at $I^-$ except for $e^{-2\beta}=\partial_\lambda r\sim -u/(4M)$. This is handled by the renormalization $\beta=\beta_M+\beta_R$ where $e^{-2\beta_M}= -u/(4M)$ and $\beta_R \rightarrow 0$ at $I^-$. In the evolution code, the singular part $\beta_M$ is analytically factored out of the Bondi equations. The regular part is given by $$e^{-2\beta_R} \sim 1 - \frac{8M^2 \partial_\lambda \rho}{u}.$$ Then, referring to Eq. (\[eq:rhoearly\]), $\beta_R$ can be initialized to $0$ on ${\cal S}^-$ with $O(\log u_-/u_-)$ accuracy.
The early time approximation breaks down before reaching the crossover points ${\cal X}$ where $\hat t \approx a$, as evident from Eqs. (\[eq:rhoearly\]) and (\[eq:jlamint\]). The way that the horizon pinches off at ${\cal X}$ in a sequence of models as $\epsilon\rightarrow 0$ is sensitive to the behavior of the model parameter $p$ along the sequence. We consider here the case $p=const$ (independent of $\epsilon$ as well as angle).
The crossover points occur at Minkowski time $\hat t_{\cal X} = a/\sqrt{1
+\epsilon\cos^2\theta} \approx a [1-(\epsilon/2) \cos^2 \theta ]$. The corresponding values $u_{\cal X}=t_{\cal X}/A$ are found from integrating Eq. (\[eq:lampr\]). The rays on the poles of the prolate spheroid, where $\sigma
=0$, do not focus. Since the horizon surface area is asymptotically constant, this lack of focusing implies that the horizon persists forever along the poles, i.e. for $-\infty<u<+\infty$ (independent of the value of $\epsilon$).
In order to investigate the location of ${\cal X}$ along the non-polar rays, consider the small $\epsilon$ behavior $$\frac {1}{\Lambda} \approx \frac {(3p^2 -5p\hat u +\hat u^2)^2}
{\hat u^2 (\hat u-p)^2}
\bigg (\frac {(5+\sqrt{13})p -2\hat u}
{(5-\sqrt{13})p -2\hat u}\bigg )^{4/\sqrt{13}},$$ where $$\hat u\approx \hat t
-a \left(1+\frac{\epsilon}{2} -\epsilon \cos^2\theta \right)
\approx \hat t -\hat t_{\cal X} -\frac{a\epsilon}{2}\sin^2\theta .$$ The dominant behavior near ${\cal X}$ is revealed by the further approximation $$\frac {1}{\Lambda} \approx \frac {9p^2}{\hat u^2}
\bigg (\frac {5+\sqrt{13}}{5-\sqrt{13}}\bigg )^{4/\sqrt{13}},$$ which is valid for $\hat u<<p$, e.g. near ${\cal X}$ where $\hat u\approx
0$. In this approximation, $$u_{\cal X}-u_- =\int_{\hat t_-}^{\hat t_{\cal X}}
\frac {d\hat t}{A\Lambda} \approx 9p^2
\bigg (\frac {5+\sqrt{13}}{5-\sqrt{13}}\bigg )^{4/\sqrt{13}}
\bigg (\frac{2}{a\epsilon\sin^2\theta}
+\frac {1}{\hat t_- -a}\bigg ),
\label{eq:pinch}$$ so that $u_{\cal X}\rightarrow\infty$ as $\epsilon\rightarrow 0$ along all rays. In this limit, the entire cross over seam on the pair-of-pants is mapped to $t=\infty$. As a result, the corresponding first order perturbation theory for this version of the close approximation is well behaved on the entire white hole horizon, not just the segment bordering the exterior space-time.
Of special physical importance is the location of the crossover surface ${\cal
X}$ where the horizon pinches off relative to the surface ${\cal B}$ defined by $\partial_\lambda r=0$. ${\cal B}$ represents a boundary for the Bondi evolution resulting from the breakdown of the surface area coordinate $r$. In the $\epsilon =0$ Schwarzschild limit, ${\cal B}$ is the $r=2M$ bifurcation sphere located at $u=0$ and ${\cal X}$ lies at $u=\infty$ (at $I^+$). In this limit, the white hole fission takes place in the infinite future. For small but nonzero $\epsilon$, the dominant $O(1/\epsilon\sin^2\theta)$ dependence in Eq. (\[eq:pinch\]), implies that ${\cal X}$ lies at a finite but large value, with the point at the equator where the white hole fissions (the crotch in the pair-of-pants picture) located at the earliest point on ${\cal X}$. However, ${\cal B}$ remains within $O(\epsilon)$ of its Schwarzschild location at $u=0$. Thus, for small $\epsilon$ the fission is “hidden” beyond ${\cal B}$ in the sense that it is not visible to observers at ${\cal I}^+$. From the time reversed view of a black hole merger, the individual black holes would merge inside a white hole horizon.
Nonlinear horizon data {#sec:mts}
======================
The close approximation results just described for a white hole fission, when reinterpreted in the time reversed sense of a black hole merger, imply that the individual black holes merge inside a white hole horizon corresponding to the marginally anti-trapped branch of the $r=2M$ Schwarzschild surface. Of prime importance in the non-perturbative regime is whether an entirely different scenario is possible in which the individual black holes form and merge without the existence of a marginally anti-trapped surface (MATS) on the event horizon. The ingoing null hypersurface which intersects the horizon in such a MATS has an extremum in its surface area. As a result, the Bondi surface area coordinate based upon an ingoing null foliation is singular at the MATS. A Bondi evolution carried out backward in time on these ingoing null hypersurfaces would terminate at the MATS. In particular, the absence of a MATS to the future of the merger is a necessary condition for a Bondi evolution backward in time throughout the entire post-merger period. (See the discussion below for more technical details.)
It is possible to avoid this problem by means of a null evolution using an affine parameter rather than a surface area parameter as radial coordinate. Null evolution codes in different gauges than the Bondi gauge have been developed [@bartnik1; @bartnik2]. However, at present such codes have not been successful in the stable evolution of dynamic horizons.
Restated from the alternative viewpoint of a white hole fission, as being pursued here, the absence of a marginally trapped surface (MTS) prior to the fission is a necessary condition for a Bondi evolution forward in time throughout the pre-fission period. Thus, the absence of a pre-fission MTS is necessary in order to carry out our strategy for obtaining the complete post-merger wave form of a binary black hole by means of a Bondi evolution.
The bifurcation sphere ${\cal B}$ in the Schwarzschild space-time is a MTS on the white hole horizon (and in this degenerate case also a MATS). As already discussed in Sec. \[sec:sminus\] and explicitly demonstrated in Sec. \[sec:results\], in the small $\epsilon$ regime of our model the corresponding white hole fissions at a very late time well beyond the MTS. This is the expected effect of a non-singular perturbation: in the $\epsilon
\rightarrow 0$ limit there is only one white hole so that the fission is hidden at $I^+$. The behavior in the non-linear regime is not so easy to predict. The following discussion of the properties of an MTS and its relation to a Bondi boundary ${\cal B}$ provides a basis for understanding the computational results of Sec. \[sec:results\].
It should first be noted that, unlike the definition of a MTS, the definition of ${\cal B}$ is foliation dependent. A MTS is a topological sphere with one non-diverging normal null direction and the other divergence-free. The Bondi boundary ${\cal B}$ on a white hole horizon ${\cal H}$ parameterized by $u$ is the earliest slice $u=u_{{\cal B}}$ whose outward null normal is not strictly diverging at all points. In the relation between $(u,\lambda, x^A)$ Sachs coordinates and $(u,r, x^A)$ Bondi coordinates, this implies that $\partial_\lambda r(u_{{\cal B}},0,x^A)=0$ at some point of ${\cal B}$. In other foliations of ${\cal H}$, ${\cal B}$ (if it exists) could occur later or earlier than in the affine foliation considered here.
In the affine foliation, a MTS on of ${\cal H}$ can be described in Sachs coordinates in the form $u+F(x^A)=0$, $\lambda=0$. The outgoing null normal to the MTS is $$L_a = -\alpha \partial_a \lambda - \partial_a (u+F)$$ where $$\alpha =\frac{1}{2 r^2} h^{AB} (\partial_A F )\partial_B F$$ which, with $n^a \partial_a =\partial_u$, defines the projection tensor $$\gamma^{ab} = g^{ab} + 2 L^{(a} n^{b)}.$$ The MTS satisfies $$\gamma^{ab} \nabla_a L_b =0$$ which has the coordinate form $$\begin{aligned}
\gamma^{ab} \nabla_a L_b &=& -\frac{1}{r^2}[D^A D_A F\nonumber \\
&+&(D^A F)\partial_\lambda g_{uA} +\alpha\partial _u (r^2)
-\partial_\lambda (r^2)
-(\partial_u h^{AB}) (D_A F)D_B F]|_{u=-F} \nonumber \\
&=& 0
\label{eq:mts}\end{aligned}$$ (where $D_A h_{BC} =0$). The MTS, if it exists, can be located by solving Eq. (\[eq:mts\]).
The following two propositions relate the Bondi coordinates to the existence of a MTS on a white hole horizon ${\cal H}$:
[*Proposition I*]{}. A Bondi cross section $u=const$ satisfying $\partial_\lambda
r =0$ is a MTS.
[*Proposition II*]{}. A MTS cannot exist in a region $u<u_{{\cal B}}$ in which $\partial_\lambda r > 0$.
The first proposition follows immediately from setting $F=0$ in Eq. (\[eq:mts\]). The second proposition follows from noting that at some point on the MTS the function $F$ would have a maximum where $D_A F =0$ and where Eq. (\[eq:mts\]) would reduce to $$D^A D_A F = \partial_\lambda (r^2) >0.
\label{eq:nmts}$$ But the inequality in Eq. (\[eq:nmts\]) precludes the existence of a maximum.
The second proposition establishes that a MTS cannot form before the Bondi boundary. Thus a Bondi evolution might terminate prematurely due to an injudicious choice of foliation of ${\cal H}$. A computational module for locating a MTS on a null hypersurface has been developed and successfully used for long term tracking of a moving black hole [@wobb]. In future work, this module will be applied to binary horizons. Here we consider only the less geometrical Bondi boundary ${\cal B}$.
As $\epsilon$ increases into the nonlinear region, the effect on ${\cal B}$ can be seen from integrating Eq. (\[eq:rlam\]) over the sphere. In doing so, we note that the terms which are divergences integrate to zero and we can apply the Gauss-Bonnet theorem to the curvature scalar term to obtain $$\partial_u \oint \partial_\lambda (r^2) dS=
-4\pi +\oint h^{AB}\omega_A\omega_B dS.
\label{eq:intrlams}$$ The term $-4\pi$ is responsible for the formation of ${\cal B}$ in the background Schwarzschild case. The nonlinear correction due to the twist is of an opposite sign and delays the formation. Thus, as $\epsilon$ increases from $0$, the formation of ${\cal B}$ is delayed while the location of ${\cal X}$ moves to earlier times. Although Eq. (\[eq:intrlams\]) only describes averaged angular behavior, it suggests that sufficient nonlinearity might cause the white hole fission, located on the equator of ${\cal X}$, to occur prior to ${\cal B}$ . (In the time reversed case, this would alow the merger of individual black holes without the necessity of a MATS and the consequent singularity in its past implied by Penrose’s theorem [@pensing].)
Numerical Results {#sec:results}
=================
The scenario hypothesized at the end of Sec. \[sec:mts\] can indeed be demonstrated by integrating the equations underlying the conformal horizon model. At an early time, the equilibrium conditions on the white hole horizon imply that $r=2M$ and $\partial_\lambda r= -u/4M >0$. As the horizon evolves, the surface area $r$ decreases along all rays but, for the axisymmetric and reflection symmetric fission considered here, it decreases fastest along the equatorial rays where the pinch-off first occurs. The outward expansion measured by $\Theta_{OUT}=2\partial_\lambda r /r$ also initially decreases along all rays, although this process can be reversed by the growth of nonlinear terms, as indicated by the ray-averaged behavior governed by Eq. (\[eq:intrlams\]). In the close approximation, the expansion goes to zero along all rays before the horizon pinches off, i.e. the crotch at the center of the pair of pants is hidden behind a MTS. The crucial question in the nonlinear regime is whether the horizon can pinch off before the formation of a MTS, i.e. whether the crotch is bare. For the related question in terms of a Bondi boundary rather than a MTS, the issue is who wins the race toward 0, the radius $r$ or the expansion $\Theta_{OUT}$ along some ray.
We conduct this race for each of a sequence of models in the range $0 \le
\epsilon \le 10^{-2}$, with the remaining parameters fixed at $M=100$, $u_-=-100$, $\hat t_- =-10$, $a=1$ and $p=(10^{-2}+10^{-5})/\sqrt{13}$ (just above the minimum value of $p$ allowed by regularity of the conformal model for this range of eccentricities).
We monitor the minimum value over the sphere of the expansion of the outgoing null rays on the horizon, and of the Bondi radius of the horizon. The results are displayed in Fig. \[fig:exp-r-race\] in terms of values of $r$ and $\Theta_{OUT}$ normalized to 1 at the initial time, so that the race starts out even.
Panel $(a)$ of the figure, for the small value $\epsilon=10^{-7}$, shows little deviation from a Schwarzschild horizon. A Bondi boundary ${\cal B}$ forms at $u\approx 0$ as a consequence of the zeroth order in $\epsilon$ term ${\cal
R}\approx 2$ in Eq. (\[eq:rlam\]) (${\cal R}= 2$ for a unit sphere), which causes $\Theta_{OUT}$ to decrease linearly with $u$. We have verified that the initial slope of $\Theta_{OUT}$ as seen in the graph corresponds to the expected Schwarzschild value.
For $\epsilon=10^{-6}$, still near the close limit, the radius of the horizon hardly changes before the Bondi horizon forms, as illustrated in panel $(b)$. However, the deviation of the expansion from a pure linear-in-time behavior is noticeable and its deviation from spherical symmetry as a function of ray is also noticeable in the full numerical data. (The angular behavior of the relevant geometrical quantities is discussed more fully below.) As $\epsilon$ increases to $10^{-5}$ in panel $(c)$, both the radius and the expansion show markedly nonlinear behavior but the expansion still readily wins the race toward zero. However, its margin of victory gets smaller with increasing $\epsilon$ as manifest in panel $(d)$ for $\epsilon=10^{-4}$ in which the race is nearly a tie.
For $\epsilon=10^{-3}$, as shown in panel $(e)$, the radius now wins the race. The fission takes place while the expansion is still significantly large, at about $90\%$ of its initial value. For a larger value of $\epsilon$, as shown in panel $(f )$, the effect is even more dramatic. The radius makes a sudden plunge to zero to win the race before the expansion has undergone any appreciable change. This is a dramatic nonlinear effect. Although the radius and expansion begin the race at the same starting point (in rescaled units), the expansion begins with a flying start (its initial slope in the figures) and gets accelerated by linear effects whereas the radius starts from rest and only gets accelerated by quadratic or higher nonlinearities.
Figures \[fig:eps-5\_frontside\] and \[fig:eps-3\] are embedding pictures of the horizon which reveal the angular behavior of the race. (The construction of the embedding pictures is explained in Ref. [@asym]). The darkened portions of the pictures indicate where the expansion has gone negative. Figure \[fig:eps-5\_frontside\] shows that when the expansion reaches zero first it does so at the pole; whereas the radius first reaches zero at the equator where the horizon pinches off to form separate white holes. Figure \[fig:eps-3\] shows that for $\epsilon = 10^{-3}$ the pinch-off occurs before the outward expansion has gone to zero along any ray.
More insight into the angular behavior is provided by surface plots as functions of $(u,\theta)$ of the quantities $r$, $\Theta_{OUT}$, the curvature scalar ${\cal R}$, the twist (as described by the normalized component $\omega_{\hat \theta} = \omega_a \hat \theta^a$, ), and the “plus” component of the outgoing shear
$$\sigma_{+, OUT} = \frac{1}{2} (\hat \theta^a \hat \theta^b
-\hat \phi^a \hat \phi^b) \nabla_al_b
= \frac{1}{4} \left[ \ln\left(\frac{K + \Re(J)}{K - \Re(J)}
\right)\right]_{,\lambda} ,$$
where $\hat
\theta^a$ and $\hat \phi^a$ are unit vectors in the $\theta$ and $\phi$ directions. (The remaining components of the twist and shear vanish because of symmetry.) Because of axial and reflection symmetry, the range $0\le \theta \le \pi /2$ suffices to display the full angular behavior.
The first set of surface plots, Figs. \[fig:rho5\]-\[fig:shearp5\], are for the mildly nonlinear case $\epsilon=10^{-5}$, corresponding to panel (c) in Fig. \[fig:exp-r-race\] in which the expansion wins the race. The evolution is traced from the starting time to the finish when the Bondi boundary forms. Figure \[fig:rho5\] shows that the radius decreases fastest at the equator, in accord with the trouser-shaped horizon picture. Figure \[fig:exp5\] shows that the expansion wins the race along a polar ray. Figure \[fig:ricci5\] shows that the curvature scalar ${\cal R}$ of the conformal horizon metric $h_{AB}$ increases from its unit sphere value ${\cal R}=2$ in the region near the poles and decreases in the region near the equator. This is what would be expected for the conformal geometry of the Minkowski time slices $\hat t =const$ of of the collapsing prolate wave front which seeds the model. However, it is important to bear in mind that the quantities in Figs. \[fig:rho5\]-\[fig:shearp5\] refer to the curved space affine time slices $t=const$. This is emphasized in Fig. \[fig:twist5\], which shows the behavior of the twist, a quantity which would vanish for the Minkowski time slices of the flat space wave front. The shear $\sigma_{+,OUT}$, as depicted in Fig. \[fig:shearp5\], is positive near the poles and negative near the equator, again as would be expected from a prolate Minkowskian wave front (according to our above polarization convention). Although the time dependence of all quantities in the model is analytic, there is an apparent “crease” at $t\approx =-80$ in the surface plots of the Ricci scalar, twist and shear which results from a rapid change in the quantity $\Lambda^{-1}=dt/d \hat t$ governing the relative Minkowski and curved space affine times. For models with larger $\epsilon$, the angular dependences are qualitatively similar but the time dependence is more dramatic.
Discussion {#sec:discussion}
==========
Expressed now in terms of a black hole merger, this work has traced the horizon structure of a head-on black hole collision from the close approximation to the nonlinear regime. It has revealed dramatic time dependence in the intrinsic and extrinsic curvature properties of the horizon in the extreme nonlinear regime. The results suggest two classes of binary black hole space-times depending upon whether the crotch in the standard trouser picture is protected, in the sense that it lies inside a marginally anti-trapped surface on the horizon, or bare. Only in the bare case is it possible that the black holes are formed by either the collapse of matter or the implosion gravitational waves (see Fig. \[fig:strategy\]) originating in an initially nonsingular space-time.
The results pave the way for an application of the PITT code to calculate the fully nonlinear wave forms emitted in the merger to ringdown phase. It remains to be seen in future work whether the dramatic time dependence in the merger stage is responsible for equally dramatic wave forms.
While the numerical results presented in this paper are for the axially symmetric case, the codes are not restricted to any symmetry. It will be interesting to see how the results for a head-on collision are modified in the inspiral and merger of spinning black holes.
This work has been partially supported by NSF PHY 9510895 and NSF PHY 9800731 to the University of Pittsburgh. We have benefited from conversations with our longtime collaborators Luis Lehner and Nigel T. Bishop. R.G. thanks the Albert-Einstein-Institut for hospitality. Computer time for this project was provided by the Pittsburgh Supercomputing Center and by NPACI.
Numerical integration {#app:numint}
=====================
Integration of $\rho$
---------------------
As explained in Sec. \[sec:confmod\], the conformal model supplies the area coordinate $r$ as well as the null data $J$, but for completeness we describe here the integration of the Raychaudhuri equation (\[eq:ruu\]) that determines $r$ from the null data in a more general setting. We integrate this equation in terms of the variable $\rho=r/r_M$, where $r_M$, defined in Eq. (\[eq:rm\]), satisfies $\dot r_M{}_{|\lambda=0}=0$ and $\ddot r_M =0$. As a result, the evolution equation for $\rho$ is identical to that for $r$, which in the spin-weighted form of Eq. (\[eq:ruus\]) becomes $$\ddot \rho =-\frac{\rho}{4} (\dot J \dot {\bar J}-\dot K^2) .$$ The initial data consists of the values of $\rho$ and $\dot \rho$ on the initial slice. Note that $\dot K = \Re(\dot J \bar J) / K$. We put the equation in first-order form, $$\dot \rho = \Pi, \quad
\dot \Pi = -4\, S \rho
\label{fosrho}$$ with $S=(\dot J \dot {\bar J}-\dot K^2)/16$.
The advantage of the first-order form is that the pair of equations (\[fosrho\]) can be discretized to second-order accuracy using only two time levels, in the same footing as the other horizon evolution equations. The time integration stencil is the midpoint rule [@recipes],
$$\begin{aligned}
\label{rhostencil}
\frac{\rho^{n+1} - \rho^{n}}{\Delta u} &=& \frac{1}{2}
(\Pi^{n+1} + \Pi^{n})
\\
\frac{\Pi^{n+1} - \Pi^{n}}{\Delta u} &=& - 2\, S^{n+\frac{1}{2}}
(\rho^{n+1} + \rho^{n})\end{aligned}$$
which can be solved simultaneously for $\rho^{n+1}$ and $\Pi^{n+1}$ to give
$$\begin{aligned}
\label{fosrhofde}
\rho^{n+1} &=& \frac{\rho^{n} (1 - S {\Delta u}^2) + \Pi^{n} {\Delta u}}
{1 + S {\Delta u}^2}
\\
\Pi^{n+1} &=& \frac{\Pi^{n} (1 - S {\Delta u}^2)
+ 4\, S \rho^{n} {\Delta u}}
{1 + S {\Delta u}^2}.\end{aligned}$$
Integration of $\omega$
-----------------------
The time dependence of $\omega$ is determined by Eq. (\[eq:omegadots\]), which we renormalize by factoring out $r_M^2$ from both sides. On the left hand side, this is accomplished by using the identity $$(r^2 \omega)\dot{} = r_M^2 ( \rho^2 \omega ) \dot {},$$ which holds on the horizon. Similarly, we re-express the first term on the right-hand side as $$r^2 \eth \left(\frac{\dot r}{r} \right) = r_M^2 \rho^2 \eth
\left( { {\dot r_M \rho + r_M \dot \rho} \over {r_M \rho} } \right) =
r_M^2 \rho^2 \left( \eth \left(\frac{\dot r_M}{r_M} \right)
+ \eth \left(\frac{\dot \rho}{\rho} \right) \right)
=r_M^2 \rho^2 \eth \left(\frac{\dot \rho}{\rho} \right).$$ Equation (\[eq:omegadots\]) then reduces to
$$\begin{aligned}
(\rho^2 \omega)\dot{} &=&
\left(\frac{1}{4} P_2 \rho + P_1 \right) \rho + P_0, \quad {\rm where}
\label{eq:dotomega} \\
P_0 &=& - \dot\rho \eth\rho \\
P_1 &=& \eth\dot\rho
+ (\dot J \bar J - \dot K K) \eth\rho
+ (J \dot K - K \dot J) \bar\eth\rho \\
P_2 &=& \bar{\dot J} \eth J - 4 \dot K \eth K
+ 2 \dot K \bar\eth J + 3 \dot J \eth \bar J
- 2 \dot J \bar\eth K + 2 J \bar\eth\dot K
+ 2 \bar J \eth\dot J - 2 K \bar\eth\dot J
- 2 K \eth\dot K .\end{aligned}$$
We use the midpoint rule to integrate Eq. (\[eq:dotomega\]), i.e. the left-hand side is evaluated as $$(\rho^2 \omega)\dot{} = \frac{1}{\Delta t} \left( (\rho^2 \omega)^{n+1}
- (\rho^2 \omega)^{n} \right)$$ and $\rho$, $J$, etc. in $P_0$, $P_1$ and $P_2$ are evaluated at $t^{n+1/2}$, [*e.g.*]{} $$J \equiv \frac{1}{2}(J^{n+1} + J^{n}), \quad
\dot J \equiv \frac{1}{\Delta t}(J^{n+1} - J^{n}) .$$ \[Since $\omega$ does not enter in the right-hand side, this is a special case of the second order Runge-Kutta scheme [@recipes], also used below in Eqs. (\[eq:rholu\]) and (\[eq:jlrho\]).\]
Integration of $\rho_{\lambda}$
-------------------------------
The time dependence of $r_{,\lambda}$, determined by Eq. (\[eq:rlams\]), is re-expressed in terms of $\rho_{,\lambda}$ using the ansatz $r=r_M \rho$, which yields, for $\lambda=0$, $$(r^2)_{,\lambda u} =
8 M^2 \rho \rho_{,\lambda u}
+ (8 M^2 \rho_{,\lambda} - 2 u \rho ) \rho_{,u} - \rho^2 .$$ Substitution into Eq. (\[eq:rlams\]) then gives $$\begin{aligned}
8 M^2 \rho \rho_{,\lambda u}
+ (8 M^2 \rho_{,\lambda} - 2 u \rho ) \rho_{,u} & = &
\Re \left[
\left( \bar\eth K - \eth \bar J \right)
\left( \frac{\eth\rho}{\rho} + \omega \right)
+ K \left( \bar\eth\omega + \frac{\bar\eth\eth\rho}{\rho}
- \frac{\eth\rho\bar\eth\rho}{\rho^2}
\right) \right. \nonumber \\
&-& \left. \bar J \left( \eth\omega + \frac{\eth^2\rho}{\rho}
- \frac{(\eth\rho)^2}{\rho^2}
\right)
- \bar J \omega^2
+ K \omega \bar\omega - \frac{1}{2}{\cal R} + \rho^2 \right],
\label{eq:rholu}\end{aligned}$$ which we integrate using a second order Runge-Kutta scheme.
Integration of $J_{\lambda}$
----------------------------
After setting $r=r_M \rho$, the evolution equation (\[eq:jlamdots\]) for $J_{,\lambda}$ becomes $$\begin{aligned}
8 M^2 \rho^2 J_{,\lambda u} - u \rho^2 \dot J
+ \left(1 + K^2 \right) \eth \omega
= - 8 M^2 \rho \left( \dot\rho J_{,\lambda} + \rho_{,\lambda} \dot J \right)
+ 4 M^2 J \left( \dot{\bar J} J_{,\lambda}
+ \dot J \bar J_{,\lambda}
- 2\, \dot K K_{,\lambda}
\right) \nonumber \\
+ \left(1 + K^2 \right)
\left( \omega^2 - 2\, \omega \frac{\eth \rho}{\rho} \right)
- \omega \left(J \bar\eth K - K \bar\eth J \right)
- \bar\omega \left(K \eth J - J \eth K \right)
\nonumber \\
+ J \left(J \bar\eth \bar\omega
- K \left( \eth\bar\omega + \bar\eth \omega \right)
+ J \bar\omega^2
- 2 K \omega \bar\omega
+ 2\, \frac{K}{\rho} \left( \omega \bar\eth\rho
+ \bar\omega \eth\rho
\right)
- 2\, J \bar\omega \frac{\bar\eth\rho}{\rho}
\right)
\label{eq:jlrho}\end{aligned}$$ which we again integrate using a second order Runge-Kutta scheme.
The case of axial symmetry {#app:axial_symmetry}
==========================
In axial symmetry the Sachs metric Eq. (\[eq:amet\]) can be written as $$ds^2 = -(W - U^2 \gamma r^{-2})du^2 - 2 du d\lambda -2 U
du\, dx + r^2\left( \frac{dx^2}{\gamma} + \gamma \,
d\varphi^2\right).
\label{eq:aximet}$$ Here $W$, $U$ and $r$ are functions of $(u,\lambda,x=\cos{\theta})$. With the dyad choice associated with Eq. (\[eq:headonj\]) the conformal horizon model gives $$\gamma =(1-x^2)\Gamma,\quad \Gamma =
\left(\frac {\hat t - r_\phi} {\hat t
-r_\theta} \right),\quad J = \frac{1}{2}\left(1/\Gamma
- \Gamma\right),
\label{eq:gamma_and_J}$$ with $\Gamma$ a well-behaved function on the sphere. For the prolate spheroidal model considered here, $\Gamma$ vanishes at the points $\hat t =r_\phi$ where the horizon pinches off and $\Gamma =O(r^2)$. Thus, in the prolate case, it is useful for numerical purposes to use $\gamma$, as opposed to $\gamma^{-1}$, as the variable to represent the metric on the two-sphere of constant $u$ and $\lambda$. The twist $\omega$ is related to the quantity $U$ by $$\omega = -\frac{1}{2} \sqrt{1 - x^2} \, U_{,\lambda} .
\label{eq:capU}$$
The Einstein equations yield the following system of PDE’s for propagating the metric variables along the horizon: $$\begin{aligned}
\partial_u (r^2 U_{,\lambda}) &=&
2\,r_{,u}\,r_{,x} - 2\,r\,r_{,xu} -
\frac{2\,r\,r_{,x}\,\gamma_{,u}}{\gamma}
- \frac{r^2\,\gamma_{,xu}}{\gamma}
\label{eq:dot_r2Ul}\\
\partial_u \partial_\lambda r^2 &=& -\frac{(r_{,x})^2\,\gamma}{r^2} +
\frac{r_{,xx}\,\gamma}{r} + \frac{(U_{,\lambda})^2 \,\gamma}{4} -
\frac{U_{,\lambda x}\,\gamma}{2} + \frac{r_{,x}\,\gamma_{,x}}{r} -
\frac{U_{,\lambda}\,\gamma_{,x}}{2} + \frac{\gamma_{,xx}}{2}
\label{eq:dot_r2l}\\
\partial_u \partial_\lambda \, \gamma &=&
\frac{r_{,x}\,U_{,\lambda}}{r^3}
+ \frac{(U_{,\lambda})^2 - 2 U_{,\lambda x} }{4 r^2}
+\frac{\gamma_{,u}\,\gamma_{,\lambda}}{\gamma}
-\frac{r_{,\lambda}\,\gamma_{,u} + r_{,u}\,\gamma_{,\lambda}}{r} .
\label{eq:dot_gammal}\end{aligned}$$
In the Schwarzschild case, $r=r_M=2M -(\lambda u/4M)$, $U=0$ and $\gamma =
(1-x^2)$. Thus, if the geometry is near Schwarzschild, all terms on the right-hand side are small except $\gamma_{xx}=-2$. In order to make Eq. (\[eq:dot\_r2l\]) numerically well behaved we subtract this term by introducing the auxiliary variable $\Delta=\partial_\lambda (r^2
- r_M^2)$. Then $\Delta$ satisfies $$\partial_u \Delta = -\frac{(r_{,x})^2\,\gamma}{r^2}
+ \frac{r_{,xx}\,\gamma}{r}
+ \frac{(U_{,\lambda})^2 \,\gamma}{4}
- \frac{U_{,\lambda x}\,\gamma}{2}
+ \frac{r_{,x}\,\gamma_{,x}}{r}
- \frac{U_{,\lambda}\,\gamma_{,x}}{2}
+ \left(\frac{\gamma_{,xx}}{2} + 1\right) .
\label{eq:dot_Delta}$$ The quantity $r_{,\lambda}$ is reconstructed as $$r_{,\lambda} = \frac{\Delta + (r_M^2)_{,\lambda}}{2 r}
= \frac{\Delta - u}{2 r}.$$
Special care is needed in order to write the right-hand sides of Eqs. (\[eq:dot\_r2Ul\]) and (\[eq:dot\_gammal\]) in manifestly regular form (for $r\neq 0$) at the axis of symmetry ($x=-1, 1$) where $\gamma= (1-x^2)\,\Gamma$ vanishes. We thus express $\gamma_{,u}/\gamma$ as $\Gamma_{,u}/\Gamma$ and $\gamma_{,x u}/\gamma$ as $$\frac{\gamma_{,x u}}{\gamma}
= \frac{\Gamma_{,ux}}{\Gamma}
- \frac{2x \Gamma_{,u} }{(1-x^2) \Gamma } .$$ Axial symmetry implies that $\Gamma_{,u}$ vanishes at the poles so that the right-hand side of Eq. (\[eq:dot\_r2Ul\]) is regular, as can be seen by differentiating the conformal data in Eq. (\[eq:gamma\_and\_J\]) to obtain $$\Gamma_{,u} =- \left( \frac{\partial \hat t}{\partial u} \right)
\frac{(r_\phi -r_\theta)}{ (\hat t -r_\theta)^2}$$ and noting that $r_\phi =r_\theta$ on the symmetry axis. Thus Eq. (\[eq:dot\_r2Ul\]) is rendered explicitly regular on the symmetry axis. At the pinch-off points at $r=0$, Eqs. (\[eq:dot\_r2Ul\]) - (\[eq:dot\_gammal\]) are singular but the numerical performance near these points can be enhanced by noting that $\gamma/r^2$ remains regular, by virtue of Eqs. (\[eq:def\_rhat\]) and (\[eq:gamma\_and\_J\]).
The data at ${\cal S}_-$ are initialized by the same prescription as for the general case discussed in Sec. \[sec:sminus\], which implies $$\partial_\lambda U_- = J_{-,x} - \frac{2 x}{1-x^2} J_- \, ,$$ in accord with Eq. (\[eq:omegaminus\]) $$\Delta_- = u_- \,\left(1 - \frac{r_-^2}{4M^2}\right) ,$$ in accord with Eq. (\[eq:rhominus\]) and $$\partial_\lambda \gamma_- = 0 .$$ in accord with Eq. (\[eq:jminus\]).
We integrate Eqs. (\[eq:dot\_r2Ul\]), (\[eq:dot\_gammal\]) and (\[eq:dot\_Delta\]) numerically to second order accuracy, using the same midpoint rule as for the general case in Appendix \[app:numint\].
Convergence tests
=================
We have verified second order convergence of the hierarchy of equations which provide horizon data for the 3D code.
We have also checked second order convergence of the axially symmetric code, and we have used it to confirm the behavior, described in Sec. \[sec:results\], of the sudden nonlinear plunge of the horizon radius to zero. Our aim is to use the axially symmetric code to generate an independent numerical solution against which to check the 3D code.
Due to its lower computational requirements, the axisymmetric calculation is significantly more efficient than the 3D one. This will be particularly useful in the approach to the caustic, where the highest resolution is needed, and where the axisymmetric code will allow us to make more detailed studies of the behavior of the horizon.
[10]{}
Luis Lehner, Nigel T. Bishop, Roberto Gómez, Bela Szilágyi, and Jeffrey Winicour, Phys. Rev. D [**60**]{}, 044005 (1999).
Sascha Husa and Jeffrey Winicour, Phys. Rev. D [**60**]{}, 084019 (1999).
Richard H. Price and Jorge Pullin, Phys. Rev. Lett. [**72**]{}, 3297 (1994).
Nigel T. Bishop, Roberto Gómez, Luis Lehner, Manoj Maharaj, and Jeffrey Winicour, Phys. Rev. D [**56**]{}, 6298 (1997).
R. Gómez, L. Lehner, R. L. Marsa, and J. Winicour, Phys. Rev. D [**57**]{}, 4778 (1997).
J. Winicour, Prog. Theor. Phys. Suppl. [**136**]{}, 57 (1999).
Manuela Campanelli, Roberto Gómez, Sascha Husa, Jeffrey Winicour, and Yosef Zlochower, Phys. Rev. D (to be published), gr-qc/0012107.
H. Bondi, M. van der Burg, and A. Metzner, Proc. R. Soc. London, [**A269**]{}, 21 (1962).
R. Sachs, Proc. R. Soc. London, [**A270**]{}, 103 (1962).
R. Sachs, J. Math. Phys. [**3**]{}, 908 (1962).
Abhay Ashtekar, Stephen Fairhurst, and Badri Krishnan, Phys. Rev. D [**62**]{}, 104025 (2000).
S. A. Hayward, Class. Quantum Grav. [**10**]{}, 773 (1993).
S. A. Hayward, Class. Quantum Grav. [**10**]{}, 779 (1993).
J. Winicour, J. Math. Phys. [**24**]{}, 1193 (1983).
J. Winicour, J. Math. Phys. [**25**]{}, 2506 (1984).
R. Gómez, L. Lehner, P. Papadopoulos, and J. Winicour, Class. Quantum Grav. [**14**]{}, 977 (1997).
R. Penrose and W. Rindler, [*Spinors and Space-Time*]{} (Cambridge University Press, Cambridge, England, 1984), Vol. 1.
R. K. Sachs, Proc. R. Soc. London [**A264**]{}, 309 (1961).
W. Israel, Phys. Rev. [**143**]{}, 1016 (1966).
R. Bartnik and A. H. Norton, in [*Computational Techniques and Applications: CTAC97*]{}, edited by B. J. Noye, M. D. Teubner, and A. W. Gill (World Scientific, Singapore, 1998), p. 91.
R. Bartnik and A. H. Norton, SIAM J. Sci. Comput. (USA) [**22**]{}, 917 (2000).
R. Penrose, Phys. Rev. Lett. [**14**]{}, 57 (1965).
W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, [*Numerical Recipes*]{} (Cambridge University Press, New York, 1986), Chap. 6.6, p. 182.
=6.0in
=6.0in
=6.0in
|
---
abstract: 'The current approaches to false discovery rate (FDR) control in multiple hypothesis testing are usually based on the null distribution of a test statistic. However, all types of null distributions, including the theoretical, permutation-based and empirical ones, have some inherent drawbacks. For example, the theoretical null might fail because of improper assumptions on the sample distribution. Here, we propose a null distribution-free approach to FDR control for large-scale two-groups hypothesis testing. This approach, named *target-decoy procedure*, simply builds on the ordering of tests by some statistic/score, the null distribution of which is not required to be known. Competitive decoy tests are constructed by permutations of original samples and are used to estimate the false target discoveries. We prove that this approach controls the FDR when the statistics are independent between different tests. Simulation demonstrates that it is more stable and powerful than two existing popular approaches. Evaluation is also made on a real dataset of gene expression microarray.'
author:
- 'Kun He$^{1,3}$, Mengjie Li$^{2,3}$, Yan Fu$^{2,3,4}$, Fuzhou Gong$^{2,3}$, Xiaoming Sun$^{1,3}$'
title: 'Null-free False Discovery Rate Control Using Decoy Permutations for Multiple Testing'
---
\[firstpage\]
False discovery rate control; Multiple testing; Target-decoy approach; Null distribution-free methods.
Introduction {#sec:intro}
============
Traditional approaches to FDR control
-------------------------------------
Multiple testing has become increasingly popular in the present big-data era. For example, a typical scenario of applying multiple testing in biomedical studies is to look for differentially expressed genes/proteins, from thousands of candidates, between two groups (i.e. cases and controls) of samples [@efron2008microarrays; @diz2011multiple]. Currently, controlling the false discovery rate (FDR), which is defined as the expected proportion of incorrect rejections among all rejections [@benjamini1995controlling], is the predominant way to do multiple testing. FDR control procedures aim at selecting a subset of rejected hypotheses such that their FDR is no more than a given level.
Because a *p*-value is typically computed from the null distribution of a test statistic in each single test, the canonical approaches to FDR control for multiple testing at present are based on the *p*-values of all tests or at least the null distribution of the test statistic. Since @benjamini1995controlling proposed the first *p*-value based sequential procedure to control the FDR (BH procedure), many FDR control approaches have been developed, e.g., [@benjamini2001control; @sarkar2002some; @storey2002direct; @storey2003positive; @benjamini2006adaptive; @basu2017weighted].
A key problem faced by these approaches is how to obtain the proper null distribution. Popular null distributions, e.g. the theoretical null, permutation null, bootstrap null and empirical null, often suffer one way or another [@efron2008microarrays; @efron2012large]. The theoretical null, though widely used, might fail in practice for many reasons, such as improper mathematical assumptions or unobserved covariates [@efron2007size; @efron2008microarrays]. For example, if the real null distribution is not normal, the *p*-values calculated by Student’s *t*-test are not uniform (0, 1) distributed for true null hypotheses. The permutation null is also widely used. There are mainly two different permutation methods, i.e., the permutation tests and the pooled permutation [@kerr2009comments]. The permutation tests are a class of widely used non-parametric tests to calculate *p*-values, and are most useful when the information about the data distribution is insufficient. However, the statistical power of permutation tests is limited by the sample size of a test [@tusher2001significance]. Instead of estimating a null distribution for each test individually, the pooled permutation in multiple testing estimates an overall null distribution for all tests [@efron2001empirical]. However, it has been found that pooling permutation null distributions across hypotheses can produce invalid *p*-values, since even true null hypotheses can have different permutation distributions [@kerr2009comments]. Bootstrap is another popular method for calculating $p$-value, but it is not applicable to the cases in which the number of tests is much larger than the sample size of a test [@fan2007many; @efron2012large; @liu2014phase].
To overcome the shortcomings of the theoretical and permutation null distributions, new methods were proposed to estimate an empirical null distribution from a large number of tests [@efron2001empirical; @efron2002empirical; @efron2008microarrays; @scott2010bayes]. For example, the empirical Bayes method estimates the empirical null distribution by decomposing the mixture of null and alternative distributions [@efron2008microarrays]. However, decomposing the mixture distribution is intrinsically a difficult problem. For example, if the empirical distribution has a strong peak, the decomposing may fail [@strimmer2008unified].
Moreover, the proportion of true null hypotheses has to be estimated either explicitly or implicitly to apply these FDR control methods. If this null proportion is ignored (e.g., assumed to be one as in the original BH procedure), the power of testing would be reduced. Since @storey2002direct proposed the first method to estimate the null proportion, estimation of the null proportion has become a key component of current FDR methods to enhance the power, such as the Bayes and the empirical Bayes methods [@storey2003positive; @storey2004strong; @benjamini2006adaptive; @efron2008microarrays; @strimmer2008unified]. More accurate estimation of the null ratio has been of great interest in the field [@langaas2005estimating; @meinshausen2006estimating; @markitsis2010censored; @yu2017parametric].
Our approach to FDR control
---------------------------
Here, we propose a new approach to FDR control, named *target-decoy procedure*, which is free of the null distribution and the null proportion. In this approach, a target score and a number of decoy scores are calculated for each test. The target score is calculated with regard to the original samples. The decoy scores are calculated with regard to randomly permutated samples. Based on the target score and decoy scores, a label and a final score are calculated for each test as follows. If the target score is more significant than half or a major proportion of the decoy scores, the test is labelled as target and the final score is set as the target score. Otherwise, if the target score is less significant than half of the decoy scores, the test is labelled as decoy and the final score is set as the decoy score with a specific rank mapped from the rank of the target score. Then the tests are sorted by their final scores and the ratio of decoy (plus one) to target tests beyond a threshold is used for FDR control. We prove that this target-decoy procedure can rigorously control the FDR when the scores are independent between tests.
Our approach is exclusively based on the scores and labels of tests. Unlike *p*-values or test statistics which have clear null distributions, the scoring function used in our approach can be any measure of the (dis)similarity of two groups of samples. Therefore, our approach is very flexible and can be more powerful than traditional approaches that are limited by the precision of *p*-values or the sample size of each test. In our approach, we obtain the labels of tests by randomly permuting the samples of each test and comparing the target score to the decoy scores. To our knowledge, the idea of using competitive permutations as decoys for FDR control is new in multiple testing.
Monte-Carlo simulations demonstrate that our approach is more stable and powerful than two popular methods, i.e., the Bayes method [@storey2002direct; @storey2003positive; @storey2004strong] and the empirical Bayes method [@efron2001empirical; @efron2002empirical; @efron2008microarrays]. The performances of the three methods were also compared on a real dataset. Because our procedure is more straightforward and can be used with arbitrary scores, we believe that it will have many practical applications.
Related work
------------
Our approach was inspired by the target-decoy database search strategy widely used to estimate the FDR of peptide identifications in proteomics [@Elias2007Target]. We proved that a modified version of this strategy (addition of one to the number of decoys) controls the FDR of peptide identifications [@he2013Multiple; @he2015theoretical]. Through decoy permutations of samples, we generalize the target-decoy strategy to multiple testing problems in this paper.
Another remarkable FDR control method free of null distribution is the knockoff filter for variable selection in regression [@barber2015controlling]. The salient idea of knockoff filter is to construct a “knockoff" variable for each real variable. The knockoff variables are not (conditionally on the original variables) associated with the response, but they compete with their real counterparts in variable selection. Thus, they could be used as the contrasts to estimate the proportion of real variables that are selected but not associated with the response. Similar to our approach, FDR control is achieved by adding one to the number of knockoffs selected. In the original paper of knockoff, @barber2015controlling considered a Gaussian linear regression model where the number of covariates is no more than the number of observations. Since then, this method has been extended to a wide range of variable selection problems in regression, such as the high dimensional setting where the number of covariates is more than the number of observations [@barber2016knockoff] and a nonparametric setting with Gaussian covariates [@candes2018panning].
The role of knockoffs is essentially very similar to that of decoys in our approach. Compared to the knockoff method, our method is different in the following three aspects. Firstly, the two methods address two different problems in statistical inference. Our approach mainly concerns about the classical multiple testing problem in case-control studies, the goal of which is to search for random variables differently distributed between cases and controls. The knockoff method mainly concerns about the variable selection problem, the goal of which is to find statistically significant associations between a response and a large set of potentially explanatory variables. Though these two problems have some related applications in practice, they are usually treated differently in mathematics. Secondly, the method of constructing knockoffs is very different from that used to construct decoys. Construction of knockoffs usually involves complex matrix computation, such as eigenvalue computation and semidefinite programming, and is only applicable in specific cases, such as the case of Gaussian covariates [@candes2018panning]. Meanwhile, it has been shown that permutation is inappropriate to be used to construct knockoffs for variable selection, because it cannot maintain the correlation between the original variables [@barber2015controlling]. In our approach, a simple permutation-based method is used to construct decoy tests. In multiple testing the impact of correlation on FDR control is greatly reduced because removing the redundant variables is not required as in variable selection. In this setting, permutation is widely used and results in good performance [@efron2012large]. Thirdly, only one knockoff copy is constructed for each covariate in current knockoff methods, and the probability of the knockoff copy or the original covariate being selected is equal (0.5) when the original covariate is not associated with the response. In our approach, multiple decoy permutations are constructed for each test, which offers us the flexibility of setting different probabilities of producing target or decoy tests for true null hypotheses (through an adaptive parameter $r$). This can enhance the power when the number of significant variables is small as we experimentally illustrated. The rest of the paper is organized as follows. Section \[sec: section 2\] gives our target-decoy approach for FDR control. Section \[sec:Problem Formulation\] discusses a general model for case-control study. The standard target-decoy procedure is presented in Section \[sec:The target-decoy approach\]. A simplified and an adaptive versions of the target-decoy procedure are provided in Sections \[sec:simplified target-decoy procedure\] and \[sec:Automatically choosing $r$\], respectively. Section \[sec:Theoretic Analysis\] establishes the theoretical foundation of our approach. Numerical results are given in Section \[sec:Simulation Studies\]. In Section \[sec:An Application\], we show an application of the target-decoy procedure on an *Arabidopsis* microarray dataset. Section \[sec:Discussion\] concludes the paper with a discussion of future works.
The target-decoy approach {#sec: section 2}
=========================
Problem formulation {#sec:Problem Formulation}
-------------------
Consider a two-groups (case and control) study involving $m$ random variables, $X_1,X_2,\cdots,\\X_m$. For each random variable $X_j$ where $j \in [m]$, there are $n$ random samples $X_{j_1},X_{j_2},\cdots,\\X_{j_n}$, in which $X_{j_1},X_{j_2},\cdots,X_{j_{n_1}}$ are from the $n_1$ cases and $X_{j_{n_1+1}},\cdots,X_{j_n}$ are from the $n_0 = n - n_1$ controls.
The goal is to search for random variables differently distributed between cases and controls. The null hypothesis for random variable $X_j$ used here is the ‘symmetrically distributed’ hypothesis $H_{j0}$: the joint distribution of $X_{j_1},X_{j_2},\cdots,X_{j_n}$ is symmetric. In other words, the joint probability density function of $X_{j_1},X_{j_2},\cdots,X_{j_n}$ (or the joint probability mass function if $X_{j_1},X_{j_2},\cdots,X_{j_n}$ are discrete) satisfies $f_{X_{j_1},\cdots,X_{j_n} } (x_{j_1},\cdots,x_{j_n} )=f_{X_{j_1},\cdots,X_{j_n} } ({\pi}_n (x_{j_1},\cdots,x_{j_n}))$ for any possible $x_{j_1},\cdots,x_{j_n}$ and any permutation ${\pi}_n$ of $x_{j_1},\cdots,x_{j_n}$. If $X_{j_1},\cdots,X_{j_n}$ are independent, this hypothesis is equivalent to that $X_{j_1},\cdots,X_{j_n}$ are identically distributed. Here we use the ‘symmetrically distributed’ hypothesis to deal with the case where $X_{j_1},\cdots,X_{j_n}$ are related but still an exchangeable sequence of random variables [@chow2012probability].
Let $S(x_1,x_2,\cdots,x_n)$ be some score to measure the difference between $x_1,x_2,\cdots,x_{n_1}$ and $x_{n_1+1},x_{n_1+2},\cdots,x_n$. Without loss of generality, we assume that larger scores are more significant and $S(x_1,\cdots,x_n )=S({\pi}_{n_1} (x_1,\cdots,x_{n_1} ),{\pi}_{n_0 }(x_{n_1+1},\cdots,x_n ))$ holds for any possible $x_1,\cdots,x_n$ where ${\pi}_{n_1} (x_1,\cdots,x_{n_1})$ is any permutation of $x_1,\cdots,x_{n_1 }$ and ${\pi}_{n_0} (x_{n_1+1},\cdots,x_n )$ is any permutation of $x_{n_1+1},\cdots,x_n$. Note that neither the null distributions of scores nor the distributions of random variables are required to be known.
The target-decoy procedure for FDR control {#sec:The target-decoy approach}
------------------------------------------
For any fixed $r \in [1, \binom{n}{n_0}]$, the target-decoy procedure is as follows.
*Algorithm 1: the target-decoy procedure*
1. For each test $j$, calculate $t$ scores including a target score and $t-1$ decoy scores. The target score is $S_j^T=S(X_{j_1},X_{j_2},\cdots,X_{j_n})$. Each decoy score is obtained by first sampling a permutation ${\pi}_n$ of $X_{j_1},X_{j_2},\cdots,X_{j_n}$ randomly and then calculating the score as $S({\pi}_n(X_{j_1},X_{j_2},\cdots,X_{j_n}))$. Sort these $t$ scores in descending order. For equal scores, sort them randomly with equal probability.
2. For each test $j$, generate a number ${\Lambda}_j=i-P_j$ where $i$ is the rank of $S_j^T$ in the $t$ scores, and $P_j$ is a random number drawn from uniform$[0,1)$ distribution. Calculate a final score $S_j$ and assign a label $L_j\in \{T,D,U\}$, where $T,D$ and $U$ stand for target, decoy and unused, respectively. If ${\Lambda}_j\leq \frac{t}{2r}$, let $L_j=T$ and $S_j=S_j^T$. If $\frac{t}{2}<{\Lambda}_j \leq t$, generate a random number ${\Lambda}_j^{'}$ drawn from uniform$\big(0,\frac{t}{2r}\big]$ distribution, and let $L_j=D$ and $S_j$ be the score ranking $\lceil{\Lambda}_j^{'}\rceil$-th. Otherwise, let $L_j=U$ and $S_j$ be some minimum score.
3. Sort the $m$ tests in descending order of the final scores. For tests with equal scores, sort them randomly with equal probability. Let $L_{(1)},L_{(2)},\cdots,L_{(m)}$ denote the sorted labels and $S_{(1)},S_{(2)},\cdots,S_{(m)}$ denote the sorted scores.
4. If the specified FDR control level is $\alpha$, let $$\label{ATD_Con}
K=\max\{k \big| \frac{1}{r}\times \frac{\# \{L_{(j)}=D,j\leq k\}+1}{\# \{L_{(j)}=T,j\leq k \}\lor 1}\leq \alpha\}$$ and reject the hypothesis with rank $j$ if $L_{(j)}=T$ and $j \leq K$.
Section \[sec:Theoretic Analysis\] will show that the target-decoy procedure controls the FDR for any fixed $r$. We introduce $r$ to enhance the power of our approach for small datasets. In practice, one can set the value of $r$ empirically or simply set $r=1$ as described in Section \[sec:simplified target-decoy procedure\]. Alternatively, an algorithm can be used to choose $r$ adaptively for a given dataset as discussed in Section \[sec:Automatically choosing $r$\].
The random permutation used in our procedures can be generated by simple random sampling either with or without replacement, just as in the permutation tests. Similarly, with larger sampling number $t - 1$, the power of our approach will become slightly stronger as shown in Section \[sec:Simulation Studies\]. We can set $t$ as $\min\{\binom{n}{n_0},\tau\}$, where $\tau$ is the maximum number of permutations we would perform.
Unlike other FDR control methods, the target-decoy approach does not depend on the null distribution. The number of permutations, $t - 1$ can be much smaller than that used in permutation tests. In our simulations, $t-1$ was set as $49$ or $1$, while in the real data experiments, it was set as $19$. Simulations demonstrate that the target-decoy approach can still control the FDR even if $t - 1$ was set as $1$, in which case little information was revealed about the null distribution.
The simplified target-decoy procedure {#sec:simplified target-decoy procedure}
-------------------------------------
Step $2$ of the target-decoy procedure can be greatly simplified as follows by setting $r = 1$.
*Algorithm 2: the simplified target-decoy procedure (Steps 1,3,4 are identical to Algorithm 1 and are omitted here.)*
1. For each test $j$, calculate a final score $S_j$ and assign it a label $L_j\in \{T,D\}$, where $T$ and $D$ stand for target and decoy, respectively. Assume that the rank of $S_j^T$ is $i$. If $i<(t+1)/2$, let $L_j=T$ and set $S_j$ as $S_j^T$. If $i>(t+1)/2$, let $L_j=D$ and set $S_j$ as the score ranking $i-\lceil t/2 \rceil$. Otherwise, $i=(t+1)/2$, let $L_j$ be $T$ or $D$ randomly and set $S_j$ as $S_j^T$.
The adaptive target-decoy procedure {#sec:Automatically choosing $r$}
-----------------------------------
The parameter $r$ is for adjusting the ratio, $\Pr(Z_{j} = -1)/\Pr(Z_{j} = 1)$, for all $1 \leq j \leq m$ satisfying $H_j = 0$. On the one hand, equation (\[ATD\_Con\]) can be too conservative for a small $r$, e.g. 1 as in the simplified target-decoy procedure, because of the addition of 1 in the numerator if there are only a few false null hypotheses. For example, assume that the total number of tests is $80$ and the FDR control level is $0.01$. If $r$ is set as $1$, no hypothesis will be rejected, because the numerator of equation (\[ATD\_Con\]) is always no less than $1$ and the fraction is greater than $1/80>0.01$. On the other hand, if $r$ is too large, many false null hypotheses will be labelled as ‘$U$, potentially decreasing the power of testing. Thus, $r$ should be set appropriately in practice to enhance the power. Below, we provide an adaptive procedure to choose a suitable $r$ for the given dataset and the FDR control level.
*Algorithm 3: the adaptive target-decoy procedure*
1. Divide the samples of each random variable into two parts as follows. Choose suitable $n_{2}$ which is smaller than $n_0$ and $n_1$ from some range, say $5 \leq n_2 \leq \min\{\lfloor n_0/2 \rfloor,\lfloor n_1/2 \rfloor\}$. For each random variable $X_j$ where $j \in [m]$, randomly choose $n_2$ random samples from $X_{j_1},X_{j_2},\cdots,X_{j_{n_1}}$ and $X_{j_{n_1+1}},\cdots,X_{j_n}$, respectively. Let $X^\text{1}_{j_{1}},X^\text{1}_{j_{2}},\cdots,X^\text{1}_{j_{2n_2}}$ be these random samples. The rest has $n_1 - n_2$ random samples from the cases and $n_0 - n_2$ random samples from the controls. Let $X^\text{2}_{j_1},X^\text{2}_{j_2},\cdots,X^\text{2}_{j_{n - 2n_2}}$ be the rest random samples.
2. Set $t$ as $\binom{2n_2}{n_2}$ and perform the target-decoy procedure on $X^\text{1}_{j_{1}},X^\text{1}_{j_{2}},\cdots,X^\text{1}_{j_{2n_2}}$ where $j\in [m]$ for some range of $r$, say $R = \{1,2,5,10,15,20,25\}$. Let $r_{max}$ be the one such that the most hypotheses are rejected by the target-decoy procedure.
3. Perform the target-decoy procedure on $X^\text{2}_{j_{1}},X^\text{2}_{j_{2}},\cdots,X^\text{2}_{j_{{n - 2n_2}}}$ where $j\in [m]$ with $r = r_{max}$ and reject corresponding hypotheses.
Control theorem {#sec:Theoretic Analysis}
---------------
In this section, we will show that the target-decoy procedure controls the FDR. Let $H_{j} = 0$ and $H_{j} = 1$ denote that the null hypothesis for test $j$ is true and false, respectively. Note that $H_{1},H_{2},\cdots,H_{m}$ are constants in the setting of hypothesis testing. Define $Z_j$ for $1\leq j \leq m$ as follows.
$L_{j} = T$ $L_{j} = D$
------------- ------------- -------------
$H_{j} = 0$ $Z_j=1$ $Z_j=-1$
$H_{j} = 1$ $Z_j=0$ $Z_j=-2$
Let $Z_{(1)},Z_{(2)},\cdots,Z_{(m)}$ denote the sorted sequence of $Z_{1},Z_{2},\cdots,Z_{m}$. Let $\vv{S}$ and $\vv{S_{\not = j}}$ denote $S_1,\cdots,S_m$ and $S_1,\cdots,S_{j-1},S_{j+1},\cdots,S_m$, respectively. Let $\vv{S_{(\cdot)}}$ and $\vv{S_{(\not = j)}}$ denote $S_{(1)},\cdots,S_{(m)}$ and $S_{(1)},\cdots,S_{(j-1)},S_{(j+1)},\cdots,S_{(m)}$, respectively. We define $\vv{s}$, $\vv{s_{\not = j}}$, $\vv{s_{(\cdot)}}$ and $\vv{s_{(\not = j)}}$ similarly. For example, we will use $\vv{s_{(\cdot)}}$ to denote a sequence of $m$ constants, $s_{(1)},\cdots,s_{(m)}$, which is one of the observed values of $S_{(\cdot)}$. We also define $\vv{L},\vv{Z},\vv{H},\vv{L_{(\not = j)}},$ etc. Then we have the following three theorems.
\[thm:equalprobability\] In the simplified target-decoy procedure, if the $m$ random variables are independent, then for any fixed $j \in [m]$ and any possible $\vv{s_{(\cdot)}}$ and $\vv{z_{(\not = j)}}$ we have $$\begin{aligned}
\Pr\Big(Z_{(j)} = -1 \big |\vv{S_{(\cdot)}} =\vv{s_{(\cdot)} },\vv{Z_{(\not = j)}}=\vv{z_{(\not = j)}}\Big) = \Pr\Big(Z_{(j)} = 1 \big |\vv{S_{(\cdot)}} =\vv{s_{(\cdot)} },\vv{Z_{(\not = j)}}=\vv{z_{(\not = j)}}\Big).
\end{aligned}$$
\[thm:ratio\] In the target-decoy procedure, if the $m$ random variables are independent, then for any fixed $j \in [m]$ and any possible $\vv{s_{(\cdot)}}$ and $\vv{z_{(\not = j)}}$ we have $$\begin{aligned}
\Pr\Big(Z_{(j)} = -1 \big |\vv{S_{(\cdot)}} =\vv{s_{(\cdot)} },\vv{Z_{(\not = j)}}=\vv{z_{(\not = j)}}\Big) = r\Pr\Big(Z_{(j)} = 1 \big |\vv{S_{(\cdot)}} =\vv{s_{(\cdot)} },\vv{Z_{(\not = j)}}=\vv{z_{(\not = j)}}\Big).
\end{aligned}$$
\[thm:fdrcontrol\] Suppose that $S_{(1)},S_{(2)},\cdots,S_{(m)}$,$Z_{(1)},Z_{(2)},\cdots,Z_{(m)}$ are random variables satisfying $S_{(1)}\geq S_{(2)}\geq\cdots\geq S_{(m)}$ and $Z_{(1)},Z_{(2)},\cdots,Z_{(m)} \in \{-2,-1,0,1\}$, and $r$ is a positive constant. For any $\alpha \in [0,1]$, define $$\label{TheoremC_10}
K=\max\{k \big| \frac{1}{r}\times\frac{\# \{Z_{(j)}<0 , j\leq k\}+1}{\# \{Z_{(j)}\geq 0,j\leq k\}\lor 1}\leq \alpha\}.$$ If there is no such $k$, let $K=0$. If for any fixed $j$ and any possible $\vv{s_{(\cdot)}}$ and $\vv{z_{(\not = j)}}$, $$\begin{aligned}\label{TheoremC_11}
\Pr\Big(Z_{(j)} = -1 \big |\vv{S_{(\cdot)}} =\vv{s_{(\cdot)} },\vv{Z_{(\not = j)}}=\vv{z_{(\not = j)}}\Big)
= r\Pr\Big(Z_{(j)} = 1 \big |\vv{S_{(\cdot)}} =\vv{s_{(\cdot)} },\vv{Z_{(\not = j)}}=\vv{z_{(\not = j)}}\Big),
\end{aligned}$$ then we have $$\mathbb{E}\bigg(\frac{\#\{Z_{(j)}=1, j\leq K\}}{\#\{Z_{(j)}\geq 0, j\leq K\}\lor 1}\bigg)< \alpha.$$
The proofs of these theorems are given in Web Appendix A. Note that Theorem \[thm:fdrcontrol\] indicates that the target-decoy procedure controls the FDR if the $m$ random variables are independent.
Specially, all of the above theorems hold for the adaptive target-decoy procedure. Recall that the null hypothesis for random variable $X_j$ used here is the ‘symmetrically distributed’ hypothesis $H_{j0}$: the joint probability density function of $X_{j_1},X_{j_2},\cdots,X_{j_n}$ satisfies $f_{X_{j_1},\cdots,X_{j_n} } (x_{j_1},\cdots,x_{j_n} )=f_{X_{j_1},\cdots,X_{j_n} } ({\pi}_n (x_{j_1},\cdots,x_{j_n}))$ for any possible $x_{j_1},\cdots,x_{j_n}$ and any permutation ${\pi}_n$ of $x_{j_1},\cdots,x_{j_n}$. If $H_{j0}$ is true, it is easy to see that $X^\text{2}_{j_1},X^\text{2}_{j_2},\cdots,X^\text{2}_{j_{n - 2n_2}}$ are also ‘symmetrically distributed’.
Simulation Studies {#sec:Simulation Studies}
==================
We used Monte-Carlo simulations to study the performance of our method. We first compared the simplified target-decoy procedure with the most remarkable multiple testing methods, including the Bayes method [@storey2002direct; @storey2003positive; @storey2004strong] and the empirical Bayes method [@efron2001empirical; @efron2002empirical; @efron2008microarrays]. To show the effectiveness of adjusting $r$, we also did a simulation on a small dataset and compared the adaptive target-decoy procedure with the simplified target-decoy procedure.
Simulation for the simplified target-decoy procedure
----------------------------------------------------
In the simulation, we considered the case-control studies in which the random variables follow the normal distribution or the gamma distribution. In addition to the normal distribution, we did simulation experiments for the gamma distribution because many random variables in real world are gamma-distributed.
Recall that the case-control study consists of $m$ random variables. For each random variable, there are $n$ random samples, $n_1$ of which are from the cases and the other $n_0 = n - n_1$ are from the controls. Let $X_{j_1},X_{j_2},\cdots,X_{j_n}$ be the $n$ random samples for random variable $X_j$.
The observation values from the normal distribution were generated in a way similar to @benjamini2006adaptive. First, let ${\zeta}_0,{\zeta}_{11}, \cdots,{\zeta}_{1n},\cdots,{\zeta}_{m1}, \cdots,{\zeta}_{mn}$ be independent and identically distributed random variables following the $N(0,1)$ distribution. Next, let $X_{ji} = \sqrt{\rho}{\zeta}_0 + \sqrt{\rho}{\zeta}_{ji} + {\mu}_{ji}$ for $j = 1,\cdots,m$ and $i = 1,\cdots,n$. We used $\rho=0, 0.4$ and $0.8$, with $\rho=0$ corresponding to independence and $\rho=0.4$ and $0.8$ corresponding to typical moderate and high correlation values estimated from real microarray data, respectively [@almudevar2006utility]. The values of ${\mu}_{ji}$ are zero for $i = n_1+1,n_1+2,\cdots,n$, the $n_0$ controls. For the $n_1$ cases where $i = 1,2,\cdots,n_1$, the values of ${\mu}_{ji}$ are also zero for $j = 1,2,\cdots,m_0$, the $m_0$ hypotheses that are true null. The values of ${\mu}_{ji}$ for $i = 1,2,\cdots,n_1$ and $j=m_0 + 1,\cdots,m$ are set as follows. We let ${\mu}_{ji} =1,2, 3$ and $4$ for $j=m_0 + 1,m_0 + 2,m_0 +3,m_0 +4$, respectively. Similarly, we let ${\mu}_{ji} =1,2, 3$ and $4$ for $j=m_0 + 5,m_0 + 6,m_0 +7,m_0 +8$, respectively. This cycle was repeated to produce ${\mu}_{(m_0 + 1)1},\cdots,{\mu}_{(m_0 + 1)n_1},\cdots,{\mu}_{m1},\cdots,{\mu}_{mn_1}$ for the false null hypotheses.
The observation values from the gamma distribution, which is characterized using shape and scale, were generated in the following way. First, let ${\Gamma}_0,{\Gamma}_{11}, \cdots,{\Gamma}_{1n},\cdots,{\Gamma}_{m1}, \cdots,{\Gamma}_{mn}$ be independent random variables where $\Gamma_0$ follows the ${\Gamma}(k_0,1)$ distribution and $\Gamma_{ji}$ follows the ${\Gamma}(k_{ji},1)$ distribution for any $j = 1,\cdots,m$ and $i = 1,\cdots,n$. Next, let $X_{ji} = \Gamma_{ji}$ for $j = 1,\cdots,m$ and $i = 1,\cdots,n$ in the simulation study for independent random variables and let $X_{ji} = \Gamma_0 + \Gamma_{ji}$ for dependent random variables. To obtain reasonable correlation values, $k_0$ was set as 4 and $k_{ji}$ was set as 1 for $i = n_1+1,n_1+2,\cdots,n$, the $n_0$ controls. For the $n_1$ cases where $i = 1,2,\cdots,n_1$, $k_{ji}$ was set as 1 for $j = 1,\cdots,m_0$, the $m_0$ hypotheses that are true null. The values of $k_{ji}$ for $i = 1,2,\cdots,n_1$ and $j=m_0 + 1,\cdots,m$ are set as follows. We let $k_{ji} =2,3, 4$ and $5$ for $j=m_0 +1,m_0 +2,m_0 +3,m_0 +4$, respectively. Similarly, we let $k_{ji} =2,3, 4$ and $5$ for $j=m_0 + 5,m_0 + 6,m_0 +7,m_0 +8$, respectively. This cycle was repeated to produce $k_{(m_0 + 1)1},\cdots,k_{(m_0 + 1)n_1},\cdots,k_{m1},\cdots,k_{mn_1}$ for the false null hypotheses.
The specified FDR control level $\alpha$ was set as 5$\%$ or 10$\%$. The total number of tests, $m$, was set as 10000. The proportion of false null hypotheses was $1\%$ or $10\%$. The total sample size, $n$, was set as 20, consisting of the same numbers of cases and controls.
Three different approaches to FDRs were compared, including the Bayes method [@storey2002direct; @storey2003positive; @storey2004strong], the empirical Bayes method [@efron2001empirical; @efron2002empirical; @efron2008microarrays] and our target-decoy approach. The Bayes method and the empirical Bayes method are among the most remarkable multiple testing methods. To compare the power of these methods, we rejected the hypotheses against the specified FDR control level $\alpha$. The rejection threshold, $s$, for the Bayes method was set as the largest $p$-value such that $q$-value($s$) is no more than $\alpha$ [@storey2002direct; @storey2003positive]. The rejection threshold, $s$, for the empirical Bayes method was set as the minimum $z$-value such that Efdr($s$) is no more than $\alpha$, where Efdr($s$) is the expected fdr of hypotheses with $z$-values no smaller than $s$ [@efron2007size; @efron2004large]. Specifically, the R packages “locfdr” version 1.1-8 [@efron2004large], and “qvalue” version 2.4.2 [@storey2003statistical] were used. Each simulation experiment was repeated for 1000 times. We calculated the mean number of rejected hypotheses to evaluate the power of each method. The FDRs of rejected hypotheses were calculated by the means of false discovery proportions (FDPs). Note that the variance of the mean of FDPs of 1000 repetitions is one thousandth of the variance of FDPs. We also estimated the standard deviation of the mean of FDPs from the sample standard deviation of FDPs.
The $p$-values of the Bayes method and the $z$-values of the empirical Bayes method were calculated with the Student’s $t$-test, Wilcoxon rank sum test, the Student’s $t$-test with permutation, or the Student’s $t$-test with bootstrap. For the permutation and bootstrap methods, we sampled the cases and the controls for each test, calculated the $z$-values for sampled data by $t$-test, and calculated the $p$-values with the null distribution of pooled $z$-values [@xie2005note; @liu2014phase]. For the bootstrap method, the resampling was within the groups. The sampling number of permutations was set as 10 [@efron2012large] and that of bootstrap was set as 200.
For our target-decoy approach, the cases and the controls of each test were permuted for 49 times or only once, and the $t$-values and the test statistics of the Wilcoxon rank sum test were used. We did the one-permutation experiments where little information about the null distributions was revealed to demonstrate that our approach does not rely on the null distribution. Because the permutation is performed inherently in our target-decoy approach, the extra permutation and bootstrap are unnecessary.
We will use abbreviations to represent the experiments. For example, *Bayes,permutation,\
Normal,10%,$\rho = 0.8$* represents the simulation experiment where the Bayes method combined with the pooled permutation is used, the random variables follow the normal distribution, the proportion of false null hypotheses is $10\%$ and the correlation values are 0.8. For our target-decoy approach, *$t$-value,49,Gamma,1%* represents the simulation experiment where the $t$-value is used as the score, 49 permutations are performed for each test, the random variables follow the gamma distribution and the proportion of false null hypotheses is as low as $1\%$.
### Independent random variables
*FDR Control.* Table \[tab:s20\_1\] shows the real FDRs of different methods with independent random variables while the specified FDR control level $\alpha$ was $5\%$ or $10\%$. The results show that the $t$-test with Bayes or empirical Bayes overestimated the FDRs for the gamma distribution. The real FDRs of pooled permutation can significantly exceed $\alpha$ when the random variables follow the gamma distribution. The Wilcoxon rank-sum test with Bayes or empirical Bayes overestimated the FDRs. The real FDRs of bootstrap were much smaller than $\alpha$. The target-decoy approach always controlled the FDR.
*Statistical power.* Table \[tab:s20\_2\] shows the statistical power of different methods with independent random variables. Bootstrap was less powerful than all the other methods, especially when the random variables followed the gamma distribution.
When the random variables followed the normal distribution, the Bayes method was a little more powerful than the target-decoy approach while $t$-test was used. However, it was much less powerful than the target-decoy approach while the Wilcoxon rank-sum test was used. The empirical Bayes method was less powerful than the Bayes method and our target-decoy approach, especially for the *Normal,10%* experiments.
When the random variables followed the gamma distribution, the target-decoy approach was much more powerful than the Bayes and empirical Bayes methods, even if only one permutation was performed. Though the pooled permutation seems to be powerful, the FDRs were not controlled.
In all the above experiments, the target-decoy approach successfully controlled the FDR and meanwhile it was remarkably powerful. Even if only one permutation was performed, many rejected hypotheses were still obtained with FDR under control.
### Dependent random variables
*FDR Control.* Table \[tab:s20\_3\] shows the real FDRs of different methods with dependent random variables while the specified FDR control level $\alpha$ was $5\%$ or $10\%$. The results show that with the Bayes method, the real FDRs of $t$-test slightly exceeded $\alpha$ in the *Normal,1%* experiments. Meanwhile, the $t$-test with Bayes or empirical Bayes overestimated the FDRs for the gamma distribution. The real FDRs of pooled permutation significantly exceeded $\alpha$ when the random variables followed the gamma distribution. The Wilcoxon rank-sum test with Bayes or empirical Bayes overestimated the FDRs. The real FDRs of bootstrap were much smaller than $\alpha$. The target-decoy approach controlled the FDR in all cases.
*Statistical power.* Table \[tab:s20\_4\] shows the statistical power of different methods with dependent random variables. Bootstrap was less powerful than all the other methods, especially when the random variables followed the gamma distribution.
When the random variables followed the normal distribution, the Bayes method was less powerful than the target-decoy approach while the Wilcoxon rank-sum test was used. Though the Bayes method seems to be a little more powerful than the target-decoy approach while the $t$-test was used, the real FDR of this method exceeded the specified FDR control level. The empirical Bayes method was less powerful than the Bayes method and our target-decoy approach in the *Normal,10%,$\rho = 0.4$* experiments.
When the random variables followed the gamma distribution, the target-decoy approach was much more powerful than the Bayes and empirical Bayes methods, even if only one permutation was performed. Though the pooled permutation seems to be powerful, the FDRs were not controlled.
Similar to the results for dependent random variables, the target-decoy approach performed significantly better than other methods for dependent random variables. It controlled the FDR in all cases without loss of statistical power.
Simulation for the adaptive procedure {#sec: Simulation for adjusting $r$}
-------------------------------------
To show the effectiveness of our adaptive target-decoy procedure for small datasets, a case-control study involving $200$ random variables was simulated. The null hypotheses of 20 random variables were true and the others were false. For each random variable, there were $20$ random samples, $10$ of which were from the cases and the other $10$ were from the controls. The observation values from the cases where the null hypotheses were false followed the $N(4,1)$ distribution, and all the other observation values followed the $N(0,1)$ distribution. All the observation values were independent. In the simulation, the cases and the controls of each test were permuted for 49 times and the $t$-values were used.
As shown in Tabel \[tab:adjust $r$\], the adaptive procedure controlled the FDR for all values of $\alpha$, and its power was much larger than the simplified target-decoy procedure for small $\alpha$.
[|r|cccccccccc|]{} $\alpha$ & $0.01$ & $0.02$ & $0.03$ & $0.04$ & $0.05$ & $0.06$ & $0.07$ & $0.08$ & $0.09$ & $0.10$\
\
FDR & $0$ & $0$ & $0$ & $0.006$ & $0.044$ & $0.044$ & $0.044$ & $0.055$ & $0.070$ & $0.087$\
Power & $0$ & $0$ & $0$ & $1$ & $21$ & $21$ & $21$ & $21$ & $22$ & $22$\
\
FDR & $0.007$ & $0.018$ & $0.0260$ & $0.032$ & $0.044$ & $0.049$ & $0.058$ & $0.069$ & $0.079$ & $0.093$\
Power & $13$ & $18$ & $18$ & $19$ & $18$ & $20$ & $21$ & $21$ & $21$ & $22$\
An Application {#sec:An Application}
==============
In this section, we apply the target-decoy approach to an *Arabidopsis* microarray dataset. To determine whether *Arabidopsis* genes respond to oncogenes encoded by the transfer-DNA (T-DNA) or to bacterial effector proteins codelivered by *Agrobacteria* into the plant cells, @lee2009agrobacterium conducted microarray experiments at $3$ h and $6$ d after inoculating wounded young *Arabidopsis* plants with two different *Agrobacterium* strains, C58 and GV3101. Strain GV3101 is a cognate of strain C58, which only lacks T-DNA, but possesses proteinaceous virulence (Vir) factors such as VirD2, VirE2, VirE3 and VirF [@vergunst2003recognition]. Wounded, but uninfected, stalks were served as control. Here we just use the 6-d postinoculation data as an example (downloaded from http://www.ncbi.nlm.nih.gov/geo/, GEO accession: GSE14106). The data consisting of 22810 genes were obtained from the C58 infected and control stalks. Both infected and control stalks were with three replicates.
Similar to the simulation experiments, the Bayes method, the empirical Bayes method and our target-decoy approach (the simplified procedure) are compared here. The $p$-values in the Bayes method and the $z$-values in the empirical Bayes method were calculated with the Student’s $t$-test, Wilcoxon rank sum test, and the Student’s $t$-test with permutation, respectively. The bootstrap method is not compared because the number of tests, 22810, is much larger than the sample size of a test, i.e., 6. For the Bayes method, two-tailed tests were used. For the empirical Bayes method, we first transformed the FDR control level to the threshold of local fdr and then identified differentially expressed genes according to the threshold. For the target-decoy approach, the absolute $t$-values and the test statistics of the Wilcoxon rank sum test were used.
[|r|cccccccccc|]{} $\alpha$ & $0.01$ & $0.02$ & $0.03$ & $0.04$ & $0.05$ & $0.06$ & $0.07$ & $0.08$ & $0.09$ & $0.10$\
\
$t$-test & $0$ & $5$ & $5$ & $171$ & $322$ & $712$ & $1108$ & $1469$ & $1875$ & $2208$\
permutation & $0$ & $0$ & $0$ & $0$ & $251$ & $1266$ & $2035$ & $2816$ & $3499$ & $4150$\
rank-sum test & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$\
\
*t*-test & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$\
permutation & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$\
rank-sum test & $\ast$ & $\ast$ & $\ast$ & $\ast$ & $\ast$ & $\ast$ & $\ast$ & $\ast$ & $\ast$ & $\ast$\
\
*t*-value & $0$ & $0$ & $0$ & $1026$ & $1481$ & $1824$ & $2204$ & $2951$ & $3506$ & $3820$\
rank-sum test & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$\
The R package ‘locfdr’ crashed while the Wilcoxon rank-sum test is used.
Because it is unknown which genes were really differentially expressed, the real FDRs cannot be computed here. The power of these methods are compared. In fairness, the sampling numbers were set as $19 = \binom{6}{3} - 1$ in all the experiments, including the pooled permutation and the target-decoy approach. That is, all possible permutations were generated for each gene.
As shown in Table $\ref{Table_real_data}$, no differentially expressed genes were found by the empirical Bayes method or the Wilcoxon rank-sum test. For the Bayes method, the $t$-test was more powerful than the pooled permutation for small $\alpha$ ($\leq 0.05$) while the pooled permutation was more powerful for large $\alpha$ ($\geq 0.06$). The target-decoy approach with $t$-test was most powerful for $0.04 \leq \alpha \leq 0.09$. The additional genes identified by the target-decoy approach are reliable, because similar numbers of genes, i.e., 785 genes for FDR 0.034, 1427 genes for FDR 0.050 and 2071 genes for FDR 0.065, were reported by a more specific analysis [@tan2014general].
Discussion {#sec:Discussion}
==========
In this paper, we proposed the target-decoy approach to FDR control, which need not estimate the null distribution or the null proportion. We theoretically proved that this approach can control the FDR for independent statistics, and experimentally demonstrated that it is more stable and powerful than two most popular methods.
Our approach can be extended to the pair-matched case-control study by adjusting Step 1 of the target-decoy procedure, i.e., randomly exchange the paired observed values just as the permutation tests for pair-matched study instead of permuting them. The other steps and analyses are the same.
In our approach, the scores are only used to determine the labels and ranks of tests, and the statistical meaning of the scores is not required. Similar to permutation tests, our approach can be used with any test statistic, regardless of whether or not its null distribution is known.
There were many studies on the FDR control in multiple testing under dependency [@benjamini2001control; @efron2007size; @ghosal2011predicting; @kang2016optimal]. In this paper we only performed experimental evaluation of our approach for dependent test statistics. The theoretic analysis under dependency will be our future work. Moreover, our control theorem is based on the ‘symmetrically distributed’ hypothesis. This null hypothesis is stronger than the more popular hypothesis that the two groups have the same means. The performance of our approach for the ‘equality of means’ hypothesis needs further studies.
Supplementary Materials {#supplementary-materials .unnumbered}
=======================
Web Appendix A referenced in Section \[sec:Theoretic Analysis\] is available with this paper at the Biometrics website on Wiley Online Library.
Almudevar, A., L. B. Klebanov, X. Qiu, P. Salzman, and A. Y. Yakovlev (2006). Utility of correlation measures in analysis of gene expression. [*3*]{}(3), 384–395.
Barber, R. F. and E. J. Cand[è]{}s (2015). Controlling the false discovery rate via knockoffs. [*43*]{}(5), 2055–2085.
Barber, R. F. and E. J. Candes (2016). A knockoff filter for high-dimensional selective inference. .
Basu, P., T. T. Cai, K. Das, and W. Sun (2017). Weighted false discovery rate control in large-scale multiple testing. [ *0*]{}(ja), 0–0.
Benjamini, Y. and Y. Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing. , 289–300.
Benjamini, Y., A. M. Krieger, and D. Yekutieli (2006). Adaptive linear step-up procedures that control the false discovery rate. [*93*]{}(3), 491–507.
Benjamini, Y. and D. Yekutieli (2001). The control of the false discovery rate in multiple testing under dependency. , 1165–1188.
Candes, E., Y. Fan, L. Janson, and J. Lv (2018). Panning for gold: model-x knockoffs for high dimensional controlled variable selection. .
Chow, Y. S. and H. Teicher (2012). . Springer Science & Business Media.
Diz, A. P., A. Carvajal-Rodr[í]{}guez, and D. O. Skibinski (2011). Multiple hypothesis testing in proteomics: a strategy for experimental work. [*10*]{}(3), M110–004374.
Efron, B. (2004). Large-scale simultaneous hypothesis testing: the choice of a null hypothesis. [ *99*]{}(465), 96–104.
Efron, B. (2007). Size, power and false discovery rates. , 1351–1377.
Efron, B. (2008). Microarrays, empirical bayes and the two-groups model. , 1–22.
Efron, B. (2012). , Volume 1. Cambridge University Press.
Efron, B. and R. Tibshirani (2002). Empirical bayes methods and false discovery rates for microarrays. [*23*]{}(1), 70–86.
Efron, B., R. Tibshirani, J. D. Storey, and V. Tusher (2001). Empirical bayes analysis of a microarray experiment. [ *96*]{}(456), 1151–1160.
Elias, J. E. and S. P. Gygi (2007). Target-decoy search strategy for increased confidence in large-scale protein identifications by mass spectrometry. [*4*]{}(3), 207–214.
Fan, J., P. Hall, and Q. Yao (2007). To how many simultaneous hypothesis tests can normal, student’s t or bootstrap calibration be applied? [ *102*]{}(480), 1282–1288.
Ghosal, S. and A. Roy (2011). Predicting false discovery proportion under dependence. [ *106*]{}(495), 1208–1218.
He, K. (2013). Multiple hypothesis testing methods for large-scale peptide identification in computational proteomics. Master’s thesis, University of Chinese Academy of Sciences. <http://dpaper.las.ac.cn/Dpaper/detail/detailNew?paperID=20015738>. \[in Chinese with English abstract\].
He, K., Y. Fu, W.-F. Zeng, L. Luo, H. Chi, C. Liu, L.-Y. Qing, R.-X. Sun, and S.-M. He (2015). A theoretical foundation of the target-decoy search strategy for false discovery rate control in proteomics. .
Kang, M. (2016). Optimal false discovery rate control with kernel density estimation in a microarray experiment. [ *45*]{}(3), 771–780.
Kerr, K. F. (2009). Comments on the analysis of unbalanced microarray data. [*25*]{}(16), 2035–2041.
Langaas, M., B. H. Lindqvist, and E. Ferkingstad (2005). Estimating the proportion of true null hypotheses, with application to dna microarray data. [*67*]{}(4), 555–572.
Lee, C.-W., M. Efetova, J. C. Engelmann, R. Kramell, C. Wasternack, J. Ludwig-M[ü]{}ller, R. Hedrich, and R. Deeken (2009). Agrobacterium tumefaciens promotes tumor induction by modulating pathogen defense in arabidopsis thaliana. [*21*]{}(9), 2948–2962.
Liu, W. and Q.-M. Shao (2014). Phase transition and regularized bootstrap in large-scale $ t $-tests with false discovery rate control. [*42*]{}(5), 2003–2025.
Markitsis, A. and Y. Lai (2010). A censored beta mixture model for the estimation of the proportion of non-differentially expressed genes. [*26*]{}(5), 640–646.
Meinshausen, N., J. Rice, et al. (2006). Estimating the proportion of false null hypotheses among a large number of independently tested hypotheses. [*34*]{}(1), 373–393.
Sarkar, S. K. (2002). Some results on false discovery rate in stepwise multiple testing procedures. , 239–257.
Scott, J. G. and J. O. Berger (2010). Bayes and empirical-bayes multiplicity adjustment in the variable-selection problem. [*38*]{}(5), 2587–2619.
Storey, J. D. (2002). A direct approach to false discovery rates. [*64*]{}(3), 479–498.
Storey, J. D. (2003). The positive false discovery rate: a bayesian interpretation and the q-value. [*31*]{}(6), 2013–2035.
Storey, J. D., J. E. Taylor, and D. Siegmund (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach. [*66*]{}(1), 187–205.
Storey, J. D. and R. Tibshirani (2003). Statistical significance for genomewide studies. [ *100*]{}(16), 9440–9445.
Strimmer, K. (2008). A unified approach to false discovery rate estimation. [*9*]{}(1), 303.
Tan, Y.-D. and H. Xu (2014). A general method for accurate estimation of false discovery rates in identification of differentially expressed genes. [*30*]{}(14), 2018–2025.
Tusher, V. G., R. Tibshirani, and G. Chu (2001). Significance analysis of microarrays applied to the ionizing radiation response. [ *98*]{}(9), 5116–5121.
Vergunst, A. C., M. C. van Lier, A. den Dulk-Ras, and P. J. Hooykaas (2003). Recognition of the agrobacterium tumefaciens vire2 translocation signal by the virb/d4 transport system does not require vire1. [*133*]{}(3), 978–988.
Xie, Y., W. Pan, and A. B. Khodursky (2005). A note on using permutation-based false discovery rate estimates to compare different analysis methods for microarray data. [*21*]{}(23), 4280–4288.
Yu, C. and D. Zelterman (2017). A parametric model to estimate the proportion from true null using a distribution for p-values. [*114*]{}, 105–118.
\[lastpage\]
|
---
abstract: 'We study theoretical foundation of model comparison for ergodic stochastic differential equation (SDE) models and an extension of the applicable scope of the conventional Bayesian information criterion. Different from previous studies, we suppose that the candidate models are possibly misspecified models, and we consider both Wiener and a pure-jump Lévy noise driven SDE. Based on the asymptotic behavior of the marginal quasi-log likelihood, the Schwarz type statistics and stepwise model selection procedure are proposed. We also prove the model selection consistency of the proposed statistics with respect to an optimal model. We conduct some numerical experiments and they support our theoretical findings.'
address: 'The Institute of Statistical Mathematics, Japan, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan'
author:
- Shoichi Eguchi and Yuma Uehara
bibliography:
- 'YU\_bibs.bib'
title: Schwartz type model selection for ergodic stochastic differential equation models
---
Introduction
============
We suppose that the data-generating process $X$ is defined on the stochastic basis $(\Omega, {\mathcal{F}}, {\mathcal{F}}_t, P)$ and it is the solution of the one-dimensional stochastic differential equation written as: $$\label{yu:tmodel}
dX_t=A(X_t)dt+C(X_{t-})dZ_t,$$ where:
- The coefficients $A$ and $C$ are Lipschitz continuous.
- The driving noise $Z$ is a standard Wiener process or pure-jump Lévy process satisfying that for any $q>0$, $$\label{yu:momcon}
E[Z_1]=0, \quad E[Z_1^2]=1, \quad E[|Z_1|^q]<\infty.$$
- The initial variable $X_0$ is independent of $Z$, and $${\mathcal{F}}_t={\sigma}(X_0)\vee{\sigma}(Z_s|s\leq t)$$
As the observations from $X$, we consider the discrete but high-frequency samples $(X_{t_j^n})_{j=0}^n$ with $$t_j^n:=jh_n, \quad T_n:=nh_n\to\infty, \quad nh_n^2\to0.$$ For $(X_{t_j^n})_{j=0}^n$, $M_1\times M_2$ candidate models are supposed to be given. Here, for each $m_{1}\in\{1,\dots, M_{1}\}$ and $m_{2}\in\{1,\dots, M_{2}\}$, the candidate model ${\mathcal{M}}_{m_1,m_2}$ is expressed as: $$\begin{aligned}
\label{yu:canmodel}
dX_t=a_{m_{2}}(X_t,{\alpha}_{m_{2}})dt+c_{m_{1}}(X_{t-},{\gamma}_{m_{1}})dZ_t,\end{aligned}$$ and the functional form of $(c_{m_1}(\cdot,\cdot),a_{m_2}(\cdot,\cdot))$ is known except for the $p_{{\gamma}_{m_1}}$ and $p_{{\alpha}_{m_2}}$-dimensional unknown parameters $\gamma_{m_{1}}$ and $\alpha_{m_{2}}$ being elements of the bounded convex domains $\Theta_{\gamma_{m_{1}}}\subset\mathbb{R}^{p_{\gamma_{m_{1}}}}$ and $\Theta_{\alpha_{m_{2}}}\subset\mathbb{R}^{p_{\alpha_{m_{2}}}}$. The main objective of this paper is to give a model selection procedure for extracting an “optimal" model ${\mathcal{M}}_{m_1^\star,m_2^\star}$ among ${\mathcal{M}}:=\{{\mathcal{M}}_{m_1,m_2}| m_{1}\in\{1,\dots, M_{1}\}, m_{2}\in\{1,\dots, M_{2}\}\}$ which reflects the feature of $X$ well.
For selecting an appropriate model from the data in hand quantitively, information criteria are one of the most convenient and powerful tools, and have widely been used in many fields. Their origin dates back to Akaike information criterion (AIC) introduced in [@Aka73; @Aka74] which puts an importance on prediction, and after that, various kinds of criteria have been produced up to the present, for their comprehensive overview, see [@BurAnd02], [@ClaHjo08], and [@KonKit08]. Among them, this paper especially sheds light on Bayesian information criterion (BIC) introduced by [@Sch78]. It is based on an approximation up to $O_p(1)$-term of log-marginal likelihood, and its original form is as follows: $$\label{BIC}
\text{BIC}_n=-2l_n(\tes^{\text{MLE}})+p\log n,$$ where $l_n$, $\tes^{\text{MLE}}$, and $p$ stand for the log-likelihood function, maximum likelihood estimator, and dimension of the parameter including the subject model. However, since the closed form of the transition density of $X$ is unavailable in general, to conduct some feasible statistical analysis, we cannot rely on its genuine likelihood; this implies that the conventional likelihood based (Bayesian) information criteria are unpractical in our setting. Such a problem often occurs when discrete time observations are obtained from a continuous time process, and to avoid it, the replacement of a genuine likelihood by some quasi-likelihood is effective not only for estimating parameters included in a subject model but also for constructing (quasi-)information criteria, for instance, see [@Uch10], [@FujUch14], [@EguMas18a] (ergodic diffusion model), [@UchYos16] (stochastic regression model), and [@FasKim17] (CARMA process). Especially, [@EguMas18a] used the Gaussian quasi-likelihood in place of the genuine likelihood, and derived quasi-Bayesian information criterion (QBIC) under the conditions: the driving noise is a standard Wiener process, and for each candidate model, there exist ${\gamma}_{m_1,0}\in\Theta_{m_1}$ and ${\alpha}_{m_2,0}\in\Theta_{m_2}$ satisfying $c_{m_1}(x,{\gamma}_{m_1,0})\equiv C(x)$ and $a_{m_2}(x,{\alpha}_{m_2,0})\equiv A(x)$, respectively. Moreover, by using the difference of the small time activity of the drift and diffusion terms, the paper also gave the two-step QBIC which selects each term separately, and reduces the computational load. In the paper, the model selection consistency of the QBIC is shown only for nested case. In such a case, by considering the (largest) model which contains all candidate models, regularized methods can also be used in the same purpose. Concerning the regularized method for SDE models, for example, see [@GreIac12] and [@MasShi17].
While when it comes to the estimation of the parameters ${\gamma}_{m_1}$ and ${\alpha}_{m_2}$, the Gaussian quasi maximum likelihood estimator (GQMLE) works well for a much broader situation: the driving noise is a standard Wiener process or pure-jump Lévy process with , and either or both of the drift and scale coefficients are misspecified. For the technical account of the GQMLE for ergodic SDE models, see [@Yos92], [@Kes97], [@UchYos11], [@UchYos12], [@Mas13-1], and [@Ueh18]. These results naturally provides us an insight that the aforementioned QBIC is also theoretically valid for the broader situation, and has the model selection consistency even if a non-nested model is contained in candidate models. In this paper, we will show that the insight is true. More specifically, we will give the QBIC building on the stochastic expansion of the log-marginal Gaussian quasi-likelihood. Although the convergence rate of the GQMLE differs in the Lévy driven or misspecified case, the form is the same as the correctly specified diffusion case, that is, a unified model selection criteria for ergodic SDE models is provided. We will also show the model selection consistency of the QBIC.
The rest of this paper is as follows: Section \[sec\_ass\] provides the notations and assumptions used throughout this paper. In Section \[sec\_res\], the main result of this paper is given. Section \[sec\_sim\] exhibits some numerical experiments. The technical proofs of the main results are summarized in Appendix.
Notations and Assumptions {#sec_ass}
=========================
For notational convenience, we previously introduce some symbols used in the rest of this paper.
- For any vector $x$, $x^{(j)}$ represents $j$-th element of $x$, and we write $x^{\otimes2}=x^\top x$ where $^\top$ denotes the transpose operator.
- ${\partial}_x$ is referred to as a differential operator with respect to any variable $x$.
- $x_n\lesssim y_n$ implies that there exists a positive constant $C$ being independent of $n$ satisfying $x_n\leq Cy_n$ for all large enough $n$.
- For any set $S$, $\bar{S}$ denotes its closure.
- We write $Y_j=Y_{t_j}$ and ${\Delta}_j Y:=Y_j-Y_{j-1}$ for any stochastic process $(Y_t)_{t\in{\mathbb{R}}^+}$.
- For any matrix valued function $f$ on ${\mathbb{R}}\times\Theta$, we write $f_s(\theta)=f(X_s,\theta)$; especially we write $f_j(\theta)=f(X_j,\theta)$.
- $I_p$ represents the $p$-dimensional identity matrix.
- $\nu_0$ represents the Lévy measure of $Z$.
- $P_t(x,\cdot)$ denotes the transition probability of $X$.
- Given a function $\rho:\mathbb{R}\to\mathbb{R}^+$ and a signed measure $m$ on a one-dimensional Borel space, we write $$\nn
||m||_\rho=\sup\left\{|m(f)|:\mbox{$f$ is $\mathbb{R}$-valued, $m$-measurable and satisfies $|f|\leq\rho$}\right\}.$$
- ${\mathcal{A}}$ and $\tilde{{\mathcal{A}}}$ stand for the infinitesimal generator and extended generator of $X$, respectively.
In the next section, we will first give the stochastic expansion of the log-marginal Gaussian quasi-likelihood for the following model: $$\label{ten:model}
dX_t=a(X_t,{\alpha})dt+c(X_{t-},{\gamma})dZ_t,$$ where similar to , the coefficients have the unknown $p_{\gamma}$-dimensional parameter ${\gamma}$ and $p_{\alpha}$-dimensional parameter ${\alpha}$. They are supposed to be elements of bounded convex domains $\Theta_{\gamma}$ and $\Theta_{\alpha}$, and for the sake of convenience, we write $\theta=({\gamma},{\alpha})$ and $\Theta_{\gamma}\times \Theta_{\alpha}:=\Theta$. We also assume that either or both of the drift and scale coefficients are possibly misspecified. Especially, we say that the model setting is semi-misspecified diffusion case when the driving noise is a standard Wiener process, the scale coefficient is correctly specified and the drift coefficient is misspecified.
Below, we table the assumptions for our main result.
\[Moments\] $Z$ is a standard Wiener process, or a pure-jump Lévy process satisfying that: $E[Z_1]=0$, $E[Z_1^2]=1$, and $E[|Z_1|^q]<\infty$ for all $q>0$. Furthermore, the Blumenthal-Getoor index (BG-index) of $Z$ is smaller than 2, that is, $$\beta:=\inf_{\gamma}\left\{{\gamma}\geq0: \int_{|z|\leq1}|z|^{\gamma}\nu_0(dz)<\infty\right\}<2.$$
\[Stability\]
1. There exists a probability measure $\pi_0$ such that for every $q>0$, we can find constants $a>0$ and $C_q>0$ for which $$\label{Ergodicity}
\sup_{t\in\mathbb{R}_{+}} \exp(at) ||P_t(x,\cdot)-\pi_0(\cdot)||_{h_q} \leq C_qh_q(x),$$ for any $x\in\mathbb{R}$ where $h_q(x):=1+|x|^q$.
2. For any $q>0$, we have $$\sup_{t\in\mathbb{R}_{+}}E[|X_t|^q]<\infty.$$
Let $\pi_1$ and $\pi_2$ be the prior densities for ${\gamma}$ and ${\alpha}$, respectively.
\[Prior\] The prior densities $\pi_1$ and $\pi_2$ are continuous, and fullfil that $$\sup_{{\gamma}\in\Theta_{\gamma}} \pi_1({\gamma}) \vee \sup_{{\alpha}\in\Theta_{\alpha}} \pi_2({\alpha}) <\infty.$$
We define an [*optimal*]{} value $\theta^\star:=({\gamma}^\star,{\alpha}^\star)$ of $\theta$ being chosen arbitrary from the sets $\displaystyle{\mathop{\rm argmax}}_{{\gamma}\in\bar{\Theta}_{\gamma}}{\mathbb{G}}_1({\gamma})$ and $\displaystyle{\mathop{\rm argmax}}_{{\alpha}\in\bar{\Theta}_{\alpha}}{\mathbb{G}}_2({\alpha})$ for ${\mathbb{R}}$-valued functions ${\mathbb{G}}_1(\cdot)$ (resp. ${\mathbb{G}}_{2}(\cdot)$) on $\Theta_{\gamma}$ (resp. $\Theta_{\alpha}$) defined by $$\begin{aligned}
&{\mathbb{G}}_1({\gamma}):=-\int_{\mathbb{R}}\left(\log c^2(x,{\gamma})+\frac{C^2(x)}{c^2(x,{\gamma})}\right)\pi_0(dx), \label{rf:con.scale}\\
&{\mathbb{G}}_2({\alpha}):=-\int_{\mathbb{R}}c(x,{\gamma}^\star)^{-2}(A(x)-a(x,{\alpha}))^2\pi_0(dx). \label{rf:con.drift}\end{aligned}$$ From the expression of ${\mathbb{G}}_1$, ${\gamma}^\star$ can be regarded as an element in $\Theta_{\gamma}$ minimizing the Stein’s loss; ${\alpha}^\star$ as an element in $\Theta_{\alpha}$ minimizing $L_2$-loss. Recall that $\Theta=\Theta_{\gamma}\times\Theta_{\alpha}$ is supposed to be a bounded convex domain. Then, we assume that:
\[Identifiability\]
- $\theta^\star$ is unique and is in $\Theta$.
- There exist positive constants $\chi_{\gamma}$ and $\chi_{\alpha}$ such that for all $({\gamma},{\alpha})\in\Theta$, $$\begin{aligned}
&{\mathbb{G}}_1({\gamma})-{\mathbb{G}}_1({\gamma}^\star)\leq-\chi_{\gamma}|{\gamma}-{\gamma}^\star|^2,\\
&{\mathbb{G}}_2({\alpha})-{\mathbb{G}}_2({\alpha}^\star)\leq-\chi_{\alpha}|{\alpha}-{\alpha}^\star|^2.\end{aligned}$$
\[Smoothness\]
1. The coefficients $A$ and $C$ are Lipschitz continuous and twice differentiable, and their first and second derivatives are of at most polynomial growth.
2. The drift coefficient $a(\cdot,{\alpha}^\star)$ and scale coefficient $c(\cdot,{\gamma}^\star)$ are Lipschitz continuous, and $c(x,{\gamma})\neq0$ for every $(x,{\gamma})$.
3. For each $i \in \left\{0,1\right\}$ and $k \in \left\{0,\dots,5\right\}$, the following conditions hold:
- The coefficients $a$ and $c$ admit extension in $\mathcal{C}(\mathbb{R}\times\bar{\Theta})$ and have the partial derivatives $(\partial_x^i \partial_\alpha^k a, \partial_x^i \partial_\gamma^k c)$ possessing extension in $\mathcal{C}(\mathbb{R}\times\bar{\Theta})$.
- There exists nonnegative constant $C_{(i,k)}$ satisfying $$\label{polynomial}
\sup_{(x,\alpha,\gamma) \in \mathbb{R} \times \bar{\Theta}_\alpha \times \bar{\Theta}_\gamma}\frac{1}{1+|x|^{C_{(i,k)}}}\left\{|\partial_x^i\partial_\alpha^ka(x,\alpha)|+|\partial_x^i\partial_\gamma^kc(x,\gamma)|+|c^{-1}(x,\gamma)|\right\}<\infty.$$
Define the $p_{\gamma}\times p_{\gamma}$-matrix ${\mathcal{I}}_{\gamma}$ and $p_{\alpha}\times p_{\alpha}$-matrix ${\mathcal{I}}_{\alpha}$ by: $$\begin{aligned}
&{\mathcal{I}}_{\gamma}=4\int_{\mathbb{R}}\frac{({\partial}_{\gamma}c(x,{\gamma}^\star))^{\otimes2}}{c^4(x,{\gamma}^\star)}C^2(x)\pi_0(dx)-2\int_{\mathbb{R}}\frac{{\partial}_{\gamma}^{\otimes2}c(x,{\gamma}^\star)c(x,{\gamma}^\star)-({\partial}_{\gamma}c(x,{\gamma}^\star))^{\otimes2}}{c^4(x,{\gamma}^\star)}(C^2(x)-c^2(x,{\gamma}^\star))\pi_0(dx),\label{yu:fish1}\\
&{\mathcal{I}}_{\alpha}=2\int_{\mathbb{R}}\frac{({\partial}_{\alpha}a(x,{\alpha}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx)-2\int_{\mathbb{R}}\frac{{\partial}_{\alpha}^{\otimes2} a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}(A(x)-a(x,{\alpha}^\star))\pi_0(dx)\label{yu:fish2}.\end{aligned}$$ In the correctly specified case, since under Assumption \[Identifiability\], we have $$c(x,{\gamma}^\star)= C(x),\quad a(x,{\alpha}^\star)=A(x), \qquad \pi_0-a.s.,$$ ${\mathcal{I}}_{\gamma}$ and ${\mathcal{I}}_{\alpha}$ are reduced to $$\begin{aligned}
&{\mathcal{I}}_{\gamma}=4\int_{\mathbb{R}}\frac{({\partial}_{\gamma}c(x,{\gamma}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx),\\
&{\mathcal{I}}_{\alpha}=2\int_{\mathbb{R}}\frac{({\partial}_{\alpha}a(x,{\alpha}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx).\end{aligned}$$
\[Fisher\] ${\mathcal{I}}_{\gamma}$ and ${\mathcal{I}}_{\alpha}$ are positive definite.
In the rest of this section, we give a brief overview of the stepwise Gaussian quasi-likelihood method for , and introduce some theoretical results under Assumptions \[Moments\]-\[Fisher\]. We consider the following stepwise Gaussian quasi-likelihood (GQL) functions ${\mathbb{G}}_{1,n}$ and ${\mathbb{G}}_{2,n}$ on $\Theta_{\gamma}$ and $\Theta_{\alpha}$: $$\begin{aligned}
&{\mathbb{G}}_{1,n}({\gamma})=-\frac{1}{h_n}\sumj \left\{h_n\log c^2_{j-1}({\gamma})+\frac{({\Delta}_j X)^2}{c^2_{j-1}({\gamma})}\right\} \label{rf:scale},\\
&{\mathbb{G}}_{2,n}({\alpha})=-\sumj \frac{({\Delta}_j X-h_na_{j-1}({\alpha}))^2}{h_nc^2_{j-1}(\ges)} \label{rf:drift}\end{aligned}$$ For such functions, we define the (stepwise) Gaussian quasi maximum likelihood estimator (GQMLE) $\tes:=(\ges,\aes)$ in the following manner: $$\begin{aligned}
&\ges\in{\mathop{\rm argmax}}_{{\gamma}\in\bar{\Theta}_{\gamma}}{\mathbb{G}}_{1,n}({\gamma}),
\nonumber\\
&\aes\in{\mathop{\rm argmax}}_{{\alpha}\in\bar{\Theta}_{\alpha}}{\mathbb{G}}_{2,n}({\alpha}).
\nonumber\end{aligned}$$
\[jointstepwise\] Different from the low frequently observed case, the stepwise type estimator exhibits the same asymptotics as the joint type estimator defined as: $$\tilde{\theta}_n\in{\mathop{\rm argmax}}_{\theta\in\bar{\Theta}} \left[-\frac{1}{h_n}\sumj \left\{h_n\log c^2_{j-1}({\gamma})+\frac{({\Delta}_j X-h_na_{j-1}({\alpha}))^2}{c^2_{j-1}({\gamma})}\right\}\right],$$ while possessing the stability of optimization for calculating the estimator. This is because the small time behavior is dominated by the scale term, and thus theoretically, $h_na_{j-1}({\alpha})$ does not affect the estimation of ${\gamma}$.
By making use of the estimates in the papers [@Yos92], [@Kes97], [@UchYos11], [@UchYos12], [@Mas13-1], and [@Ueh18], the following asymptotic results about $\tes$ can be obtained (or directly follows) under Assumption \[Moments\]-\[Fisher\]: let $A_n:={\mathop{\rm diag}}\{a_n I_{p_{\gamma}}, \sqrt{T_n} I_{p_{\alpha}}\}$ where $a_n=\sqrt{n}$ in the correctly specified or semi-misspecified diffusion case, and otherwise, $a_n=\sqrt{T_n}$. Then we have
- Tail probability estimates: for any $L>0$ and $r>0$, there exists a positive constant $C_L$ such that $$\label{eq: TPE}
\sup_{n\in{\mathbb{N}}} P\left(\left|A_n(\tes-\theta^\star)\right|>r\right)\leq \frac{C_L}{r^L}.$$
- Asymptotic normality: $$A_n(\tes-\theta^{\star}){\xrightarrow{{\mathcal{L}}}}N(0, {\mathcal{I}}^{-1}{\Sigma}({\mathcal{I}}^{-1})^\top),$$ where ${\mathcal{I}}=\begin{pmatrix}{\mathcal{I}}_{\gamma}& O \\ {\mathcal{I}}_{{\alpha}{\gamma}} & {\mathcal{I}}_{{\alpha}}\end{pmatrix}$ with $${\mathcal{I}}_{{\alpha}{\gamma}}=2\int_{\mathbb{R}}{\partial}_{\alpha}a(x,{\alpha}^\star){\partial}_{\gamma}^\top c^{-2}(x,{\gamma}^\star)(a(x,{\alpha}^\star)-A(x)) \pi_0(dx),$$ and the form of ${\Sigma}:=\begin{pmatrix}{\Sigma}_{\gamma}&{\Sigma}_{{\alpha}{\gamma}}\\{\Sigma}_{{\alpha}{\gamma}}^\top&{\Sigma}_{{\alpha}}\end{pmatrix}$ is given as follows:
1. Correctly specified diffusion case: $${\Sigma}=2{\mathcal{I}}=2{\mathop{\rm diag}}\{{\mathcal{I}}_{\gamma}, {\mathcal{I}}_{\alpha}\}=\begin{pmatrix}8\int_{\mathbb{R}}\frac{({\partial}_{\gamma}c(x,{\gamma}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx) & O\\ O &4\int_{\mathbb{R}}\frac{({\partial}_{\alpha}a(x,{\alpha}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx) \end{pmatrix},$$ hence the asymptotic variance can be simply written as $${\mathcal{I}}^{-1}{\Sigma}({\mathcal{I}}^{-1})^\top=2 {\mathcal{I}}^{-1}=\begin{pmatrix}\frac{1}{2}\int_{\mathbb{R}}\frac{({\partial}_{\gamma}c(x,{\gamma}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx) & O\\ O &\int_{\mathbb{R}}\frac{({\partial}_{\alpha}a(x,{\alpha}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx) \end{pmatrix}.$$
2. Semi-misspecified diffusion case: $$\begin{aligned}
&{\Sigma}_{\gamma}=8\int_{\mathbb{R}}\frac{({\partial}_{\gamma}c(x,{\gamma}^\star))^{\otimes2}}{c^2(x,{\gamma}^\star)}\pi_0(dx) \\
&{\Sigma}_{{\alpha}{\gamma}}=0\\
&{\Sigma}_{\alpha}=4\int \left[\left(\frac{{\partial}_{\alpha}a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}-{\partial}_x f(x)\right)C(x)\right]^{\otimes 2}\pi_0(dx)\end{aligned}$$ where the function $f$ is the solution of the following Poisson equation: $$\begin{aligned}
{\mathcal{A}}f^{(j)}(x)&=\frac{{\partial}_{{\alpha}^{(j)}} a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}(A(x)-a(x,{\alpha}^\star)),\end{aligned}$$ for $j\in\{1,\dots, p_{\alpha}\}$.
3. Misspecified diffusion case: $$\begin{aligned}
&{\Sigma}_{\gamma}=4\int ({\partial}_x f_1(x) C(x))^{\otimes 2}\pi_0(dx)\\
&{\Sigma}_{{\alpha}{\gamma}}=4\int \left(\frac{{\partial}_{\alpha}a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}-{\partial}_x f_2(x)\right)C^2(x)({\partial}_x f_1(x))^\top \pi_0(dx)\\
&{\Sigma}_{\alpha}=4\int \left[\left(\frac{{\partial}_{\alpha}a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}-{\partial}_x f_2(x)\right)C(x)\right]^{\otimes 2}\pi_0(dx)\end{aligned}$$ where the functions $f_1$ and $f_2$ are the solution of the following Poisson equations: $$\begin{aligned}
{\mathcal{A}}f_1^{(j_1)}(x)&=\frac{{\partial}_{{\gamma}^{(j_1)}} c(x,{\gamma}^\star)}{c^3(x,{\gamma}^\star)}(c^2(x,{\gamma}^\star)-C^2(x)), \\
{\mathcal{A}}f_2^{(j_2)}(x)&=\frac{{\partial}_{{\alpha}^{(j_2)}} a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}(A(x)-a(x,{\alpha}^\star)),\end{aligned}$$ for $j_1\in\{1,\dots, p_{\gamma}\}$ and $j_2\in\{1,\dots, p_{\alpha}\}$.
4. Lévy driven case (both correctly specified and misspecified case): $$\begin{aligned}
&{\Sigma}_{\gamma}=4\int_{\mathbb{R}}\int_{\mathbb{R}}\left(\frac{{\partial}_{\gamma}c(x,{\gamma}^\star)}{c^3(x,{\gamma}^\star)}C^2(x)z^2+g_1(x+C(x)z)-g_1(x)\right)^{\otimes2}\pi_0(dx)\nu_0(dz),\\
&{\Sigma}_{{\alpha}{\gamma}}=-4\int_{\mathbb{R}}\int_{\mathbb{R}}\left(\frac{{\partial}_{\gamma}c(x,{\gamma}^\star)}{c^3(x,{\gamma}^\star)}C^2(x)z^2+g_1(x+C(x)z)-g_1(x)\right)\\
&\qquad \qquad \quad\left(\frac{{\partial}_{\alpha}a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}C(x)z+g_2(x+C(x)z)-g_2(x)\right)^\top\pi_0(dx)\nu_0(dz),\\
&{\Sigma}_{\alpha}=4\int_{\mathbb{R}}\int_{\mathbb{R}}\left(\frac{{\partial}_{\alpha}a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}C(x)z+g_2(x+C(x)z)-g_2(x)\right)^{\otimes2}\pi_0(dx)\nu_0(dz),\end{aligned}$$ where the functions $g_1$ and $g_2$ are the solution of the following extended Poisson equations: $$\begin{aligned}
\tilde{{\mathcal{A}}} g_1^{(j_1)}(x)&=-\frac{{\partial}_{{\gamma}^{(j_1)}} c(x,{\gamma}^\star)}{c^3(x,{\gamma}^\star)}(c^2(x,{\gamma}^\star)-C^2(x)), \\
\tilde{{\mathcal{A}}} g_2^{(j_2)}(x)&=-\frac{{\partial}_{{\alpha}^{(j_2)}} a(x,{\alpha}^\star)}{c^2(x,{\gamma}^\star)}(A(x)-a(x,{\alpha}^\star)),\end{aligned}$$ for $j_1\in\{1,\dots, p_{\gamma}\}$ and $j_2\in\{1,\dots, p_{\alpha}\}$ (In the correctly specified case, $g_1$ and $g_2$ are identically 0).
We note that in the (semi-)misspecified diffusion case, the asymptotic results on the stepwise GQMLE is not verified. However, by taking the same route to [@Ueh18 Theorem 3.1], we can easily show the tail probability estimates. Concerning asymptotic normality, it can also be derived from the argument of Remark \[jointstepwise\].
The theory of Poisson equation and extended Poisson equation plays an important role for dealing with the misspecification effect. The former equation corresponds the generator of diffusion processes, and the latter one does the extended generator of Feller Markov processes. The existence and regularity conditions for their solutions are discussed in [@ParVer01], [@VerKul11], and [@Ueh18], and in the diffusion case, [@UchYos11 Remark 2.2] provides the explicit form of $g_1$ and $g_2$ under $p_{\gamma}=p_{\alpha}=1$.
Main results {#sec_res}
============
Building on the stepwise Gaussian quasi-likelihood ${\mathbb{G}}_{1,n}$ and ${\mathbb{G}}_{2,n}$, the next theorem gives the stochastic expansion of the log-marginal quasi-likelihood which is the main result of this paper:
\[YU:se\] If Assumptions \[Moments\]-\[Fisher\] are satisfied for the statistical model , we have $$\begin{aligned}
&\log\left(\int_{\Theta_{\gamma}}\exp\left({\mathbb{G}}_{1,n}({\gamma})\right)\pi_1({\gamma})d{\gamma}\right)={\mathbb{G}}_{1,n}(\ges)-\frac{1}{2}p_{\gamma}\log n+\log\pi_1\left({\gamma}^\star\right)+\frac{p_{\gamma}}{2}\log 2\pi-\frac{1}{2}\log \det {\mathcal{I}}_{\gamma}+o_p\left(1\right),\\
&\log\left(\int_{\Theta_{\alpha}}\exp\left({\mathbb{G}}_{2,n}({\alpha})\right)\pi_2({\alpha})d{\alpha}\right)={\mathbb{G}}_{2,n}(\aes)-\frac{1}{2}p_{\alpha}\log T_n+\log \pi_2({\alpha}^\star)+\frac{p_{\alpha}}{2}\log 2\pi-\frac{1}{2}\log\det {\mathcal{I}}_{\alpha}+o_p(1).\end{aligned}$$
In the present settings, although the scale estimator has the two kinds of convergence rates depending on the model setups, Theorem \[YU:se\] holds regardless of the convergence rate of the scale estimator. By ignoring the $O_p(1)$ terms in each expansion, we define the two-step quasi-Bayesian information criteria (QBIC) by $$\begin{aligned}
&\mbox{QBIC}_{1,n}={\mathbb{G}}_{1,n}(\ges)-\frac{1}{2}p_{\gamma}\log n,\\
&\mbox{QBIC}_{2,n}={\mathbb{G}}_{2,n}(\aes)-\frac{1}{2}p_{\alpha}\log(T_n).\end{aligned}$$
Next, we consider model selection consistency of the proposed information criteria. Suppose that candidates for drift and scale coefficients are given as $$\begin{aligned}
& c_{1}(x,\gamma_{1}),\ldots,c_{M_{1}}(x,\gamma_{M_{1}}), \label{se:ms.c} \\
& a_{1}(x,\alpha_{1}),\dots,a_{M_{2}}(x,\alpha_{M_{2}}), \label{se:ms.a} \end{aligned}$$ where $\gamma_{m_{1}}\in\Theta_{\gamma_{m_{1}}}\subset\mathbb{R}^{p_{\gamma_{m_{1}}}}$ for any $m_{1}\leq M_{1}$ and $\alpha_{m_{2}}\in\Theta_{\alpha_{m_{2}}}\subset\mathbb{R}^{p_{\alpha_{m_{2}}}}$ for any $m_{2}\leq M_{2}$. Then, each candidate model $\mathcal{M}_{m_{1},m_{2}}$ is given by $$\begin{aligned}
dX_t=a_{m_{2}}(X_t,{\alpha}_{m_{2}})dt+c_{m_{1}}(X_{t-},{\gamma}_{m_{1}})dZ_t.\end{aligned}$$ In each candidate model $\mathcal{M}_{m_{1},m_{2}}$, the functions and are denoted by $G_{1,n}^{(m_{1})}$ and $G_{1}^{(m_{1})}$, respectively. The functions $G_{2,n}^{(m_{2}|m_{1})}$ and $G_{2}^{(m_{2}|m_{1})}$ correspond to and with $\gamma_{m_{1}}$. Using the QBIC, we propose the stepwise model selection as follows.
- We select the best scale coefficient $c_{\hat{m}_{1,n}}$ among , where $\hat{m}_{1,n}$ satisfies $\{\hat{m}_{1,n}\}={\mathop{\rm argmax}}_{m_{1}}\mathrm{QBIC}_{1,n}^{(m_{1})}$ with $$\begin{aligned}
&\mathrm{QBIC}_{1,n}^{(m_{1})}=G_{1,n}^{(m_{1})}(\hat{\gamma}_{m_{1},n})-p_{\gamma_{m_{1}}}\log n,\\
&\hat{\gamma}_{m_{1},n}\in{\mathop{\rm argmax}}_{{\gamma}_{m_{1}}\in\bar{\Theta}_{{\gamma}_{m_{1}}}}{\mathbb{G}}_{1,n}^{(m_{1})}({\gamma}_{m_{1}}).\end{aligned}$$
- Under $c_{\hat{m}_{1,n}}$ and $\hat{\gamma}_{\hat{m}_{1,n},n}$, we select the best drift coefficient with index $\hat{m}_{2,n}$ such that $\{\hat{m}_{2,n}\}={\mathop{\rm argmax}}_{m_{2}}\mathrm{QBIC}_{2,n}^{(m_{2}|\hat{m}_{1,n})}$, where $$\begin{aligned}
&\mathrm{QBIC}_{2,n}^{(m_{2})}=G_{2,n}^{(m_{2}|\hat{m}_{1,n})}(\hat{\alpha}_{m_{2},n})-p_{\alpha_{m_{2}}}\log(T_{n}),\\
&\hat{\alpha}_{m_{2},n}\in{\mathop{\rm argmax}}_{\alpha_{m_{2}}\in\bar{\Theta}_{\alpha_{m_{2}}}}{\mathbb{G}}_{2,n}^{(m_{2}|\hat{m}_{1,n})}(\alpha_{m_{2}}).\end{aligned}$$
Through this procedure, we can obtain the model ${\mathcal{M}}_{\hat{m}_{1,n},\hat{m}_{2,n}}$ as the final best model among the candidates described by and .
The [*optimal value*]{} $\theta_{m_{1},m_{2}}^{\star}=(\alpha_{m_{2}}^{\star},\gamma_{m_{1}}^{\star})$ of $\mathcal{M}_{m_{1},m_{2}}$ is defined in a similar manner as the previous section. We assume that the model indexes $m_{1}^{\star}$ and $m_{2}^{\star}$ are uniquely given as follows: $$\begin{aligned}
\{m_{1}^{\star}\}&= {\operatornamewithlimits {argmin}}_{m_{1}\in\mathfrak{M}_{1}}\dim(\Theta_{\gamma_{m_{1}}}),\\
\{m_{2}^{\star}\}&= {\operatornamewithlimits {argmin}}_{m_{2}\in\mathfrak{M}_{2}}\dim(\Theta_{\alpha_{m_{2}}}),\end{aligned}$$ where $\mathfrak{M}_{1}={\mathop{\rm argmax}}_{1\leq m_{1}\leq M_{1}}G_{1}^{(m_{1})}(\gamma_{m_{1}}^{\star})$ and $\mathfrak{M}_{2}={\mathop{\rm argmax}}_{1\leq m_{2}\leq M_{2}}G_{2}^{(m_{2}|m_{1}^{\star})}(\alpha_{m_{2}}^{\star})$. Then, we say that $\mathcal{M}_{m_{1}^{\star},m_{2}^{\star}}$ is the [*optimal model*]{}. That is, the optimal model consists of the elements of optimal model sets $\mathfrak{M}_{1}$ and $\mathfrak{M}_{2}$ which have the smallest dimension. The following theorem means that the proposed criteria and model selection method have the model selection consistency.
\[thm:mod.cons\] Suppose that Assumptions \[Moments\]-\[Fisher\] hold for the all candidate models and that these exists a $\mathcal{M}_{m_{1}^{\star},m_{2}^{\star}}$ is the optimal model. Let $m_{1}\in\{1,\ldots,M_{1}\}\backslash\{m_{1}^{\star}\}$ and $m_{2}\in\{1,\ldots,M_{2}\}\backslash\{m_{2}^{\star}\}$. Then the model selection consistency of the proposed QBIC hold in the following senses. $$\begin{aligned}
& \lim_{n\to\infty}\mathbb{P}\left(\mathrm{QBIC}_{1,n}^{(m_{1}^{\star})}-\mathrm{QBIC}_{1,n}^{(m_{1})}>0\right) =1, \\
& \lim_{n\to\infty}\mathbb{P}\left(\mathrm{QBIC}_{2,n}^{(m_{2}^{\star}|\hat{m}_{1,n})}-\mathrm{QBIC}_{2,n}^{(m_{2}|\hat{m}_{1,n})}>0\right)=1.\end{aligned}$$ \[se:thm.modcon\]
We here consider the case where there are several optimal models. Then, we define the optimal model index sets $\mathfrak{M}_{1}^{\star}$ and $\mathfrak{M}_{2}^{\star}$ by $$\begin{aligned}
\mathfrak{M}_{1}^{\star}&= {\operatornamewithlimits {argmin}}_{m_{1}\in\mathfrak{M}_{1}}\dim(\Theta_{\gamma_{m_{1}}}),\\
\mathfrak{M}_{2}^{\star}&= {\operatornamewithlimits {argmin}}_{m_{2}\in\mathfrak{M}_{2}}\dim(\Theta_{\alpha_{m_{2}}}),\end{aligned}$$ respectively. Applying the proof of Theorem \[se:thm.modcon\] for each elements of $\mathfrak{M}_{1}^{\star}$ and $\mathfrak{M}_{2}^{\star}$, we can show the model selection consistency with respect to the optimal model sets.
Numerical experiments {#sec_sim}
=====================
In this section, we present simulation results to observe finite-sample performance of the proposed QBIC. We use [yuima]{} package on R (see [@YUIMA14]) for generating data. In the examples below, all the Monte Carlo trials are based on 1000 independent sample paths, and the simulations are done for $(h_{n},T_{n})=(0.01,10), (0.005,10), (0.01,50)$, and $(0.005,50)$ (hence in each case, $n=1000, 2000, 5000$, and $10000$). We simulate the model selection frequencies by using proposed QBIC and compute the model weight $w_{m_{1},m_{2}}$ ([@BurAnd02 Section 6.4.5]) defined by $$\begin{aligned}
\begin{split}
w_{m_{1},m_{2}}&=\frac{\ds{\exp\Big\{-\frac{1}{2}\big(\mathrm{QBIC}_{1,n}^{(m_{1})}-\mathrm{QBIC}_{1,n}^{(\hat{m}_{1,n})}\big)\Big\}}}{\ds{\sum_{k=1}^{M_{1}}\exp\Big\{-\frac{1}{2}\big(\mathrm{QBIC}_{1,n}^{(k)}-\mathrm{QBIC}_{1,n}^{(\hat{m}_{1,n})}\big)\Big\}}} \\
&\quad\times\frac{\ds{\exp\Big\{-\frac{1}{2}\big(\mathrm{QBIC}_{2,n}^{(m_{2}|m_{1})}-\mathrm{QBIC}_{n}^{(\hat{m}_{2,n}|m_{1})}\big)\Big\}}}{\ds{\sum_{\ell=1}^{M_{2}}\exp\Big\{-\frac{1}{2}\big(\mathrm{QBIC}_{2,n}^{(\ell|m_{1})}-\mathrm{QBIC}_{2,n}^{(\hat{m}_{2,n}|m_{1})}\big)\Big\}}}\times100.
\label{def:weight}
\end{split}\end{aligned}$$ The model weight can be used to empirically quantify relative frequency (percentage) of the model selection from a single data set. The model which has the highest $w_{m_{1},m_{2}}$ value is regarded as the most probable model. From its definition , $w_{m_{1},m_{2}}$ satisfies the equation $\sum_{k=1}^{M_{1}}\sum_{\ell=1}^{M_{2}}w_{k,\ell}=100$.
Ergodic diffusion model {#subsec_sim1}
-----------------------
Suppose that we have a sample ${\mathbf{X}}_{n}=(X_{t_{j}})_{j=0}^{n}$ with $t_{j}=jh_{n}$ from the true model $$\begin{aligned}
dX_{t}=-\frac{1}{2}X_{t}dt+dw_{t},\quad t\in[0,T_{n}],\quad X_{0}=0,\end{aligned}$$ where $T_{n}=nh_{n}$, and $w$ is a one-dimensional standard Wiener process. We consider the following scale (Scale) and drift (Drift) coefficients: $$\begin{aligned}
&\;{\bf Scale}\;{\bf 1:} c_{1}(x,\gamma_{1})=\exp\left\{\frac{\gamma_{1,1}+\gamma_{1,2}x+x^{2}}{1+x^{2}}\right\};
\;{\bf Scale}\;{\bf 2:} c_{2}(x,\gamma_{2})=\exp\left\{\frac{\gamma_{2,1}+x+\gamma_{2,3}x^{2}}{1+x^{2}}\right\}; \\
&\;{\bf Scale}\;{\bf 3:} c_{3}(x,\gamma_{3})=\exp\left\{\frac{1+\gamma_{3,2}x+\gamma_{3,3}x^{2}}{1+x^{2}}\right\};
\;{\bf Scale}\;{\bf 4:} c_{4}(x,\gamma_{4})=\exp\left\{\frac{1+\gamma_{4,2}x}{1+x^{2}}\right\}; \\
&\;{\bf Scale}\;{\bf 5:} c_{5}(x,\gamma_{5})=\exp\left\{\frac{1+\gamma_{5,3}x^{2}}{1+x^{2}}\right\};
\;{\bf Scale}\;{\bf 6:} c_{6}(x,\gamma_{6})=\exp\left\{\frac{\gamma_{6,2}x+x^{2}}{1+x^{2}}\right\}; \\
&\;{\bf Scale}\;{\bf 7:} c_{7}(x,\gamma_{7})=\exp\left\{\frac{x+\gamma_{7,3}x^{2}}{1+x^{2}}\right\},\end{aligned}$$ and $$\begin{aligned}
{\bf Drift}\;{\bf 1:}\; a_{1}(x,\alpha_{1})=-\alpha_{1}(x-1);
\;{\bf Drift}\;{\bf 2:}\; a_{2}(x,\alpha_{2})=-\alpha_{2}x-1;
\;{\bf Drift}\;{\bf 3:}\; a_{3}(x,\alpha_{3})=-\alpha_{3}.\end{aligned}$$ Each candidate model is given by a combination of these diffusion and drift coefficients; for example, in the case of Scale 1 and Drift 1, we consider the statistical model $$\begin{aligned}
dX_{t}=-\alpha_{1}(X_{t}-1)dt+\exp\left\{\frac{\gamma_{1,1}+\gamma_{1,2}X_{t}+X_{t}^{2}}{1+X_{t}^{2}}\right\}dw_{t}.\end{aligned}$$ In this example, although the candidate models do not include the true model, the optimal parameter $(\gamma_{m_{1}}^{\star},\alpha_{m_{2}}^{\star})$ and optimal model indices $m_{1}^{\star}$ and $m_{2}^{\star}$ can be obtained by the functions $$\begin{aligned}
G_{1}^{(m_{1})}(\gamma_{m_{1}})&=-\int_{{\mathbb{R}}}\left\{\log c_{m_{1}}(x,\gamma_{m_{1}})^{2}+\frac{1}{c_{m_{1}}(x,\gamma_{m_{1}})^{2}}\right\}\pi_{0}(dx), \\
G_{2}^{(m_{2}|m_{1}^{\star})}(\alpha_{m_{2}})&=-\int_{{\mathbb{R}}}c_{m_{1}^{\star}}(x,\gamma_{m_{1}^{\star}}^{\star})^{-2}\left\{-\frac{x}{2}-a_{m_{2}}(x,\alpha_{m_{2}})\right\}^{2}\pi_{0}(dx),\end{aligned}$$ where $\pi_{0}(dx)=\frac{1}{\sqrt{2\pi}}\exp(-x^{2}/2)dx$. The definition of the optimal model, Tables \[simu:tab1\], and \[simu:tab2\] provide that the optimal model consists of Scale 1 and Drift 1.
Table \[simu:tab3\] summarizes the comparison results of model selection frequency and the mean of $w_{m_{1},m_{2}}$. The indicator of the optimal model defined by Scale 1 and Drift 1 is given by $w_{1,1}$. For all cases, the optimal model is frequently selected, and the value of $w_{1,1}$ is the highest. Also observed is that the frequencies that the optimal model is selected and the value of $w_{1,1}$ become higher as $n$ increases. This result demonstrates the model selection consistency of proposed QBIC.
Ergodic Lévy driven SDE model
-----------------------------
The sample data ${\mathbf{X}}_{n}=(X_{t_{j}})_{j=0}^{n}$ with $t_{j}=jh_{n}$ is obtained from $$\begin{aligned}
dX_{t}=-\frac{1}{2}X_{t}dt+\frac{1}{1+X_{t-}^{2}}dZ_{t},\quad t\in[0,T_{n}],\quad X_{0}=0,\end{aligned}$$ where $T_{n}=nh_{n}$, the driving noise process is the normal inverse Gaussian Lévy process satisfying $\mathcal{L}(Z_{t})=NIG(3,0,3t,0)$. In this example, we consider the following candidate scale (Scale) and drift (Drift) coefficients: $$\begin{aligned}
&\;{\bf Scale}\;{\bf 1:} c_{1}(x,\gamma_{1})=\gamma_{1};
\;{\bf Scale}\;{\bf 2:} c_{2}(x,\gamma_{2})=\exp\left\{\frac{1}{2}(\gamma_{2,1}\cos x +\gamma_{2,2}\sin x)\right\}; \\
&\;{\bf Scale}\;{\bf 3:} c_{3}(x,\gamma_{3})=\frac{\gamma_{3}}{1+x^{2}};
\;{\bf Scale}\;{\bf 4:} c_{4}(x,\gamma_{4})=\frac{1+\gamma_{4}x^{2}}{1+x^{2}},\end{aligned}$$ and $$\begin{aligned}
{\bf Drift}\;{\bf 1:}\; a_{1}(x,\alpha_{1})=-\alpha_{1,1}x-\alpha_{1,2};
\;{\bf Drift}\;{\bf 2:}\; a_{2}(x,\alpha_{2})=-\alpha_{2}x;
\;{\bf Drift}\;{\bf 3:}\; a_{3}(x,\alpha_{3})=-\alpha_{3}.\end{aligned}$$ Each candidate model is constructed in a similar manner as Section \[subsec\_sim1\]. Then, the true model consists of Scale 3 and Drift 2 with $(\gamma_{3},\alpha_{2})=(1,-\frac{1}{2})$. Note that Scale 1, 2, and Drift 3 are misspecified coefficients. From Table \[simu:tab4\], we can show that the tendencies of model selection frequency and model weight are analogous to Section \[subsec\_sim1\].
Appendix
========
**Proof of Theorem \[YU:se\]** Since in the correctly specified and semi-misspecified diffusion cases, Theorem \[YU:se\] can be shown in a similar way as [@EguMas18a], we consider the cases where the rate of convergence for scale estimator is $\sqrt{T_{n}}$. In the following, we also consider the zero-extended version of ${\mathbb{G}}_{1,n}({\gamma})$ and $\pi_1({\gamma})$ just for the simplicity of the following discussion. Applying the change of variable, we have $$\begin{aligned}
&\log\left(\int_{\Theta_{\gamma}}\exp\left({\mathbb{G}}_{1,n}({\gamma})\right)\pi_1({\gamma})d{\gamma}\right)\\
&={\mathbb{G}}_{1,n}(\ges)-\frac{p_{\gamma}}{2}\log n+\log\left(\int_{{\mathbb{R}}^p}\exp\left\{{\mathbb{G}}_{1,n}\left(\ges+\frac{t}{\sqrt{n}}\right)-{\mathbb{G}}_{1,n}(\ges)\right\}\pi_1\left(\ges+\frac{t}{\sqrt{n}}\right)dt\right).\end{aligned}$$ Below we show that $$\log\left(\int_{{\mathbb{R}}^p}\exp\left\{{\mathbb{G}}_{1,n}\left(\ges+\frac{t}{\sqrt{n}}\right)-{\mathbb{G}}_{1,n}(\ges)\right\}\pi_1\left(\ges+\frac{t}{\sqrt{n}}\right)dt\right)=\log\pi_1\left({\gamma}^\star\right)+\frac{p_{\gamma}}{2}\log 2\pi-\frac{1}{2}\log \det {\mathcal{I}}_{\gamma}+o_p(1).$$ For a fixed positive constant ${\delta}$, we divide ${\mathbb{R}}^{p_{\gamma}}$ into $$\begin{aligned}
&D_{1,n}:=\left\{t\in{\mathbb{R}}^{p_{\gamma}}: |t|< {\delta}n^{\frac{1}{2}}\right\}, \\
&D_{2,n}:=\left\{t\in{\mathbb{R}}^{p_{\gamma}}: |t|\geq {\delta}n^{\frac{1}{2}}\right\}.\end{aligned}$$ First we look at the integration on $D_{1,n}$. Taylor’s expansion around $\ges$ gives $${\mathbb{G}}_{1,n}\left(\ges+\frac{t}{\sqrt{n}}\right)-{\mathbb{G}}_{1,n}(\ges)=\frac{1}{2n}{\partial}^2_{\gamma}{\mathbb{G}}_{1,n}(\ges)[t,t]+\frac{1}{6n\sqrt{n}}\int_0^1 {\partial}^3_{\gamma}{\mathbb{G}}_{1,n}\left(\ges+\frac{t}{\sqrt{n}}u\right)du[t,t,t].$$ Here, for any $t\in D_{1,n}$, the second term of the right-hand-side is bounded by $${\delta}\left|\frac{1}{6n}\int_0^1 {\partial}^3_{\gamma}{\mathbb{G}}_{1,n}\left(\ges+\frac{t}{\sqrt{n}}u\right)du\right||t|^2.$$ It follows from [@Fer96 THEOREM 2 (d)] and the estimates of the GQL and GQMLE given in the papers [@Yos92], [@Kes97], [@UchYos11], [@UchYos12], [@Mas13-1], and [@Ueh18] that for any subsequence $\{n_j\}\subset \{n\}$, we can pick a subsubsequence $\{n_{k_j}\}\subset\{n_j\}$ fulfilling that for any ${\epsilon}\in\left(0,\frac{1}{2}\right)$ $$\begin{aligned}
&T_n^{\epsilon}(\hat{{\gamma}}_{n_{k_j}}-{\gamma}^\star){\xrightarrow{a.s.}}0,\label{yu:conve1}\\
&\frac{1}{n_{k_j}}{\partial}^2_{\gamma}{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}\right){\xrightarrow{a.s.}}-{\mathcal{I}}_{\gamma}, \\
&\frac{1}{n_{k_j}} \int_0^1 {\partial}^3_{\gamma}{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}u\right)du{\xrightarrow{a.s.}}^\exists \tilde{{\mathbb{G}}}<\infty,\\
& \sup_{t\in {\mathbb{R}}}\left|\pi_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-\pi_1\left({\gamma}^\star\right)\right|{\xrightarrow{a.s.}}0,\\
& \sup_{{\gamma}\in\Theta_{\gamma}}\left|\frac{1}{n_{k_j}}{\mathbb{G}}_{n_{k_j}}({\gamma})-{\mathbb{G}}({\gamma})\right|{\xrightarrow{a.s.}}0,\\
&\left|{\mathbb{G}}_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_1\left({\gamma}^\star+\frac{t}{\sqrt{n_{k_j}}}\right)\right|+\left|{\mathbb{G}}_1(\hat{{\gamma}}_{n_{k_j}})-{\mathbb{G}}_1({\gamma}^\star)\right|{\xrightarrow{a.s.}}0. \label{yu:conve6}\end{aligned}$$ We hereafter write the set $E\subset\Omega$ on which - hold. For simplicity, we write $$R_{n_{k_j}}=\left|\frac{1}{2n_{k_j}}{\partial}^2_{\gamma}{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}\right)+\frac{1}{2}{\mathcal{I}}_{\gamma}\right|+{\delta}\left|\frac{1}{6n_{k_j}}\int_0^1 {\partial}^3_{\gamma}{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+u \left({\gamma}^\star-\hat{{\gamma}}_{n_{k_j}}\right)\right)du\right|.$$ For any $\omega\in E$ and ${\epsilon}>0$, - enable us to pick $N(\omega)\in{\mathbb{N}}$ and small enough $\zeta(\omega)>0$ satisfying that for all $n_{k_j}\geq N(\omega)$ and ${\delta}<\zeta(\omega)$, $R_{n_{k_j}}(\omega)\leq {\epsilon}$. For any set $A$, we define the indicator function ${\mathbbm{1}}_{A}(\cdot)$ by: $$\begin{aligned}
{\mathbbm{1}}_{A}(t)=\begin{cases}1,&t\in A,\\0,& \text{otherwise}.\end{cases}\end{aligned}$$ Then, for all ${\epsilon}'>0$ and $\omega\in E$, we can choose $N(\omega)\leq N'(\omega)\in{\mathbb{N}}$ such that for all $n_{k_j}\geq N(\omega)$, $$\begin{aligned}
&\left|\exp\left\{{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}\right)\right\}\pi_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-\exp\left(-\frac{1}{2}{\mathcal{I}}_{\gamma}[t,t]\right)\pi_1\left({\gamma}^\star\right)\right|{\mathbbm{1}}_{D_{1,{n_{k_j}}}}(t)\\
&\leq \exp\left(-\frac{1}{2}{\mathcal{I}}_{\gamma}[t,t]\right)\left(\sup_{t\in {\mathbb{R}}}\left|\pi_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-\pi_1\left({\gamma}^\star\right)\right|+\sup_{{\gamma}\in\Theta_{\gamma}}\pi_1\left({\gamma}\right)\left|\exp\left(|t|^2R_{n_{k_j}}\right)-1\right| \right)\\
&\leq {\epsilon}'.\end{aligned}$$ From the positive definiteness of ${\mathcal{I}}_{\gamma}$ and above estimates, for all large enough $n_{k_j}$, we have $$\exp\left\{{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}\right)\right\}\pi_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right){\mathbbm{1}}_{D_{1,n}}(t)\leq \sup_{{\gamma}\in\Theta_{\gamma}} \pi_1({\gamma}) \exp\left\{-\frac{1}{4}{\mathcal{I}}_{\gamma}[t,t]\right\},$$ almost surely, and the right-hand-side is integrable over ${\mathbb{R}}^{p_{\gamma}}$. Thus it follows from $D_{1,n}\to{\mathbb{R}}^{p_{\gamma}}$ and the dominated convergence theorem that $$\begin{aligned}
&\log\left(\int_{{\mathbb{R}}^{p_{\gamma}}} \exp\left\{{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}\right)\right\}\pi_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right){\mathbbm{1}}_{D_{1,n_{k_j}}}(t)dt\right)\\
&{\xrightarrow{a.s.}}\log\left(\int_{{\mathbb{R}}^{p_{\gamma}}}\exp\left(-\frac{1}{2}{\mathcal{I}}_{\gamma}[t,t]\right)\pi_1\left({\gamma}^\star\right)dt\right)\\
&=\log\pi_1\left({\gamma}^\star\right)+\frac{p_{\gamma}}{2}\log 2\pi-\frac{1}{2}\log \det {\mathcal{I}}_{\gamma}.\end{aligned}$$ Now we move on to the evaluation on $D_{2,n}$. Since $$\begin{aligned}
&\frac{1}{n_{k_j}}\left({\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_{1,n_{k_j}}(\hat{{\gamma}}_{n_{k_j}})\right)\\
&\leq 2\sup_{{\gamma}\in\Theta_{\gamma}}\left|\frac{1}{n_{k_j}}{\mathbb{G}}_{1,n_{k_j}}({\gamma})-{\mathbb{G}}_1({\gamma})\right|+{\mathbb{G}}_1\left({\gamma}^\star+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_1({\gamma}^\star)\\
&+{\mathbb{G}}_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_1\left({\gamma}^\star+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_1(\hat{{\gamma}}_{n_{k_j}})+{\mathbb{G}}_1({\gamma}^\star),\end{aligned}$$ on $D_{2,n}$, the identifiability condition and the almost sure convergence of each ingredient imply that on $D_{2,n}$ and for all large enough $n_{k_j}$, there exists a positive constant ${\epsilon}''$ satisfying $$\label{ep}
\frac{1}{n_{k_j}}\left({\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}\right)\right)<-{\epsilon}'',$$ almost surely. Thus we arrive at $$\begin{aligned}
&\int_{{\mathbb{R}}^p_{\gamma}} \exp\left\{{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)-{\mathbb{G}}_{1,n_{k_j}}\left(\hat{{\gamma}}_{n_{k_j}}\right)\right\}\pi_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right){\mathbbm{1}}_{D_{2,n_{k_j}}}(t)dt\\
&\leq \exp\left(-n_{k_j}{\epsilon}''\right)\int_{{\mathbb{R}}^p_{\gamma}}\pi_1\left(\hat{{\gamma}}_{n_{k_j}}+\frac{t}{\sqrt{n_{k_j}}}\right)dt {\xrightarrow{a.s.}}0.\end{aligned}$$ Again applying the converse of [@Fer96 Theorem 2(d)], we get the convergence in probability, and in turn the desired result. As for $\log\left(\int_{\Theta_{\alpha}}\exp\left({\mathbb{G}}_{2,n}({\alpha})\right)\pi_2({\alpha})d{\alpha}\right)$, the proof is similar, thus we omit its details.
**Proof of Theorem \[se:thm.modcon\]** For proof of Theorem \[se:thm.modcon\], we consider the nested model selection case and non-nested model selection case. The (non-)nested model means that the candidate models (do not) include the optimal model. In a similar way as [@EguMas18b Theorems 3.3] and [@EguMas18a Theorems 5.5], we can prove that Theorem \[se:thm.modcon\] is established for the nested model. Below, we will deal with the non-nested model selection case.
In the non-nested model selection case, because of the definition of the optimal model, we have $G_{1}^{(m_{1}^{\star})}(\gamma_{m_{1}^{\star}}^{\star})>G_{1}^{(m_{1})}(\gamma_{m_{1}}^{\star})$ a.s. for every $m_{1}\neq m_{1}^{\star}$. Further, assumptions give the equations $$\begin{aligned}
\frac{1}{n}G_{1,n}^{(m_{1})}(\hat{\gamma}_{m_{1},n})&=\frac{1}{n}G_{1,n}^{(m_{1})}(\gamma_{m_{1}}^{\star})+o_{p}(1)=G_{1}^{(m_{1})}(\gamma_{m_{1}}^{\star})+o_{p}(1), \\
\frac{1}{n}G_{1,n}^{(m_{1}^{\star})}(\hat{\gamma}_{m_{1}^{\star},n})&=\frac{1}{n}G_{1,n}^{(m_{1}^{\star})}(\gamma_{m_{1}^{\star}}^{\star})+o_{p}(1)=G_{1}^{(m_{1}^{\star})}(\gamma_{m_{1}^{\star}}^{\star})+o_{p}(1).\end{aligned}$$ Hence, for any $m_{1}\in\{1,\ldots,M_{1}\}\backslash\{m_{1}^{\star}\}$, $$\begin{aligned}
\mathbb{P}\left(\mathrm{QBIC}_{1,n}^{(m_{1}^{\star})}-\mathrm{QBIC}_{1,n}^{(m_{1})}>0\right)
&=\mathbb{P}\left\{\frac{1}{n}\left(G_{1,n}^{(m_{1}^{\star})}(\hat{\gamma}_{m_{1}^{\star},n})-G_{1,n}^{(m_{1})}(\hat{\gamma}_{m_{1},n})\right)>\left(p_{\gamma_{m_{1}^{\star}}}-p_{\gamma_{m_{1}}}\right)\frac{\log n}{n}\right\} \nn\\
&=\mathbb{P}\left\{G_{1}^{(m_{1}^{\star})}(\gamma_{m_{1}^{\star}}^{\star})-G_{1}^{(m_{1})}(\gamma_{m_{1}}^{\star})>o_{p}(1)\right\} \nn\\
&=\mathbb{P}\left\{G_{1}^{(m_{1}^{\star})}(\gamma_{m_{1}^{\star}}^{\star})-G_{1}^{(m_{1})}(\gamma_{m_{1}}^{\star})>0\right\}+o(1) \nn\\
&\to1 \label{se:prf.thm.modcon1}\end{aligned}$$ as $n\to\infty$. As with , we can show that for any $m_{2}\in\{1,\ldots,M_{2}\}\backslash\{m_{2}^{\star}\}$, $$\begin{aligned}
\mathbb{P}\left(\mathrm{QBIC}_{2,n}^{(m_{2}^{\star}|m_{1}^{\star})}-\mathrm{QBIC}_{2,n}^{(m_{2}|m_{1}^{\star})}>0\right)
&\to1. \label{se:prf.thm.modcon2}\end{aligned}$$ From and , we have $$\begin{aligned}
\mathbb{P}\left(\mathrm{QBIC}_{2,n}^{(m_{2}^{\star}|\hat{m}_{1,n})}-\mathrm{QBIC}_{2,n}^{(m_{2}|\hat{m}_{1,n})}>0\right)&=\mathbb{P}\left(\mathrm{QBIC}_{2,n}^{(m_{2}^{\star}|\hat{m}_{1,n})}-\mathrm{QBIC}_{2,n}^{(m_{2}|\hat{m}_{1,n})}>0, \hat{m}_{1,n}=m_{1}^{\star}\right) \\
&\quad+\mathbb{P}\left(\mathrm{QBIC}_{2,n}^{(m_{2}^{\star}|\hat{m}_{1,n})}-\mathrm{QBIC}_{2,n}^{(m_{2}|\hat{m}_{1,n})}>0, \hat{m}_{1,n}\neq m_{1}^{\star}\right) \\
&\leq\mathbb{P}\left(\mathrm{QBIC}_{2,n}^{(m_{2}^{\star}|m_{1}^{\star})}-\mathrm{QBIC}_{2,n}^{(m_{2}|m_{1}^{\star})}>0\right) \\
&\quad+\mathbb{P}\left(\hat{m}_{1,n}\neq m_{1}^{\star}\right) \\
&=\mathbb{P}\left(\mathrm{QBIC}_{2,n}^{(m_{2}^{\star}|m_{1}^{\star})}-\mathrm{QBIC}_{2,n}^{(m_{2}|m_{1}^{\star})}>0\right) \\
&\quad+\mathbb{P}\left(\mathrm{QBIC}_{1,n}^{(m_{1}^{\star})}<\mathrm{QBIC}_{1,n}^{(m_{1}^{\star})}\right) \\
&\to1+0=1.\end{aligned}$$ The proof of Theorem \[se:thm.modcon\] is complete.
Acknowledgement {#acknowledgement .unnumbered}
---------------
This work was supported by JST CREST Grant Number JPMJCR14D7, Japan.
Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Scale 7
---------------------------------------------------- -- --------- --------- --------- --------- --------- --------- ---------
\[-3mm\] $G_{1}^{(m_{1})}(\gamma_{m_{1}}^{\star})$ -1.2089 -1.2822 -1.4833 -1.6225 -1.4833 -1.2602 -3.2860
\[1mm\]
: The values of $G_{1}^{(m_{1})}(\gamma_{m_{1}}^{\star})$ for each candidate diffusion coefficient.
\[simu:tab1\]
Drift 1 Drift 2 Drift 3
------------------------------------------------------------------ -- --------- --------- ---------
\[-3mm\] $G_{2}^{(m_{2}|m_{1}^{\star})}(\alpha_{m_{2}}^{\star})$ -0.0624 -0.8193 -0.0979
\[1mm\]
: The values of $G_{2}^{(m_{2}|m_{1}^{\star})}(\alpha_{m_{2}}^{\star})$ for each candidate drift coefficient.
\[simu:tab2\]
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$T_{n}$ $h_{n}$ Scale $1^{\ast}$ Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Scale 7
--------- --------- ------------------ ----------- ---------------------------------------------------------------------------------- --------- --------- --------- --------- --------- ---------
10 0.01 Drift $1^{\ast}$ frequency **[409]{} & 72 & 5 & 1 & 5 & 95 & 70\
& & weight & **[30.27]{} & 7.26 & 0.41 & 0.04 & 0.41 & 7.57 & 5.38\
& & Drift 2 & frequency & 60 & 84 & 2 & 0 & 0 & 31 & 22\
& & & weight & 5.94 & 6.67 & 0.13 & 0.01 & 0.02 & 2.64 & 1.98\
& & Drift 3 & frequency & 125 & 5 & 0 & 0 & 0 & 3 & 11\
& & & weight & 22.91 & 2.50 & 0.15 & 0.04 & 0.10 & 3.06 & 2.51\
10& 0.005 & Drift $1^{\ast}$ & frequency & **[449]{} & 86 & 6 & 0 & 4 & 73 & 45\
& & weight & **[33.19]{} & 8.07 & 0.53 & 0.02 & 0.30 & 5.61 & 3.51\
& & Drift 2 & frequency & 64 & 96 & 3 & 0 & 0 & 26 & 8\
& & & weight & 6.61 & 7.65 & 0.19 & 0.00 & 0.01 & 1.95 & 0.89\
& & Drift 3 & frequency & 129 & 4 & 1 & 0 & 0 & 2 & 4\
& & & weight & 24.63 & 2.94 & 0.26 & 0.02 & 0.07 & 2.07 & 1.48\
50 & 0.01 &Drift $1^{\ast}$ & frequency & **[832]{} & 58 & 2 & 0 & 1 & 1 & 12\
& & weight & **[62.59]{} & 5.19 & 0.19 & 0.00 & 0.10 & 0.08 & 0.99\
& & Drift 2 & frequency & 2 & 13 & 0 & 0 & 0 & 0 & 0\
& & & weight & 0.29 & 1.12 & 0.00 & 0.00 & 0.00 & 0.00 & 0.04\
& & Drift 3 & frequency & 79 & 0 & 0 & 0 & 0 & 0 & 0\
& & & weight & 28.43 & 0.74 & 0.01 & 0.00 & 0.00 & 0.01 & 0.21\
50 & 0.005 & Drift $1^{\ast}$ & frequency & **[841]{} & 59 & 3 & 0 & 2 & 0 & 7\
& & weight & **[62.80]{} & 5.30 & 0.30 & 0.00 & 0.19 & 0.00 & 0.59\
& & Drift 2 & frequency & 3 & 13 & 0 & 0 & 0 & 0 & 0\
& & & weight & 0.31 & 1.15 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\
& & Drift 3 & frequency & 72 & 0 & 0 & 0 & 0 & 0 & 0\
& & & weight & 28.46 & 0.76 & 0.01 & 0.00 & 0.00 & 0.00 & 0.12\
****************
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: The mean of model weight $w_{m_{1},m_{2}}$ and model selection frequencies for various situations. The optimal model consists of Scale 1 and Drift 1.
\[simu:tab3\]
$T_{n}$ $h_{n}$ Scale 1 Scale 2 Scale $3^{\ast}$ Scale 4
--------- --------- ------------------ ----------- --------- --------- ------------------ ---------
10 0.01 Drift 1 frequency 3 38 169 27
weight 0.54 5.07 25.09 7.94
Drift $2^{\ast}$ frequency 12 72 [**548**]{} 131
weight 0.91 5.55 [**39.94**]{} 13.28
Drift 3 frequency 0 0 0 0
weight 0.10 0.50 0.78 0.30
10 0.005 Drift 1 frequency 1 36 174 28
weight 0.38 4.84 26.64 6.94
Drift $2^{\ast}$ frequency 11 70 [**557**]{} 123
weight 0.81 5.29 [**42.01**]{} 11.51
Drift 3 frequency 0 0 0 0
weight 0.07 0.45 0.82 0.24
50 0.01 Drift 1 frequency 0 0 68 24
weight 0.00 0.01 14.44 6.70
Drift $2^{\ast}$ frequency 0 1 [**659**]{} 248
weight 0.00 0.09 [**54.09**]{} 24.66
Drift 3 frequency 0 0 0 0
weight 0.00 0.00 0.00 0.00
50 0.005 Drift 1 frequency 0 0 69 20
weight 0.00 0.01 15.31 5.84
Drift $2^{\ast}$ frequency 0 1 [**684**]{} 226
weight 0.00 0.09 [**57.41**]{} 21.35
Drift 3 frequency 0 0 0 0
weight 0.00 0.00 0.00 0.00
: The mean of model weight $w_{m_{1},m_{2}}$ and model selection frequencies for various situations. The true model consists of Scale 3 and Drift 2.
\[simu:tab4\]
[99]{}
H. Akaike. Information theory and an extension of the maximum likelihood principle. In [*Second [I]{}nternational [S]{}ymposium on [I]{}nformation [T]{}heory ([T]{}sahkadsor, 1971)*]{}, pages 267–281. Akadémiai Kiadó, Budapest, 1973.
H. Akaike. A new look at the statistical model identification. , 19(6):716–723, 1974.
A. Brouste, M. Fukasawa, H. Hino, S. M. Iacus, K. Kamatani, Y. Koike, H. Masuda, R. Nomura, T. Ogihara, Y. Shimizu, M. Uchida, and N. Yoshida. The yuima project: A computational framework for simulation and inference of stochastic differential equations. , 57(4):1–51, 2014.
K. P. Burnham and D. R. Anderson. . Springer-Verlag, New York, second edition, 2002. A practical information-theoretic approach.
G. Claeskens and N. L. Hjort. , volume 27 of [ *Cambridge Series in Statistical and Probabilistic Mathematics*]{}. Cambridge University Press, Cambridge, 2008.
A. De Gregorio and S. M. Iacus. Adaptive [LASSO]{}-type estimation for multivariate diffusion processes. , 28(4):838–860, 2012.
S. Eguchi and H. Masuda. Data driven time scale in gaussian quasi-likelihood inference. , 2018.
S. Eguchi, H. Masuda, et al. Schwarz type model comparison for laq models. , 24(3):2278–2327, 2018.
V. Fasen and S. Kimmig. Information criteria for multivariate [CARMA]{} processes. , 23(4A):2860–2886, 2017.
T. S. Ferguson. . Texts in Statistical Science Series. Chapman & Hall, London, 1996.
T. Fujii and M. Uchida. A[IC]{} type statistics for discretely observed ergodic diffusion processes. , 17(3):267–282, 2014.
M. Kessler. Estimation of an ergodic diffusion from discrete observations. , 24(2):211–229, 1997.
S. Konishi and G. Kitagawa. . Springer Series in Statistics. Springer, New York, 2008.
H. Masuda. Convergence of gaussian quasi-likelihood random fields for ergodic [L]{}évy driven [SDE]{} observed at high frequency. , 41(3):1593–1641, 2013.
H. Masuda and Y. Shimizu. Moment convergence in regularized estimation under multiple and mixed-rates asymptotics. , 26(2):81–110, 2017.
E. Pardoux and A. Y. Veretennikov. On the [P]{}oisson equation and diffusion approximation. [I]{}. , 29(3):1061–1085, 2001.
G. Schwarz et al. Estimating the dimension of a model. , 6(2):461–464, 1978.
M. Uchida. Contrast-based information criterion for ergodic diffusion processes from discrete observations. , 62(1):161–187, 2010.
M. Uchida and N. Yoshida. Estimation for misspecified ergodic diffusion processes from discrete observations. , 15:270–290, 2011.
M. Uchida and N. Yoshida. Adaptive estimation of an ergodic diffusion process based on sampled data. , 122(8):2885–2924, 2012.
M. Uchida and N. Yoshida. Model selection for volatility prediction. In [*The fascination of probability, statistics and their applications*]{}, pages 343–360. Springer, Cham, 2016.
Y. Uehara. Statistical inference for misspecified ergodic l[é]{}vy driven stochastic differential equation models. , 2018.
A. Y. Veretennikov and A. M. Kulik. The extended [P]{}oisson equation for weakly ergodic [M]{}arkov processes. , (85):22–38, 2011.
N. Yoshida. Estimation for diffusion processes from discrete observation. , 41(2):220–242, 1992.
|
---
abstract: 'Given a graph $G$ we consider sequentially placing dimers on it, namely choosing a maximal independent subset of edges, i.e. edges that do not share common vertices. We study the number of vertices that do not belong to any edge found in the maximal set. We prove a CLT result for this model in the case when the underlying graph is $\Z^d$.'
author:
- 'Jacob J. Kagan'
title: Domino Tile Placing on Graphs
---
Introduction {#introduction .unnumbered}
============
Dimer models arise naturally in several problems. Dimer coverings of graphs were extensively considered in the frame of perfect matchings and therefore have considerable interest in various areas. In particular, planar graph dimer coverings were extensively studied, see [@kenyon2009lectures] for an overview.
We consider a different dimer-grpah model, instead of considering coverings we consider dimer placements on a graph and we are interested in the fraction of vertices that such a process leaves uncovered. The general problem of placing dimers and monomers on a graph, is known to be notoriously hard, and we are not aware of any previous results in this direction [^1].
The motivation for our model comes from chemistry. It has been experimentally observed that in the reaction of release of halogens from cobalt chains a non-negligible fraction of halogens was not released during heating[^2]. This suggested a cooperative mechanism of release, which gave rise to the 1 dimensional model we consider in the following section.
Unlike the more general setting which we consider in later sections the 1-dimensional case can be analysed using generating functions, and it gives some explicit predictions that are given in the work of Tulchinsky et al. [@tulchinsky2017reversible]. We begin by reproducing this result for the sake of completeness.
1-dimensional model {#sec:lin}
===================
Consider a row of $n$ sites and a process of placing “dimers" that is tiles that cover two neighbouring cites with no tile overlaps allowed. We construct the process as follows: at each step we uniformly choose a pair of adjacent sites, if both sites are free we cover them with a tile, otherwise we do nothing. We repeat this process until no more tiles can be placed.
We encode the process by considering $n$ zeros in a row, covering a pair with a tile corresponds to replacing the corresponding zeros with a pair of ones. We consider the number of zeros that remain after there are no more admissible substitutions. This corresponds to calculating the number of holes (we sometimes refer to them as “monomers") the tilling process leaves.
Consider the first step of the process. Let us denote the pair as $(k+1,k+2)$, thus $k$ may take values $k\in \{0,\ldots, n-2 \}$. This step can be represented by the following picture:
$$\begin{gathered}
\label{eq:rep}
\underbrace{\overbrace{0\ldots 0}^{k} \cdot 0\cdot 0 \cdot\overbrace{0\ldots 0}^{n-k-2}}_n \\
\\
\Downarrow\\
\underbrace{\overbrace{0\ldots 0}^{k} \cdot 1\cdot 1 \cdot\overbrace{0\ldots 0}^{n-k-2}}_n\end{gathered}$$
Let us introduce a random variables $X_n$ which denotes the number of zeros left at the end of the process in an interval of length $n$ and $Y_1$ which denotes the place of the tile picked at step $1$. The above suggests the following equation:
$$\BE[X_n|Y_1 = k] = \BE[X_k|Y_1 = k] + \BE [X_{n-k-2}|Y_1 = k]$$
It is important to note that $X_k$ and $X_{n-k-2}$ are independent and are independent of $Y_1$. We will use this fact extensively in the following analysis.
An equation for the expectation.
--------------------------------
An interval of $n$ sites contains $n-1$ gaps. Note that each gap corresponds to a possible choice of a dimer (the site on the left and the site from the right). Assuming the dimers are uniformly chosen we can take the expectation of the recurrence equation and obtain:
$$\begin{aligned}
\BE X_n &= \frac{1}{n-1}\sum_{k = 0}^{n-2}\BE(X_k) + \BE(X_{n-k-2}) = \frac{2}{n-1}\sum_{k = 0}^{n-2}\BE(X_k)\end{aligned}$$
Let us denote $e_n =\BE X_n $ and rewrite the above equation $$\label{eq:E[X_n]}
e_n = \frac{2}{n-1}\sum_{k = 0}^{n-2}e_k$$ This equation can be simplified, by getting rid of the sum on the right hand side. Note that: $$(n+1)\cdot e_{n+2} = 2\sum_{k=0}^{n}e_k$$ $$n\cdot e_{n+1} = 2\sum_{k=0}^{n-1}e_k$$ taking the difference we obtain
$$\label{eq:rec}
(n+1)\cdot e_{n+2} -n\cdot e_{n+1}= 2\cdot e_{n}$$
with the initial conditions $e_0 = 0$ (and $e_1 = 1$)
solving the recurrence
----------------------
Let us introduce the generating function
$$A(x) = \sum_{n=0}^\infty e_n\cdot x^n$$ we can obtain a differential equation for $A(x)$ from the difference equation (\[eq:rec\]). We do this by first multiplying our equation by $x^n$
$$(n+1)e_{n+2}x^{n} -n e_{n}x^n = 2e_n x^n$$ and then manipulating to adjust the powers and the indexes which results in the equation $$\frac{1}{x}(n+2)e_{n+2}x^{n+1} -(n+1)e_{n+1}x^n = 2e_n x^n+\frac{1}{x^2}e_{n+2}x^{n+2} -\frac{1}{x}e_{n+1}x^{n+1}$$ Summing it over $n$ and noting
$$\begin{aligned}
&\frac{1}{x}\sum_{n=0}^{\infty}(n+2)e_{n+2}\cdot x^{n+1} = \frac{1}{x}(A(x)' -e_1), & &\frac{1}{x^2}\sum_{n=0}^{\infty}e_{n+2}\cdot x^{n+2} = \frac{1}{x^2}(A(x) - e_1x -e_0)\\
& \sum_{n = 0}^{\infty} (n+1)\cdot e_{n+1}\cdot x^{n} = A(x)', & &\frac{1}{x}\sum_{n=0}^{\infty} e_{n+1}x^{n+1} = \frac{1}{x}(A(x) - e_0)\end{aligned}$$
transforms equation (\[eq:rec\]) to an ODE for the generating function $A(x)$: $$\frac{1}{x}(A(x)' -e_1) - A(x)' = 2A(x) + \frac{1}{x^2}(A(x) - e_1x -e_0) - \frac{1}{x}(A(x) - e_0)$$ using $e_0 = 0$ the ODE for A(x) reads: $$\begin{aligned}
\begin{cases}
A(x)'(x - x^2) = A(x)(2x^2 + 1 - x) \\
A(0) = e_0 = 0
\end{cases}\end{aligned}$$ This can be solved by separating the variables $$(\log A(x))'= \frac{A(x)'}{A(x)}= \frac{-2x^2+x-1}{x^2-x} = (-2x + \log x - 2\log (x-1))'$$ using $A(0) = 0$ the solution is given by $$A(x) = \frac{x}{(x-1)^2} e^{-2x}$$
To extract the $e_n$ series recall the following Taylor expansions: $$\begin{aligned}
\frac{x}{(x-1)^2} = \sum_{n=0}^\infty n\cdot x^n & & e^{-2x} = \sum_{n=0}^\infty \frac{(-2x)^n}{n!}\end{aligned}$$
plugging the expansions in
$$\begin{aligned}
A(x) = \sum_{n=0}^\infty e_n\cdot x^n &= \sum_{i = 0}^{\infty} i\cdot x^i \cdot\sum_{j = 0}^\infty \frac{(-2x)^j}{j!} \\
&= \sum_{n=0}^\infty x^n\sum_{i+j = n}i\cdot\frac{(-2)^j}{j!} \\
&= \sum_{n=0}^\infty x^n\sum_{k=0}^n (n-k)\cdot\frac{(-2)^k}{k!}\end{aligned}$$
We obtain $$\BE X_n = \sum_{k=0}^n (n-k)\cdot\frac{(-2)^k}{k!} = n\sum_{k=0}^n\frac{(-2)^k}{k!} -2\sum_{k=0}^{n-1} \frac{(-2)^k}{k!} = \frac{n}{e^2}-\frac{2}{e^2} + o(1/n)$$ This implies a non vanishing concentration of monomers $$\lim_{n\to\infty}\frac{\BE X_n}{n} = \frac{1}{e^2}$$ substituting the numerical value ($ e^2 \approx 7.389$ ) gives a concentration of 13.5%.
Variance
--------
Let us write down the equation for the variance of $X_n$. By definition: $$\begin{aligned}
\label{eq:var_def}
\var X_n = \BE X^2_n -(\BE X_n)^2 = \BE [ \BE [X^2_n|Y_1] ] -(\BE X_n)^2\end{aligned}$$ using the recurrence relation we have $$\begin{aligned}
\BE [ \BE [X^2_n|Y_1] ] &= \frac{1}{n-1}\sum_{k = 0}^{n-2} \BE(\BE[X_k|Y_1] + \BE [X_{n-2-k}|Y_1])^2 \\
&=\frac{1}{n-1}\sum_{k = 0}^{n-2} \BE[X^2_k] + \BE[X^2_{n-2-k}] + 2\BE[X_k]\BE[X_{n-2-k}]&- \BE^2[X_k] -\BE^2[X_{n-2-k}] \\
& &+ \BE^2[X_k] +\BE^2[X_{n-2-k}]\\
& = \frac{2}{n-1}\sum_{k = 0}^{n-2} \var[X_k] + \frac{1}{n-1}\sum_{k = 0}^{n-2} \BE^2[X_k + X_{n-2-k}]\\
& = \frac{2}{n-1}\sum_{k = 0}^{n-2} \var[X_k] + \frac{1}{n-1}\sum_{k = 0}^{n-2} \BE^2 [X_n]\\
& = \frac{2}{n-1}\sum_{k = 0}^{n-2} \var[X_k] + \BE^2 [X_n]\end{aligned}$$ where we used the independence of $X_k$, $X_{n-2-k}$ and $Y_1$ in the second line, and the recurrence for expectation in the third line.
Plugging this result into the definition of the variance \[eq:var\_def\], we obtain an equation identical to the one for the expectation (\[eq:E\[X\_n\]\]): $$\var[X_n] = \frac{2}{n-1}\sum_{k = 0}^{n-2} \var[X_k]$$
Thus, for the variance we also have $$\lim_{n\to \infty} \frac{\var X_n}{n} = e^{-2}$$
Generating Function
-------------------
We conclude this part by a derivation of explicit expressions for the generating functions sums. We are interested in an equation of the form $$X_n = X_k + X_{n-k-2}$$
where $X_k$ and $X_{n-k-2}$ are independent given $k$ as well as independent of $k$ itself. We introduce the generating function $f_n(\lambda) = \BE[e^{\lambda \cdot X_n}]$, and recall that the generating function of a sum of independent variables is a product of the generating functions. $$(n-1)f_n(\lambda) = \sum_{k=0}^{n-2} f_k(\lambda)\cdot f_{n-k-2}(\lambda)$$ to make it more transparent, let us define $m = n-2$ and rewrite the the above equation
$$\label{eq:conv}
(m+1)f_{m+2}(\lambda) = \sum_{k=0}^{m} f_k(\lambda)\cdot f_{m-k}(\lambda)$$
It is now clear that the right hand side is a convolution. This motivates the definition $$g(\lambda,t) = \sum_{m=0}^\infty f_m(\lambda)t^m$$ we use equation \[eq:conv\] to obtain an ODE for $g$ $$\frac{1}{t}[(m+2)f_{m+2}t^{m+1}] -\frac{1}{t^2}[f_{m+2}(\lambda)t^{m+2}] = t^{m}\sum_{k=0}^{m} f_k(\lambda)\cdot f_{m-k}(\lambda)$$ summing over $m$ gives: $$\begin{aligned}
\frac{1}{t}[\sum_{m = -1}^\infty (m+2)f_{m+2}(\lambda)\cdot t^{m+1} - f_1(\lambda)] -\frac{1}{t^2}[\sum_{m = -2}^{\infty} f_{m+2}(\lambda)t^{m+2} - f_1(\lambda)\cdot t -f_0(\lambda)] \\
= \sum_{m = 0}^{\infty} t^{m}\sum_{k=0}^{m} f_k(\lambda)\cdot f_{m-k}(\lambda)\end{aligned}$$ Rearranging it we obtain $$\frac{1}{t}[\dd_t g(\lambda,t) - f_1(\lambda)] -\frac{1}{t^2}[g(\lambda,t) - f_1(\lambda)\cdot t -f_0(\lambda)] = g^2(\lambda,t)$$ Plugging $f_0(\lambda) = \BE[\exp(\lambda\cdot X_0)] = 1$ and multiplying by $t$ gives: $$\dd_t g(\lambda,t)- \frac{1}{t}[g(\lambda,t) -1] - t\cdot g^2(\lambda,t) = 0$$
This is a variant of Riccati’s equation. Looking for a solution of the form:
$$g(\lambda,t) = -\frac{y'}{y\cdot t }$$ where $y' = \dd_t y(\lambda, t)$, we obtain the following equation for $y$: $$- \frac{y''(ty)-y'(y+ty')}{t^2y^2} - \frac{1}{t}[-\frac{y'}{ty}-1] - t\frac{(y')^2}{t^2y^2} = 0$$ simplifying this we obtain
$$-y''+\frac{2}{t}y'+y = 0$$ it can be easily verified that this equation has a solution
$$y = C_1(1-t)e^t +C_2(1+t)e^{-t}$$ plugging this back into the definition of $y$ we obtain for $g$ $$g(\lambda,t) = \frac{C_1e^t+C_2e^{-t}}{ C_1(1-t)e^t +C_2(1+t)e^{-t}}$$ Importantly $g$ depends only on the ratio $C_1/C_2$. Note that for $t = 0$ we obtain $g(\lambda,0)\equiv 1$ (this is simply the requirement that the equation exist). Differentiating $g$ we obtain $$\dd_t g(\lambda,t) = \frac{C_1e^t-C_2e^{-t}}{ C_1(1-t)e^t +C_2(1+t)e^{-t}} +t\cdot\bigg(\frac{C_1(\lambda)e^t-C_2(\lambda)e^{-t}}{ C_1(1-t)e^t +C_2(1+t)e^{-t}}\bigg)^2$$ Setting $t = 0$ and recalling that $f_1 = e^\lambda $ we obtain $$\frac{C_1}{C_2} = \frac{1+e^{\lambda}}{1-e^{\lambda}} = \coth (\lambda/2)$$ We choose $C_1 = \cosh (\lambda/2) $ and $C_2 = \sinh(\lambda /2)$
It is more convenient to work with $y$ instead of $g$. By definition $$t\cdot g(t,\lambda) = \sum_{n=0}^\infty f_n(\lambda)t^{n+1} = -\dd_t (\ln y)$$ Therefore $$f_n(\lambda) = -\frac{1}{(n+1)!}\dd_t^{(n+2)}( \ln y)$$
Faá di Bruno’s formula gives $$\begin{aligned}
\frac{d^n}{dx^n}\phi(h(x)) &= \sum \frac{n!}{m_1!m_2! \ldots m_n!}\cdot \phi^{(m_1+\ldots m_n)}\cdot \prod_{j=1}^n\bigg(\frac{h^{(j)}(x)}{j!} \bigg)^{m_j}\\
& = \sum_{k = 1}^n \phi^{(k)}\cdot B_{n,k}(h', h'',\ldots, h^{(n-k+1)}) \end{aligned}$$ where $B_{n,k}$ are the Bell polynomials given by
$$\begin{aligned}
B_{n,k}(x_1, x_2,\ldots, x_{n-k+1}) & = \sum \frac{n!}{j_1!j_2!\ldots j_{n-k+1}!}\left( \frac{x_1}{1!}\right)^{j_1}\left( \frac{x_2}{2!}\right)^{j_2}\left( \frac{x_{n-k+1}}{(n-k+1)!}\right)^{j_{n-k+1}}\\
\\
j_1+j_2+\ldots j_{n-k+1} & = k\\
j_1+2j_2+\ldots +(n-k+1)j_{n-k+1} & = n\end{aligned}$$
This explicit expression is unfortunately impractical for large $n$s.
General construction
=====================
In the previous section we considered dimers on finite segment of sites, studying its limit as $n \to \infty$. We can try a different approach, namely, to construct the process directly on the line. The dimer-placement model definition is clearest in the abstract setting of a graph. We therefore present it in a setting slightly more general than intuitive. Let $G = (V,E)$ be a graph, for simplicity, assume it is of bounded degree $d$. We associate with each edge $e$ an independent random variable $\tau_e$ distributed uniformly on $[0,1]$. We think of $\tau_e$ as a “wakeup time" for the edge. When the wakeup time occurs, we cover the edge and both the vertices it contains if none of them were previously covered (corresponding to adding the edge to $E(t)$. If either vertex is already covered we do nothing).
First, let us illustrate this is indeed a generalization of the model we studied in the previous section. To see that note that the underlying graph is a segment of length $n$ and the edges are the natural edges which correspond to the gaps between sites. The fact that $\tau_e$ are i.i.d gives a uniform distribution on the order in which the edges are picked.
Formally, the process we study defines a family of sets with the following property:
\[constuction\_on\_graph\] Given $\{ \tau_e\}_e$ define the family of sets $E(t)$ for $t\in [0,1]$ such that
$$\begin{aligned}
E(t) &= E(t_{-})\cup \{ e \;| \text{ $\tau_e = t$ and $e\cap E(t_{-}) = \emptyset$ } \}
%\\ V(t) &= V(t_{-}) \cup \{ v_1,v_2\in e \text{ if $e$ was added to $E(t)$ } \}\end{aligned}$$
where $$\begin{aligned}
E(t_{-}) = \cup_{s<t}E(s), \: E(0) = \emptyset
%E(t) &= E(t_{-})\cup \{ e \;| \text{ $\tau_e = t$ and $e\cap E(t_{-}) = \emptyset$ } \}
%\\ V(t) &= V(t_{-}) \cup \{ v_1,v_2\in e \text{ if $e$ was added to $E(t)$ } \}\end{aligned}$$
It is easy to see this a set family with this property, if it exists, generates a maximal independent set of edges at time 1. Namely every edge $e\in G$ is either in the set $E(t)$ for $t> \tau_e$ or it contains a vertex which belongs to an edge already in $E(t)$ at the time $t<\tau_e$. Indeed, if the edge $e$ does not belong to $E(t)$ it will be added to the family at time $\tau_e$, unless one of its vertices belong to an edge previously added to $E(t)$.
Our next goal is to establish the existence of $E(t)$. We will show that for almost all $\{\tau_e \}_e$ the family $E(t)$ exists. Uniqueness follows from the construction.
Intuitively, the reason a set is ill-defined would be some elements for which it is not clear if they belong to the set or not. To address this let us consider for every edge $e$ the function
$$\begin{aligned}
1_e(t) =
\begin{cases}
1 \text{ if $e\in E(t)$ at time $t$ }\\
0 \text{ otherwise}
\end{cases}\end{aligned}$$
and the truncated functions
$$\begin{aligned}
1^{(r)}_e(t) =
\begin{cases}
1 \text{ if $e\in E_r(t)$ at time $t$ when considering only the edges in $B_r(e)$ }\\
0 \text{ otherwise}
\end{cases}\end{aligned}$$
where we define $B_r(e)$, the ball of radius $r$ around an edge $e = (u,v)$ to be $$B_r(e) = B_r(u)\cup B_r(v)$$ ($B_r(v)$ is the the sub graph of $G$ which is at graph distance at most $r$ from $v$).
Theorem \[thm:well\_def\] shows that a.s. the truncated functions converge to a limit which we identify with $1_e(t)$ $$\begin{aligned}
1^{(r)}_e \stackrel{r\to\infty}{\longrightarrow} 1_e(t)\end{aligned}$$
The reason $1^{(r)}_e$ could fluctuate as $r$ grows is a “cascade of tiles" that changes the state of an edge, making it impossible to determine from a finite ball around it whether the edge is present in the cover. If such fluctuations do not subside for arbitrarily large $r$s the event of an edge belonging to $E(t)$ unmeasurable in the graph topology and the process ill defined.
We start with a definition that captures the intuition of a cascade of tiles:
Let $ \gamma = (e_1,e_2,\ldots, e_n)$ where $e_i$ edges of $G$ be a path in $G$. We call $\gamma$ *monotone* if for all $i<j $ and $e_i,e_j\in \gamma$ holds $\tau_{e_i} > \tau_{e_j}$.
Note that there is a natural partial order by inclusion for the set of monotone paths $\gamma_1\leq \gamma_2$ if $\gamma_1\subset \gamma_2$. Unfortunately, a monotone path needs not to be simple, however it is clear from the definition that any monotone $\gamma$ cannot contain the same edge twice.
For an edge $e$ with time $\tau_e$ it is clear from the definition of our process, that only tiles with times preceding $\tau_e$ can influence the event of laying the tile corresponding to $e$. Therefore only edges lying on monotone path starting at $e$ can influence the event $e\in E(t)$ and thus the functions $1^{(r)}_e(t)$.
The following lemma gives an estimate on the length of a monotone path.
Let $\gamma$ be a path of length $n$.
$$\pr (\gamma \text{ is monotone}) = \frac{1}{n!}$$
\[lem:path\]
Let $\gamma = (e_1, e_2, \ldots, e_n)$. Consider the event that $\gamma$ is monotone, this is given by the event $$A = \{\tau_{e_1}> \tau_{e_2}> \ldots > \tau_{e_n}\}$$
This is a decreasing sequence of i.i.d. random variables. The probability of this event is the same as that of any random permutation $$\pr (\tau_{e_1}> \tau_{e_2}> \ldots > \tau_{e_n} ) = \pr(\tau_{\sigma(e_1)}> \tau_{\sigma (e_2)}> \ldots > \tau_{\sigma(e_n)})$$ The number of permutations on $n$ elements is $n!$.
The previous discussion implies the following natural definition of the set of monotone paths having a first edge $e$, $$\Gamma_e = \{ \gamma|\; \gamma = (e_1,e_2,\ldots , e_n) \text{ is monotone and } e_1 = e \}$$
We claim that $\Gamma_e$ contains only finite paths a.s. Actually, an even more general statement is true, all monotone paths in a bounded degree graph are finite.
\[cor:finite\_rad\] Almost surely, there exists $r<\infty$ such that $\Gamma_e \subset B_r(e)$.
$$\pr( \Gamma_e \nsubseteq B_r(v)) \leq \pr( \text{exists a monotone path $\gamma$ starting with $e$ s.t. }|\gamma|>r )$$ Plugging in our assumption of the uniform bound $d$ on the vertex degree, we can bound the number of paths of length $r$ beginning with $e$ by $d^r$ and thus obtain a union bound of this event: $$\pr( \Gamma_e \nsubseteq B_r(e)) \leq \frac{d^r}{r!}$$ Note that the probabilities of the events $\Gamma_e \nsubseteq B_r(v)$ have a finite sum $$\sum_r \pr( \Gamma_e \nsubseteq B_r(e)) < \sum_r \frac{d^r}{r!} = e^d <\infty$$ therefore by the Borel Cantelli lemma only finitely many of them happen a.s. Therefore $\Gamma_e\subset B_r(e)$ for some finite random $r$ a.s.
Armed with this result we can now prove
\[thm:well\_def\] For almost all $\{ \tau_e\}_{e\in G}$ the family $E(t)$ defined by property \[constuction\_on\_graph\] exists and is unique.
It is enough to show that for any edge $e\in E$ the functions $1^{(r)}_e(t)$ converge a.s. This will in particular settle the question of the existence of $E(t)$ by giving an explicit construction of it.
By lemma \[cor:finite\_rad\] for each edge $e$ almost surely there exists a finite radius $r_0$ such that $\Gamma \subset B_{r_0}(e)$. Thus for any $r_1,r_2>r_0$ holds
$$1^{(r_1)}_e(t) = 1^{(r_2)}_e(t)$$ This proves the convergence of the sequence $\{ 1^{(r)}_e(t)\}_r$ establishing the uniqueness.
Monomer concentration results {#sec:monomer}
===============================
This section is dedicated to establishing results regarding the limiting behaviour of monomers. We begin with generic results, which easily generalize to bounded degree graphs, and then continue with results and proofs which are special to $\Z^d$. Since the generalization to bounded degree graphs is fairly clear when possible, we see no reason to complicate the notation for the sake of seemingly added generality therefore we work throughout with the graph of $\Z^d$. For the sake of clarity, we separate the $\Z^d$ specific results from the more generic ones.
Consider a box of side-length $n$ which we denote by $\Lambda_n$. Denote by $$\begin{aligned}
S_n = \frac{1}{|\Lambda_n|}\sum_{v\in\Lambda_n} (X_v -\BE X_v)\end{aligned}$$ We evaluate the probability $$\pr(|S_n| >\epsilon)$$ Let us define a truncated event $Y_v^{r}$ to be the event $v$ is not covered by a dimer when considering only dimers in a ball or radius $r$ around $v$. Using the truncated variables we can bound this probability:
$$\begin{aligned}
\pr(|S_n| >\epsilon) & \leq \pr(\left| \sum_{v\in\Lambda_n} Y^r_v - |\Lambda_n|\cdot\BE Y^r_v \right| > \frac{\epsilon}{3} \cdot |\Lambda_n|) + \\
& \pr (\sum_{\Lambda_n} |X_v -Y^r_v| > \frac{\epsilon}{3} \cdot |\Lambda_n|) + \pr(|\sum_{v\in\Lambda_n} |\BE Y^r_v -\BE X_v| > \frac{\epsilon}{3} \cdot |\Lambda_n|)\end{aligned}$$
From the proof of lemma \[cor:finite\_rad\] we have $$\pr(X_v\neq Y_v^{r}) \leq \frac{d^r}{r!}$$ Thus, $$\begin{aligned}
\BE (|\sum_{\Lambda_n} X_v - Y_v^{(r)}|) &\leq \BE (\sum_{\Lambda_n} |X_v - Y_v^{(r)}|)\\
& = \sum_{\Lambda_n} \BE(|X_v - Y_v^{(r)}|) \leq |\Lambda_n|\cdot \frac{d^r}{r!}\end{aligned}$$ This bounds the last summand, in particular by
$$\begin{aligned}
\pr (|\sum_{v\in\Lambda_n} |\BE Y^r_v -\BE X_v| > \epsilon \cdot |\Lambda_n|) \leq 1_{\frac{d^r}{r!}\leq \frac{\epsilon}{3} }\end{aligned}$$
By Markov’s inequality we obtain $$\begin{aligned}
\pr (\sum_{\Lambda_n} |X_v -Y^r_v| > \frac{\epsilon}{3} \cdot |\Lambda_n|) \leq \frac{3}{\epsilon} \cdot \frac{d^r}{r!}\end{aligned}$$ which bounds the second summand. Note that these bounds are uniform in $n$ and tend to 0 as $r\to\infty$.
For the first summand we note that $$\begin{aligned}
\sum_{\Lambda_n} Y^r_v - |\Lambda_n|\cdot\BE Y^r_v = \sum_{i\in (0,r)^d}\sum_{\substack{v\in\Lambda_n \\ v\equiv i\mod(r)}} Y^r_v - |\Lambda_n|\cdot\BE Y^r_v\end{aligned}$$ The inner sum is a sum of i.i.d. Bernoulli random variables therefore by the Chernoff bound $$\begin{aligned}
\pr(\sum_{\substack{v\in\Lambda_n \\ v\equiv i\mod(r)}} Y^r_v - |\Lambda_{\frac{n}{r}}|\cdot\BE Y^r_v > \frac{\epsilon}{3}\cdot |\Lambda_{\frac{n}{r}}|) \leq e^{- (\epsilon /3)^2 \cdot |\Lambda_{\frac{n}{r}}|}\end{aligned}$$ taking a union bound we obtain $$\begin{aligned}
\pr (\sum_{\Lambda_n} |X_v -Y^r_v| >\frac{\epsilon}{3}\cdot |\Lambda_n|)\leq r^d \cdot e^{-(\epsilon/3)^2\cdot |\Lambda_{\frac{n}{r}}|}\end{aligned}$$ summing the estimates we obtain $$\begin{aligned}
\pr (|S_n|>\epsilon)\leq r^d \cdot e^{-(\epsilon/3)^2\cdot |\Lambda_{\frac{n}{r}}|} + \frac{3}{\epsilon}\cdot \frac{d^r}{r!} + 1_{\frac{d^r}{r!}\leq \frac{\epsilon}{3}} \end{aligned}$$
Taking $r=\log n$ for $n$ large enough (so as $ 1_{\frac{d^r}{r!}\leq \frac{\epsilon}{3}} = 0$) we use Stirling’s formula to obtain
$$\begin{aligned}
\pr (|S_n|>\epsilon)\leq \log^d n \cdot e^{-\epsilon^2\cdot n^{d-1}} + (\frac{d}{\log n})^{\log n} \end{aligned}$$
a CLT result for $\Z^d$
-----------------------
First let us show how for the $\Z^d$ case the stationarity of our process implies a positive concentration of monomers. This is a direct consequence of Templeman’s ergodic theorem for $\Z^d$ actions (see [@sarig2009lecture chapter 2, theorem 2.6]) which we cite here for completeness:
\[thm:Tempelman\] Let $T_1,\ldots,T_d$ be d-commuting probability preserving maps on a probability space, and suppose $\{I_r\}_{r\geq 1}$ is an increasing sequence of boxes which tends to $\Z^d_+$. If $f\in L^1$, then $$\begin{aligned}
\frac{1}{|I_r|} \sum_{v \in I_r} f\circ T^{v} \stackrel{r\to\infty}{\longrightarrow}\BE(f|Inv(T_1)\cap\ldots\cap Inv(T_d)) \; a.s.\end{aligned}$$
where we sum over $v$ as vectors in $\Z^d$ namely $T^v = T_1^{v_1}\circ\ldots\circ T_d^{v_d}$ where $v_i$ is the i-th coordinate of $v$. $Inv(T)$ is the space of invariant functions under $T$ and $\BE(f|Inv(...))$ is the conditional expectation.
Using this we obtain $$\begin{aligned}
S_n \stackrel{n\to\infty}{\longrightarrow} 0 \end{aligned}$$
or in other words, $$\begin{aligned}
\frac{1}{|\Lambda_n|} \sum_{v\in \Lambda_n} X_v \stackrel{n\to\infty}{\longrightarrow} \BE X_0 = \pr(X_0 = 1)\end{aligned}$$
Our goal is to establish a CLT result for $S_n$ namely we would like to show that
$$\begin{aligned}
\pr (\sqrt{|\Lambda_n|}S_n) \sim N(0, \sigma). \end{aligned}$$
Consider $X_u$ and $X_v$, and denote the distance beween them by $r = ||u-v||$. For these variables there is a mixing property by lemma \[cor:finite\_rad\], namely
$$\begin{aligned}
|\pr (X_u\cap X_v) - \pr(X_u)\pr(X_v)| \leq \frac{d^r}{r!}\end{aligned}$$
(this follows from noticing that if $u\notin \max \Gamma_v $ makes $X_u,X_v$ independent, and plugging the estimate on the radius of $\max \Gamma_v$.)
A central limit theorem for our process follows from the theorem of Bolthausen [@bolthausen1982central] which we cite for completeness: Let $X_\rho$ be a real valued stationary random field with $\rho\in \Z^d$, i.e. the $X_\rho$ are real random variables and the joint laws are shift invariant.
If $\Lambda\subset\Z^d$ let $\CA_{\Lambda}$ be the $\sigma$-algebra generated by $X_\rho$, $\rho\in\Lambda$. If $\Lambda_1,\Lambda_2\subset \Z^d$, let $d(\Lambda_1,\Lambda_2) = \inf\{ d(\rho_1,\rho_2): \rho_1\in \Lambda_1,\rho_2\in\Lambda_2 \}$. The mixing coefficients we use are defined as follows, if $n\in\BN$,$k,l\in\BN\cup\{ \infty\}$ $$\begin{aligned}
\alpha_{k,l}(n) &= \sup \{|\pr (A_1\cap A_2) - \pr(A_1)\pr(A_2)| : A_i\in \CA_{\Lambda_i},\: |\Lambda_1|\leq k, |\Lambda_2|\leq l, d(\Lambda_1,\Lambda_2)\geq n \}\\
\rho(n) &= \sup \{ |\Cov(Y_1,Y_2)| : Y_i\in L_2(\CA_{\{ \rho_i \} }),\: \Vert Y_i\Vert_2\leq 1,\: d(\rho_1,\rho_2)\geq n \}\end{aligned}$$
If $\sum_{m=1}^\infty m^{d-1}\alpha_{k,l}(m) <\infty$ for $k+l \leq 4$, $\alpha_{1,\infty}(m) = o(m^{-d})$ and if $$\begin{aligned}
\sum_{m=1}^\infty m^{d-1}\rho(m) < \infty\end{aligned}$$ or $$\begin{aligned}
\text{for some } \delta >0 \; \Vert X_{\rho}\Vert_{2+\delta}<\infty \text{ and } \sum_{m=1}^\infty m^{d-1}\alpha_{1,1}(m)^{\delta/(2+\delta)} < \infty\end{aligned}$$ Then $\sum_{\rho\in\Z^d} | \Cov (X_0, X_{\rho})|<\infty $ and if $\sigma^2 = \sum_{\rho} \Cov(X_0,X_\rho)>0$, then the laws of $ |\Lambda_n|^{1/2} \cdot S_n/\sigma $ converge to the standard normal one.
The finiteness of the sums involving the mixing coefficients is an immediate consequence of lemma \[cor:finite\_rad\]. It asserts that the sums contains only a finite number of summands.
The only non trivial matter is checking $\sigma^2 = \sum_{\rho} \Cov(X_0,X_\rho)>0$. The idea is seen most clearly for $\Z^2$, therefore we present it here for the plane grid, however it generalizes for arbitrary dimension $d$.
We call a “cage" the event when for a rectangle frame enclosing a $4 \times 1$ segment is filled with tiles before the inside is being filled. See figure \[fig:cage\] for illustration.
![a cage \[fig:cage\]](cage){height="8cm"}
Our next observation is that the sites within the cage are covered independently of the sites outside the cage. Using Templeman’s theorem \[thm:Tempelman\] we can see there is a positive density of cages. Using this we can bound $\sigma$ from 0: $$\begin{aligned}
\var[|\Lambda_n|\cdot S_n] & \geq \BE [ \var[\sum_{v\in \Lambda_n} x_v|\tau] ] \\
& = \BE [ \var [ \sum_{v \text{ in cage}} x_v+ \sum_{v \text{ out of cage}} x_v | \tau ] ]\\
& \geq \BE [ \var [ \sum_{v \text{ in cage}} x_v | \tau ] ]\\
& \geq c \cdot \BE [\text{ \# of cages } ]\end{aligned}$$ where $\tau$ is the the $\sigma$-field created by conditioning the positions of cages in a box of size $n$. Note that by Tempelman’s theorem, we know there is a positive density of cages. This bounds the sum of covariances from 0, allowing us to apply Bolthousen’s theorem.
Questions and discussion
========================
There is another setting which should be solvable using a recurrence equation approach, this is a $d$-regular tree, which can be obtained as a limit of trees chopped at depth $n$. The recurrence in this case is more complicated than the recurrence we obtained for $\Z$, but the independence of the different parts of the graph holds in this case as well, which was the key element in the approach.
This gives rise to a couple of questions: The CLT result for our model followed from Bolthousen’s theorem which uses the $\Z^d$ structure, and moreover in bounding the variance from 0 we used Templeman’s theorem which heavily uses the $\Z^d$ structure. However, the essential ingredient for a CLT type of theorem is mixing or near independence, which we obtain from lemma \[cor:finite\_rad\] in our model.
It would be interesting to see in what generality the CLT result is valid? In particular for Cayley graphs of what groups can such a result be obtained?
One cannot expect to be able to establish these results for a general Cayley graph using the tools of ergodic theory. To apply ergodic theorems ameanability is obviously needed, however it is not clear such a result cannot be established by other means.
The tail estimates we show in section \[sec:monomer\] lead us to conjecture that the result should be valid at least for uniformly bounded degree graphs.
[^1]: A problem in the spirit of our model has been considered in [@krapivsky2016kinetics], there the authors consider a kinetic model of adsorption.
[^2]: See [@tulchinsky2017reversible] for a detailed account of the chemical aspects. In a private discussion with the first author of [@tulchinsky2017reversible], it seems that the reported number of 20% was inaccurate and a better estimate is 14.5%, which is in remarkable agreement with the prediction of the model we consider, especially having in mind its simplicity.
|
---
author:
- 'E. Werner, B. Mehlig'
title: |
[Supplemental material for]{}\
Scaling regimes of a semi-flexible polymer in a rectangular channel
---
We summarise our predictions for the scalings for the statistics of the extension, and the free energy of confinement in Table \[tab:1\]. Where numerical prefactors are known, they have been included in the table. In this table, we also include previous results for the scalings, at very strong confinement (regime III of Fig. 1), and for a polymer confined to a slit (the limit ${D_{\rm W}}\to \infty$). These results are briefly discussed below.
### Odijk regime
If both ${D_{\rm H}}\ll {\ell_{\rm P}}$ and ${D_{\rm W}}\ll {\ell_{\rm P}}$, then it is impossible for the polymer to turn around completely. Instead, the polymer must stay almost parallel to the channel axis, undulating slightly from side to side. In this regime the statistics of the extension and the free energy of confinement have been studied in detail both theoretically and numerically [@burkhardt2010; @chen2013].
### Backfolded Odijk regime
Odijk [@odijk2008] has predicted the existence of a regime intermediate between the Odijk regime and the extended de Gennes regime, for very slender semi-flexible polymers. For square channels, this regime was studied by Muralidhar [*et al.*]{} [@muralidhar2014a], who gave it the name “backfolded Odijk regime”. In this regime the size of the channel is such that backfolds are possible but rare. Odijk defines the *global persistence length* $g$ as the orientational correlation length of the corresponding ideal polymer. The backfolded Odijk regime requires that ${\ell_{\rm P}}\ll g\ll {l_{\rm cc}}$. This condition can only be satisfied for very thin polymers [@muralidhar2014a]. Assuming that a polymer section that is free of backfolds follows the statistics of the Odijk regime, one can predict scaling relations for the extension *in terms of the global persistence length* $g$. However, no theory for how $g$ itself depends on the channel size exists for relevant channel sizes [@muralidhar2014a]. Note also that the statistics of the Odijk regime are not derived under the assumption that the chain does not backfold, but under the stricter assumption that it is almost completely parallel with the channel axis. Since such strong alignment prohibits backfolding, the agreement with Odijk statistics is only approximate in this regime.
The predictions for the backfolded Odijk regime in the square channel [@odijk2008; @muralidhar2014a] are straightforward to generalise to rectangular channels. The results are shown in Table \[tab:1\]. The prediction for the scaling of the extension was derived by Odijk [@odijk2008].
### Slit regimes
For completeness, we include in Table \[tab:1\] the scaling regimes for infinite channel aspect ratio, i.e. for the polymer confined in a slit. The results for these regimes are quoted from recent studies by Dai [*et al.*]{} [@dai2012], Taloni [*et al.*]{} [@taloni2013], and Tree [*et al.*]{} [@tree2014]. For polymers that are too short to exhibit the asymptotic relation $R\propto L$ (see main text), these scalings also describe the statistics in regimes Ia-c in Fig. 1 in the main text.
[llllllll]{} && &\
(lr)[3-5]{} (lr)[6-8]{} Regime & Name & ${D_{\rm H}}$ & ${D_{\rm W}}$ [^1] & $L$ & $R/L$ & $\sigma_R^2/L$ & ${F_{\rm c}}/L$\
Ia & de Gennes & $\gg{\ell_{\rm K}}^2/w$ & — & $\gg \Big(\frac{{D_{\rm W}}^4 {D_{\rm H}}} {{\ell_{\rm K}}w}\Big)^{\frac{1}{3}}$ & $\approx \Big(\frac{{\ell_{\rm K}}w}{{D_{\rm H}}{D_{\rm W}}}\Big)^{\frac{1}{3}}$ [@benkova2015a] & $\approx \Big(\frac{{\ell_{\rm K}}w {D_{\rm W}}^2}{{D_{\rm H}}}\Big)^{\frac{1}{3}}$ [@werner2015]& $\approx \Big(\frac{{\ell_{\rm K}}w}{{D_{\rm H}}^5}\Big)^{\frac{1}{3}}$ [@werner2015]\
Ib & — & ${\ell_{\rm K}}\ll {D_{\rm H}}\ll \frac{{\ell_{\rm K}}^2}{w}$ & ${D_{\rm W}}^2 \gg {D_{\rm H}}{\ell_{\rm K}}^2/w$ & $\gg \Big(\frac{{D_{\rm W}}^4 {D_{\rm H}}} {{\ell_{\rm K}}w}\Big)^{\frac{1}{3}}$ & $\approx \Big(\frac{{\ell_{\rm K}}w}{{D_{\rm H}}{D_{\rm W}}}\Big)^{\frac{1}{3}}$ [@werner2015] & $\approx \Big(\frac{{\ell_{\rm K}}w {D_{\rm W}}^2}{{D_{\rm H}}}\Big)^{\frac{1}{3}}$ [@werner2015]& $=\frac{\pi^2}{6} {\ell_{\rm K}}{D_{\rm H}}^{-2}$ [@casassa1967]\
Ic & — & $w\le{D_{\rm H}}\ll{\ell_{\rm K}}$ [^2] & ${D_{\rm W}}^2 \gg {D_{\rm H}}{\ell_{\rm K}}^2/w$ & $\gg \Big(\frac{{D_{\rm W}}^4 {D_{\rm H}}} {{\ell_{\rm K}}w}\Big)^{\frac{1}{3}}$ & $\approx \Big(\frac{{\ell_{\rm K}}w}{{D_{\rm H}}{D_{\rm W}}}\Big)^{\frac{1}{3}}$ [@werner2015]& $\approx \Big(\frac{{\ell_{\rm K}}w {D_{\rm W}}^2}{{D_{\rm H}}}\Big)^{\frac{1}{3}}$ [@werner2015]& $= 1.1032(1) {\ell_{\rm P}}^{-\frac{1}{3}} {D_{\rm H}}^{-\frac{2}{3}}$ [@chen2013][^3]\
IIa & extended de Gennes & ${\ell_{\rm K}}\ll {D_{\rm H}}\ll \frac{{\ell_{\rm K}}^2}{w}$ & ${D_{\rm W}}^2 \ll {D_{\rm H}}{\ell_{\rm K}}^2/w$ & $\gg\Big(\frac{{D_{\rm H}}^2{D_{\rm W}}^2{\ell_{\rm K}}}{w^2}\Big)^{\frac{1}{3}}$ & $= 0.9338(84)\Big(\frac{{\ell_{\rm K}}w} {{D_{\rm H}}{D_{\rm W}}}\Big)^\frac{1}{3}$ [@werner2014] & $=0.13(1) {\ell_{\rm K}}$ [@werner2014] & $=\frac{\pi^2}{6} {\ell_{\rm K}}({D_{\rm H}}^{-2} + {D_{\rm W}}^{-2})$ [@casassa1967]\
IIb & — & $w\ll{D_{\rm H}}\ll{\ell_{\rm K}}$ & ${\ell_{\rm K}}^2\ll{D_{\rm W}}^2 \ll \frac{{D_{\rm H}}{\ell_{\rm K}}^2}{w}$ & $\gg\Big(\frac{{D_{\rm H}}^2{D_{\rm W}}^2{\ell_{\rm K}}}{w^2}\Big)^{\frac{1}{3}}$ & $\approx \Big(\frac{{\ell_{\rm K}}w}{{D_{\rm H}}{D_{\rm W}}}\Big)^{\frac{1}{3}}$ [@odijk2008] & $ = 0.20(2) {\ell_{\rm K}}$ [@werner2014][^4] & $= 1.1032(1) {\ell_{\rm P}}^{-\frac{1}{3}} {D_{\rm H}}^{-\frac{2}{3}}$ [@chen2013]\
IIIa & Odijk & ${D_{\rm H}}\ll {\ell_{\rm P}}$ & ${D_{\rm W}}\ll {\ell_{\rm P}}$ & $ \gg {\ell_{\rm P}}^{\frac{1}{3}}{D_{\rm W}}^{\frac{2}{3}}$ & $= 1\! - 0.09137(7) \frac{{D_{\rm H}}^{\frac{2}{3}} + {D_{\rm W}}^{\frac{2}{3}} }{{\ell_{\rm P}}^{\frac{2}{3}}}\! $ [@burkhardt2010] & $= 0.00478(10) \frac{{D_{\rm H}}^2 + {D_{\rm W}}^2}{{\ell_{\rm P}}}\! $ [@burkhardt2010] & $= 1.1032(1) {\ell_{\rm P}}^{-\frac{1}{3}} ({D_{\rm H}}^{-\frac{2}{3}}\! + {D_{\rm W}}^{-\frac{2}{3}})\! $ [@chen2013]\
IIIb & backfolded Odijk & & $\gg \Big(\frac{g^{\frac{3}{2}}{\ell_{\rm P}}{D_{\rm H}}^3{D_{\rm W}}^{2}}{w^3}\Big)^\frac{2}{9}$ & $\approx \Big(\frac{g w}{{\ell_{\rm P}}^{\frac{1}{3}}{D_{\rm W}}^{\frac{2}{3}}{D_{\rm H}}}\Big)^{\frac{1}{3}}$ [@odijk2008] & $\approx g$ [@muralidhar2014a] & $\approx {\ell_{\rm P}}^{-\frac{1}{3}} {D_{\rm H}}^{-\frac{2}{3}}$ [@muralidhar2014a]\
\
Sa & de Gennes & $\gg{\ell_{\rm K}}^2/w$ & — & $\gg \Big(\frac{{D_{\rm H}}^5}{{\ell_{\rm K}}w}\Big)^{\frac{1}{3}}$ & $\approx \Big(\frac{{\ell_{\rm K}}w}{L{D_{\rm H}}}\Big)^\frac{1}{4}$ [@dai2012] & $\approx \Big(\frac{L{\ell_{\rm K}}w}{{D_{\rm H}}}\Big)^{\frac{1}{2}}$ [@taloni2013] & $\approx \Big(\frac{{\ell_{\rm K}}w}{{D_{\rm H}}^5}\Big)^{\frac{1}{3}}$ [@taloni2013]\
Sb & extended de Gennes & ${\ell_{\rm K}}\ll {D_{\rm H}}\ll \frac{{\ell_{\rm K}}^2}{w}$ & — & $\gg \frac{{\ell_{\rm K}}{D_{\rm H}}}{w}$ & $\approx \Big(\frac{{\ell_{\rm K}}w}{L{D_{\rm H}}}\Big)^\frac{1}{4}$ [@dai2012] & $\approx \Big(\frac{L{\ell_{\rm K}}w}{{D_{\rm H}}}\Big)^{\frac{1}{2}}$ [@taloni2013]& $=\frac{\pi^2}{6} {\ell_{\rm K}}{D_{\rm H}}^{-2}$ [@casassa1967]\
Sc & Odijk-Flory & $w <{D_{\rm H}}\ll {\ell_{\rm K}}$ & — & $\gg \frac{{\ell_{\rm K}}{D_{\rm H}}}{w}$ & $\approx \Big(\frac{{\ell_{\rm K}}w}{L{D_{\rm H}}}\Big)^\frac{1}{4}$ [@odijk2008] & $\approx \Big(\frac{L{\ell_{\rm K}}w}{{D_{\rm H}}}\Big)^{\frac{1}{2}}$ [@taloni2013] & $= 1.1032(1) {\ell_{\rm P}}^{-\frac{1}{3}} {D_{\rm H}}^{-\frac{2}{3}}$ [@chen2013]\
[11]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, , , **** ().
, ****, ().
, ****, ().
, , , ****, ().
, , , , ****, ().
, , , ****, ().
, , , ****, ().
, , , (), <http://dx.doi.org/10.1039/C4SM02382J>.
, **.
, ****, ().
, ****, ().
[^1]: We assume throughout that ${D_{\rm W}}\ge{D_{\rm H}}$.
[^2]:
[^3]:
[^4]: The step variance $\sigma_0^2$ [@werner2014] obeys $\sigma_0^2= {\ell_{\rm K}}^2/2$ in this regime.
|
=1 v Ł[[L]{}]{}
ł
=1
\
\
\
[**Diffeomorphisms, Noether Charges and Canonical Formalism in 2D Dilaton Gravity**]{}
.5cm
José Navarro-Salas$^{1,2}$, Miguel Navarro$^{3,}$[^1]\
and César F. Talavera$^{1,2}$
.5cm
1. Departamento de Física Teórica, Burjassot-46100, Valencia, Spain.
2. IFIC, Centro Mixto Universidad de Valencia-CSIC, Burjassot-46100, Valencia, Spain.
3. The Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom.
**Abstract**
We carry out a parallel study of the covariant phase space and the conservation laws of local symmetries in two-dimensional dilaton gravity. Our analysis is based on the fact that the Lagrangian can be brought to a form that vanishes on-shell giving rise to a well-defined covariant potential for the symplectic current. We explicitly compute the symplectic structure and its potential and show that the requirement to be finite and independent of the Cauchy surface restricts the asymptotic symmetries.
1cm
PACS number(s): 04.20.Cv, 04.20.Fy, 12.25.+e
Introduction.
=============
Great efforts have been developed recently on the study of two-dimensional dilaton gravity theories. The reason for this interest is that these theories serve as toy models in which we can develop and test techniques and methods to be further applied to more realistic (higher dimensional) gravity theories. Remarkably, the string inspired model (CGHS-model) of Ref. [@CGHS] (see also [@Witten]) admits black hole solutions and, therefore, provides an interesting toy model to study black hole issues.
One of the aims of this paper is to study the reduced phase space of the CGHS-model. This point could be of great interest for a non-perturbative canonical quantization of the theory. Our work is based on the covariant phase-space formalism [@Crnkovic]-[@Nosotrosii] and extends the results of a previous paper [@Navarro]. The covariant formalism has already been applied to the CGHS-model in Ref. [@Mikovic; @Soh] although their results are valid for the case of a closed space only.
Moreover, for Lagrangians vanishing on-shell, the Noether’s procedure can be incorporated, in a rather natural way, to the covariant canonical formalism. Therefore, we shall also study, in a parallel way, the covariant phase space and the conservation laws associated with diffeomorphism invariance. Our analysis will shed new insight on the controversy about the notion of mass in 2D dilaton gravity (see [@Bilal]-[@Bak]).
In Section 2 we present briefly the covariant phase space formalism pointing out the fact that, for Lagrangians vanishing on-shell, the space of solutions can be endowed with a natural potential for the symplectic structure. The Noether charge technique is naturally incorporated in this scheme. In Section 3 we study in a systematic way the conservation laws associated with the diffeomorphism invariance and, in particular, with the asymptotic (Poincaré) symmetries of the CGHS model. In Section 4 we determine the symplectic potential of the CGHS model. The condition of having a well-defined potential (i.e. finite and independent of the Cauchy surface) will restrict the allowed asymptotic symmetries. The Lorentz symmetry break down and the spatial translation turns out to be a gauge-type transformation. This will permit to understand the results of Section 3. We shall also consider, in Section 5, the case of spherically symmetric $3+1$ Einstein gravity, which can also be regarded as a 2D dilaton gravity model (see [@Kuchar] for a related perspective). Although the stringy and Schwarzschild black holes have the same canonical structure they differ in the form of the potential. As a byproduct, this accounts for the numerical factors in the Komar-type formulas for the mass in gravity models. We state our conclusions in Section 6.
Covariant phase space and conservation laws
===========================================
Given a field theory with dynamical fields $\Psi^\alpha(x)$ and action $S=S(\Psi^\alpha(x))$, the phase space can be defined, in a covariant way, as the space of solutions of the classical equations of motion. The standard formula S(X\^) = \_[M]{} [S \^]{} X\^+ \_j\^(\^, X\^) \[dosi\] can be interpreted now as the exterior derivative of $S$, on the covariant phase space, acting on a tangent vector $X^\alpha$ (which solves the linearized equations of motion). In contrast with the variational calculus which takes the variation $X^\alpha$ vanishing on the boundary of $\cal M$, it is now the first term of the r.h.s. of (\[dosi\]) which vanishes automatically. Therefore, the covariant phase space can be equipped with a presymplectic two-form = \_j\^d\_ , \[dosii\] where $\Sigma$ is a Cauchy hypersurface and $\delta$ stands for the exterior derivative operator. Due to the fact that the symplectic current $\omega^\mu = \delta j^\mu$ is conserved, the presymplectic form (\[dosii\]) is, in general grounds, independent of the Cauchy surface with a suitable choice of boundary conditions.
From the above expression it is clear that the one-form = \_j\^d\_\[dosiii\] could serve as a potential for the presymplectic form (\[dosii\]). However, $j^\mu$ is not, in general, conserved and hence $\theta$ is not well-defined.
Now, let us suppose that the presymplectic potential current $j^\alpha$ is itself conserved, \_j\^\_[|\_[sol]{}]{}=0 . \[c1\] Then, for any field $X\sim \delta \Psi^a$ satisfying the linearized equations of motion, we will have that $J^\alpha_X=\hi_X j^\alpha$ is a conserved current: \_J\^\_X[\_[|\_[sol]{}]{}]{}=0 . \[c2\]
What is the condition for a presymplectic potential current to be conserved? On solutions we have \_j\^\_[|\_[sol]{}]{} = \_[|\_[sol]{}]{} . \[c3\] Therefore, it is enough that the Lagrangian vanishes on the covariant phase space. In this situation the one-form (\[dosiii\]) is well defined (with appropriate boundary conditions), $J^\alpha_X = j^\alpha(X)$ coincides with the Noether current and $\theta(X)$ is the corresponding Noether charge.
Energy-momentum conservation in the CGHS model. {#a}
===============================================
The action of the CGHS model is: S\_[CGHS]{} &=& \_[M]{}d\^2x . \[CGHS1\] By doing $\varphi=\e^{-\Phi}$ we obtain S\_[CGHS]{} &=& \_[M]{}d\^2x , \[CGHS2\] which, for our purposes, is a form of the action more easy to deal with.
Now, it is convenient to define a new metric $\g_{\nu\mu}$ by means of g=\^[-2]{} , \[E2\] in term of which the action takes a remarkably simpler form: S\_[CGHS]{}=12\_[M]{} \^2 x , \[CGHS3\] The new variable $\widehat g$, which allows to eliminate the kinetic term in the action, also emerges in the gauge-theoretical formulation [@Cangemi] of the theory, and in more general models [@Louis].
The equations of motions are given by: R=0 &,& \^2=4ł\^2 , \_i = 0 , \[hatEM\]\[mov1\]\
\_\_\^2 &=& \^2 + 12(12(\_i)\^2)\_- 12\_\_i\_\_i , \[mov2\] and, if we add a convenient total divergence to the action of the CGHS model in (\[CGHS3\]) we can easily bring it to a form vanishing on-shell S\_[CGHS]{}=\_[M]{}Ł\_[CGHS]{} = 12\_[M]{} \^2x\[CGHS4\] .
The symplectic potential associated to the above Lagrangian is: j\^&=&12.
It can be shown by direct computation, and using the equations of motions, that the above symplectic potential is preserved actually.
The conserved current associated to a diffeomorphism generated by a vector field $X_f = f^\mu\frac{\p}{\p x^\mu}$, defined on the configuration space of the theory as (g)\_=\_f\_+\_f\_, =f\^\_ , \[c8\] can be written in the form: J\_f\^&=& 12 { \_+ \_(\^2\[\^f\^-\^f\^\]) .\
& & . + 12 \_(f\^\_i \^\_i - f\^\_i \^\_i) } . \[10\] It has, therefore, the form of the divergence of an antisymmetric tensor and is, because of that, identically preserved (notice, however, that arriving at eq. (\[10\]) requires to use the equations of motion). The conserved charge associated to $\widehat{J}^\alpha_f$ can be made explicit by noticing that the divergence of an antisymmetric tensor $F^{\mu\nu}$ can be written, in 2D, as \_F\^ &=& \_=\
&=& \_= \^ \_K , \[tresi\] with K = -12 \^ F\_ . \[tresii\] Therefore, \^\_f = \^ \_ , \[tresiii\] with = -12 \^ ( f\_\_\^2 + \^2 \_f\_+ 12 f\_\_i \_\_i ) . \[tresiv\]
In terms of the physical metric $g_{\mu\nu}$ the conserved current is given by J\^\_f &=& 12 \_\
&=& \^ \_K , \[tresv\] with the charge K = -12 \^ ( \^2 \_f\_+ 12 f\_\_i \_\_i ) . \[tresvi\]
It is interesting to compare (\[tresv\]) with Komar’s formula for the conserved current in 4D [@Bak; @Komar], and to notice that the presence of the matter term in (\[tresv\]) has its origin in the total divergence terms added to the Lagrangian. On the other hand, these total divergence terms in the Lagrangian are the reason why eq. (\[tresv\]) differs from other expressions for $J^\alpha_f$ given in the literature [@Iyer] and, as we will see, they contribute to make $K$ finite, under appropriate asymptotic conditions.
From the above expressions it is not difficult to obtain, by choosing $f^\mu=\varepsilon^{\mu\nu} x_\nu + a^\mu$ and following the generalized Belinfante procedure [@Bak], a symmetric energy-momentum pseudotensor for the CGHS model: \^[ab]{}= 12\_\_(\^2) . \[12\]
On the other hand, in the absence of matter, any solution of the equations of motion can be brought, by means of a diffeomorphism, to the form s\^2= -( ł- ł\^2 x\^+ x\^- )\^[-1]{} x\^+x\^- , \^2=ł-ł\^2 x\^+x\^- , \[solkruskal\] where $x^+, x^-$ can be considered as the null Kruskal coordinates. The spacetime has four regions which can be characterised by the sign of the Kruskal coordinates. The asymptotic flat regions are characterised by $-\l^2x^+x^->0$. In the region $I$ ($x^+>0$, $x^-<0$) the metric can be written in a static asymptotically-flat form: s\^2&=& -(1+\^[-2]{})\^[-1]{}\^+\^- \[metricaflat\] ,\
\^[-2]{}&=&+\^[2]{} \[vpflat\] , by means of the coordinate change x\^+ &=& e\^[(+)]{} , \[cambioi\]\
x\^- &=& -e\^[-(-)]{} . \[cambioii\]
In the other asymptotically flat region $II$ ($x^+<0\>,\>x^->0$) the static metric can be achieved by the change x\^+ &=& - e\^[(+)]{} , \[cambioiii\]\
x\^- &=& e\^[-(-)]{} . \[cambioiv\]
If we calculate the energy of the basic solution of the CGHS model, eq. (\[metricaflat\],\[vpflat\]), by means of this E-M pseudotensor we will, surprisingly, not find any sensible result. In fact, the resulting expression is divergent and even do not involve the constant $m$. In the next sections, we will find the explanation for this result: the construction of a symmetrized E-M pseudotensor requires the theory to be invariant under asymptotic Lorentz transformations. We will show, however, that in order to have a well defined physical theory, we can not allow pure-Lorentz asymptotic rotations.
Going back to eq. (\[tresv\]), the contribution to the conserved charge for the basic solution in (\[metricaflat\]-\[vpflat\]) is: K\_I &=&12{ ([m]{}+e\^[2]{}) \_f\^+ ([m]{}+e\^[2]{}) \_f\^+ 2mf\^}\_[|\_[+]{}]{} , \[c30\] where the subindex $I$ refers to the region in which the above current has been evaluated. With the asymptotic fall-off conditions e\^[2]{} \_f\^ && 0 ,\
e\^[2]{} \_f\^ && 0 , \[c30i\] the Noether charge associated with the Killing time translation ($f^\tau \stackrel{\sigma\rightarrow\infty}{\sim} 1$) is $K_I = m$. Terms like $\lambda e^{2\lambda\sigma} f^\tau$, that would appear in the expression for the Noether charge had we started with Lagrangian (\[CGHS2\]), cancel out in (\[c30\]). It is just the Lagrangian (\[CGHS4\]) which gives directly the finite terms only. The reason is that the Noether charge (\[c30\]) can be seen as the result of contracting the presymplectic potential with the infinitesimal diffeomorphism $X$ associated with the asymptotic time translation (in region I). Both quantities are well defined in the covariant phase space, as we will see in the next section. The charges associated with the asymptotic spatial translations and Lorentz transformations are zero and divergent, respectively.
Moreover we also want to stress that the Noether charge (\[c30\]) just gives the mass of the black hole without the discrepant factor $\frac12$, as happens in the Komar’s formula for energy in General Relativity. We shall also understand this fact in the context of the canonical formalism.
Canonical structure and asymptotic symmetries of the CGHS model. {#E}
================================================================
Let us begin our analysis of the canonical structure of the CGHS model by writing the general classical solution of the theory without matter. It is well known that any solution is equivalent under diffeomorphisms to the solution s\^2 = -x\^+x\^- ,\^2=ł-ł\^2 x\^+x\^- . \[E5\] The solutions are characterized by an unique diffeomorphism invariant parameter, $m$, and therefore the variable canonically conjugate to $m$ should be “hidden” in the group of diffeomorphisms. The situation is somewhat similar to the trivial example of the free particle. Any solution is equivalent, under the Galileo group, to the one with the particle lying at rest and, therefore, the canonical degrees of freedom of the system are found in the symmetry (Galileo) group.
Our aim now is to find the degrees of freedom of the theory that are “hidden” in the group of diffeomorphisms. To this end we shall compute explicitly the two-form (\[dosii\]) (more precisely, the potential one-form (\[dosiii\]) ). This requires to adjust the boundary condition adequately for the potential form to be finite and independent of the spacelike Cauchy surface. Therefore, we shall assume the metric $g_{\mu\nu}$ to be flat at spatial infinity with a specific fall-off behaviour.
Let us apply a general diffeomorphism to the basic solution (\[E5\]). We find s\^2 &=& -PM , \[E9\]\
\^2 &=& ł-ł\^2 PM , \[E10\] where $P$ and $M$ are two arbitrary functions $P,M\>:{\cal M}\rightarrow \R$; $x^+ = P(\tau,\sigma)$, $x^- = M(\tau,\sigma)$.
We have \_ &=& -12(\_P\_M + \_M \_P) , \[E11\]\
&=& 12\^\_P\_M , \[E12\]\
\^ &=& -1[()\^2]{}\^ \^\_ , \[E13\]\
\_ &=& -12, \[E14\]\
\^\_ &=& 1[2]{} \^ . \[E15\] Obviously we also have \_\_M = 0 ,\_\_P = 0 ,a,b , \[E16\]and, therefore, for the metric parametrized as in eq. (\[E11\]): \_ &=& -12(\_M\_P +\_P\_M .\
&&.+\_P\_M +\_M\_P)\
&=& -12\[E17\]\
&=& \_h\_+\_h\_, where the one-form $h_\mu$ is given by: h\_= -12(\_PM + \_MP) h\^= -12\^(\_PM + \_MP) . \[E18\] We can easily see that, with the one-form $h^\alpha$ defined above, we can write as well: \_\^2 = \_(h\^\_\^2) ,. \[E18b\] So, to get the symplectic potential for the general solution given in (\[E9\]-\[E10\]), it is enough to replace in eq. (\[10\]) the diffeomorphism $f^\mu$ by the quantities $h^\mu$ as defined in (\[E18\]).
The symplectic potential will therefore be given by the divergence of an antisymmetric tensor ($K$ is now a one-form) \^= \^ \_K , \[E19\] and the symplectic form will be a pure-boundary term, thus implying that the theory has a finite number of degrees of freedom.
Conditions of flatness at spatial infinity. {#f}
-------------------------------------------
The condition for the metric to be flat at spacelike infinity means: g\_ = -12 \_ , \[f1\] where $\eta_{\mu\nu}=\left( \begin{array}{cc} -1 & 0 \\
0 & 1 \end{array} \right)$ and $\bar{m} = \frac{m}{\lambda}$. In region I ($P \equiv x^+ > 0$ and $M \equiv x^- < 0$) we can make P = \^[C]{} ,-M = \^[R]{} . \[f5\] Using (\[f1\]) and because of $-PM \spacelike +\infty$, we arrive at CR -\^2 + \^2 , \[f6\] or, what is the same, && CR -1 ,\
&& C’ +C’R 0 ,\
&& C’R’1 , requirements whose solution can be written in the form: C(,) &=& \^+ + A + U(,) , \[f7\]\
R(,) &=& -1\^- - B + V(,) , \[f8\] where $\alpha$, $A$ and $B$ are real numbers, $\sigma^+ = \tau + \sigma$, $\sigma^- = \tau - \sigma$, and U,V 0 . \[f9\]
The interpretation of (\[f7\]-\[f8\]) on the light of (\[f6\]) is obvious: the only allowed diffeomorphisms $(C,R)$ are those that asymptotically are Poincaré transformations in the coordinates $\tau$, $\sigma$. Surprisingly we will find additional constraints on the asymptotic transformations in the computation of the (on-shell) symplectic potential.
Symplectic potential.
---------------------
The symplectic current potential is given by \^= \^ \_K , \[cuatrodosi\] with $\widehat K$, formally, given by (\[tresiv\]).
The first consequence of the above formulas is that the symplectic potential reads as = \_\_ x\^ = (i\^0\_R) - (i\^0\_L) , \[f13i\] where $\Sigma$ is an arbitrary Cauchy surface (see Fig. I). We have to stress that $\Sigma$ is not required to intersect the bifurcation point of the horizon as it was in Ref. [@Navarro]. The point now is to show that the one-form $K$ can have well defined values in the right and left spatial infinities. In fact we shall find that not all the asymptotic Poincaré transformations are permitted in order to have a well defined result for $\theta$ (i. e., independent of the Cauchy surface).
=1
Replacing the “diffeomorphism” $h^\mu$ by its expression in eq. (\[E18\]) we find after a bit of algebra: (P,M,m) &=& -12 \^2 (PM - MP)\
&& -1[4]{} \^2 PM\^[ł]{}(\_P\_łM +\_M\_łP)\
&& + \^[ł]{}(\_P\_łM +\_M\_łP) , and, after having made use of (\[f5\]), we find (C,R,m) &=& -1[4]{} \^2 PM\^[ł]{} (\_C\_łR + \_R\_łC)\
&& +\^[ł]{} (\_C\_łR +\_R\_łC)\
&& -2(R-C) , where $\chi$ is given by: = 12\^[ł]{}\_łC\_R 1 . And after the replacements in (\[f7\],\[f8\]) we find &=& 1[4]{}(|[m]{} - \^2 PM) \^[ł]{}(\_U\_łV +\_V\_łU) \[f14\]\
&& + 1[2]{}(|[m]{} - \^2 PM) ( U + V)\
&& - 1[2]{} (|[m]{} - \^2 PM) (\_+ U + \_- V )\
&& - (V-U)\
&& + 1[2]{}(|[m]{} - \^2 PM) 2 \[f16\]\
&& + 2(1\^- + \^+) \[f17\]\
&& + m () . \[f18\]
It is easy to realize from the last expression above that for the symplectic potential to be finite and independent of the spacelike Cauchy surface (i.e, independent of $\tau$), requires first that $\alpha=1$. That is to say, the Lorentz transformations are not allowed. So that, we are left with: &=& 1[4]{}(|[m]{} - \^2 PM) \^[ł]{}(\_U\_łV +\_V\_łU)\
&& + 1[2]{}(|[m]{} - \^2 PM) (U + V) \[f20\]\
&& + m () . Moreover, to find a finite resulting expression for (\[f20\]) we have to require an appropriate asymptotic fall-off for the functions $U$ and $V$. From a close inspection of eq. (\[f20\]), and taking into account the asymptotic behaviour of $-PM$, it is not difficult to realize that the most natural requirement in order to have a sensible reduced phase space is e\^[2]{} , e\^[2]{} && 0 ,\
e\^[2]{} U’, e\^[2]{} V’ && 0 . \[f21ii\] Therefore we have arrived at: (i\^0\_R) = m ( ) , \[f21iii\] where $\frac{A+B}{2} \equiv f(i^0_R)$ is the Killing time translation at right spatial infinity.
In the other asymptotically flat region $x^+ = P(\tau,\sigma) < 0$, $x^- = M(\tau,\sigma) > 0$, we should write -P = e\^[C]{} ,M = e\^[R]{} \[f21iv\] instead of (\[f5\]), where the asymptotic flatness requires that (the asymptotic Lorentz transformation has already been neglected) C(,) &=& + + A + U(,) ,\
R(,) &=& -(- ) - B + V(,) ,\
& & U, V 0 . \[f21v\]
Proceeding in the same way as in the region I we obtain (i\^0\_L) = m ( ) , \[f21vi\] where now $\frac{A+B}{2} \equiv f(i^0_L)$ stands for the Killing time translation at left infinity.
Taking into account (\[f21iii\]) and (\[f21vi\]) we obtain the final expression for the symplectic potential = m ( f(i\^0\_R) - f(i\^0\_L) ) . \[f21vii\]
Diffeomorphisms in the presence of matter. {#g}
------------------------------------------
When matter is present, the procedure applied above is much more complicated. This is so because we would not be able to write the symplectic form as a pure boundary term. The model has an infinite number of degrees of freedom and, because of that, the symplectic form has, unavoidably, a bulk term. Intuitively we expect, however, that diffeomorphisms should be “almost” pure gauge. In the covariant formalism, this means that the presymplectic two-form (\[dosii\]) should be degenerated along the directions that corresponds to the gauge transformations of the theory. We can arrive at this result by contracting the symplectic two-form with the generator of a diffeomorphism: (g)\_=\_f\_+\_f\_,=f\^\_,\_i=f\^\_\_i . \[g1\]
The only linearized equation of motion which is not trivial to obtain is: \_\^\_ - \_\^\_=0 . \[g3\] and, after a long computation, we arrive at: \_[X\_f]{} j\^= \_T\^ , i.e. $\hi_{X_f} \omega$ is a pure boundary term, with T\^ = -T\^ &=& 12 { \^2\[g4\]\
&&-\^2(\^f\^- \^f\^)\
&&+2(f\^\^\^2- f\^\^\^2)\
&&+(f\^\_\^2g\^ -f\^\_\^2g\^)\
&&. +(f\^\^\_i\_i - f\^\^\_i\_i) } .
It is convenient now to rewrite the expressions above in terms of the physical metric, which has a better behaviour at spacelike infinity. We find: 2T\^&=& \^2{-(\^f\^-\^f\^).\
&&-(f\^\^\^2- f\^\^\^2)\
&&+(g\^\_f\^- g\^\_f\^)\
&&-12(g\^\_\^2 f\^- g\^\_\^2f\^)\[g8\]\
&&+12(\^\^2 f\_g\^ -\^\^2 f\_g\^)\
&&-(\^f\^-\^f\^)\
&&.+2(f\^\^\^2- f\^\^\^2)}\
&&+(f\^\^\_i\_i- f\^\^\_i\_i).
If we take, as it appears the most natural, boundary conditions such that $\phi_i\spacelike0$, the expressions above indicates clearly that the analysis of the model with matter reduces itself to the case without matter. Therefore, the contribution of diffeomorphisms to the reduced phase space of the theory is the same when there are matter fields as when there are not. For instance, if we take into account that $\vp^2\sspacelike\e^{2\lambda\sigma}$, we see that the leading term in (\[g8\]) behaves as $-\delta\log{\vp^2}\vp^2\left(\nabla^\lambda f^\alpha
-\nabla^\alpha f^\lambda\right)$. The finiteness of this term implies $\varepsilon^{\mu\nu}\p_\mu f_\nu\spacelike0$, thus forbidding as symmetries of the theory those diffeomorphisms that are asymptotically Lorentz transformations.
Symplectic potential of Schwarzschild black holes.
==================================================
The symplectic current potential of general relativity in vacuum is given by j\^= ( g\^ \^\_ - g\^ \^\_ ) , \[cincoi\] and due to the Hilbert-Einstein Lagrangian vanishes on-shell the current (\[cincoi\]) is conserved. In this section we shall work out the symplectic potential associated with the Schwarzschild black hole solutions. Instead of starting with the basic solution and acting on it with a general diffeomorphism we shall assume that the relevant asymptotic symmetry is the Killing time translation. Therefore we can write the general solution in regions I and II as follows ds\^2 = -( 1- ) d(t+f(t,r))\^2 + ( 1- )\^[-1]{} dr\^2 + r\^2 ( d\^2 + \^2 d\^2 ) . \[cincoii\] In addition, we shall choose the Cauchy surface in such a way that it connects the spatial infinities through the asymptotically flat regions I and II.
The symplectic potential is the integral of an exact three-form and, therefore, it receives contribution from the two-spheres $S^2_{R,L}$ at infinity only &=& \_[S\^2\_R]{} d \[ - f’ f + r(r-2m) f’ .\
&& . - 2 f m + 2 m f \]\
&& - \_[S\^2\_L]{} d \[ - f’ f + r(r-2m) f’ .\
&& . - 2 f m + 2 m f \] \[cincoiii\] .
To obtain a well-defined result we have to assume the following fall-off behaviour: r\^2 f’, 0 . \[cincoiv\] With the prescribed fall-off the integral (\[cincoiii\]) turns out to be = 12 . \[cincovi\]
Conclusions and final comments.
===============================
On the light of the result of Secs. 3-4 we observe that the asymptotic fall-off behaviour of the diffeomorphisms entering in the symplectic potential (\[c30i\]) are similar to that required to have a well-defined Noether charge (\[f21ii\]). This is a consequence of the closed relationship between the canonical formalism and the Noether theorem outlined in Sec. 2.
Using the covariant phase space picture we have determined the canonical structure of the CGHS model in the absence of matter, and the character of the asymptotic symmetries, without any a priori assumption on the dilaton asymptotic behaviour. The requirements made in 4.1-2 on the metric are enough to arrive at a clear result. The difference of Killing time translations at spatial infinities turns out to be the conjugate variable to the black hole mass. The asymptotic spatial translations are “gauge”-type symmetries: they decouple in the symplectic potential and leads to trivial Noether charge. The asymptotic Lorentz transformation breaks down (it cannot be permitted to have a well-defined symplectic form) and leads to a divergent Noether charge. This results are closely related. On general grounds, the action of a Lorentz transformation gives linear momentum to the system. In the CGHS model it breaks down and, therefore, the linear momentum vanishes identically, in accord with the “gauge” nature of the spatial translations for the model. This provides an explanation for the failure of the symmetric energy-momentum pseudotensor. The definition of this quantity requires the theory to be invariant under asymptotic Lorentz transformations and we have shown that this is not the case for the CGHS model.
As a byproduct of our study we also provide an explanation of the well-known factor 2 in the Komar formula for the mass in General Relativity. Although both the stringy and Schwarzschild black holes have the same symplectic structure = = m ( f(i\^0\_R) - f(i\^0\_L) ) , \[seisi\] they differ in the form of the symplectic potential. For the CGHS black hole the potential contains only the term with $\delta\left( f(i^0_R) - f(i^0_L) \right)$ \_[CGHS]{} = m ( f(i\^0\_R) - f(i\^0\_L) ) . \[seisii\] The corresponding Noether charge associated with a (right) asymptotically Killing time translation is just the black hole mass \_[CGHS]{} ( ) = m . \[seisiii\]
In the case of Schwarzschild black hole the symplectic potential contains a term with $\delta m$ as well. So that the Noether charge cannot coincide exactly with the mass. Since the potential is symmetric in $m$ and $f(i^0_R) - f(i^0_L)$ \_[Sch]{} = 12 ( m ( f(i\^0\_R) - f(i\^0\_L) ) - ( f(i\^0\_R) - f(i\^0\_L) ) m ) , \[seisiv\] the Noether charge is actually one half of the mass \_[Sch]{} ( ) = . \[seisv\]
Acknowledgments {#acknowledgments .unnumbered}
===============
M. N. would like to thanks L. J. Garay for valuable discussions. M. N. acknowledges the MEC for financial support. C. F. T. is grateful to the Generalitat Valenciana for a FPI grant. This work was partially supported by the CICYT and the DGICYT.
[99]{}
C. G. Callan, S. B. Giddings, J. A. Harvey and A. Strominger, [*Phys. Rev.*]{} [**D**]{} 45 R (1992) 1005
E. Witten, [*Phys. Rev.*]{} [**D**]{} 44 (1991) 314
C. Crnkovi' c and E. Witten, in [*Three Hundred Years of Gravitation*]{}, eds. S. W. Hawking and W. Israel (Cambridge, 1987), p. 676
J. Lee and R. M. Wald, [*J. Math. Phys.*]{} 31 (1990) 725
A. Ashtekar, L. Bombelli and O Reula, in [*Mechanics, Analysis and Geometry: 200 Years after Lagrange*]{}, ed. M. Francavigilia (ESP, 1991), p. 417
V. Aldaya, J. Navarro-Salas and M. Navarro, [*Phys. Lett.*]{} [**B**]{} 287 (1992) 109
J. Navarro-Salas, M. Navarro, C. F. Talavera and V. Aldaya, [*Phys. Rev.*]{} [**D**]{} 50 (1994) 901
J. Navarro-Salas, M. Navarro and C. F. Talavera, [*Phys. Lett*]{} [**B**]{} 335 (1994) 334
A. Mikovic and M. Navarro, [*Phys. Lett.*]{} [**B**]{} 315 (1993) 267
K. S. Soh, [*Phys. Rev.*]{} [**D**]{} 49 (1994) 1906
A. Bilal and I. J. Kogan, [*Phys. Rev.*]{} [**D**]{} 47 (1993) 5408
S. P. de Alwis, [*Phys. Rev.*]{} [**D**]{} 49 (1994) 941
D. Bak, D. Cangemi and R. Jackiw, [*Phys. Rev.*]{} [**D**]{} 49 (1994) 5173
K. Kuchar, [*Geometrodinamycs of Schwarzschild Black Holes*]{}, preprint UU-REL-94-3-1
D. Cangemi and R. Jackiw, [*Phys. Rev. Lett.*]{} 69 (1992) 233
D. Louis-Martinez, J. Gegenberg and G. Kunstatter, [*Phys. Lett.*]{} B 321 (1994) 193
A. Komar, [*Phys. Rev.*]{} 113 (1959) 934
V. Iyer and R. M. Wald, [*Phys. Rev.*]{} [**D**]{} 50 (1994) 846
=1
Figure Captions {#figure-captions .unnumbered}
===============
Figure I: Kruskal diagram for black hole spacetime. $\Sigma$ is an arbitrary Cauchy surface.
FIGURE I
[^1]: On leave of absence from \[2\] and [*Instituto Carlos I de Física Teórica y Computacional, CSIC, and Facultad de Ciencias, Universidad de Granada, Campus de Fuentenueva, 18002, Granada, Spain.*]{}
|
---
abstract: 'We report the magnetotransport properties of HoSb, a semimetal with antiferromagnetic ground state. HoSb shows extremely large magnetoresistance (XMR) and Shubnikov-de Haas (SdH) oscillation at low temperature and high magnetic field. Different from previous reports in other rare earth monopnictides, kinks in $\rho(B)$ and $\rho_{xy}(B)$ curves and the field dependent resistivity plateau are observed in HoSb, which result from the magnetic phase transitions. The fast Fourier transform analysis of the SdH oscillation reveals the split of Fermi surfaces induced by the nonsymmetric spin-orbit interaction. The Berry phase extracted from SdH oscillation indicates the possible nontrivial electronic structure of HoSb in the presence of magnetic field. The Hall measurements suggest that the XMR originates from the electron-hole compensation and high mobility.'
author:
- 'Yi-Yan Wang'
- 'Lin-Lin Sun'
- Sheng Xu
- Yuan Su
- 'Tian-Long Xia'
bibliography:
- 'bibtex.bib'
title: Unusual magnetotransport in holmium monoantimonide
---
[^1]
[^2]
Introduction
============
In recent years, rare earth monopnictides LnX (Ln=rare earth elements; X=N, P, As, Sb, Bi) have been widely studied for their novel physical properties[@zeng2015topological; @tafti2015resistivity; @PhysRevLett.117.127204; @Ban2017observation; @PhysRevB.96.081112; @PhysRevB.96.125112; @sun2016large; @PhysRevB.93.241106; @PhysRevB.94.081108; @PhysRevB.95.115140; @nayak2017multiple; @Singha2017Fermi; @PhysRevMaterials.2.024201; @PhysRevB.97.155153; @tafti2016temperature; @PhysRevB.93.235142; @PhysRevB.94.165163; @ghimire2016magnetotransport; @Yu2017Magnetoresistance; @pavlosiuk2016giant; @PhysRevLett.117.267201; @PhysRevB.96.075159; @alidoust2016new; @guo2017possible; @PhysRevB.97.081108; @PhysRevB.96.041120; @PhysRevLett.120.086402; @PhysRevB.93.205152; @neupane2016observation; @PhysRevB.97.115133; @PhysRevB.97.085137; @PhysRevB.96.125122; @PhysRevB.96.035134; @duan2018tunable; @li2017predicted]. Remarkably, extremely large magnetoresistance (XMR) and resistivity plateau are often observed in these materials. The electron-hole compensation is a prevalent explanation for the origin of XMR[@ali2014large; @PhysRevB.93.235142; @Yu2017Magnetoresistance; @PhysRevB.97.085137]. Conventional nonmagnetic metals only have a small magnetoresistance (MR). However, in the case of electron-hole compensation, high mobility of carriers will result in the emergence of XMR based on the semiclassical two-band model. Other mechanisms, such as the removed suppression of backscattering induced by the breaking of topological protection[@liang2015ultrahigh] or the change of spin texture[@PhysRevLett.115.166601], have also been proposed to explain the XMR. The occurrence of resistivity plateau is universal in XMR materials[@tafti2015resistivity; @PhysRevB.94.041103], and it has been suggested to originate from the nearly invariable carrier concentration and mobility at low temperature[@PhysRevB.93.235142; @PhysRevB.96.125112; @PhysRevB.97.085137]. Usually, the field-dependent MR of nonmagnetic compensated semimetals exhibits unsaturated quadratic behavior. However, in antiferromagnets CeSb and NdSb, the field-induced metamagnetic transition from the antiferromagnetic (AFM) state to ferromagnetic (FM) state gives rise to unusual transport phenomena. Due to the influence of spin scattering, the $\rho$(*B*) curve of NdSb shows a kink at high field and specific temperature range[@PhysRevB.93.205152]. The situation in CeSb is more complex, where the authors claim there exist several magnetic phases in the magnetic phase diagram, and the kinks are different at different temperatures[@PhysRevB.97.081108]. Consequently, it is interesting to study the transport properties in magnetic LnX materials.
Since LaX (X=N, P, As, Sb, Bi) are predicted to be topological semimetals or topological insulators[@zeng2015topological], the topological property of rare earth monopnictides has attracted much attention. For the nonmagnetic materials LaSb/LaBi/YSb, only LaBi is considered to hold nontrivial band structure[@PhysRevB.95.115140; @nayak2017multiple; @PhysRevB.97.155153], while LaSb[@PhysRevLett.117.127204; @PhysRevB.96.041120] and YSb[@PhysRevLett.117.267201; @Yu2017Magnetoresistance] are trivial semimetals. The topological transition from a trivial to a nontrivial phase in CeX (X=P, As, Sb, Bi) further emphasizes the importance of strong spin-orbit-coupling (SOC) effect[@PhysRevLett.120.086402]. For the paramagnetic (PM) TmSb and PrSb, topologically trivial characteristic of the band has been revealed by the trivial Berry phase or the study of electronic structure[@PhysRevB.97.085137; @PhysRevB.96.125122]. The situation is different in CeSb and NdSb with magnetic phase transition. Although the band inversion is absent in the PM state of CeSb[@PhysRevB.97.081108; @PhysRevB.96.041120], the presence of Weyl fermion is suggested to be possible in the FM state of CeSb as supported by the observation of negative longitudinal MR and the electronic structure calculations[@guo2017possible]. The existence of Dirac semimetal state is suggested in the AFM state of NdSb, where the negative longitudinal MR has been observed and attributed to the chiral anomaly[@PhysRevB.97.115133]. The ferromagnetic GdSb is also predicted to hold Weyl fermions[@li2017predicted]. The topological properties of magnetic LnX materials still deserve further exploration.
HoSb possesses an AFM ground state. The magnetic structure of HoSb changes from MnO-type AFM arrangement to HoP-type arrangement, then to FM arrangement under the application of field[@PhysRev.131.922; @brun1980quadropolar]. The magnetic transition makes HoSb a good platform to investigate the influence of magnetic interaction on the magnetotransport and topological properties. In this work, we grew the single crystals of HoSb and studied the magnetotransport properties. The observation of the kinks in $\rho(B)$ and $\rho_{xy}(B)$ curves and the field-dependent resistivity plateau indicate that the magnetotransport properties of HoSb is different from those of other rare earth monopnictides. The unusual properties are ascribed to the change of magnetic structure. At 2.65 K $\&$ 14 T, the transverse MR of HoSb reaches 2.42$\times$10$^4$$\%$. The XMR is suggested to originate from the electron-hole compensation and high mobility. The split of Fermi surfaces and the nontrivial Berry phase of the electron bands have been observed/extracted from the Shubnikov-de Haas (SdH) oscillation, which can be attributed to the nonsymmetric spin-orbit interaction and the possible nontrivial electronic structure of HoSb in the magnetic field, respectively.
Experimental methods and crystal structure
==========================================
Single crystals of HoSb were grown from the Sb flux. Starting materials of Ho and Sb with a molar ratio of 1:6 were put into an alumina crucible and then sealed in a quartz tube. The quartz tube was heated to 1150$^{\circ}$C and held for 10 h. After that, it was cooled to 750$^{\circ}$C at a rate of 1$^{\circ}$C/h, where the excess Sb flux was removed with a centrifuge. Finally, the cubic crystals were obtained. The atomic proportion determined by energy dispersive x-ray spectroscopy (EDS, Oxford X-Max 50) was consistent with 1:1 for Ho:Sb. Single crystal and powder x-ray diffraction (XRD) patterns were obtained with a Bruker D8 Advance x-ray diffractometer using Cu K$_{\alpha}$ radiation. TOPAS-4.2 was employed for the refinement. Transport measurements were carried out on a Quantum Design physical property measurement system (QD PPMS-14 T). The magnetic properties were measured on a Quantum Design magnetic property measurement system (MPMS3).
The single crystal XRD pattern of HoSb presented in Fig. 1(a) reveals that the (*0 0 2l*) plane is the surface of the crystal. HoSb crystallizes in a simple rock-salt structure, as shown in the inset of Fig. 1(a). Fig. 1(b) shows the powder XRD pattern of HoSb. The reflections are well indexed in the space group *Fm*-3*m* and the refined lattice parameter a is 6.13(1)[Å]{}. A selected cubic single crystal of HoSb is presented in the inset of Fig. 1(b).
{width="48.00000%"}
Results
=======
Fig. 2(a) shows the temperature-dependent resistivity of HoSb at various magnetic fields. The applied magnetic field is along \[001\] direction, perpendicular to the direction of the electric current. The inset shows the zero field resistivity in the low-temperature region. With decreasing temperature, the zero field resistivity decreases above the N$\acute{e}$el temperature $T_N$ = 5.7 K while below $T_N$ it drops suddenly. The anomaly observed around $T_N$ corresponds to the transition from PM to AFM, which is suppressed gradually under field. When a moderate field is applied, an upturn appears in the resistivity curve as the temperature decreases. Apart from the upturn, a resistivity plateau is observed with the further decrease of the temperature, which can be seen clearly from Fig. 2(b) where the temperature is plotted in a logarithmic scale. Similar behavior has also been observed in other XMR materials, such as the isostructural compounds La(Sb/Bi)[@tafti2015resistivity; @sun2016large] and YSb[@Yu2017Magnetoresistance]. Remarkably, the temperature where the plateau starts increases with increasing field in HoSb. Fig. 2(c) plots the temperature dependence of *$\partial\rho/\partial T$* at different fields. Two characteristic temperatures *T$_m$* and *T$_i$* can be identified from the *$\partial\rho/\partial T$* curves, as shown in the inset of Fig. 2(c). *T$_m$* is defined as the temperature where the sign of *$\partial\rho/\partial T$* changes from positive to negative and *T$_i$* is defined as the temperature where the valley appears. The resistivity reaches the minimum at *T$_m$* and a resistivity plateau emerges below *T$_i$*. Fig. 2(d) displays the *T$_m$* and *T$_i$* derived from Fig. 2(c) as a function of field. *T$_m$* and *T$_i$* both increase with increasing field. The increasing *T$_i$* differs greatly from the nearly field-independent *T$_i$* in other XMR materials[@PhysRevB.94.041103; @tafti2015resistivity; @sun2016large; @Lv2016Extremely].
{width="48.00000%"}
Magnetoresistance describes the change of the resistivity under magnetic field and can be obtained by the formula $MR = (\rho(B)-\rho(0))/\rho(0)$. Figure. 3(a) shows the transverse MR of HoSb plotted as a function of field at various temperatures. The value of MR reaches 2.42$\times$10$^4$$\%$ at 2.65 K and 14 T without any sign of saturation and decreases with increasing temperature. In addition, obvious SdH oscillation is also observed at low temperature and high field. The inset of Fig. 3(a) shows the oscillatory component of resistivity ($\Delta\rho_{xx}=\rho_{xx}-<\rho_{xx}>$) after subtracting a smooth background. Figure. 3(b) plots the fast Fourier transform (FFT) spectra of the oscillation, which reveal the frequencies. Considering the analysis of angle-dependent SdH oscillation (not presented here) and previous studies on the FSs of rare earth monopnictides, we identified the frequencies and drew the possible projection of FSs along $k_x$ (the inset of Fig. 3(b)). There are several pairs of peaks in the FFT spectra. Since the frequency $F$ is proportional to the extremal cross-sectional area $A$ of FS normal to the field according to the Onsager relation $F=(\phi_0/2\pi^2)A=(\hbar/2\pi e)A$, the feature that the frequencies appear in pairs indicates the split of FSs. The split of FSs has also been observed in TmSb and attributed to the nonsymmetric spin-orbit interaction[@PhysRevB.97.085137]. In the measurement, the current *I* and field *B* are parallel to *x* axis and *z* axis, respectively. The elliptical FSs can be divided into three categories: $\alpha^{\prime}$, $\alpha^{\prime\prime}$ and $\alpha^{\prime\prime\prime}$. The absence of the frequency from $\alpha^{\prime\prime\prime}$ may be related to the low mobility along the long axis of the elliptical FSs, which has been derived in YSb and LaSb[@PhysRevB.96.075159; @PhysRevB.96.125112].
{width="48.00000%"}
In general, the Lifshitz-Kosevich (LK) formula[@shoenberg1984magnetic; @PhysRevMaterials.2.021201] $$\label{equ1}
\Delta\rho\propto\frac{\lambda T}{sinh(\lambda T)}e^{-\lambda
T_D}cos[2\pi\times(\frac{F}{B}-\frac{1}{2}+\beta+\delta)]$$ is employed to describe the amplitude of SdH oscillation. In the formula, $R_T=(\lambda T)/sinh(\lambda T)$ is the thermal factor, where $\lambda= (2\pi^2k_{B}m^*)/(\hbar eB)$ ($m^*$ and $k_B$ are the effective mass of carrier and the Boltzmann constant, respectively). $T_D$ is the Dingle temperature. $2\pi \beta$ is the Berry phase. $\delta$ is a phase shift. For the 2D system, the value of $\delta$ is 0. For the 3D system, $\delta$ takes $+1/8$ (hole pocket) or $-1/8$ (electron pocket)[@shoenberg1984magnetic]. Figure. 3(c) plots the temperature dependent FFT amplitude. The solid lines are fittings using the thermal factor $R_T$. As shown in Table I, the obtained effective masses are comparable with that of LaSb and TmSb[@tafti2015resistivity; @PhysRevB.97.085137]. Berry phase can be extracted from the SdH oscillation and used to estimate the topological property of materials roughly. For such a multifrequency oscillation, it is difficult to map the Landau level index fan diagram, so we use the multiband LK formula to fit the oscillation pattern (Fig. 3(d)). The obtained Berry phase and Dingle temperature are listed in Table I. The values of Berry phase corresponding to the electron pocket (including the frequencies of $\alpha^{\prime}_1$, $\alpha^{\prime}_2$, $\alpha^{\prime\prime}_1$ and $\alpha^{\prime\prime}_2$) are close to the nontrivial value $\pi$, while the others are far away from $\pi$. However, a significant gap at *X* point in HoSb has been suggested by the ARPES experiments performed in the absence of magnetic field[@PhysRevB.96.035134]. It is possible that the Weyl fermions may exist in the FM state of HoSb. The similar situation has also been suggested in CeSb[@guo2017possible] by the transport evidence and in GdSb[@li2017predicted] by the first-principles calculations. In addition, it should be noted that the $\pi$ Berry phase cannot be viewed as a smoking gun of topologically nontrivial material[@PhysRevX.8.011027]. Further study is needed to check the topological property of HoSb in the presence of the magnetic field.
$F$ (T) A (${\AA}^{-2}$) $k_F$ (${\AA}^{-1}$) $m^*/m_e$ $T_D$ (K) $2\pi\beta$
----------------------------- --------- ------------------ ---------------------- ----------- ----------- -------------
$\alpha^{\prime}_{1}$ 235.3 0.022 0.085 0.156 13.2 1.05$\pi$
$\alpha^{\prime}_{2}$ 416.3 0.040 0.112 0.164 17.5 0.89$\pi$
$\alpha^{\prime\prime}_{1}$ 657.2 0.063 0.141 0.173 29.3 1.17$\pi$
$\alpha^{\prime\prime}_{2}$ 733.7 0.070 0.149 0.191 27.4 0.83$\pi$
$\beta_{1}$ 839.9 0.080 0.160 0.250 17.9 0.14$\pi$
$\beta_{2}$ 986.5 0.094 0.173 0.414 14.7 0.48$\pi$
$\gamma_{1}$ 1532.0 0.146 0.216 0.311 20.6 0.09$\pi$
$\gamma_{2}$ 1570.7 0.150 0.218 0.336 18.3 0.34$\pi$
: Parameters obtained from the analysis of SdH oscillation. $F$, oscillation frequency; A, extremal cross-sectional area of FS normal to the field; $k_F$, Fermi vector; $m^*$, effective mass of carrier; $T_D$, Dingle temperature; $2\pi\beta$, Berry phase ($\delta$ has been included).[]{data-label="oscillations"}
{width="\textwidth"}
Hall measurements are performed to reveal the origin of XMR in HoSb. As shown in Fig. 4(a), the field dependent Hall resistivity $\rho_{xy}=[\rho_{xy}(+B)-\rho_{xy}(-B)]/2$ exhibits nonlinear behavior, indicating the coexistence of electron and hole. With the increase of temperature, the dominant carrier changes from electron to hole. Unlike the Hall resistivity of usual two-component systems, there is a kink in the $\rho_{xy}(B)$ of HoSb, which will be discussed later. Figure. 4(b) shows the $\rho_{xy}(B)$ at 3 K. Apart from the imperfect part induced by the kink, the curve can be well fitted by the semiclassical two-band model, $$\label{equ2}
\rho_{xy}=\frac{B}{e}\frac{(n_h \mu_h^2-n_e \mu_e^2)+(n_h-n_e)(\mu_h \mu_e)^2 B^2}{(n_h \mu_h+n_e \mu_e)^2+(n_h-n_e)^2 (\mu_h \mu_e)^2 B^2},$$ where $n_h(n_e)$ and $\mu_h(\mu_e)$ represent the hole (electron) concentration and hole (electron) mobility, respectively. Figures. 4(c) and 4(d) show the obtained concentration and mobility, respectively. At 3 K, $n_e=1.06\times10^{21}cm^{-3}$, $n_h=0.88\times10^{21}cm^{-3}$, $\mu_e=4.47\times10^3cm^2V^{-1}s^{-1}$ and $\mu_h=7.07\times10^3cm^2V^{-1}s^{-1}$. The ratio $n_h/n_e\approx0.83$ indicates the compensation of carriers in HoSb. From the field dependent resistivity $$\label{equ3}
\rho (B)=\frac{(n_h \mu_h+n_e \mu_e)+(n_h \mu_e+n_e \mu_h)\mu_h \mu_e B^2}{e (n_h \mu_h+n_e \mu_e)^2+e (n_h-n_e)^2 (\mu_h \mu_e)^2 B^2},$$ the relation MR=$\mu_e\mu_h$*B*$^2$ can be obtained for the perfect electron-hole compensation ($n_e$=$n_h$). Consequently, high mobility will lead to the large and unsaturated MR. If the ratio $n_h/n_e$ deviates from 1 slightly, MR will deviate from quadratic behavior slightly and remain unsaturated up to a high field[@PhysRevB.97.085137]. The XMR in HoSb originates from the electron-hole compensation and high mobility.
{width="70.00000%"}
The field dependent MR has been examined to investigate the origin of the kink. As shown in Fig. 4(e), although the curve can be roughly fitted by MR$\propto$*B*$^n$ and the obtained $n$=1.996 is close to 2, the fit is not perfect and the kink can be seen clearly as that in the Hall data. However, the data in high field can be well fitted (the inset of Fig. 4(e), $n$=1.84), indicating that the kink may originate from some transitions under low magnetic field. Then the field dependent magnetization was measured to examine the effect of the magnetic structure. As shown in Figs. 4(f) and 4(g), the magnetic moments of HoSb adopt MnO-type antiferromagnetic arrangement at low temperature, and the increase of field makes the arrangement change to HoP-type magnetic structure (manifested as the kink in the magnetization curve) and then to FM state[@PhysRev.131.922; @brun1980quadropolar]. In the process of the transitions, the emergent disorder reduces the relaxation time $\tau$. Since $\mu=e\tau/m^*$, the decrease of relaxation time will cause the decrease of mobility, resulting in the formation of the kinks in the field dependent Hall resistivity and MR.
Figure 5 shows the study on the magnetic properties of HoSb. The magnetization curves exhibit a transition from PM to AFM at low temperature and low field (Fig. 5(a)). The transition temperature of 5.9 K is slightly larger than the value obtained from resistivity, which may be from the difference of samples. Under a high field, the AFM order is suppressed and the phase changes from PM to FM (Fig. 5(b)). The detailed measurements of the temperature-dependent magnetization at different fields have been performed to map the magnetic phase diagram. As shown in Fig. 5(c), the N$\acute{e}$el temperature decreases with the increase of field. In addition, an unexpected cusp at the temperature below $T_N$ has been found in a specific range of field, which was not reported in previous studies. Such an anomaly has also been observed in CuMnSb and attributed to the canted AFM structure without uniform magnetic moment[@PhysRevMaterials.2.054413]. Figure 5(d) plots the magnetic field and temperature phase diagram of HoSb, in which three regimes (PM, AFM, and FM) can be distinguished. With the increase of magnetic field, the AFM order is gradually suppressed and FM order gradually appears.
Discussions
===========
Since the temperature $T_i$ in other nonmagnetic XMR materials is nearly field-independent[@tafti2015resistivity; @PhysRevB.94.041103], the observation of field-dependent $T_i$ in HoSb is interesting. Previously, the XMR and resistivity plateau in LaSb were attributed to the field-induced metal-insulator transition and the saturation of insulating bulk resistance induced by the metallic surface conductance, respectively[@tafti2015resistivity]. Later, the possibility of field-induced metal-insulator transition was excluded[@tafti2016temperature; @PhysRevB.96.125112]. Guo *et al.* pointed out that the resistivity plateau comes from the nearly invariable carrier concentration and mobility at low temperature in compensated semimetals[@PhysRevB.93.235142]. Han *et al.* suggested that the resistivity plateau originates from the temperature-insensitive resistivity at zero field if the MR can be scaled by Kohler’s rule[@PhysRevB.96.125112]. However, for a two-component system that satisfies $n_e=n_h$ and $\mu_e=\mu_h$, the Kohler’s rule can be easily derived from the two band model (Eq. 3). The temperature-insensitive resistivity at zero field is related to the nearly invariable carrier concentration and mobility at low temperature.
Different from nonmagnetic LaSb/LaBi/YSb and paramagnetic TmSb/PrSb, HoSb is an antiferromagnet and the magnetic structure can be changed by field. The increased disorder in the process of phase transitions reduces the mobility and leads to the kinks in $\rho(B)$ and $\rho_{xy}(B)$. Under different fields, the temperature where the FM phase appears is different. Meanwhile, the resistivity plateau always starts in the FM state. This indicates that the change of mobility induced by phase transitions has affected the emergence of resistivity plateau. It is suggested that the field dependence of $T_i$ is related to the field-dependent mobility in phase transitions.
Summary
=======
In summary, single crystals of HoSb are grown and the magnetotransport properties have been studied. The unusual magnetotransport properties, including the *field-dependent resistivity plateau* and the *kinks in $\rho(B)$ and $\rho_{xy}(B)$ curves*, can be attributed to the magnetic phase transitions. HoSb also exhibits XMR and SdH oscillation at low temperature and high field. The FFT spectra of SdH oscillations reveal the split of Fermi surfaces. The nontrivial Berry phase of the electron bands indicates that HoSb may have topologically nontrivial electronic structure under the field. The electron-hole compensation and high mobility are suggested to be responsible for the observation of XMR.
Acknowledgments
===============
We thank Hechang Lei and Lingxiao Zhao for helpful discussions. This work is supported by the National Natural Science Foundation of China (No.11574391), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (No. 14XNLQ07, No. 18XNLG14).
[^1]: These authors contributed equally to this paper
[^2]: These authors contributed equally to this paper
|
---
author:
- 'Jeffrey D. P. Kenney & Rebecca A. Koopmann'
title: Ongoing Gas Stripping in the Virgo Cluster Spiral NGC 4522
---
Introduction
============
Among the many types of galaxy interactions which are posited to occur in clusters, one of the most important may be the stripping of gas from the interstellar media (ISM) of galaxies due to interactions with the gas in the intracluster medium (ICM) (Gunn & Gott 1972). This process of ICM-ISM stripping is likely to significantly affect the morphology and evolution of cluster spiral galaxies, and may be one of the factors which explains the morphology-density relationship (Dressler 1997). It remains the best explanation for several observations: the large number of HI deficient spiral galaxies observed in clusters (Giovanelli & Haynes 1983), spiral galaxies with truncated HI disks yet relatively undisturbed stellar disks (Warmels 1988; Cayatte 1990) and cluster spirals with vigorous star formation in their central regions, but very little throughout most of their outer disks (van den Bergh 1990; Koopmann & Kenney 1998a,b). Yet despite widespread indirect evidence for the ICM-ISM stripping of cluster spiral galaxies, there are few clear examples of gas actively being stripped from the disks of spirals by the ICM.
Instead, the best cases so far for ongoing ICM-ISM stripping are elliptical galaxies. Evidence seems quite strong that stripping of the outer hot halo gas occurs in the Virgo ellipticals NGC 4406 (M86; Forman 1979; White 1991), NGC 4472 (M49; Irwin & Sarazin 1996) and perhaps NGC 4636 (Trinchieri 1994). The ease and consequences of ICM-ISM stripping for ellipticals are different from those for spirals, primarily because the hot ISM in ellipticals is a halo component, which has a much lower density than the disk gas of spirals. This hot gas does not efficiently form stars, so its loss does not greatly effect the elliptical’s future evolution. However, the loss of star-forming disk gas from spirals would have a greater evolutionary consequence, perhaps accelerating their evolution to a lenticular morphology.
Perhaps the best spiral/irregular candidates for ongoing ICM-ISM stripping are three galaxies (97073, 97079, 97087) with peculiar H$\alpha$ and non-thermal radio continuum properties located in the outskirts of the cluster A1367 (Gavazzi 1995). These galaxies have radio continuum emission extended well beyond the optical disks, with head-tail morphologies, and strongly enhanced star formation rates, including bright HII regions, which in at least one case lies in an arc in the outer disk of the galaxy. Among more nearby spiral galaxies, there are several with peculiarities that have been attributed to ongoing ISM-ICM stripping, although for most of these galaxies the evidence is less compelling. The group galaxy NGC 7421 has a bow-shock shaped HI morphology, suggestive of an ISM-ICM interaction (Ryder 1997). The cluster or group spiral galaxies NGC 1961, NGC 2276, NGC 3312, NGC 4438, NGC 4654, and NGC 5291 (respectively Shostak 1982; Mulchaey 1993; Gallagher 1978; Kotanyi 1983; Phookun & Mundy 1995; Malphrus 1997) have all been cited as possible examples of ICM-ISM stripping in action. However, for each of them, a tidal or other origin for their peculiarities has also been suggested (Pence & Rots 1997; Gruendl 1993; McMahon 1992; Kenney 1995; Cayatte 1990; Malphrus 1997). In each of these systems, the stellar as well as the gaseous distribution appears to be disturbed, thus in none of these cases is it clear that ICM-ISM stripping is responsible for the galaxy peculiarity. In contrast to the many spiral galaxies which are widely recognized as being in various stages of gravitational interactions and mergers, there are no generally recognized “smoking guns” for ICM-ISM stripping in spirals.
In this paper, we present broadband BVR and narrowband H$\alpha$ images of the relatively obscure, highly inclined Virgo cluster spiral galaxy NGC 4522. Its relatively normal stellar disk and selectively disturbed ISM strongly suggest that its ISM is actively being stripped by an ICM-ISM interaction, and we consider NGC 4522 an excellent nearby spiral candidate for ongoing ICM-ISM stripping.
The Galaxy NGC 4522
===================
The optical peculiarities of NGC 4522 are reflected in its classification of SBcd:(sp) (RC3) and Sc/Sb: (Binggeli, Sandage, & Tammann 1985). The uncertain Sc/Sb Hubble classification probably results from the small bulge and the relatively high star formation rate in the inner region (like Sc), in combination with the lack of star formation in the outer disk (more like Sb). The galaxy is highly inclined (i=75$\pm$5$\deg$), making it possible to see extraplanar features. The “spindle” designation from the RC3 probably arises from the extraplanar dust and HII regions, which are discussed below. A photograph in Sandage & Bedke (1994) shows an “arm fragment” emerging from the disk, which we show to be composed primarily of HII regions. It is significant that all the known peculiarities in NGC 4522 are associated with dust, gas, or HII regions, and not older stars.
Although it was not mapped in HI in either the Westerbork (Warmels 1988) or VLA (Cayatte 1990) Virgo cluster HI surveys, it has been detected in HI with the Arecibo telescope (Helou 1984). It has an HI deficiency of 0.6, meaning that NGC4522 contains only $\sim$1/4 the amount of atomic gas that a normal late type spiral of its size would contain (Giovanelli & Haynes 1983; Kenney & Young 1989). It is 3.3$\deg$ from M87, which is at the center of the main galaxy concentration in Virgo, and only 1.5$\deg$ from M49, which is at center of subcluster B (Binggeli, Popescu, & Tammann 1993). Binggeli, Sandage, & Tammann (1985) consider it a member of the Virgo cluster, and Yasuda (1997) find a Tully-Fisher distance consistent with Virgo cluster membership. This part of the Virgo cluster contains many HI-deficient galaxies (Haynes & Giovanelli 1986), as well as x-ray emission from hot intracluster gas (Bohringer 1994). The ROSAT map shows weak extended x-ray emission at the projected location of NGC 4522 (Böhringer 1994), although NGC 4522 itself does not appear to be a strong localized source of x-ray emission (Fabbiano 1992; Böhringer 1994; Snowden 1997). A compilation of galaxy properties is given in Table 1.
Observations
============
Exposures of NGC 4522 were taken in BVR and narrowband H$\alpha$+\[N II\] filters on the 3.5 m WIYN telescope$\footnote{
The WIYN Observatory is a joint facility of the University of Wisconsin-Madison,
Indiana University, Yale University, and the National Optical Astronomy
Observatories.}$ at KPNO in April 1997. A 2048$\times$2048 S2KB CCD with a plate scale of 0.2$''$/pixel was used, giving a field of view of 6.8$'$ (31 kpc). The narrowband filter had a bandwidth of 70Å centered on 6625Å, and included the redshifted H$\alpha$ and \[N II\] lines. Integration times totalled 10 minutes for B and V, 20 minutes for R, and 30 minutes for H$\alpha$+\[N II\], and were divided up into 3 or more exposures per filter. The seeing was 1.1$''$. We used the IRAF package in a standard way to bias-subtract, flat-field, register, and combine the images for each filter. Cosmic rays were removed using the pixel rejection routines in the combine task. After sky subtraction, the R-band image was scaled and then subtracted from the narrowband filter image to obtain a continuum-free H$\alpha$+\[N II\] (hereafter referred to as an H$\alpha$) image.
The Peculiar H$\alpha$ Morphology of NGC 4522 and Possible Explanations
=======================================================================
The B-band and H$\alpha$ images of NGC 4522 are shown in Figures 1 and 2, and contour maps produced from R-band and H$\alpha$ images smoothed to 3$''$ resolution are shown in Figure 3. There is no evidence from any of the broadband images or contour maps of any significant peculiarity in the distribution of older stars. The presence of dust and star-forming regions in the inner galaxy strongly affects the observed light distribution in the central r$\simeq$60$''$, but there is less dust and little or no massive star formation beyond this radius. R-band surface photometry shows that the stellar light distribution beyond r=20$''$ and out to at least r=170$''$ is well fit by an exponential with a scale length of 30$\pm$3$''$, and that the bulge contribution is very small (Koopmann, Kenney, and Young 1998).
In contrast, the H$\alpha$ morphology is distinctly peculiar. Ten percent of the total H$\alpha$ emission arises from a one-sided, extraplanar distribution, predominantly organized into filaments which extend more than 35$''$ = 3 kpc from the outer edge of a truncated H$\alpha$ disk. The extraplanar H$\alpha$ emission arises from both HII regions (Fig 2), and diffuse emission (Fig.3a). The H$\alpha$ morphology is reminiscent of a bow shock, and, when coupled with a normal stellar disk, strongly indicates that the ISM of NGC 4522 is being selectively disturbed.
The outermost bright HII complex in the southwestern part of the disk, at a projected radius of 40$''$=3.1 kpc, has an interesting bubble morphology in H$\alpha$, as shown in Fig. 2. This relatively uniform surface brightness shell with a diameter of 5$''$=400 pc extends over 180$\deg$, and appears to emerge from two bright HII regions at its base The regular morphology of the shell suggests that a wind from supernovae and OB stars is expanding into a relatively uniform medium, perhaps the ICM.
What internal processes could selectively disturb gas in galaxies? Starbursts and cooling flows produce distinctive ionized gas morphologies which clearly differ from that in NGC 4522. Intense starbursts have outflowing gas with double bubble (Jogee 1998) or biconical (Heckman 1990) extraplanar H$\alpha$ distributions. Extraplanar gas is generally produced on both sides of the disk, and originates from a location which is near the galaxy center and inside a region of active star formation. “Cooling flow” galaxies often exhibit filamentary H$\alpha$ morphologies which extend from all sides of the galaxy center (e.g. Lynds 1970). Some edge-on spiral galaxies contain extraplanar H$\alpha$ filaments or diffuse emission, probably caused by massive star formation within the disk (Rand, Kulkarni, & Hester 1990; Pildis, Bregman, & Schombert 1994; Rand 1997). These filaments are generally smaller and less luminous than those in NGC 4522, and are not selectively located near the outer edge of the star-forming disk, as are those in NGC 4522.
Galaxy-galaxy gravitational interactions are also unlikely to explain the morphology of NGC 4522. While gravitational interactions affect gas differently than stars, both stars and gas are disturbed (e.g., Barnes & Hernquist 1992). Given the degree of morphological peculiarities in H$\alpha$, one would expect to see some disturbance in the stars of NGC 4522, such as a tail or warp. For example, while the peculiar Virgo cluster galaxy NGC 4438 has a one-sided set of extraplanar ionized gas filaments (Kenney 1995) somewhat like those in NGC 4522, the stellar disk of NGC 4438 is highly disturbed, indicating that NGC 4438 has experienced a high-velocity galaxy-galaxy interaction(Kenney 1995; Moore 1996).
The distinctive H$\alpha$ morphology of NGC 4522, together with the normal-appearing stellar disk, not only seems inconsistent with vigorous disk star formation, a central starburst, a cooling flow, or a galaxy-galaxy interaction, but is what is generally expected for ICM-ISM stripping. Moreover, the extraplanar optical emission in edge-on star-forming galaxies, starbursts, cooling flows, and NGC 4438 have line ratios and morphologies indicating shock-excited, diffuse gas (Heckman 1989; Heckman 1990; Kenney 1995; Rand 1997), rather than HII regions, as is observed in NGC 4522. We therefore propose that the ISM of NGC 4522 is being stripped by the gas pressure of the intracluster medium (ICM).
Given the large number of HI-deficient galaxies in clusters, it is reasonable to wonder why NGC 4522 should be such a good case for observing ongoing stripping. We believe that both intrinsic properties and favorable viewing circumstances are responsible. The galaxy is especially susceptible to stripping, due to its high speed and low mass. Its radial velocity of 2330 $\kms$ is on the high end of the distribution for Virgo galaxies, implying that it has a velocity of at least 1280 $\kms$ with respect to the mean cluster velocity of 1050 $\kms$ (Binggeli, Popescu, & Tammann 1993). There must also be a significant velocity component in the plane of the sky to cause the extraplanar filaments in this nearly edge-on galaxy. The high velocity ensures a strong ICM-ISM interaction, since ram pressure is proportional to the square of the velocity. Long-slit H$\alpha$ spectroscopy along the disk plane shows a fairly normal, rising rotation curve, with V$_{\rm rot}$=103 $\kms$ at a radius of 42$''$ (Rubin, Waterman & Kenney 1998). The fairly low mass of the galaxy indicated by this rotational speed implies that the ISM of NGC 4522 will be less tightly bound to the galaxy than in brighter, more massive spirals. NGC 4522 also has a favorable viewing angle. Since it is close to edge-on, it is easier to see extraplanar gas. In addition, we are likely to be viewing it at a particularly favorable time, perhaps on its first passage through the dense part of the ICM.
Star Formation in NGC 4522
==========================
There has been much discussion in the literature of galaxies with star formation rates enhanced due to triggering by ICM-ISM interactions (Gavazzi 1995; Dressler & Gunn 1983). The star formation properties of NGC 4522 can be assessed with Figure 4, which shows normalized H$\alpha$ luminosities versus central light concentrations for samples of Virgo cluster (bottom) and isolated (top) spiral galaxies. In this figure, which is adapted from Koopmann & Kenney (1998a), the concentration parameter C30 is the flux ratio of the R-band light within 0.3R$_{24}$ to that within R$_{24}$, and is an objective tracer of the stellar bulge-to-disk ratio. The H$\alpha$ luminosity normalized by the the R-band luminosity is a measure of the present-day massive star formation rate divided by the past-average star formation rate. Figures 4a and b show the [*global*]{} H$\alpha$ to R luminosity ratio, Figures 4c and d show this ratio for the [*inner disk*]{}, i.e., within 0.3R$_{24}$, and Figures 4e and f show this ratio for the [*outer disk*]{}, i.e., the annulus from 0.3-1.0R$_{24}$. The dotted lines in each panel bound the values found for the isolated spirals (but not S0’s).
Figure 4b shows that the global L(H$\alpha$)/L(R) for NGC 4522 is slightly below average, compared with both isolated and Virgo spirals of the same central light concentration. Since the H$\alpha$ luminosities are not corrected for internal extinction, and this correction should be higher for highly inclined galaxies like NGC 4522, the true global NMSFR for NGC 4522 is probably close to average. Its modest infrared-to-optical luminosity ratio L$_{\rm FIR}$/L$_{\rm opt}$=0.5 (Table 1) suggests that the extinction correction is not too large. Figure 4d shows that the *central L(H$\alpha$)/L(R) for NGC 4522 is somewhat above average, compared with both isolated and Virgo spirals. Correcting for extinction would presumably elevate NGC 4522 even higher above the average. Thus it is plausible that the star formation rate in the center of NGC 4522 has been modestly enhanced by a factor of $\sim$2 by an ICM-ISM interaction. The other Virgo galaxies with similar or higher central star formation rates show evidence for some kind of interaction (Koopmann & Kenney 1998b). Figure 4f shows that the outer disk L(H$\alpha$)/L(R) for NGC 4522 is well below average. Thus a lack of star formation in the outer disk, together with moderately enhanced star formation in the inner disk, combine to produce an average global NMSFR.*
Approximately 10% of the global H$\alpha$ flux arises from HII regions which appear to be located above the disk plane, in gas presumably stripped from the disk. This corresponds to a total extraplanar star formation rate of 0.01 M$\solar$ yr$^{-1}$, according to the assumptions of Kennicutt (1983). The stars which form in this outflowing gas will enter the galaxy halo or intracluster space, depending on whether the gas is bound to the galaxy at the time when the stars form. This may be a source of newly formed intracluster stars, which is relevant in light of the recent discovery of intracluster stars in the Virgo cluster (Ferguson, Tanvir & von Hippel 1998).
This star formation rate is probably much lower than the gas stripping rate. NGC 4522 has an HI mass of 0.4$\times$10$^9$ M$\solar$, compared to the average value of 1.4$\times$10$^9$ M$\solar$ for an isolated spiral galaxy of the same optical diameter (Giovanelli & Haynes 1983), suggesting that $\sim$10$^9$ M$\solar$ of gas been stripped from the galaxy. If this gas is stripped over a cluster crossing time of 10$^9$ yrs, this corresponds to an average stripping rate of 1 M$\solar$ yr$^{-1}$. This is significantly larger than both the global SFR of 0.1 M$\solar$ yr$^{-1}$, and the extraplanar SFR of 0.01 M$\solar$ yr$^{-1}$. Thus most of the stripped gas is not converted into stars by the ICM-ISM interaction. While in galaxy disks gas can continually and quickly cycle through phases and return to dense star-forming gas clouds, this probably doesn’t happen in the stripped gas. Instead, most of the stripped gas likely escapes the galaxy, where it is heated by and then joins the hot intracluster medium, enriching its abundance of heavy elements. The effect of this ICM-ISM interaction on galaxy evolution is that stellar diskbuilding is truncated in time in the outer disk, but is normal or perhaps even somewhat accelerated in the inner disk, if the inner star formation rate is enhanced. Integrated over time, the outer disks never achieve their maximum stellar mass surface density, whereas inner disks are little affected.
Is Molecular Gas Stripped?
==========================
The absence of HII regions in the outer disk, and the existence of HII regions in the extraplanar gaseous filaments suggests that even molecular gas has been removed from the outer disk as a consequence of stripping. According to the ram pressure stripping criterion of Gunn & Gott (1972), gas is stripped if the ram pressure $\rho$v$^2$ exceeds the gravitational force per unit area G$\sigma _{\rm gas}$ $\sigma _{\rm tot}$/2$\pi$ binding gas to the disk. Thus gas at a surface density of $\sigma _{\rm gas}$=20 M$_{\sun}$ pc$^{-2}$ would be stripped from the disk of NGC 4522 at r=5 kpc, if the galaxy is moving at v=1500 km s$^{-1}$ wrt the ICM, and the ICM density is $\rho$=10$^{-4}$ cm$^{-3}$ (Fabricant & Gorenstein 1983). This assumes a spherical galaxy mass distribution, and a rotation speed (in the plane of the galaxy) of 103 km s$^{-1}$ at r=5 kpc (Rubin 1998).
The value of $\sigma _{\rm gas}$=20 M$_{\sun}$ pc$^{-2}$ resulting from this simple calculation is nearly an order of magnitude lower than the gas (H$_2$+He) surface densities of 170 M$_{\sun }$ pc$^{-2}$ found for inner disk (1$\leq$R$\leq$8 kpc) Milky Way GMCs (Larson 1981; Solomon 1987), but within a factor of 4 of the value of 80 M$_{\sun }$ pc$^{-2}$ found for the outer disk (8$\leq$R$\leq$24 kpc) Milky Way clouds described by Heyer (1998). Molecular cloud surface densities are unlikely to be a universal constant, since they are clearly much higher than 170 M$_{\sun }$ pc$^{-2}$ in the centers of many galaxies (Kenney 1993; Scoville 1997), and may be lower than this in the outer disks of galaxies. We cannot say whether the stripped gas surface density is as low as 20 M$_{\sun}$ pc$^{-2}$ in NGC 4522, but other effects may allow the effective stripped surface density to be higher than this.
One possibile effect is that NGC 4522 is currently passing through a region with above average ICM density, although neither Einstein nor ROSAT maps show any evidence for enhanced x-ray emission near NGC 4522 (Fabbiano 1991; Snowden 1997).
It might be possible to strip clouds with surface densitities of 100-200 M$_{\sun }$ pc$^{-2}$ with the same ram pressure, if the effective area for stripping were much larger than an individual molecular cloud, so that the appropriate gas surface density in the ram pressure stripping equation were correspondingly lower. This might occur, for example, if magnetic fields coupled GMCs to the surrounding intercloud medium. If the effective length scale for stripping were 1 kpc instead of 100 pc, then the strippable gas surface density might be an order of magnitude lower.
It also may be possible to strip the outer disk of all gas and star formation, even without directly stripping GMCs. GMCs may be short-lived, and the gas may be stripped when the gas is in a non-molecular, lower density phase. GMCs which form massive stars are likely to be short-lived, with lifetimes of $\sim$10$^7$ yrs (Elmegreen 1991; Larson 1994), due to the enormous energy input into the cloud from massive stars and supernovae. If most of the GMC mass is cycled through a low surface density phase over timescales shorter than a galaxy rotation period (5$\times$10$^7$ yrs at r=5 kpc in NGC 4522), then most of the ISM at a given radius can be stripped, if the ram pressure stripping equation holds for the minimum surface density through which most of the ISM cycles. While gas cycling is surely an important factor affecting the response of a galaxy’s ISM to an interaction with the ICM, this cannot be the sole factor in the stripping of NGC 4522’s star-forming clouds, unless the HII regions which exist in the extraplanar gas formed from gas which became dense after being stripped.
Although most HI-deficient Virgo cluster (Kenney & Young 1989; Kenney 1990) and the 3 peculiar A1367 cluster spirals (Gavazzi 1995) have relatively normal CO luminosities (Boselli 1994) , only the outer disk is stripped of gas in spiral galaxies, meaning that only a modest fraction of the total H$_2$ is typically lost due to stripping. Stripping only outer disk gas changes the global CO luminosity of spirals by only a modest amount. There are many post-stripped spirals in Virgo with relatively normal massive star formation rates in the inner disk, and virtually no massive star formation in the outer disks (Koopmann & Kenney 1998b). Such galaxies could have modest reductions in CO luminosity, without being noticeable. The most severely stripped cluster galaxies could be classified as S0’s, and the CO normalcy of this heterogeneous group is unknown.
Comparisons between NGC 4522 and Other Candidate Strippers
==========================================================
NGC 4522 exhibits several interesting differences from other ICM-ISM stripping candidates, particularly the three candidates in A1367. In NGC 4522, HII regions are clearly located above the disk plane in the wake of the interaction. While the global SFR in NGC 4522 is normal to reduced, the three A1367 stripping candidates have strongly enhanced SFRs (Gavazzi 1995). NGC 4522 is HI deficient, while the three A1367 galaxies have nearly normal HI content (Dickey & Gavazzi 1991). Their HI distributions seem to be off-center, although the existing HI maps have low S/N (Dickey & Gavazzi 1991). These galaxies, particularly 97073, resemble the Virgo Cluster spiral NGC 4654 more closely than NGC 4522. While NGC 4654 has an HI content and star formation rate typical for a late-type spiral, its HI distribution has a head-tail morphology, and on the opposite side of galaxy from the HI tail there is a curved ridge of bright HII regions forming the edge of its outer disk (Phookun & Mundy 1995). Perhaps the main difference between NGC 4654 and the three A1367 galaxies is the ICM density, which is nearly an order of magnitude larger in A1367 (Gavazzi 1995). This may account for the strongly enhanced SFRs in the A1367 galaxies, compared to the modest enhancements in the SFRs of NGC 4654 and other Virgo galaxies.
NGC 4522 is already HI-deficient and has a shrunken star forming disk, suggesting that stripping has been going on for a longer time in NGC 4522 than NGC 4654 and the three A1367 galaxies. There are a number of Virgo spiral galaxies with truncated star forming disks, which are likely to be undergoing or to have undergone ICM-ISM stripping: NGC 4064, NGC 4380, NGC 4405, NGC 4413, NGC 4580, and IC 3392 (Koopmann & Kenney 1998a,b). NGC 4522 has a higher star formation rate than the other truncated spirals identified in Koopmann & Kenney (1998b), suggesting that it might be at an intermediate evolutionary phase in ICM stripping, more advanced than the 3 A1367 galaxies and NGC 4654, yet less advanced than the Virgo spirals with severely truncated star forming disks.
Future Work
===========
While its optical morphology alone strongly suggests that NGC 4522 is experiencing ICM-ISM stripping, HI mapping will clearly reveal much more about this galaxy’s interaction with its environment. One question among many is whether there is any evidence for a shock at the ICM-ISM interface. There is no strong x-ray emission from NGC 4522 or any other non-Seyfert Virgo spiral, or from any of the 3 A1367 cluster spirals (Gavazzi 1995), suggesting that strong x-ray emission is not produced by ICM-ISM stripping in spirals.
Acknowledgements
=================
We are grateful to Wendy Hughes and Orion Peck for processing some of the optical images, Mark Heyer and Bev Smith for helpful discussions, and Steve Snowden for providing a ROSAT x-ray map prior to publication. Thanks especially to the WIYN staff for their excellent support at the telescope. This work has been partially supported by NSF grant AST-9322779.
Barnes, J. E., & Hernquist, L. 1992, ARAA, 30, 705 Binggeli, B., Sandage, A., & Tammann, G.A. 1985, AJ, 90, 1681 Binggeli, B., Popescu, & Tammann, G. A. 1993, AAS, 98, 275 Böhringer, H., Briel, U. G., Schwarz, R. A., Voges, W., Hartner, G., & Trümper, J. 1994, Nature, 368, 828 Boselli, A., Gavazzi, G., Combes, F., Lequeux, J., & Casoli, F., 1994, AA, 285, 69 Cayatte, V., van Gorkom, J. H., Balkowski, C., & Kotanyi, C. 1990, AJ, 100, 604 Dickey, J. M., & Gavazzi, G. 1991, ApJ 373, 347 Dressel, L. L. & Condon, J. J. 1976, ApJS, 31, 187 deVaucouleurs, G., deVaucouleurs, A., Corwin, H. G., Buta, R. J., Paturel, G., Fouqué, P. 1991, [*Third Reference Catalog of Bright Galaxies*]{}, (New York: Springer-Verlag) (RC3) Dressler, A., Oemler, A., Couch, W. J., Smail, I., Ellis, R. S., Barger, A., Butcher, H., Poggianti, B. M., & Sharples, R. M. 1997, ApJ, 490, 577 Elmegreen, B. G. 1991, in The Physics of Star Formation and Early Stellar Evolution, eds., C. J. Lada & N. D. Kylafis, (Kluwer, Dordrecht), p. 35 Fabbiano, G., Kim, D.-W., & Trinchieri, G. 1992, ApJS, 80, 531 Fabricant, D. & Gorenstein, P. 1983, ApJ, 267, 535 Ferguson, H. C., Tanvir, N. R., & von Hippel, T. 1998, Nature, 391, 461 Forman, W., Schwarz, J., Jones, C., Liller, W., & Fabian, A. C. 1979, ApJ, 234, L27 Freedman, W. L. 1994, Nature, 371, 757 Gallagher, J. S. 1978, ApJ, 223, 386 Gavazzi, G., Conttursi, A., Carrasco, L., Boselli, A., Kennicutt, R., Scodeggio, M., & Jaffe, W. 1995, AA 304, 325 Giovanelli, R. & Haynes, M. P., 1983, AJ, 88, 881 Gruendl, R. A., Vogel, S. N., Davis, D. S., & Mulchaey, J. S. 1993, ApJ, 413, L81 Gunn, J. E., & Gott, J. R., 1972, ApJ, 176, 1 Haynes, M. P., & Giovanelli, R. 1986, ApJ, 306, 466 Heckman, T. M., Armus, L., & Miley, G. K. 1990, ApJS, 574, 833 Heckman, T. M., Baum, S. A., van Breugel, W. J. M., & McCarthy, P. 1989, ApJ, 338, 48 Helou, G., Hoffman,G. L. & Salpeter, E. E. 1984, ApJS, 55, 433 Heyer, M. H., Brunt, C., Snell, R. L., Howe, J. E., Schloerb, F. P., & Carpenter, J. M. 1998, ApJS, 115, 241 Irwin, J. A., & Sarazin, C. L. 1996, ApJ, 471, 683 Jacoby, G. H. 1992, PASP, 104, 599 Jogee, S., Kenney, J. D. P., & Smith, B. 1998, ApJ, 494, L185 Kenney, J. D. P. 1990, in [*The Interstellar Medium in Galaxies*]{}, H. A. Thronson & Shull, J.M. (eds.) (Dordrecht: Kluwer), p. 151 Kenney, J. D. P., Carlstrom, J., & Young, J. S. 1993, ApJ, 418, 687 Kenney, J. D. P., Rubin, V. C., Planesas, P., & Young, J. S. 1995, ApJ, 438, 135 Kenney, J. D. P., & Young, J. S. 1989, ApJ, 344, 171 Kennicutt, R. C. 1983, AJ, 88, 483 Koopmann, R. A. & Kenney, J. D. P. 1998a, ApJ, 497, L75 Koopmann, R. A. & Kenney, J. D. P. 1998b, in prep. Koopmann, R. A., Kenney, J. D. P., Young, J. 1998, in prep. Kotanyi, C. G. 1980, AAS, 41, 421 Kotanyi, C. G., van Gorkom, J. H., & Ekers, R. 1983, ApJ, 273, L7 Larson, R. B. 1981, MNRAS, 194, 809 Larson, R. B. 1994, in Structure and Content of Molecular Clouds, eds. T. L. Wilson & K. J. Johnston (Springer-Verlag, Berlin), p. 13 Lynds, R. 1970, ApJ, 159, L151 Malphrus, B.. K., Simpson, C. E., Gottesman, S. T., & Hawarden, T. G. 1997, AJ, 114, 1427 McMahon, P. M., Richter, O.-G., van Gorkom, J. H., & Ferguson, H. C. 1992, AJ, 103, 399 Moore, B., Katz, N., Lake, G., Dressler, A., & Oemler, A., Jr. 1996, Nature, 379, 613 Mulchaey, J. S., Davis, D. S., Mushotzky, R. F., & Burstein, D. 1993, ApJ, 404, L9 Pence, W. D., & Rots, A. H., 1997, ApJ, 478, 107 Phookun, B., & Mundy, L. G. 1995, ApJ, 453, 154 Pildis, R. A., Bregman, J. N., & Schombert, J. M. 1994, ApJ, 427, 160 Rand, R. J., 1997, in The Interstellar Medium in Galaxies, ed. J. M. van der Hulst, (Kluwer: Dordrecht), p. 105 Rand, R. J., Kulkarni, S. R., & Hester, J. J. 1990, ApJ, 352, L1 Rubin, V. C., Waterman, A. H., & Kenney, J. D. P. 1998, ApJ, in prep Ryder, S. D., Purcell, G., Davis, D., & Andersen, V. 1997, Publ.Astron.Soc.Aust. 14, 81 Sandage, A. & Bedke, J. 1994, [*The Carnegie Atlas of Galaxies*]{}, (Washington: Carnegie) Scoville, N. Z., Yun, M. S., & Bryant, P. M. 1997, ApJ, 484, 702 Shostak, G. S., Hummel, E., Shaver, P. A., van der Hulst, J. M., & van der Kruit, P. C. 1982, AA, 115, 293 Smith, B. J. & Madden, S. C. 1997, 114, 138 Snowden, S. 1997, private communication Solomon, P. M., Rivolo, A. R., Barrett, J., & Yahil, A. 1987, ApJ, 319, 730 Trinchieri, G., Kim, D.-W., Fabbiano, G., & Canizares, C. R. C. 1994, ApJ, 428, 555 van den Bergh, S., Pierce, M., & Tully, R. B. 1990, ApJ, 359, 4 Warmels, R., 1988, AASupp, 72, 19 White, D. A., Fabian, A. C., Forman, W., Jones, C., & Stern, C. 1991, ApJ, 375, 35 Yasuda, N., Fukugita, M., & Okamura, S. 1997, ApJS, 108, 417
|
---
abstract: 'The taxicab number, $1729$, is the smallest number that can be written as a sum of two cubes in two different ways. It also has the following property: if we add its digits we obtain $19$. The number obtained from $19$ reversing the order of its digits is $91$. If we multiply $19$ by $91$ we obtain again $1729$. In the paper we study various generalizations of this property.'
---
\[theorem\][Corollary]{} \[theorem\][Lemma]{} \[theorem\][Proposition]{}
\[theorem\][Definition]{} \[theorem\][Question]{} \[theorem\][Example]{} \[theorem\][Conjecture]{}
\[theorem\][Remark]{}
Viorel Niţică\
Department of Mathematics\
West Chester University of Pennsylvania\
West Chester, PA 19383\
USA\
<vnitica@wcupa.edu>\
.2 in
Introduction {#sec:1}
============
The *taxicab number,* $1729$, became well known due to a discussion between Hardy and Ramanujan [@Hardy]. It is the smallest positive integer that can be written in two ways as a sum of two cubes: $1^3+12^3$ and $9^3+10^3$. The number $1729$ also has a less well known property: if we add its digits we obtain $19$; multiplying $19$ by $91$, the number obtained from 19 reversing the order of its digits, we obtain again $1729$. It is not hard to show that the set of integers with this property is finite and equal to $\{1, 81, 1458, 1729\}$.
In a conversation that the author had with his colleague, Professor Shiv Gupta, Shiv asked if the second property can be generalized. One replaces the sum of the digits of an integer by the sum of the digits times an integer multiplier and then multiplies the product by the number obtained reversing the order of the digits in the product. The taxicab number becomes a particular example with multiplier 1. A computer search produced a large number of examples with larger multiplier. There are 23 integers less than 10000 having this property; see sequence [[](http://oeis.org/A305131)]{} in the OEIS [@online]. For example, 2268 has multiplier 2. The sum of the digits is $18$, one has $18\times 2=36$, and $36\times 63=2268$.
One may replace the last product in the above procedure by a sum. A computer search showed that there are numbers that have the property for sums. There are 264 integers less than 10000 having the property; see sequence [[](http://oeis.org/A305130)]{} in the OEIS [@online]. For example, $121212$ has multiplier $6734$. The sum of the digits is $9$, one has $9\times 6732=60606$, and $60606+60606=121212$.
The paper is dedicated to the study of these properties. After the paper was submitted for publication we learned from the editor that our work may be related to the study of Niven (or Harshad) numbers. These are numbers divisible by the sum of their decimal digits. Niven numbers have been extensively studied as one can see for instance from Cai [@C], Cooper and Kennedy [@CK], De Koninck and Doyon [@KD], Grundman [@G]. One of the classes of integers we study, that of multiplicative Ramanujan-Hardy numbers, is a subclass of the class of Niven numbers. Of interest are also $q$-Niven numbers, which are numbers divisible by the sum of their base $q$ digits. See, for example, Fredricksen, Ionaşcu, Luca, and Stănică [@FILS]. Some other variants of Niven numbers can be found in Boscaro [@B1] and Bloem [@B2].
Statements of the main results {#sec:1-bis}
==============================
In what follows let $b\ge 2$ be an arbitrary numeration base.
If $N$ is a positive integer written in base $b$, we call *reversal* of $N$ and let $N^R$ denote the integer obtained from $N$ by writing its digits in reverse order.
We observe that addition and multiplication are independent of the numeration base. The operation of taking the reversal is not.
Let $s_b(N)$ denote the sum, done in base 10, of the base $b$ digits of an integer $N$.
\[def:1\] A positive integer $N$ written in base $b$ is called *$b$-additive Ramanujan-Hardy number,* or simply $b$-ARH number, if there exists a positive integer $M$, called *additive multiplier*, such that $$\label{eq:1}
N=Ms_b(N)+(Ms_b(N))^R,$$ where $(Ms_b(N))^R$ is the reversal of base $b$-representation of $Ms_b(N)$.
\[def:2\] A positive integer $N$ written in base $b$ is called *$b$-multiplicative Ramanujan-Hardy number,* or simply $b$-MRH number, if there exists a positive integer $M$, called *multiplicative multiplier*, such that $$\label{eq:2}
N=Ms_b(N)\cdot (Ms_b(n))^R,$$ where $(Ms_b(N))^R$ is the reversal of base $b$-representation of $Ms_b(N)$.
To simplify the notation, let $s(N)$, ARH, MRH denote $s_{10}(N)$, 10-ARH, 10-MRH.
While $b$-MRH numbers are $b$-Niven numbers, $b$-Niven numbers are not necessarily $b$-MRH numbers.
The number $[144]_7$ is a $7$-Niven number but not a $7$-MRH number.
We observe that the notions of $b$-ARH and $b$-MRH numbers are dependent on the base.
The number $[12]_{10}$ is an ARH number, but $[12]_{9}$ is not a $9$-ARH number. The number $[81]_{10}$ is an MRH number, but $[81]_{9}$ is not a $9$-MRH number.
Once these notions are introduced, several natural questions arise.
\[q:1\] Do there exist infinitely many $b$-ARH numbers?
\[q:2\] Do there exist infinitely many $b$-MRH numbers?
\[q:3\] Do there exist infinitely many additive multipliers?
\[q:4\] Do there exist infinitely many multiplicative multipliers?
In what follows, if $x$ is a string of digits, we let $(x)^{\land k}$ denote the base 10 integer obtained by repeating $x$ $k$-times. We also let $[x]_b$ denote the value of the string $x$ in base $b$.
The following example gives an explicit positive answer to Question \[q:1\] if $b=10$.
\[thm:1\] Consider the numbers $$\label{eq:3}
N_k=(12)^{\land 3^k},$$ where $k$ is a positive integer. All numbers $N_k$ are ARH numbers and Niven numbers. In particular, there exist infinitely many Niven numbers with no digit equal to zero.
\[r:1\] If we allow zero digits an infinity of $b$-MRH numbers is given by $\{[1(0)^{\land k}]_b\vert k\in \mathbb{N}\}$. Last example has the unpleasant feature that the apparent multiplicative multiplier of each $b$-MRH numbers is the number itself and the search for other multipliers is dependent on the base. In order to avoid trivial considerations, we consider from now on only examples of $b$-ARH and $b$-MRH numbers that have many digits different from zero.
It follows from the proof of Example \[thm:1\] that $Ms(N_k)=(Ms(N_k))^R$. The following theorem gives an example in which it is clear from the proof that $Ms_b(N_k)\not =(Ms_b(N_k))^R$ for an arbitrary even base $b$. One can read from the proof the explicit base $b$ expansion of the multipliers. Counting the multipliers shows that the set of multipliers of a $b$-ARH number $N$ can grow exponentially in terms of the number of digits of $N$.
\[thm:11\] Consider the numbers $$\label{eq:3**}
N_k=[(1)^{\land k}]_b,$$ where $b$ is even, $k=[1(0)^{\land p}]_b, p\ge 1$, $p$ an arbitrary natural number. All numbers $N_k$ are $b$-ARH numbers and not $b$-Niven numbers.
Each $N_k$ has a subset of additive multipliers of cardinality $2^{\frac{k-2p}{2}}$ consisting of all integers $[(1)^{\land p}I]_b$, where $I$ is a sequence of $0$ and $1$ of length $k-2p$ in which no two digits symmetric about the center of the sequence are identical.
We show an example that illustrates the results in Theorem \[thm:11\]. Assume that $b=2$, $k=16=[10000]_2$, and $p=4$. Then $N_{16}=[(1)^{\land 16}]_2$ and $s_2(N_{16})=2^4=[10000]_2$. The following $16=2^{\frac{16-2\cdot 4}{2}}$ numbers are additive multipliers of $N_{16}$: $$\begin{gathered}
\ [111100001111]_2,\ [111100010111]_2,\ [111100101011]_2,\ [111100111100]_2,\\
\ [111101001101]_2,\ [111101010101]_2,\ [111101101001]_2,\ [111101110001]_2,\\
\ [111110001110]_2,\ [111110010110]_2,\ [111110101010]_2,\ [111110110010]_2,\\
\ [111111001100]_2,\ [111111010100]_2,\ [111111101000]_2,\ [111111110000]_2.
\end{gathered}$$
The numbers $N_k$ may have other multipliers, besides those listed in Theorem \[thm:11\]. The growth of the set of multipliers can be larger than that shown in Theorem \[thm:11\] and depends on the numeration base; see Theorem \[thm:larger-growth\]. Nevertheless, for $b=2$ there are no other multipliers of $N_k$ besides those listed in Theorem \[thm:11\]. We observe that the numbers $N_k$ from Theorem \[thm:11\] have an even number of digits and the numbers $N_k$ from Theorem \[thm:larger-growth\] have an odd number of digits.
\[thm:larger-growth\] Consider the numbers $$\label{eq:3larger-growth}
N_k=[(1)^{\land p}(10)^{\land k-2p}0(1)^{\land p}]_b,$$ where $b$ is even and $k=[1(0)^{\land p}]_b, p\ge 1,$ $p$ arbitrary natural number. All numbers $N_k$ are $b$-ARH numbers and not $b$-Niven numbers.
For each $N_k$ the set of additive multipliers has cardinality $(b-1)^\frac{k-2p}{2}$ and consists of all integers $[(1)^{\land p}I0]_b$, where $I$ is a concatenation of $k-2p$ two digits strings of type $0\alpha, \alpha\not = 0$, in which any pair of nonzero digits symmetric about the center of $I0$ have their sum equal to $b$.
We show an example that illustrates the results in Theorem \[thm:larger-growth\]. Assume that $b=4$, $k=4=[10]_4$, and $p=1$. Then $N_{4}=[1101001]_4$ and $s_4(N_{4})=4=[10]_4$. The following $3=3^\frac{4-2\cdot 1}{2}$ numbers are additive multipliers of $N_{4}$: $$\ [102020]_4, \ [101030]_4, \ [103010]_4.$$
The following corollary of Theorem \[thm:2\] gives a partial answer to Question \[q:3\].
If $b$ is even there exist infinitely many additive multipliers. Moreover, there exists infinitely many $b$-ARH numbers that have at least two additive multipliers.
The numbers $N_k$ from Theorems \[thm:11\] and \[thm:larger-growth\] are not $b$-MRH numbers.
Do there exist infinitely many $b$-MRH numbers that have at least two multiplicative multipliers?
If $b$ is even there exist infinitely many $b$-ARH numbers that are not $b$-MRH.
Motivated by the results in Theorems \[thm:11\] and \[thm:larger-growth\] and by the examples of ARH and MRH numbers shown in Sections \[sec:7\] and \[sec:8\], we introduce the following notions.
If $N$ is a $b$-ARH number, let the *multiplicity* of $N$ be the cardinality of the corresponding set of additive multipliers.
If $N$ is a $b$-MRH number, let the *multiplicity* of $N$ be the cardinality of the corresponding set of multiplicative multipliers.
Theorem \[thm:11\] has the following corollary.
The multiplicity of $b$-ARH numbers is unbounded for any even base.
\[q:13\] Is the multiplicity of $b$-MRH numbers bounded?
\[r:2\] For Questions \[q:2\] and \[q:4\] we do not have an answer with $b$-MRH numbers having all digits different from zero. See Theorem \[thm:mrhexample\] for an infinity of $b$-MRH numbers with half of the digits different from zero. No prime number can be an MRH number. Note that no integer with two prime factors in the prime factorization can be an MRH number. Such an MRH number has the multiplier equal to $1$ and among the MRH numbers with multiplier $1$ none has two factors in the prime factorization.
The following theorem shows an infinity of $b$-Niven numbers that are not $b$-MRH numbers.
\[thm:niven-not-mrh\] Let $b\ge 2$ be a numeration base. For $n$ not divisible by $b-1$ define $$R_n=\frac{b^n-1}{b-1}=[(1)^{\land n}]_{b}, n\ge 1.$$ Then $(b-1)nR_n$ is a $b$-Niven number that is not a $b$-MRH number.
For $b$-ARH numbers one has the following result.
\[thm:noARH\] There exist infinitely many integers that are not $b$-ARH numbers.
The following Theorem gives a partial answer to Question \[q:2\].
\[thm:mrhexample\] Let $b$ odd and $k\ge 2$. Then the numbers $$\label{eq:mrhex}
N_k=[(b-1)^{\land 2^{k-1}-1}(b-2)(0)^{\land 2^{k-1}-1}1]_b$$ are $b$-MRH numbers and $s_b(\sqrt{N_k})=s_b(N_k)$.
Moreover, if $b\equiv 3 \pmod {4}$ then $\sqrt{N_k}$ is itself a $b$-Niven number.
We illustrate the result in Theorem \[thm:mrhexample\].
- For $b=3, k=2$ we get $N_2=[2101]_3$ which is a $3$-MRH number. Then $\sqrt{[2101]_3}=[22]_3$, $s_3([2101]_3)=s_3([22]_3)=4$ and $[22]_3$ is a $3$-Niven number.
- For $b=5, k=2$ we get $N_2=[4301]_5$ which is a $5$-MRH number. Then $\sqrt{[4301]_5}=[44]_5$, $s_5([4301]_5)=s([44]_5)=8$ and $[44]_5$ is a $5$-Niven number.
- For $b=17, k=5$, $N_5$ is a $17$-MRH number, but $\sqrt{N_5}$ is not a $17$-Niven number.
- For $b=7, k=2$ we get $N_2=[6501]_7$ which is a $7$-MRH number. Then $\sqrt{[6501]_7}=[66]_7$, $s_7([6501]_7)=s_7([66]_7)=12$ and $[66]_7$ is a $7$-Niven number.
Third item shows that the congruence condition in Theorem \[thm:mrhexample\] is necessary. Second item shows that $\sqrt{N_k}$ may be a $b$-Niven number even without this condition.
The following corollary of Theorem \[thm:mrhexample\] gives a partial positive answer to Question \[q:4\].
If $b$ is odd there exist infinitely many multiplicative multipliers.
We show two unexpected corollaries of the proof of Theorem \[thm:mrhexample\].
If $b$ is odd there exist infinitely many $b$-MRH numbers that are perfect squares.
\[coro:number\] If $b\equiv 3\pmod{4}$ there exists an infinity of $b$-MRH numbers $N$ for which $\sqrt{N}$ is a $b$-Niven number and for which $s_b(N)=s_b(\sqrt{N})$.
The following notions of high degree $b$-Niven numbers are motivated by Corollary \[coro:number\], which provides plenty of examples.
An integer $N$ is called *quadratic $b$-Niven number* if $N$ and $N^2$ are $b$-Niven numbers. If in addition $s_b(N)=s_b(N^2)$ then $N$ is called *strongly quadratic $b$-Niven number*.
The study of high degree $b$-Niven numbers is continued in Niţică [@N]. We show that for each degree there exists an infinity of bases in which $b$-Niven numbers of that degree appear.
We show in Sections \[sec:7\] that $6$ is not an additive multiplier for base $10$ and ARH numbers without zero digits, and that $9$ is not an additive multiplier for base $10$. We show in Section \[sec:8\] that $3$ is not a multiplicative multiplier for base $10$. We do not know how to answer the following questions for any base.
\[q:6\] Do there exist infinitely many integers that are not additive multipliers?
\[q:7\] Do there exist infinitely many integers that are not multiplicative multipliers?
In what follows let $\lfloor x \rfloor$ denote the integer part, let $\ln x$ denote the natural logarithm and let $\log_b x$ denote base $b$ logarithm of the positive real number $x$.
The following theorems give bounds for the number of digits in a $b$-ARH number in terms of the multiplier.
\[thm:2\] Let $N$ be a $b$-ARH number with $k$ digits and additive multiplier $M$. Then $$k\leq \begin{cases}
M+2, & \text{ if } b\ge 4;\\
M+3, & \text{ if } b=2 \text{ or } b=3.
\end{cases}$$
For fixed additive multiplier $M$ and base $b$, the set of $b$-ARH numbers with multiplier $M$ is finite.
\[thm:2-strong\] Let $N$ be a $b$-ARH number with $k$ digits and additive multiplier $M$. Under any of the following assumptions:
- $b \ge 10$ and $M\ge b^6;$
- $3\le b \le 9$ and $M\ge b^7;$
- $b=2$ and $M\ge b^8,$
one has $$\label{eq:4-Strong}
\begin{gathered}
k\le 2\lfloor \log_b M \rfloor.
\end{gathered}$$
The following theorems give bounds for the number of digits in a $b$-MRH number in terms of the multiplier.
\[thm:3\] Let $N$ be a $b$-MRH number with $k$ digits and multiplicative multiplier $M$. Then $$k\leq \begin{cases}
M+4, & \text{ if } b\ge 6;\\
M+5, & \text{ if } b= 5;\\
M+7, & \text{if } 2\le b\le 4.
\end{cases}$$
Theorem \[thm:3\] shows that a MRH number with multiplicity $1$ can have at most 5 digits. A computer search shows that the set of all such numbers is indeed $\{1,81, 1458, 1729\}$.
For fixed multiplicative multiplier $M$ and base $b$, the set of $b$-MRH numbers with multiplier $M$ is finite.
\[thm:3-new\] Let $N$ be a $b$-MRH number with $k$ digits and multiplicative multiplier $M$. Under any of the following assumptions:
- $b \ge 9$ and $M\ge b^9;$
- $5\le b \le 8$ and $M\ge b^{10};$
- $b=4$ and $M\ge b^{11};$
- $b=3$ and $M\ge b^{12};$
- $b=2$ and $M\ge b^{16};$
one has $$\label{eq:5-new}
k\le 3\lfloor \log_b M\rfloor.$$
We summarize the rest of the paper. Example \[thm:1\] is proved in Section \[sec:2\], Theorem \[thm:11\] is proved in Section \[sec:2bis\], Theorem \[eq:3larger-growth\] is proved in Section \[sec:3larger-growth\], Theorem \[thm:noARH\] is proved in Section \[sec:noARH\], Theorem \[thm:niven-not-mrh\] is proved in Section \[sec:niven-not-mrh\], Theorem \[thm:mrhexample\] is proved in Section \[sec:mrhexample\], Theorem \[thm:2\] is proved in Section \[sec:3\], Theorem \[thm:2-strong\] is proved in Section \[sec:4\], Theorem \[thm:3\] is proved in Section \[sec:5\], and Theorem \[thm:3-new\] is proved in Section \[sec:6\]. In Section \[sec:7\] we show examples of ARH numbers and ask additional questions and in Section \[sec:8\] we show examples of MRH numbers and ask additional questions. In Section \[sec:9\] we describe an approach to Question \[q:2\] if $b=10$.
Proof of Example \[thm:1\] {#sec:2}
==========================
One obtains a formula for $N_k$ by adding two geometric series. $$\label{eq:5}
\begin{gathered}
N_k=10^{2\cdot 3^k-1}+10^{2\cdot 3^k-3}+\ldots +10\\
+2(10^{2\cdot 3^k-2}+10^{2\cdot 3^k-4}+\ldots +1)\\
=12\cdot \frac{10^{2\cdot 3^k}-1}{99}=4\cdot \frac{10^{2\cdot 3^k}-1}{33}.
\end{gathered}$$
Note that $s(N_k)=3^{k+1}$. We show by induction that $s(N_k)$ divides $N_k$. The case $k=0$ gives $s(N_0)=3$ which divides $N_0=12$. Assume that for fixed $k$, $s(N_k)$ divides $N_k$.
$$\label{eq:6}
\begin{gathered}
N_{k+1}=4\cdot \frac{10^{2\cdot 3^{k+1}}-1}{33}=4\cdot \frac{(10^{2\cdot 3^k})^3-1^3}{33}\\
=4\cdot \frac{10^{2\cdot 3^k}-1}{33}\cdot (10^{4\cdot 3^k}+10^{2\cdot 3^k}+1)=N_k\cdot (10^{4\cdot 3^k}+10^{2\cdot 3^k}+1),
\end{gathered}$$
which is clearly divisible by $s(N_{k+1})=3^{k+2}$ due to $N_k$ divisible by $s(N_k)=3^{k+1}$ and $10^{4\cdot 3^k}+10^{2\cdot 3^k}+1$ divisible by $3$. Therefore $s(N_k)$ divides $N_k$ and $N_k$ is a Niven number.
Observe now that $N_k/2=(N_k/2)^R$. It follows from and the fact that $N_k$ is divisible by $s(N_k)=3^{k+1}$ that $N_k/2$ is divisible by $s(N_k)$. We conclude that $N_k$ is an ARH number with additive multiplier $M=N_k/(2s(N_k))$.
Proof of Theorem \[thm:11\] {#sec:2bis}
===========================
Let $N_k=[(1)^{\land k}]_b$ where $k$ is even and $k=[1(0)^{\land p}]_b, p\ge 1$, $p$ arbitrary natural number. Then $s_b(N_k)=[1(0)^{\land p}]_b$. Let $M=[(1)^{\land p}I]_b$, where $I$ is a string of $0$ and $1$ of length $k-2p$ in which no two digits symmetric about the center of the sequence are identical. Note that $M^R=[(I)^R(1)^{\land p}]_b$. The following calculation shows that $N_k$ is a $b$-ARH number. Note that $I+(I)^R=[(1)^{\land k-2p}]_b$.
$$\begin{gathered}
s_b(N_k)\cdot M+(s_b(N_k)\cdot M)^R\\
=[1(0)^{\land p}]_b\cdot [(1)^{\land p}I]_b+([1(0)^{\land p}]_b\cdot [(1)^{\land p}I]_b)^R\\
=[(1)^{\land p}I(0)^{\land p}]_b+([(1)^{\land p}I(0)^{\land p}]_b)^R\\
=[(1)^{\land p}I(0)^{\land p}]_b+[(0)^{\land p}(I)^R(1)^{\land p}]_b=[(1)^{\land k}]_b=N_k.
\end{gathered}$$
In order to count the multipliers, observe that the length of the string $I$ is $k-2p$. If we know half of its digits we can find the other half using that no two digits symmetric about the center of the string are identical. The number of strings of $0$ and $1$ of length $\frac{k-2p}{2}$ is $2^{\frac{k-2p}{2}}$. Finally, observe that $N_k$ is not divisible by $s_b(N_k)$, so $N_k$ is not a $b$-Niven number..
Proof of Theorem \[thm:larger-growth\] {#sec:3larger-growth}
======================================
Let $N_k=[(1)^{\land p}(10)^{\land k-2p}0(1)^{\land p}]_b$ where $b$ is even and $k=[1(0)^{\land p}]_b, p\ge 1$. Then $s_b(N_k)=[1(0)^{\land p}]_b$. Let $M=[(1)^{\land p}I0]_b$. Note that $M^R=[0(I)^R(1)^{\land p}]_b$. The following calculation shows that $N_k$ is a $b$-ARH number. Note that $I0+0(I)^R=[(10)^{\land k-2p}0]_b$.
$$\begin{gathered}
s_b(N_k)\cdot M+(s_b(N_k)\cdot M)^R\\
=[1(0)^{\land p}]_b\cdot [(1)^{\land p}I0]_b+([1(0)^{\land p}]_b\cdot [(1)^{\land p}I0]_b)^R\\
=[(1)^{\land p}I0(0)^{\land p}]_b+([(1)^{\land p}I0(0)^{\land p}]_b)^R\\
=[(1)^{\land p}I0(0)^{\land p}]_b+[(0)^{\land p}0(I)^R(1)^{\land p}]_b=[(1)^{\land p}(10)^{\land k-2p}0(1)^{\land p}]_b=N_k.
\end{gathered}$$
In order to count the multipliers, observe that the number of nonzero digits in the string $I0$ is $k-2p$. If we know half of the nonzero digits we can find the other half using that no two digits symmetric about the center of the string $I0$ are identical. There are $\frac{k-2p}{2}$ positions to be filled and each one can be filled in $b-1$ ways. To show that there are no other multiplier it is enough to prove, using induction on length, that the string $[(10)^{\land k-2p}0]_b$ cannot be written as a sum of a string $J$ and its reversal except if $J=I0$, where $I$ is as above. Finally, observe that $N_k$ is not divisible by $s_b(N_k)$, so $N_k$ is not a $b$-Niven number.
Proof of Theorem \[thm:niven-not-mrh\] {#sec:niven-not-mrh}
======================================
McDaniels proved [@D Theorem 2] that if $b=10$ and $m \le 9R_n$ then $s(9mR_n)=9n$. The proof is valid in any base $b$ and follows readily upon writing $m$ as: $$m=\sum_{i=0}^k a_ib^i, k<n.$$ It gives that if $m\le (b-1)R_n$ then $s_b((b-1)mR_n)=(b-1)n$. If $m=n$ one has $s_b((b-1)nR_n)=(b-1)n$, which shows that $(b-1)nR_n$ is a $b$-Niven number. By contradiction, assume that $(b-1)nR_n$ is a $b$-MRH number with multiplier $M$. It follows that: $$\label{eq:contrad}
(b-1)nM((b-1)nM)^R=(b-1)nR_n.$$
We recall that a base $b$ number is divisible by $b-1$ if the sum of its base $b$ digits is divisible by $b-1$. The divisibility test and $b-1\not \vert n$ imply that $b-1\not\vert R_n$, but $b-1\vert ((b-1)nM)^R$. As $b-1\not\vert n$, there are at least two factors of $b-1$ in the factorization of the left hand side of and only one factor of $b-1$ in the right hand side of . This gives a contradiction.
Proof of Theorem \[thm:noARH\] {#sec:noARH}
==============================
A $b$-ARH number is a sum of an integer and its reversal. In order to prove the theorem it is enough to show that there exist infinitely many integers that are not a sum of an integer and its reversal. There are $b^{k}-b^{k-1}=b^{k-1}(b-1)$ base $b$ $k$-digit numbers. Those of type $N+N^R$, either have $N=[a_ka_{k-1}\cdots a_2a_1]_b$ with $a_k+a_1\le b-1$, or have $N$ with $k-1$ digits. There are $\frac{b(b-1)}{2}\cdot b^{k-2}$ $k$-digit numbers with $a_k+a_1\le b-1$ and there are $b^{k-1}-b^{k-2}$ $(k-1)$-digit numbers. Overall, we have $$\frac{b(b-1)}{2}\cdot b^{k-2}+(b^{k-1}-b^{k-2})=b^{k-1}\left ( \frac{b+1}{2}\right )-b^{k-2}$$ $k$-digit numbers of type $N+N^R$. Hence there are $$\label{eq:new-add1}
b^k-b^{k-1}-\left ( b^{k-1}\left ( \frac{b+1}{2}\right )-b^{k-2}\right )=b^{k-1}\left ( \frac{b-3}{2}\right )+b^{k-2}$$ $k$-digit numbers that are not of type $N+N^R$. The right hand side of equation has limit $\infty$ as $k\to \infty$ for $b\ge3$ and this ends the proof if $b\ge 3$. If $b=2$, consider the numbers $[(1)^{\land k}]_2$. These are not ARH-numbers if $k$ is odd.
Proof of Theorem \[thm:mrhexample\] {#sec:mrhexample}
===================================
As $\gcd(b,2)=1$ Euler’s Theorem implies that $2^k$ divides $b^{\phi(2^k)}-1$. Clearly $b-1$ also divides $b^{\phi(2^k)}-1$. Assume that $\text{gcd}(2^k,b-1)=2^\ell$. Then $2^{k-\ell}(b-1)$ divides $b^{\phi(2^k)}-1=b^{2^{k-1}}-1$. Consider the product $$(b^{2^{k-1}}-1)^2=b^{2\cdot2^{k-1}}-2b^{2^{k-1}}+1.$$ The product is divisible by $2^{k-1}(b-1)$, written in base $b$ equals $N_k$, and $s_b(N_k)=2^{k-1}(b-1)$. We conclude that $N_k$ is a $b$-MRH number.
To finish the proof of the theorem observe that if $b\equiv 3\pmod{4}$ then $\text{gcd}(2^k,b-1)=2$. Therefore $2^{k-1}(b-1)$ divides $b^{2^k}-1=[(b-1)^{2^k-1}]_b$. Finally $$s_b(\sqrt{N_k})=s_b([(b-1)^{2^k-1}]_b)=2^{k-1}(b-1)\vert b^{2^k}-1=\sqrt{N_k}.$$
Proof of Theorem \[thm:2\] {#sec:3}
==========================
As $N$ has $k$ digits one has that: $$\label{eq:7}
N\ge b^{k-1}.$$
The largest possible value for $s_b(N)$ is $(b-1)k$. We observe that reversing the order of the digits in an integer increases its value by at most $b$ times. One has that: $$\label{eq:8}
Ms_b(N)+(Ms_b(N))^R\le (b^2-1)kM.$$
Combining equations , , one has that: $$\label{eq:9}
b^{k-1} \le (b^2-1)kM.$$
We prove by induction on the variable $k$ that: $$\label{eq:10}
b^{k-1} > (b^2-1)kM, \text{ for }k\ge M+3, M\ge 1, b\ge 4,$$ which combined with gives a contradiction and ends the proof of Theorem \[thm:2\] for $b\ge 4$.
In the first step $k=M+3$. The statement in becomes $$\label{eq:11}
b^{M+2} > (b^2-1)(M^2+3M), \text{ for }M\ge 1, b\ge 4.$$
We prove by induction on the variable $M$. In the initial step $M=1$ and one has $$b^3 > 4(b^2-1) \Leftrightarrow b^2(b-4)+4 > 0,$$ which is clearly true for $b\ge 4$.
Now assume that is true for $M$ and prove it for $M+1$. Using the induction hypothesis one has that: $$\label{eq:12}
b^{M+3}=b\cdot b^{M+2}> b\cdot (b^2-1)(M^2+3M).$$
In order to finish the proof by induction, we still need to check that: $$\label{eq:13}
b\cdot (b^2-1)(M^2+3M)\ge (b^2-1)\left ( (M+1)^2+3(M+1)\right ).$$
After simplifications, becomes $$\label{eq:14}
(b-1)M^2+(3b-5)M-4\ge 0.$$
As the left hand side of is larger than $M^2+4M-4$, which is positive if $M\ge 2$, we conclude that is true for all $M\ge 1$ and finish the proof of .
We continue with the general step in the proof of . By induction: $$\label{eqn:ind1}
b^k=b\cdot b^{k-1}>b(b^2-1)kM.$$
We still need to check that $$\label{eqn:ind2}
b(b^2-1)kM\ge (b^2-1)k(M+1),$$ which is obvious and finishes the proof of and that of Theorem \[thm:2\] for base $b\ge 4$.
Now assume $b=3$. Equation is still valid.
We prove by induction on the variable $k$ that: $$\label{eq:some-new-1}
b^{k-1} > (b^2-1)kM, \text{ for }k\ge M+4, M\ge 1.$$ Equations and give a contradiction that finishes the proof of the theorem.
If $k=M+4$ one has that: $$\label{eq:some-new-2}
b^{M+3} > (b^2-1)(M^2+4M), \text{ for } M\ge 1,$$ which we prove by induction on $M$.
The case $M=1$ is true. We assume true for $M$ and prove it for $M+1$. By induction one has that: $$b^{M+4}=b\cdot b^{M+3} > b(b^2-1)(M^2+4M).$$
To finish the proof of we still need to check that: $$\label{eq:1corec}
b(b^2-1)(M^2+4M)\ge (b^2-1)\left ( (M+1)^2+4(M+1)\right ),$$ which simplifies to $(b-1)M^2+(4b-6)M-5\ge 0$ and is true for $M\ge 1, b=3$.
The rest of the proof of follows from and .
Assume $b=2$. Equation is still valid.
We prove by induction on the variable $k$ that: $$\label{eq:some-new-6}
b^{k-1} > (b^2-1)kM, \text{ for }k\ge M+4, M\ge 3.$$ Equations and give a contradiction that ends the proof of the theorem for $b=2, M\ge 3$.
If $k=M+4$ one has that: $$\label{eq:some-new-60}
2^{M+3} > 3(M^2+4M), \text{ for } M\ge 3,$$ which we prove by induction on $M$.
The case $M=3$ is true. Assume now that is true for $M$ and prove it for $M+1$.
$$2^{M+4}=2\cdot 2^{M+3} > 6(M^2+4M).$$
To finish the proof we still need to check that: $$6(M^2+4M)\ge 3\big ( (M+1)^2+4M\big ).$$ The equation simplifies to $M^2+2M-1\ge 0$ and it is true for $M\ge 3$.
To finish the proof of the theorem if $b=2$, it remains to discuss the cases $M=1, M=2$.
Let $M=1$. If $k\le 4$ the theorem is trivially true, so assume $k\ge 5$. Let $N$ be a $2$-ARH number with $k$ digits and $M=1$. Then $s_2(N)\le k$ and $N\ge 2^{k-1}$. This implies $$2^{k-1}\le 3k.$$ One shows that $3k<2^{k-1}$ for $k\ge 5$ and gets a contradiction.
Let $M=2$. If $k\le 4$ the theorem is trivially true, so assume $k\ge 5$. Let $N$ be a $2$-ARH number with $k$ digits and $M=2$. Then $s_2(N)\le k$ and $N\ge 2^{k-1}$. This implies $$2^{k-1}\le 6k.$$ One shows that $6k<2^{k-1}$ for $k\ge 5$ and gets a contradiction.
Proof of Theorem \[thm:2-strong\] {#sec:4}
=================================
It follows from formula in the proof of Theorem \[thm:2\] that: $$\label{eq:1-strong}
b^{k-1}\le (b^2-1)kM.$$
We show by induction on the variable $k$ that: $$\label{eq:2-strong}
b^{k-1}>(b^2-1)kM\text{ if }M\ge b^6, k\ge 2\lfloor \log_b M \rfloor+1, b\ge 10$$ which together with ends the proof of Theorem \[thm:2-strong\] for base $b\ge 10$.
First we show by induction on the variable $M$ that: $$\label{eq:3-strong}
M >2 b^2(b^2-1)\log_b M+b^2(b^2-1) \text{ if }M\ge b^5, b\ge 10.$$
If $M=b^6$ is equivalent to $$\label{equ:crucial}
b^3+13b(1-b^2)> 0,$$ which is true if $b\ge 10$.
Now assume that is true for a fixed $M$. One has $$M+1> 2b^2(b^2-1)\log_b M+b^(b^2-1)+1.$$
To finish the proof of we still need to check that: $$2b^2(b^2-1)\log_b M+b^2(b^2-1)+1\ge 2b^2(b^2-1)\log_b (M+1)+b^2(b^2-1),$$ which after simplifications becomes $$1\ge 2b^2(b^2-1)\left ( \log_b(M+1)-\log_b M\right ),$$ which is true due to $M\ge b^5$ and the Mean Value Theorem.
We start the proof of . In the first step $k=2\lfloor\log_b M\rfloor+1$ and becomes $$\label{eq:4-strong}
b^{2\lfloor \log_b M\rfloor}>(b^2-1)M(2\lfloor\log_b M\rfloor+1).$$
Due to $\log_b M-1\le \lfloor \log_b M\rfloor \le \log_b M$ one has $$\label{eq:5-strong}
\begin{gathered}
b^{2\lfloor\log_b M\rfloor}\ge b^{2(\log_b M-1)}\\
(b^2-1)M(2\lfloor\log_b M\rfloor+1)\le (b^2-1)M(2\log_b M+1).
\end{gathered}$$
In order to prove it is enough to show that $$b^{2(\log_b M-1)}> (b^2-1)M(2\log_b M+1),$$ which is equivalent to . This ends the proof of the first induction step.
Now assume that is true for fixed $k$ and show that it is true for $k+1$. Due to the induction hypothesis one has that: $$b^k\ge b\cdot (b^2-1)kM.$$
To finish the proof of we still need to check that $$b\cdot (b^2-1)kM > b(b^2-1)(k+1)M,$$ which is obviously true.
The proofs of the other cases are similar. The only significant difference appears in . If $3\le b\le 9$, becomes $b^4-15(b^2-1)\ge 0$, which is true. If $b=2$ becomes $b^6>17(b^2-1)$, which is true.
Proof of Theorem \[thm:3\] {#sec:5}
==========================
As $N$ has $k$ digits one has that: $$\label{eq:15}
N\ge b^{k-1}.$$
The largest possible value for $s_b(N)$ is $(b-1)k$. Reversing the order of the digits in an integer increases its value by at most $b$ times. One has that: $$\label{eq:16}
Ms_b(N)\cdot (Ms_b(N))^R\le b(b-1)^2k^2M^2.$$
Combining equations , , one has that: $$\label{eq:17}
b^{k-1}\le b(b-1)^2k^2M^2.$$
Now we prove by induction on the variable $k$ that: $$\label{eq:18}
b^{k-1} > b(b-1)^2k^2M^2, \text{ for }k\ge M+5, M\ge 1, b\ge 6$$ which combined with ends the proof of Theorem \[thm:3\] for $b\ge 6$.
In the initial induction step $k=M+5$. The statement in becomes $$\label{eq:19}
b^{M+4} > b(b-1)^2(M+5)^2M^2, \text{ for }M\ge 1, b\ge 6.$$
We prove by induction on the variable $M$. If $M=1$ becomes $b^5 > 36b(b-1)^2$, which is true if $b\ge 6$.
Now we assume that is true for $M$ and prove it for $M+1$. From the induction hypothesis one has that: $$\label{eq:20}
b^{M+5}=b\cdot b^{M+4} > b\cdot b(b-1)^2(M+5)^2M^2.$$
In order to finish the proof, we still need to check that: $$\label{eq:21}
b\cdot b(b-1)^2(M+5)^2M^2\ge b(b-1)^2(M+6)^2(M+1)^2$$ for $M\ge 1$.
After simplifications, becomes $$\label{eq:22}
(b-1)M^4+(10b-14)M^3+(25b-61)M^2-84M-36\ge 0,$$ which is true for $M\ge 1$ and $b\ge 6$.
This finishes the proof of .
We continue with the general step in the proof of . By induction $$b^k=b\cdot b^{k-1}>b\cdot b(b-1)^2k^2M^2.$$
To finish the proof of we still need to check that $$b\cdot b(b-1)^2k^2M^2\ge b(b-1)^2(k+1)^2M^2,$$ which after simplifications becomes $(b-1)k^2-2k-1\ge 0$. This is true if $k\ge 1$ and $b\ge 6$.
This finishes the proof of Theorem \[thm:3\] for $b\ge 6$.
The proof of the case $b=5$ is similar. The only significant changes appear in and in . Equation simplifies to $b^5 > 49(b-1)^2$, which is true for $b= 5$, and becomes $$(b-1)M^4+(12b-16)M^3+(36b-78)M^2-112M-49\ge 0,$$ which is true if $b=5$.
If $2\le b\le 4$, using $M\ge 1$, the statement in the theorem is true if $k\le 8$. We can assume $k\ge 9$.
If $b=4$ is still true and gives $4^{k-2}\le 9k^2M^2$. It is easy to show by induction that for $k\ge 9$ one has $4^{k-2}>9k^2(k-8)^2$. If $k\ge M+8$ this implies $4^{k-2}> 9k^2M^2$, a contradiction.
If $b=3$, is still true and gives $3^{k-2}\le 4k^2M^2$. It is easy to show by induction that for $k\ge 9$ one has $3^{k-2}>4k^2(k-8)^2$. If $k\ge M+8$ this implies $3^{k-2}> 8k^2M^2$, a contradiction.
If $b=2$, is still true and gives $2^{k-2}\le k^2M^2$. It is easy to show by induction that for $k\ge 9$ one has $2^{k-2}>k^2(k-8)^2$. If $k\ge M+8$ this implies $2^{k-2}> 8k^2M^2$, a contradiction.
Proof of Theorem \[thm:3-new\] {#sec:6}
==============================
It follows from formula in the proof of Theorem \[thm:3\] that: $$\label{eq:1*}
b^{k-1}\le b (b-1)^2k^2M^2.$$
We prove by induction on the variable $k$ that: $$\label{eq:2*}
b^{k-1}> b(b-1)^2k^2M^2\text{ for }M\ge b^9, k\ge 3\lfloor \log_b M\rfloor +1, b\ge 9,$$ which combined with finishes the proof of Theorem \[thm:3-new\].
We start showing by induction on $M$ that: $$\label{eq:3*}
M> (b-1)^2b^4(3\log_b M+1)^2\text{for }M\ge b^9, b\ge 9
.$$
If $M=b^9$ becomes, after cancellations, $$\label{eq:changes13}
b^5>28^2(b-1)^2$$ which is true for $b\ge 9$.
We assume now that is true for fixed $M$. We show that it is true for $M+1$. From the induction hypothesis one has that: $$M+1> (b-1)^2b^4(3\log_b M+1)^2+1.$$
To finish the proof of , one still needs to check that: $$(b-1)^2b^4(3\log_b M+1)^2+1\ge (b-1)^2b^4\left ( 3\log_b(M+1)+1\right )^2,$$ which after algebraic manipulations becomes $$\label{eq:4*}
1\ge b^4(b-1)^2(3\log_b(M+1)-3\log_b M)(3\log_b (M+1)M+2).$$
Due to the Mean Value Theorem, follows if we show that: $$\label{eq:4*bis}
1\ge 3 b^4(b-1)^2\cdot\frac{1}{M}\left ( 3\log_b(M^2+M)+2\right ).$$
Consider the function $g(M)=\frac{1}{M}[3\log_b(M^2+M)+2]$, with derivative: $$g'(M)=\frac{\frac{1}{\ln b}\cdot\frac{3}{M^2+M}\cdot(2M+1)M-\left ( 3\log_b(M^2+M)+2\right )}{M^2}.$$
For $M\ge b^9$ the first term in the numerator of $g'(M)$ is $\le 6$ and the absolute value of the second term is $\ge 30$. We conclude that $g'(M)$ is negative and $g(M)$ is decreasing on the interval $[b^9,+\infty)$. The value of $$\label{eq:lt1}
3\cdot b^4(b-1)^2\cdot g(M)$$ for $M=b^9$ is larger than $\frac{168(b-1)^2}{b^5}$, which shows that is true if $b\ge 9$. Consequently and are true.
We start the proof of . In the first step $k=3\lfloor \log_b M\rfloor+1$. Equation becomes $$\label{eq:11*}
b^{3\lfloor \log_b M\rfloor}>(b-1)^2b(3\lfloor \log_b M\rfloor+1)^2M^2.$$
Due to $\log_b M-1\le \lfloor \log_b M\rfloor \le \log_b M$, follows if we prove that $$\label{eq:12*}
b^{3(\log_b M-1)}\ge (b-1)^2bM^2(3\log_b M+1)^2\text{ for }M\ge b^9.$$
After algebraic manipulations is exactly , so it is true. Now we show the general induction step for . Assume valid for fixed $M$. Then one has that: $$b^k\ge b\cdot b^{k-1}\ge b\cdot (b-1)^2bk^2M^2.$$
To finish we still need to check that: $$b\cdot (b-1)^2bk^2M^2\ge (b-1)^2b(k+1)^2M^2,$$ which after simplifications becomes $bk^2\ge k^2+2k+1$ which is true for $k\ge 1$. This ends the proof of the case $b\ge 9$.
The proofs of the other cases are similar and the only significant changes appear in checking the equation and checking that expression is less than $1$. Due to our assumptions the equation remains valid and the expression is less than $1$.
Examples of ARH numbers {#sec:7}
=======================
$M$ $N$
----- ----------------------------
1 18, 99
2 12, 33, 66, 99
3 99
4 99
5 11,22,33,44,55,66,77,88,99
6
7 747
: ARH numbers with multipliers 1, 2, 3, 4, 5, 7 and without zero digits.[]{data-label="t:1"}
We list in Table \[t:1\] small additive multipliers $M$ and the corresponding ARH numbers $N$ without zero digits. Theorem \[thm:2\] shows that an ARH number with multiplier $6$ has at most $8$ digits. A computer search through all integers with at most 8 digits and all digits different from zero, shows that $6$ is not an additive multiplier for numbers with all digits different from zero. If we allow for zero digits one finds that $909$ is an ARH number with multiplier $6$. A computer search through all integers with at most $11$ digits shows that $9$ is not an additive multiplier. These observations motivate Question \[q:6\].
We observe that certain ARH numbers, for example $99$, have several additive multipliers, respectively $1,2,3,4,5$. We also observe that certain multipliers, for example $5$, have associated several ARH numbers, respectively $11,22,33,44,55,66,77,88,99$. The last observation motivates the following definition and questions.
If $M$ is an additive multiplier in a base $b$, let the *multiplicity* of $M$ be the cardinality of the corresponding set of $b$-ARH numbers.
\[q:9\] If we fix the multiplicity and the base, is the set of additive multipliers infinite?
\[q:10-bis\] If we fix the base, is the multiplicity of additive multipliers bounded?
Examples of MRH numbers {#sec:8}
=======================
We list in Table \[t:2\] small multiplicative multipliers $M$ and the corresponding MRH numbers $N$. Theorem \[thm:3\] shows that a MRH number with multiplier $3$ has at most $7$ digits. A computer search through all integers with at most 7 digits shows that $3$ is not a multiplicative multiplier. This motivates Question \[q:7\].
$M$ $N$
----- -------------------
1 1, 18, 1458, 1729
2 2268, 736
3
4 1944, 7744
5 71685
: MRH numbers with multipliers $1,2,3,4,5$ and without zero digits.[]{data-label="t:2"}
One can also arrange the data as in Table \[t:3\], where, for small values of $k$, we list multiplicative multipliers $M$ and the corresponding MRH numbers $N$ with $k$ digits.
$k$ $M$ $N$ $k$ $M$ $N$
----- ----- ---------------- ----- ----- --------------------
1 1 1 7 22 9379678
2 1 81 28 6527836
3 2 736 29 9253987
4 1 1458, 1729 32 2892672
2 2268 33 8673885
4 1944, 7744 34 7526716
5 5 71685 38 3773932, 6362226
7 23632 39 5673564
8 94528 41 2187391
9 42282 49 4274613, 8239644
14 51142 63 1821771
23 78246 72 7651584
6 12 132192 73 2895472
14 188356, 247324 82 7651584
19 161595 84 3252312
21 433755, 496692 8 37 13184839
22 234256 46 11361448
23 685584 48 14292288
26 258778 53 15437628
27 332424 61 15178752
29 679354 66 15995232
31 122512 89 7331464
33 176418 66 15995232
34 132192, 751842 68 11715516
36 271188 71 16746912
37 215821 74 12419568, 15478432
38 332424 75 19348875
39 145314 76 17433792
44 235224 77 19552995
78 12661272, 22694256
79 11437225
86 21371688
89 12918439
: MRH numbers with $1,2,3,4,5,6, 7, 8$ digits and no zero digits.[]{data-label="t:3"}
We observe from Table \[t:3\] that certain MRH numbers, for example, $332424$, $132192$, and $3252312$, have several multipliers (respectively $\{27, 38\}$, $\{12, 34\}$, $\{72, 82\})$. We also observe from Table \[t:2\] that certain multipliers, for example $4$, have associated several MRH numbers, respectively $1944, 7744$. The last observation motivates the following definition and questions.
If $M$ is a multiplicative multiplier in base $b$, let the *multiplicity* of $M$ be the cardinality of the corresponding set of $b$-MRH numbers.
\[q:12\] If we fix the multiplicity and the base, is the set of multiplicative multipliers infinite?
\[q:13-bis\] If we fix the base, is the multiplicity of multiplicative multipliers bounded?
Conclusion {#sec:9}
==========
In this paper for any numeration base $b$ we introduce two new classes of integers, $b$-ARH numbers and $b$-MRH numbers. They have properties that generalize a property of the taxicab number $1729$. The second class is a subclass of the class of $b$-Niven numbers. We ask several natural questions about these classes and partially answer some of them. In particular, we show that the class of $b$-ARH numbers is infinite if $b$ is even and that the class of $b$-MRH numbers is infinite if $b$ is odd.
Among the questions left open, the most intriguing is if the set of MRH numbers with all digits different from zero is infinite. One way to attack it is to find an infinity of integers $N$ such that $N=N^R$, $N$ is divisible by $s(N^2)$, and $N^2$ has no digit equal to zero. Then the squares are an infinity of MRH numbers with nonzero digits. Our data shows some examples of such integers.
- $N^2=188356=434^2, s(N^2)=31| 434$,
- $N^2=234256=484^2, s(N^2)=22| 484$,
- $N^2=685584=828^2, s(N^2)=36| 828$.
Acknowledgments
===============
The author would like to thank the editor and the referee for valuable comments that helped him write a better paper. He also thanks the OEIS Wiki community for help with posting the sequences A305130 and A305131 on OEIS.
[99]{}
S. Boscaro, Nivenmorphic integers, [*J. Rec. Math.*]{}, [**28**]{} (1996–1997), 201–205.
E. Bloem, Harshad numbers, [*J. Rec. Math.*]{}, [**34**]{} (2005), 128.
T. Cai, On $2$-Niven numbers and $3$-Niven numbers, [*Fibonacci Quart.*]{}, [**34**]{} (1996), 118–120.
C. N. Cooper and R. E. Kennedy, On consecutive Niven numbers, [*Fibonacci Quart.*]{}, [**21**]{} (1993), 146–151.
W. L. McDaniel, The existence of infinitely many $k$-Smith numbers, [*Fibonacci Quart.*]{}, [**25**]{} (1987) 76–80.
J. M. De Koninck and N. Doyon, Large and small gaps between consecutive Niven numbers, [*J. Integer Seq.*]{}, [**6**]{} (2003), Article 03.2.5.
H. G. Grundman, Sequences of consecutive Niven numbers, [*Fibonacci Quart.*]{}, [**32**]{} (1994), 174–175.
H. Fredricksen, E. J. Ionaşcu, F. Luca, and P. Stănică, Remarks on a sequence of minimal Niven numbers, In: [*Sequences, Subsequences, and Consequences*]{}, Lec. Notes in Comp. Sci., Vol 4893. Springer, 2007, pp. 162–168.
G. H. Hardy, [*Ramanujan: Three Lectures on Subjects Suggested by his Life and Work*]{}, Chelsea, 1999.
N. J. A. Sloane, [*The On-Line Encyclopedia of Integer Sequences*]{}, <http://oeis.org>.
V. Niţică, High degree $b$-Niven numbers, preprint, 2018, <http://arxiv.org/abs/1807.02573>.
------------------------------------------------------------------------
2010 [*Mathematics Subject Classification*]{}: Primary 11B83; Secondary 11B99.
*Keywords:* base, $b$-Niven number, reversal, additive $b$-Ramanujan-Hardy number, multiplicative $b$-Ramanujan-Hardy number, high degree $b$-Niven number.
------------------------------------------------------------------------
(Concerned with sequences [[](http://oeis.org/A005349)]{}, [[](http://oeis.org/A067030)]{}, [[](http://oeis.org/A305130)]{}, and [[](http://oeis.org/A305131)]{}.)
------------------------------------------------------------------------
Received 2018; revised version received . Published in [*Journal of Integer Sequences*]{},
------------------------------------------------------------------------
Return to . .1in
|
---
abstract: |
In this article we construct three new families of surfaces of general type with $p_g=q =0, K^2=6 $, and seven new families of surfaces of general type with $p_g=q =1, K^2=6 $, realizing 10 new fundamental groups. We also show that these families correspond to pairwise distinct irreducible connected components of the Gieseker moduli space of surfaces of general type.
We achieve this using two different main ingredients. First we introduce a new class of surfaces, called generalized Burniat type surfaces, and we completely classify them (and the connected components of the moduli space containing them). Second, we introduce the notion of Bagnera-de Franchis varieties: these are the free quotients of an Abelian variety by a cyclic group (not consisting only of translations). For these we develop some basic results.
author:
- 'Ingrid Bauer, Fabrizio Catanese, Davide Frapporti'
title: 'Generalized Burniat Type surfaces and Bagnera-de Franchis varieties'
---
[^1]
addtoreset[equation]{}[section]{}
[*Dedicated to the memory of Kunihiko Kodaira with great admiration.*]{}
Introduction {#introduction .unnumbered}
============
The present paper continues, with new inputs, a research developed in a series of articles ([@bacat], [@bcg], [@BC10burniat2], [@BC11burniat1], [@keumnaie], [@BC12], [@4names], [@BC13burniat3], [@BC13]) and dedicated to the discovery of new surfaces of general type with geometric genus $p_g = 0$, to their classification, and to the description of their moduli spaces (see the survey article [@survey] for an account of what is known about surfaces wit $p_g=0$, related conjectures and results).
Indeed, in this article, we consider the more general case of surfaces of general type with $\chi = 1$, i.e., with $p_g = q$.
In the first part we focus again on the construction method originally due to Burniat (singular bidouble coverings), but in the reformulation done by Inoue (quotients by Abelian groups of exponent two), presenting it in a rather general fashion which shows how topological methods allow to describe explicitly connected components of moduli spaces. A first novelty here is a refined analysis of pencils of Del Pezzo surfaces admitting a certain group of symmetries, as we shall now explain.
In a more general approach (cf. [@BC13]) we consider quotients (cf. [@BC12] for the case of a free action, treated there in an even greater generality), by some group $G$ of the form $(\mathbb{Z}/m)^r$, of varieties $\hat{X}$ contained in a product of curves $\Pi_i C_i$, where each $C_i$ is a maximal Abelian cover of the projective line with Galois group of exponent $m$ and with fixed branch locus.
In the case $m=2$ there is a connection with the Burniat surfaces: these are surfaces of general type with invariants $p_g=0$ and $K^2 = 6,5,4,3,2$, whose birational models were constructed by Pol Burniat (cf. [@burniat]) in 1966 as singular bidouble covers of the projective plane. Later these surfaces were reconstructed by Inoue (cf. [@inoue]) as $G:=(\mathbb{Z}/2\mathbb{Z})^3$-quotients of a ($G$-invariant) hypersurface $\hat{X}$ of multidegree $(2,2,2)$ in a product of three elliptic curves.
While Inoue writes the (affine) equation of $\hat{X}$ in terms of the uniformizing parameters of the respective elliptic curves using a variant of the Weierstrass’ function (a Legendre function), we found it much more useful to write the elliptic curves as the complete intersection of two diagonal quadrics in three space.
This algebraic and systematic approach allows us, also with the aid of computer algebra, to find all the possible such constructions.
Our situation is as follows: we consider first the following diagram of quotient morphisms:
$$\label{burniatdiagr}
\xymatrix{
E_1 \times E_2 \times E_3 \ar[dd]_{\mathcal{H}':=(\mathbb{Z}/2)^3}^{\pi'}& E_1: = \{x_1^2 + x_2^2 + x_3^2 = 0, \ x_0^2 = a_1x_1^2 + a_2x_2^2\}\\
&E_2: = \{ u_1^2 + u_2^2 + u_3^2 = 0, \ u_0^2 = b_1u_1^2 + b_2u_2^2 \}\\
P_1:=\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1\ar[dd]^{\pi }_{\mathcal{H}:=((\mathbb{Z}/2)^2)^3} & E_3: = \{z_1^2 +z_2^2 +z_3^2 = 0, \ z_0^2 = c_1 z_1^2 + c_2z_2^2 \}\\
&\\
P_2:=\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1
}$$
where the map $\pi'$ is given by “forgetting” the variables $x_0,\, u_0,\, z_0$,\
the map $\pi$ is given by setting $x_j^2=y_j$, $u_j^2=v_j$, $z_j^2=w_j$, $j=1,2,3$, and where we view $P_2\subset (\mathbb P^2)^3$ as the subvariety defined by the equations $$y_1+y_2+y_3=0\, ,\, v_1+v_2+v_3=0\, ,\, w_1+w_2+w_3=0\,.$$ The Galois group for $\pi \circ \pi'$ is rather large, it is indeed $({\ensuremath{\mathbb{Z}}}/2 {\ensuremath{\mathbb{Z}}})^9 \cong \{ \pm 1\}^9$.
We consider then $P_1$ with homogeneous coordinates $((s_1:t_1),(s_2:t_2),(s_3:t_3))$ and for each ${\lambda}:=(\lambda_1, \ldots, \lambda_8)\in \mathbb C^8\setminus \{0\}$ we consider the hypersurface $Y_{\lambda}$ of multidegree $(1,1,1)$ in $P_1$ given by the multihomogeneous equation $$\begin{aligned}
\lambda_1 s_1s_2s_3+\lambda_2 s_1s_2t_3+\lambda_3 s_1t_2s_3+\lambda_4 s_1t_2t_3+\\
\lambda_5 t_1s_2s_3+\lambda_6 t_1s_2t_3+\lambda_7 t_1t_2s_3+\lambda_8 t_1t_2t_3=0.\nonumber\end{aligned}$$
We then classify the subgroups $H_1$ (resp. $H_0$) of $\mathcal{H} \cong ((\mathbb{Z}/ 2 \mathbb{Z})^2)^3$ which are isomorphic to $(\mathbb{Z}/ 2 \mathbb{Z})^2$ (resp. to $(\mathbb{Z}/ 2 \mathbb{Z})^3$) and satisfy the property that there is an irreducible Del Pezzo surface $Y_{\lambda}$ invariant under $H_1$ (resp. $H_0$).
We consider then $\hat{X}_{\lambda}:= (\pi')^{-1}(Y_{\lambda})$, which is then invariant under the subgroup $\mathcal G_1 \cong (\mathbb{Z} /2 \mathbb{Z})^5 \subset (\mathbb{Z} /2 \mathbb{Z})^9$ inverse image of $H_1$ (resp. $\mathcal G_0 \cong (\mathbb{Z} /2 \mathbb{Z})^6$). We determine in this article all the subgroups $G \cong (\mathbb{Z}/2\mathbb{Z})^3 \subset \mathcal G_1$ (resp. $\mathcal G_0$), having the property that $G$ acts freely on $\hat{X}_{\lambda}$.
This leads us to introduce a class of surfaces of general type, described by the following
Let $G \cong (\mathbb{Z} / 2 \mathbb{Z})^3 \leq \mathcal G_1$ (resp. $\mathcal G_0$) be such that $G$ acts freely on $\hat{X}_{\lambda}$. Then the minimal resolution $S$ of $X_{\lambda}:=\hat{X}_{\lambda} /G$ is called a [*generalized Burniat type surface*]{}.
With the help of the computer algebra system MAGMA (cf. [@MAGMA]) we can classify all generalized Burniat type surfaces (=GBT surfaces for short) and can prove the following (see Proposition \[onefam\] and Theorem \[fundgroup\])
1. There are 16 irreducible families of GBT surfaces. These have $K^2 = 6$ and $0 \leq p_g=q \leq 3$. The families are listed in Tables \[q0\] - \[q3\], and the dimension of the irreducible family is 4 in cases $\mathcal S_1$ and $\mathcal S_2$, and 3 otherwise.
2. Among the 16 families of generalized Burniat type surfaces four have $p_g=q=0$ (Table \[q0\]), eight have $p_g=q=1$ (Table \[q1\]), three have $p_g=q=2$ (Table \[q2\]) and one has $p_g=q=3$ (Table \[q3\]). Family $\mathcal S_2$ is the family of primary Burniat surfaces (the one due to Pol Burniat).
3. The fundamental groups of these families are pairwise non isomorphic, except that $\pi_1(S_{11})\cong\pi_1(S_{12})$ and $\pi_1(S_{14})\cong\pi_1(S_{15})$, where $S_j$ is in the family $\mathcal S_j$
4. The surfaces in the families $\mathcal S_1$, $\mathcal S_3$ and $\mathcal S_4$ realize new (i.e., up to now unknown) fundamental groups of surfaces with $p_g=0, K^2 = 6$, while the surfaces in the families $\mathcal S_5$-$\mathcal S_{11}$ realize new fundamental groups for surfaces with $p_g=q=1, K^2 = 6$.
5. In cases $\mathcal S_1$-$\mathcal S_{10}$, each family of generalized Burniat type surfaces maps with a generically finite morphism onto an irreducible connected component of the Gieseker moduli space of surfaces of general type.
We use indeed the techniques developed in [@BC12] to determine the irreducible connected components of the moduli space containing the generalized Burniat type surfaces. We do not spell out all the details in the cases $\mathcal S_{13}$-$\mathcal S_{16}$, since the surfaces that we obtain in this way are not new and have already been classified by other authors.
In cases $\mathcal S_1$-$\mathcal S_{10}$ we can apply the general results of [@BC12] concerning classical diagonal Inoue type varieties in order to describe the connected components of the moduli space containing the generalized Burniat type surfaces.
We then show that it is no coincidence that the fundamental groups of the families $\mathcal S_{11}$ and $\mathcal S_{12}$ in Table \[q1\] are isomorphic. These families of surfaces are shown to be contained in a larger irreducible family, which corresponds to another realization as Inoue type varieties. This is done via the concept of a Bagnera-de Franchis variety, which we define simply as the quotient of an Abelian variety $A$ by a nontrivial finite cyclic group $G$ acting freely on $A$ and not containing any translation.
We obtain in this way the following theorem
Define a Sicilian surface to be any minimal surface of general type $S$ such that
- $S$ has invariants $K_S^2 = 6$, $p_g(S) =q(S) = 1$,
- there exists an unramified double cover $ \hat{S} \rightarrow S$ with $ q (\hat{S}) = 3$,
- the Albanese morphism $ \hat{\alpha} \colon \hat{S} \rightarrow A = \operatorname{Alb}(\hat{S})$ is birational onto its image $Z$, a divisor in $A$ with $ Z^3 = 12$.
1\) Then the canonical model of $\hat{S}$ is isomorphic to $Z$, and the canonical model of $S$ is isomorphic to $Y = Z / (\mathbb{Z}/2 \mathbb{Z})$. $Y$ is a divisor in a Bagnera-de Franchis threefold $ X: = A/ G$, where $A = (A_1 \times A_2) / T$, $ G \cong T \cong \mathbb{Z}/2 \mathbb{Z}$, and where the action is as in (\[BCF\]).
2\) Sicilian surfaces exist, have an irreducible four dimensional moduli space, and their Albanese map $\alpha \colon S \rightarrow A_1 = A_1/ A_1[2]$ has general fibre a non hyperelliptic curve of genus $g=3$.
3\) A GBT surface is a Sicilian surface if and only if it is in the family $\mathcal S_{11}$ or $\mathcal S_{12}$.
4\) Any surface homotopically equivalent to a Sicilian surface is a Sicilian surface.
Indeed, one can replace the above assumption of homotopy equivalence by a weaker one, see Corollary \[he\].
In Section \[bdf\] we discuss the basic results of the theory of Bagnera-de Franchis varieties, and show how to describe concretely the effective divisors on them, thus solving in a special case one of the main technical difficulties in the general theory of Inoue type varieties, developed in [@BC12].
Inoue’s description of Burniat surfaces {#Inouedescription}
=======================================
We briefly recall the description of (primary) *Burniat surfaces* (those constructed by P. Burniat in [@burniat]) given by Inoue in [@inoue].
Inoue considers, for $j\in \{1,2,3\}$, a complex elliptic curve $E_j:=\mathbb{C}/\langle 1,\tau_j\rangle$ with uniformizing parameter $z_j$, and then the following three commuting involutions on the Abelian variety $A^0:=E_1\times E_2 \times E_3$:
$$\begin{array}{rcc}
g_1(z_1,z_2,z_3)= (-z_1+\frac{1}{2},z_2+\frac{1}{2}, z_3)\,, \\
g_2(z_1,z_2,z_3)= (z_1, -z_2+\frac{1}{2}, z_3+\frac{1}{2})\,, \\
g_3(z_1,z_2,z_3)= (z_1+\frac{1}{2},z_2, -z_3+\frac{1}{2})\,.
\end{array}$$
Note that $G:=\langle g_1, g_2,g_3\rangle\cong (\mathbb Z/ 2\mathbb Z)^3$.
Let $\mathcal{L}_j$, for $j=1,2,3$, be a Legendre function for $E_j$: $\mathcal{L}_j\colon E_j\rightarrow \mathbb{P}^1$, a meromorphic function which makes $E_j$ a double cover of ${\ensuremath{\mathbb{P}}}^1$ branched over the four distinct points: $\pm 1, \pm a_j\in \mathbb{P}^1\setminus \{0, \infty\}$.
It is well known that the following statements hold (see [@inoue Lemma 3-2] and [@BC11burniat1 Section 1] for an algebraic treatment):
- $\mathcal{L}_j(0)=1$, $\mathcal{L}_j(\frac{1}{2})=-1$, $\mathcal{L}_j(\frac{\tau_j}{2})=a_j$, $\mathcal{L}_j(\frac{\tau_j+1}{2})=-a_j$;
- set $b_j:=\mathcal{L}_j(\frac{\tau_j}{4})$: then $b_j^2=a_j$;
- $\frac{\mathrm d\mathcal L_j}{\mathrm d z_j}(z_j)=0$ if and only if $z_j\in\{0,\frac 12, \frac {\tau_j}2, \frac {\tau_j+1}2\}$ since these are the ramification points of $\mathcal{L}_j$.
Moreover, $$\begin{aligned}
&\mathcal{L}_j(z_j)=\mathcal{L}_j(z_j+1)=\mathcal{L}_j(z_j+\tau_j)=\mathcal{L}_j(-z_j)=
-\mathcal{L}_j\bigg(z_j+\dfrac 12\bigg),\nonumber \\
&\mathcal{L}_j\bigg(z_j+\dfrac{\tau_j}2\bigg)= \dfrac{a_j} {\mathcal L_j(z_j)}\,.\nonumber\end{aligned}$$
For $c \in \mathbb{C}\setminus\{0\}$, Inoue considers the surface $$\hat X_c:=\{[z_1,z_2,z_3]\in A^0 \mid \mathcal{L}_1(z_1)\mathcal{L}_2(z_2)\mathcal{L}_3(z_3)=c\}\,$$ inside the Abelian variety $A^0$. Then he shows:
- $\hat X_c$ is a hypersurface in $A^0$ of multidegree $(2,2,2)$ and is invariant under the action of $G$, $\forall c$.
- For a general choice of $c$, $\hat X_c$ is smooth, and $G$ acts freely on $\hat X_c$, whence $X_c:=\hat X_c/ G$ is a smooth minimal surface of general type with $p_g=0$ and $K^2=6$.
- For special values of $c$, the hypersurface $\hat X_c$ has $4,8,12,16$ nodes, which are isolated fixed points of $G$; in these cases the minimal resolution of singularities of $X_c:=\hat X_c/ G$ is a minimal surface of general type with $p_g=0$ and $K^2=5,4,3,2$.
The minimal resolution of singularities $S_c$ of $X_c$ is called a [*Burniat surface*]{}. If $X_c$ is already smooth, or equivalently if $K_{S_c}^2 = 6$, then $S_c$ is called a [*primary*]{} Burniat surface. For an extensive treatment of Burniat surfaces and their moduli spaces we refer to [@BC11burniat1], [@BC10burniat2], [@BC13burniat3].
Intersection of diagonal quadrics and [$(\mathbb Z/ 2\mathbb Z)^n$]{}-actions {#intersection}
==============================================================================
As already in [@BC13 Section 3], we exhibit $A^0$ as a Galois covering of $({\ensuremath{\mathbb{P}}}^1)^3$ with Galois group $ \cong ({\ensuremath{\mathbb{Z}}}/2)^9$. This is done via the following diagram.
The main purpose of this section is to find irreducible Del Pezzo surfaces in $P_1$ which are left invariant under large subgroups of the group ${{\mathcal H}}\cong ({\ensuremath{\mathbb{Z}}}/2)^6$.
$$\label{diag1}
\xymatrix{
E_1\times E_2\times E_3 \ar[dd]^{\pi'}_{\mathcal H':=(\mathbb Z/ 2\mathbb Z)^3} & E_1:=\{ x_1^2+x_2^2+x_3^2=0,
\quad x_0^2=a_1x_1^2+a_2x_2^2\}\\
& E_2:= \{u_1^2+u_2^2+u_3^2=0, \quad u_0^2=b_1u_1^2+b_2u_2^2\} \\
P_1:=\mathbb P^1\times\mathbb P^1\times\mathbb P^1 \ar[dd]^{\pi}_{\mathcal H:=((\mathbb Z/ 2\mathbb Z)^2)^3}&
E_3: = \{z_1^2+z_2^2+z_3^2=0, \quad z_0^2=c_1z_1^2+c_2z_2^2\}\\
&\\
P_2:=\mathbb P^1\times\mathbb P^1\times\mathbb P^1\\
}$$
The map $\pi'$ is given by “forgetting” the variables $x_0,\, u_0,\, z_0$, whereas the map $\pi$ is given by setting $x_j^2=y_j$, $u_j^2=v_j$, $z_j^2=w_j$, $j=1,2,3$, and viewing $P_2\subset (\mathbb P^2)^3$ as the subvariety defined by the equations $$y_1+y_2+y_3=0\, ,\, v_1+v_2+v_3=0\, ,\, w_1+w_2+w_3=0\,.$$ The Galois group for $\pi \circ \pi'$, is $({\ensuremath{\mathbb{Z}}}/2 {\ensuremath{\mathbb{Z}}})^9 \cong \{ \pm 1\}^9$.
Restricting diagram (\[diag1\]) to one (w.l.o.g. the first) factor we get:
$$\label{diag2}
\xymatrix{
E_1=E \ar[d]_{\mathbb Z/ 2\mathbb Z}&& \\
\mathbb P^1\ar[d]_{(\mathbb Z/ 2\mathbb Z)^2} &= &\{x_1^2+x_2^2+x_3^2=0\}=:C\subset \mathbb P^2\\
\mathbb P^1&= &\{y_1+y_2+y_3=0\}\subset \mathbb P^2
}$$
Since $$x_1^2+x_2^2+x_3^2=0 \Longleftrightarrow \det \left ( \begin{array}{cc}
x_1+ix_2 & -x_3\\
x_3 & x_1-i x_2
\end{array}\right)=0\,,$$ we get an isomorphism of $C$ with ${\ensuremath{\mathbb{P}}}^1$: $$(s:t)= (x_1+ix_2: x_3)= (-x_3: x_1-i x_2)\,$$ and a parametrization of $C$ $$(x_1 : x_2 : x_3) = (i (s^2 - t^2) :(s^2 + t^2) : 2 i st).$$
With this parametrization, we can rewrite the action of $(\mathbb Z/ 2\mathbb Z)^2$ on $\mathbb P^1$ in the following way (on the left hand side we use the convenient notation by which all variables not mentioned in a transformation are left unchanged by the transformation):
- $x_1\mapsto -x_1$ corresponds to $A_1\colon (s:t)\mapsto(t:s)$;
- $x_2\mapsto -x_2$ corresponds to $A_{-1}\colon (s:t)\mapsto(-t:s)$;
- $x_3\mapsto -x_3$ corresponds to $B\colon (s:t)\mapsto(s:-t)$.
The fixed points of these three involutions are respectively:
- $s=\pm t$, equivalently, $ x_1=x_3\pm i x_2=0$;
- $s=\pm i t$, equivalently, $ x_2=x_3\pm i x_1=0$;
- $st=0$, equivalently, $ x_3=x_1\pm i x_2=0$.
For each ${\lambda}:=(\lambda_1, \ldots, \lambda_8)\in \mathbb C^8\setminus \{0\}$ we consider the hypersurface $Y_{\lambda}$ of multidegree $(1,1,1)$ in $P_1 = \mathbb P^1_{(s_1:t_1)}\times\mathbb P^1_{(s_2:t_2)}\times\mathbb P^1_{(s_3:t_3)} $ given by the multihomogeneous equation $$\begin{aligned}
\lambda_1 s_1s_2s_3+\lambda_2 s_1s_2t_3+\lambda_3 s_1t_2s_3+\lambda_4 s_1t_2t_3+\\
\lambda_5 t_1s_2s_3+\lambda_6 t_1s_2t_3+\lambda_7 t_1t_2s_3+\lambda_8 t_1t_2t_3=0.\nonumber\end{aligned}$$
Clearly, $Y_{\lambda}$ is a Del Pezzo surface of degree 6. Since we shall be looking for Del Pezzo surfaces $Y_{\lambda}$ which are left invariant by certain subgroups of $\mathcal H$ (the Galois group of $\pi$), we first need to establish conditions ensuring that the hypersurface $Y_{\lambda}$ is left invariant by an element $h=(h_1,h_2,h_3)\in \mathcal H$.
This is done in the next lemma, which is easy to verify and which takes care of the normal form of a transformation $(h_1,h_2,h_3) \in {{\mathcal H}}$, taken up to a permutation of the three factors (here $\operatorname{Id}$ is the identity map of $\mathbb P^1$, while $A_1$, $A_{-1}$ and $B$ are the maps defined above).
Let $h=(h_1,h_2,h_3)\in \mathcal H \setminus \{\operatorname{Id}\}$ be one of the transformations listed in the first column of Table \[tab1\].
Then $Y_{\lambda}$ is $h$-invariant if and only if the coefficients ${\lambda}_j$ satisfy the linear conditions listed in Table \[tab1\].
$h$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $\lambda_6$ $\lambda_7$ $\lambda_8$ $c^2$
---------------------------------------------------- ------------- -------------- ---------------------- --------------- ------------------------------ ----------------------- ---------------------- --------------- ----------------------------
$\operatorname{Id},\operatorname{Id},A_{\alpha_3}$ $c\lambda_1$ $c\lambda_3$ $c\lambda_5$ $c\lambda_7$ $\alpha_3$
0 0 0 0
0 0 0 0
$\operatorname{Id},A_{\alpha_2},A_{\alpha_3}$ $c\alpha_3\lambda_2$ $c\lambda_1$ $c\alpha_3\lambda_6$ $c\lambda_5$ $\alpha_2\alpha_3$
$\operatorname{Id},A_{\alpha_2},B$ $c\lambda_1$ $-c\lambda_2$ $c\lambda_5$ $-c\lambda_6$ $\alpha_2$
0 0 0 0
0 0 0 0
$A_{\alpha_1},A_{\alpha_2},A_{\alpha_3}$ $c\alpha_2\alpha_3\lambda_4$ $c\alpha_2\lambda_3$ $c\alpha_3\lambda_2$ $c\lambda_1$ $\alpha_1\alpha_2\alpha_3$
$A_{\alpha_1},A_{\alpha_2},B $ $c\alpha_2\lambda_3$ $-c\alpha_2\lambda_4$ $c\lambda_1$ $-c\lambda_2$ $\alpha_1\alpha_2$
$A_{\alpha_1},B,B$ $c\lambda_1$ $-c\lambda_2$ $-c\lambda_3$ $c\lambda_4$ $\alpha_1$
0 0 0 0
0 0 0 0
: []{data-label="tab1"}
Note that in Table \[tab1\], the numbers $\alpha_i \in \{ \pm 1 \}$, since they are labelling $A_1$ and $A_{-1}$. If for a given case there appear two rows, this means that there are two alternatives, one for each row.
\[ibb\] Consider the following matrices: $$\Gamma_1:=\left(\begin{array}{cc}1&1\\1&-1\\\end{array}\right) \qquad
\Gamma_{-1}:=\left(\begin{array}{cc}i&i\\-1&1\\ \end{array}\right) \,,$$ and denote by $f_1$, respectively $f_{-1}$, the induced projectivities in $\mathrm{Aut}(\mathbb P^1)$ (observe that $f_1 = f_1^{-1}$).
It is straightforward to verify the following conjugacies
- $B= f_1^{-1} \circ A_1\circ f_1 = f_{-1} ^{-1} \circ A_{-1} \circ f_{-1}$,
- $ A_1= f_1^{-1} \circ B\circ f_1 = f_{-1} ^{-1} \circ B\circ f_{-1}$,
- $A_{-1}= f_1^{-1} \circ A_{-1}\circ f_1 = f_{-1} ^{-1} \circ A_1\circ f_{-1}$.
\[not-irr\]
If $Y_{{\lambda}}$ is invariant under $h=(\operatorname{Id},\operatorname{Id},A_\alpha) $ ($\alpha=\pm 1$), or under $h=(\operatorname{Id},\operatorname{Id},B) $ then the equation of $Y_{{\lambda}}$ is reducible. Since these projectivities are conjugate, it suffices to consider the case $h=(\operatorname{Id},\operatorname{Id},B) $, when the equation of $Y_{{\lambda}}$ is $$\begin{aligned}
s_3 ( \lambda_1s_1s_2+\lambda_3s_1t_2+\lambda_5t_1s_2+\lambda_7t_1t_2)=0 &\mbox{ or } \\
t_3 ( \lambda_2s_1s_2+\lambda_4s_1t_2+\lambda_6t_1s_2+\lambda_8t_1t_2)=0 \end{aligned}$$
The above enable us to prove the following:
\[cc2\] Let ${\lambda} \in \mathbb C^8 \setminus \{0\}$ be such that $Y_{{\lambda}}$ is irreducible. Assume moreover that there is a subgroup $H_1\cong (\mathbb Z/ 2\mathbb Z)^2$ of $\mathcal H$, such that $Y_{{\lambda}}$ is $H_1$-invariant. Then, up to the action of ${\ensuremath{\mathbb{P}}}\operatorname{GL}(2, {\ensuremath{\mathbb{C}}})^3$ and up to a permutation of the factors of $(\mathbb P^1)^3$, there are exactly two possibilities:
- $H_1=\langle (A_1,A_1,A_1), (\operatorname{Id},B,B) \rangle$, or
- $H_1=\langle (\operatorname{Id},B,B), (B,B,\operatorname{Id}) \rangle$.
Let $H_1=\langle h,h' \rangle$. Then, by Remarks \[not-irr\] and \[ibb\], after possibly changing the coordinates of $(\mathbb P^1)^3$, we may assume that $h=(B, B, B) $ or $=(\operatorname{Id},B,B)$.
in this case $h'\in\{ (\operatorname{Id},B,B), (A_{\alpha_1},B,B)\}$ implies that $(B, \operatorname{Id}, \operatorname{Id}) \in H_1$ or $(A_{\alpha_1}, \operatorname{Id}, \operatorname{Id}) \in H_1$, contradicting the irreducibility of $Y_{{\lambda}}$ (cf. Remark \[not-irr\]).
If we assume that $h'\in\{ (\operatorname{Id},A_{\alpha_2},B), (A_{\alpha_1},A_{\alpha_2},A_{\alpha_3})\}$, $\alpha_i \in \{ \pm 1\}$, then we see (cf. Table \[tab1\]) that the invariance of $Y_{{\lambda}}$ under $h$ and $h'$ implies that ${{\lambda}}=0$: this is a contradiction.
Assuming instead that $h'=(\operatorname{Id},A_{\alpha_2},A_{\alpha_3})$, then conjugating $h'$ by $(f_1, f_{\alpha_2}, f_{\alpha_3})$, we see that in the new coordinates we have: $$h=(f_1^{-1}\,B\,f_1, f_{\alpha_2}^{-1}\,B \,f_{\alpha_2}, f_{\alpha_3}^{-1}\,B\,
f_{\alpha_3})=(A_1,A_1,A_1)$$ and $$h'=(f_1^{-1}\,\operatorname{Id}\,f_1, f_{\alpha_2}^{-1}\,A_{\alpha_2}\, f_{\alpha_2}, f_{\alpha_3}^{-1}\,A_{\alpha_3}\,
f_{\alpha_3})=(\operatorname{Id},B,B)\,,$$ i.e., we are in case i).
Assume finally that $h'=(A_{\pm 1},A_{\pm 1}, B)$. Then $h\cdot h' = (A_{\mp 1},A_{\mp 1}, \operatorname{Id})$ and we reduce to the previous case showing that we are in case i).
in this case if $h'= (\operatorname{Id},A_{\alpha_2},A_{\alpha_3})$, the equation of $Y_{{\lambda}}$ is (cf. Table \[tab1\]): $$(\lambda_1s_1+\lambda_5t_1)(s_2s_3+ct_2t_3)=0,$$ contradicting the irreducibility of $Y_{{\lambda}}$.
If $h'\in\{(B,B,B),(\operatorname{Id},A_{\alpha_2},B), (\operatorname{Id},B,A_{\alpha_3}),(A_{\alpha_1},B,B)\}$, we obtain that $Y_{{\lambda}}$ is not irreducible by Remark \[not-irr\].
Assume that $h'\in\{ (A_{\alpha_1},\operatorname{Id},A_{\alpha_3}), (A_{\alpha_1},A_{\alpha_2},\operatorname{Id}), (B,A_{\alpha_2},\operatorname{Id}), (B,\operatorname{Id},A_{\alpha_3}), \\ (A_{\alpha_1},A_{\alpha_2},B), (A_{\alpha_1},B,A_{\alpha_3}), (B,A_{\alpha_2},B), (B,B,A_{\alpha_3})\}$. Then one checks easily, consulting Table \[tab1\], that ${{\lambda}}=0$, hence also these cases can be excluded.
If $h'\in\{ (A_{\alpha},\operatorname{Id},B), (A_{\alpha},B,\operatorname{Id})\}$, $\alpha \in \{ \pm 1\}$, after changing the coordinates by $(f_\alpha, \operatorname{Id},\operatorname{Id})$ we get $H_1=\langle(\operatorname{Id},B,B), (B,\operatorname{Id},B)\rangle$, hence we are in case ii).
Assume now that $h'=(A_{\alpha_1},A_{\alpha_2}, A_{\alpha_3})$. Changing coordinates by conjugating with $(\gamma_1, \gamma_2, \gamma_3)$, where $\gamma_j:= \operatorname{Id}$ if $\alpha_j=1$ and $\gamma_j:=( f_{-1}\circ f_1)$ if $\alpha_j=-1$ and using the fact that $$( f_{-1}\circ f_1)^{-1}\circ B\circ ( f_{-1}\circ f_1)=B, \ \
( f_{-1}\circ f_1)^{-1}\circ A_{-1}\circ ( f_{-1}\circ f_1)=A_1,$$ we see that (in the new coordinates) we are in case i). If $h'=(B,A_{\alpha_2},A_{\alpha_3})$, then changing the coordinates by conjugating with $(f_1, \gamma_2, \gamma_3)$, where $\gamma_j$ is defined as above, we are in case i). Finally, if $h'\in\{ (B,\operatorname{Id},B), (B,B,\operatorname{Id})\}$, then we are in case ii).
It is seen immediately that in case i) each Del Pezzo surface $Y_{{\lambda}}=\{\lambda_1s_1s_2s_3+\lambda_8 t_1t_2t_3=0\}$ is invariant under $H_1$, whereas in case ii) each surface $Y_{{\lambda}}=\{\lambda_1(s_1s_2s_3+t_1t_2t_3)+ \lambda_4(s_1t_2t_3+t_1s_2s_3)=0\}$ is invariant under $H_1$. In particular, in both respective cases i) and ii), we obtain a linear action of $H_1$ on the vector space $V:= H^0((\mathbb P^1)^3, \mathcal O_{(\mathbb P^1)^3} (1,1,1))$, which is independent of the chosen invariant surface in the pencil (see proposition \[linearization\]).
\[char1\] With the same notation as in Proposition \[cc2\], the respective decompositions of $V$ in character spaces with respect to the above action of $H_1 \cong (\mathbb Z/ 2 \mathbb Z)^2$ are as follows:
[*i) $H_1=\langle (A_1,A_1,A_1), (\operatorname{Id},B,B)\rangle$:*]{}
- $V^{++} = \{\lambda_1(s_1s_2s_3+t_1t_2t_3)+ \lambda_4(s_1t_2t_3+t_1s_2s_3) \mid \lambda_1, \lambda_4 \in \mathbb C \} \cong \mathbb C^2$;
- $V^{+-} = \{\lambda_2(s_1s_2t_3+t_1t_2s_3)+ \lambda_3(s_1t_2s_3+t_1s_2t_3) \mid \lambda_2, \lambda_3 \in \mathbb C \}$;
- $V^{-+} = \{\lambda_1(s_1s_2s_3-t_1t_2t_3)+ \lambda_4(s_1t_2t_3-t_1s_2s_3) \mid \lambda_1, \lambda_4 \in \mathbb C \} $;
- $V^{--} = \{\lambda_2(s_1s_2t_3-t_1t_2s_3)+ \lambda_3(s_1t_2s_3-t_1s_2t_3) \mid \lambda_2, \lambda_3 \in \mathbb C \}$.
[*ii) $H_1=\langle (\operatorname{Id},B,B), (B,B,\operatorname{Id})\rangle$:*]{}
- $V^{++} = \{\lambda_1s_1s_2s_3+\lambda_8 t_1t_2t_3 \mid \lambda_1, \lambda_8 \in \mathbb C \} \cong \mathbb C^2$;
- $V^{+-} = \{\lambda_4s_1t_2t_3+\lambda_5 t_1s_2s_3 \mid \lambda_4, \lambda_5 \in \mathbb C \}$;
- $V^{-+} = \{\lambda_2s_1s_2t_3+\lambda_7 t_1t_2s_3 \mid \lambda_2, \lambda_7 \in \mathbb C \} $;
- $V^{--} = \{\lambda_3s_1t_2s_3+\lambda_6 t_1s_2t_3 \mid \lambda_3, \lambda_6 \in \mathbb C \}$.
This is a simple calculation using Table \[tab1\].
The same arguments as in the proof of Proposition \[cc2\] yield the following statement:
\[cc3\] Let ${\lambda} \in \mathbb C^8 \setminus \{0\}$ be such that $Y_{{\lambda}}$ is irreducible. Assume moreover that there is a subgroup $H_0 \cong (\mathbb Z/ 2\mathbb Z)^3$ of $\mathcal H$, such that $Y_{{\lambda}}$ is $H_0$-invariant. Then, up to the action of ${\ensuremath{\mathbb{P}}}\operatorname{GL}(2, {\ensuremath{\mathbb{C}}})^3$ and up to a permutation of the factors of $(\mathbb P^1)^3$, we have: $$H_0=\langle (\operatorname{Id},B,B),(A_1,A_1,A_1), (B,B,\operatorname{Id})\rangle \,.$$
Again we see immediately that the Del Pezzo surface $Y_{{\lambda}}=\{ s_1s_2s_3 + t_1t_2t_3\}$ is invariant under $H_0$, hence we get again a linear action of $H_0$ on the vector space $V:= H^0((\mathbb P^1)^3, \mathcal O_{(\mathbb P^1)^3} (1,1,1))$.
\[char2\] Use the same notation as in Proposition \[cc3\]; then $V$ decomposes in $8$ one-dimensional character spaces for the action of $H_0 \cong (\mathbb Z/ 2 \mathbb Z)^3$, as follows:
- $V^{+++} = \{\lambda(s_1s_2s_3+ t_1t_2t_3 ) \mid \lambda \in \mathbb C \} $;
- $V^{+-+} = \{\lambda(s_1s_2s_3- t_1t_2t_3 ) \mid \lambda \in \mathbb C \}$;
- $V^{++-} = \{\lambda(s_1t_2t_3+ t_1s_2s_3 ) \mid \lambda \in \mathbb C \}$;
- $V^{+--} = \{\lambda(s_1t_2t_3- t_1s_2s_3 ) \mid \lambda \in \mathbb C \}$;
- $V^{-++} = \{\lambda(s_1s_2t_3+ t_1t_2s_3 ) \mid \lambda \in \mathbb C \}$;
- $V^{--+} = \{\lambda(s_1s_2t_3- t_1t_2s_3 ) \mid \lambda \in \mathbb C \}$;
- $V^{-+-} = \{\lambda(t_1s_2t_3+ s_1t_2s_3 ) \mid \lambda \in \mathbb C \}$;
- $V^{---} = \{\lambda(t_1s_2t_3- s_1t_2s_3 ) \mid \lambda \in \mathbb C \}$.
\[invDP\] Case 1): $$H_1: =\langle (\operatorname{Id},B,B),(A_1,A_1,A_1) \rangle \cong (\mathbb Z/ 2\mathbb Z)^2\ \triangleleft\mathcal H.$$ Then there are four pencils of Del Pezzo surfaces, which are left invariant by $H_1$ (cf. Proposition \[char1\]); their inverse images under $\pi'$ (see (\[diag1\])) $\pi'^{-1}(Y_\nu)$ (resp. $\pi'^{-1}(Y'_\nu)$, $\pi'^{-1}(Y''_\nu)$, $\pi'^{-1}(Y'''_\nu)$) are pencils of hypersurfaces of multidegree $(2,2,2)$ in $A^0= E_1 \times E_2 \times E_3$ invariant under $\mathcal G'_1\cong (\mathbb Z/ 2\mathbb Z)^5\subset (\mathbb Z/ 2\mathbb Z)^9$.
We list now the four pencils (${\nu}=(\nu_1:\nu_2) \in \mathbb P^1$): $$\label{eqDp1}
Y_{\nu}:=\{\nu_1(s_1s_2s_3+ t_1t_2t_3)+ \nu_2(s_1t_2t_3+t_1s_2s_3)=0\}\,,$$ $$Y'_{\nu}:=\{\nu_1(s_1s_2t_3+ t_1t_2s_3)+ \nu_2(s_1t_2s_3+t_1s_2t_3)=0\}\,,$$ $$Y''_{\nu}:=\{\nu_1(s_1s_2s_3- t_1t_2t_3)+ \nu_2(s_1t_2t_3-t_1s_2s_3)=0\}\,,$$ $$Y'''_{\nu}:=\{\nu_1(s_1s_2t_3- t_1t_2s_3)+ \nu_2(s_1t_2s_3-t_1s_2t_3)=0\}\,.$$
It is immediate to see that the $4$ pencils are transformed to each other by the elements of the group $ {{\mathcal H}}= (({\ensuremath{\mathbb{Z}}}/2)^2)^3$ (for instance we pass from the first to the second via $s_3 \leftrightarrow t_3$, from the first to the third via $t_1 \leftrightarrow - t_1$, and so on). Therefore, in the future we shall only consider the first pencil: (\[eqDp1\]).
Case 2): $$H_1: =\langle (\operatorname{Id},B,B), (B,B,\operatorname{Id})\rangle \cong (\mathbb Z/ 2\mathbb Z)^2\ \triangleleft\mathcal H.$$ Then there are four pencils of Del Pezzo surfaces, which are left invariant by $H_1$; their respective inverse images under $\pi'$ yield four pencils, invariant under $\mathcal G_1\cong (\mathbb Z/ 2\mathbb Z)^5\subset (\mathbb Z/ 2\mathbb Z)^9$.
The four pencils are given by the following equations ($\mu\in \mathbb C, \mu \neq 0$): $$\label{eqDp2}
Y_\mu:=\{s_1s_2s_3+\mu\, t_1t_2t_3=0\}\,,$$ $$Y'_\mu:=\{s_1t_2s_3+\mu\, t_1s_2t_3=0\}\,,$$ $$Y''_\mu:=\{s_1t_2t_3+\mu\, t_1s_2s_3=0\}\,,$$ $$Y'''_\mu:=\{s_1s_2t_3+\mu\, t_1t_2s_3=0\}\,.$$
Also in this case the $4$ pencils are transformed to each other by the elements of the group $ {{\mathcal H}}= (({\ensuremath{\mathbb{Z}}}/2)^2)^3$, hence in the future we shall only consider the first pencil: (\[eqDp2\]).
Case 3): $$H_0: =\langle (\operatorname{Id},B,B), (A_1,A_1,A_1), (B,B,\operatorname{Id})\rangle \cong ((\mathbb Z/ 2\mathbb Z))^3 \triangleleft\mathcal H.$$ Then there are eight Del Pezzo surfaces which are are left invariant by $H_0$; their respective inverse images under $\pi'$ are invariant under $\mathcal G_0\cong (\mathbb Z/ 2\mathbb Z)^6\subset (\mathbb Z/ 2\mathbb Z)^9$.
Their respective equations are the following ones: $$\label{eqDp3}
Y_{1}:=\{s_1s_2s_3+ t_1t_2t_3 = 0\}, \ \ Y_{-1}:=\{s_1s_2s_3- t_1t_2t_3 = 0\},$$ $$Y_{1}':=\{s_1t_2t_3+ t_1s_2s_3 = 0\}, \ \ Y_{-1}':=\{s_1t_2t_3- t_1s_2s_3 = 0\},$$ $$Y_{1}'':=\{s_1s_2t_3+ t_1t_2s_3 = 0\}, \ \ Y_{-1}'':=\{s_1s_2t_3- t_1t_2s_3 = 0\},$$ $$Y_{1}''':=\{t_1s_2t_3+ s_1t_2s_3 = 0\}, \ \ Y_{-1}''':=\{t_1s_2t_3- s_1t_2s_3 = 0\}.$$
Also here the $8$ hypersurfaces are transformed to each other by the elements of the group $ {{\mathcal H}}= (({\ensuremath{\mathbb{Z}}}/2)^2)^3$, hence in the future we shall only consider the first one: (\[eqDp3\]).
\[BurniatHyp\] Let $\hat{X}$ be an irreducible hypersurface, in the product of three smooth elliptic curves $A^0 := E_1 \times E_2 \times E_3$, which is the inverse image under $\pi'$ of a Del Pezzo surface $Y$ of degree 6, invariant under a subgroup $H \cong (\mathbb Z/ 2\mathbb Z)^2\ \triangleleft\mathcal H$.\
Then we call $\hat X$ a *Burniat hypersurface in $A^0$*.
\[stabs\] Let $\mathcal G_0\cong (\mathbb Z/ 2\mathbb Z)^6\triangleleft (\mathbb Z/ 2\mathbb Z)^3\times
(\mathbb Z/ 2\mathbb Z)^3\times(\mathbb Z/ 2\mathbb Z)^3$ be the group: $$\mathcal G_0:=\{(\epsilon_0,\eta_1,\epsilon_1,\eta_0,\epsilon_2,\zeta_0,\epsilon_3)
\subset \{\pm 1\}^7 \cong (\mathbb Z/ 2\mathbb Z)^7 \mid \epsilon_1\epsilon_2\epsilon_3=1\}\,,$$ which acts on $E_1\times E_2\times E_3$ by: $$\begin{array}{lll}x_0\mapsto \epsilon_0x_0\,,& u_0 \mapsto \eta_0 u_0\,,
&z_0\mapsto \zeta_0 z_0\,, \\
x_3\mapsto \epsilon_1 x_3\,, &u_3 \mapsto \epsilon_2 u_3\,, &
z_3 \mapsto \epsilon_3 z_3\,
\end{array}\quad
\mbox{ and }\quad
\begin{pmatrix}x_1\\u_1\\z_1\end{pmatrix}\mapsto \eta_1
\begin{pmatrix}x_1\\u_1\\z_1\end{pmatrix}\,.$$
With the same notation as in Remark \[invDP\]:
1. $\pi'^{-1}(Y_\nu)$ is invariant under the group $$\mathcal G'_1:=\{(\epsilon_0,\eta_1,1,\eta_0,\epsilon_2,\zeta_0,\epsilon_3) \mid \epsilon_2\epsilon_3=1\}\cong (\mathbb Z/ 2\mathbb Z)^5\triangleleft \mathcal G_0\,.$$
2. $\pi'^{-1}(Y_\mu)$ is invariant under the group $$\mathcal G_1:=\{(\epsilon_0,1,\epsilon_1,\eta_0,\epsilon_2,\zeta_0,\epsilon_3) \mid
\epsilon_1\epsilon_2\epsilon_3=1\}\cong (\mathbb Z/ 2\mathbb Z)^5\triangleleft \mathcal G_0\,.$$
3. If $\mu=\pm 1$, then $\pi'^{-1}(Y_{\mu})$ is invariant under $\mathcal G_0$.
Just note that multiplication of $(x_1,u_1,z_1)$ by $-1$ corresponds to $(s_j:t_j) \mapsto(t_j:s_j)$ for each $j=1,2,3$.
Fixed points
------------
In order to systematically search for all the subgroups $G\cong (\mathbb Z/ 2\mathbb Z)^3\triangleleft \mathcal G_0$ acting freely on a Burniat hypersurface in $A^0:=E_1\times E_2\times E_3$, we need to determine which elements of $\mathcal G_0$ have fixed points on $A^0$ .
Fix $a_1,a_2 \in \mathbb C$ pairwise distinct such that the curve $$E:=\{x_1^2+x_2^2+x_3^3=0, \, x_0^2=a_1x_1^2+a_2x_2^2 \}\subset \mathbb P^2$$ is smooth. Then $$g(x_0:x_1:x_2:x_3):=(\alpha_0x_0: \alpha_1x_1:x_2:\alpha_3x_3)\,, \quad \alpha_j\in \{\pm 1\}\,$$ has fixed points on $E$ if and only if
- either $\alpha_0=\alpha_1=\alpha_3=-1$, or
- exactly one $\alpha_i=-1$ and the others are equal to 1.
From now on, we change to an additive notation in which $\mathbb Z /2 \mathbb Z$ is the additive group $\{0,1\}$.
Let $g\in \mathcal G_0$ be an element fixing points on $ A^0$. By [@BC13 Proposition 3.3], $g$ is an element in Table \[FixEl\].
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
--------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$\epsilon_0$ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $0$ $0$ $ 0 $ $ 1 $ $ 1 $
$\eta_1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$\epsilon_1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $
$\eta_0$ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $
$\epsilon_2$ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $
$\zeta_0$ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $
$\epsilon_3$ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $ $ 0 $ $ 1 $ $ 1 $ $ 0 $
: The elements of $\mathcal G_0$ having fixed points on $A^0$, written additively![]{data-label="FixEl"}
\[fixingpts\] 1) Let $\hat X:=\pi'^{-1}(Y_{\pm 1})$. In Table \[FixEl\], the elements 1-3 fix pointwise a surface $S\subset A^0$. Each element 4-9 fixes pointwise a curve $C\subset A^0$ and its fixed locus has non trivial intersection with $\hat X$ since $\hat X\subset A^0$ is an ample divisor. Finally, the elements 10-17 have isolated fixed points on $A^0$; arguing as in [@BC13 Proposition 3.3] one proves that the elements 11-17 have fixed points on $\hat X$, while the fixed locus of element 10 intersects $\hat X$ only for special choices of the three elliptic curves.
2\) The same holds for $\hat X:=\pi'^{-1}(Y_{\nu})$ (resp. $\pi'^{-1}(Y_\mu)$), considering only the elements 1-7,10,11,14,15 (resp. 1-13), i.e. the ones belonging to $\mathcal G'_1$ (resp. $\mathcal G_1$). In particular, the fixed locus of element 10 intersects $\hat X$ only for special choices of the three elliptic curves and of the parameter $\nu$ (resp. $\mu$).
Description in terms of Legendre families
------------------------------------------
We now describe the families of Burniat hypersurfaces in $A^0$ in terms of Legendre functions $\mathcal{L}$ (see Section \[Inouedescription\]).
To this purpose, we consider the following 1-parameter family of intersections of two quadrics:
$$E (b) :=\{ x_1^2+x_2^2+x_3^2=0,
\quad x_0^2= (b^2+1)^2 x_1^2+( b^2-1)^2x_2^2\}\\,$$ where $ b \in {\ensuremath{\mathbb{C}}}\setminus \{ 0, 1, -1 , i, - i \}$.
We set $$\xi : = \frac{bs}{t}\,,$$ and in this way the family of genus one curves $E(b)$ is the Legendre family of elliptic curves in Legendre normal affine form: $$y^2 = (\xi^2-1) (\xi^2-a^2), \ \ a : = b^2\,.$$
In fact, $$\begin{aligned}
x_0^2&=& (b^2+1)^2 x_1^2+( b^2-1)^2x_2^2 =- (a+1)^2 (s^2-t^2)^2+( a -1)^2(s^2 + t^2) ^2 = \\
&=& 4 [(a^2 + 1) s^2 t^2 - a (t^4 + s^4)] = 4 t^4 \left[ (a^2 + 1)\left( \frac{\xi} {b}\right)^2 - a \left( 1 + \left( \frac{\xi} {b}\right)^4\right)\right ] = \\
& =& - 4 t^4 \frac{1}{b^2} [ - (a^2 + 1) \xi^2 + (a^2 + \xi^4) = \frac{- 4 t^4}{b^2} [ (\xi^2-1) (\xi^2-a^2)]
\end{aligned}$$ and it suffices to set $$y : = \frac{i b x_0}{2 t^2}\,.$$
The group $({\ensuremath{\mathbb{Z}}}/ 2)^3$ acts fibrewise on the family $E(b)$ via the commuting involutions: $$x_0 \longleftrightarrow - x_0, \ \ x_3 \longleftrightarrow - x_3,\ \ x_1 \longleftrightarrow - x_1,$$ which on the birational model given by the Legendre family act as $$y \longleftrightarrow - y, \ \ \xi \longleftrightarrow - \xi, \ \ \xi \longleftrightarrow \frac{a}{\xi}\,.$$
Consider the subgroup $${\Gamma}_{2,4} : = \left\{ \left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right)
\in {\ensuremath{\mathbb{P}}}SL (2 , {\ensuremath{\mathbb{Z}}}) \biggm| \begin{array}{cc}
\alpha \equiv 1 \mod 4, & \beta \equiv 0 \mod 4, \\\gamma \equiv 0 \mod 2, &\delta \equiv 1 \mod 2
\end{array}
\right\}$$ a subgroup of index $2$ of the congruence subgroup $${\Gamma}_2 : = \left\{ \left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right)
\in {\ensuremath{\mathbb{P}}}SL (2 , {\ensuremath{\mathbb{Z}}}) \biggm| \begin{array}{cc}
\alpha \equiv 1 \mod 2, & \beta \equiv 0 \mod 2, \\\gamma \equiv 0 \mod 2, &\delta \equiv 1 \mod 2
\end{array}
\right\} .$$
To the chain of inclusions $${\Gamma}_{2,4} < {\Gamma}_2 < {\ensuremath{\mathbb{P}}}SL (2 , {\ensuremath{\mathbb{Z}}})$$ corresponds a chain of fields of invariants $${\ensuremath{\mathbb{C}}}(j) \subset {\ensuremath{\mathbb{C}}}({\lambda}) = {\ensuremath{\mathbb{C}}}(\tau)^{{\Gamma}_2} \subset {\ensuremath{\mathbb{C}}}(\tau)^{{\Gamma}_{2,4}}\,,$$ where the respective degrees of the extensions are 6, 2.
Here, ${\lambda}$ is the cross-ratio of the four points $\mathfrak p (0), \mathfrak p (\frac{1}{2}), \mathfrak p (\frac{\tau}{2}), \mathfrak p (\frac{1 + \tau}{2}), $ where $\mathfrak p $ is the Weierstrass function, and $j ({\lambda}) = \frac{({\lambda}^2 - {\lambda}+ 1)^3}{{\lambda}^2 ({\lambda}-1)^2}$ is the $j$-invariant.
If ${\lambda}(a)$ is the cross ratio of the four points $1,-1, a, - a$, ${\lambda}(a) = \frac{(a-1)^2}{( a+1)^2 }$, thus ${\ensuremath{\mathbb{C}}}(a) = {\ensuremath{\mathbb{C}}}(\sqrt {\lambda})$ is a quadratic extension and there are two values of $a$ for which we get a Legendre function for the elliptic curve. Setting $b : = {{\mathcal L}}(\frac{\tau}{4} )$, we have that $a = b^2$, hence ${\ensuremath{\mathbb{C}}}(b)$ is a quadratic extension of ${\ensuremath{\mathbb{C}}}(\tau)^{{\Gamma}_{2,4}}$.
In other words, the parameter $ b \in {\ensuremath{\mathbb{C}}}\setminus \{ 0, 1, -1 , i, - i \}$ yields an unramified covering of degree $4$ of $ {\lambda}\in {\ensuremath{\mathbb{C}}}\setminus \{ 0, 1\}$ , hence the field ${\ensuremath{\mathbb{C}}}(b)$ is the invariant field for a subgroup [ ${\Gamma}_{2,8}$]{} of index $2$ in ${\Gamma}_{2,4}$.
[ By [@Bianchi §182], $b$ is invariant under the subgroup of ${\Gamma}_2$ given by the transformation such that $\alpha^2+\alpha\beta \equiv 1 \mod 8$. Since $\alpha \equiv 1 \mod 2$, this equation is equivalent to require that $\beta \equiv 0 \mod 8$, i.e.: $${\Gamma}_{2,8} : = \left\{ \left( \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array} \right)
\in {\ensuremath{\mathbb{P}}}SL (2 , {\ensuremath{\mathbb{Z}}}) \biggm| \begin{array}{cc}
\alpha \equiv 1 \mod 4, & \beta \equiv 0 \mod 8, \\\gamma \equiv 0 \mod 2, &\delta \equiv 1 \mod 2
\end{array}
\right\} .$$ ]{}
Consider now the following family $${{\mathcal A}}^0 = E(b_1) \times E(b_2) \times E(b_3).$$ It is the family of products of three elliptic curves with a [ ${\Gamma}_{2,8}$]{}-level structure: ${{\mathcal A}}^0$ is the quotient of $({\ensuremath{\mathbb{C}}}\times {\ensuremath{\mathbb{H}}})^3$, with coordinates $((z_1, \tau_1),(z_2, \tau_2),(z_3, \tau_3))$, by the action of the group (a semidirect product ) generated by $({\ensuremath{\mathbb{Z}}}^2)^3$ which acts by $$((m_1, n_1),(m_2, n_2),(m_3, n_3)) \circ ((z_1, \tau_1),(z_2, \tau_2),(z_3, \tau_3))=$$ $$= ((z_1 + m_1 + n_1 \tau_1, \tau_1),(z_2 + m_2 + n_2 \tau_2, \tau_2),(z_3+ m_3 + n_3 \tau_3, \tau_3))$$ and by $ {{} {\Gamma}_{2,8}} ^3 \subset {\ensuremath{\mathbb{P}}}SL (2 , {\ensuremath{\mathbb{Z}}})^3$.
The fibre of $f : {{\mathcal A}}^0 {\ensuremath{\rightarrow}}{{\mathcal E}}: = ( {\ensuremath{\mathbb{H}}})^3 / {{} {\Gamma}_{2,8}} ^3)$ is the product of the three elliptic curves, for $k= 1,2,3$, $E_k: =\mathbb C/\langle 1, \tau_k\rangle$.
Let $\mathcal{L}_k\colon E_k\rightarrow \mathbb P^1$ be a Legendre function for $E_k$. We have seen that the relation between $\mathcal L_k( z_k)$ and the coordinates $(s_k:t_k)$ of $\mathbb P^1$ is $$\dfrac{\mathcal{L}_k(z_k)}{b_k}= \dfrac{s_k}{t_k}$$ where $b_k:=\mathcal L_k(\frac{\tau_k}{4})$. A basis for the $(\mathbb{Z}/2 \mathbb{Z})^3$-action on $E_k$, $k=1,2,3$, is given by: $$\label{Z23-action}
\begin{array}{ccccc}
x_0 \mapsto - x_0 &\hat{=} & (z_k\mapsto -z_k)&\hat{=}&(1,0,0)\\
x_1 \mapsto - x_1 &\hat{=} & (z_k\mapsto -z_k+\frac{\tau_k}{2})&\hat{=}&(0,1,0)\\
x_3 \mapsto - x_3 &\hat{=} & (z_k\mapsto -z_k+\frac{1}{2})&\hat{=}&(0,0,1)
\end{array}$$
The above formulae define an action of $(({\ensuremath{\mathbb{Z}}}/2)^3)^3$ on the fibration $f : {{\mathcal A}}^0 {\ensuremath{\rightarrow}}{{\mathcal E}}$, which acts trivially on the basis.
It follows from (\[eqDp1\], \[eqDp2\], \[eqDp3\]) that it suffices to consider only the families of Burniat hypersurfaces defined by: $$\label{righteq}
\begin{array}{l}
\hat {{\mathcal X}}_\nu= \{([(z_1, \tau_1),(z_2, \tau_2),(z_3, \tau_3)], (\nu_1 : \nu_2)) \in {{\mathcal A}}^0 \times {\ensuremath{\mathbb{P}}}^1 \mid\\ \\
\nu_1(\mathcal L_1(z_1)\mathcal L_2(z_2)\mathcal L_3(z_3) + b_1b_2b_3)+
\nu_2(\mathcal L_1(z_1) b_2b_3 + b_1\mathcal L_2(z_2)\mathcal L_3(z_3))=0 \} \,,
\end{array}$$ $$\hat {{\mathcal X}}_{\mu}= \{([(z_1, \tau_1),(z_2, \tau_2),(z_3, \tau_3)], \mu ) \in {{\mathcal A}}^0 \times {\ensuremath{\mathbb{C}}}^* \mid \mathcal L_1(z_1)\mathcal L_2(z_2)\mathcal L_3(z_3)=\mu \}\,,$$ $$\hat {{\mathcal X}}_b= \{[(z_1, \tau_1),(z_2, \tau_2),(z_3, \tau_3)] \in {{\mathcal A}}^0\mid \mathcal L_1(z_1)\mathcal L_2(z_2)\mathcal L_3(z_3)=b_1b_2b_3\}\,,$$
where the meaning of the subscript is to refer to the variables: ${\nu}=(\nu_1:\nu_2) \in \mathbb P^1 , \ \mu\in {\ensuremath{\mathbb{C}}}^*, \ \ b:=b_1b_2b_3$.
\[1fam\] There is also an obvious action of the symmetric group $\mathfrak S_3$ on the family $f : {{\mathcal A}}^0 {\ensuremath{\rightarrow}}{{\mathcal E}}$.
Let $\hat X$ be a Burniat hypersurface in $A^0$ (see Definition \[BurniatHyp\]). An explicit calculation using the above equations shows that $\hat X$ has at most finitely many nodes as singularities.
Let $\epsilon\colon X' \rightarrow \hat X$ be the minimal resolution of its singularities. Since $\hat{X}$ has at most canonical singularities, $K_{X'}=\epsilon^* K_{\hat X}$ and $X'$ is a minimal surface of general type with $K^2_{X'}=48$ and $\chi(X')=8$ (cf. [@inoue]).
Generalized Burniat type surfaces
=================================
Using the notation introduced in the previous sections, we give the following definition.
Let $\hat X$ be a Burniat hypersurface in $A^0:=E_1 \times E_2 \times E_3$, let $G\cong (\mathbb Z/ 2\mathbb Z)^3$ be a subgroup of $\mathcal G_0$ acting freely on $\hat X$.
The minimal resolution $S$ of the quotient surface $ X:= \hat X/G$ is called a *generalized Burniat type (GBT) surface*. We call $ X$ the *quotient model of $S$* (indeed, we easily see that $X$ is the canonical model of $S$).
1\) Since $G$ acts freely and $\hat X$ has at most nodes as singularities (we assume $Y$, hence also $\hat{X}$, to be irreducible!), a generalized Burniat type surface $S$ is a smooth minimal surface of general type with $K^2_S=6$ and $\chi(S)=1$.
2\) If $G\triangleleft \mathcal G_1 $ or $G\triangleleft \mathcal G'_1 $, then there is a pencil of Burniat hypersurfaces which are left invariant by the $G$-action, and the family of quotients of the hypersurfaces on which the action is free is then a one parameter family of GBT $G$-quotient surfaces (if we vary also $E_1 , E_2 ,E_3$ we obtain a four dimensional family).
Let ${\Delta}$ be the subgroup of $\operatorname{Aut}(( (\mathbb Z/ 2\mathbb Z)^3)^3 ) $ generated by:\
$$\begin{aligned}
l_1(g_1,g_2,g_3) &=& (g_2,g_1,g_3)\\
l_2(g_1,g_2,g_3) &=& (g_3,g_2,g_1)\end{aligned}$$
$$\begin{aligned}
h_1(g_1,g_2,g_3) &=& (f(g_1), f(g_2), g_3)\\
h_2(g_1,g_2,g_3) &=& (f(g_1), g_2, f(g_3))\\
h_3(g_1,g_2,g_3) &=& (g_1, f(g_2), f(g_3))\end{aligned}$$
where $g_j \in (\mathbb Z/ 2\mathbb Z)^3 \, (j\in\{1,2,3\})$ and where $f$ is defined by: $$\begin{array}{ccc}
f\colon (\mathbb Z/ 2\mathbb Z)^3&\longrightarrow &(\mathbb Z/ 2\mathbb Z)^3 \\[6pt]
f\colon (a,b,c)& \longmapsto &(a+b,b,b+c)
\end{array}$$
1\) It is easy to see that ${\Delta}(\mathcal{G}_0) = (\mathcal{G}_0)$.
2\) We claim now that, as it can be verified, for each ${\delta}\in {\Delta}$, ${\delta}(g)$ is conjugate to $g$ via an element $\phi$ of the group of automorphisms of ${{\mathcal A}}^0$.
For example, let $E:=\mathbb C/\langle 1, \tau \rangle$ be a complex elliptic curve and let $\tau':=\tau+1$. Then $E=\mathbb C/\langle 1, \tau'\rangle$ and the $(\mathbb Z/ 2\mathbb Z)^3$-action, defined in (\[Z23-action\]), is: $$\begin{array}{ccc}
(z\mapsto -z)&=&(1,0,0)\\
(z\mapsto -z+\frac{\tau'}{2})=(z\mapsto -z+\frac{\tau+1}{2})&=& (1,1,1)\\
(z\mapsto -z+\frac{1}{2})&=&(0,0,1)
\end{array}$$ This shows that the groups $G$ and $G':=h_j(G)\subset \mathcal G_0\,, j=1,2,3$ are conjugate via an automorphism of ${{\mathcal A}}^0$.
It follows then that $g\in \mathcal G_0$ acts freely on one of the families $\hat {{\mathcal X}}$ if and only if ${\delta}(g)$ acts freely on the transformed family $\phi (\hat {{\mathcal X}})$.
It follows also that two groups in the same ${\Delta}$-orbit yield isomorphic families of GBT surfaces, hence we can restrict our attention to a single representative for each ${\Delta}$-orbit.
\[onefam\]
1. There are exactly 16 irreducible families of generalized Burniat type surfaces, listed in tables \[q0\]-\[q3\].
2. The family of generalized Burniat type surfaces has dimension 4 in cases $\mathcal S_1$ and $\mathcal S_2$, and dimension 3 otherwise.
1\) The MAGMA script below searches for subgroups of $G \leq \mathcal{G}_0$, which satisfy the following
- $G \cong ({\ensuremath{\mathbb{Z}}}/ 2{\ensuremath{\mathbb{Z}}})^3$;
- $G$ does not contain the elements 1-9, 11-17 of table \[FixEl\].
The 161 groups of the output therefore act freely on $\hat{X}_b \subset E(b_1) \times E(b_2) \times E(b_3)$, except for a finite number of values of $b_1, b_2, b_3 \in {\ensuremath{\mathbb{C}}}$ (cf. Remark \[fixingpts\]).
The following script moreover proves that the 161 groups $G$ belong to exactly 16 ${\Delta}$-orbits.
F:=FiniteField(2); V6:=VectorSpace(F,6);
V3:=VectorSpace(F,3); H3:=Hom(V6,V3);
U:={ V6![0,0,0,0,0,1], V6![0,0,0,1,0,0], V6![1,0,0,0,0,0],
V6![1,0,0,0,1,0], V6![0,1,0,0,0,0], V6![0,1,0,1,1,1],
V6![0,0,1,1,0,0], V6![0,0,1,0,1,1], V6![1,1,1,0,0,1],
V6![1,1,1,1,1,0], V6![0,0,0,0,1,0], V6![0,0,0,1,0,1],
V6![0,0,1,0,0,0], V6![1,0,0,0,0,1], V6![0,0,1,0,1,0],
V6![1,0,0,1,0,0]};
S3:={Kernel(f): f in H3 | Dimension(Kernel(f)) eq 3};
M:=[**];
for k in S3 do K:=Set(k);
if #(K meet U) eq 0 then Append(~M, k ); end if;
end for;
#M;
161
P:={1..#M}; Q:={}; // Delta-action
g1:=hom<V6->V6| V6![0,0,0,1,0,0],V6![0,1,0,0,0,0],V6![0,0,0,0,1,0],
V6![1,0,0,0,0,0],V6![0,0,1,0,0,0], V6![0,0,0,0,0,1]>;
g2:=hom<V6->V6| V6![0,0,0,0,0,1],V6![0,1,0,0,0,0], V6![0,0,1,0,0,0],
V6![0,0,0,1,0,0],V6![0,0,1,0,1,0], V6![1,0,0,0,0,0]>;
f1:=hom<V6->V6| V6![1,0,0,0,0,0],V6![1,1,1,1,1,0], V6![0,0,1,0,0,0],
V6![0,0,0,1,0,0],V6![0,0,0,0,1,0], V6![0,0,0,0,0,1]>;
f2:=hom<V6->V6| V6![1,0,0,0,0,0],V6![1,1,1,0,0,1], V6![0,0,1,0,0,0],
V6![0,0,0,1,0,0],V6![0,0,0,0,1,0], V6![0,0,0,0,0,1]>;
f3:=hom<V6->V6| V6![1,0,0,0,0,0],V6![0,1,0,1,1,1], V6![0,0,1,0,0,0],
V6![0,0,0,1,0,0],V6![0,0,0,0,1,0], V6![0,0,0,0,0,1]>;
L1:=Transpose(Matrix([g1(x): x in Basis(V6)]));
L2:=Transpose(Matrix([g2(x): x in Basis(V6)]));
H1:=Transpose(Matrix([f1(x): x in Basis(V6)]));
H2:=Transpose(Matrix([f2(x): x in Basis(V6)]));
H3:=Transpose(Matrix([f3(x): x in Basis(V6)]));
GL6:=GeneralLinearGroup(6,F); PG:=sub<GL6|L1,L2,H1,H2,H3>;
while not IsEmpty(P) do
i:=Rep( P ); Exclude(~P,i); Include(~Q,i);
for m in PG do
f:=map<V6->V6| x:->[(m[1],x),(m[2],x),(m[3],x),
(m[4],x),(m[5],x),(m[6],x)]>;
test:=sub<V6|f(Set(M[i]))>;
if exists(x){x: x in P | M[x] eq test } then
Exclude(~P, x); end if;
end for; end while;
#Q;
16
This proves the first assertion.
2\) In tables \[q0\]-\[q3\] we list one representative $G$ for each of the 16 ${\Delta}$-orbits.
Observe that the dimension of the family is 3 (the number of moduli of the three elliptic curves $E(b_1) \times E(b_2) \times E(b_3)$) if and only if the group $G$ stabilizes only a finite number of Burniat hypersurfaces, equivalently iff $G$ is neither contained in $\mathcal{G}_1$ nor in $\mathcal{G}'_1$. Otherwise $G$ is contained in $\mathcal{G}_1$ or in $\mathcal{G}'_1$ and, by Lemma \[stabs\], fixes a pencil of Burniat hypersurfaces. Therefore in this case the dimension of the family of generalized Burniat type surfaces is 4.
It is now easy to verify that in case $\mathcal S_1$ (of Table \[q0\]) $G \subset \mathcal{G}'_1$, in case $\mathcal S_2$ $G \subset \mathcal{G}_1$, whereas in cases $\mathcal S_3$-$\mathcal S_{16}$ of Tables \[q0\]-\[q3\] $G$ is not contained in any of the two groups $\mathcal{G}_1, \mathcal{G}'_1$.
The fundamental groups
----------------------
To determine the fundamental group of a GBT surface $S\rightarrow X =\hat X/G$, we preliminarily observe that, by van Kampen’s theorem and since $X$ has only nodes as singularities, $ \pi_1(X)= \pi_1(S)$. Then we argue as follows.
Let $E_j=\mathbb{ C}/ \langle e_j, \tau_j e_j\rangle$, $j=1,2,3$ and denote by $\Lambda$ the fundamental group of $A^0:=E_1\times E_2 \times E_3$. In particular, $\Lambda= \Lambda_1\oplus \Lambda_2\oplus \Lambda_3$, where $\Lambda_j= \langle e_j, \tau_j e_j\rangle$.
Consider the affine group $$\Gamma:=\langle\gamma_1,\gamma_2,\gamma_3, e_1, \tau_1 e_1, e_2,\tau_2 e_2, e_3,\tau_3 e_3
\rangle\leq \mathbb{A}(3,\mathbb{C})\,,$$ generated by $\Lambda$ and by lifts $\gamma_j$ of the generators $g_j$ of $G$ as affine transformations.
Then $\Gamma=\pi_1(X)= \pi_1(S)$.
Observe that by the Lefschetz’ hyperplane section theorem (see [@milnor_morse Theorem 7.4]) follows that $\pi_1(\hat X)\cong \pi_1(A^0)= \Lambda$.
The universal covering $\tilde{X}$ of $\hat X\subset A^0$ has then a natural inclusion $\tilde{X}\subset \mathbb C^3$ and the affine group ${\Gamma}$ acts on $\mathbb{C}^3$ leaving $\tilde{X}$ invariant. Since the action of ${\Gamma}$ on $\tilde{X}$ is free, and $X= \hat X/G= \tilde X/\Gamma$ we conclude that $\Gamma=\pi_1(X)= \pi_1(S)$.
The following MAGMA script, which is an extended version of the previous one, computes the fundamental group of each GBT surface. Observe that the fundamental group does only depend on $G$: since it does not change within the same connected family, and since each group $G$ determines an irreducible family.
V9:=VectorSpace(F,9); T:=[* *];
h:=hom<V6->V9| V9![1,0,0,0,0,0,0,0,0], V9![0,1,0,0,1,0,0,1,0],
V9![0,0,1,0,0,0,0,0,1], V9![0,0,0,1,0,0,0,0,0],
V9![0,0,0,0,0,1,0,0,1], V9![0,0,0,0,0,0,1,0,0]>;
G1:=DirectProduct([CyclicGroup(2),CyclicGroup(2),CyclicGroup(2)]);
G2:=DirectProduct([CyclicGroup(2),CyclicGroup(2),CyclicGroup(2)]);
G3:=DirectProduct([CyclicGroup(2),CyclicGroup(2),CyclicGroup(2)]);
H:=DirectProduct([G1,G2,G3]);
PolyGroup:=func<seq|Group<a1,a2,a3,a4|
a1^seq[1], a2^seq[2],a3^seq[3],a4^seq[4], a1*a2*a3*a4>>;
P1:=PolyGroup([2,2,2,2]);
P2:=PolyGroup([2,2,2,2]);
P3:=PolyGroup([2,2,2,2]);
P:=DirectProduct([P1,P2,P3]);
f:=hom<P->H | H!(1,2),H!(3,4),H!(5,6),H!(1,2)(3,4)(5,6),
H!(7,8),H!(9,10),H!(11,12),H!(7,8)(9,10)(11,12),
H!(13,14),H!(15,16),H!(17,18),H!(13,14)(15,16)(17,18)>;
for i in Q do G:=h(M[i]);
s:=[]; for j in [1..3] do s[j]:=Id(H); end for;
for i in {1..3} do
for j in [1..9] do
if (G.i)[j] eq 1 then s[i]:=s[i]* H!(2*j-1,2*j);
end if; end for; end for;
GG1:=sub<H|s>;
Pi1:=Simplify(Rewrite(P,GG1@@f));
Append(~T, [* G, Pi1, AbelianQuotient(Pi1) *]);
end for;
Since the fundamental groups are infinite and the presentations given as output are quite long, we only list the respective first homology groups for the $16$ families of surfaces in Tables \[q0\], \[q1\], \[q2\] and \[q3\].
It is not obvious, from the presentation given as output of the MAGMA script, whether two of these fundamental groups are isomorphic or not. To check whether two different families have different fundamental groups, we can compare the number of normal subgroups of the fundamental group of index $k \leq m$ (in our case $m=6$). This can be done easily using the MAGMA function: `LowIndexNormalSubgroups(H, m)` which returns a sequence containing the normal subgroups of the finitely presented group $H$ of index $k \leq m$. This allows us to see that the fundamental groups of the families we constructed are pairwise non-isomorphic, except for two pairs: $(\mathcal S_{11}, \mathcal S_{12})$ and $(\mathcal S_{14}, \mathcal S_{15})$.\
Indeed in these cases, the fundamental groups are isomorphic. We verified this using the MAGMA function `SearchForIsomorphism(H, K, n)` which attempts to find an isomorphism of the finitely presented group $H$ with the finitely presented group $K$. The search is restricted to those homomorphisms for which the sum of the word-lengths of the images of the generators of $H$ in $K$ is at most $n$ (in our case $n=8$). The answer is given as follows: if an isomorphism $\phi$ is found, then the output is the triple (true, $\phi$, $\phi^{-1}$); otherwise, the output is ‘false’.
That these isomorphisms exist is no coincidence: we shall in fact show later that in both cases we have two families of surfaces which are contained in a larger irreducible family (see Sections \[modspace\] and \[BdFthree\]).
Since for a smooth projective surface $S$ it holds $q(S)=\frac12 \operatorname{rk}H_1(S, \mathbb Z)$, we have proved the following:
\[fundgroup\] Among the 16 families of generalized Burniat type surfaces four have $p_g=q=0$ (Table \[q0\]), eight have $p_g=q=1$ (Table \[q1\]), three have $p_g=q=2$ (Table \[q2\]) and one has $p_g=q=3$ (Table \[q3\]).\
Moreover, the fundamental groups of these families are pairwise non isomorphic, except for $\pi_1(S_{11})\cong\pi_1(S_{12})$ and $\pi_1(S_{14})\cong\pi_1(S_{15})$, where $S_j$ is in the family $\mathcal S_j$.
\[eqGBT\] We observe that the family $S_2$ in Table \[q0\] corresponds to the family of [*primary Burniat surfaces*]{} (cf. Section \[Inouedescription\]).
The moduli space of generalized Burniat type surfaces {#modspace}
=====================================================
The aim of this section is to describe the connected components of the Gieseker moduli space of surfaces of general type containing the isomorphism classes of the generalized Burniat type surfaces.
First we shall prove the following result:
Let $X$ be the canonical model of a generalized Burniat type surface $S$. Then the base of the Kuranishi family of $X$ is smooth.
Recall that $X$ is the quotient model of a generalized Burniat type surface $S$, and let $\hat{X} \rightarrow X$ be the canonical $G \cong (\mathbb Z / 2 \mathbb Z)^3$-cover. Then $\hat{X} \subset A^0=E_1 \times E_2 \times E_3$ is a hypersurface of multidegree $(2,2,2)$ having at most nodes as singularities.
It suffices to show (cf. [@superficial Proposition 4.5]) that the base of the Kuranishi family of $\hat{X}$ is smooth (since the base of Kuranishi family of $X$ is given by the $G$-invariant part of the base of the Kuranishi family of $\hat{X}$).
Now, since $\hat X$ moves in a family with smooth base of dimension $13=6+7$, it it is enough to show that $$\dim \operatorname{Ext}^1_{\mathcal{O}_{\hat X}} (\Omega^1_{\hat{X}},\mathcal{O}_{\hat X}) = 13.$$ Moreover, the Kodaira-Spencer map of the above family is a bijection, but we omit the verification here.
Indeed $\hat X \subset A^0$ is an ample divisor, and it suffices to apply the following lemma.
\[divisorinppav\] Let $A$ be an Abelian variety of dimension $n$ and let $D \subset A$ be an ample divisor. Then: $$\dim \operatorname{Ext}^1_{\mathcal{O}_D} (\Omega^1_D,\mathcal{O}_D) = \frac 12 n(n+1) + \dim |D|.$$
Consider the exact sequence $$0 \rightarrow \mathcal O_D (- D) \rightarrow \Omega_{A}^1 \otimes \mathcal O_D \cong \mathcal O_D^{\oplus n} \rightarrow \Omega^1_D \rightarrow 0.$$ Applying the functor $\operatorname{Hom}_{\mathcal O_D}(_-,\mathcal O_D)$, we obtain the long exact sequence: $$\label{ext}
\begin{array}{rl}
0 &\rightarrow \operatorname{Hom}(\Omega^1_{D},\mathcal O_{D})=0 \rightarrow \operatorname{Hom}(\mathcal O_{D}^{\oplus n},\mathcal O_{D}) \rightarrow \operatorname{Hom}(\mathcal O_{D} (-D),\mathcal O_{D}) \rightarrow \\
&\rightarrow \operatorname{Ext}^1(\Omega^1_{D},\mathcal O_{D}) \rightarrow \operatorname{Ext}^1(\mathcal O_{D}^{\oplus n},\mathcal O_{ D}) \rightarrow \operatorname{Ext}^1(\mathcal O_{D} (-D),\mathcal O_{D}) \rightarrow \\
&\rightarrow \operatorname{Ext}^2(\Omega^1_{D},\mathcal O_{D}) \rightarrow \operatorname{Ext}^2(\mathcal O_{D}^{\oplus n},\mathcal O_{D}) \rightarrow \operatorname{Ext}^2(\mathcal O_{ D} (- D),\mathcal O_{D}) \rightarrow \ldots .
\end{array}$$
We have that
- $\operatorname{Ext}^i(\mathcal O_{D}^{\oplus n},\mathcal O_{D}) = H^i(D, \mathcal O_{D})^{\oplus n}$;
- $\omega_{D} = \omega_{A} \otimes \mathcal O_{D} (D) = \mathcal O_{D} (D)$;
- $\operatorname{Ext}^i(\mathcal O_{ D} (- D),\mathcal O_{D}) \cong \operatorname{Ext}^i(\mathcal O_{ D},\mathcal O_{D}(D)) = \operatorname{Ext}^i(\mathcal O_{ D},\omega_{D}) \cong$\
$H^{n-1-i}(D,\mathcal O_{ D})^*$, where the last equality holds by Serre duality.
Next, we consider the short exact sequence:
$$0 \rightarrow \mathcal O_A (- D) \rightarrow \mathcal O_A \rightarrow \mathcal O_{ D} \rightarrow 0$$ and the associated long cohomology sequence
$$\begin{gathered}
\label{coh}
0 \rightarrow H^0(\mathcal O_A) \rightarrow H^0(\mathcal O_{ D}) \rightarrow H^1(\mathcal O_A (- D) )\rightarrow H^1(\mathcal O_A) \rightarrow \\\rightarrow H^1(\mathcal O_{ D})
\rightarrow H^2(\mathcal O_A (- D) )\rightarrow H^2(\mathcal O_A) \rightarrow \ldots
\rightarrow H^{n-1}(\mathcal O_{ D}) \rightarrow \\
\rightarrow H^n(\mathcal O_A (- D) )\rightarrow H^n(\mathcal O_A) \rightarrow 0.\end{gathered}$$
Note that by Serre duality $H^i(\mathcal O_A (- D) ) \cong H^{n-i}(\mathcal O_A (D) )^*$, and since $D \subset A$ is an ample divisor, we get that these cohomology groups are trivial for $i \leq n-1$ by the Kodaira vanishing theorem.
This implies that
- $\dim H^i(\mathcal{O}_D) = \dim H^i(\mathcal{O}_A) = \binom {n} {i}$, for $0 \leq i \leq n-2$,
- $\dim H^{n-1} (\mathcal{O}_D) = \dim |D| + n$.
Inserting this information in the long exact sequence (\[ext\]), we see that $$\dim \operatorname{Ext}^1_{\mathcal{O}_D} (\Omega^1_{D},\mathcal{O}_D) = \frac 12 n(n+1) + \dim |D| \,,$$ once we show that $$\varphi \colon \operatorname{Ext}^1(\mathcal O_{D}^{\oplus n},\mathcal O_{ D}) \rightarrow \operatorname{Ext}^1(\mathcal O_{D} (-D),\mathcal O_{D})$$ is surjective.
But $$\operatorname{Ext}^1(\mathcal O_{D}^{\oplus n},\mathcal O_{ D}) \cong H^1(\mathcal O_{ D})^{\oplus n} \cong H^1(\mathcal O_A^{\oplus n}) \cong H^1(\Theta_A)$$ and $$\operatorname{Ext}^1(\mathcal O_{D} (-D),\mathcal O_{D}) \cong H^{n-1-1}(\mathcal O_{ D})^* \cong H^{n-2}(\mathcal O_A)^* \cong H^2(\mathcal O_A),$$ where the first and third equality follow from Serre duality.
Composing with these isomorphisms, $\varphi$ becomes $$H^1(\Theta_A) \rightarrow H^2(\mathcal O_A),$$ the contraction with the first Chern class of $D$, an element of $H^1(A,\Omega^1_A)$, which is represented by a non degenerate alternating form. Hence the surjectivity of $\varphi$ follows.
Surfaces in $\mathcal S_j$ with $j \leq 10$, i.e., with $p_g=q\leq 1$.
----------------------------------------------------------------------
Recall the following definition.
\[DCIT\] A complex projective manifold $X$ is said to be a *diagonal classical Inoue-type manifold* if
1. $\dim(X)\geq 2$;
2. there is a finite group $G$ and a Galois étale $G$-covering $\hat X\rightarrow X\,(=\hat X/G)$ such that:
3. $\hat X$ is an ample divisor inside a $K(\Gamma,1)$-projective manifold $Z$ (hence by Lefschetz $\pi_1(\hat X) \cong \pi_1(Z)\cong \Gamma$) and moreover
4. the action of $G$ on $\hat X$ yields a faithful action on $\pi_1(\hat X) \cong \Gamma$: in other words the exact sequence $$1 \rightarrow \Gamma\cong \pi_1(\hat X) \rightarrow \pi_1(X) \rightarrow G \rightarrow 1$$ gives an injection $G\rightarrow \mathrm{Out}(\Gamma)$, defined by conjugation;
5. $Z=(A_1\times \ldots \times A_r)\times (C_1 \times \ldots \times C_s)$ where each $A_j$ is an Abelian variety and each $C_k$ is a curve of genus $g(C_k)\geq 2$;
6. the action of $G$ on $\hat X$ is induced by a diagonal action on $Z$;
7. the faithful action on $\pi_1(\hat X) \cong \Gamma$, induced by conjugation by lifts of elements of $G$, has the Schur property: $$\tag{SP}\label{SP}
\mathrm{Hom}(V_j,V_k)^G=0 \,, \quad \forall k \neq j\,,$$ where $V_j:=\Lambda_j \otimes \mathbb Q$, being $\Lambda_j:=\pi_1(A_j)$ (it suffices to verify that, for each $\Lambda_j$, there is a subgroup $H_j$ of $G$ for which $\mathrm{Hom}(V_j,V_k)^{H_j}=0 \,, \forall k \neq j$).
We say instead that $X$ is a *diagonal classical Inoue-type variety* if we replace the assumption of smoothness of $X$ by the assumption that $X$ has canonical singularities.
To fix the notation, let us call a surface $S$ a [*generalized Burniat type (GBT) surface of type $j$*]{} if $S$ belongs to the family $\mathcal S_j$ in Tables \[q0\]-\[q3\].
Let $X_j$ be the canonical model of a GBT surface $S_j$ of type $j$. Then the embedding $\hat X_j \subset A^0 = E_1 \times E_2 \times E_3$ realizes $X_j$ as a diagonal classical Inoue-type variety if and only if $1\leq j\leq 10$.
It is trivial to see that the canonical model of a generalized Burniat type surface $X_j=\hat X/G_j$ satisfies conditions (1-6) in Definition \[DCIT\]. Hence there remains only to determine which surfaces fulfill the Schur Property (\[SP\]).\
To verify the Schur Property one has to find, for each pair $j\neq k \in\{1,2,3\}$ an element $g\in G$ such that, $d g$ being the derivative of $g$, $\mathrm d g_{|E_j} \cdot \mathrm d g_{|E_k}= -1$. Let $j=1$ and $g=(0,1,0,1,1,0,1,1,0)\in G_1$: then $\mathrm d g_{|E_1}= -1$, $\mathrm d g_{|E_2}=\mathrm d g_{|E_3}= 1$, while for $g'=(0,0,0,0,0,1,1,0,1)\in G_1$ one has $\mathrm d g'_{|E_2}=-1$ and $\mathrm d g'_{|E_3}= 1$. Hence $X_1$ satisfies (\[SP\]). Considering a suitable pair of generators of $G_j$, one can prove in the same way that $X_j$ satisfies (\[SP\]) for $j=2,\ldots, 10$.
Consider now the case $j=11$ and let $g$ be one of the three generators of $G_{11}$ in Table \[q1\]. Then $\mathrm d g_{|E_1}= \mathrm d g_{|E_3}= -1$ and $\mathrm d g_{|E_2}=1$; this means that $\mathrm{Hom}(V_1,V_3)^G\neq0$, hence $X_{11}$ does not fulfill the Schur property.
In the same way one can show that $S_j$ does not fulfill (\[SP\]) for $j=12,\ldots, 16$.
We are now in the position to prove the following result.
Let $S$ be a smooth projective surface homotopically equivalent to a GBT surface $S_j$ of type $j$ with $1\leq j \leq 10$. Then $S$ is a GBT surface of type $j$, i.e., contained in the same irreducible family as $S_j$.
Assume that $S$ is homotopically equivalent to $S_j$ ($1 \leq j\leq 10$), hence in particular $S$ has the same fundamental group as $S_j$. Consider the étale $G_j \cong (\mathbb Z / 2 \mathbb Z)^3$-cover $\hat S \rightarrow S$. Then by [@BC12 Theorem 0.5] we have a splitting of the Albanese variety and an Albanese map $f \colon \hat S \rightarrow E_1 \times E_2 \times E_3$ which is generically finite onto its image $W$. By loc. cit. Lemma 1.2, $G_j$ acts on $E_1 \times E_2 \times E_3$ with the same action as for a GBT surface of type $j$. It is now easy to verify that there is no effective $G_j$-invariant divisor of numerical type $(1,1,1)$ on $E_1 \times E_2 \times E_3$, hence $W$ has class $2F_1 + 2F_2 + 2F_3$, where $F_i$ is the class of a fibre of the projection of $E_1 \times E_2 \times E_3$ on the $j$-th factor. Therefore $f$ is birational onto its image and one verifies as in loc. cit. that $W$ has at most rational double points as singularities and is therefore the canonical model $\hat X$ of $\hat S$.
[**Claim:**]{} $W$ is the pull-back of a Del Pezzo surface in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ for a suitable degree $({\ensuremath{\mathbb{Z}}}/2 {\ensuremath{\mathbb{Z}}})^3$-covering $\pi'$.
[*Proof of the claim.*]{} The pull back of a divisor of multidegree $(1,1,1)$ on $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ under any $$\pi' : E_1 \times E_2 \times E_3 \rightarrow \mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$$ is a divisor which has the same class as $W$: hence the two divisors are linearly equivalent to a translate of each other. Since the corresponding linear systems have the same dimension we infer that $W$ is the translate of such an effective divisor. Changing the origin of the Abelian variety $A^0$ we obtain another action of $({\ensuremath{\mathbb{Z}}}/2 {\ensuremath{\mathbb{Z}}})^3$ such that $W$ is invariant; the claim is thus proven.
We have therefore seen that $S$ is a GBT surface and has the same fundamental group as $S_j$. Thus by our classification $S$ is in the same irreducible family as $S_j$, whence $S$ is a GBT surface of type $j$.
The same conclusion holds under the weaker assumptions:
1\) $\pi_1(S) \cong \pi_1 ( S_j)$
2\) the corresponding covering $\hat{S}$, whose Albanese is a product of $3$ elliptic curves because of the Schur property, satisfies that the image of the Albanese map has class $(2,2,2)$.
We can now summarize our results in the following theorem
\[moduli\] The connected component $\mathfrak N_j$ of the Gieseker moduli space $\mathfrak M_{1,6}^{can}$ corresponding to generalized Burniat type surfaces of type $j$ ($1 \leq j \leq 10$) is irreducible, normal and unirational, of dimension 4 if $j=1$ or $2$, else of dimension 3.
We have shown that the Kuranishi family is smooth, hence the moduli space is normal. By the previous theorem each family of GBT surfaces with $ j \leq 10$ surjects onto a connected component of the Gieseker moduli space: since the family has a rational base (a projective bundle over a rational variety), follows the assertion about the unirationality.
Surfaces in $\mathcal S_j$ with $j=11,12$, having $p_g=q=1$.
------------------------------------------------------------
\
Since these surfaces do not fulfill the Schur property, the family constructed as $(\mathbb{Z}/2 \mathbb{Z})^3$-quotient of a Burniat hypersurface in a product of three elliptic curves is not complete. We will study these surfaces in greater generality in Section \[bdf\] and Section \[BdFthree\]. In fact, it turns out that the families $\mathcal S_{11}$, $\mathcal S_{12}$ yield two irreducible subsets each of codimension one in an irreducible connected component of dimension 4 of the moduli space of surfaces of general type with $p_g=q=1$, $K^2 =6$.
Surfaces in $\mathcal S_j$ with $j=13,14,15$, i.e., those with $p_g=q=2$
------------------------------------------------------------------------
\
We have three families (each of dimension 3, the number of moduli of the triple of elliptic curves) of GBT surfaces with $p_g = q =2$. We have already observed that the embedding $ \hat X_j \subset A^0= E_1\times E_2 \times E_3$ does not fulfill the Schur property. In fact, it is not difficult to show that each of the three families is a subfamily of a four dimensional irreducibile family, where the product of the two elliptic curves on which the projection of $G_j$ acts freely deforms to an Abelian surface $A_2$. In this case the embedding $ \hat X_j \subset E_1\times A_2$ fulfills the Schur property and we can show that we obtain in this way exactly two irreducible connected components of the moduli space of surfaces of general type.
We do not give more details here since these surfaces have already been classified in [@penpol].
Observe in fact the following:
Let $S$ be a GBT surface with $p_g(S) = q(S) =2$. Then $S$ is of Albanese general type and the Albanese map is generically of degree 2.
Assume $S$ to be of type $13$ (the proof in the other two cases is exactly the same) and consider the following diagram:
$$\xymatrix{
&\hat X\subset E_1\times E_2 \times E_3 \ar[d]_{G_{13}}\ar[r]^{\quad p_{23}}&E_2 \times E_3 \ar[d]^{p_{23}(G_{13})}\\
S\ar[r]&X \ar[r]^{a\qquad}& (E_2 \times E_3) /p_{23}(G_{13}) \
}$$ Since $p_{23} \colon \hat X \rightarrow E_2 \times E_3$ is generically finite of degree 2 (as $\hat X$ is a divisor of multidegree $(2,2,2)$), and since $G_{13} \cong p_{23}(G_{13})$, one sees immediately that $a$ is generically finite of degree 2 and that $\operatorname{Alb}(S) = (E_2 \times E_3) /p_{23}(G_{13})$.
We recall the following result due to Penegini and Polizzi:
\[albdc\] Let $\mathcal{M}$ be the moduli space of minimal surfaces $S$ of general type with $p_g=q=2$, $K_S^2 =6$ and Albanese map of degree 2. Then the following holds:
- $\mathcal{M}$ is the union of three irreducible connected components, namely $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$ and $\mathcal{M}_{II}$.
- $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$ and $\mathcal{M}_{II}$ are generically smooth of respective dimensions 4, 4, 3.
- The general surface in $\mathcal{M}_{Ia}$ and $\mathcal{M}_{Ib}$ has ample canonical class; all surfaces in $\mathcal{M}_{II}$ have ample canonical class.
It is immediately clear that the subset of the moduli space corresponding to GBT surfaces with $p_g=q=2$ cannot be contained in $\mathcal M_{II}$, since it has dimension 3, while the families $13,14,15$ yield irreducible families of dimension at least four. We have the following
\[genus\] Let $S_j$ be a GBT surface of type $j$ with $p_g= q= 2$, i.e., $j \in \{13,14,15\}$. Consider the pencil $f_j \colon S_j \rightarrow \mathbb P^1 \cong E_k/p_k(G_j)$, where $k=1$ for $S_{13}$ and $k=3$ for $j=14,15$. Then the general fibre of $f_j$ is a smooth curve of genus 5 if $j=13$ and of genus 3 if $j=14,15$.
Consider the diagram $$\xymatrix{
&\hat X_j\subset E_1\times E_2 \times E_3 \ar[d]_{G_{j}\cong (\mathbb Z / 2)^3}\ar[r]^{\qquad p_k}&
E_k \ar[d]^{p_k(G_{j})}\\
S_j\ar[r]&X_j \ar[r]^{f_j}& E_k /p_k(G_j) \
}$$ Note that the general fibre of $p_k$ is a divisor of bidegree $(2,2)$ in the product of two elliptic curves, whence has genus 5. Since $p_1(G_{13}) \cong (\mathbb Z/ 2 \mathbb Z)^3$, the genus of a general fibre of $f_{13}$ is 5, whereas $p_3(G_{14}), p_3(G_{15}) \cong (\mathbb Z/ 2 \mathbb Z)^2$, whence the genus of a general fibre of $f_{14}$ and $f_{15}$ is 3.
This allows us to conclude the following:
Let $S_j$ be a GBT surface of type $j$ with $p_g= q= 2$. Then the point of the Gieseker moduli space corresponding to $S_{13}$ lies in $\mathcal M_{Ia}$, whereas the point corresponding to $S_{14}$, resp. $S_{15}$, lies in $\mathcal M_{Ib}$.
In particular, GBT surfaces with $p_g=q=2$ of type 13 (resp. 14, 15) form a three dimensional subset of the four dimensional connected component $\mathcal M_{Ia}$ (resp. $\mathcal M_{Ib}$).
This follows from Lemma \[genus\] and [@penpol Proposition 22].
Consider for $j=13$ (the same holds also for $j=14,15$) the irrational pencil $f \colon S_{13} \rightarrow E_2 / p_2(G_{13})$. Observe that $E_2 / p_2(G_{13})$ is an elliptic curve and that the genus of the fibres of $f$ is $3$. This implies that $f$ is not isotrivial (otherwise it would be contained in the table of [@penegini]). This contradicts Theorem A of [@zucconi].
Surfaces in $\mathcal S_{16}$, i.e., those with $p_g=q=3$.
----------------------------------------------------------
\
Minimal surfaces of general type with $p_g=q=3$ are completely classified by the work of several authors (cf. [@CCML98; @Pir02; @HP02]).
A minimal surface of general type with $p_g = q = 3$ has $K^2 = 6$ or $K^2 = 8$ and, more precisely:
- if $K^2 = 6$, $S$ is the minimal resolution of the symmetric square of a curve of genus $3$;
- otherwise $S = (C_2 \times C_3)/\sigma$, where $C_g$ denotes a curve of genus $g$ and $\sigma$ is an involution of product type acting on $C_2$ as an elliptic involution (i.e., with elliptic quotient), and on $C_3$ as a fixed point free involution.
In particular, the moduli space of minimal surfaces of general type with $p_g = q = 3$ is the disjoint union of $\mathcal M_{6,3,3}$ and $\mathcal M_{8,3,3}$, which are both irreducible of respective dimension 6 and 5.
We get:
Generalized Burniat type surfaces with $p_g=q=3$ (i.e. of type 16) form a three dimensional subset of the six dimensional connected component $\mathcal M_{6,3,3}$.
Bagnera-de Franchis varieties {#bdf}
=============================
A [*Generalized Hyperelliptic Variety (GHV)*]{} $X$ is defined to be a quotient $X= A/G$ of an Abelian Variety $A$ by a nontrivial finite group $G$ acting freely, and with the property that $G$ contains no translations.
Remark that, if $G$ is any group acting freely on $A$, which is not a subgroup of the group of translations, then the quotient $X= A/G$ is a GHV. Because the subgroup $G_T$ of translations in $G$ is a normal subgroup of $G$, and, if we denote $G' = G/G_T$, then $A/G = A' / G'$, where $A'$ is the Abelian variety $ A' : = A/G_T$.
1\) A [*Bagnera-de Franchis variety*]{} (for short: BdF variety) is a GHV $ X= A/G $ such that $G \cong \mathbb{Z}/m \mathbb{Z}$ is a cyclic group.
2\) A [*Bagnera-de Franchis variety of product type*]{} is a Bagnera-de Franchis variety $ X= A/G $, where $A = (A_1 \times A_2)$, $A_1, A_2$ are Abelian Varieties, and $G \cong \mathbb{Z}/m \mathbb{Z}$ is generated by an automorphism of the form $$g(a_1, a_2 ) = ( a_1 + \beta_1, \alpha_2 (a_2)),$$
where $\beta_1 \in A_1[m]$ is an element of order exactly $m$, and similarly $\alpha_2 : A_2 \rightarrow A_2$ is a linear automorphism of order exactly $m$ without $1$ as eigenvalue (these conditions guarantee that the action is free).
3\) If moreover all eigenvalues of $\alpha_2 $ are primitive $m$-th roots of $1$, we shall say that $ X= A/G $ is a [*primary Bagnera-de Franchis variety*]{}.
1\) One can give a similar definition of Bagnera-de Franchis manifolds, requiring only that $A, A_1, A_2$ be complex tori.
2\) If $A$ has dimension $n=2$, the Bagnera-de Franchis manifolds coincide with the Generalized Hyperelliptic varieties, due to the classification result of Bagnera-de Franchis in [@BdF].
We have the following proposition, giving a characterization of Bagnera-de Franchis varieties.
\[quotprodtype\] Every Bagnera-de Franchis variety $ X= A/G$ is the quotient of a Bagnera-de Franchis variety of product type, $(A_1 \times A_2)/ G$ by any finite subgroup $T$ of $(A_1 \times A_2)$ which satisfies the following properties:
1. $T$ is the graph of an isomorphism between two respective subgroups $ T_1 \subset A_1, \ T_2 \subset A_2$,
2. $(\alpha_2 - \operatorname{Id}) T_2 = 0$,
3. if $ g (a_1, a_2 ) = ( a_1 + \beta_1, \alpha_2 (a_2)) ,$ then the subgroup of order $m$ generated by $\beta_1$ intersects $T_1$ only in $\{0\}$.
In particular, we may write $X$ as the quotient $X = (A_1 \times A_2)/ (G \times T)$ of $ A_1 \times A_2 $ by the Abelian group $G \times T$.
We refer to [@FabTop].
Actions of a finite group on an Abelian variety
-----------------------------------------------
Assume that we have the action of a finite group $G$ on a complex torus $ A = V / \Lambda$. Since every holomorphic map between complex tori lifts to a complex affine map of the respective universal covers, we can attach to the group $G$ the group of affine transformations $\Gamma$, which consists of all affine maps of $V$ which lift transformations of $G$. Then $\Gamma$ fits into an exact sequence: $$1 \longrightarrow \Lambda \longrightarrow \Gamma \longrightarrow G \longrightarrow 1\,.$$
The following is a slight improvement of [@BC12 Lemma 1.2]:
The group $\Gamma$ determines the real affine type of the action of $\Gamma$ on $V$ (respectively: the rational affine type of the action of $\Gamma$ on $
\Lambda \otimes \mathbb{Q}$), in particular the above exact sequence determines the action of $G$ up to real affine isomorphism of $A$ (resp.: rational affine isomorphism of $
( \Lambda \otimes \mathbb{Q} )/ \Lambda$).
It is clear that $ V = \Lambda \otimes_{\mathbb{Z}} \mathbb{R}$ as a real vector space, and we denote by $V_{\mathbb{Q}}: =
\Lambda \otimes \mathbb{Q}$. Let $$\Lambda' : = \ker ( \alpha_L \colon \Gamma \rightarrow \operatorname{GL}(V_{\mathbb{Q}}) \subset \operatorname{GL}(V)) ,$$ $$\overline{G}_1 : = \operatorname{im}( \alpha_L \colon \Gamma \rightarrow \operatorname{GL}(V_{\mathbb{Q}}))\,.$$ The group $\Lambda'$ is obviously Abelian, contains $\Lambda$, and maps isomorphically onto a lattice $\Lambda' \subset V$.
In turn $ V = \Lambda' \otimes_{\mathbb{Z}} \mathbb{R}$, and, if $ G' : = \Gamma / \Lambda'$, then $ G' \cong \overline{G}_1 $ and the exact sequence $$1 \longrightarrow \Lambda' \longrightarrow \Gamma \longrightarrow G' \longrightarrow 1\, ,$$ since we have an embedding $ G' \subset \operatorname{GL}(\Lambda')$, shows that the affine group $ \Gamma \subset \operatorname{Aff}(\Lambda') \subset \operatorname{Aff}(V) $ is uniquely determined ($\Gamma$ is the inverse image of $G'$ under $\operatorname{Aff}(\Lambda') \rightarrow \operatorname{GL}(\Lambda')$).
There remains only to show that $\Lambda'$ is determined by $\Gamma$ as an abstract group, independently of the exact sequence we started with. In fact, one property of $\Lambda'$ is that it is a maximal Abelian subgroup, normal and of finite index.
Assume that $\Lambda''$ has the same property: then their intersection $\Lambda^0 : = \Lambda' \cap \Lambda''$ is a normal subgroup of finite index, in particular $\Lambda^0 \otimes_{\mathbb{Z}} \mathbb{R} = \Lambda' \otimes_{\mathbb{Z}} \mathbb{R} = V $; hence $\Lambda'' \subset \ker ( \alpha_L \colon \Gamma \rightarrow \operatorname{GL}(V))= \Lambda'$, where $\alpha_L$ is induced by conjugation on $\Lambda^0$.
By maximality $\Lambda' = \Lambda''$.
Observe that, in order to obtain the structure of a complex torus on $V / \Lambda'$, we must give a complex structure on $V$ which makes the action of $G' \cong \overline{G}_1$ complex linear.
In order to study the moduli spaces of the associated complex manifolds, we introduce therefore a further invariant, called the Hodge type, according to the following definition.
Given a faithful representation $ G \rightarrow \operatorname{Aut}(\Lambda)$, where $\Lambda$ is a free Abelian group of even rank $2n$, a [*$G$-Hodge decomposition*]{} is a $G$-invariant decomposition $$\Lambda \otimes \mathbb{C} = H^{1,0} \oplus H^{0,1}, \ H^{0,1} = \overline{H^{1,0}}.$$ Write $\Lambda \otimes \mathbb{C}$ as the sum of isotypical components $$\Lambda \otimes \mathbb{C} = \oplus_{\chi \in \operatorname{Irr}(G)}
U_{ \chi}.$$
Write also $U_{ \chi} = W_{ \chi} \otimes M_{ \chi}$, where $W_{ \chi} $ is the given irreducible representation, and $ M_{ \chi} $ is a trivial representation of dimension $n_{ \chi} $.
Then $V : = H^{1,0} = \oplus_{\chi \in \operatorname{Irr}(G)}
V_{ \chi},$ where $V_{ \chi} = W_{ \chi} \otimes M^{1,0}_{ \chi}$ and $M^{1,0}_{ \chi}$ is a subspace of $M_{ \chi}$. The [*Hodge type*]{} of the decomposition is the datum of the dimensions $$\nu ( \chi): = \dim_{\mathbb{C}} M^{1,0}_{ \chi}$$ corresponding to the Hodge summands for non real representations (observe in fact that one must have: $\nu ( \chi) + \nu ( \bar{\chi}) = \dim ( M_{ \chi})$).
Given a faithful representation $ G \rightarrow \operatorname{Aut}(\Lambda)$, where $\Lambda$ is a free Abelian group of even rank $2n$, all the $G$-Hodge decompositions of a fixed Hodge type are parametrized by an open set in a product of Grassmannians. Since, for a non real irreducible representation $\chi$ one may simply choose $M^{1,0}_{ \chi}$ to be a complex subspace of dimension $\nu ( \chi)$ of $ M_{ \chi}$, and for $M_{ \chi} = \overline{(M_{ \chi})}$, one simply chooses a complex subspace $M^{1,0}_{ \chi}$ of half dimension. Then the open condition is just that (since $ M^{0,1}_{ \chi} : = \overline {M^{1,0}_{ \chi} }$) we want $ M_{ \chi} = (M^{1,0}_{ \chi} ) \oplus (M^{0,1}_{ \chi} ) $, or, equivalently, $M_{ \chi} = (M^{1,0}_{ \chi}) \oplus \overline{(M^{1,0}_{\bar{ \chi}} )}$.
Bagnera-de Franchis varieties of small dimension
------------------------------------------------
We have shown that a Bagnera-de Franchis variety $X=A/G$ can be seen as the quotient of one of product type $(A_1 \times A_2)/G$ by a finite subgroup $T$ of $A_1 \times A_2$, satisfying the properties stated in Proposition \[quotprodtype\].
Dealing with appropriate choices of $T$ is the easy part, since, as we saw, the points $t_2$ of $T_2$ satisfy, by property (2), $ \alpha_2 (t_2) = t_2$.
It suffices then to choose $T_2 \subset A_2[*]:= \ker (\alpha_2 - \operatorname{Id}_{A_2})$, which is a finite subgroup of $A_2$, and then to pick an isomorphism $\psi \colon T_2 \rightarrow T_1 \subset A_1$, such that $ T_1 : = \operatorname{im}(\psi) \cap \langle \langle \beta_1 \rangle \rangle = \{ 0\}$.
We therefore restrict ourselves from now on to [*Bagnera-de Franchis varieties of product type*]{} and we show now how to further reduce to the investigation of [*primary Bagnera-de Franchis varieties*]{}.
In fact, in the case of a BdF variety of product type, $\Lambda_2$ is a $G$-module, hence a module over the group ring $$R : = R(m) : = \mathbb{Z}[G] \cong \mathbb{Z}[x] / (x^m - 1).$$
The ring $R$ is in general far from being an integral domain, since indeed it can be written as a direct sum of cyclotomic rings, which are the integral domains defined as $R_k: = \mathbb{Z}[x] / (P_k(x))$. Here $P_k(x)$ is the $k$-th cyclotomic polynomial $$P_k(x) = \prod_{0 < j < k,\ (k,j)=1} (x - \epsilon^j)\,,$$ where $\epsilon = \exp ( 2 \pi i /k)$. Then $$R (m) = \oplus_ { k | m} R_k \,.$$
The following elementary lemma, together with the splitting of the vector space $V$ as a direct sum of eigenspaces for $g$, yields a decomposition of $A_2$ as a direct product $ A_2 = \oplus_{ k | m} A_{2,k} $ of $G$ -invariant Abelian subvarieties $A_{2,k} $ on which $g$ acts with eigenvalues of order precisely $k$.
Assume that $M$ is a module over a ring $R = \oplus_k R_k$. Then there is a unique direct sum decomposition $$M = \oplus_k M_k,$$ such that
- $M_k$ is an $R_k$-module, and
- the $R$-module structure of $M$ is obtained through the projections $ R \rightarrow R_k$.
We can write the identity in $R$ as a sum of idempotents $1 = \Sigma_k e_k$, where $e_k$ is the identity of $R_k$, and $ e_k e_j = 0$ for $ j \neq k$.
Then each element $w \in M$ can be written as $$w = 1 w = ( \Sigma_k e_k) w = \Sigma_k e_k w = : \Sigma_k w_k.$$
Hence $ M_k$ is defined as $ e_k M$.
1\) If we have a primary Bagnera-de Franchis variety, then $\Lambda_2$ is a module over the integral domain $R : = R_m: = \mathbb{Z}[x] / (P_m(x))$.
Since $\Lambda_2$ is a projective $R$-module, $\Lambda_2$ splits as the direct sum $\Lambda_2 = R^r \oplus I $ of a free module with an ideal $ I \subset R$ (see [@Milnor Lemmas 1.5 and 1.6]), and $\Lambda_2$ is indeed free if the class number $h(R)=1$. The integers $m$ for which this occurs are listed in the table on [@washington page 353].
2\) To give a complex structure to $A_2 : = (\Lambda_2 \otimes_{\mathbb{Z}} \mathbb{R} )/ \Lambda_2$ it suffices to give a decomposition $\Lambda_2 \otimes_{\mathbb{Z}} \mathbb{C} = V \oplus \overline{V}$, such that the action of $x$ is holomorphic. This is equivalent to asking that $V$ is a direct sum of eigenspaces $ V_{\lambda}$, for $\lambda = \epsilon^j$ a primitive $m$-th root of unity.
Writing $U: = \Lambda_2 \otimes_{\mathbb{Z}} \mathbb{C} = \oplus U_{\lambda}$, the desired decomposition is obtained by choosing, for each eigenvalue $\lambda$, a decomposition $ U_{\lambda} = U_{\lambda}^{1,0} \oplus U_{\lambda}^{0,1} $ such that $ \overline{U_{\lambda}^{1,0}} = U^{0,1}_{\overline{\lambda}}$.
The simplest case (see [@cacicetraro] for more details) is the one where $I =0, r=1$, hence $ \dim (U_{\lambda} ) = 1$. Therefore we have only a finite number of complex structures, depending on the choice of the $\frac{\varphi(m)}{2}$ indices $j$ such that $ U_{\epsilon^j} = U^{1,0}_{\epsilon^j}$ (here $\varphi(m)$ is the [*Euler function*]{}).
Observe that the classification of BdF varieties in small dimension is possible thanks to the observation that the $\mathbb{Z}$-rank of $R$ (or of any ideal $I \subset R$) cannot exceed the real dimension of $A_2$: in other words we have $$\varphi(m) \leq 2 (n-1),$$ where $\varphi(m)$ is the Euler function, which is multiplicative for relatively prime numbers, and satisfies $\varphi(p^r) = (p-1) p^{r-1}$, if $p$ is a prime number.
For instance, if $n \leq 3$, then $\varphi(m) \leq 4$. Observe that $\varphi(p^r) \leq 4$ iff
- $p=3$, $5$ and $r=1$, or
- $p=2$, $r \leq 3$.
Hence, for $n \leq 3$, the only possibilities for $m$ are
- $\varphi(m) =1$: $m=2$;
- $\varphi(m) =2$: $m = 3,4,6$;
- $\varphi(m) =4$: $m = 5, 8, 10, 12$.
The classification is then also made easier by the fact that, in the above range for $m$, $R_m$ is a P.I.D., hence every torsion free module is free. In particular $\Lambda_2$ is a free $R$-module.
The classification for $n = 4$, since we must have $\varphi(m) \leq 6$, is going to include also the case $m=7, 9$.
We state now a result which will be useful in Section \[BdFthree\].
The Albanese variety of a Bagnera-de Franchis variety $X = A/G$ is the quotient $A_1 / (T_1 + \langle \langle \beta_1 \rangle \rangle)$.
Observe that the Albanese variety $H^0(\Omega^1_X)^{\vee}/ \operatorname{im}(H_1(X, \mathbb{Z}))$ of $X = A/G$ is a quotient of the vector space $V_1$ by the image of the fundamental group of $X$ (actually of its abelianization, the first homology group $H_1(X, \mathbb{Z})$): since the dual of $V_1$ is the space of $G$-invariant forms on $A$, $H^0(\Omega^1_A)^G \cong H^0(\Omega^1_X)$.
We also observe that there is a well defined map $X \rightarrow A_1 / (T_1 + \langle \langle \beta_1 \rangle \rangle)$, since $T_1$ is the first projection of $T$. The image of the fundamental group of $X$ contains the image of $\Lambda$, which is precisely the extension of $\Lambda_1$ by the image of $T$, namely $T_1$. Since we have the exact sequence $$1 \longrightarrow \Lambda = \pi_1 (A) \longrightarrow \pi_1 (X) \longrightarrow G \longrightarrow 1$$ the image of the fundamental group of $X$ is generated by the image of $\Lambda$ and by the image of the transformation $g$, which however acts on $A_1$ by translation by $\beta_1 = [b_1]$.
Unlike the case of complex dimension $n=2$, there are Bagnera-de Franchis varieties $X = A/G$ with trivial canonical divisor, for instance an elementary example is given by any BdF variety which is standard (i.e., has $m=2$) and is such that $A_2$ has even dimension.
Line bundles on quotients and linearizations
--------------------------------------------
Recall the following well known result (see Mumford’s books [@abvar], [@GIT]).
\[linearization\] Let $ Y = X/G$ be a quotient algebraic variety and let $p \colon X \rightarrow Y$ be the quotient map. Then:
1. there is a functor between
- line bundles $\mathcal{L}'$ on $Y$ and
- $G$-linearized line bundles $\mathcal{L}$,
associating to $\mathcal{L}'$ its pull back $p^* (\mathcal{L}')$.
2. The functor $\mathcal{L} \mapsto p_*(\mathcal{L})^G$ is a right inverse to the previous one, and $p_*(\mathcal{L})^G$ is invertible if the action is free, or if $Y$ is smooth.
3. Given a line bundle $\mathcal{L}$ on $X$, it admits a $G$-linearization if and only if there is a Cartier divisor $D$ on $X$, which is $G$-invariant and such that $\mathcal{L} \cong \mathcal{O}_X(D) = \{ f \in \mathbb{C}(X) | \operatorname{div}(f) + D \geq 0\}.$
4. A necessary condition for the existence of a $G$-linearization on a line bundle $\mathcal{L}$ on $X$ is that $$\label{neccond}
\forall g \in G, \ g^* ( \mathcal{L} ) \cong \mathcal{L}.$$
If condition (\[neccond\]) holds for $(\mathcal{L},G)$, one defines the [*Theta group*]{} of $\mathcal{L}$ as: $$\Theta (\mathcal{L}, G) : = \{ (\psi , g ) | g \in G, \ \psi : g^* ( \mathcal{L} ) \rightarrow \mathcal{L} \ {\rm is \ an \ isomorphism} \},$$ and there is an exact sequence $$\label{theta}
1 \longrightarrow \mathbb{C}^* \longrightarrow \Theta (\mathcal{L}, G) \longrightarrow G \longrightarrow 1.$$
- The splittings of the above sequence correspond to the $G$-linearizations of $\mathcal{L}$.
- If the sequence splits, the linearizations are a principal homogeneous space over the dual group $ \operatorname{Hom}(G, \mathbb{C}^*) = : G^*$ of $G$ (namely, each linearization is obtained from a fixed one by multiplying with an arbitrary element in $ \operatorname{Hom}(G, \mathbb{C}^*) = : G^*$).
Thus, the question of the existence of a $G$-linearization on a line bundle $\mathcal{L}$ is reduced to the algebraic question of the splitting of the central extension (\[theta\]) given by the Theta group. This question is addressed by group cohomology theory, as follows (for details see [@BAII]).
Let $\mathcal{L}$ be an invertible sheaf on $X$, whose class in $\operatorname{Pic}(X)$ is $G$-invariant. Then there exists a $G$-linearization of $\mathcal{L}$ if and only if the extension class $ [\psi] \in H^2(G, \mathbb{C}^*)$ of the exact sequence (\[theta\]) induced by the Theta group $\Theta (G, \mathcal{L})$ is trivial.
The group $H^2(G, \mathbb{C}^*)$ is the group of [*Schur multipliers*]{} (see again [@BAII page 369]).
Schur multipliers occur naturally when we have a projective representation of a group $G$. Since, if we have a homomorphism $ \varphi \colon G \rightarrow \mathbb{P} \operatorname{GL}(r, \mathbb{C})$, we can pull back the central extension $$1 \longrightarrow \mathbb{C}^* \longrightarrow \operatorname{GL}(r, \mathbb{C}) \longrightarrow \mathbb{P} GL(r, \mathbb{C}) \longrightarrow 1$$ via $\varphi$, we obtain an exact sequence $$1 \longrightarrow \mathbb{C}^* \longrightarrow \hat{G} \longrightarrow G\longrightarrow 1,$$ and the extension class $[\psi] \in H^2(G, \mathbb{C}^*)$ is the obstruction to lifting the projective representation to a linear representation $ G \rightarrow \operatorname{GL}(r, \mathbb{C})$.
It is an important remark that, if the group $G$ is finite, and $ n = \mathrm{ord} (G)$, then the cocycles take values in the group of roots of unity $\mu_n : = \{ z \in \mathbb{C}^* | z^n = 1 \}$.
1\) Let $E$ be an elliptic curve with origin $O$, and let $G$ be the group of $2$-torsion points $G : = E[2] \cong (\mathbb{Z}/2 \mathbb{Z})^2$, acting by translations on $E$. The divisor class of $2 O$ is never represented by a $G$-invariant divisor, since all the $G$-orbits consist of $4$ points, and the degree of $2O$ is not divisible by $4$. Hence, $\mathcal{L} : = \mathcal{O}_E (2O)$ does not admit a $G$-linearization. However, we have a projective representation on $\mathbb{P}^1 = \mathbb{P} (H^0( \mathcal{O}_E (2O)))$, where each non zero element $\eta_1$ of the group fixes 2 divisors: the sum of the two points corresponding to $\pm \frac{\eta_1}{2}$, and its translate by another element $\eta_2 \in E[2]$.
The two group generators yield two linear transformations, which act on $ V : = H^0( \mathcal{O}_E (2O))= \mathbb{C} x_0 \oplus \mathbb{C} x_1$ as follows: $$\eta_1(x_0) = x_1, \eta_1(x_1) = x_0, \ \eta_2 (x_j) = (-1)^j x_j.$$ The linear group generated is however $D_4 \neq G$, since $$\eta_1\eta_2 (x_0) = x_1, \ \eta_1\eta_2 (x_1) = - x_0.$$ 2) The previous example is indeed a special case of the [*Heisenberg extension*]{}, and $V $ generalizes to the [*Stone-von Neumann representation*]{} associated to an Abelian group $G$.
This is simply the space $ V : = L^2 (G, \mathbb{C})$ of square integrable functions on $G$ (see [@igusa],[@abvar]):
- $G$ acts on $ V : = L^2 (G, \mathbb{C})$ by translation $f (x) \mapsto f (x - g)$,
- $G^*$ acts on $V$ by multiplication with the given character $ f (x) \mapsto f (x) \cdot \chi (x)$, and
- the commutator $[g, \chi ]$ acts on $V$ by the scalar multiplication with the constant $ \chi (g)$.
The Heisenberg group is the group of automorphisms of $V$ generated by $G$, $G^*$ and by $\mathbb{C}^*$ acting by scalar multiplication. Then there is a central extension $$1 \longrightarrow \mathbb{C}^* \longrightarrow \operatorname{Heis}(G) \longrightarrow G \times G^* \longrightarrow 1 ,$$ whose class in $H^2 ( G \times G^*, \mathbb{C}^*)$ is given by the $\mathbb{C}^*$-valued bilinear form $$\beta \colon (g, \chi) \mapsto \chi (g) \in \Lambda^2 ( \operatorname{Hom}( G \times G^*, \mathbb{C}^*)) \subset H^2 ( G \times G^*, \mathbb{C}^*).$$
The relation with Abelian varieties $ A = V / \Lambda$ is through the Theta group associated to an ample divisor $L$.
In fact, by the theorem of Frobenius the alternating form $ c_1(L) \in H^2(A, \mathbb{Z}) \cong \wedge^2 ( \operatorname{Hom}(\Lambda, \mathbb{Z}))$ admits, in a suitable basis of $\Lambda$, the normal form $$D: = \begin{pmatrix}
0 & D' \\
- D' & 0
\end{pmatrix} ,$$ where $D' : = \operatorname{diag}(d_1, d_2, \dots , d_g)$, $d_1\mid d_2 \mid \dots \mid d_g$.
If one sets $G : = \mathbb{Z}^g/ D' \mathbb{Z}^g$, then $L$ is invariant under $G \times G^* \cong G \times G \subset A$, acting by translation, and the Theta group of $L$ is just isomorphic to the Heisenberg group $\operatorname{Heis}(G)$.
The nice part of the story is the following very useful result, which was used by Atiyah in the case of elliptic curves to study vector bundles on these (cf. [@atiyah]). We give a proof even if the result is well known.
\[Heisenberg\] Let $G$ be a finite Abelian group, and let $ V : = L^2 (G, \mathbb{C})$ be the Stone-von Neumann representation. Then $V \otimes V^{\vee}$ is a representation of $G \times G^*$ and splits as the direct sum of all the characters of $G \times G^*$.
Since the centre $\mathbb{C}^*$ of the Heisenberg group $ \operatorname{Heis}(G)$ acts trivially on $V \otimes V^{\vee}$, we have that $V \otimes V^{\vee}$ is a representation of $G \times G^*$. Observe that $G \times G^*$ is equal to its group of characters, and its cardinality equals the dimension of $V \otimes V^{\vee}$, hence it suffices (and it will also be useful for applications) to write for each character of $G \times G^*$ an explicit eigenvector.
We shall use the letters $g, h, k$ for elements of $G$, and the greek letters $\chi, \eta, \xi$ for elements in the dual group. Observe that $V$ has two bases, one given by $\{ g \in G\}$, and the other given by the characters $\{ \chi \in G^* \}$. The [*Fourier transform*]{} $\mathcal{F}$ yields an isomorphism of the vector spaces $ V : = L^2 (G, \mathbb{C})$ and $ W : = L^2 (G^*, \mathbb{C})$: $$\mathcal{F} (f) := \hat{f}, \ \hat{f}(\chi) : = \int f(g) ( \chi, g) \ \mathrm{d}g.$$ The action of $h \in G$ on $V$ sends $ f(g) \mapsto f (g-h)$, hence for the characteristic functions in $\mathbb{C}[G]$, $h\in G$ acts as $ g \mapsto g + h$. Instead $\eta \in G^*$ sends $ f \mapsto f \cdot \eta$, hence $ \chi \mapsto \chi + \eta$. Note that we use the additive notation also for the group of characters.
Restricting $V$ to the [*finite Heisenberg group*]{}, which is a central extension of $ G \times G^*$ by $\mu_n$, we get a unitary representation, hence we identify $V^{\vee}$ with $\bar{V}$. Then a basis of $V \otimes \bar{V}$ is given by the set $\{ g \otimes \bar{\chi}\}$.
Given a vector $w:= \sum_{g, \chi} a_{g, \chi} (g \otimes \bar{\chi} ) \in V \otimes \bar{V}$, then the action by $h \in G$ is given by $$h(w) = \sum_{g, \chi} (\chi, h) a_{g -h, \chi} (g \otimes \bar{\chi} ),$$ while the action by $ \eta \in G^*$ is given by $$\eta(w) = \sum_{g, \chi} (\eta, g) a_{g, \chi - \eta} (g \otimes \bar{\chi} ).$$
Hence one verifies right away that $$F_{k, \xi} : = \sum_{g, \chi} ( \chi - \xi, g - k) (g \otimes \bar{\chi} )$$ is an eigenvector with character $ (\xi, h) (\eta, k)$ for $ (h, \eta) \in ( G \times G^*)$.
A surface in a Bagnera-de Franchis threefold {#BdFthree}
============================================
Let $A_1$ be an elliptic curve, and let $A_2$ be an Abelian surface together with a line bundle $L_2$ yielding a polarization of type $(1,2)$.
Take on $A_1$ the line bundle $L_1=\mathcal{O}_{A_1} ( 2 O)$, and let $L$ be the line bundle on $A' : = A_1 \times A_2$, obtained as the exterior tensor product of $L_1$ and $L_2$, so that $$H^0 (A', L) = H^0 (A_1, L_1) \otimes H^0 (A_2, L_2) .$$ Moreover, we choose the origin in $A_2$ such that the space of sections $H^0 (A_2, L_2) $ consists only of [*even*]{} sections (hence, we shall no longer be free to further change the origin by an arbitrary translation).
We want to construct a Bagnera-de Franchis variety $X: = A/ G$, where
- $A = (A_1 \times A_2) / T$, and $G \cong T \cong \mathbb{Z}/2 \mathbb{Z}$, such that
- there is a $G\times T$ invariant divisor $D \in |L|$, whence we get a surface $S =D/(T \times G) \subset X$, with $K_S^2 = \frac{1}{4} K_D^2 =
\frac{1}{4} D^3 = 6$.
Write as usual $A_1 = \mathbb{C} / \mathbb{Z} \oplus \mathbb{Z} \tau$, and let $A_2 = \mathbb{C}^2 / \Lambda_2$. Suppose moreover, that $\lambda_1,\lambda_2, \lambda_3, \lambda_4$ is a basis of $\Lambda_2$ such that with respect to this basis the Chern class of $L_2$ is in Frobenius normal form. Let then $G=\langle g \rangle \cong \mathbb{Z}/ 2 \mathbb{Z}$ act on $A_1 \times A_2$ by $$\label{BCF} g(a_1, a_2 ) : = (a_1 + \frac{\tau}{2}, - a_2 + \frac{\lambda_2}{2}),$$ and define $T : = ( \mathbb{Z}/2 \mathbb{Z}) ( \frac 12 ,\frac{\lambda_4}{2})$.
Now, $G \times T$ surjects onto the group of two torsion points $A_1[2]$ of the elliptic curve, and also on the subgroup $ ( \mathbb{Z}/2 \mathbb{Z}) ( \lambda_2 / 2) \oplus ( \mathbb{Z}/2 ) ( \lambda_4 / 2) \subset A_2[2]$. Moreover, both $H^0 (A_1, L_1) $ and $ H^0 (A_2, L_2)$ are the Stone-von Neumann representation of the finite Heisenberg group of $G$, which is a central $\mathbb{Z}/2\mathbb{Z}$ extension of $ G \times T$.
By Proposition \[Heisenberg\], since in this case $ V \cong \overline{V}$ (the only roots of unity occurring are just $\pm 1$), we conclude that there are exactly 4 divisors in $|L|$, invariant by:
- $ (a_1, a_2) \mapsto (a_1, - a_2)$ (since the sections of $L_2$ are even),
- $(a_1, a_2) \mapsto (a_1 + \frac{\tau}{2}, a_2 + \frac{\lambda_2}{2})$, and
- $(a_1, a_2) \mapsto (a_1 + \frac 12, a_2 + \frac{\lambda_4}{2})$.
Hence these four divisors descend to give four surfaces $S_i \subset X$, $i\in \{1,2,3,4\}$.
\[BCF2\] Let $S$ be a minimal surface of general type with invariants $K_S^2 = 6$, $p_g(S) =q(S) = 1$ such that
- there exists an unramified double cover $ \hat{S} \rightarrow S$ with $ q (\hat{S}) = 3$, and such that
- the Albanese morphism $ \hat{\alpha} \colon \hat{S} \rightarrow A = \operatorname{Alb}(\hat{S})$ is birational onto its image $Z$, a divisor in $A$ with $ Z^3 = 12$.
1\) Then the canonical model of $\hat{S}$ is isomorphic to $Z$, and the canonical model of $S$ is isomorphic to $Y = Z / (\mathbb{Z}/2 \mathbb{Z})$, which a divisor in a Bagnera-de Franchis threefold $ X: = A/ G$, where $A = (A_1 \times A_2) / T$, $ G \cong T \cong \mathbb{Z}/2 \mathbb{Z}$, and where the action is as in (\[BCF\]).
2\) These surfaces exist, have an irreducible four dimensional moduli space, and their Albanese map $\alpha \colon S \rightarrow A_1 = A_1/ A_1[2]$ has general fibre a non hyperelliptic curve of genus $g=3$.
By assumption the Albanese map $ \hat{\alpha} \colon \hat{S} \rightarrow A$ is birational onto $Z$, and we have $ K_{ \hat{S}}^2 = 12 = K_Z^2$, since $\mathcal{O}_Z(Z)$ is the dualizing sheaf of $Z$.
We shall argue similarly to [@BC12 Step 4 of Theorem 0.5, page 31]. Denote by $W$ the canonical model of $\hat{S}$, and observe that by adjunction (see loc. cit.) we have $ K_W = \hat{\alpha}^* (K_Z ) - \mathfrak A$, where $\mathfrak A$ is an effective $\mathbb{Q}$-Cartier divisor.
We observe now that $K_Z$ and $K_W$ are ample, hence we have an inequality, $$12 = K_W^2 = (\hat{\alpha}^* (K_Z ) - \mathfrak A)^2 = K_Z^2 - (\hat{\alpha}^* (K_Z ) \cdot \mathfrak A) - (K_W \cdot \mathfrak A) \geq K_Z^2 = 12,$$ and since both terms are equal to $12$, we conclude that $\mathfrak A= 0$, which means that $K_Z$ pulls back to $K_W$, whence $W$ is isomorphic to $Z$. We have a covering involution $ \iota \colon \hat{S} \rightarrow \hat{S}$, such that $ S = \hat{S} / \iota$. Since the action of $\mathbb{Z}/2\mathbb{Z}$ is free on $\hat{S}$, $\mathbb{Z}/2\mathbb{Z}$ also acts freely on $Z$.
Since $Z^3 = 12$, $Z$ is a divisor of type $(1,1,2)$ in $A$. The covering involution $ \iota \colon \hat{S} \rightarrow \hat{S}$ can be lifted to an involution $g$ of $A$, which we write as an affine transformation $ g (a) = \alpha a + \beta$.
We have now Abelian subvarieties $A_1 = \ker (\alpha - \operatorname{Id})$, $A_2 = \ker (\alpha + \operatorname{Id})$, and since the irregularity of $S$ equals $1$, $A_1$ has dimension $1$, and $A_2$ has dimension $2$.
We observe preliminarly that $g$ is fixed point free: since otherwise the fixed point locus would be non empty of dimension one (as there is exactly one eigenvalue equal to $1$), so it would intersect the ample divisor $Z$, contradicting that $ \iota \colon Z \rightarrow Z$ acts freely.
Therefore $Y = Z / \iota $ is a divisor in the Bagnera-de Franchis threefold $ X = A / G$, where $G$ is the group of order two generated by $g$.
We can then write the Abelian threefold $A$ as $ (A_1 \times A_2) / T$, and since $\beta_1 \notin T_1$ (cf. Proposition \[quotprodtype\]) we have only two possible cases:
- $T = 0$, or
- $T \cong \mathbb{Z}/2\mathbb{Z}$.
We further observe that, since the divisor $ Z$ is $g$-invariant, its polarization is $\alpha$ invariant, in particular its Chern class $c \in \wedge ^2 ( \operatorname{Hom}(\Lambda, \mathbb{Z}))$, where $A = V / \Lambda$. Since $ T = \Lambda / ( \Lambda_1 \oplus \Lambda_2)$, $c$ pulls back to $$c' \in \wedge ^2 ( \operatorname{Hom}( \Lambda_1 \oplus \Lambda_2, \mathbb{Z})) = \wedge ^2 ( \Lambda_1^{\vee}) \oplus \wedge ^2 ( \Lambda_2^{\vee}) \oplus (\Lambda_1^{\vee})\otimes (\Lambda_2^{\vee}) ,$$ and by invariance $c' = (c'_1 \oplus c'_2 ) \in \wedge ^2 ( \Lambda_1^{\vee}) \oplus \wedge ^2 ( \Lambda_2^{\vee})$. So Case 0) bifurcates in the following cases:
- $c'_1$ is of type $(1)$, $c'_2$ is of type $(1,2)$;
- $c'_1$ is of type $(2)$, $c'_2$ is of type $(1,1)$.
Both cases can be discarded, since they lead to the same contradiction. Setting $D: = Z$, then $D$ is the divisor of zeros on $A = A_1 \times A_2$ of a section of a line bundle $L$ which is an exterior tensor product of $L_1$ and $L_2$. Since $$H^0 (A, L) = H^0 (A_1, L_1) \otimes H^0 (A_2, L_2) ,$$ and $H^0 (A_1, L_1)$ has dimension one in case 0-I), while $H^0 (A_2, L_2)$ has dimension one in case 0-II), we conclude that $D$ is a reducible divisor, a contradiction, since $D$ is smooth and connected.
In case 1), we denote $A' : = A_1 \times A_2$, and we let $D$ be the inverse image of $Z$ inside $A'$. Again $D$ is smooth and connected, since $\pi_1(\hat{S})$ surjects onto $\Lambda$. Now $ D^2 = 24$, so the Pfaffian of $c'$ equals $4$, and there are a priori several possibilities:
- $c'_1$ is of type $(1)$;
- $c'_2$ is of type $(1,1)$;
- $c'_1$ is of type $(2)$, $c'_2$ is of type $(1,2)$.
The cases 1-I) and 1-II) can be excluded as case 0), since $D$ would then be reducible.
We are then left only with case 1-III), and we may, without loss of generality, assume that $H^0 (A_1, L_1)= H^0 (A_1, \mathcal{O}_{A_1} (2 O))$. Moreover, we have already assumed that we have chosen the origin so that all the sections of $H^0 (A_2, L_2)$ are even.
We have $ A = A' / T $, and we may write the generator of $T$ as $t_1 \oplus t_2$, and write $ g (a_1 \oplus a_2 ) = (a_1 + \beta_1) \oplus ( a_2 - \beta_2)$.
By the description of Bagnera-de Franchis varieties (cf. Proposition \[quotprodtype\]) we have that $t_1$ and $\beta_1$ are a basis of the group of $2$ torsion points of the elliptic curve $A_1$.
Since all sections of $L_2$ are even, the divisor $D$ is $ G \times T$-invariant if and only if it is invariant under $T$ and under translation by $\beta$.
This condition however implies that translation of $L_2$ by $\beta_2$ is isomorphic to $L_2$, and similarly for $t_2$. It follows that $\beta_2, t_2$ form a basis of $K_2:= \ker (\phi_{L_2}\colon A_2 \rightarrow \operatorname{Pic}^0 (A_2))$, where $\phi(y) = t_yL_2 \otimes L_1^{-1}$. The isomorphism of $G \times T$ with both $K_1 : = A_1 [2]$ and $K_2$ allows to identify both $ H^0 (A_1, L_1) $ and $ H^0 (A_2, L_2)$ with the Stone-von Neumann representation $L^2 (T,\mathbb{C})$: observe in fact that there is only one alternating function $(G \times T) \rightarrow \mathbb{Z}/2\mathbb{Z}$, independent of the chosen basis.
Therefore, there are exactly $4$ invariant divisors in the linear system $|L|$. Explicitly, if $ H^0 (A_1, L_1) $ has basis $x_0, x_1$ and $ H^0 (A_2, L_2) $ has basis $y_0, y_1$, then the invariant divisors correspond to the four eigenvectors $$x_0 y_0 + x_1 y_1\,, \quad x_0 y_0 - x_1 y_1\,, \quad x_0 y_1 + x_1 y_0\,, \quad x_0 y_1 - x_1 y_0\,.$$
To prove irreducibility of the above family of surfaces, it suffices to show that all the four invariant divisors occur in the same connected family.
To this purpose, we just observe that the monodromy of the family of elliptic curves $E_{\tau} : = \mathbb{C} / ( \mathbb{Z} \oplus \mathbb{Z} \tau)$ on the upper half plane has the effect that a transformation in $\operatorname{SL}( 2 , \mathbb{Z})$ acts on the subgroup $E_{\tau} [2]$ of points of $2$-torsion by its image matrix in $\operatorname{GL}( 2 , \mathbb{Z}/2\mathbb{Z})$, and in turn the effect on the Stone-von Neumann representation is the one of twisting it by a character of $E_{\tau} [2]$.
This concludes the proof that the moduli space is irreducible of dimension $4$, since the moduli space of elliptic curves, respectively the moduli space of Abelian surfaces with a polarization of type $(1,2)$, are irreducible, of respective dimensions $1$, $3$.
The final assertion is a consequence of the fact that $\operatorname{Alb}(S) = A_1 / (T_1 + \langle \langle \beta_1 \rangle \rangle)$, so that the fibres of the Albanese map are just divisors in $A_2$ of type $(1,2)$. Their self intersection equals $4 = 2 (g-1)$, hence $g=3$.
In order to establish that the general curve is non hyperelliptic, it suffices to prove the following lemma.
Let $A_2$ be an Abelian surface, endowed with a divisor $L$ of type $(1,2)$, so that there is an isogeny of degree two $f \colon A_2 \rightarrow A'$ onto a principally polarised Abelian surface, and $L = f^*(\Theta)$. Then the only curves $C \in |L|$ which are hyperelliptic are contained in the pull backs of a translate of $\Theta$ by a point of order $2$ for a suitable such isogeny $f' \colon A_2 \rightarrow A''$. In particular, the general curve $C \in |L|$ is not hyperelliptic.
Note that $A'$ is the quotient of $A$ by an involution, given by translation with a two torsion element $t \in A[2]$. Let $C \in |L|$, and consider $D : = f_* (C) \in | 2 \Theta|$. There are two cases:
- $ C + t = C$;
- $ C + t \neq C$.
In case I) $D = 2 B$, where $B$ has genus $2$, so that $ C = f^* (B)$, hence, since $ 2B \equiv 2 \Theta$, $B$ is a translate of $\Theta$ by a point of order $2$. There are exactly two such curves, and for them $ C \rightarrow B$ is étale.
In case II) the map $C \rightarrow D$ is birational, $f^* (D) = C \cup (C + t)$. Now, $C+t$ is also linearly equivalent to $ L$, hence $C$ and $C+t$ intersect in the $4$ base points of the pencil $|L|$. Hence $D$ has two double points and geometric genus equal to $3$. These double points are the intersection points of $\Theta$ and a translate of $\Theta$ by a point of order $2$, and are points of $2$-torsion.
The sections of $H^0(\mathcal{O}_{A'} (2 \Theta))$ are all even and $ | 2 \Theta|$ is the pull-back of the space of hyperplane sections of the Kummer surface $\mathcal{K} \subset \mathbb{P}^3$, the quotient $\mathcal{K} = A' / \{\pm 1\}$.
Therefore the image $E'$ of each such curve $D$ lies in the pencil of planes through $2$ nodes of $\mathcal{K}$.
$E'$ is a plane quartic, hence $E'$ has geometric genus $1$, and we conclude that $C$ admits an involution $\sigma$ with quotient an elliptic curve $E$ (normalization of $E'$), and the double cover is branched in $4$ points.
Assume that $C$ is hyperelliptic, and denote by $h$ the hyperelliptic involution, which lies in the centre of $\operatorname{Aut}(C)$. Hence we have $(\mathbb{Z}/2\mathbb{Z})^2$ acting on $C$, with quotient $\mathbb{P}^1$. We easily see that there are exactly six branch points, two being the branch points of $ C/h \rightarrow \mathbb{P}^1$, four being the branch points of $E \rightarrow \mathbb{P}^1$. It follows that there is an étale quotient $ C \rightarrow B$ , where $B$ is the genus $2$ curve, double cover of $\mathbb{P}^1$ branched on the six points.
Now, the inclusion $ C \subset A_2$ and the degree $2$ map $ C \rightarrow B$ induce a degree two isogeny $ A_2 \rightarrow J(B)$, and $C$ is the pull back of the Theta divisor of $J(B)$, thus it cannot be a general curve.
This ends the proof of Theorem \[BCF2\].
We shall give the surfaces of Theorem \[BCF2\] a name.
A minimal surface $S$ of general type with invariants $K_S^2 = 6$, $p_g(S) =q(S) = 1$ such that
- there exists an unramified double cover $ \hat{S} \rightarrow S$ with $ q (\hat{S}) = 3$, and such that
- the Albanese morphism $ \hat{\alpha} \colon \hat{S} \rightarrow A = \operatorname{Alb}(\hat{S})$ is birational onto its image $Z$, a divisor in $A$ with $ Z^3 = 12$,
is called a [*Sicilian surface with $q(S)=p_g(S)=1$*]{}.
We have seen that the canonical model of a Sicilian surface $S$ is an ample divisor in a Bagnera-de Franchis threefold $X =A/G$, where $G =\langle g \rangle \cong \mathbb{Z}/ 2 \mathbb{Z}$. Hence the fundamental group of $S$ is isomorphic to the fundamental group $\Gamma$ of $X$. Moreover, $\Gamma$ fits into the exact sequence $$1 \longrightarrow \Lambda \longrightarrow \Gamma \longrightarrow G = \mathbb{Z}/2\mathbb{Z} \longrightarrow 1\,,$$ and is generated by the union of the set $\{ g , t\}$ with the set of translations by the elements of a basis $\lambda_1, \lambda_2, \lambda_3, \lambda_4$ of $\Lambda_2$, where $$g ( v_1 \oplus v_2 ) = ( v_1 + \frac{\tau}{2} ) \oplus ( - v_2 + \frac{\lambda_2}{2} )$$ $$t ( v_1 \oplus v_2 ) = ( v_1 + \frac 12 ) \oplus (v_2 + \frac{\lambda_4}{2} ).$$
$\Gamma$ is therefore a semidirect product of $\mathbb{Z}^5 = \Lambda_2 \oplus \mathbb{Z} t$ with the infinite cyclic group generated by $g$: conjugation by $g$ acts as $-1$ on $\Lambda_2$, and it sends $t \mapsto t - \lambda_4$ (hence $ 2 t - \lambda_4$ is an eigenvector for the eigenvalue $1$).
We shall now give a topological characterization of Sicilian surfaces with $q=p_g=1$, following the lines of [@BC12].
Observe in this respect that $X$ is a $K(\Gamma, 1)$-space, so that its cohomology and homology are just group cohomology, respectively homology, of the group $\Gamma$.
\[he\] A Sicilian surface $S$ with $q(S)=p_g(S)=1$ is characterized by the following properties:
1. $K_S^2 = 6$,
2. $ \chi(S) = 1$,
3. $\pi_1(S) \cong \Gamma$, where $\Gamma$ is as above,
4. the classifying map $f \colon S \rightarrow X$, where $X$ is the Bagnera-de Franchis threefold which is a classifying space for $\Gamma$, has the property that $ f_* [S] = :B$ satisfies $ B^3=6$.
In particular, any surface homotopically equivalent to a Sicilian surface is a Sicilian surface, and we get a connected component of the moduli space of surfaces of general type which is stable under the action of the absolute Galois group $Gal (\bar{{\ensuremath{\mathbb{Q}}}}, {\ensuremath{\mathbb{Q}}})$.
Since $\pi_1(S) \cong \Gamma$, first of all $ q(S) =1$, hence also $p_g(S) = 1$. By the same token there is a double étale cover $\hat{S} \rightarrow S$ such that $ q(\hat{S} ) = 3$, and the Albanese image of $\hat{S} $, counted with multiplicity, is the inverse image of $B$, therefore $Z^3 = 12$. From this, it follows that $\hat{S} \rightarrow Z$ is birational, since the class of $Z$ is indivisible.
We may now apply the previous Theorem \[BCF2\] in order to obtain the classification.
Observe finally that the condition $(\hat{\alpha}_* \hat{S})^3 = 12$ is not only a topological condition, it is also invariant under $Gal (\bar{{\ensuremath{\mathbb{Q}}}}, {\ensuremath{\mathbb{Q}}})$.
Proof of the main theorems
==========================
We conclude in this last short section the proofs of Main Theorem 1 and Main Theorem 2.
Statements 1), 2) and 3) summarize the contents of Proposition \[onefam\] and Theorem \[fundgroup\].
4\) We observe preliminarly that our fundamental groups are virtually Abelian of rank 6 (i.e., they have a normal subgroup of finite index $\cong {\ensuremath{\mathbb{Z}}}^6$). By the results of [@4names], the fundamental group of (the minimal resolution of) a product-quotient surface has a finite index normal subgroup which is the product of at most two fundamental groups of curves. Therefore if it is virtually Abelian it has rank 2 or 4.
This argument excludes rightaway that our fundamental groups may be isomorphic to the fundamental groups of some product-quotient surfaces.
The only remaining case for $p_g=0$ is the Kulikov surface, whose first homology group has $3$-torsion. [^2]
The known surfaces with $p_g=q=1$ and $K^2=6$ are either product-quotient surfaces (cf. [@pol09]) or mixed quasi-étale surfaces, which are constructed in [@FP14]. Comparing Table 2 from loc. cit with our Table \[q1\], we see that they have different homology groups from ours.
5\) is proved in Theorem \[moduli\].
The assertions 1) and 2) are contained in Theorem \[BCF2\].
4\) is contained in Corollary \[he\].
3\) Observe that in cases $\mathcal S_{11}$ and $\mathcal S_{12}$ of Table \[q1\] there is a subgroup $H \cong ({\ensuremath{\mathbb{Z}}}/2 {\ensuremath{\mathbb{Z}}})^2$ acting by translations on $E_1\times E_2 \times E_3$. Denote by $\hat S$ the quotient of the Burniat hypersurface by $H$. Then $\hat S$ is an étale double cover of the GBT $S$, which satisfies the defining property of Sicilian surfaces.
There remains to show that the other GBT surfaces (with $p_g=q=1$) are not Sicilian surfaces. This is now obvious since they have fundamental groups non-isomorphic to $\pi_1(S_{11})$, where $S_{11}$ belongs to the family $\mathcal S_{11}$ .
[CCML98]{}
M. F. Atiyah. Vector bundles over an elliptic curve. , 7:414–452, 1957.
I. Bauer and F. Catanese. Some new surfaces with $p_g = q = 0$. In Turin Univ. Torino, editor, [*The Fano Conference*]{}, pages 123–142, 2004.
I. Bauer and F. Catanese. Burniat surfaces. [II]{}. [S]{}econdary [B]{}urniat surfaces form three connected components of the moduli space. , 180(3):559–588, 2010.
I. Bauer and F. Catanese. Burniat surfaces [I]{}: fundamental groups and moduli of primary [B]{}urniat surfaces. In [*Classification of algebraic varieties*]{}, EMS Ser. Congr. Rep., pages 49–76. Eur. Math. Soc., Zürich, 2011.
I. Bauer and F. Catanese. The moduli space of [K]{}eum-[N]{}aie surfaces. , 5(2):231–250, 2011.
I. Bauer and F. Catanese. Inoue type manifolds and [I]{}noue surfaces: a connected component of the moduli space of surfaces with [$K^2=7$]{}, [$p_g=0$]{}. In [*Geometry and arithmetic*]{}, EMS Ser. Congr. Rep., pages 23–56. Eur. Math. Soc., Zürich, 2012.
I. Bauer and F. Catanese. Burniat surfaces [III]{}: deformations of automorphisms and extended [B]{}urniat surfaces. , 18:1089–1136, 2013.
I. Bauer and F. Catanese. Burniat-type surfaces and a new family of surfaces with [$p_g=0$, $K^2=3$]{}. , 62(1):37–60, 2013.
I. Bauer, F. Catanese, and F. Grunewald. The classification of surfaces with $p_g=q=0$ isogenous to a product of curves. , 4(2):547–586, 2008.
I. Bauer, F. Catanese, F. Grunewald, and R. Pignatelli. Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups. , 134(4):993–1049, 2012.
W. Bosma, J. Cannon, and C. Playoust. The [M]{}agma algebra system. [I]{}. [T]{}he user language. , 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).
I. Bauer, F. Catanese, and R. Pignatelli. Surfaces of general type with geometric genus zero: A survey. In [*Complex and Differential Geometry*]{}, volume 8, pages 1–48. Springer Proceedings in Mathematics, 2011.
G. Bagnera and M. de Franchis. Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti. , 15:251–343, 1908.
L. Bianchi. . Spoerri, [P]{}isa, 1916. 2a edizione.
P. Burniat. Sur les surfaces de genre ${P}_{12}>1$. , 71(4):1–24, 1966.
F. Catanese. A superficial working guide to deformations and moduli. In [*Handbook of moduli. [V]{}ol. [I]{}*]{}, volume 24 of [*Adv. Lect. Math. (ALM)*]{}, pages 161–215. Int. Press, Somerville, MA, 2013.
F. Catanese. Topological methods in moduli theory. preliminary version, 2014.
F. Catanese and C. Ciliberto. On the irregularity of cyclic coverings of algebraic surfaces. In [*Geometry of complex projective varieties ([C]{}etraro, 1990)*]{}, volume 9 of [*Sem. Conf.*]{}, pages 89–115. Mediterranean, Rende, 1993.
F. Catanese, C. Ciliberto, and M Mendes Lopes. On the classification of irregular surfaces of general type with nonbirational bicanonical map. , 350(1):275–308, 1998.
D. [Frapporti]{} and R. [Pignatelli]{}. . , 2014. doi:10.1017/S0017089514000184.
C. D. Hacon and R. Pardini. Surfaces with [$p_g=q=3$]{}. , 354(7):2631–2638 (electronic), 2002.
J. Igusa. . Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 194.
M. Inoue. Some new surfaces of general type. , 17(2):295–319, 1994.
N. Jacobson. . W. H. Freeman and Co., San Francisco, Calif., 1980.
J. Milnor. . Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51. Princeton University Press, Princeton, N.J., 1963.
J. Milnor. . Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72.
D. Mumford. . Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34. Springer-Verlag, Berlin-New York, 1965.
D. Mumford. . Tata Institute of Fundamental Research Studies in Mathematics, No. 5. Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970.
M. Penegini. The classification of isotrivially fibred surfaces with [$p_g=q=2$]{}. , 62(3):239–274, 2011. With an appendix by S[ö]{}nke Rollenske.
Pirola. Surfaces with [$p_g=q=3$]{}. , 108(2):163–170, 2002.
F. Polizzi. Standard isotrivial fibrations with $p_g =q = 1$. , 321(6):1600–1631, 2009.
M. Penegini and F. Polizzi. Surfaces with [$p_g=q=2$]{}, [$K^2=6$]{}, and [A]{}lbanese map of degree [$2$]{}. , 65(1):195–221, 2013.
L. C. Washington. , volume 83 of [*Graduate Texts in Mathematics*]{}. Springer-Verlag, New York, second edition, 1997.
F. Zucconi. Surfaces with $p_g = q = 2$ and an irrational pencil. , 55(3):649–672, 2003.
[**Authors’ Adresses:**]{}\
[*Ingrid Bauer, Fabrizio Catanese, Davide Frapporti*]{}\
Lehrstuhl Mathematik VIII\
Mathematisches Institut der Universität Bayreuth, NW II\
Universitätsstr. 30; D-95447 Bayreuth, Germany.
[**Email Adresses:**]{}\
Ingrid.Bauer@uni-bayreuth.de\
Fabrizio.Catanese@uni-bayreuth.de\
Davide.Frapporti@uni-bayreuth.de
Tables
======
$\epsilon_0$ $\eta_1$ $\epsilon_1$ $\eta_0$ $\eta_1$ $\epsilon_2$ $\zeta_0$ $\eta_1$ $\epsilon_3$ $H_1$
-- -------------- ---------- -------------- ---------- ---------- -------------- ----------- ---------- -------------- -------
1 0 0 1 0 0 1 0 0
0 1 0 1 1 0 1 1 0
0 0 0 0 0 1 1 0 1
1 0 0 0 0 1 1 0 1
0 0 1 0 0 0 1 0 1
0 0 0 1 0 1 0 0 1
1 0 0 0 0 1 1 0 1
0 1 0 0 1 0 1 1 0
0 0 1 1 0 1 1 0 0
1 0 1 0 0 1 1 0 0
0 1 0 0 1 0 1 1 0
0 0 0 1 0 1 1 0 1
: $q=0$[]{data-label="q0"}
$\epsilon_0$ $\eta_1$ $\epsilon_1$ $\eta_0$ $\eta_1$ $\epsilon_2$ $\zeta_0$ $\eta_1$ $\epsilon_3$ $H_1$ $\pi_1$
-- -------------- ---------- -------------- ---------- ---------- -------------- ----------- ---------- -------------- ------- ---------
1 0 1 0 0 0 1 0 1
0 1 0 0 1 0 1 1 0
0 0 0 0 0 1 1 0 1
0 1 0 1 1 0 1 1 0
0 0 1 1 0 0 1 0 1
0 0 0 0 0 1 1 0 1
1 0 0 0 0 1 1 0 1
0 1 0 0 1 0 1 1 0
0 0 1 1 0 1 0 0 0
1 0 0 0 0 1 1 0 1
0 1 0 1 1 0 1 1 0
0 0 1 1 0 0 1 0 1
1 0 0 1 0 1 1 0 1
0 1 0 0 1 0 1 1 0
0 0 1 1 0 1 0 0 0
1 0 1 1 0 0 1 0 1
0 1 0 1 1 0 1 1 0
0 0 0 0 0 1 1 0 1
1 0 0 1 0 1 0 0 1
0 1 0 1 1 0 0 1 0
0 0 1 1 0 1 1 0 0
1 0 1 0 0 0 1 0 1
0 1 0 0 1 0 1 1 0
0 0 0 1 0 1 1 0 1
: $q=1$[]{data-label="q1"}
$\epsilon_0$ $\eta_1$ $\epsilon_1$ $\eta_0$ $\eta_1$ $\epsilon_2$ $\zeta_0$ $\eta_1$ $\epsilon_3$ $H_1$ $\pi_1$
-- -------------- ---------- -------------- ---------- ---------- -------------- ----------- ---------- -------------- ------- ---------
1 0 0 1 0 1 1 0 1
0 1 0 1 1 0 1 1 0
0 0 1 1 0 1 0 0 0
1 0 1 0 0 0 0 0 1
0 1 1 0 1 1 0 1 0
0 0 0 1 0 1 0 0 1
1 0 1 0 0 0 1 0 1
0 1 1 0 1 1 0 1 0
0 0 0 1 0 1 1 0 1
: $q=2$[]{data-label="q2"}
$\epsilon_0$ $\eta_1$ $\epsilon_1$ $\eta_0$ $\eta_1$ $\epsilon_2$ $\zeta_0$ $\eta_1$ $\epsilon_3$ $\pi_1$
-- -------------- ---------- -------------- ---------- ---------- -------------- ----------- ---------- -------------- ---------
1 0 1 0 0 0 1 0 1
0 1 1 0 1 1 1 1 0
0 0 0 1 0 1 1 0 1
: $q=3$[]{data-label="q3"}
[^1]: The present work took mainly place in the realm of the DFG Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds”.\
The second author also acknowledges support of the ERC-advanced Grant 340258-TADMICAMT
[^2]: Disclaimer: the fundamental group of the Inoue surface with $p_g=0$, $K^2=6$ has not yet been calculated and we do not claim it is different from ours.
|
---
abstract: |
In this note we demonstrate that the algebra associated with coordinate transformations might contain the origins of a scalar field that can behave as an inflaton and/or a source for dark energy. We will call this particular scalar field the [*diffeomorphism*]{} scalar field. In one dimension, the algebra of coordinate transformations is the Virasoro algebra while the algebra of gauge transformations is the Kac-Moody algebra. An interesting representation of these algebras corresponds to certain field theories that have meaning in any dimension. In particular the so called Kac-Moody sector corresponds to Yang-Mills theories and the Virasoro sector corresponds to the diffeomorphism field theory that contains the scalar field and a rank-two symmetric, traceless tensor. We will focus on the contributions of the diffeomorphism scalar field to cosmology. We show that this scalar field can, qualitatively, act as a phantom dark energy, an inflaton, a dark matter source, and the cosmological constant $\Lambda $.\
\
Keywords: coadjoint representation; dark energy; dark matter; cosmology
author:
- |
Vincent G.J. Rodgers[^1] and Takeshi Yasuda[^2]\
Department of Physics and Astronomy\
The University of Iowa\
Iowa City, IA 52242 USA
title: General Coordinate Transformations as the Origins of Dark Energy
---
Introduction
============
Recently experimental cosmology has brought new challenges to the theoretical understanding of the universe. Until now, the standard hot big-bang cosmology has been very successful in explaining four important observed facts: (1) the expansion of Universe, (2) the origin of the cosmic background radiation, (3) the nucleosynthesis of the light elements, (4) the formation of galaxies and the large scale structure of Universe. However, recent discoveries from Supernova Cosmology Project \[\[SNCP\]\] and WMAP \[\[WM\]\] strongly suggest that the standard cosmology needs to be revised. This new standard cosmology must now explain$$\begin{aligned}
&&\ \circ \ \ \mbox{Flat and accelerating Universe} \\
&&\ \circ \ \ \mbox{Rapid expansion of Universe in early period (Inflation)%
} \\
&&\ \circ \ \ \mbox{Composition of Universe: 73\% dark energy; 23\% dark
matter; 4\% baryons\thinspace }\end{aligned}$$The theoretical search for this new standard cosmology is very active and there is an overwhelming amount of literature published where most of them have the same theme: scalar fields. In this work we ask if there is a guiding principle that might account for the presence of new fields in cosmology. We posit that there is a symmetry that can accompany general relativity in providing an explanation for some of the new features found in cosmology. This symmetry is not new and is already exploited in general relativity through general coordinate covariance. However we will realize the symmetry using an approach used in string theory. Since string theories are two dimensional field theories, direct contributions to gravity coming from curvature and the Einstein-Hilbert action are diminished because of the low dimension. In fact in two dimensions the Einstein-Hilbert action can at most distinguish different topological structures. Our starting principle will be the algebra of Lie derivatives which exists in any dimension including one dimension. We will show that this algebra admits a field theoretic representation analogous to the Yang-Mills gauge potentials that arise from the algebra of gauge transformations.
In one dimension, this field theory comes from the [*coadjoint representation*]{} \[\[Ki1\],\[Ki2\],\[Wi2\],\[DvNR\]\] of the one dimensional algebra of Lie derivatives called the Virasoro algebra. In string theory this representation is used to understand the geometric origin of gravitational anomalies. As we will review here, one can understand the origins of two dimensional gauge and gravitational anomalies through the [*Geometric Action*]{} \[\[AS\],\[W1\],\[LR1\],\[LR2\],\[RR\]\] as well as establishes the Lagrangian for the fields living in the coadjoint representation called the [*Transverse Action*]{} \[\[BLR\],\[BRY\]\]. For this work, the importance feature of the Transverse Action as compared to the Geometric Action is that the Transverse Action can exist in any dimension. We will review the construction of the Transverse Action with both the algebra of gauge and coordinate transformations. We show that there is a naturally occurring scalar fields called the [*diffeomorphism scalar field*]{} (diff scalar) that arises naturally from the algebra of associated with coordinate transformations.
As a matter of completeness we will keep this paper somewhat self contained and outline of this paper as follows. We will first review the Robertson-Walker cosmology and as well as four of the more popular scalar field theories discussed in the literature. This will introduce our notation to the reader and put this work in a more familiar setting. In the next section we discuss the method we use at arriving at the Lagrangian for the diff scalar. We are then able to discuss the cosmological implications of the diffeomorphism scalar field by combining the action for the diff scalar with the Einstein-Hilbert action in the next section. We show that the diff scalar field can, qualitatively, act as phantom dark energy, an inflaton, a dark matter source, and the cosmological constant $\Lambda $.
Standard Cosmology Review
=========================
The standard cosmology is based on two postulates that Universe is homogeneous and isotropic at large scales \[\[Pe\],\[We1\],\[D1\]\]. These postulates are strongly supported by the cosmic microwave background (CMB) radiation coming from various regions of sky: the temperature of the CMB radiation from different parts of sky are almost identical. These postulates are called the *cosmological principle*. With proper choice of coordinates these symmetry postulates thus allow us to write the metric in the form of Robertson-Walker metric:$$ds^{2}=-dt^{2}+a^{2}\left( t\right) \left\{ {dr^{2}\over 1-kr^{2}}
+r^{2}\left( d\theta ^{2}+\sin ^{2}\theta \mbox{ }d\phi ^{2}\right) \right\}$$Here we used the convention: $\hbar =c=1$. The function $a\left( t\right) $ is called the scale factor of the universe (*cosmic scale factor*). Rescaling this factor appropriately, the constant $k$ can be always taken to be $0$, $+1$, or $-1$: $$\begin{aligned}
&&\ k=+1:\mbox{ closed universe with positive spatial curvature} \\
&&\ k=0:\mbox{ flat universe with zero spatial curvature} \\
&&\ k=-1:\mbox{ open universe with negative spatial curvature}\end{aligned}$$ Since current observations seems to prefers a flat universe, we will adopt $k=0$ throughout this paper. The cosmological principle also demands that all cosmic tensors are maximally form-invariant with respect to the spatial coordinates. As a consequence of this constraint the most general form-invariant tensor for the energy-momentum tensor is given by:$$T_{00}=\rho \left( t\right) ,\ \ \ T_{0i}=0,\ \ \ T_{ij}=p\left( t\right)
g_{ij},\ \ \ \ \ \ i,j=1,2,3$$where $\rho \left( t\right) $ and $p\left( t\right) $ are arbitrary functions that depend only on time. Rewriting the energy-momentum tensor as$$T_{\mu \nu }=\left( \rho +p\right) U_{\mu }U_{\nu }+pg_{\mu \nu }$$where $U_{\mu }$ is a “four-velocity vector"$$U^{0}=1,\mbox{ \thinspace \thinspace \thinspace \thinspace \thinspace
\thinspace \thinspace \thinspace \thinspace \thinspace \thinspace }U^{i}=0%
\mbox{ \thinspace }\left( i=1,2,3\right)$$we identify $\rho \left( t\right) $ and $p\left( t\right) $ as with energy density and pressure of a perfect fluid.
Using this form of the energy-momentum tensor and the Robertson-Walker metric, we obtain the *Friedmann equations* from Einstein’s field equations:
$$\begin{aligned}
H^{2}\left( t\right) &=&{\rho \left( t\right) \over 3M_{pl}^{2}}+{%
\Lambda \over 3}-{k \over a^{2}\left( t\right) } \\
\dot{H}\left( t\right) &=&{dH\left( t\right) \over dt}=-{1 \over 2M_{\rm pl}^{2}%
}\left( \rho \left( t\right) +p\left( t\right) \right) +{k \over a^{2}\left(
t\right) }\end{aligned}$$
where the Hubble parameter $H(t)$ is defined as$$H(t) ={\dot{a}\left( t\right) \over a\left( t\right) }$$and$$M_{\rm pl}=\mbox{\mbox{reduced Planck mass}}=\left( 8\pi G\right) ^{-1/2}.$$From the energy-momentum conservation condition $\nabla _{\mu }T^{\mu 0}=0$ we find$$\dot{\rho}+3H\left( \rho +p\right) =0.$$This equation and the Friedmann equations are not independent from each other. The equation of state, the pressure-to-energy density ratio, is defined as$$w={p \over \r }.$$Typical values of $w$ are$$\begin{aligned}
\mbox{Radiation}\mbox{:\ \ \ } &&w=\frac{1}{3} \\
\mbox{Matter}\mbox{:\ \ \ } &&w=0 \\
\mbox{Cosmological constant}\mbox{:\ \ \ } &&w=-1.\end{aligned}$$
The condition for the accelerating expansion of Universe can be given in several ways:$$\begin{aligned}
\mbox{the cosmic scale factor is accelerating} &\mbox{:}&\mbox{\ \ \ \ \ }
\ddot{a}>0 \nonumber \\
\mbox{the comoving Hubble length is decreasing} &\mbox{:}&\mbox{\ \ \ \ }
{d\over dt}\left({H^{-1}(t) \over a(t) }\right)<0 \label{inf}
\\
\mbox{negative Pressure\ } &\mbox{:}&\mbox{\ \ \ }\rho \left(t\right) +3p\left( t\right) <0. \nonumber\end{aligned}$$The last condition can be also expressed by using the equation of state:$$w<-\frac{1}{3}.$$
Single-Field Inflation Model
----------------------------
The inflation models assume an era of inflation during the Big Bang, in which the energy density of the Universe was dominated by the potential energy of the scalar fields (inflatons). The inflation is defined as any epoch during which the conditions Eq.\[inf\] are satisfied and used for the events in early Universe. The negative pressure condition for the inflation model motivates one to introduce a scalar field (inflaton) with its Lagrangian density given by$$\mathcal{L}_{\phi }=\frac{1}{2}g^{\mu \nu }\left( \partial _{\mu }\phi
\right) \left( \partial _{\nu }\phi \right) -V\left( \phi \right) .
\label{Intro028}$$The term $V\left( \phi \right) $ is the potential of the scalar field and the choice of its form characterizes the types of inflationary models. Using the above Lagrangian density we have$$T^{\mu \nu }=\left( \partial ^{\mu }\phi \right) \left( \partial ^{\nu }\phi
\right) -g^{\mu \nu }\left[ \frac{1}{2}\left( \partial ^{\lambda }\phi
\right) \left( \partial _{\lambda }\phi \right) +V\left( \phi \right) \right]
\label{Intro029}$$or$$\rho =\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\left( \nabla_{i}\phi
\right) ^{2}+V\left( \phi \right) \label{Intro030}$$and$$p=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{6}\left( \nabla_{i}\phi \right)
^{2}-V\left( \phi \right) . \label{Intro031}$$The equation of motion for the scalar field is given by$$\ddot{\phi}+3H\dot{\phi}-{1 \over a^{2}\left( t\right) }\vec{\nabla}%
^{2}\phi =-{\partial V \over \partial \phi }. \label{Intro032}$$
Quintessence
------------
In quintessence $Q$ is a scalar field that is introduced to explain the accelerating Universe. Quintessence is a dynamical field and generally has a time-varying, spatially inhomogeneous negative pressure \[\[Ca\]\]. The Lagrangian for the quintessence is given by$$\mathcal{L}=\frac{1}{2}\partial _{\mu }Q\partial ^{\mu }Q-V\left( Q\right) .$$The energy density and pressure of quintessence are, then, respectively, $$\rho =\frac{1}{2}\dot{Q}^{2}+V\left( Q\right) ,\ \ \ \ p=\frac{1}{2}\dot{Q}%
^{2}-V\left( Q\right) .$$It is easy to see that the ratio of the pressure to the energy density, $w$, is $-1<w\leq 0$ for quintessence. A particular class of quintessence modelshave a tracker behavior. In these models the density of quintessence closely tracks the radiation density until the equality between matter and radiation is established. Once the matter-radiation equality is established the quintessence start to behave as dark energy. k-essence and phantom energy are the special cases of quintessence.
k-essence
---------
The idea of k-essence (or k-inflation) was originally introduced as a possible model for inflation \[\[ADM\]\]. In this model an inflationary evolution of the early universe is driven by non-quadratic kinetic energy terms starting from rather arbitrary initial conditions instead of the potential energy term $V\left( \varphi \right) $. When the idea of k-essence is applied to the coincidence problem[^3], it was shown that \[\[AM\], \[AMS\]\] k-essence can act as a dynamical attractor at the onset of matter domination period and introduce the cosmic acceleration at present time. A purely kinetic k-essence model was investigated in \[\[Sc1\]\]. In this investigation the author showed that k-essence in this model can evolve like the sum of a dark matter component and a dark energy component if the Lagrangian for the k-essence has a local extremum.
The Lagrangian for a typical k-essence model is given as the pressure of the k-essence:$$\mathcal{L}=P=V\left( \phi \right) F\left( X\right) ,$$where $\phi $ is the scalar field representing k-essence, and $X$ is defined as$$X=\frac{1}{2}\partial _{\mu }\phi \partial ^{\mu }\phi .$$The pressure in this model is given by the Lagrangian itself, while the energy density is given by$$\rho =V\left( \phi \right) \left[ 2XF_{X}-F\right]$$where$$F_{x}={dF\over dX}.$$Using these pressure and energy density, we have the ratio of the pressure to the energy density, $w$, as$$w={F \over 2XF_{X}-F}.$$The behavior of this ratio is determined by the form of $V\left( \phi
\right) $ and $F\left( X\right) $.
Phantom Dark Energy
-------------------
The phantom dark matter is a special case of quintessence characterized by the ratio of the pressure to the energy density, $w<-1$. Though the dark energy with $w<-1$ will violate all energy conditions \[\[Wa\]\], the dark energy models with $w<-1$ that are consistent with observed data can be constructed \[\[Ca1\]\]. A phenomenological model of phantom dark energy can be constructed to give new insight to the coincident problem through the interaction of the phantom dark energy with dark energy and dark matter \[[CW]{}\].
The Origin of the Diffeomorphism Field
======================================
In this work we posit that there exist a scalar field, the diffeomorphism scalar field, that has its origins in one of the most primitive symmetries used in physics; general coordinate covariance. When coupled to a Robertson-Walker metric, this scalar field qualitatively exhibits the behavior of an inflaton, phantom dark energy and the cosmological constant \[\[RY\]\]. Here we review the construction of the Lagrangian for the diffeomorphism field. We will marry the general coordinate transformations with gauge transformations to show the analogy with the vector potential in Yang-Mills theories.
Recall that under a general coordinate transformation, tensors transform as $$\begin{aligned}
A'_a(x') &=& \left({\partial x^b \over \partial x'^a}\right) A_b(x) \\
A'^a(x') &=& \left({\partial x'^a \over \partial x'^b}\right) A^b(x).\end{aligned}$$ Their infinitesimal counterpart, i.e. $x'^a=x^a+\xi^a$, where $\xi$ is vector field, determine the Lie derivative; $$\begin{aligned}
{\cal L}_\xi A_a(x) &=& -\xi(x)^b \partial_b A_a(x) - A_b(x)\partial_a \xi^b(x) \\
{\cal L}_\xi A^a(x) &=& -\xi(x)^b \partial_b A^a(x) + A^b(x)\partial_b \xi^a(x).\end{aligned}$$ It is important to note that for tensors the Lie derivative is independent of the connection, $\nabla_a$. Thus the field theory associated with diffeomorphisms will be independent of Riemannian curvature.
The analogue of the above for finite and infinitesimal transformations for gauge transformations on a (non-Abelian) vector potential are $$V'_a(x) = U(x) V_a(x) U(x)^{-1} + i U(x)\partial_a U(x)^{-1}$$ and $$\delta_\Lambda V_a(x) = i \left[\Lambda(x), V_a(x)\right] - \partial_a \Lambda(x),$$ where $U(x)=\exp{i \Lambda(x)}$ is the finite gauge transformation generated by $\Lambda(x)$.
The Adjoint Representation
--------------------------
The vector fields ($\xi^a$) and gauge parameters ($\L$) act on themselves and form the adjoint representation. The gauge and coordinate transformations in one dimension are particularly interesting because the [*dual*]{} representation of the adjoint representation, called the coadjoint representation, reveals the gauge and gravitational anomalies that appear in two dimensional field theories and string theories. Interestingly enough, the coadjoint representation appears to come from a field theory that need not be constrained to two dimensions. In the case of the gauge transformations, that field theory is Yang-Mills.
To see how these field theories comes about, lets begin by marrying the one dimensional infinitesimal coordinate transformations (Lie derivatives) and infinitesimal gauge transformations together. This marriage is the semi-direct product of the Virasoro algebra \[\[Vi1\]\], which corresponds to Lie derivatives of one dimensional vector fields, and an SU(N) Kac-Moody algebra, which corresponds to SU(N) gauge transformations on a circle or a line. This algebra admits central extensions (or phases at the group level) so that the basic elements of this algebra contains the Lie derivative generated by the one dimensional vector field $\xi^a(\theta)$, a gauge parameter $\Lambda(\theta)$, and a constant, say $a$ so we can have a three-tuple $\left({\xi}; \L; a \right)$. Then the commutation relations, $[[*,*]]$ for the adjoint representation is $$[[({\xi}; \,\L; \,a), ({\eta}; \X; \,b) ]] =({\xi \circ
\eta}; \,- \xi^b \partial_b \X + \eta^b \partial_b \L + [\L,\X];\, \left\{\L,\X,\xi,\eta\right\}),$$ where the central extensions $\left\{\L,\X,\xi,\eta\right\}$ is given by $$\begin{aligned}
\left\{\L,\X,\xi,\eta\right\}&=&\frac{k}{2\pi} \int (\L
\partial_a \X - \X \partial_a \L )d\,\theta^a +
\frac{c}{2\pi} \int (\xi^a \nabla_a \nabla_b \nabla_c
\eta^c)\,d\theta^b \,\,\,\,\\
&+& \frac{h}{2\pi} \int (\xi^a \nabla_a \eta_b)\, d\theta^b - (\xi
\leftrightarrow \eta). \nonumber
\end{aligned}$$ The parameter $h$,$k$, and $c$ are arbitrary constants that define the central extension. The new vector field $(\xi \circ \eta)^a$ is defined through the Lie derivative, $$(\xi \circ \eta)^a \equiv {\cal L}_{\xi} \eta^a = -\xi^b \partial_b \eta^a + \eta^b \partial_b
\xi^a.$$
The Coadjoint Representation
----------------------------
Associated with the adjoint representation, we can define its dual by writing an invariant product \[\[Ki1\],\[Wi2\]\] . Let $\left(\eta; \X; \,b\right)$ be an element of the adjoint and ${\cal B}=\left( D, A, \mu \right)$ be an element of the algebra’s dual. Then the one dimensional line integral gives an invariant pairing: $$\left\langle \left( D, A, \mu \right) \mid \left(\eta; \X; \,b\right) \right\rangle = \int\left(D_{a b}(\theta) \xi^a(\theta) +A_b(\theta) \L(\theta)\right)\,d\theta^b + b \,\mu.$$
Demanding that the pairing to be invariant under gauge and coordinate transformations is equivalent to acting on the pairing with an adjoint element and getting zero. Consider the action of the adjoint element ${\cal F}=\left( \xi \left( \theta \right) ,\Lambda
\left( \theta \right) ,a\right) $ on the pairing. Then the invariance of the pairing $$\delta_{\cal F} \left\langle \left( D, A, \mu \right) \mid \left(\eta; \X; \,b\right) \right\rangle = 0$$ and Leibnitz rule implies that $$\delta _{\mathcal{F}}B=\left( \delta_{\cal F}
D\left( \theta \right) ,\delta_{\cal F} A\left( \theta \right) ,0\right)$$where $$\delta_{\cal F} {\rm D}\left( \theta \right) =\;\stackrel{\rm shift\; in\; the \;diff\;coadjoint\; element}{%
{}{}{\overbrace{\stackunder{coordinate\ trans}{\underbrace{2\xi ^{^{\prime
}}{\rm D}+{\rm D}^{^{\prime }}\xi + \frac {c \mu}{2\pi}\xi^{\prime \prime
\prime } + \frac {h \mu}{2 \pi} \xi^{\prime}}}-%
\stackunder{gauge\ trans}{\underbrace{Tr\left( {\rm A}\Lambda^{\prime }\right)
}}}}}$$ and $$\delta_{\cal F} {\rm A(\theta )}=\;\stackrel{\rm shift\; in\; the\; gauge\; coadjoint\; element}{\,\overbrace{%
\stackunder{ coord\ trans}{\underbrace{{\rm A}^{\prime }\xi +\xi^{\prime }{\rm A}}}-\stackunder{ gauge\ trans}{\,\underbrace{[\Lambda
\,{\rm A}-{\rm A\,}\Lambda ]+k\,\mu \,\Lambda^{\prime }}}}}.$$ We have suppressed the tensor indices for the fields $D_{a b}$ and $A_b$ and $^{\prime }$ denotes $\partial _\theta$.\
The Geometric Action
--------------------
The fields $A$ in the coadjoint representation correspond to the spatial components of a gauge field while $D$ is the space-space component of a rank two tensor called the diffeomorphism field. The “$\m$” is the central extension. Now there are two distinct Lagrangians that one can construct using the structure of the coadjoint representation. The first is called the [*Geometric Lagrangian*]{}.
In the geometric Lagrangian or geometric action \[\[Ki3\], [\[PS1\]]{}\] the coadjoint elements serve as background fields while the coordinate and group transformations become the dynamical fields. There one considers a two parameter family of coordinate and gauge transformations that can act on a fixed element of the coadjoint representation, say ${\cal B}=\left(
D\left( \theta \right) ,A\left( \theta \right) ,\tilde{\mu}\right)$. The two parameters, say $\lambda$ and $\tau$, correspond to a natural coordinatization of the [*orbit*]{} of ${\cal B}$. The orbit of ${\cal B}$ is defined to simply be all gauge fields and diffeomorphism fields that are related to ${\cal B}$ via gauge and coordinate transformation.
On the orbit of any coadjoint element, there exist a natural symplectic structure \[\[Ar2\]\] corresponding to a rank two antisymmetric tensor \[\[Ki3\],\[Ki4\]\]. When integrated on a two-manifold this will give the [*geometric action*]{}. To find the geometric action for our case, namely the semi-direct product of the Virasoro and Kac-Moody groups, we define two adjoint vectors corresponding to changes in directions corresponding to coordinates $\l$ and $\t$ on the orbit. These adjoint vectors are $u_{\lambda}$ and $u_{\tau}$. Now let $g$ represent the group action of both an arbitrary gauge transformation, $U(\theta, \l,\t)$, and coordinate transformation, $s(\theta,\l,\t)$. We denote by $Ad(g)$ the action of these gauge and coordinate transformations of the adjoint representation and $Ad^{\ast }(g)$ the group action on coadjoint elements. Then using two adjoint elements corresponding to changes in the $\l$ and $\t$ directions and a coadjoint representation $w_{g}$ defined by $$\begin{aligned}
u_{\lambda } &=&\left( \xi _{\lambda }\left( \theta \right) , \Lambda
_{\lambda }\left( \theta \right) , a_{\lambda }\right) \nonumber \\
u_{\tau } &=&\left( \xi _{\tau }\left( \theta \right) , \Lambda _{\tau
}\left( \theta \right) , a_{\tau }\right) \\
w_{g} &=&Ad^{\ast }\left( g\right) w = Ad^{\ast }\left( g\right) \left(
D\left( \theta \right) , A\left( \theta \right) , \tilde{\mu}\right). \nonumber\end{aligned}$$We then pair these elements via $\left\langle w_{g}\ |\ \left[ u_{\lambda },u_{\tau }\right]
\right\rangle $ to define a natural symplectic two form in the coordinates of $\l$ and $\t$. This is given by$$\left\langle w_{g}\ |\ \left[u_{\lambda },u_{\tau }\right] \right\rangle
=\left\langle Ad^{\ast }\left( g\right) w\ |\ \left[u_{\lambda },u_{\tau }%
\right]\right\rangle =\left\langle w\ |\ \left[ Ad\left( g^{-1}\right)
u_{\lambda },Ad\left( g^{-1}\right) u_{\tau }\right] \right\rangle
\label{VKM024}$$Concretely we have$$\xi _{\lambda }\left( \theta \right) =\left( \partial _{\lambda }s\left(
\theta \right) \right) \circ s^{-1}\ \ \ \ \ \ \mbox{and}\ \ \ \ \ \ \Lambda
_{\lambda }\left( \theta \right) =\left( \partial _{\lambda }U\right) \circ
U^{-1}, \label{VKM025}$$and we find$$\begin{aligned}
Ad\left( g^{-1}\right) \xi _{\lambda }\left( \theta \right) &=&s^{-1}\circ
\left( \partial _{\lambda }s\left( \theta \right) \right) =\partial
_{\lambda }\theta =\frac{\partial _{\lambda }s}{\partial _{\theta }s}
\label{VKM026} \\
Ad\left( g^{-1}\right) \Lambda _{\lambda }\left( \theta \right)
&=&U^{-1}\circ \left( \partial _{\lambda }U\right) =U^{-1}\partial _{\lambda
}U. \label{VKM027}\end{aligned}$$Then in Eq.(\[VKM024\]) we can write $$\begin{aligned}
&&\left[ Ad\left( g^{-1}\right) u_{\lambda },Ad\left( g^{-1}\right) u_{\tau }%
\right] = \nonumber \\
&&\left[ \left( \frac{\partial _{\lambda }s}{\partial _{\theta }s}%
,U^{-1}\partial _{\lambda }U,a_{\lambda }\right) ,\left( \frac{\partial
_{\tau }s}{\partial _{\theta }s},U^{-1}\partial _{\tau }U,a_{\tau }\right)%
\right] =\\
&&\left( \left[ \frac{\partial _{\lambda }s}{\partial _{\theta }s},\frac{%
\partial _{\tau }s}{\partial _{\theta }s}\right],\left[ \frac{\partial
_{\lambda }s}{\partial _{\theta }s},U^{-1}\partial _{\tau }U\right] +\left[
U^{-1}\partial _{\lambda }U,\frac{\partial _{\tau }s}{\partial _{\theta }s}%
\right] +\left[ U^{-1}\partial _{\lambda }U,U^{-1}\partial _{\tau }U\right]
, c_{V}+c_{K}\right) \nonumber\end{aligned}$$where $c_{V}$ and $c_{K}$ are the central terms due to the commutation relations for the Virasoro and Kac-Moody algebras, respectively. Choosing $c=-h$ \[\[KR\]\] , the central term for the Virasoro commutator is given by$$c_{V}=\frac{c}{24\pi }\int_{0}^{2\pi }d\sigma u^{\prime }v^{\prime \prime }
\label{VKM029}$$where $u=\frac{\partial _{\lambda }s}{\partial _{\theta }s}$ and $v=\frac{%
\partial _{\tau }s}{\partial _{\theta}s}$, whereas the Kac-Moody commutator gives$$c_{k}=\frac{k}{2\pi }\int_{0}^{2\pi }d\sigma Tr\left( u v'\right) ,$$where $u=U^{-1}\partial _{\lambda }U$ and $v=U^{-1}\partial _{\tau }U$. Hence, after some calculations, the geometric action $$S_{V\otimes KM}
=\int d\lambda d\tau \left\langle w\ |\ \left[ Ad\left( g^{-1}\right)
u_{\lambda },Ad\left( g^{-1}\right) u_{\tau }\right] \,\right\rangle$$ becomes \[\[AS\],\[RR\],\[LR1\],\[LR2\]\] $$\begin{aligned}
&S& =\frac 1{2\pi }\stackunder{\mbox{\small{Metric \,Coupling}}}
{\underbrace{%
\int d^{\;2}\theta \;{\rm D}\left( \theta \right) \left( \frac{%
\partial _\tau s}{\partial _\theta s}\right) }} \nonumber \\
&+&\frac 1{2\pi }{%
\int d^{\;3}\theta \;\mbox{Tr} {\rm A}\left( \theta \right)
\left( \frac{\partial _\lambda s}{\partial _\theta s}\partial _\theta \left(
U^{-1}\partial _\tau U\right) \right)} \label{VKM031} \\
&-&\frac 1{2\pi }\int d^{\;3}\theta \;\mbox{Tr} {\rm A}(\theta){\left(\frac{\partial _\tau s}{\partial _\theta s}%
\partial _\theta \left( U^{-1}\partial _\lambda U\right) +\left[
U^{-1}\partial _\lambda U,U^{-1}\partial _\tau U\right] \right) }
\nonumber \\
&-&{\beta c \over 48\pi }
{\int
d^{\;2}\theta \;\left( {\partial _\theta ^2s \over \left( \partial _\theta
s\right) ^2}\partial _\tau \partial _\theta s-{\left( \partial _\theta
^2s\right) ^2 \over\left( \partial _\theta s\right) ^3}\partial _\tau s\right)
\label{3.7} } \nonumber \\
&-&{\beta k \over 4\pi }
{\int d^{\;3}\theta \;{\rm Tr}\left( U^{-1}\partial _\theta UU^{-1}\partial _\tau
U\right) } \nonumber \\
&+&{{\beta k \over 4\pi }\int d^{\;3}\theta \;\mbox{Tr}\left(
\left[ U^{-1}\partial _\theta U,U^{-1}\partial _\lambda U\right]
U^{-1}\partial _\tau U\right) \nonumber },\end{aligned}$$ where the measures $d^{\;2}\theta=d\theta d\tau$ and $d^{\;3}\theta=d\theta d\tau d\lambda$.
The fourth summand is recognized as the Liouville-Polyakov action \[\[P1\]\] while the fifth and sixth terms are the Wess-Zumino-Novikov-Witten actions \[\[Wi1\]\]. In the summand called the “Metric Coupling", one sees that the diffeomorphism field interacts with the metric through $${\rm D}\left( \theta \right) \left( \frac{
\partial _\tau s}{\partial _\theta s}\right) \rightarrow \sqrt{h} D_{a b}h^{a b},$$ where $h_{ab}$ is the two dimensional induced metric. [*This interaction is the first ingredient we will need to discuss cosmology.*]{} When the trace of $D_{ab}$ becomes constant, this term in the Lagrangian will appear as a cosmological constant.
Transverse Actions
------------------
Every point on the coadjoint orbit of ${\cal B}=\left( D, A, \mu \right)$ corresponds to a group element, $s(\theta)$ and $U(\theta)$. There are, however, group elements that do not appear on the orbit because they do not change ${\cal B}=\left( D, A, \mu \right)$. These group elements are generated by the [*isotropy algebra*]{} of the coadjoint element, ${\cal B}=\left( D, A, \mu \right)$. An adjoint element ${\cal F}$ belongs to the isotropy algebra of ${\cal B}=\left( D, A, \mu \right)$ when $$\delta_{\cal F} {\rm D}\left( \theta \right) =\;{%
{}{}{2\xi ^{^{\prime
}}{\rm D}+{\rm D}^{^{\prime }}\xi + \frac {c \mu}{2\pi}\xi^{\prime \prime
\prime } + \frac {h \mu}{2 \pi} \xi^{\prime}-
{{\rm Tr}\left( {\rm A}\Lambda^{\prime }\right)
}}}=0$$ and $$\delta_{\cal F} {\rm A(\theta )}=\;{\rm A}^{\prime }\xi +\xi^{\prime }{\rm A}-{[\Lambda
\,{\rm A}-{\rm A\,}\Lambda ]+k\,\mu \,\Lambda^{\prime }}=0.$$ These adjoint elements are candidates for the conjugate momenta of the fields $D$ and $A$ that appear in ${\cal B}=\left( D, A, \mu \right)$. To see this recall that in Yang-Mills theories the finite gauge transformations are given by$$\begin{aligned}
A_{i}\left( x\right) &\rightarrow &U\left( x\right) A_{i}\left( x\right)
U^{-1}\left( x\right) -\left( \partial _{i}U\left( x\right)
\right) U^{-1}\left( x\right) \label{YM001} \\
E^{i}\left( x\right) &\rightarrow &U\left( x\right) E^{i}\left( x\right)
U^{-1}\left( x\right) . \label{YM002}\end{aligned}$$In temporal gauge$$A_{0}=0, \label{YM003}$$ the residual gauge transformation are time-independent gauge transformations. The infinitesimal residual gauge transformations are:$$\begin{aligned}
\delta A_{ai}\left( x\right) &=&\left[ \Lambda \left( x\right) ,A\left(
x\right) \right] _{ai}-\partial _{i}\Lambda _{a}\left( x\right)
\label{YM004} \\
\delta E_{a}^{i}\left( x\right) &=&\left[ \Lambda \left( x\right) ,E\left(
x\right) \right] _{a}^{i}. \label{YM005}\end{aligned}$$ The *isotropy algebras* are given by setting these equations to zero:$$\begin{aligned}
\delta_\Lambda A_{ai}\left( x\right) &=&\left[ \Lambda \left( x\right) ,A\left(
x\right) \right] _{ai}-\partial _{i}\Lambda _{a}\left( x\right) =0
\label{YM006} \\
\delta_\Lambda E_{a}^{i}\left( x\right) &=&\left[ \Lambda \left( x\right) ,E\left(
x\right) \right] _{a}^{i}=0. \label{YM007}\end{aligned}$$ In Eq.(\[YM001\]) and Eq.(\[YM002\]), the transformation laws reveal that in one space and one time dimensions, the gauge potential is in the coadjoint representation while its conjugate momentum, the electric field, is in the adjoint representation. Because $E_i$ is in the adjoint algebra it can gauge transform $A_i$. The Gauss law constraints$$\nabla _{i}E_{a}^{i}\left( x\right) =\partial _{i}E_{a}^{i}\left( x\right)
+ \left[ A_{i}\left( x\right) ,E^{i}\left( x\right) \right] _{a}=0,
\label{YM008}$$forbids this from happening. From Eq.(\[YM006\]) we see that the [*conjugate momentum to the coadjoint element is an element of the isotropy algebra*]{}. With this one now has a way of finding a Lagrangian that will give dynamics to the fields in ${\cal B}$. We construct a Lagrangian where one of the field equations reduces to a Gauss Law constraint which corresponds to the isotropy equation in two dimensions. The Lagrangian can exist in any dimension. For the Kac-Moody sector the Lagrangian is the familiar: $$S_{YM}=-\frac{1}{2}\int d^{n}x\;{\rm Tr}\left( F_{\mu \nu }\left( x\right) F^{\mu
\nu }\left( x\right) \right). \label{YM021}$$
![The Foliation of the Dual of the Algebra[]{data-label="fig0"}](geo)
This approach was applied to the isotropy equation obtained from the Virasoro algebra, $$\delta _{\xi }D=2D^{\prime }\xi +D\xi ^{\prime }+q\xi ^{\prime \prime \prime
}+\beta \xi ^{\prime }=0 \label{TAV002}$$where$$q=\frac{c\mu }{2\pi }\ \ \ \ \mbox{and\ \ \ \ }\beta =\frac{h\mu }{2\pi }.$$Following the Yang-Mills case, we assume that this equation is obtained after enforcing the temporal gauge and on finds the [*transverse action*]{}, $$\begin{aligned}
S_{\mbox{\tiny diff}}=&-&\int d^nx \sqrt{g}~\alpha
\left( X^{l m r}~{\rm D}^a{}_r
X_{m l a} +2 X^{l m r} {\rm D}_{l a}
X^a{}_{r m}\right)\\
&-&\int d^nx \sqrt{g}\left(q
X^{a b}{}_b {} \nabla_l \nabla_m{} X^{l m }{}_a+ \frac{\b}2 X^{b g a}
X_{b g a}-\frac{1}{2}X_{m n r }\left( x\right) X^{m n r }\left(
x\right) \right) \nonumber\end{aligned}$$ where ${\rm X}^{m n r} = \nabla^r D^{m n}$. The term proportional to $q$ arises from the central extension in the algebra. Algebraically the central extension exists only in one dimension. However, in the construction of the transverse action, this contribution is seen as coming from a derivative interaction of the connection with the diffeomorphism field. We expect this term to be relevant in the early universe when there is only time dependence in the system. As we will see later, after time evolution, the $q$ dependence of the system becomes negligible. This could be the signature that the symmetries in the cosmological principle are broken and that spatial dependence becomes relevant.
A pictorial interpretation of the geometric and transverse action can be seen in Fig.(\[fig0\]). The geometric actions is the physics of the coordinates and gauge fields (collective coordinates) and is represented by horizontal slices on the foliation of the dual of the algebra. These horizontal slices are equivalence classes of fields and any physics there would constitute anomalies in the gauge and diffeomorphism symmetries. One the other hand, the transverse actions move the fields from one equivalence class to another and do not contain spurious gauge and coordinate degrees of freedom because of the Gauss law constraints. It is for this reason that we say that the transverse actions are transverse to the geometric actions. In $n$ dimensions the mass dimensions for the coupling constants and the diff field are:$$\left[ \alpha \right] =\frac{2-n}{2},\ \ \ \left[ \beta \right] =0,\ \ \ %
\left[ q\right] =-2,\ {\mbox{\rm and}}\ \ \ \left[
D_{\mu \nu }\right] =\frac{n-2}{2}.$$
To see that the diffeomorphism action recovers Eq.(\[TAV002\]) in two dimensions, we vary the action with respect to the space-time component ${\rm D}_{i 0}$. Then by setting ${\rm D}_{i 0}=0$, the field equation becomes $$X^{l m 0} \partial_i {\rm D}^{l m} - \partial_m (X^{m l 0} {\rm D}_{l i}) -\partial_l(X^{m l 0} {\rm D}_{m i}) -
q{~}\partial_i \partial_l \partial_m X^{l m 0}= 0. \label{diffeq1}$$ In $1+1$ this corresponds to the isotropy equation found on the coadjoint orbit where D corresponds to to space-space component, $D_{11}$. The field equations in 1+1 dimensions reduce to $$X {\rm D}' + 2 X' {\rm D} + q{~} X''' =0 \,\, \rightarrow \,\,
\xi {\rm D}' + 2 \xi' {\rm D} + q{~} \xi''' =0,$$ where the adjoint element $\xi$ corresponds to the conjugate momentum, $X \equiv X^{1 1 0}$, of ${\rm
D}\equiv{\rm D}_{1 1}$. This action is exactly analogous to Yang-Mills theory.
Cosmology and the Diffeomorphism Field
======================================
We have seen that there are two types of physical action naturally arise from the coadjoint representations of the Virasoro algebras (diffeomorphisms) and Kac-Moody algebra (affine Lie algebra). One action is the geometric action that comes from the coadjoint orbits of algebras. The other action is from the phase space that is transverse to the coadjoint orbits: the transverse action. The transverse action of the Kac-Moody algebra is the Yang-Mills action in two dimensions and it is also valid in higher dimensions. The transverse action for the Virasoro algebra can be constructed by following the analogy of the construction for the Yang-Mills theory. The action associated with the algebra of diffeomorphisms is also valid in higher dimensions as its origins are Lie Derivatives.
The Virasoro algebra is the symmetry under coordinate transformation on the unit circle, i.e., diffeomorphisms. There an important lesson from string theory is that diffeomorphism should play an active role in the theory of gravitation. Indeed, we have already seen important implication of diffeomorphism in the geometric action for the Virasoro algebra Eq.\[\[VKM031\]\]:$$\frac{1}{2\pi }\int d\tau d\theta D\left(\theta \right) \left(\frac{
\partial _{\tau }s}{\partial _{\theta }s}\right). \label{C001}$$If we rewrite this term in coordinate invariant notion, we have$$\int \sqrt{h}D_{ab}h^{ab}dx \label{C002}$$where $h^{ab}$ is the inverse metric on the two manifold. Our interpretation of the diffeomorphism field $D$ is that it is a non-Riemannian contribution to gravitation: it appears already in two dimensions where the Einstein-Hilbert action only offers topological information. It is easy to observe that when the trace of $D_{ab}$ takes a constant value, this term appears as a natural source for the cosmological constant. Such a term in the action does not depend on the two dimensional structure.
In this section we will apply the transverse action of the Virasoro algebra to cosmology. More specifically, we will study the coupling of the trace part of the diffeomorphism field $D$ to the Robertson-Walker metric for the flat Universe ($k=1$) and study the cosmological implications of the diff field. Many of the calculations in this section are done by Mathematica with MathTensor package \[\[PC\],\[WO\]\]. The action used to study the cosmological effect of the diff field $D$ has two parts:$$S=S_{\mbox{\tiny HE}}+S_{\mbox{\tiny diff}} \label{C003}$$where $S_{\mbox{\tiny HE}}$ is the Hilbert-Einstein action without cosmological constant and $S_{\mbox{\tiny diff}}$ is the transverse action for diff field:$$S_{\mbox{\tiny HE}}=-{1\over 16\pi G}\int dx^{4}\sqrt{g\left( x\right) }R\left( x\right)
,\ \ \ \ \ x=\left( \vec{x},t\right) \label{C004}$$and$$\begin{aligned}
S_{\mbox{\tiny diff}}=&-&\int d^nx \sqrt{g}~\alpha
\left( X^{l m r}~{\rm D}^a{}_r
X_{m l a} +2 X^{l m r} {\rm D}_{l a}
X^a{}_{r m}\right) \nonumber \\
&-&\int d^nx \sqrt{g}\left(q
X^{a b}{}_b {} \nabla_l \nabla_m{} X^{l m }{}_a+ \frac{\b}2 X^{b g a}
X_{b g a}\right) \label{C005}\\
&-& \frac{1}{2}\int d^nx \,\sqrt{g}X_{m n r } X^{m n r } + \lambda \int d^nx \sqrt{g}\,D_{\ m }^{m }.\nonumber\end{aligned}$$ We note that $\alpha $ term is from Gauss’ law constraint, $q$ and $\beta $ terms are from the central extension of the algebra. $\lambda $ term is analogous to Eq.\[C002\], i.e., the cosmological term. Since we will set the ordinary cosmological constant $\Lambda $to zero, there will be no ambiguity. Again the mass dimensions in $n$ dimensions are$$\left[ \alpha \right] =\frac{2-n}{2},\ \ \ \left[ \beta \right] =0,\ \ \ %
\left[ q\right] =-2,\ \ \ \left[ \lambda \right] =\frac{n+2}{2},\ \ \ \left[
D_{\mu \nu }\right] =\frac{n-2}{2}.$$The equations of motion and the energy momentum tensor for the diff field $%
D_{\mu \nu }$ are obtained by varying the diff action $S_{\mbox{\tiny diff}}$ with respect to $D_{\mu \nu }$ and $g_{\mu \nu }$, respectively:$$\mbox{equations of motion:\ \ \ }{\delta S_{\mbox{\tiny diff}} \over \delta D_{\mu \nu }}%
=0$$and$$\mbox{energy-momentum tensor:\ \ \ }\delta S_{\mbox{\tiny diff}}=\frac{1}{2}\int d^{4}x%
\sqrt{g\left( x\right) }\,T^{\mu \nu }\left( x\right) \delta g_{\mu \nu }.$$ The trace of the diff field $D_{\mu \nu }$ is defined through the decomposition$$D_{\mu \nu }\left( x\right) =\frac{1}{n}\psi \left( x\right) g_{\mu \nu
}+W_{\mu \nu }\left( x\right)$$where $\psi \left( x\right) $ is a scalar field, $W_{\mu \nu }\left(
x\right) $ is a traceless symmetric tensor field, and $n$ is the dimension of the spacetime. Since we are interested only in the trace part of the diff field $D_{\mu \nu }$, we set the traceless part to zero: $W_{\mu \nu
}\left( x\right) =0$. The equations of motion for $D_{\mu \nu }\left(
x\right) $, reduce to \[\[YT\]\] $$\begin{aligned}
&&\left( 4-{4\beta \over n}-2\alpha\left( {2\over n^{2}}+{1\over n}\right)
\psi \left( x\right) \right) \nabla ^{\mu }\nabla _{\mu }\psi \left(
x\right) \\
&&\ \ \ -\alpha \left( {2\over n^{2}}+{1\over n}\right) \nabla ^{\mu }\psi
\left( x\right) \nabla _{\mu }\psi \left( x\right) -{1\over n}q\nabla ^{\mu
}\nabla ^{\nu }\nabla _{\mu }\nabla _{\nu }\psi \left( x\right) +n\lambda =0.
\nonumber\end{aligned}$$where $\nabla _{\mu }$ is the covariant derivative. The energy-momentum tensor for $D_{\mu \nu }$ is reduced to \[\[YT\]\] $$\begin{aligned}
T^{\mu \nu } &=&2\left( {2\over n}-{2\beta \over n^{2}}-\left( {1 \over%
n^{2}}+{2 \over n^{3}}\right) \alpha \psi \left( x\right) \right) \nabla
^{\mu }\psi \left( x\right) \nabla ^{\nu }\psi \left( x\right) \nonumber \\
&&+{q\over n^{2}}\left(
\begin{array}{c}
\nabla ^{\sigma }\nabla _{\sigma }\psi \left( x\right) \nabla ^{\mu }\nabla
^{\nu }\psi \left( x\right) +\nabla _{\sigma }\psi \left( x\right) \nabla
^{\sigma }\nabla ^{\mu }\nabla ^{\nu }\psi \left( x\right) \\
\\
-\nabla ^{\mu }\psi \left( x\right) \nabla ^{\sigma }\nabla ^{\nu }\nabla
_{\sigma }\psi \left( x\right) -\nabla ^{\nu }\psi \left( x\right) \nabla
^{\sigma }\nabla ^{\mu }\nabla _{\sigma }\psi \left( x\right)%
\end{array}%
\right) \nonumber \\
&&-\left( {2\over n}-{2\beta \over n^{2}}-\left({1\over n^{2}}+{2 \over %
n^{3}}\right) \alpha \psi \left( x\right) \right) \nabla ^{\sigma }\psi
\left( x\right) \nabla _{\sigma }\psi \left( x\right) g^{\mu \nu } \nonumber \\
&&-{q\over 2n^{2}}\nabla ^{\sigma }\nabla ^{\tau }\psi \left( x\right)
\nabla _{\sigma }\nabla _{\tau }\psi \left( x\right) g^{\mu \nu } \nonumber \\
&&+\lambda \psi \left( x\right) g^{\mu \nu }.\end{aligned}$$
Applying the cosmological principle to the diffeomorphism field, we write, in four dimensions,$$D_{\mu \nu }\left( t\right) =\frac{1}{4}f(t) g_{\mu \nu }$$where $f(t) =\psi \left( \vec{x},t\right) $ is a scalar function that only depends on time. Using this form of diff field, we find the equation of motion for the diff field as$$\begin{array}{c}
\left( 1-\frac{\beta }{4}-\frac{3}{16}\alpha f\left( t\right) \right)
f^{\prime \prime }\left( t\right) +3H\left[ \left( 1-\frac{\beta }{4}-\frac{3%
}{16}\alpha f\left( t\right) \right) f^{\prime }\left( t\right) +\frac{1}{16}%
qf^{\left( 3\right) }\left( t\right) \right] \\
\\
+\lambda -\frac{3}{32}\alpha f^{\prime }(t) ^{2} +\frac{1}{16}qf^{\left( 4\right) }(t) +\frac{3}{8} q H^{2}f^{^{\prime \prime }}f\\
\\
-\frac{9}{16}qH^{3}f^{\prime
}(t) +\frac{3}{16}qH^{\prime }f^{\prime \prime }(t) -%
\frac{3}{8}qf^{\prime }(t) H^{\prime }H=0%
\end{array}
\label{EQMs01}$$Here we used ‘ $^{\prime }$ ’ as a time derivative and $H=H\left( t\right) $. The energy density and pressure of the diff field are also found, respectively, as$$\begin{aligned}
\rho _{\mbox{\tiny diff}}\left( t\right) &=&\frac{1}{2}\left( 1-\frac{\beta }{4}-\frac{3}{%
16}\alpha f\left( t\right) \right) f^{\prime }\left( t\right) ^{2}+\lambda
f\left( t\right) \\
&&+\frac{7}{32}qH^{2}f^{\prime }\left( t\right) ^{2}+\frac{1}{8}qHf^{\prime
}\left( t\right) f^{\prime \prime }\left( t\right) -\frac{1}{32}qf^{^{\prime
\prime }}\left( t\right) ^{2} \nonumber\end{aligned}$$and$$\begin{aligned}
p_{\mbox{\tiny diff}}\left( t\right) &=&\frac{1}{2}\left( 1-\frac{\beta }{4}-\frac{3}{16}%
\alpha f\left( t\right) \right) f^{\prime }\left( t\right) ^{2}-\lambda
f\left( t\right) \\
&-&\frac{9}{32}qH^{2}f^{\prime }\left( t\right) ^{2}+\frac{3}{16}qHf^{\prime
}\left( t\right) f^{\prime \prime }\left( t\right) -\frac{1}{32}qf^{^{\prime
\prime }}\left( t\right) ^{2}+\frac{1}{16}qf^{\prime }\left( t\right)
f^{\left( 3\right) }\left( t\right) . \nonumber\end{aligned}$$Note that the equation of motion for the diff field depends on both $H$ and $%
H^{\prime }$, where as the energy density and the pressure only depend on $H$. The equation of state, the pressure-to-energy density ratio, $w\left(
t\right) $, for the diff field is given by$$w\left( t\right) ={p_{\mbox{\tiny diff}}\left( t\right) \over \rho _{\mbox{\tiny diff}}\left(
t\right) }$$
To study the properties of the diff field in the early universe, we assume that the contributions from the other sources are negligible. In other words, we set the energy densities and pressures for other matters (including radiation) to zero. Under this assumption the Friedmann equations can be written in terms of the scalar field $f\left(t\right) $ and its derivatives in time:$$\begin{aligned}
H^{2}(t) &=&{1\over 3M_{\rm pl}^{2}}\left[
\begin{array}{c}
\frac{1}{2}\left( 1-\frac{\beta }{4}-\frac{3}{16}\alpha f\left( t\right)
\right) f^{\prime }\left( t\right) ^{2}+\lambda f\left( t\right) \\
\\
+\frac{7}{32}qH^{2}f^{\prime \prime }\left( t\right) +\frac{1}{8}qHf^{\prime
}\left( t\right) f^{\prime \prime }\left( t\right) -\frac{1}{32}qf^{^{\prime
\prime }}\left( t\right) ^{2}%
\end{array}%
\right] \label{FrieS01} \\
&& \nonumber \\
H^{\prime }\left( t\right) &=&-{1\over 2M_{\mbox pl}^{2}}\left[
\begin{array}{c}
\left( 1-\frac{\beta }{4}-\frac{3}{16}\alpha f\left( t\right) \right)
f^{\prime }\left( t\right) ^{2} \\
\\
-{1 \over 16}qf^{^{\prime \prime }}\left( t\right) ^{2}+\frac{1}{16}%
qf^{\prime }\left( t\right) f^{\left( 3\right) }\left( t\right) \\
\\
-\frac{1}{16}qH^{2}f^{\prime }\left( t\right) ^{2}+\frac{5}{16}qHf^{\prime
}\left( t\right) f^{\prime \prime }\left( t\right)%
\end{array}%
\right] \label{FrieS02} \\
&& \nonumber\end{aligned}$$The expression of Hubble parameter $H\left( t\right) $ can be obtained from Eq.\[FrieS01\]. Since it is in a quadratic form in $H$, we have two possibilities:$$H_{\pm}(t) ={1 \over 96M_{\mbox pl}^{2}+9qf^{\prime }\left( t\right) ^{2}}%
\left[ 3qf^{\prime }\left( t\right) f^{\prime \prime }\left( t\right) \pm
\sqrt{Q}\right] \label{Hubs01}$$where$$\begin{aligned}
Q &=&9q^{2}f^{\prime }\left( t\right) ^{2}f^{\prime \prime }\left( t\right)
^{2} \label{Hubs02} \\
&&-\left( 96M_{pl}^{2}+9qf^{\prime }\left( t\right) ^{2}\right) \left(
\begin{array}{c}
4\left( \beta -4\right) f^{\prime }\left( t\right) ^{2}+f\left( t\right)
\left( 32\lambda +3\alpha f^{\prime }\left( t\right) ^{2}\right) \\
+qf^{\prime \prime }\left( t\right) ^{2}-2qf^{\prime }\left( t\right)
f^{\left( 3\right) }\left( t\right)%
\end{array}%
\right) . \nonumber\end{aligned}$$
The Friedmann equations, Eq.\[FrieS01\] and Eq.\[FrieS02\], also allow us to simplify the equation of motion for $D_{\mu \nu }$, Eq.\[EQMs01\]: these two equations can be used to eliminate $H$ and $H^{\prime }$ in the equation of motion for $D_{\mu \nu }$.
Numerical Results
=================
We will numerically solve the equation of motion for the trace part of $D_{\mu \nu }$ in the Robertson-Walker metric. We use the numerical solution of $f\left( t\right) $ to plot the Hubble parameter $H\left(
t\right) $, energy-density $\rho \left( t\right) $, pressure $p\left(
t\right) $, the equation of state $w\left( t\right) $ and $\rho \left(
t\right) +3p\left( t\right) $. Since the equation of motionthat we are studying is very complicated, we will limit our analysis to the basic behavior of each quantities under given set of parameters and initial conditions. Our main goal in this numerical computation is not to obtain the results that agree with the current observation, but to investigate the effects of the trace part of diff field $D_{\mu \nu }$ on the quantities mentioned above to study the nature of the scalar field $f\left( t\right) $. We use the explicit Runge-Kutta method for this numerical computation.
Parameters
----------
Eq.\[Hubs01\] suggests that the Hubble parameter $H\left( t\right) $ is expected to increase its magnitude very rapidly as$$f^{\prime }\left( t\right) ^{2}\rightarrow -\frac{32}{3q}M_{\mbox pl}^{2}.$$The above condition also implies that the value of $q$ should be negative. In this limit it is not difficult see that the positive solution (the solution with $+\sqrt{Q}$) has a simpler behavior near the “peak". Hence, in the following, we will study the positive solution case. Noting that the function of parameter $\beta $ is basically to shift the value of kinetic term, we set this parameter to zero. Noting that the mass dimension scalar field $f\left( t\right) $ is one, so it’s natural to use the planck mass $%
M_{pl}$ for this scalar field, i.e., we set $M_{pl}$ to one. Hence undetermined parameters are $\alpha $, $q$, and $\lambda $.
Trial Initial Conditions
------------------------
The equation of motion for the trace part of $D_{\mu \nu }$ in the Robertson-Walker metric is a fourth order nonlinear differential equation with three parameters (after setting $\beta =0$ and $M_{pl}=1$). To solve these differential equations numerically, we need to specify trial values for the three parameters and four initial conditions. In this numerical computation the initial conditions for $f\left( t\right) $, $f^{\prime
}\left( t\right) $, $f^{\prime \prime }\left( t\right) $, and $f^{\left(
3\right) }\left( t\right) $ are used. A priori, we do not know how to choose these parameters and initial conditions. To overcome this, we use the physical condition on the Hubble parameter as a guide for choosing the trial values. We search for a set of the parameters and the initial conditions that give a positive and finite value for the Hubble parameter. With this as a criterion, a choice for the trial of parameters and the initial conditions become $$\begin{aligned}
&&\alpha =-1,\ \ \ q=-6,\ \ \ \lambda =0.05, \nonumber \\
f\left( 0\right) &=&-10,\ f^{\prime }\left( 0\right) =0,\ f^{\prime \prime
}(0)=-1,\ f^{\left( 3\right) }=-1. \label{ref}\end{aligned}$$This set of trial values give the Hubble parameter plot shown in Fig.\[fig:hubble\].
![The Trial Hubble Parameter Plot[]{data-label="fig:hubble"}](hubble)
Note that we used$\ k=0$ (flat Universe), $\Lambda =0$ (no cosmological constant), $\beta =0$, and $M_{\mbox pl}=1$.
Computation
-----------
As we have noted in Chapter 1, the nature of scalar fields are characterized by the equation of state $w\left( t\right) $, and this value is defined as the ratio of the pressure $p\left( t\right) $ to the energy density $\rho
\left( t\right) $. The acceleration of Universe can be studied through the quantity $\rho \left( t\right) +3p\left( t\right) $. We, thus, compute $%
f\left( t\right) $, $H\left( t\right) $, $\rho \left( t\right) $, $p\left(
t\right) $, $w\left( t\right) $, and $\rho \left( t\right) +3p\left(
t\right) $ for the reference set and study the effect of variations in two parameters ($q$ and $\lambda $) and four initial conditions of the scalar field $f\left( t\right) $ ($f\left( 0\right) $, $f^{\prime }\left( 0\right) $, $f^{\prime \prime }\left( 0\right) $, and $f^{\left( 3\right) }\left(
0\right) $) on these quantities. The following is the sets of two parameters ($q$ and $\lambda $) and four initial conditions of the scalar field $%
f\left( t\right) $ ($f\left( 0\right) $, $f^{\prime }\left( 0\right) $, $%
f^{\prime \prime }\left( 0\right) $, and $f^{\left( 3\right) }\left(
0\right) $) used in this numerical computation
Results
-------
The effect of changes in two parameters ($q$ and $\lambda $) and four initial conditions on the scalar field $f\left( t\right) $ ($f\left(
0\right) $, $f^{\prime }\left( 0\right) $, $f^{\prime \prime }\left(
0\right) $, and $f^{\left( 3\right) }\left( 0\right) $) are studied. Each parameter/initial condition is varied with respect to the corresponding starting value. The results are plotted for the scalar field $f\left(
t\right) $, Hubble parameter $H\left( t\right) $, energy density $\rho
\left( t\right) $, pressure $P\left( t\right) $, and the equation of state (pressure-to-energy density ratio) $w\left( t\right) $. The unspecified values are the same as that of the reference set Eq \[ref\]. Here are our results.
Effect of Changes in $q$
------------------------
Red $(\cdots)$ $q=-2$ , Green $(^{\underline{\,\,\,\,\,\,\,}})$ $q=-6$,Blue (- -) $q=-8$
![Scalar field $f(t)$ obtained with $q=-2$ (red), $-6$ (green), $-8$ (blue)[]{data-label="fig:ff01"}](ff01)
![Hubble parameter $H(t)$ obtained with $q=-2$ (red), $-6$ (green), $-8$ (blue)[]{data-label="fig:hh01"}](hh01)
![Energy density $\rho(t)$ with $q=-2$ (red), $-6$ (green), $-8$ (blue)[]{data-label="fig:rr01"}](rr01)
![Pressure $p(t)$ obtained with $q=-2$ (red), $-6$ (green), $-8$ (blue)[]{data-label="fig:pp01"}](pp01)
![Equation of State $w(t)$ obtained with $q=-2$ (red), $-6$ (green), $-8$ (blue)[]{data-label="fig:ww01"}](ww01)
![$\rho(t)+3 p(t)$ obtained with $q=-2$ (red), $-6$ (green), $-8$ (blue)[]{data-label="fig:rp01"}](rp01)
Effect of Change in $\lambda $
------------------------------
Red $(\cdots)$ $\l=0.08$ , Green $(^{\underline{\,\,\,\,\,\,\,}})$ $\l=0.05$,Blue (- -) $\l=0.02$
![$f(t)$ obtained with $\l=0.08$ (red), $0.05$ (green), $0.02$ (blue)[]{data-label="fig:ff02"}](ff02)
![$H(t)$ obtained with $\l=0.08$ (red), $0.05$ (green), $0.02$ (blue)[]{data-label="fig:hh02"}](hh02)
![$\r(t)$ obtained with $\l=0.08$ (red), $0.05$ (green), $0.02$ (blue)[]{data-label="fig:rr02"}](rr02)
![$p(t)$ obtained with $\l=0.08$ (red), $0.05$ (green), $0.02$ (blue)[]{data-label="fig:pp02"}](pp02)
![Equation of State $w(t)$ obtained with $\l=0.08$ (red), $0.05$ (green), $0.02$ (blue)[]{data-label="fig:ww02"}](ww02)
![$\r(t)+3p(t)$ obtained with $\l=0.08$ (red), $0.05$ (green), $0.02$ (blue)[]{data-label="fig:rp02"}](rp02)
Effect of Change in Initial Value of $f\left( t_{0}\right) $
------------------------------------------------------------
Red $(\cdots)$ $f(t_0)=-8$ , Green $(^{\underline{\,\,\,\,\,\,\,}})$ $f(t_0)=-10$,Blue (- -) $f(t_0)=-12$
![Scalar field $f(t)$ obtained with $f(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:ff03"}](ff03)
![Hubble parameter $H(t)$ obtained with $f(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:hh03"}](hh03)
![Energy Density $\r(t)$ obtained with $f(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:rr03"}](rr03)
![Pressure $p(t)$ obtained with $f(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:pp03"}](pp03)
![Equation of State $w(t)$ obtained with $f(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:ww03"}](ww03)
![$\r(t)+3 p(t)$ obtained with $f(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:rp03"}](rp03)
Effect of Change in Initial Value of $f^{\prime }\left( t_{0}\right) $
----------------------------------------------------------------------
Red $(\cdots)$ $f^{\prime }\left( t_{0}\right)=-0.5$ , Green $(^{\underline{\,\,\,\,\,\,\,}})$ $f^{\prime }\left( t_{0}\right)=0$,Blue (- -) $f^{\prime }\left( t_{0}\right)=0.5$
![Scalar field $f(t)$ obtained with $f^{\prime}(t_0)=-0.52$ (red), $0$ (green), $0.5$ (blue)[]{data-label="fig:ff04"}](ff04)
![Scalar field $H(t)$ obtained with $f^{\prime}(t_0)=-0.52$ (red), $0$ (green), $0.5$ (blue)[]{data-label="fig:hh04"}](hh04)
![Energy density $\r(t)$ obtained with $f^{\prime}(t_0)=-0.52$ (red), $0$ (green), $0.5$ (blue)[]{data-label="fig:rr04"}](rr04)
![Pressure $p(t)$ obtained with $f^{\prime}(t_0)=-0.52$ (red), $0$ (green), $0.5$ (blue)[]{data-label="fig:pp04"}](pp04)
![Equation of state $w(t)$ obtained with $f^{\prime}(t_0)=-0.52$ (red), $0$ (green), $0.5$ (blue)[]{data-label="fig:ww04"}](ww04)
![$\r(t)+3 p(t)$ obtained with $f^{\prime}(t_0)=-0.52$ (red), $0$ (green), $0.5$ (blue)[]{data-label="fig:rp04"}](rp04)
Effect of Change in Initial Value of $f^{\prime \prime }\left( t_{0}\right) $
-----------------------------------------------------------------------------
Red $(\cdots)$ $f^{\prime \prime }(t_0)=-0.5$ , Green $(^{\underline{\,\,\,\,\,\,\,}})$ $f^{\prime \prime }(t_0)=-1$,Blue (- -) $f^{\prime \prime }(t_0)=-1.5$
![Scalar field $f(t)$ with $f^{\prime \prime}(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:ff05"}](ff05)
![Hubble parameter $H(t)$ with $f^{\prime \prime}(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:hh05"}](hh05)
![Energy density $\r(t)$ with $f^{\prime \prime}(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:rr05"}](rr05)
![Pressure $p(t)$ with $f^{\prime \prime}(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:pp05"}](pp05)
![Equation of state $w(t)$ with $f^{\prime \prime}(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:ww05"}](ww05)
![$\r(t)+3 p(t)$ with $f^{\prime \prime}(t_0)=-8$ (red), $-10$ (green), $-12$ (blue)[]{data-label="fig:rp05"}](rp05)
Effect of Change in Initial Value of $f^{\left( 3\right) }\left(
t_{0}\right) $
----------------------------------------------------------------
Red $(\cdots)$ $f^{\left(3\right)}=-0.5$ , Green $(^{\underline{\,\,\,\,\,\,\,}})$ $f^{\left(3\right)}=-1$,Blue (- -) $f^{\left(3\right)}=-1.5$
![Scalar field $f(t)$ obtained with $f^{(3)}(t_0)=-0.5$ (red), $-1$ (green), $-1.5$ (blue)[]{data-label="fig:ff06"}](ff06)
![Hubble parameter $H(t)$ obtained with $f^{(3)}(t_0)=-0.5$ (red), $-1$ (green), $-1.5$ (blue)[]{data-label="fig:hh06"}](hh06)
![Energy density $\r(t)$ obtained with $f^{(3)}(t_0)=-0.5$ (red), $-1$ (green), $-1.5$ (blue)[]{data-label="fig:rr06"}](rr06)
![Pressure $p(t)$ obtained with $f^{(3)}(t_0)=-0.5$ (red), $-1$ (green), $-1.5$ (blue)[]{data-label="fig:pp06"}](pp06)
![Equation of state $w(t)$ obtained with $f^{(3)}(t_0)=-0.5$ (red), $-1$ (green), $-1.5$ (blue)[]{data-label="fig:ww06"}](ww06)
![$\r(t)+3 p(t)$ obtained with $f^{(3)}(t_0)=-0.5$ (red), $-1$ (green), $-1.5$ (blue)[]{data-label="fig:rp06"}](rp06)
Conclusion
==========
From the plots obtained in the last section we find that the equation of state $w\left( t\right) $ can take, at least, values between $-2$ and $0$. Most interestingly it always converges to $-1$ after some time for all the initial conditions we used. It means that the trace part of the diff field $%
D_{\mu \nu }$ acts like a cosmological constant $\Lambda $ in later times. This scalar field spends most of the time before $w=-1$ as a phantom dark energy. This is a time-dependent phantom dark energy and its effects on Universe will be different from the time-independent cases \[\[Ca1\], [CW]{}, \[Gr\]\]. Since the equation of state $w$ can be $0$ in the very early Universe, it behaves as a dark matter when it happens. The fact that the value of $w$ never gets close to $1/3$ is very desirable because otherwise the radiation-like behavior of the scalar field at early times would influence already well explained primordial nucleosynthesis \[[BKLMS]{}\].
The effects on the scalar field on the Hubble parameter are also notable. It is easy to see that there are expected peaks \[Eq.\[Hubs01\]\]. Figure \[fig:hh01\], Figure \[fig:hh03\], Figure \[fig:hh04\], and Figure \[fig:hh05\] show shoulder-like regions where the value of Hubble parameter stay almost constant, i.e., the universe is expanding exponentially over this region:$$H=\frac{\dot{a}}{a}={\mbox{constant}} \rightarrow a\left(
t\right) \sim e^{Ht}.$$This scalar field, hence, acts like an inflaton over these regions. The constant Hubble parameter in later time can be interpreted as the current expansion of the Universe due to the cosmological constant like behavior of the scalar field. The last important observation is that the value of $\rho
\left( t\right) +3p\left( t\right) $ is almost always negative. That is, this scalar contributes to the acceleration of Universe for most of time.
Though our results are qualitative, they are showing very interesting cosmological implications of the diffeomorphism field $D_{\mu \nu }$: the trace part of the diffeomorphism field $D_{\mu \nu }$ can produce the inflation-like behavior of the Universe in early time and the accelerating Universe in later time qualitatively. It is also observed that the time-dependent phantom dark energy plays a crucial role in this model. Our results, however, lack the realistic prediction: $e$-foldings $N$,$$N\left( t\right) =\ln \frac{a\left( t_{\mbox{\tiny end}}\right) }{a\left( t\right) }%
=\int_{t}^{t_{end}}H\left( t\right) dt,$$during “inflation" predicted by this model is about $N\sim 2$, whereas any realistic model for inflation requires $N\sim 70$. This suggest that more research in the study the set of parameters and the initial values necessary to make sensible predictions. Furthermore the understanding of the time-dependent phantom dark energy is essential in future research.
Acknowledgements {#acknowledgements .unnumbered}
================
VGJR would like to thank A.P. Balachandran, Y. Meurice, V.P. Nair, and P. Ramond for discussion. This work was supported by NSF grant PHY 02-44377.
[99]{} \[ADM\]C, Armendariz-Picón, T. Damour, V. Mukhanov, *hep-th/9904075* v1
\[AM\]C, Armendariz-Picón and V. Mukhanov, *Phys. Rev. D* **63**, 103510 (2001)
\[AMS\]C, Armendariz-Picón, V. Mukhanov, and P.J. Steinhardt, *Phys. Rev. Lett.* **85**, 4438 (2000)
\[AS\]A. Alekseev and S. Shatashvili, *Nucl. Phys.* **B323** 719 (1989)
\[Ar2\]V.I. Arnold, *Mathematical Methods of Classical Mechanics*, 2nd. ed. Springer-Verlag, (1989)
\[BLR\]T.Branson, R.P. Lano and V.G.J. Rodgers, *Phys. Lett.* *B*** 412,** 253 (1997)
\[BKLMS\]V. Barger, J.P. Kneller, H.-S. Lee, D. Marfatia, and G. Steigman, *Phys. Lett B* **566**, 8 (2003)
\[BRY\]T.P. Branson, V.G.J. Rodgers, T. Yasuda,*Int. J. Mod. Phys. A* Vol. **15,** 3549 (2000)
\[Ca\]R.R. Caldwell, *Brazilian Journal of Physics*, vol **30**, no 2 215
\[Ca1\]R.R. Caldwell, *astro-ph/9908168 v2*
\[CW\]R. -G. Cai and A. Wnag, *hep-th/0411025*
\[D1\]S. Dodelson, *Modern Cosmology*, Academic Press (2003)
\[DvNR\]G.W. Delius, P. van Nieuwenhuizen and V.G.J. Rodgers, *Int. J. Mod. Phys.* **A5** 3943 (1990)
\[Gr\]A. Gruzinov, *astro-ph/0405096 v1*
\[Ki1\]A. A. Kirillov, *Unitary representations of nilpotent Lie groups*, Uspekhi Mat. Nauk **17** (1962)
\[Ki2\]A. A. Kirillov, *Elements of the Theory of Reperesentations*, Springer-Verlag (1975)
\[Ki3\]A.A. Kirillov, *Lect. Notes in Math.* **970** 101, Springer-Verlag (Berlin, 1982)
\[Ki4\]A. A. Kirillov, Lectures on the Orbit Method, AMS (2000)
\[KR\]V.G. Kac and A.K. Raina, *Bombay Lectures on Highest Weight Representaions of infinite Dimensional Lie Algebras* World Scientific Publishing (1987)
\[LR1\]R.P. Lano and V.G.J. Rodgers, *Nucl. Phys.* **B437** (1995) 45
\[LR2\]R.P. Lano and V.G.J. Rodgers, *Mod. Phys. Lett.* **A7** (1992) 1725
\[P1\]A.M. Polyakov, *Mod. phys. Lett.* **A2** No.11 (1987) 893
\[PC\]L. Parker and S.M. Christensen, *MathTensor*, Addison-Wesley Publishing Company (1994)
\[Pe\]P.J. Peebles, Principles of Physical Cosmology, Princeton University Press (Princeton, 1993)
\[PS1\]A. Pressley and G Segal, *Loop Groups*, Oxford Univ. Press (Oxford,1986)
\[RR\]B. Rai and V.G.J. Rodgers, *Nucl. Phys.* **B341** (1990) 119
\[RY\]V.G.J. Rodgers, T. Yasuda, *Mod.Phys.Lett. A***18**, 2467 (2003)
\[Sc1\]R.J. Scherrer, *astro-ph/0402316v3*
\[SNCP\]http://panisse.lbl.gov/public/
\[Vi1\]M.A. Virasoro, *Phys. Rev. Lett*. **22** (1969) 37
\[W1\]P.B. Wiegmann, *Nucl. Phys.* **B323** (1989) 311
\[Wa\]R.M. Wald, *General Relativity*, The University of Chicago Press (1984)
\[We1\]S. Weinberg, *Gravitation and Cosmology*, John Wiley & Sons, Inc. (1972)
\[Wi1\]E. Witten, *Comm. Math. Phys.* **92** (1984) 455
\[Wi2\]E. Witten, *Comm. Math. Phys.* **114** (1988) 1
\[WM\]http//lambda.gsfc.nasa.gov/
\[WO\]S. Wolfram, *The MATHEMATICA Book, Version 3*, Cambridge University Press (1996)
\[YT\]Takeshi Yasuda, *The Cosmological Implications of Diffeomorphsims* Ph.D. thesis, The University of Iowa (2006)
[^1]: vincent-rodgers@uiowa.edu
[^2]: takeshi-yasuda@uiowa.edu
[^3]: This is the problem related to the precision required for the initial value of the total energy density of the Universe so that the spatial curvature of the Universe is almost zero: the flatness problem.
|
---
abstract: 'We introduce the Kronecker factored online Laplace approximation for overcoming catastrophic forgetting in neural networks. The method is grounded in a Bayesian online learning framework, where we recursively approximate the posterior after every task with a Gaussian, leading to a quadratic penalty on changes to the weights. The Laplace approximation requires calculating the Hessian around a mode, which is typically intractable for modern architectures. In order to make our method scalable, we leverage recent block-diagonal Kronecker factored approximations to the curvature. Our algorithm achieves over $90\%$ test accuracy across a sequence of $50$ instantiations of the permuted MNIST dataset, substantially outperforming related methods for overcoming catastrophic forgetting.'
author:
- |
Hippolyt Ritter[^1]\
University College London Aleksandar Botev\
University College London David Barber\
University College London\
& Alan Turing Institute
bibliography:
- 'references.bib'
title: Online Structured Laplace Approximations For Overcoming Catastrophic Forgetting
---
Introduction {#sec:intro}
============
Creating an agent that performs well across multiple tasks and continuously incorporates new knowledge has been a longstanding goal of research on artificial intelligence. When training on a sequence of tasks, however, the performance of many machine learning algorithms, including neural networks, decreases on older tasks when learning new ones. This phenomenon has been termed ‘catastrophic forgetting’ [@catastrophic:forgetting:connectionist; @catastrophic:interference; @connectionist:models:memory] and has recently received attention in the context of deep learning [@empirical:cf; @ewc]. Catastrophic forgetting cannot be overcome by simply initializing the parameters for a new task with optimal ones from the old task and hoping that stochastic gradient descent will stay sufficiently close to the original values to maintain good performance on previous datasets [@empirical:cf].
Bayesian learning provides an elegant solution to this problem. It combines the current data with prior information to find an optimal trade-off in our belief about the parameters. In the sequential setting, such information is readily available: the posterior over the parameters given all previous datasets. It follows from Bayes’ rule that we can use the posterior over the parameters after training on one task as our prior for the next one. As the posterior over the weights of a neural network is typically intractable, we need to approximate it. This type of Bayesian online learning has been studied extensively in the literature [@bayesian:online; @online:variational:bayes:zoubin; @online:variational:bayes:honkela].
In this work, we combine Bayesian online learning [@bayesian:online] with the Kronecker factored Laplace approximation [@scalable:laplace] to update a quadratic penalty for every new task. The block-diagonal Kronecker factored approximation of the Hessian [@kfac; @kfra] allows for an expressive scalable posterior that takes interactions between weights within the same layer into account. In our experiments we show that this principled approximation of the posterior leads to substantial gains in performance over simpler diagonal methods, in particular for long sequences of tasks.
Bayesian online learning for neural networks {#sec:ewc}
============================================
We are interested in optimizing the parameters $\theta$ of a single neural network to perform well across multiple tasks ${\mathcal{D}}_1, \ldots, {\mathcal{D}}_T$, specifically finding a MAP estimate ${\theta}^* = \operatorname*{arg\,max}_{\theta}p({\theta}|{\mathcal{D}}_1, \ldots, {\mathcal{D}}_T)$. However, the datasets arrive sequentially and we can only train on one of them at a time.
In the following, we first discuss how Bayesian online learning solves this problem and introduce an approximate procedure for neural networks. We then review recent Kronecker factored approximations to the curvature of neural networks and how to use them to obtain a better fit to the posterior. Finally, we introduce a hyperparameter that acts as a regularizer on the approximation to the posterior.
Bayesian online learning
------------------------
Bayesian online learning [@bayesian:online], or Assumed Density Filtering [@smec], is a framework for updating an approximate posterior when data arrive sequentially. Using Bayes’ rule we would like to simply incorporate the most recent dataset ${\mathcal{D}}_{t+1}$ into the posterior as:
$$p({\theta}|{\mathcal{D}}_{1:t+1})
= \frac{p({\mathcal{D}}_{t+1}|{\theta}) p({\theta}|{\mathcal{D}}_{1:t})}
{\int d{\theta}' p({\mathcal{D}}_{t+1}|{\theta}') p({\theta}'|{\mathcal{D}}_{1:t})}$$
where we use the posterior $p({\theta}|{\mathcal{D}}_{1:t})$ from the previously observed tasks as the prior over the parameters for the most recent task. As the posterior given the previous datasets is typically intractable, Bayesian online learning formulates a parametric approximate posterior $q$ with parameters $\phi_t$, which it iteratively updates in two steps:
#### Update step
In the update step, the approximate posterior $q$ with parameters $\phi_t$ from the previous task is used as a prior to find the new posterior given the most recent data:
$$p({\theta}|{\mathcal{D}}_{t+1}, \phi_t)
= \frac{p({\mathcal{D}}_{t+1}|{\theta}) q({\theta}|\phi_t)}
{\int d{\theta}' p({\mathcal{D}}_{t+1}|{\theta}') q({\theta}'|\phi_t)}
\label{eq:bo:update}$$
#### Projection step
The projection step finds the distribution within the parametric family of the approximation that most closely resembles this posterior, i.e. sets $\phi_{t+1}$ such that:
$$q({\theta}|\phi_{t+1}) \approx p({\theta}|{\mathcal{D}}_{t+1}, \phi_t)$$
@bayesian:online suggest minimizing the KL-divergence between the approximate and the true posterior, however this is mostly appropriate for models where the update-step posterior and a solution to the KL-divergence are available in closed form. In the following, we therefore propose using a Laplace approximation to make Bayesian online learning tractable for neural networks.
The online Laplace approximation
--------------------------------
Neural networks have found wide-spread success and adoption by performing simple MAP inference, i.e. finding a mode of the posterior:
$${\theta}^*
= \operatorname*{arg\,max}_{\theta}\; \log p({\theta}|{\mathcal{D}})
= \operatorname*{arg\,max}_\theta \; \log p({\mathcal{D}}|{\theta}) + \log p({\theta})
\label{eq:post}$$
where $p({\mathcal{D}}|{\theta})$ is the likelihood of the data and $p({\theta})$ the prior. Most commonly used loss functions and regularizers fit into this framework, e.g. using a categorical cross-entropy with $L_2$-regularization corresponds to modeling the data with a categorical distribution and placing a zero-mean Gaussian prior on the network parameters. A local mode of this objective function can easily be found using standard gradient-based optimizers.
Around a mode, the posterior can be locally approximated using a second-order Taylor expansion, resulting in a Normal distribution with the MAP parameters as the mean and the Hessian of the negative log posterior around them as the precision. Using a Laplace approximation for neural networks was pioneered by @mackay:laplace.
We therefore proceed in two iterative steps similar to Bayesian online learning, using a Gaussian approximate posterior for $q$, such that $\phi_t = \{\mu_t, \Lambda_t\}$ consists of a mean $\mu$ and a precision matrix $\Lambda$:
#### Update step
As the posterior of a neural network is intractable for all but the simplest architectures, we will work with the unnormalized posterior. The normalization constant is not needed for finding a mode or calculating the Hessian. The Gaussian approximate posterior results in a quadratic penalty encouraging the parameters to stay close to the mean of the previous approximate posterior:
$$\begin{split}
\log p({\theta}|{\mathcal{D}}_{t+1}, \phi_t)
&\propto \log p({\mathcal{D}}_{t+1}|{\theta}) + \log q({\theta}|\phi_t)\\
&\propto \log p({\mathcal{D}}_{t+1}|{\theta}) - \frac{1}{2} ({\theta}- \mu_t)^\top \Lambda_t ({\theta}- \mu_t)
\label{eq:objective}
\end{split}$$
#### Projection step
In the projection step we approximate the posterior with a Gaussian. We first update the mean of the approximation to a mode of the new posterior:
$$\mu_{t+1} = \operatorname*{arg\,max}_{\theta}\; \log p({\mathcal{D}}_{t+1}|{\theta}) + \log q({\theta}|\phi_t)$$
and then perform a quadratic approximation around it, which requires calculating the Hessian of the negative objective. This leads to a recursive update to the precision with the Hessian of the most recent log likelihood, as the Hessian of the negative log approximate posterior is its precision:
$$\Lambda_{t+1} = {H}_{t+1}(\mu_{t+1}) + \Lambda_t$$
where ${H}_{t+1}(\mu_{t+1}) = -{\left.{\frac{\partial^2 \log p({\mathcal{D}}_{t+1}|{\theta})}{\partial {\theta}\partial {\theta}}}\right\rvert_{{\theta}=\mu_{t+1}}}$ is the Hessian of the newest negative log likelihood around the mode. The precision of a Gaussian is required to be positive semi-definite, which is the case for the Hessian at a mode. In order to numerically guarantee this in practice, we use the Fisher Information as an approximation [@fisher] that is positive semi-definite by construction.
The recursion is initialized with the Hessian of the log prior, which is typically constant. For a zero-mean isotropic Gaussian prior, corresponding to an $L_2$-regularizer, it is simply the identity matrix times the prior precision.[^2]
A desirable property of the Laplace approximation is that the approximate posterior becomes peaked around its current mode as we observe more data. This becomes particularly clear if we think of the precision matrix as the product of the number of data points and the average precision. By becoming increasingly peaked, the approximate posterior will naturally allow the parameters to change less for later tasks. At the same time, even though the Laplace method is a local approximation, we would expect it to leave sufficient flexibility for the parameters to adapt to new tasks, as the Hessian of neural networks has been observed to be flat in most directions [@hessian:empirical].
We will also compare to fitting the true posterior with a new Gaussian at every task for which we compute the Hessian of all tasks around the most recent MAP estimate:
$$\Lambda_{t+1} = {H}_{prior} + \sum_{i=1}^{t+1} {H}_i(\mu_{t+1})
\label{eq:approximate}$$
This procedure differs from the online Laplace approximation only in evaluating all Hessians at the most recent MAP parameters instead of the respective task’s ones. Technically, this is not a valid Laplace approximation, as we only optimize an approximation to the posterior. Hence the optimal parameters for the approximate objective will not exactly correspond to a mode of the exact posterior. However, as we will use a positive semi-definite approximation to the Hessian, this will only introduce a small additional approximation error.
Calculating the Hessian across all datasets requires relaxing the sequential learning setting to allowing access to previous data ‘offline’, i.e. between tasks. We use this baseline to check if there is any loss of information in using estimates of the curvature at previous parameter values.
Kronecker factored approximation of the Hessian
-----------------------------------------------
Modern networks typically have millions of parameters, so the size of the Hessian is several terabytes. An approximation that is simple to implement with automatic differentiation frameworks is the diagonal of the Fisher matrix, i.e. the expected square of the gradients, where the expectation is over the datapoints and the conditional distribution defined by the model. While this approximation has been used successfully [@ewc], it ignores interactions between the parameters.
Recent works on second-order optimization [@kfac; @kfra] have developed block-diagonal approximations to the Hessian. They exploit that, for a single data point, the diagonal blocks of the Hessian of a feedforward network — corresponding to the weights of a single layer — are Kronecker factored, i.e. a product of two relatively small matrices.
We denote a neural network as taking an input ${a}_0{=}x$ and producing an output ${h}_L$. The input is passed through layers $1,\ldots,L$ as the linear pre-activations ${h}_l{=}{W}_l {a}_{l-1}$ and the activations ${a}_l{=}f_l({h}_l)$, where $f_l$ is a non-linear elementwise function. The outputs then parameterize the log likelihood of the data, and, using the chain rule, we can write the Hessian w.r.t. the weights of a single layer as:
$${H}_l = {\frac{\partial^2 \log p({\mathcal{D}}|{h}_L)}{\partial \operatorname{vec}({W}_l) \partial \operatorname{vec}({W}_l)}} = {\mathcal{Q}}_l \otimes {\mathcal{H}}_l \label{eq:kf}$$
where $\operatorname{vec}({W}_l)$ is the weight matrix of layer $l$ stacked into a vector and we define ${\mathcal{Q}}_l = a_{l-1} a_{l-1}^\top$ as the covariance of the inputs to the layer. ${\mathcal{H}}_l = {\frac{\partial^2 \log p({\mathcal{D}}|{\theta})}{\partial h_l \partial h_l}}$ is the pre-activation Hessian, i.e. the second derivative w.r.t. the pre-activations $h_l$ of the layer. We provide the basic derivation of [Eq. (\[eq:kf\])]{} and the recursive formula for calculating ${\mathcal{H}}_l$ in [Appendix \[app:kf\]]{}. To maintain the Kronecker factorization in expectation, i.e. for an entire dataset, [@kfac] and [@kfra] assume the two factors to be independent and approximate the expected Kronecker product by the Kronecker product of the expected factors.
The block-diagonal approximation splits the Hessian-vector product in the quadratic penalty across the layers. Due to the Kronecker factored approximation, it can be calculated efficiently for each layer using the following well-known identity:
$$({\mathcal{Q}}_l \otimes {\mathcal{H}}_l) \operatorname{vec}({W}_l - {W}_l^*)
= \operatorname{vec}({\mathcal{H}}_l \left({W}_l - {W}_l^*\right)^\top {\mathcal{Q}}_l)$$
where $\operatorname{vec}$ stacks the columns of a matrix into a vector and we use that ${\mathcal{H}}$ is symmetric.
The block-diagonal Kronecker factored approximation corresponds to assuming independence between the layers and factorizing the covariance between the weights of a layer into the covariance of the columns and rows, resulting in a matrix normal distribution [@matrixnormal]. The same approximation has been used recently to sample from the predictive posterior [@scalable:laplace; @bayesian:maml]. While it still makes some independence assumptions about the weights, the most important interactions — the ones within the same layer — are accounted for. In order to guarantee for the curvature being positive semi-definite, we approximate the Hessian with the Fisher Information as in [@kfac] throughout our experiments.
Regularizing the approximate posterior {#sec:hyperparam}
--------------------------------------
@ewc, who develop a similar method inspired by the Laplace approximation, suggest using a multiplier $\lambda$ on the quadratic penalty in [Eq. (\[eq:objective\])]{}. This hyperparameter provides a way of trading off retaining performance on previous tasks against having sufficient flexibility for learning a new one. As modifying the objective would propagate into the recursion for the precision matrix, we instead place the multiplier on the Hessian of each log likelihood and update the precision as:
$$\Lambda_{t+1} = \lambda {H}_{t+1}(\mu_{t+1}) + \Lambda_t$$
The multiplier affects the width of the approximate posterior and thus the location of the next MAP estimate. As it acts directly on the parameter of a probability distribution, its optimal value can inform us about the quality of our approximation: if it strongly deviates from its natural value of $1$, our approximation is a poor one and over- or underestimates the uncertainty about the parameters. We visualize the effect of $\lambda$ in [Fig. \[fig:contours\]]{} in [Appendix \[app:reg:vis\]]{}.
Related work {#sec:related}
============
Our method is closely related to Bayesian online learning [@bayesian:online] and to Laplace propagation [@laplace:prop]. In contrast to Bayesian online learning, as we cannot update the posterior over the weights in closed form, we use gradient-based methods to find a mode and perform a quadratic approximation around it, resulting in a Gaussian approximation. Laplace propagation, similar to expectation propagation [@ep], maintains a factor for every task, but approximates each of them with a Gaussian. It performs multiple updates, whereas we use each dataset only once to update the approximation to the posterior.
The most similar method to ours for overcoming catastrophic forgetting is Elastic Weight Consolidation (EWC) [@ewc]. EWC approximates the posterior after the first task with a Gaussian. However, it continues to add a penalty for every new task [@reply:ferenc]. This is more closely related to Laplace propagation, but may be overcounting early tasks [@quadratic:penalties] and does not approximate the posterior. Furthermore, EWC uses a simple diagonal approximation to the Hessian. @imm approximate the posterior around the mode for each dataset with a diagonal Gaussian in addition to a similar approximation of the overall posterior. They update this approximation to the posterior as the Gaussian that minimizes the KL divergence with the individual posterior approximations. @vcl implement online variational learning [@online:variational:bayes:zoubin; @online:variational:bayes:honkela], which fits an approximation to the posterior through the variational lower bound and then uses this approximation as the prior on the next task. Their Gaussian is fully factorized, hence they do not take weight interactions into account either.
[@scalable:laplace] and [@bayesian:maml] have independently proposed the use of block-diagonal Kronecker factored curvature approximations [@kfac; @kfra] to sample from an approximate Gaussian posterior over the weights of a neural network. They find that this requires adding a multiple of the identity to their curvature factors as an ad-hoc regularizer, which is not necessary for our method. In our work, we use an approximate posterior with the same Kronecker factored covariance structure as a prior for subsequent tasks. We iteratively update this approximation for every new dataset. The curvature factors that we accumulate throughout training could be used on top of our method to approximate the predictive posterior similar to [@scalable:laplace; @bayesian:maml]. However, both the curvature factors and the mode that our method finds will be different to performing a Laplace approximation in batch mode. Our work links the Kronecker factored Laplace approximation [@scalable:laplace] to Bayesian online learning [@bayesian:online] similar to how Variational Continual Learning [@vcl] connects Online Variational Learning [@online:variational:bayes:zoubin; @online:variational:bayes:honkela] to Bayes-by-Backprop [@weight:uncertainty].
We discuss additional related methods without a Bayesian motivation in [Appendix \[app:related\]]{}.
Experiments {#sec:experiments}
===========
In our experiments we compare our online Laplace approximation to the approximate Laplace approximation of [Eq. (\[eq:approximate\])]{} as well as EWC [@ewc] and Synaptic Intelligence (SI) [@synaptic:intelligence], both of which also add quadratic regularizers to the objective. Further, we investigate the effect of using a block-diagonal Kronecker factored approximation to the curvature over a diagonal one. We also run EWC with a Kronecker factored approximation, even though the original method is based on a diagonal one. We implement our experiments using Theano [@theano] and Lasagne [@lasagne] software libraries.
Permuted MNIST
--------------
![Mean test accuracy on a sequence of permuted MNIST datasets for different methods for overcoming catastrophic forgetting. The x-axis marks the number of datasets that have been observed so far for training and testing. We categorize SI as a diagonal method, as it only has a single measure of importance per weight and does not take interactions between the parameters into account. The dotted black line shows the performance of a single network trained on all observed data at each task.[]{data-label="fig:pmnist:accs"}](permutedmnist_mean_accs){width="\linewidth"}
[0.475]{} {width="\textwidth"}
[0.475]{} {width="\textwidth"}
![Effect of $\lambda$ for different curvature approximations for permuted MNIST. Each plot shows the mean, minimum and maximum across the tasks observed so far, as well as the accuracy on the first and most recent task.[]{data-label="fig:pmnist:hyper"}](permutedmnist_mean_accs){width="\linewidth"}
![Effect of $\lambda$ for different curvature approximations for permuted MNIST. Each plot shows the mean, minimum and maximum across the tasks observed so far, as well as the accuracy on the first and most recent task.[]{data-label="fig:pmnist:hyper"}](pmnist_kf_persistent){width="\textwidth"}
\
![Effect of $\lambda$ for different curvature approximations for permuted MNIST. Each plot shows the mean, minimum and maximum across the tasks observed so far, as well as the accuracy on the first and most recent task.[]{data-label="fig:pmnist:hyper"}](pmnist_diagonal_persistent){width="\textwidth"}
As a first experiment, we test on a sequence of permutations of the MNIST dataset [@mnist]. Each instantiation consists of the $28{\times}28$ grey-scale images and labels from the original dataset with a fixed random permutation of the pixels. This makes the individual data distributions mostly independent of each other, testing the ability of each method to fully utilize the model’s capacity.
We train a feed-forward network with two hidden layers of $100$ units and ReLU nonlinearities on a sequence of $50$ versions of permuted MNIST. Every one of these datasets is equally difficult for a fully connected network due to its permutation invariance to the input. We stress that our network is smaller than in previous works as the limited capacity of the network makes the task more challenging. Further, we train on a longer sequence of datasets. Optimization details are in [Appendix \[app:details\]]{}.
[Fig. \[fig:pmnist:accs\]]{} shows the mean test accuracy as new datasets are observed for the optimal hyperparameters of each method. We refer to the online Laplace approximation as ‘Online Laplace’, to the Laplace approximation around an approximate mode as ‘Approximate Laplace’ and to adding a quadratic penalty for every set of MAP parameters as in [@ewc] as ‘Per-task Laplace’. The per-task Laplace method with a diagonal approximation to the Hessian corresponds to EWC.
We find our online Laplace approximation to maintain higher test accuracy throughout training than placing a quadratic penalty around the MAP parameters of every task, in particular when using a simple diagonal approximation to the Hessian. However, the main difference between the methods lies in using a Kronecker factored approximation of the curvature over a diagonal one. Using this approximation, we achieve over $90\%$ average test accuracy across $50$ tasks, almost matching the performance of a network trained jointly on all observed data. Recalculating the curvature for each task instead of retaining previous estimates does not significantly affect performance.
Beyond simple average performance, we investigate different values of the hyperparameter $\lambda$ on the permuted MNIST sequence of datasets for our online Laplace approximation. The goal is to visualize how it affects the trade-off between remembering previous tasks and being able to learn new ones for the two approximations of the curvature that we consider. [Fig. \[fig:pmnist:hyper\]]{} shows various statistics of the accuracy on the test set for the smallest and largest value of the hyperparameter on the quadratic penalty that we tested, as well as the one that optimizes the validation error. We are particularly interested in the performance on the first dataset and the most recent one, as a measure for memory and flexibility respectively. For all displayed values of the hyperparameter, the Kronecker factored approximation ([Fig. \[fig:pmnist:hyper:kf\]]{}) has higher test accuracy than the diagonal approximation ([Fig. \[fig:pmnist:hyper:diag\]]{}) on both the most recent and the first task, as well as on average. For the natural choice of $\lambda=1$ (leftmost subfigure respectively), the network’s performance decays for the first task for both curvature approximations, yet it is able to learn the most recent task well. The performance on the first task decays more slowly, however, for the more expressive Kronecker factored approximation of the curvature. Increasing the hyperparameter, corresponding to making the prior more narrow as discussed in [Section \[sec:hyperparam\]]{}, leads to the network remembering the first task much better at the cost of not being able to achieve optimal performance on the most recently added task. Using $\lambda=3$ (central subfigure), the value that achieves optimal validation error in our experiments, the Kronecker factored approximation leads to the network performing similarly on the most recent and first tasks. This coincides with optimal average test accuracy. We are not able to find such an ideal trade-off for the diagonal Hessian approximation, resulting in worse average performance and suggesting that the posterior cannot be matched well without accounting for interactions between the weights. Using a large value of $\lambda=100$ (rightmost subfigure) reverts the order of performance between the most recent and the first task for both approximations: while for small $\lambda$ the first task is ‘forgotten’, the network’s performance now stays at a high level — for the Kronecker factored approximation it remembers it perfectly — which comes at the cost of being unable to learn new tasks well.
We conclude from our results that the online Laplace approximation overestimates the uncertainty in the approximate posterior about the parameters for the permuted MNIST task, in particular with a diagonal approximation to the Hessian. Overestimating the uncertainty leads to a need for regularization in the form of reducing the width of the approximate posterior, as the value that optimizes the validation error is $\lambda=3$. Only when regularizing too strongly the approximate posterior underestimates the uncertainty about the weights, leading to reduced performance on new tasks for large values of $\lambda$. Using a better approximation to the posterior leads to a drastic increase in performance and a reduced need for regularization in the subsequent experiments. We note that some regularization is still necessary, suggesting that even the Kronecker factored approximation overestimates the variance in the posterior, and a better approximation could lead to further improvements. However, it is also possible that the Laplace approximation as such requires a large amount of data to estimate the interaction between the parameters sufficiently well; hence it might be best suited for settings where plenty of data are available.
Disjoint MNIST
--------------
[r]{}[0.35]{} {width="\linewidth"}
We further experiment with the disjoint MNIST task, which splits the MNIST dataset into one part containing the digits ‘$0$’ to ‘$4$’, and a second part containing ‘$5$’ to ‘$9$’ and training a ten-way classifier on each set separately. Previous work [@imm] has found this problem to be challenging for EWC, as during the first half of training the network is encouraged to set the bias terms for the second set of labels to highly negative values. This setup makes it difficult to balance out the biases for the two sets of classes after the first task without overcorrecting and setting the biases for the first set of classes to highly negative values. @imm report just over $50\%$ test accuracy for EWC, which corresponds to either completely forgetting the first task or being unable to learn the second one, as each task individually can be solved with around $99\%$ accuracy.
We use an identical network architecture to the previous section and found stronger regularization of the approximate posterior to be necessary. For the Laplace methods, we tested values of $\lambda \in \{1, 3, 10, \ldots, 3{\times}10^5,10^6\}$, and $c \in \{0.1, 0.3, 1, \ldots, 3{\times}10^4,10^5\}$ for SI. We train using Nesterov momentum with a learning rate of $0.1$ and momentum of $0.9$ and decay the learning rate by a factor of $10$ every $1000$ parameter updates using a batch size of $250$. We decay the initial learning rate for the second task depending on the hyperparameter to prevent the objective from diverging. We test various decay factors for each hyperparameter, but as a rule of thumb found $\frac{\lambda}{10}$ to perform well for the Kronecker factored, and $\frac{\lambda}{1000}$ for the diagonal approximation. The results are averaged across ten independent runs.
[Fig. \[fig:disjoint\]]{} shows the test accuracy for various hyperparameter values for a Kronecker factored and a diagonal approximation of the curvature as well as SI. As there are only two datasets, the three Laplace-based methods are identical, therefore we focus on the impact of the curvature approximation. Approximating the Hessian with a diagonal corresponds to EWC. While we do not match the performance of the method developed in [@imm], we find the Laplace approximation to work significantly better than reported by the authors. The Kronecker factored approximation gives a small improvement over the diagonal one and requires weaker regularization, which further suggests that it better fits the true posterior. It also outperforms SI.
Vision datasets
---------------
(input) at (0,0) [Input]{}; at (0, 0.75) [Shared Parameters]{}; (net) at (0,1.5) [Network]{};
at (-1.3, 2.25) [${W}_L^{(1)}$]{}; at (1.3, 2.25) [${W}_L^{(T)}$]{};
(out1) at (-1.7,3) [$h_L^{(1)}$]{}; at (0,3) [$\hdots$]{}; (outT) at (1.7,3) [$h_L^{(T)}$]{};
(\[xshift=-1.7cm\] input.north) – (\[xshift=-1.7cm\] net.south); (\[xshift=1.7cm\] input.north) – (\[xshift=1.7cm\] net.south); (\[xshift=-1.7cm\] input.north) – (\[xshift=-1.7cm\] net.south) – (\[xshift=1.7cm\] net.south) – (\[xshift=1.7cm\] input.north);
(\[xshift=-1.7cm\] net.north) – (out1.south); (\[xshift=1.7cm\] net.north) – (out1.south east); (\[xshift=-1.7cm\] net.north) – (out1.south) to \[out=0,in=-135\] (out1.south east) – (\[xshift=1.7cm\] net.north);
(\[xshift=-1.7cm\] net.north) – (outT.south west); (\[xshift=1.7cm\] net.north) – (outT.south); (\[xshift=1.7cm\] net.north) – (outT.south) to \[out=180,in=140\] (outT.south west) – (\[xshift=-1.7cm\] net.north);
{width="\textwidth"}
As a final experiment, we test our method on a suite of related vision datasets. Specifically, we train and test on MNIST [@mnist], notMNIST[^3], Fashion MNIST [@fashionmnist], SVHN [@svhn] and CIFAR10 [@cifar] in this order. All five datasets contain around $50,000$ training images from $10$ different classes. MNIST contains hand-written digits from ‘$0$’ to ‘$9$’, notMNIST the letters ‘A’ to ‘J’ in different computer fonts, Fashion MNIST different categories of clothing, SVHN the digits ‘0’ to ‘9’ on street signs and CIFAR10 ten different categories of natural images. We zero-pad the images of the MNIST-like datasets to be of size $32{\times}32$ and replicate their intensity values over three channels, such that all images have the same format.
We train a LeNet-like architecture [@mnist] with two convolutional layers with $5{\times}5$ convolutions with $20$ and $50$ channels respectively and a fully connected hidden layer with $500$ units. We use ReLU nonlinearities and perform a $2{\times}2$ max-pooling operation after each convolutional layer with stride $2$. An extension of the Kronecker factored curvature approximations to convolutional neural networks is presented in [@kfac:conv]. As the meaning of the classes in each dataset is different, we keep the weights of the final layer separate for each task. We optimize the networks as in the permuted MNIST experiment and compare to five baseline networks with the same architecture trained on each task separately.
Overall, the online Laplace approximation in conjunction with a Kronecker factored approximation of the curvature achieves the highest test accuracy across all five tasks (see [Appendix \[app:numerical:vision\]]{} for the numerical results). However, the difference between the three Laplace-based methods is small in comparison to the improvement stemming from the better approximation to the Hessian. We therefore plot the test accuracy curves through training only for the online Laplace approximation in the main text in [Fig. \[fig:lenet\]]{} to show the difference to SI and between the two curvature approximations. The corresponding figures for having a separate quadratic penalty for each task and the approximate Laplace approximation are in [Appendix \[app:figures:vision\]]{}.
Using a diagonal Hessian approximation for the Laplace approximation, the network mostly remembers the first three tasks, but has difficulties learning the fifth one. SI, in contrast, shows decaying performance on the initial tasks, but learns the fifth task almost as well as our method with a Kronecker factored approximation of the Hessian. However, using the Kronecker factored approximation, the network achieves good performance relative to the individual networks across all five tasks. In particular, it remembers the easier early tasks almost perfectly while being sufficiently flexible to learn the more difficult later tasks better than the diagonal methods, which suffer from forgetting.
Conclusion {#sec:conclusion}
==========
We proposed the online Laplace approximation, a Bayesian online learning method for overcoming catastrophic forgetting in neural networks. By formulating a principled approximation to the posterior, we were able to substantially improve over EWC [@ewc] and SI [@synaptic:intelligence], two recent methods that also add a quadratic regularizer to the objective for new tasks. By further taking interactions between the parameters into account, we achieved considerable increases in test accuracy on the problems that we investigated, in particular for long sequences of datasets. Our results demonstrate the importance of going beyond diagonal approximation methods which only measure the sensitivity of individual parameters. Dealing with the complex interaction and correlation between parameters is necessary in moving towards a more complete response to the challenge of continual learning.
Derivation of the Kronecker factorization of the diagonal blocks of the Hessian {#app:kf}
===============================================================================
@kfac and @kfra both develop block-diagonal Kronecker factored approximations to the Fisher and Gauss-Newton matrix of fully connected neural networks respectively, which in turn both are positive semi-definite approximations of the Hessian. Both use their approximations for optimization, hence the positive semi-definiteness is crucial in order to prevent parameter updates that increase the loss. We require this property as well, as we perform a Laplace approximation and the Normal distribution requires its covariance to be positive semi-definite.
In the following, we provide the basic derivation for the diagonal blocks of the Hessian being Kronecker factored as developed in [@kfra] and state the recursion for calculating the pre-activation Hessian.
We denote a neural network as taking an input ${a}_0 = x$ and producing an output ${h}_L$. The input is passed through layers $l=1,\ldots,L$ as linear pre-activations ${h}_l = {W}_l {a}_{l-1}$ and non-linear activations ${a}_l = f_l({h}_l)$, where ${W}_l$ denotes the weight matrix and $f_l$ the elementwise activation function. Bias terms can be absorbed into ${W}_l$ by appending a $1$ to every ${a}_l$. The weights are optimized w.r.t. an error function ${E}(y, {h}_L)$, which can usually be expressed as a negative log likelihood.
Using the chain rule, the gradient of the error function w.r.t. an individual weight can be calculated as:
$${{\frac{\partial {E}}{\partial {W}_{a,b}^l}}}
= \sum_i {\frac{\partial {h}^l_i}{\partial {W}^l_{a,b}}} {{\frac{\partial {E}}{\partial {h}^l_i}}}
= {a}^{l-1}_b {{\frac{\partial {E}}{\partial {h}^l_a}}}$$
Differentiating again w.r.t. another weight within the same layer gives:
$$\left[ {H}_l \right]_{(a,b), (c,d)}
\equiv {{\frac{\partial^2 {E}}{\partial {W}_{a,b} \partial {W}_{c,d}}}}
= {a}^{l-1}_b {a}^{l-1}_d \left[ {\mathcal{H}}_l \right]_{(a,c)}$$
where
$$\left[ {\mathcal{H}}_l \right]_{a,b} \equiv {{\frac{\partial^2 {E}}{\partial {h}^l_a \partial {h}^l_b}}}$$
is defined to be the pre-activation Hessian.
This can also be expressed in matrix notation as a Kronecker product:
$${H}_l
= {{\frac{\partial^2 {E}}{\partial \operatorname{vec}({W}^l) \partial \operatorname{vec}({W}^l)}}}
= \left( {a}_{l-1} {a}_{l-1}^\top \right) \otimes {\mathcal{H}}_l$$
Similar to backpropagation, the pre-activation Hessian can be calculated as:
$${\mathcal{H}}_l = B_l {W}_{l+1}^\top {\mathcal{H}}_{l+1} {W}_{l+1} B_l + D_l$$
where the diagonal matrices $B_l$ and $D_l$ are defined as
$$\begin{aligned}
B_l &= \operatorname{diag}(f_l'({h}_l))\\
D_l &= \operatorname{diag}(f_l''({h}_l) {{\frac{\partial {E}}{\partial {a}_l}}})\end{aligned}$$
$f'$ and $f''$ denote the first and second derivative of $f$. The recursion for ${\mathcal{H}}$ is initialized with the Hessian of the error w.r.t. the network outputs, i.e. ${\mathcal{H}}_L \equiv {{\frac{\partial^2 {E}}{\partial h_L \partial h_L}}}$. For the derivation of the recursion and how to calculate the diagonal blocks of the Gauss-Newton matrix, we refer the reader to [@kfra], and to [@kfac] for the Fisher matrix.
Visualization of the effect of $\lambda$ for a Gaussian prior and posterior {#app:reg:vis}
===========================================================================
![Contours of a Gaussian likelihood (dashed blue) and prior (shades of purple) for different values of $\lambda$. Values smaller than $1$ shift the joint maximum $\theta^*$, marked by a ‘${\times}$’,towards that of the likelihood, i.e. the new task, for values greater than $1$ it moves towards the prior, i.e. previous tasks.[]{data-label="fig:contours"}](contours){width="0.5\linewidth"}
A small $\lambda$ resulting in high uncertainty shifts the mode towards that of the likelihood, i.e. enables the network to learn the new task well even if our posterior approximation underestimates the uncertainty. Vice versa, increasing $\lambda$ moves the joint mode towards the prior mode, improving how well the previous parameters are remembered. The optimal choice depends on the true posterior and how closely it is approximated.
In principle, it would be possible to use a different value $\lambda_t$ for every dataset. In our experiments, we keep the value of $\lambda$ the same across all tasks as the family of posterior approximation is the same throughout training. Furthermore, using a separate hyperparameter for each task would let the number of hyperparameters grow linearly in the number of tasks, which would make tuning them costly.
Additional related work {#app:related}
=======================
Various methods for overcoming catastrophic forgetting without a Bayesian motivation have also been proposed over the past year. @synaptic:intelligence develop ‘Synaptic Intelligence’ (SI), another quadratic penalty on deviations from previous parameter values where the importance of each weight is heuristically measured as the path length of the updates on the previous task. @gem formulate a quadratic program to project the gradients such that the gradients on previous tasks do not point in a direction that decreases performance; however, this requires keeping some previous data in memory. @deep:generative:replay suggest a dual architecture including a generative model that acts as a memory for data observed in previous tasks. Other approaches that tackle the problem at the level of the model architecture include [@progessive:nns], which augments the model for every new task, and [@pathnet], which trains randomly selected paths through a network. @hard:attention propose sharing a set of weights and modifying them in a learnable manner for each task. @conceptor introduce conceptor-aided backpropagation to shield gradients against reducing performance on previous tasks.
Optimization details {#app:details}
====================
For the permuted MNIST experiment, we found the performance of the methods that we compared to mildly depend on the choice of optimizer. Therefore, we optimize all techniques with Adam [@adam] for $20$ epochs per dataset and a learning rate of $10^{-3}$ as in [@synaptic:intelligence], SGD with momentum [@momentum] with an initial learning rate of $10^{-2}$ and $0.95$ momentum, and Nesterov momentum [@nag] with an initial learning rate of $0.1$, which we divide by $10$ every $5$ epochs, and $0.9$ momentum. For the momentum based methods, we train for at least $10$ epochs and early-stop once the validation error does not improve for $5$ epochs. Furthermore, we decay the initial learning rate with a factor of $\frac{1}{1 + k t}$ for the momentum-based optimizers, where $t$ is the index of the task and $k$ a decay constant. We set $k$ using a coarse grid search for each value of the hyperparameter $\lambda$ in order to prevent the objective from diverging towards the end of training, in particular with the Kronecker factored curvature approximation. For the Laplace approximation based methods, we consider $\lambda \in \{1, 3, 10, 30, 100\}$; for SI we try $c \in \{0.01, 0.03, 0.1, 0.3, 1\}$. We ultimately pick the combination of optimizer, hyperparameter and decay rate that gives the best validation error across all tasks at the end of training. For the Laplace-based methods, we found momentum based optimizers to lead to better performance, whereas Adam gave better results for SI.
Numerical results of the vision experiment {#app:numerical:vision}
==========================================
-------- -------------------- ------- -------- -------- ------- ------- -------
Method Approximation MNIST nMNIST fMNIST SVHN C10 Avg.
SI n/a 87.27 79.12 84.61 77.44 57.61 77.21
PTL Diagonal (EWC) 97.83 94.73 89.13 79.80 53.29 82.96
Kronecker factored 97.85 94.92 89.31 85.75 58.78 85.32
AL Diagonal 96.56 92.33 89.27 78.00 56.57 82.55
Kronecker factored 97.90 94.88 90.08 85.24 58.63 85.35
OL Diagonal 96.48 93.41 88.09 81.79 53.80 82.71
Kronecker factored 97.17 94.78 90.36 85.59 59.11 85.40
-------- -------------------- ------- -------- -------- ------- ------- -------
: Per dataset test accuracy at the end of training on the suite of vision datasets. SI is Synaptic Intelligence [@synaptic:intelligence] and EWC Elastic Weight Consolidation [@ewc]. We abbreviate Per-Task Laplace (one penalty per task) as PTL, Approximate Laplace (Laplace approximation of the full posterior at the mode of the approximate objective) and our Online Laplace approximation as OL. nMNIST refers to notMNIST, fMNIST to FashionMNIST and C10 to CIFAR10.
\[tab:vision:test\]
Additional figures for the vision experiment {#app:figures:vision}
============================================
Hyperparameter comparison on permuted MNIST for approximate and per-task Laplace {#app:figures:hyperparam}
--------------------------------------------------------------------------------
[0.45]{} {width="\textwidth"}
[0.45]{} {width="\textwidth"}
[0.45]{} {width="\textwidth"}
[0.45]{} {width="\textwidth"}
Vision datasets with approximate and per-task Laplace {#app:figures:vision}
-----------------------------------------------------
{width="\textwidth"}
\
{width="\textwidth"}
[^1]: Corresponding author: `j.ritter@cs.ucl.ac.uk`
[^2]: @quadratic:penalties recently discussed a similar recursive Laplace approximation for online learning, however with limited experimental results and in the context of using a diagonal approximation to the Hessian.
[^3]: Originally published at <http://yaroslavvb.blogspot.co.uk/2011/09/notmnist-dataset.html> and downloaded from <https://github.com/davidflanagan/notMNIST-to-MNIST>
|
---
abstract: |
A new general relativistic magnetohydrodynamics ([**GRMHD**]{}) code “RAISHIN” used to simulate jet generation by rotating and non-rotating black holes with a geometrically thin Keplarian accretion disk finds that the jet develops a spine-sheath structure in the rotating black hole case. Spine-sheath structure and strong magnetic fields significantly modify the Kelvin-Helmholtz ([**KH**]{}) velocity shear driven instability. The RAISHIN code has been used in its relativistic magnetohydrodynamic ([**RMHD**]{}) configuration to study the effects of strong magnetic fields and weakly relativistic sheath motion, $c/2$, on the KH instability associated with a relativistic, $\gamma = 2.5$, jet spine-sheath interaction. In the simulations sound speeds up to $ \sim c/\sqrt 3$ and Alfvén wave speeds up to $\sim
0.56~c$ are considered. Numerical simulation results are compared to theoretical predictions from a new normal mode analysis of the RMHD equations. Increased stability of a weakly magnetized system resulting from $c/2$ sheath speeds and stabilization of a strongly magnetized system resulting from $c/2$ sheath speeds is found.
author:
- 'Philip Hardee, Yosuke Mizuno, & Ken-Ichi Nishikawa'
title: 'GRMHD/RMHD Simulations & Stability of Magnetized Spine-Sheath Relativistic Jets'
---
Introduction {#sec:intro}
============
Relativistic jets are associated with active galactic nuclei and quasars ([**AGN**]{}), with black hole binary systems ([**microquasars**]{}), and are thought responsible for the gamma-ray bursts ([**GRBs**]{}). The observed proper motions in AGN and microquasar jets imply speeds from $\sim 0.9~c$ (e.g., Mirabel & Rodriquez 1999) up to $\sim 0.999~c$ (e.g., the 3C345 jet Zensus et al. 1995; Steffen et al. 1995), and the inferred speeds for GRBs are $\sim 0.99999~c$ (e.g., Piran 2005).
Jets at the larger scales may be kinetically dominated and contain relatively weak magnetic fields, but stronger magnetic fields exist closer to the acceleration and collimation region. Here GRMHD simulations of jet formation (e.g., Koide et al. 2000; Nishikawa et al. 2005; De Villiers et al. 2003, 2005; Hawley & Krolik 2006; McKinney & Gammie 2004; McKinney 2006; Mizuno et al. 2006) and earlier theoretical work (e.g., Lovelace 1976; Blandford 1976; Blandford & Znajek 1977; Blandford & Payne 1982) invoke strong magnetic fields. Additionally, the GRMHD simulations suggest that jets driven by magnetic fields threading the ergosphere can reside within a broader sheath outflow driven by the magnetic fields anchored in the accretion disk (e.g., McKinney 2006; Hawley & Krolik 2006; Mizuno et al. 2006), or less collimated accretion disk wind (e.g., Nishikawa et al. 2005).
Recent observations of QSO winds with speeds, $\sim 0.1 - 0.4c$, also indicate that a jet could reside in a high speed sheath (Chartas et al. 2002, 2003; Pounds et al. 2003a, 2003b; Reeves et al.2003). Circumstantial evidence such as the requirement for large Lorentz factors suggested by the TeV BL Lacs when contrasted with much slower observed motions has been used to suggest the presence of a spine-sheath morphology (Ghisellini et al. 2005), and Siemignowska et al. (2007) have proposed a spine-sheath model for the PKS 1127-145 jet. Spine-sheath structure has also been proposed based on theoretical arguments (e.g., Sol et al. 1989; Henri & Pelletier 1991; Laing 1996; Meier 2003) and has been investigated in the context of GRB jets (e.g., Rossi et al. 2002; Lazzatti & Begelman 2005; Zhang et al. 2003, 2004; Morsony et al. 2006).
In §2 we illustrate the spine-sheath configuration found by our GRMHD jet generation simulations. Previous relativistic fluid dynamical (RHD) simulation and theoretical work has shown the importance of spine-sheath structure to KH instability (Hardee & Hughes 2003). In §3 we report on numerical results that extend this previous investigation numerically and in §4 theoretically to the strongly magnetized RMHD regime.
GRMHD Jet Spine-Sheath Generation {#sec:1}
=================================
In order to study the formation of relativistic jets from a geometrically thin Keplerian disk, we use a 2.5-dimensional GRMHD code with Boyer-Lindquist coordinates $(r, \theta, \phi)$. The method is based on a 3+1 formalism of the general relativistic conservation laws of particle number and energy momentum, Maxwell equations, and Ohm’s law with no electrical resistance (ideal MHD condition) in a curved spacetime. In the simulations presented here we use minmod slope limiter reconstruction, HLL approximate Riemann solver, flux-CT scheme and Noble’s 2D method (see Mizuno et al. 2006 and references therein).
A geometrically thin Keplerian disk rotates around a black hole (non-rotating, $a=0.0$ or rapidly co-rotating, $a=0.95$, here $a$ is black hole spin parameter), where the disk density is 100 times the coronal density. The thickness of the disk is $H/r \sim 0.06$. The background corona is free-falling, and the initial magnetic field is uniform and parallel to the rotational axis. Simulations are normalized by the speed of light, $c$, and the Schwarzschild radius, $r_{\rm S}$, with timescale, $\tau_{\rm S} \equiv r_{\rm S}/c$. Values of the magnetic field strength and gas pressure depend on the normalized density, $\rho_{0}$. In these simulations the magnetic field strength, $B_{0}$, is set to $0.05 \sqrt{\rho_{0} c^{2}})$. The $128 \times 128$ computational grid with logarithmic spacing in the radial direction spans the region $1.1 r_{\rm S} \le r \le 20.0 r_{\rm
S}$ (non-rotating black hole) and $0.75 r_{\rm S}\le r \le 20.0 r_{\rm
S}$ (rapidly rotating black hole) and $0.03 \le \theta \le \pi/2$ where we assume axisymmetry with respect to the $z$-axis and mirror symmetry with respect to the equatorial plane. We employ a free boundary condition at the inner and outer boundaries in the radial direction.
Figure 1 shows snapshots of the density (panels (a) and (b)), plasma beta ($\beta=p_{\rm gas}/p_{\rm mag}$) distribution (panels (c) and (d)), and total velocity (panels (e) and (f)) for the non-rotating black hole, $a=0.0$ (left panels); and the rapidly rotating black hole, $a=0.95$ (right panels); at each simulation’s terminal time (non-rotating: $t = 275\tau_{\rm S}$ and rotating: $t = 200\tau_{\rm
S}$). At the marginally stable circular orbit ($r = 3r_{\rm S}$) the disk orbits the black hole in about $40 \tau_{\rm S}$. The total velocity distribution of non-rotating and rapidly rotating black hole cases are shown in Figs. 1e and 1f.
![Snapshots of the non-rotating black hole ([*a, c, e*]{}) and the rapidly rotating black hole ([*b, d, f*]{}) at the applicable terminal simulation time. The color scales show the logarithm of density (upper panels), plasma beta ($\beta = p_{\rm gas} / p_{\rm
mag}$; middle panels) and total velocity (lower panels). A negative velocity indicates inflow towards the black hole. The white lines indicate magnetic field lines (contour of the poloidal vector potential; upper panels) and contours of the toroidal magnetic field strength (middle panels). Arrows depict the poloidal velocities normalized to light speed, as indicated above each panel by the arrow.[]{data-label="fig:1"}](phfig01b.eps){width="48.00000%"}
The jets in both cases have speeds greater than $0.4~c$ (mildly relativistic) that are comparable to the Alfvén speeds. In the jets, toroidal velocity is the dominant velocity component. In the rapidly rotating black hole case, the velocity distribution indicates a two-component jet with the inner jet not seen in the non-rotating black hole case. The inner jet is faster than the outer jet (over $0.5c$).
RMHD Spine-Sheath Simulations {#sec:2}
=============================
In these simulations a “preexisting” jet is established across a computational domain of $6 R_{j} \times 6 R_{j} \times 60 R_{j}$ with $60 \times 60 \times 600$ zones. The jet is in total pressure balance with a lower-density magnetized sheath with $\rho_{j}/\rho_{e}=2.0$, where $\rho$ is the mass density in the proper frame. The jet speed is $u_{j}=0.9165~c$ and $\gamma_j \equiv (1 - u_j^{2})^{-1/2}=2.5$. The initial magnetic field is uniform and parallel to the jet flow. A precessional perturbation is applied at the inflow by imposing a transverse component of velocity with $u_{\bot}=0.01u_{j}$. Here we show simulations with a precessional perturbation of angular frequency $\omega R_{j}/u_{j}=0.93$. In order to investigate the effect of an external wind, we have performed a no wind case ($u_{e}=0$) and a relativistic wind case ($u_{e}=0.5~c$). Simulations are halted after $\sim 60$ light crossing times of the jet radius (see Mizuno et al. 2007 for details).
We have performed weakly magnetized simulations with sound speeds $a_{e} \sim 0.57~c$ and $a_{j} \sim 0.51~c$, and Alfvén speeds $v_{Ae} \sim 0.07~c$ and $v_{Aj} \sim 0.06~c$. The stabilizing effect of a sheath wind is revealed in Figure 2. Here we see considerable reduction
![3D isovolume density image of the weakly magnetized case with no wind (top) and a c/2 wind (bottom). Magnetic field lines in white. []{data-label="fig:2a"}](phfig02ab.eps){width="48.00000%"}
![3D isovolume density image of the weakly magnetized case with no wind (top) and a c/2 wind (bottom). Magnetic field lines in white. []{data-label="fig:2a"}](phfig02bb.eps){width="48.00000%"}
in transverse structure and the jet spine reaches a larger distance before disruption in the presence of a wind.
We have also performed strongly magnetized simulations with Alfvén speeds $v_{Ae} \sim 0.56~c$ and $v_{Aj} \sim 0.45~c$, and sound speeds $a_{e} \sim 0.30~c$ and $a_{j} \sim 0.23~c$. The stabilizing influence of a magnetic field and the stabilization of the jet spine in the presence of a magnetized sheath wind is shown in Figure 3.
![3D isovolume density image of the strongly magnetized case with no wind (top) and a c/2 wind (bottom). Magnetic field lines in white. []{data-label="fig:3"}](phfig03ab.eps){width="48.00000%"}
![3D isovolume density image of the strongly magnetized case with no wind (top) and a c/2 wind (bottom). Magnetic field lines in white. []{data-label="fig:3"}](phfig03bb.eps){width="48.00000%"}
Here we see that the presence of the strong magnetic field has stabilized the jet spine even more than occured for the weakly magnetized wind case and the initial helical perturbation is damped in the presence of the strongly magnetized sheath wind.
More quantitatively we can analyse the growth or damping of the initial perturbation via 1D cuts in the radial velocity as shown in Figure 4.
![Radial velocity ($v_x$) along one dimensional cuts parallel to the jet axis and located at $x/R_J =$ 0.2 (solid line), 0.5 (dotted line) and 0.8 (dashed line) for the weakly magnetized (top) and strongly magnetized cases (bottom).[]{data-label="fig:4"}](phfig04b.eps){width="48.00000%"}
Measurable reduction in transverse motion is seen for the weakly magnetized wind case, significant reduction occurs for the strongly magnetized no wind case and stabilization occurs in the strongly magnetized wind case.
RMHD Spine-Sheath Stability {#sec:3}
===========================
Stability of a jet spine-sheath configuration can be analyzed by modeling the jet/spine as a cylinder of radius R embedded in an infinite sheath. A dispersion relation describing the growth or damping of the normal modes can be derived assuming uniform conditions within the spine, e.g., a uniform proper density, $\rho _{j}$, axial magnetic field, $B_{j}=B_{j,z}$, and velocity, $\mathbf{u}_{j}=u_{j,z}$, and assuming uniform conditions in the external sheath, e.g., a uniform proper density, $ \rho _{e}$, axial magnetic field, $B_{e}=B_{e,z}$, and velocity $\mathbf{u}_{e}=u_{e,z}$ (see Hardee 2007).
Each normal mode consists of a single fundamental and multiple body wave solutions to the dispersion relation. In the low frequency limit the helical fundamental mode has an analytical solution given by $$\frac{\omega }{k}=\frac{\left[ \eta u_{j}+u_{e}\right] \pm i\eta
^{1/2}\left[ \left( u_{j}-u_{e}\right) ^{2}-V_{As}^{2}/\gamma
_{j}^{2}\gamma _{e}^{2} \right] ^{1/2}}{(1+V_{Ae}^{2}/\gamma
_{e}^{2}c^{2})+\eta (1+V_{Aj}^{2}/\gamma _{j}^{2}c^{2})}
\label{eq1}
\vspace{-0.3cm}$$ where $\eta \equiv \gamma _{j}^{2}W_{j}\left/ \gamma
_{e}^{2}W_{e}\right.$, $V_{A}^2 \equiv B^2/4 \pi W$, $W\equiv \rho
+\left[ \Gamma /\left( \Gamma -1\right) \right] P/c^{2}$ is the enthalpy, and $$V_{As}^{2}\equiv \left( \gamma _{Aj}^{2}W_{j}+\gamma
_{Ae}^{2}W_{e}\right) \frac{B_{j}^{2}+B_{e}^{2}}{4\pi W_{j}W_{e}}~.
\label{eq2}
\vspace{-0.3cm}$$ In equation (2) $\gamma _{Aj,e}\equiv (1-v_{Aj,e}^{2}/c^{2})^{-1/2}$ is the Alfvén Lorentz factor. The jet is stable when $$\left( u_{j}-u_{e}\right)^{2}-V_{As}^{2}/\gamma _{j}^{2}\gamma
_{e}^{2} < 0~.
\label{eq3}
\vspace{-0.3cm}$$ In the low frequency limit the real part of the first helical body mode has an analytical solution given by $$kR\approx k^{\min }R\equiv \frac{5}{4} \pi \left[ \frac{
v_{msj}^2u_{j}^{2}-v_{Aj}^2a_{j}^{2}}{\gamma
_{j}^{2}(u_{j}^{2}-a_{j}^{2})(u_{j}^{2}-v_{Aj}^{2})}\right]^{1/2}.
\label{eq4}
\vspace{-0.3cm}$$ In equation (4) $v_{ms}$ is a magnetosonic speed defined by $v_{ms}
\equiv \left[a^2/\gamma_A^2 + v_A^2\right]^{1/2}$ where $a$ is the sound speed, and $v_{A}$ is the Alfvén wave speed.
Equations (1 & 4) provide estimates for the helical fundamental and first body modes that can be followed by root finding techniques to higher frequencies. The results of numerical solution to the dispersion relation for the parameters appropriate to the numerical simulations shown in §3 are displayed in Figure 5.
![Solutions for helical fundamental (red lines) and first body (green lines) modes for weakly magnetized ($a_{j,e} \gg v_{Aj,e}$) and strongly magnetized ($ v_{Aj,e} \sim 2a_{j,e}$) jet simulations with no wind ($u_e = 0$) and with a c/2 wind ($u_e = 0.5$). Solutions show the real, $k_rR_j$, (dashed lines) and imaginary, $k_iR_j$, (dash-dot lines) parts of the wavenumber as a function of the angular frequency, $\omega R_j/ u_j$. Where the imaginary part of the wavenumber is shown in blue, the solution is damped. Immediately under the solutions for fundamental ([**S**]{}) and first body ([**B1**]{}) modes is a panel that shows the wavelength, $\lambda/R_j$, (dash-dot lines) and wave speed, $v_w/c$, (dotted lines). The simulation precession frequency $\omega
2 = 0.93$ is indicated by the vertical solid line. []{data-label="fig:5"}](phfig05b.eps){width="48.00000%"}
In the weakly magnetized cases fundamental ([**S**]{}) mode solutions consist of a growing (shown) and damped (not shown) solution pair (see eq. 1) and first body ([**B1**]{}) mode solutions consist of a real and growing or damped solution pair. The presence of the external wind flow leads to reduced growth of the S mode and weak damping of the B1 mode.
In the strongly magnetized no wind case S mode solutions again consist of a growing and damped solution pair. However, we now find multiple growing solutions associated with the B1 mode at lower frequencies, and a modest damping rate accompanies the crossing of the multiple body mode solutions. At higher frequencies the B1 mode is similar to the weakly magnetized case.
In the strongly magnetized wind case weak growth is associated with the slower, $S_{s}$, moving shorter wavelength solution and weak damping is associated with the faster, $S_{f}$, moving longer wavelength solution. At frequencies, $\lesssim \omega 2$, the growth rate is larger than the damping rate but at higher frequencies the damping rate is larger than growth rate for the S mode solution pair. In general the B1 mode is damped.
Conclusions {#sec:concld}
===========
Increased stability of the weakly-magnetized system with mildly relativistic sheath flow and stabilization of the strongly-magnetized system with mildly relativistic sheath flow is in agreement with theoretical results. In the fluid limit the present results confirm earlier results obtained by Hardee & Hughes (2003), who found that the development of sheath flow around a relativistic jet spine explained the partial stabilization of the jets in their numerical simulations.
The simulation results agree with theoretically predicted wavelengths and wave speeds. On the other hand, growth rates and spatial growth lengths obtained from the linearized equations or from the present relatively low resolution simulations only provide guidelines to the rate at which perturbations grow or damp.
A rapid decline in perturbation amplitudes in the sheath as a function of radius, governed by a Hankel function in the dispersion relation, suggests that the present results will apply to sheaths more than about three times the spine radius in thickness.
Where flow and magentic fields are parallel, current driven ([**CD**]{}) modes are stable (Isotomin & Pariev 1994, 1996). However, we expect magnetic fields to have a significant toroidal component. Provided radial gradients are not too large we expect the present results to remain approximately valid where $u_{j,e}$ and $B_{j,e}$ refer to poloidal velocity and magnetic field components.
In the helically twisted magnetic and flow field regime likely to be relevant to many astrophysical jets CD modes (Lyubarskii 1999) and/or KH modes could be unstable. While both CD and KH instability produce helically twisted structure, the conditions for instability, the radial structure, the growth rate and the pattern motions are different. These differences may serve to identify the source of helical structure on relativistic jet flows and allow determination of jet properties near to the central engine.
Resarch supported by NASA/MSFC cooperative agreement NCC8-256 and NSF award AST-0506666 to UA (P. Hardee), the NASA/MSFC postdoctoral program administered by ORAU (Y. Mizuno), and by NASA awards NNG-05GK73G, HST-AR-10966.01-A and NSF award AST-0506719 to UAH (K.Nishikawa). The numerical simulations were performed on the IBM p690 at NCSA and the Altix3700 BX2 at YITP.
Blandford, R. D. 1976, , 176, 465
Blandford, R.D., & Payne, D.G. 1982, , 199, 883
Blandford, R.D., & Znajek, R.L. 1977, , 179, 433
Chartas, G., Brandt, W N., & Gallagher, S.C. 2003, , 595, 85
Chartas, G., Brandt, W.N., Gallagher, S. C., & Garmire, G.P. 2002, , 579, 169
De Villiers, J.-P., Hawley, J.F., & Krolik, J.H. 2003, , 599, 1238
De Villiers, J.-P., Hawley, J.F., Krolik, J.H., & Hirose, S. 2005, , 620, 878
Ghisellini, G., Tavecchio, F., & Chiaberge, M. 2005, , 432, 401
Hardee, P.E. 2007, , 663 (in press)
Hardee, P.E., & Hughes, P.A. 2003, , 583, 116
Hawley, J.F., & Krolik, J.H. 2006, , 641, 103
Henri, G., & Pelletier, G. 1991, , 383, L7
Istomin, Y.N., & Pariev, V.I. 1994, , 267, 629
Istomin, Y. N., & Pariev, V.I. 1996, , 281, 1
Koide, S., Meier, D.L., Shibata, K., & Kudoh, T. 2000, , 536, 668
Laing, R. A. 1996, ASP Conf. Series 100: Energy Transport in Radio Galaxies and Quasars, eds. P.E. Hardee, A.H. Bridle & A. Zensus (San Francisco: ASP), 241
Lazzati, D., & Begelman, M.C. 2005, , 629, 903
Lovelace, R.V.E. 1976, , 262, 649
Lyubarskii, Y.E. 1999, , 308, 1006
McKinney, J.C. 2006, , 368, 1561
McKinney, J.C. & Gammie, C.F. 2004, , 977
Meier, D.L. 2003, New A Rev., 47, 667
Mirabel, I.F., & Rodriguez, L.F. 1999, ARAA, 37, 409
Mizuno, Y., Nishikawa, K.-I., Koide, S., Hardee, P., & Fishman, G.J. 2006, PoS (http://pos.sissa.it), MQW6, 045
Mizuno, Y., Hardee, P.E., & Nishikawa, K.-I. 2007, , 662, 835
Morsony, B.J., Lazzati, D., & Begelman, M C. 2006, (astro-ph/0609254)
Nishikawa, K.-I., Richardson, G., Koide, S., Shibata, K., Kudoh, T., Hardee, P., & Fishman, G.J. 2005, , 625, 60
Piran, T. 2005, Rev. Mod. Phys., 76, 1143
Pounds, K.A., King, A.R., Page, K.L., & O’Brien, P.T. 2003a, , 346, 1025
Pounds, K.A., Reeves, J.N., King, A.R., Page, K.L., O’Brien, P.T., & Turner, M.J.L. 2003b, , 345, 705
Reeves, J.N., O’Brien, P.T., & Ward, M. J. 2003, , 593, L65
Rossi, E., Lazzati, D., & Rees, M.J. 2002, , 332, 945
Siemiginowska, A., Stawarz, L., Cheung, C. C., Harris, D. E., Sikora, M., Aldcroft, T. L., & Bechtold, J. 2007, , 657, 145
Sol, H., Pelletier, G., & Assero, E. 1989, , 237, 411
Steffen, W., Zensus, J.A., Krichbaum, T.P., Witzel, A., & Qian, S.J. 1995, , 302, 335
Zhang, W., Woosley, S.E., & Heger, A. 2004, , 608, 365
Zhang, W., Woosley, S.E., & MacFadyen, A. I. 2003, , 586, 356
Zensus, J. A., Cohen, M. H., & Unwin, S. C. 1995, , 443, 35
|
---
abstract: '[In a cross-sectional study, adolescent and young adult females were asked to recall the time of menarche, if experienced. Some respondents recalled the date exactly, some recalled only the month or the year of the event, and some were unable to recall anything. We consider estimation of the menarcheal age distribution from this interval censored data. A complicated interplay between age-at-event and calendar time, together with the evident fact of memory fading with time, makes the censoring informative. We propose a model where the probabilities of various types of recall would depend on the time since menarche. For parametric estimation we model these probabilities using multinomial regression function. Establishing consistency and asymptotic normality of the parametric MLE requires a bit of tweaking of the standard asymptotic theory, as the data format varies from case to case. We also provide a non-parametric MLE, propose a computationally simpler approximation, and establish the consistency of both these estimators under mild conditions. We study the small sample performance of the parametric and non-parametric estimators through Monte Carlo simulations. Moreover, we provide a graphical check of the assumption of the multinomial model for the recall probabilities, which appears to hold for the menarcheal data set. Our analysis shows that the use of the partially recalled part of the data indeed leads to smaller confidence intervals of the survival function.]{} [Interval censoring, Informative censoring, Maximum likelihood estimation, Retrospective study, Current status data, Self consistency.]{}'
author:
- |
Sedigheh Mirzaei Salehabadi$^\ast$, Debasis Sengupta, Rahul Ghosal\
*St. Jude Children’s Research Hospital, Memphis, USA\
Indian Statistical Institute, Kolkata, India\
North Carolina State University, Raleigh, USA*\
\[2pt\] [Sedigheh.Mirzaei@stjude.org]{}
bibliography:
- 'refs.bib'
title: Estimating menarcheal age distribution from partially recalled data
---
[S. Mirzaei S. and others]{} [Estimating menarcheal age distribution from partially recalled data]{}
Introduction {#intro}
============
In a recent survey conducted by the Indian Statistical Institute (ISI) in and around the city of Kolkata [@Dasgupta_2015], over four thousand randomly selected individuals, aged between 7 and 21 years, were sampled. In this retrospective and cross-sectional study, the subjects were interviewed on or around their birthdays. The data on female subjects contains age, menarcheal status, some physical measurements and information on some socioeconomic variables. If a subject had already experienced menarche, she was asked to recall the date of the onset of her menarche.
We considered a subset of the original data, consisting of respondents who came from a general caste family with monthly expenditure greater than or equal to Rs. 15000 and both parents graduate. Among the 289 females represented in the data set, 45 individuals did not have menarche, 68 individuals recalled the exact date of the onset of menarche, 43 and 30 individuals recalled the calendar month and the calendar year of the onset, respectively, and 103 individuals could not recall any range of dates. Thus, the data are interval-censored. A major goal of this study is to estimate the distribution of the age at onset of menarche. This problem should be of interest to anyone working with incompletely recalled time-to-event data, of which there are many examples in the literature. The key variables in these studies include age at onset of menarche in adolescent and young adult females [@Koo_1997], time-to-pregnancy [@Joffe_1995], time-to-weaning from breastfeeding [@Gillespie_2006], time-to-injury for victims injured during a year [@Harel_1994], time-to-employment [@Mathiowetza_1988], and so on. In these studies, estimation of the time-to-event distribution is important for building a standard for individuals, comparing two populations or assessing the effect of a covariate. There is a possibility that the recalled time-to-event is inaccurate [@Koo_1997; @Mathiowetza_1988]. In the ISI study, this problem was somewhat circumvented by allowing the respondents to report an interval in lieu of the exact age-at-menarche. The recalled intervals generally happened to be in terms of calendar months and years. We refer to this special type of incompleteness as partial recall.
![Cumulative proportion of decreasing degrees of recall for different age ranges in menarcheal data[]{data-label="fig_1"}](cumprob-data1.pdf){height="2.8in" width="3.6in"}
Figure \[fig\_1\] shows the cumulative proportions of successively less precise recall in different groups of ages at interview, for the respondents of the ISI study. It is seen that the lines do not cross and the age group order is preserved. Also, there is greater precision of recall at lower age group, i.e., memory fades with time. Thus, two subjects interviewed at the same age would have different chances of recalling their age at menarche, depending on which of them had experienced the event earlier. In other words, the censoring mechanism underlying such recall-based data is inherently informative. The natural question is: how can one model the different degrees of partial recall, so that the distribution of menarcheal age can be estimated?
There is no suitable model and method in the literature for estimating the time-to-event distribution from partially recalled data, though such data abound in various fields. Apart from the informative nature of censoring, the problem is complicated by the mismatch of the time scales of the partial recall information (expressed through calendar time) and the time-to-event (measured from a respondent-specific starting time, e.g., birth). [@Mirzaei_2015] and [@Mirzaei_2016] addressed the first issue by proposing a model for this type of informative censoring, but they bypassed the second issue by clubbing all the cases of partial recall with the cases of no recall.
In this paper we propose a realistic censoring model for estimating the time-to-event distribution from partially recalled data. We present our modelling framework in Section \[likelihood\], and derive the appropriate likelihood under the proposed model. In Section \[Param\] we express the likelihood as a product of densities in an appropriate space, and discuss asymptotic properties of a parametric maximum likelihood estimator (MLE). In Section \[nonparametric\] we derive the non-parametric maximum likelihood estimator (NPMLE) and an approximate MLE (AMLE), and also establish consistency of both these estimators. In Sections \[Simulation\] and \[adequacy\] we report the results of Monte Carlo simulations of small sample performance of the MLE and the AMLE, and present some diagnostic checks of adequacy of the model. We analyze the real data set in Section \[DataAnalysis\]. We conclude with some discussion and indications of possible future extensions in Section \[s:discuss\]. The proofs of all the theorems and the results of additional simulations and data analysis are given in the supplementary material.
Model and Likelihood {#likelihood}
====================
Consider a set of $n$ subjects having ages at occurrence of landmark events $T_1,\ldots,T_n$, which are samples from the distribution $F$, with density $f$. Let these subjects be interviewed at ages $S_1,\ldots,S_n$, respectively. Suppose the $S_i$s are samples from another distribution and are independent of the $T_i$s. Let $\delta_i$ be the indicator of $T_i\le S_i$. This inequality means that the event for the $i$th subject had occurred on or before the time of interview.
In the case of current status data, one only observes $(S_i,\delta_i), i=1,2,\ldots, n$. The corresponding likelihood, conditional on the times of interview, is $$\prod_{i=1}^{n}[F(S_i)]^{\delta_i}[\bar{F}(S_i)]^{1-\delta_i}, \label{YN}$$ where $\bar{F}(S_i)=1-F(S_i)$. For properties of the MLE based on the above likelihood, see [@Lee_2003].
The structure of recalled data is generally more complicated. [@Mirzaei_2015] proposed a simplistic model, where the subject may either recall the time of the event exactly or not remember it at all. They used an indicator, $\varepsilon_i$, to record whether an exact recall is possible. As the chance of recall may depend on the time elapsed since the event, they modeled the non-recall probability as a function of this time. According to this model, $$P(\varepsilon_i=0|S_i=s,T_i=t)=\pi(s-t) \qquad \mbox{for} \quad 0<t<s, \label{pifirst}$$ for some non-recall probability function $\pi$. Thus, the likelihood is $$\prod_{i=1}^{n}\left[\left(\int_{0}^{S_i} f(u)\pi(S_i-u) du\right)^{1-\varepsilon_i}\left[f(T_i)(1-\pi(S_i-T_i))\right]^{\varepsilon_i}\right]^{\delta_i}
[\bar{F}(S_i)]^{1-\delta_i}. \label{MirSen}$$
Let us now consider the possibility that the $i$th subject can recall the date of the event only up to a calendar month or a calendar year, and define the recall status variable $\varepsilon_i$ for the $i$th subject as $$\varepsilon_i= \left\{
\begin{array}{ll}
0 & \text {if there is exact recall},\\
1 & \text {if the date is recalled up to the calendar month},\\
2 & \text {if the date is recalled up to the calendar year},\\
%3 & \text {if the event has not happened or the date is not recalled}.
3 & \text {if the date is not recalled}.
\end{array} \right. \label{epsilon}$$ The value of $\varepsilon_i$ concerns the state of recall. When $\delta_i=1$, $\varepsilon_i=0$ means that the exact date of the event is recalled. When $\delta_i=0$, $\varepsilon_i$ may be assigned the value 0, as no recall failure is expected in case the event is reported not to have happened.
We regard the four scenarios as outcomes of a multinomial selection, where allocation probabilities depend on the time elapsed since the occurrence of the event. Thus, for $0<t<s$, we model the allocation probabilities as $$\begin{array}{l}
P(\varepsilon_i=0|S_i=s,T_i=t)=\pi^{(0)}(s-t),\\
P(\varepsilon_i=1|S_i=s,T_i=t)=\pi^{(1)}(s-t),\\
P(\varepsilon_i=2|S_i=s,T_i=t)=\pi^{(2)}(s-t),\\
P(\varepsilon_i=3|S_i=s,T_i=t)=\pi^{(3)}(s-t).
\end{array} \label{pifunc}$$ where $\sum_{k=0}^{3}\pi^{(k)}(s-t)=1$.
We refer to the set-up described in the first paragraph of this section, together with and as the proposed model. According to this model, contributions to the likelihood in different cases are as follows.
[Case]{}
: \(i) When $\delta_i=0$ (the event has not occurred till the time of observation), the contribution of the $i$th individual to the likelihood is $\bar{F}(S_i)$.
[Case]{}
: (ii): When $\delta_i=1$ and $\varepsilon_i=0$ (the event has occurred and the $i$th individual can remember the time), the contribution of the individual to the likelihood is $f(T_i)\pi^{(0)}(S_i-T_i)$.
[Case]{}
: (iii): When $\delta_i=1$ and $\varepsilon_i=1$ (the event has occurred but the $i$th individual can only recall the calendar month of the event), the contribution of the individual to the likelihood is $\int_{M_{i1}}^{M_{i2}} f(u)\pi^{(1)}(S_i-u) du$, where $M_{i1}$ and $M_{i2}$ are the ages of the individual at the beginning and the end of the calendar month recalled by the individual.
[Case]{}
: (iv): When $\delta_i=1$ and $\varepsilon_i=2$ (the event has occurred but the $i$th individual can only recall the calendar year of the event), the contribution of the individual to the likelihood is $\int_{Y_{i1}}^{Y_{i2}} f(u)\pi^{(2)}(S_i-u) du$, where $Y_{i1}$ and $Y_{i2}$ are the ages of the individual at the beginning and the end of the calendar year recalled by the individual.
[Case]{}
: (v): When $\delta_i=1$ and $\varepsilon_i=3$ (the event has occurred but the $i$th individual cannot recall the time at all), the contribution of the individual to the likelihood is $\int_{0}^{S_i}f(u)\pi^{(3)}(S_i-u) du$.
Therefore, the overall likelihood is $$\begin{aligned}
\prod_{i=1}^{n}[\bar{F}(S_i)]&^{1-\delta_i}
\Biggl[\left(f(T_i)\pi^{(0)}(S_i-T_i)\right)^{I_{(\varepsilon_i=0)}} \left(\int_{M_{i1}}^{M_{i2}}f(u)\pi^{(1)}(S_i-u) du\right)^{I_{(\varepsilon_i=1)}} \times \notag \\ &\left(\int_{Y_{i1}}^{Y_{i2}}f(u)\pi^{(2)}(S_i-u) du\right)^{I_{(\varepsilon_i=2)}} \left(\int_{0}^{S_i} f(u)\pi^{(3)}(S_i-u) du\right)^{I_{(\varepsilon_i=3)}}\Biggr]^{\delta_i}.
\label{ourM1}\end{aligned}$$ Note that when $\pi^{(1)}=\pi^{(2)}=0$, the likelihood reduces to . When $\pi^{(1)}=\pi^{(2)}=0$ and $\pi^{(0)}$ is a constant, it becomes a constant multiple of the likelihood corresponding to non-informatively interval censored data. If $\pi^{(0)}=\pi^{(1)}=\pi^{(2)}=0$, it reduces to the current status likelihood .
While the proposed model is specific to the data at hand, it can easily be adjusted for arbitrary types of recall, which need not even be ordered.
The factors in the product likelihood (\[ourM1\]) have different forms in different cases. We now show that they can be expressed as the common density of some random vector with respect to a suitable dominating measure.
The main challenge to obtaining a common format of the data lies in the fact that $M_{i1}$, $M_{i2}$, $Y_{i1}$ and $Y_{i2}$ are the ages of the $i$th individual at specified calendar times. We make use of the fact that these observables are functions of $T_i$ and the date of birth of the $i$th individual. Specifically, for the $i$th subject, let $m_i$ be the serial number of the month of birth within the year of birth and $d_i$ be the time (measured in years) from the beginning of the month of birth till the event of birth. For the sake of simplicity, we assume that every year has duration $1$ and every month has duration $1/12$.
When $\epsilon_i=1$, i.e., the month of the event is recalled, we write $$\begin{array}{l}
M_{i1} = \lfloor 12(d_i + T_i) \rfloor / 12-d_i, \\
M_{i2} = M_{i1}+1/12,
\end{array} \label{Month}$$ where $\lfloor\cdot\rfloor$ is the integer part of its argument. Thus, the variables $\lfloor 12(d_i + T_i) \rfloor$, $m_i$ and $d_i$ can be obtained from $M_{i1}$, $M_{i2}$, $m_i$ and $d_i$ and vice versa. Likewise, when $\epsilon_i=2$, i.e., the year of the event is recalled, we write $$\begin{array}{l}
Y_{i1}=\lfloor \big(T_i + d_i + (m_i-1)/12 \big) \rfloor-\big(d_i+(m_i-1)/12\big),\\
Y_{i2} = Y_{i1}+1.
\end{array} \label{Year}$$ It is clear that the variables $\lfloor \big(T_i + d_i + (m_i-1)/12 \big) \rfloor$, $m_i$ and $d_i$ are equivalent to $Y_{i1}$, $Y_{i2}$, $m_i$ and $d_i$. Therefore, we define the variable $$V_i= \left\{
\begin{array}{ll}
T_i & \text {if $\varepsilon_i=0$ , $\delta_i=1$,}\\
\lfloor 12(d_i + T_i) \rfloor / 12 & \text {if $\varepsilon_i=1$ , $\delta_i=1$,}\\
\lfloor \big(T_i + d_i + (m_i-1)/12 \big) \rfloor & \text {if $\varepsilon_i=2$ , $\delta_i=1$,}\\
0 & \text {if $\varepsilon_i=3$, $\delta_i=1$, or if $\delta_i=0$,}
% \lfloor(T_i+m_i+d_i)/C_2\rfloor & \text {if $\varepsilon_i=3$},\\
% \lfloor T_i/C_3\rfloor & \text {if $\varepsilon_i=0$},
\end{array} \right. \label{W}$$ which captures the essential part of the occasionally observable variables $T_i$, $M_{i1}$, $M_{i2}$, $Y_{i1}$ and $Y_{i2}$, and subsequently work with the observable vector $$Y_i=(S_i,V_i,\varepsilon_i,\delta_i,m_i,d_i). \label{Y}$$ We have already assumed that the $T_i$s (time-to-event) are samples from the distribution $F$ and the $S_i$s (ages on interview date) are samples from another distribution. We now denote by $G_1$, $G_2$ and $G_3$ the distributions of $S_i$, $m_i$ and $d_i$, respectively, for every $i$. The distribution $G_2$ is defined over the set $\{1,2,\ldots, 12\}$, and $G_3$ is defined over the interval $[0,1/12]$. The latter assumption disregards the fact that $d_i$ is known only up to days (measured as fixed fractions of a year), to keep the description simple.
Theorem \[Thm1\] presented below gives the density of $Y_i$, after the subscript $i$ is dropped for simplicity. The dominating probability measure used for defining this density is $\mu=\vartheta_1\times\vartheta_2\times\vartheta_3\times\vartheta_4\times\vartheta_5\times\vartheta_6$ where $\vartheta_1$ is the measure with respect to which $G_1$ has a density (e.g., the counting or the Lebesgue measure, depending on whether $G_1$ is discrete or continuous), $\vartheta_2$ is the sum of the counting and the Lebesgue measures, each of $\vartheta_3,\vartheta_4$ and $\vartheta_5$ is the counting measure and $\vartheta_6$ is the Lebesgue measure [@Ash_2000].
\[Thm1\] The density of $Y=(S,V,\varepsilon,\delta,m,d)$ with respect to the measure $\mu$ is $$\begin{aligned}
\mbox{}\hskip-10pt&&\hskip-25pt h(s,v,\varepsilon,\delta,m,d)\nonumber\\
\mbox{}\hskip-10pt&=&\left\{
\begin{array}{ll}
g_1(s)g_2(m)g_3(d)\bar{F}(s) & \text {if $\delta=0$},\\
g_1(s)g_2(m)g_3(d)f(v)\pi^{(0)}(s-v)I_{(v<s)} & \text {if $\varepsilon=0$ and $\delta=1$},\\
g_1(s)g_2(m)g_3(d)\int_{v-d}^{min(s,v+\frac{1}{12}-d} f(u)\pi^{(1))}(s-u) du & \text {if $\varepsilon=1$ and $\delta=1$},\\
g_1(s)g_2(m)g_3(d)\int_{v-d-\frac{m-1}{12}}^{min(s,v+1-d-\frac{m-1}{12})} f(u)\pi^{(2)}(s-u) du & \text {if $\varepsilon=2$ and $\delta=1$},\\
g_1(s)g_2(m)g_3(d)\int_{0}^{s} f(u)\pi^{(3)}(s-u) du & \text {if $\varepsilon=3$ and $\delta=1$},
\end{array} \right. \label{densitY}\end{aligned}$$ where $g_1$, $g_2$ and $g_3$ are the densities of $G_1$, $G_2$ and $G_3$ with respect to the measures $\vartheta_1$, $\vartheta_5$ and $\vartheta_6$, respectively.
Theorem \[Thm1\] implies that the likelihood can be written as $$\begin{aligned}
&\prod_{i=1}^{n}[\bar{F}(S_i)]^{1-\delta_i}
\Biggl[\left(f(V_i)\pi^{(0)}(S_i\!-\!V_i)\right)^{I_{(\varepsilon_i=0)}} \!\left(\int_{V_i-d_i}^{V_i-d_i+\frac{1}{12}}\!\!f(u)\pi^{(1)}(S_i\!-\!u) du\!\right)^{I_{(\varepsilon_i=1)}} \notag \\ & \times\! \left(\int_{V_i-d_i-\frac{m_i-1}{12}}^{V_i-d_i-\frac{m_i-1}{12}+1}\!\!f(u)\pi^{(2)}(S_i\!-\!u) du\!\right)^{I_{(\varepsilon_i=2)}} \!\!\!\left(\int_{0}^{S_i} \!\!f(u)\pi^{(3)}(S_i\!-\!u) du\!\right)^{I_{(\varepsilon_i=3)}} \Biggr]^{\delta_i}\!\!, \notag \\
& \qquad \qquad =\frac{\prod_{i=1}^{n}h(S_i,V_i,\varepsilon_i,\delta_i,m_i,d_i)}{\prod_{i=1}^{n}g_1(S_i)g_2(m_i)g_3(d_i)},
\label{likedensity}\end{aligned}$$ where the $i$th factor is the conditional density of $(V_i,\varepsilon_i,\delta_i)$ given $(S_i,m_i,d_i)$.
Parametric estimation {#Param}
=====================
Suppose the forms of the functions $\bar{F}$, $f$, $\pi^{(0)}$, $\pi^{(1)}$, $\pi^{(2)}$ and $\pi^{(3)}$ in the likelihood are known up to a few parameters, and accordingly they are written as $\bar{F}_\theta$, $f_\theta$, $\pi^{(0)}_{\eta}$, $\pi^{(1)}_{\eta}$, $\pi^{(2)}_{\eta}$ and $\pi^{(3)}_{\eta}$, respectively. The MLE of the (possibly vector) parameters $\theta$ and $\eta$ are obtained by maximizing .
Since the equivalent likelihood (\[likedensity\]) is identified as a product of conditional densities, standard results for consistency and asymptotic normality of the MLE become applicable. The regularity conditions for these results would then be specified in terms of the density of $Y_i$. In the first section of the supplementary material, we provide easily verifiable sufficient conditions that involve the density $f_\theta$ (the density of $T_i$) and the functions $\pi_{\eta}^{(0)},\pi_{\eta}^{(1)},\pi_{\eta}^{(2)}$ and $\pi_{\eta}^{(3)}$, which define the conditional probability distribution of the random variable $\varepsilon_i$ given $T_i$ and $S_i$.
Non-parametric estimation {#nonparametric}
=========================
Non-parametric MLE {#NPMLE}
------------------
Before embarking on the task of estimation, we establish the following result on the issue of identifiability.
\[Thm2\] The distribution functions $G_1$ and $F$, and recall probabilities $\pi_{}^{(k)}$, $k=0,1,2,3$ are identifiable from $h$ in .
The likelihood is difficult to maximize because of the integrals contained in the expression. In order to simplify it, we assume that the function $\pi^{(l)}$ in is piecewise constant, having the form $\pi_{}^{(l)}(x)=b_{l1} I(x_1<x\leq x_2)+b_{l2}I(x_2<x\leq x_3)+\ldots+b_{lL}I(x_L<x <\infty)$, $l=0,1,2,3$, where $0=x_1<x_2<\cdots < x_L$ are a chosen set of time-points and $b_{l1},b_{l2},\ldots,b_{lL}$ are unspecified parameters taking values in the range $[0,1]$ such that $\sum_{l=0}^{3} b_{lj}=1$ for $j=1,2,\ldots,L$. Then the likelihood (\[ourM1\]) reduces to $$\begin{aligned}
L=&\prod_{i=1}^{n}[\bar{F}(S_i)]^{1-\delta_i}
\Biggl[\left\{f(T_i) \left(\sum_{l=1}^L b_{0l} I\big(W_{l+1}(S_i)<T_i\leq W_l(S_i)\big)\right)\right\}^{I_{(\varepsilon_i=0)}} \notag \\ &\hskip-20pt \times\!\left\{\sum_{\substack{l=1 \\ [W_{l+1}(S_i),W_l(S_i)]\cap [M_{i1},M_{i2}]\ne \phi }}^L b_{1l} \Big(F\big(\min(W_l(S_i),M_{i2})\big)\!-\!F\big(\max(W_{l+1}(S_i),M_{i1})\big)\Big)\right\}^{I_{(\varepsilon_i=1)}} \notag \\
&\hskip-20pt \times\!\left\{\sum_{\substack{l=1 \\ [W_{l+1}(S_i),W_l(S_i)]\cap [Y_{i1},Y_{i2}]\ne \phi }}^L b_{2l} \Big(F\big(\min(W_{l}(S_i),Y_{i2})\big)\!-\!F\big(\max(W_{l+1}(S_i),Y_{i1})\big)\Big)\right\}^{I_{(\varepsilon_i=2)}} \notag \\
&\hskip-20pt \times\!\left\{\sum_{l=1}^L b_{3l} \Big(F(W_l(S_i))-F(W_{l+1}(S_i))\Big)\right\}^{I_{(\varepsilon_i=3)}}\Biggr]^{\delta_i}, \label{ourMINT}\end{aligned}$$ where $W_l(S_i)=(S_i-x_l)\vee t_{min}$ for $l=1,\ldots,L$ and $W_{L+1}(S_i)=t_{min}, i=1,2,\ldots,n$. The likelihood involves probabilities assigned to intervals of the type $[t,t_{max}]$ or $(t,t_{max}]$, as per the baseline probability distribution. Since these intervals have overlap, we try to write them as unions of some disjoint intervals. Let ${\cal I}_1$, ${\cal I}_2$, ${\cal I}_3$, ${\cal I}_4$ and ${\cal I}_5$ be sets of indices $i$ (between 1 and $n$) that satisfy the conditions $\delta_i=0$, $\delta_i\varepsilon_i=1$, $\delta_i(1-\varepsilon_i)=1$, $\delta_i\varepsilon_i=2$ and $\delta_i\varepsilon_i=3$. respectively. Consider the intervals $$\begin{array}{r@{\hskip3pt}c@{\hskip3pt}ll}
A_i&=&(S_i,t_{max}] & \mbox{for }i\in{\cal I}_1,\\[.4ex]
A_i&=&[T_i,t_{max}] & \mbox{for }i\in {\cal I}_2,\\[.4ex]
A_i'&=&(T_i,t_{max}]& \mbox{for }i\in{\cal I}_2,\\[.5ex]
A_{il}&=& \left\{ \begin{matrix}(W_l(S_i),t_{max}],& l=1,\ldots,L,\\
[W_l(S_i),t_{max}],&l=k+1,\ \ \ \,\\ \end{matrix} \right.& \mbox{for}\ i\in{\cal I}_2 \cup{\cal I}_3,\\[2ex]
B_{il}&=&[W_{l+1}(S_i) \vee M_{i1},W_{l+1}(S_i) \wedge M_{i1}] & \mbox{for }i\in {\cal I}_4\ \& \ l=1,\ldots,L,\\[.4ex]
C_{il}&=&[W_{l+1}(S_i) \vee Y_{i1},W_{l+1}(S_i) \wedge Y_{i1}] & \mbox{for }i\in {\cal I}_5\ \& \ l=1,\ldots,L,\\
\end{array}\label{Ai}$$ and the sets $$\begin{array}{r@{\hskip3pt}c@{\hskip3pt}l}
{\cal A}_1&=&\{A_i: \ \ i\in{\cal I}_1\},\\[.25ex]
{\cal A}_2&=&\{A_i\setminus A_i': \ \ i\in {\cal I}_2\},\\[.25ex]
{\cal A}_3&=&\{A_i':\ \ i\in{\cal I}_2\},\\[.25ex]
{\cal A}_4&=&\{A_{i(l+1)}\setminus A_{il}:\ \ 1\le l\le L \mbox{ and}\ i\in{\cal I}_3\},\\[.25ex]
{\cal A}_5&=&\{ B_{il}:\ \ 1\le l\le L \mbox{ and}\ i\in{\cal I}_4\},\\[.25ex]
{\cal A}_6&=&\{ C_{il}:\ \ 1\le l\le L \mbox{ and}\ i\in{\cal I}_5\}.\\
\end{array}\label{CalA}$$ As $F$ is absolutely continuous, the elements of ${\cal A}_2$ and ${\cal A}_3$ are distinct with probability 1. Let $n_i$ be the cardinality of ${\cal I}_i$, $i=1,2,3,4,5$. We arrange the singleton elements of ${\cal A}_2$ in increasing order, and denote them as $B_1, B_2,\ldots, B_{n_2}$. We also arrange the elements of ${\cal A}_3$ in the corresponding order and denote them as $B_{n_2+1},B_{n_2+2},\ldots,B_{2n_2}$. We then collect the unique elements of ${\cal A}_1\cup {\cal A}_4\cup {\cal A}_5\cup {\cal A}_6$ that are distinct from $B_1,B_2,\ldots,B_{2n_2}$, and denote them as $B_{2n_2+1},B_{2n_2+2},\ldots, B_M$. Observe that the collection $B_1,B_2,\ldots,B_M$ consists of the distinct elements of $\bigcup_{i=1}^{6}{\cal A}_i$, arranged in a particular order. Denote the non-empty subsets of the index set $\{1,2,\ldots,M\}$ by $s_1,s_2,\ldots,s_{2^M-1}$. Define $$I_r=\left\{\bigcap_{i\in
s_r}B_i\right\}\bigcap\left\{\bigcap_{i\notin
s_r}B_i^c\right\}\qquad \mbox{for }r=1,2,\ldots,2^M-1.\label{Ir}$$ Some of the $I_r$s may be empty sets, denoted here by $\phi$. Let $$\begin{aligned}
{\cal C}&=&\{s_r:\,I_r\ne\phi,\,1\le r\le 2^M-1\}, \label{C}\\
{\cal A}&=&\{I_r:\,I_r\ne\phi,\,1\le r\le 2^M-1\}. \label{A}\end{aligned}$$ It can be verified that the elements of ${\cal A}$ are distinct and disjoint.
Note that each of the intervals $B_1,\ldots,B_M$ is a union of disjoint sets that are members of ${\cal A}$. For any Borel set $A$, suppose $P(A)$ is the probability assigned to $A$ as per the probability distribution $F$. Let $p_r=P(I_r)$, for $I_r \in \cal A$. Then the likelihood reduces to $$\begin{aligned}
L=& \prod_{i\in {\cal I}_1} \!\!\left(\sum\limits_{\substack{r:I_r \subseteq A_i \\ s_r\in {\cal C} }} \!\!p_r\!\right)\!\times\!\!
\prod_{i\in {\cal I}_2}\!\!\left(\!1-\sum_{l=1}^{L} (b_{1l}+b_{2l}+b_{3l}) I_{(T_i \in A_{i(l+1)}\backslash A_{il})}\!\right)\!\cdot\! \left(\sum\limits_{\substack{r:I_r \subseteq A_{i}\backslash A_{i'}\\ s_r\in {\cal C}}}\!\!p_r\!\right)\notag\\
&\hskip-20pt \times\! \prod_{i\in {\cal I}_3}\!\left[\sum_{l=1}^{L} b_{3l}\!\!\left(\!\sum\limits_{\substack{r:I_r \subseteq A_{i(l+1)}\backslash A_{il}\\ s_r\in {\cal C}}}\!\!p_r\!\right)\!\right]\!\times\! \prod_{i\in {\cal I}_4}\!\left[\sum\limits_{\substack {l=1 \\ [W_{l+1}(S_i),W_l(S_i)]\cap [M_{i1},M_{i2}]\ne \phi }}^{L} \!\!b_{1l}\!\!\left(\sum\limits_{\substack{r:I_r \subseteq B_{il}\\ s_r\in {\cal C}}}\!\!p_r\!\right)\!\right]\notag\\
&\hskip-20pt \times \prod_{i\in {\cal I}_5}\left[\sum\limits_{\substack {l=1 \\ [W_{l+1}(S_i),W_l(S_i)]\cap [Y_{i1},Y_{i2}]\ne \phi }}^{L} b_{2l}\!\left(\sum\limits_{\substack{r:I_r \subseteq C_{il}\\ s_r\in {\cal C}}}\!p_r\!\right)\right]. \label{ourM_int1}\end{aligned}$$ Thus, maximizing the likelihood amounts to maximizing with respect to $p_r$ for $s_r \in \cal C$.
There is a partial order among the members of ${\cal C}$ in the sense that some sets are contained in others. Consider the following subsets of ${\cal C}$. $$\begin{aligned}
{\cal C}_1&=&\{s:\,s\in{\cal C};\ \mbox{there is another element $s'\in{\cal C}$, such that $s\subset s'$}\}, \nonumber\\
{\cal C}_2&=&\{s:\,s\in{\cal C};\ \mbox{there is another element $s'\in{\cal C}$, such that} \nonumber\\
&& s'\backslash(s\cap s') \mbox{ consists of a singleton $j$ and }s\backslash(s\cap s')=\{j+n_2\}\}, \nonumber\\
{\cal C}_0&=&{\cal C}\backslash({\cal C}_1\cup{\cal C}_2).
\label{C0}\end{aligned}$$ We now present a result which shows that maximization of the likelihood can be restricted to ${\cal C}_0$.
\[Thm3\] Maximizing the likelihood with respect to $p_r$ for $s_r \in \cal C$ is equivalent to maximizing it with respect to $p_r$ for $s_r \in {\cal C}_0$, i.e., $$\mathop{\max\limits_{p_r:p_r
\in [0,1], \sum_{s_r \in \cal C}p_r=1}} L\ \ =\ \
\displaystyle\mathop{\max\limits_{p_r:p_r \in [0,1], \sum_{s_r \in
{\cal C}_0}p_r=1}} L$$
It transpires from the above theorem that the likelihood has the same maximum value, irrespective of whether $s_r$ is chosen from the class ${\cal C}$ or ${\cal C}_0$. Therefore, we can replace ${\cal C}$ by ${\cal C}_0$ in .
Let us relabel the intervals $I_j,$ $s_j\in{\cal C}_0$, by $J_1,J_2,\ldots,J_\nu$. Further, let ${\cal A}_0=\{J_1,J_2,\ldots,J_\nu\}$ and $q_j=P(J_j)$ for $j=1,2,\ldots,\nu$. If the likelihood is rewritten with the condition $s_r\in{\cal C}$ replaced by the equivalent condition $I_r\in{\cal A}$, then Theorem \[Thm3\] shows that the latter condition can be replaced by $I_r\in{\cal A}_0$. In other words, maximizing the likelihood is equivalent to maximizing $$\begin{aligned}
&L(p,\eta)\notag\\
&=
\prod_{i\in {\cal I}_1}\!\! \left(\sum_{j:J_j \subseteq A_i } \!\!q_j\!\!\right)\!\!\times\!\!
\prod_{i\in {\cal I}_2}\!\!\left(\!1-\sum_{l=1}^{L} (b_{1l}+b_{2l}+b_{3l}) I_{(T_i \in A_{i(l+1)}\backslash A_{il})}\!\right)\!\cdot\! \left(\sum_{j: J_j \subseteq A_{i}\backslash A_{i'}}\!\!q_j\!\!\right)\notag\\
&\times\!\! \prod_{i\in {\cal I}_3}\!\!\left[\sum_{l=1}^{L} \!b_{3l}\!\!\left(\sum_{j: J_j\subseteq A_{i(l+1)}\backslash A_{il}}\!\!q_j\!\!\right)\!\!\right]\!\!\times\!\! \prod_{i\in {\cal I}_4}\!\!\left[\sum\limits_{\substack {l=1 \\ [W_{l+1}(S_i),W_l(S_i)]\cap [M_{i1},M_{i2}]\ne \phi }}^{L} \! b_{1l}\!\!\left(\sum_{j:J_j \subseteq B_{il}}\!\!q_j\!\right)\!\!\right]\notag\\
&\times \prod_{i\in {\cal I}_5}\left[\sum\limits_{\substack {l=1 \\ [W_{l+1}(S_i),W_l(S_i)]\cap [Y_{i1},Y_{i2}]\ne \phi }}^{L} b_{2l}\left(\sum_{j:J_j \subseteq C_{il}} q_j\right)\right]=\prod_{i=1}^n\left(\sum_{j=1}^v \alpha_{ij}q_j\right),\label{ourMRed}\end{aligned}$$ with respect to the vector parameters $p=(q_1,q_2,\ldots,q_\nu)^T$ and $\eta=(b_{11},\ldots,$ $b_{1L},b_{21},\ldots,b_{2L},b_{31},\ldots,b_{3L})^T$, subject to the restrictions $\sum_{j=1}^\nu q_j=1$, $0\le q_1,\ldots,q_\nu\leq 1$, where $$\alpha_{ij} = \left\{
\begin{array}{ll}
I_{(J_j\subseteq A_{i})} & \mbox{if}~~ i\in{\cal I}_1,\\
\left(1-\sum_{l=1}^{L} (b_{1l}+b_{2l}+b_{3l}) I_{(T_i \in A_{i(l+1)}\backslash A_{il})}\right).I_{(J_j\subseteq A_{i}\backslash A'_{i})} & \mbox{if}~~ i\in {\cal I}_2,\\
\sum_{l=1}^{L} b_{1l} .I_{(J_j\subseteq A_{i(l+1)}\backslash A_{il})} & \mbox{if}~~i\in {\cal I}_3,\\
\sum_{l=1}^{L} b_{2l} .I_{(J_j\subseteq B_{il})} & \mbox{if}~~i\in {\cal I}_4,\\
\sum_{l=1}^{L} b_{3l} .I_{(J_j\subseteq C_{il})} & \mbox{if}~~i\in {\cal I}_5,\\
\end{array} \right. \label{alpha}$$ for $i=1,\ldots,n,$ and $j=1,\ldots,\nu$.
Now consider the set ${\cal A}_2=\left\{\{T_i\},\ i\in{\cal I}_2\right\}$ defined in , with cardinality set $n_2$ (defined after ). The task of maximization is simplified further through the following result, which is interesting by its own right.
\[Thm4\] The set ${\cal A}_2$ is contained in the set ${\cal A}_0$ almost surely. Further, if $G$ is a discrete distribution with finite support, then the probability of ${\cal A}_0$ being equal to ${\cal
A}_2$ goes to one as $n\rightarrow \infty$.
We are now ready for the next result regarding the existence and uniqueness of the NPMLE. The uniqueness is established probabilistically under the condition that $n_2$, the number of cases with exact recall, goes to infinity.
\[Thm5\] The likelihood has a maximum. Further, if $G$ is a discrete distribution with finite support, then the probability that it has a unique maximum goes to one, as $n_2\rightarrow \infty$.
Self-consistency approach for estimation {#selfconsistency}
----------------------------------------
We follow the footsteps of [@Efron_1967] and [@Turnbull_1976] to obtain the NPMLE through the self consistency approach. For $\ i=1,2,\ldots, n,$ let $$L_{ij} = \left\{
\begin{array}{ll}
1 & \mbox{if}~~ T_i\in J_j,\\
0 &\mbox{otherwise},\\
\end{array} \right.$$ When $i\in {\cal I}_2$, the value of $L_{ij}$ is known. If $i\notin {\cal I}_2$, its expectation with respect to the probability vector $p$ is given by $$E(L_{ij})=\frac{\alpha_{ij}q_j}{\sum\limits_{j=1}^\nu
\alpha_{ij}q_j}=\mu_{ij}(p),\hspace{.4cm} \mbox{say.}
\label{mu1}$$ Thus, $\mu_{ij}(p)$ represents the probability that the $i$-th observation lies in $J_j$. The average of these probabilities across the $n$ individuals, $$\frac{1}{n}\sum\limits_{i=1}^n\mu_{ij}(p)= \mathcal{\pi}_j(p),\hspace{.4cm} \mbox{say,} \label{pi}$$ should indicate the probability of the interval $J_j$. Thus, it is reasonable to expect that the vector $p$ would satisfy the equation $$q_j= \mathcal{\pi}_j(p) \hspace{.4cm} \mbox{for}\quad 1\leq j \leq \nu. \label{sc}$$ An estimator of $p$ may be called self consistent if it satisfies . The form of these equations suggests the following iterative procedure.
[Step I.]{}
: Obtain a set of initial estimates $q^{(0)}_j\hspace{.2cm} (1\leq j\leq m)$.
[Step II.]{}
: At the $n$th stage of iteration, use current estimate, ${p}^{(n)}$, to evaluate $\mu_{ij}(p^{(n)})$ for $ i=1,2,\ldots, n,\ j=1,2,\ldots, \nu$ and $\mathcal{\pi}_j({p}^{(n)})$ for $j=1,2,\ldots, \nu$ from and , respectively.
[Step III.]{}
: Obtain updated estimates ${p}^{(n+1)}$ by setting $q^{(n+1)}_j= \mathcal{\pi}_j({p}^{(n)})$.
[Step IV.]{}
: Return to Step II with ${p}^{(n+1)}$ replacing ${p}^{(n)}$.
[Step V.]{}
: Iterate; stop when the required accuracy has been achieved.
The following theorem shows that equation defining a self consistent estimator must be satisfied by an NPML estimator of $p$.
\[Thm6\] An NPML estimator of ${p}$ must be self consistent.
A computationally simpler estimator {#Simple}
-----------------------------------
The computational complexity of the NPMLE depends on the number of segments $(k)$ used in the piecewise constant formulation of the function $\pi_\eta$. One can conceive of a computational simplification on the basis of Theorem \[Thm3\]. According to this theorem, the NPMLE has mass only at points of exact recall of the event, when $n$ is large. In such a case, the likelihood involves $J_j$s that are singletons only. Formally, let $t_1,\ldots,t_{n_2}$ be the ordered set of distinct ages at event that have been perfectly recalled, and $q_1^*,\ldots,q_{n_2}^*$ be the probability masses allocated to them. The likelihood , subject to the constraint that $q_j=0$ whenever $J_j \notin {\cal A}_2$, is equivalent to the unconstrained maximization of $$L(p^*,\eta)=\prod_{i=1}^{n} \left[\sum_{j=1}^{n_2} \alpha_{ij}q^*_j \right], \label{aproxour}$$ with respect to the parameters $p^*=(q^*_1,\ldots,q^*_{n_2})^T$ and $\eta$, over the set $$\Re^*=\left\{(p^*,\eta) |\sum_{j=1}^{n_2} q^*_j=1,\quad
0\le q^*_1,\ldots,q^*_{n_2}\leq 1,\ 0\le b_1\le\cdots\le b_k\le 1
\right\}.$$ Let the likelihood be maximized at $(\hat p^*,\hat{\eta}^*)$, where $\hat p^*=(\hat{q}^*_1,\ldots,\hat{q}^*_{n_2})^T$. We define an approximate NPMLE (AMLE) of $F$ as $$\tilde{F}_n(t) = \sum_{j: t_j\leq t} \hat{q}^*_j. \label{Ftilde}$$ Both NPMLE and AMLE depend on $L$, the number of line segments in the descriptions of recall probabilities. One can use successively higher values of $L$ (e.g., higher powers of 2) and choose a value after which further increase does not add substantially to the details. A data analytic illustration of this principle in given in Section \[DataAnalysis\].
Consistency of the estimators {#Consistency}
-----------------------------
Let $\Theta$ be the set of all distribution functions over $[t_{min},t_{max}]$, i.e., $$\begin{aligned}
\Theta =&\{F\,:\,[t_{min},t_{max}]\rightarrow[0,1];\,F\mbox{ right continuous, nondecreasing};\\ \nonumber
&\hskip2.1inF(t_{min})=0;\,F(t_{max})=1\}.\end{aligned}$$ and $\overline{\Theta}$ be the set of all sub-distribution functions, i.e., $$\begin{aligned}
\overline{\Theta} =&\{F\,:\,[t_{min},t_{max}]\rightarrow[0,1];\,F\mbox{ right continuous, nondecreasing};\\ \nonumber
&\hskip2.1inF(t_{min})=0;\,F(t_{max})\le1\}.\end{aligned}$$ Note that, with respect to the topology of vague convergence, $\overline{\Theta}$ is compact by Helley’s selection theorem. Further, let $F_0$ denote the true distribution of the time of occurrence of landmark events with density $f_0$, and $F_0(t_{min})=0$. For any given distribution $F\in\Theta$ having masses restricted to the set $\{t_1,\ldots,t_{n_2}\}$, the log of the likelihood (\[aproxour\]) can be rewritten as a function of $F$ (instead $q^*_1,\ldots,q^*_{n_2}$) as $$\ell(F)=\sum_{i=1}^{n} \log\left[\sum_{j=1}^{n_2}
\alpha_{ij}\left\{F(t_j)-F(t_{j^-})\right\} \right].
\label{ouraprox2}$$ Define the set $${\cal E}=\{F\,:\,F\in \Theta,\,E[\ell(F)-\ell(F_0)]=0\}, \label{class}$$ which is an equivalence class of the true distribution $F_0$.
Strong consistency of the AMLE and weak consistency of the NPMLE are established by the following theorems.
\[Thm7\] In the above set-up, the AMLE $\{\tilde{F_n}\}$ converges almost surely to the equivalence class ${\cal E}$ of the true distribution $F_0$, in the topology of vague convergence.
\[Thm8\] [In the set-up described before Theorem \[Thm7\], the NPMLE $\{\hat{F_n}\}$ converges in probability to the equivalence class ${\cal E}$ of the true distribution $F_0$, in terms of the Lévy distance.]{}
Simulation of performance {#Simulation}
=========================
Parametric estimation {#parsim}
---------------------
We consider the MLEs based on the current status likelihood (\[YN\]) (described here as Current Status MLE), the likelihood based on binary recall (described here as Binary Recall MLE) and the likelihood (\[ourM1\]) based on partial recall (described here as Partial Recall MLE). Computation of the three MLEs is done through numerical optimization of likelihood using the Quasi-Newton method [@Nocedal_2006].
For the purpose of simulation, we generate samples of time-to-event from the Weibull distribution with shape and scale parameters $\theta_1$ and $\theta_2$, respectively. Thus, $\theta=(\theta_1,\theta_2)$. We generate the recall probabilities through the multinomial logistic model, $\log\Big(\pi_{\eta}^{(k)}(u)/\pi_{\eta}^{(0)}(u)\Big)=\alpha_k+\beta_ku$, $k=1,2,3$. Since $\sum_{k=0}^{3}\pi_{\eta}^{(k)}(u)=1,$ the probabilities can be written as $$\begin{array}{l}
\pi_{\eta}^{(0)}(u)=1/\big(1+\sum_{k=1}^{3}e^{\alpha_k+\beta_ku}\big),\\
\pi_{\eta}^{(k)}(u)=e^{\alpha_k+\beta_ku}/\left(1+\sum_{k=1}^{3}e^{\alpha_k+\beta_ku}\right), \quad \ k=1,2,3,
\end{array} \label{pifuncfinal}$$ where $\eta=(\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\beta_3)$. Further, we generate the ‘age at interview’ from the discrete uniform distribution over \[8,21\].
We use the following sets of values of the parameters.
1. $\theta=(10,12)$ and $\eta=(-0.05,-0.05,-0.05,0.01,0.01,0.01)$,
2. $\theta=(10,12)$ and $\eta=(-2,-1,-0.4,0.05,0.3,0.02)$,
3. $\theta=(10,12)$ and $\eta=(-2,-0.7,-1,0.5,0.06,0.2)$,
4. $\theta=(10,12)$ and $\eta=(-2,-2,-2,0.3,0.08,0.08)$.
Note that for the chosen value of $\theta$, the median of the Weibull distribution turns out to be 11.6, which is in line with the median estimated from the data described in Section \[intro\] under a simpler model [@Mirzaei_2015]. Also, the chosen values of $\eta$ correspond to the following probabilities of different types of recall, five years after the event.
1. $\pi_{\eta}^{(0)}(5)=\pi_{\eta}^{(1)}(5)=\pi_{\eta}^{(2)}(5)=\pi_{\eta}^{(3)}(5)=0.25$,
2. $\pi_{\eta}^{(0)}(5)=0.28$, $\pi_{\eta}^{(1)}(5)=0.46, \pi_{\eta}^{(2)}(5)=0.21, \pi_{\eta}^{(3)}(5)=0.05$,
3. $\pi_{\eta}^{(0)}(5)=0.23$, $\pi_{\eta}^{(1)}(5)=0.15, \pi_{\eta}^{(2)}(5)=0.23, \pi_{\eta}^{(3)}(5)=0.38$,
4. $\pi_{\eta}^{(0)}(5)=0.5$, $\pi_{\eta}^{(1)}(5)=0.1, \pi_{\eta}^{(2)}(5)=0.1, \pi_{\eta}^{(3)}(5)=0.3$.
Choice (iv) is meant to favour the Binary Recall MLE, as the chances of partial recall are slim. Choice (ii) should favour the Partial Recall MLE. Choice (iii), with a high probability attached to ‘no recall’, gives Current Status MLE its best chance. Choice (i) does not favour any single method.
While computing the Binary Recall MLE, we assume the following form of the non-recall probability function $\pi_\eta$: $$\log\Big(\pi_\eta(u)/1-\pi_\eta(u)\Big)=\alpha+\beta u.$$
We run 1000 simulations for each of the above combinations of parameters, for sample size $n=100,300,1000$, to compute the empirical bias, the standard deviation (Stdev) and the mean squared error (MSE) for the MLEs of the parameter $\theta=(\theta_1,\theta_2)$, the median time-to-event, and $\pi_{\eta}^{(0)}(5)$ (the exact recall probability 5 years after the event), based on the three likelihoods. These indicators of performance, for the combinations of parameter values given in case (i) to case (iv), are reported in Table \[t:tableone\] for $n=100$.
In cases (i)–(iii), it is found that the bias and the standard deviation (and consequently the MSE) of the Partial Recall MLE is generally less than (and sometimes comparable to) those of the other two estimators and its performance improves with increasing sample size. The Current Status MLE, which uses the least amount of information from the data, has the poorest performance even in case (iii), where a substantial proportion of the subjects are designed to have no recollection of the event date. The substantial gap between the performance of the Binary Recall MLE and the Partial Recall MLE shows that the later estimator is able to utilize the additional information available from partial recall data. Similar tables for $n=300$ and 1000 are given in the supplementary material, to save space. The conclusions are similar, though all the methods perform better when the sample size increases.
For sensitivity analysis, we consider the following mixture model for the time-to-event distribution $$\gamma \log \mbox{Normal} + (1- \gamma) \mbox{Weibull},$$ with the parameters of $\log$ Normal $(\mu=2.45, \sigma^2=0.07)$ and $\gamma=0.2$ and 0.5. Note that for the chosen values of $\mu$ and $\sigma^2$, the median of the time-to-event distribution remains 11.6. The rest of the simulation set-up also remain the same as before. The sensitivity analysis is done for the sample size of $n=300$ with 1000 simulations runs, under the assumption $\gamma=0$, and reported in the supplementary material. The summary of the findings is that the miss-specification does not alter the relative order of the performances of the three estimators when $\gamma=0.2$. When $\gamma=0.5$, Partial Recall MLE has smaller MSE than Binary Recall MLE, as before, but both of these estimators are outperformed by the Current Status MLE.
Non-parametric estimation {#nonparsim}
-------------------------
We generate sample times-to-event ($T$) from the Weibull distribution with shape and scale parameters $\theta=(10,12)$ as before, but truncate the generated samples to the interval \[8,16\]. This truncated distribution has median of 11.6. The corresponding ‘time of interview’ ($S$) is generated from the discrete uniform distribution over $\{8,\ldots,21\}$. These choices are in line with the data set described in Section \[intro\], and lead to about 29% cases of no-occurrence of the event till the time of interview ($S<T$). As for the recall probabilities, we use with $L=4$, $x_1=0$, $x_2=3$, $x_3=6$, $x_4=9$ and three sets of values of the parameters, described bellow.
Case (a) $b_0=(0.15,0.10,0.08,0.05)$, $b_1=(0.28,0.2,0.15,0.1)$, $b_2=(0.22$, $0.25,0.17,0.1)$, $b_3=(0.35,0.45,0.6,0.75)$, which correspond to overall probabilities of exact recall $E[\pi^{(0)}(S{-}T)|S{>}T]=0.10$, recall up to calendar month $E[\pi^{(1)}(S{-}T)|S{>}T]=0.20$, recall up to calendar year $E[\pi^{(2)}(S{-}T)|S{>}T]=0.20$ and no recall $E[\pi^{(3)}(S{-}T)|S{>}T]=0.50$.
Case (b) $b_0=(0.69,0.55,0.49,0.31)$, $b_1=b_2=(0.08,0.05,0.03,0.02)$, $b_3=(0.15,0.35,0.45,0.65)$ which correspond to overall exact recall probability 0.55, calendar month recall probability 0.05, calendar year recall probability 0.05 and no-recall probability 0.35.
Case (c) $b_0=1-(b_1+b_2+b_3)$, $b_1=b_2=b_3=(0.25,0.25,0.25,0.25)$, which correspond to equal probability (0.25) of each type of recall.
It has been observed by [@Mirzaei_2016] that in the special case of binary recall, the performances of AMLE and NPMLE are comparable. Therefore, we choose not to run simulation for NPMLE, which involves heavier computation. Instead, we compare the performance of the AMLE estimated from (described here as Partial Recall AMLE) with those of the AMLE based on , proposed by [@Mirzaei_2016] (described here as Binary Recall AMLE), and the empirical estimate of $F$ (described here as EDF). The EDF is used only as a hypothetical benchmark of performance that could have been achieved with complete data.
The Partial Recall AMLE is implemented by using the correct value of $L,x_1,x_2,\ldots$, $x_L$ in , while the likelihood is recursively maximized alternately with respect to the probability parameter $p^*$ and the nuisance parameter $\eta=(b_0,b_1,b_2,b_3)^T$.
Figure \[fig\_2\] shows plots of the bias, the variance and the mean square error (MSE) of the three estimators for different ages, when $n=100$ and parameters of the recall functions are chosen as in Cases (a), (b) and (c). The Partial Recall AMLE is found to have smaller bias, variance and MSE than the Binary Recall AMLE, although its performance is expectedly poorer than that of the EDF.
Plots similar to Figure \[fig\_2\] for $n=300$ and 1000 are given in the supplementary material. At those sample sizes, the performance parameters of partial AMLE are found to be closer to those of EDF than those of binary AMLE.
![Comparison of bias, variance and MSE of the estimator for $n=100$ in cases (a) (top panel), (b) (middle panel) and (c) (bottom panel)[]{data-label="fig_2"}](BVM100.pdf){height="8in" width="7.5in"}
Adequacy of the Model {#adequacy}
=====================
One can use the chi-square goodness of fit test to check how well the assumed parametric model actually fits the data. For this purpose, the data may be transformed to the vector $Y=(S,V,\varepsilon,\delta,m,d)$, and the support of the distribution of this vector may be appropriately partitioned, depending on the availability of data. An example is given in the next section.
Modeling of the recall probability functions is a critical issue. One has to choose suitable functional forms, and also strike a balance between a flexible model and a parsimonious one with fewer parameters. We provide below an exploratory technique for selecting the functional forms.
As we have seen in Section \[nonparametric\], use of the piecewise constant form of the recall probabilities reduces the likelihood (\[ourM1\]) to the likelihood . If the distribution of $T$ is known, one can obtain the MLE of the parameters $ b_{l1}, b_{l2} ,\ldots , b_{lk},\ l=0,1,2,3$. The conditional MLE of the piecewise constant functions $\pi_{}^{(1)},\pi_{}^{(2)},\pi_{}^{(3)}$ and $\pi_{}^{(0)}$, for any given $F_\theta$ can be obtained iteratively. By using a candidate parametric form $\pi_{\eta}^{(1)},\pi_{\eta}^{(2)},\pi_{\eta}^{(3)}$ and $\pi_{\eta}^{(0)}$, one can first estimate the MLEs $\hat{\theta}$ and $\hat\eta$ and then compare the plots of $\hat\pi_{\eta}^{(1)},\hat\pi_{\eta}^{(2)},\hat\pi_{\eta}^{(3)}$ and $\hat\pi_{\eta}^{(0)}$ with the plots of the conditional MLE of the piecewise constant versions of $\pi_{}^{(1)},\pi_{}^{(2)},\pi_{}^{(3)}$ and $\pi_{}^{(0)}$, with $F_\theta$ held fixed at $F_{\hat{\theta}}$. An example of this graphical check is given in the next section.
In addition, comparative plots of $F_{\hat{\theta}}$ computed for an assumed form of the recall probability functions and the piecewise constant forms mentioned in Section \[NPMLE\], can also serve as a graphical check of the adequacy of that assumed form. An example of this graphical check for the data set of next section is given in the supplementary material.
Data Analysis {#DataAnalysis}
=============
For the data set described in Section \[intro\], the landmark event is the onset of menarche in young and adolescent females. In a parametric analysis, we used the Weibull model for menarcheal age and the multinomial logistic model for the recall probabilities $\pi_{\eta}^{(0)}, \pi_{\eta}^{(1)}, \pi_{\eta}^{(2)}$ and $\pi_{\eta}^{(3)}$, as in Section \[parsim\]. We used the three different methods mentioned in Section \[parsim\] for estimating the parameters $\theta_1$ and $\theta_2$ as well as the median of the age at menarche. Table \[t:tabletwo\] gives a summary of the findings. The Partial Recall MLEs have smaller standard errors than those of the other two estimators.
\~
Figure \[fig\_3\] shows the survival functions estimated from the three parametric methods, the Partial Recall AMLE presented in Section \[Simple\] (with knot points of the recall probability functions chosen as in the first paragraph of Section \[nonparsim\]) and Binary Recall AMLE (with the same knot points). The parametric MLEs are not very far from the non-parametric AMLEs. Though there appears to be little difference between the Partial Recall and Binary Recall MLEs, their standard errors are different (check Figure 3 of supplementary material).
In order to formally check how well the assumed parametric model fits the data, we use the chi-square goodness of fit test, by discretizing the range of the hexatuple $(S,V,\varepsilon,\delta,m,d)$. The range of $S$ is split into the intervals $[7,14]$ and $(14,21]$, the range of $d$ is split into the intervals $[0,1/24]$ and $(1/24,1/12]$, while the range of $V$ is split into the sets $[0,11.84]$ and $(11.84,21]$ ($11.84$ being the median of the observed non-zero values of $V$). The ranges of $\varepsilon$ and $\delta$ have four points ($0$, $1$, $2$ and $3$) and two points ($0$ and $1$), respectively, none of which are clubbed. The range of $m$ is the interval $[0,11]$, which is not split. When $\delta=0$, the value of $\varepsilon$ is irrelevant and $V=0$, i.e., there are four bins for the two groups of values of $S$ and two groups of $d$. When $\delta=1$ and $\varepsilon=3$, $V$ can only be zero and again there are only four bins. When $\delta=1$ and $\varepsilon=0,1$ or 2, in each case there are eight bins arising from two groups of values of $S$ and two groups of non-zero values of $V$ and $d$. Thus, we have a total of 32 bins.
In order to avoid small expected frequency in some cells we merge some bins where expected frequency is less than $5$. After this pruning, we have a reduced total of 21 bins. There are 8 parameters to estimate. Thus, the null distribution should be $\chi^2$ with 12 degrees of freedom. The p–value of the test statistic for the given data happens to be 0.169. Therefore, violation of the chosen model is not indicated.
We now check the adequacy of the functional form of the $\pi_{\eta}^{(l)}$s by comparing the $\pi_{\hat\eta}^{(k)}$s with the conditional MLE of the corresponding piecewise constant function in , as indicated in the last section. For the given data, the largest value of $S_i-T_i$ in a perfectly recalled case happens to be 10.88 years. Therefore, we consider recall functions over the interval 0 to 12 years. With $F$ chosen as Weibull and $\theta_1$ and $\theta_2$ fixed at the values reported in the last row of Table \[t:tabletwo\], we obtained the conditional MLE of the values of $\pi_{\eta}^{(0)}$, $\pi_{\eta}^{(1)}$, $\pi_{\eta}^{(2)},\pi_{\eta}^{(3)}$ in different segments of equal length. Figure \[fig\_4\](a) shows the plots of the estimated recall probabilities under the logistic and the piecewise constant models, with number of segments $L=4$. The estimated functions are found to be close to each other for $l=0,1,2,3$. Figure \[fig\_4\](b) shows the same plots for $L=8$. The finer partition seems unnecessary. As another check of the functional form of the recall probability, we estimated the survival functions of time-to-event from the proposed parametric method using the multiple logistic regression model presented in Section \[parsim\] and the piecewise constant recall probability model introduced in Section \[NPMLE\] (with knot points of the recall probability functions chosen as in the first paragraph of Section \[nonparsim\]). Figure 4 of supplementary material shows the two estimates of the survival function, which happen to be very close to each other.
We have seen the cumulative proportions of decreasing degrees of recall for different age ranges in the case of the menarcheal data in Figure \[fig\_1\]. As an additional check for the assumed model, we consider the model based estimates of these cumulative proportions for ages $s=11,14,17$ and 20 (i.e., at the middle of the respective age intervals). We used the Partial Recall MLE of parameters $\hat\theta$ and $\hat\eta$ to calculate $f_{\hat\theta}$ and $\pi_{\hat\eta}^{(j)}$ for $j=0,1,2,3$ and then computed the requisite probabilities through numerical integration. Figure \[fig\_5\] shows the cumulative proportions in different age groups (solid lines) along with the corresponding model based estimates (dashed lines). The estimated probabilities are quite close to the empirical proportions.
Concluding Remarks {#s:discuss}
==================
The aim of this paper has been to offer a realistic model for time-to-event based on partial recall information through an informative censoring model, where the range of relevant dates may depend on calendar time (rather than time elapsed since the event). The simulations and the data analysis of the menarcheal data set show that there is much to be gained from partial recall information in the form of the event falling in a calendar month or a calendar year. Many other forms of partial recall information may be handled in a similar way. As the simulations reported in Section \[Simulation\] show, a particular category of partial recall (eg. recall up to a calendar month or year) is justified if that category is not very rare in the data.
The recalled time-to-event can sometimes be erroneous. Grouping of the uncertainly recalled event date by the calendar month or year may reduce the error to some extent. If one adopts this solution, the method presented in this paper provides a viable method of analysis. [@Skinner_1999], while working with data without instances of non-recall, has modeled erroneously recalled time-to-event as $t'_i=t_ik_i$, where $t_i$ is the correct time-to-event and $k_i$ is a multiplicative error of recall that is independent of $t_i$. Since $k_i$s are unobservable, they have used a mixed-effects regression model to account for erroneous recalls. One may investigate whether a similar adjustment in the term $f_{\theta}(T_i)$ of the likelihood , improves the analysis.
The Cox regression model has been adapted to the retrospective recall model for binary recall data [@MirSen_2014], and an adaptation to partial recall would be interesting. The multiple logistic regression model provides a framework for incorporating covariate effect on the recall probabilities also. These problems will be taken up in future.
Software {#sec9}
========
Software in the form of R code, together with the data set and complete documentation is available at GitHub (<https://github.com/rahulfrodo/PartialRecall>).
Supplementary Material {#sec10}
======================
Supplementary material is available online at <http://biostatistics.oxfordjournals.org>.
Acknowledgements
================
This research is partially sponsored by the project “Physical growth, body composition and nutritional status of the Bengal school aged children, adolscents, and young adults of Calcutta, India: Effects of socioeconomic factors on secular trends”, funded by the Neys Van Hoogstraten Foundation of the Netherlands. The authors thank Professor Parasmani Dasgupta of the Biological Anthropology Unit of ISI, for making the data available for this research. The authors thank an anonymous referee and an associate editor for suggesting useful changes that improved the content and the presentation of the paper.
|
---
abstract: 'In this paper we consider a class of connected closed $G$-manifolds with a non-empty finite fixed point set, each $M$ of which is totally non-homologous to zero in $M_G$ (or $G$-equivariantly formal), where $G={\Bbb Z}_2$. With the help of the equivariant index, we give an explicit description of the equivariant cohomology of such a $G$-manifold in terms of algebra, so that we can obtain analytic descriptions of ring isomorphisms among equivariant cohomology rings of such $G$-manifolds, and a necessary and sufficient condition that the equivariant cohomology rings of such two $G$-manifolds are isomorphic. This also leads us to analyze how many there are equivariant cohomology rings up to isomorphism for such $G$-manifolds in 2- and 3-dimensional cases.'
address:
- 'School of Mathematical Science, Fudan University, Shanghai, 200433, People’s Republic of China.'
- 'Institute of Mathematics, School of Mathematical Science, Fudan University, Shanghai, 200433, People’s Republic of China.'
author:
- Bo Chen and Zhi Lü
title: ' **Equivariant cohomology and analytic descriptions of ring isomorphisms** '
---
\[section\] \[section\] \[section\] \[section\] \[section\]
\[thm\][Definition]{} \[thm\][Notation]{} \[thm\][Example]{} \[thm\][Conjecture]{} \[thm\][Problem]{}
\#1 \#1 \#1 \#1 \#1 \#1 \#1 \#1 \#1 \#1
Introduction
============
Throughout this paper, assume that $G={\Bbb Z}_2$ unless stated otherwise. Let $EG\longrightarrow BG$ be the universal principal $G$-bundle, where $BG=EG/G={\Bbb R}P^\infty$ is the classifying space of $G$. It is well-known that $H^*(BG;{\Bbb Z}_2)={\Bbb
Z}_2[t]$ with the $t$ one-dimensional generator.
.2cm
Let $X$ be a $G$-space. Then $X_G:=EG\times_GX$—–the orbit space of the diagonal action on the product $EG\times X$—–is the total space of the bundle $X\longrightarrow X_G\longrightarrow BG$ associated to the universal principal bundle $G\longrightarrow
EG\longrightarrow BG$. The space $X_G=EG\times_GX$ is called the [*Borel construction*]{} on the $G$-space $X$. Applying cohomology with coefficients ${\Bbb Z}_2$ to $X_G$ gives the [*equivariant cohomology*]{} $H^*_G(X;{\Bbb Z}_2):=H^*(X_G;{\Bbb Z}_2).$ It is well-known that equivariant cohomologies $H^*_G(X;{\Bbb Z}_2)$ and $H^*_G(X^G;{\Bbb Z}_2)$ are $H^*(BG;{\Bbb Z}_2)$-modules; in particular, $H^*_G(X^G;{\Bbb Z}_2)$ is a free $H^*(BG;{\Bbb
Z}_2)$-module.
.2cm
Suppose that $M$ is a connected closed manifold and admits a $G$-action with $M^G$ a non-empty finite fixed set. For the fibration $M\longrightarrow M_G\longrightarrow BG$, if the restriction to a typical fiber $$H^*_G(M;{\Bbb Z}_2)\longrightarrow
H^*(M;{\Bbb Z}_2)$$ is an epimorphism, then $M$ is called [*totally non-homologous to zero*]{} in $M_G$ (see [@b]). Under this condition, $M$ is also called [*$G$-equivariantly formal*]{} (cf. [@gkm]). Note that when the action group on $M$ is replaced by a 2-torus $({\Bbb Z}_2)^k$ with $k\geq 1$ and each component of $M^K$ has dimension at most 1 for $K< ({\Bbb Z}_2)^k$ a corank-1 2-torus, if $M$ is $({\Bbb Z}_2)^k$-equivariantly formal (i.e., the edge morphism $H^*_{({\Bbb Z}_2)^k}(M;{\Bbb Z}_2)\longrightarrow
H^*(M;{\Bbb Z}_2)$ is surjective), then there is a mod 2 GKM theory (corresponding to GKM theory, see [@gkm], [@gz]), indicating that the $({\Bbb Z}_2)^k$-equivariant cohomology of $M$ can be explicitly expressed in terms of its associated graph $(\Gamma_M,
\alpha)$ (cf. [@bgh], [@l]). In particular, when $k=1$, $M$ is naturally restricted to have dimension at most 1, so this means that the GKM theory can be carried out only for at most 1-dimensional $G$-equivariantly formal manifolds. In this paper, we shall give explicit descriptions of equivariant cohomology rings of $G$-equivariantly formal manifolds at any dimension, and these descriptions are of algebra rather than combinatorics.
.2cm
Let $\Lambda_n$ denote the set of all $n$-dimensional connected closed $G$-manifolds with a non-empty finite fixed point set, each of which is $G$-equivariantly formal. Note that obviously $\Lambda_1$ contains a unique 1-dimensional closed manifold, i.e., a circle $S^1$. Taking a $M$ in $\Lambda_n$, by Conner and Floyd [@cf], one knows that $|M^G|$ must be even. Let $r$ be a positive integer, and write $\Lambda_n^{2r}=\big\{M\in\Lambda_n\big|
|M^G|=2r\big\}$. Then $\Lambda_n=\bigcup\limits_{r\geq
1}\Lambda_n^{2r}$. Given a $M$ in $\Lambda_n$, we know from [@ap] and [@b] that the following conditions are equivalent
1. $M$ is $G$-equivariantly formal;
2. $|M^G|=\sum\limits_{i=0}^nb_i$ where $b_i$ is the $i$-th mod 2 Betti number of $M$;
3. $H_G^*(M;{\Bbb Z}_2)$ is a free $H^*(BG;{\Bbb Z}_2)$-module;
4. The inclusion $i:M^G\hookrightarrow M$ induces a monomorphism $i^*: H_G^*(M;{\Bbb Z}_2)\longrightarrow
H_G^*(M^G;{\Bbb Z}_2)$.
If $|M^G|=2r$, since $H^*_G(M^G;{\Bbb Z}_2)=\bigoplus\limits_{p\in
M^G}H^*_G(\{p\};{\Bbb Z}_2)$ and the equivariant cohomology of a point is isomorphic to $H^*(BG;{\Bbb Z}_2)={\Bbb Z}_2[t]$, we have that $H_G^*(M^G;{\Bbb Z}_2)\cong ({\Bbb Z}_2)^{2r}[t]$ is a polynomial ring (or algebra). Thus we obtain a monomorphism from $H_G^*(M;{\Bbb Z}_2)$ into $({\Bbb Z}_2)^{2r}[t]$, also denoted by $i^*$, so $H_G^*(M;{\Bbb Z}_2)$ may be identified with a subring (or subalgebra) of $({\Bbb Z}_2)^{2r}[t]$.
.2cm
Using the equivariant index, we shall give an explicit description of $H_G^*(M; {\Bbb Z}_2)$ in $({\Bbb Z}_2)^{2r}[t]$ (see Theorem \[ring str\]). Then we consider the following questions:
1. If the equivariant cohomology rings of two $G$-manifolds in $\Lambda_n$ are isomorphic, then can the isomorphism between them be explicitly expressed?
2. When are the equivariant cohomology rings of two $G$-manifolds in $\Lambda_n
$ isomorphic?
3. How many are there equivariant cohomology rings (up to isomorphism) of $G$-manifolds in $\Lambda_n$?
We completely answer (Q1) and (Q2). An interesting thing is that we do not only find an explicit description for the isomorphism between equivariant cohomology rings of two $G$-manifolds in $\Lambda_n$, but such a description is also analytic (see Theorem \[analyze\]), so that we may obtain a necessary and sufficient condition that the equivariant cohomology rings of such two $G$-manifolds are isomorphic in terms of algebra (see Theorem \[ns\]). As for (Q3), the question is answered completely in the case $n=2$. When $n=3$, we find an upper bound of the number for the equivariant cohomology rings (up to isomorphism) of $G$-manifolds in $\Lambda_3^{2r}$ (see Proposition \[number\]).
.2cm
The paper is organized as follows. In Section 2 we review the localization theorem and reformulate the equivariant index from the work of Allday and Puppe [@ap]. In Section 3 we study the structure of equivariant cohomology of a $G$-manifold in $\Lambda_n$ and obtain an explicit description in terms of algebra. Then we completely answer (Q2) and (Q3) in the case $n=2$ in Section 4. In Section 5, we give an analytic description for the isomorphism between equivariant cohomology rings of two $G$-manifolds in $\Lambda_n$, so that we may obtain a necessary and sufficient condition that the equivariant cohomology rings of two $G$-manifolds in $\Lambda_n$ are isomorphic. In section 6, we discuss the number of the equivariant cohomology rings (up to isomorphism) of $G$-manifolds in $\Lambda_3^{2r}$, and obtain an upper bound of the number.
.2cm The authors express their thanks to Shengzhi Xu for helpful conversation in the argument of Lemma \[bound\]. The authors also would like to express their gratitude to Professor Volker Puppe, who informed of us that there is an essential relationship between the equivariant cohomology rings and the coding theory (see [@p]), and Theorem 3 in [@kp] implies that the map $g$ is surjective in the remark \[vp\] of our paper.
Localization theorem and equivariant index
==========================================
Suppose that $M$ is an $n$-dimensional $G$-manifold with $M^G$ a non-empty finite set. Let $S$ be the subset of $H^*(BG;{\Bbb Z}_2)$ generated multiplicatively by nonzero elements in $H^1(BG;{\Bbb
Z}_2)$. Then one has the following well-known localization theorem (see [@ap], [@h]).
\[Localization theorem\]\[local\] $$S^{-1}i^*:
S^{-1}H^*_G(M;{\Bbb Z}_2)\longrightarrow S^{-1}H^*_G(M^G;{\Bbb
Z}_2)$$ is an isomorphism of $S^{-1}H^*(BG;{\Bbb Z}_2)$-algebras, where $i$ is the inclusion of from $M^G$ into $M$.
Take an isolated point $p\in M^G$. Let $i_p$ be the inclusion of from $p$ into $M$, then one has the equivariant Gysin homomorphism $$i_{p!}: H^*_G(\{p\};{\Bbb Z}_2)\longrightarrow H^{*+n}_G(M;{\Bbb
Z}_2).$$ On the other hand, one has also a natural induced homomorphism $$i^*_p: H^*_G(M;{\Bbb Z}_2)\longrightarrow H^*_G(\{p\};{\Bbb
Z}_2)$$ and in particular, it is easy to check that $i^*=\bigoplus\limits_{p\in M^G}i_p^*$. Furthermore, one knows that the equivariant Euler class at $p$ is $$\chi_G(p)=i^*_pi_{p!}(1_p)\in H^n_G(\{p\};{\Bbb Z}_2)= H^n(BG;{\Bbb Z}_2)={\Bbb Z}_2t^n,$$ which is equal to that of the real $G$-representation at $p$, where $1_p\in H^*_G(\{p\};{\Bbb Z}_2)$ is the identity and ${\Bbb
Z}_2t^n=\{at^n|a\in{\Bbb Z}_2\}$. Thus, we may write $\chi_G(p)=t^n$. Write $\theta_p=i_{p!}(1_p)$. Then $\theta_p\in
H^n_G(M;{\Bbb Z}_2)$ and $i^*_p(\theta_p)=\chi_G(p)$.
\[in\] All elements $\theta_p, p\in M^G$ are linearly independent over $H^*(BG;{\Bbb Z}_2)$.
Let $\sum_{p\in M^G}l_p\theta_p=0$, where $l_p\in H^*(BG;{\Bbb
Z}_2)$. From [@ap Proposition 5.3.14(2)], one knows that $i^*_q(\theta_p)=0$ for $q\not=p$ in $M^G$, so $$i^*_q(\sum_{p\in M^G}l_p\theta_p)=\sum_{p\in
M^G}l_pi^*_q(\theta_p)=l_qi^*_q(\theta_q)=l_q\chi_G(q)=0.$$ Since $\chi_G(q)=t^n$ is a unit in $S^{-1}H^*_G(\{q\};{\Bbb Z}_2)\cong
S^{-1}H^*(BG;{\Bbb Z}_2)$, one has $l_q=0$.
\[l1\] Let $\alpha\in S^{-1}H^*_G(M;{\Bbb Z}_2)$. Then $$\alpha=\sum_{p\in M^G}{{f_p\theta_p}\over{t^n}}$$ where $f_p=S^{-1}i^*_p(\alpha)\in S^{-1}H^*(BG;{\Bbb Z}_2)$.
By [@ap Proposition 5.3.18(1)], one has that $$\alpha=\sum_{p\in M^G}S^{-1}i_{p!}(S^{-1}i^*_p(\alpha)/\chi_G(p)).$$ Since $f_p=S^{-1}i^*_p(\alpha)\in S^{-1}H^*(BG;{\Bbb Z}_2)$, one has that ${{f_p}\over{\chi_G(p)}}={{f_p}\over{t^n}}\in
S^{-1}H^*(BG;{\Bbb Z}_2)$. Since $S^{-1}i_{p!}$ is a $S^{-1}H^*(BG;{\Bbb Z}_2)$-algebra homomorphism, one has $$S^{-1}i_{p!}(S^{-1}i^*_p(\alpha)/\chi_G(p))=S^{-1}i_{p!}({{f_p}\over{t^n}})={{f_p}\over{t^n}}S^{-1}
i_{p!}(1_p)={{f_p}\over{t^n}}i_{p!}(1_p)={{f_p\theta_p}\over{t^n}}$$ so $\alpha=\sum\limits_{p\in M^G}{{f_p\theta_p}\over{t^n}}.$
\[loc\] (i) By Lemma \[in\], one sees from the formula of Lemma \[l1\] that $\{{{\theta_p}\over {t^n}}|p\in M^G\}$ forms a basis of $S^{-1}H^*_G(M;{\Bbb Z}_2)$ as a $S^{-1}H^*(BG;{\Bbb Z}_2)$-algebra.
.2cm (ii) In some sense, the formula $\alpha=\sum\limits_{p\in M^G}{{f_p\theta_p}\over{t^n}}$ explicitly indicates the isomorphism $S^{-1}i^*:S^{-1}H^*_G(M;{\Bbb Z}_2)\longrightarrow
S^{-1}H^*_G(M^G;{\Bbb Z}_2)$ in Theorem \[local\], which is given by mapping $\alpha=\sum\limits_{p\in M^G}{{f_p\theta_p}\over{t^n}}$ to $\bigoplus\limits_{p\in M^G}{{f_p}\over{t^n}}$.
The equivariant Gysin homomorphism of collapsing $M$ to a point gives the $G$-index of $M$, i.e., $$\text{Ind}_G: H^*_G(M;{\Bbb Z}_2)\longrightarrow H^{*-n}(BG;{\Bbb
Z}_2).$$
\[formula\] For any $\alpha\in S^{-1}H^*_G(M;{\Bbb Z}_2)$, $$S^{-1}\text{\rm Ind}_G(\alpha)=\sum_{p\in M^G}{{f_p}\over{t^n}}$$ where $f_p=S^{-1}i^*_p(\alpha)\in S^{-1}H^*(BG;{\Bbb Z}_2)$. In particular, if $\alpha\in H^*_G(M;{\Bbb Z}_2)$, then $f_p=i^*_p(\alpha)\in H^*(BG;{\Bbb Z}_2)$ and $$\label{ind}
\text{\rm Ind}_G(\alpha)=\sum_{p\in M^G}{{f_p}\over{t^n}}\in
H^*(BG;{\Bbb Z}_2)={\Bbb Z}_2[t].$$
By [@ap Lemma 5.3.19], one has that $\text{Ind}_G(\theta_p)=1_p$, so by Lemma \[l1\] $$S^{-1}\text{Ind}_G(\alpha)=\sum_{p\in
M^G}{{f_pS^{-1}\text{Ind}_G(\theta_p)}\over{t^n}}=\sum_{p\in
M^G}{{f_p\cdot 1_p}\over{t^n}}=\sum_{p\in M^G}{{f_p}\over{t^n}}.$$ The last part of Theorem \[formula\] follows immediately since $H^*(BG;{\Bbb Z}_2)\longrightarrow S^{-1}H^*(BG;{\Bbb Z}_2)$ is injective.
a\) It should be pointed out that all arguments in this section can still be carried out if the action group $G$ is a 2-torus $({\Bbb Z}_2)^k$ of rank $k>1$. In this case, $H^*(BG;{\Bbb
Z}_2)$ is a polynomial algebra ${\Bbb Z}_2[t_1,...,t_k]$ where the $t_i$’s are one-dimensional generators in $H^1(BG;{\Bbb Z}_2)$, so that the formula (\[ind\]) becomes $$\label{ind1}
\text{Ind}_G(\alpha)=\sum_{p\in M^G}{{f_p}\over{\chi_G(p)}}\in {\Bbb
Z}_2[t_1,...,t_k]$$ where $\chi_G(p)$ is a polynomial of degree $n$ in ${\Bbb Z}_2[t_1,...,t_k]$. Note that related results can also be found in [@d] and [@ks].
b\) The formula (\[ind1\]) is an analogue of the Atiyah-Bott-Berlin-Vergne formula for the case $G=T$ (i.e., a torus), see [@ab] and [@bv].
Equivariant cohomology structure
================================
In this section, our task is to study the structures of equivariant cohomology rings of $G$-manifolds in $\Lambda_n$.
\[dim1\] Let $M\in \Lambda_n^{2r}$. Then $$\dim_{{\Bbb Z}_2}H^i_G(M;{\Bbb Z}_2)=\begin{cases}
\sum\limits_{j=0}^ib_j & \text{ if $i\leq n-1$}\\
2r & \text{ if $i\geq n$.}
\end{cases}$$
Let $$P_s(M_G)=\sum_{i=0}^{\infty}\dim_{{\Bbb Z}_2}H^i_G(M;{\Bbb
Z}_2)s^i$$ be the equivariant Poincaré polynomial of $H^*_G(X;{\Bbb Z}_2)$. Since $H^*_G(M;{\Bbb Z}_2)$ is a free $H^*(BG;{\Bbb Z}_2)$-module, one has that $H^*_G(M;{\Bbb
Z}_2)=H^*(M;{\Bbb Z}_2)\otimes_{{\Bbb Z}_2}H^*(BG;{\Bbb Z}_2)$ so $$P_s(M_G)=\sum_{i=0}^{\infty}\dim_{{\Bbb Z}_2}H^i_G(M;{\Bbb Z}_2)s^i={1\over {1-s}}\sum_{i=0}^n\dim_{{\Bbb
Z}_2}H^i(M;{\Bbb Z}_2)s^i.$$ Write $b_i=\dim_{{\Bbb Z}_2}H^i(M;{\Bbb
Z}_2)$. Note that $b_i=b_{n-i}$ by Poincaré duality and $b_0=b_n=1$ since $M$ is connected. Then $$\begin{aligned}
& \quad P_s(M_G)=\sum\limits_{i=0}^{\infty}\dim_{{\Bbb
Z}_2}H^i_G(M;{\Bbb Z}_2)s^i={1\over
{1-s}}\sum\limits_{i=0}^n\dim_{{\Bbb
Z}_2}H^i(M;{\Bbb Z}_2)s^i\\
&= b_0+(b_0+b_1)s+\cdots+
(b_0+b_1+\cdots+b_{n-1})s^{n-1}+(b_0+b_1+\cdots+b_n)(s^n+\cdots)
\\
&=b_0+(b_0+b_1)s+\cdots+
(b_0+b_1+\cdots+b_{n-1})s^{n-1}+2r(s^n+\cdots).\end{aligned}$$ The lemma then follows from this.
Let $x=(x_1,...,x_{2r})$ and $y=(y_1,...,y_{2r})$ be two vectors in $({\Bbb Z}_2)^{2r}$. Define $x\circ y$ by $$x\circ y=(x_1y_1,...,x_{2r}y_{2r}).$$ Then $({\Bbb Z}_2)^{2r}$ forms a commutative ring with respect to two operations $+$ and $\circ$. Let $$\mathcal{V}_{2r}=\big\{x=(x_1,...,x_{2r})^\top\in ({\Bbb
Z}_2)^{2r}\big| |x|=\sum\limits_{i=1}^{2r}x_i=0\big\}.$$ Then it is easy to see that $\mathcal{V}_{2r}$ is a $(2r-1)$-dimensional subspace of $({\Bbb Z}_2)^{2r}$, and there is only such a subspace in $({\Bbb Z}_2)^{2r}$. However, generally the operation $\circ$ in $\mathcal{V}_{2r}$ is obviously not closed.
.2cm
Given a $M\in \Lambda_n^{2r}$, one then has that the inclusion $i:
M^G\hookrightarrow M$ induces a monomorphism $$i^*: H_G^*(M;{\Bbb Z}_2)\longrightarrow ({\Bbb Z}_2)^{2r}[t].$$ By Lemma \[dim1\], there are subspaces $V^M_i$ with $\dim
V^M_i=\sum\limits_{j=0}^ib_j$ $ (i=0,..., n-1)$ of $({\Bbb
Z}_2)^{2r}$ such that $$H^i_G(M;{\Bbb Z}_2)\cong i^*(H^i_G(M;{\Bbb Z}_2))=\begin{cases}
V_i^Mt^i & \text{ if } i\leq n-1\\
({\Bbb Z}_2)^{2r}t^i & \text{ if } i\geq n \end{cases}$$ where $V_i^Mt^i=\{vt^i| v\in V_i^M\}$.
\[p\] There are the following properties:
1. ${\Bbb Z}_2\cong V^M_0\subset V_1^M\subset\cdots\subset V_{n-2}^M\subset V_{n-1}^M=\mathcal{V}_{2r}$, where $V^M_0$ is generated by $(1,...,1)^\top\in ({\Bbb Z}_2)^{2r}$;
2. For $d=\sum\limits_{i=0}^{n-1}i d_i<n$ with each $d_i\geq0$, $v_{\omega_{d_0}}\circ\cdots\circ
v_{\omega_{d_{n-1}}}\in V^M_{d}$, where $v_{\omega_{d_i}}=v^{(i)}_1\circ\cdots\circ v^{(i)}_{d_i}$ with each $v^{(i)}_j\in V^M_i$.
For an element $\alpha\in H_G^*(M;{\Bbb Z}_2)$ of degree $d$, one has that $i^*(\alpha)=vt^d$ where $v\in ({\Bbb Z}_2)^{2r}$. Since $i^*=\bigoplus\limits_{p\in M^G}i^*_p$, by Theorem \[formula\] one has that $$\text{Ind}_G(\alpha)=\sum_{p\in
M^G}{{i^*_p(\alpha)}\over{t^n}}={1\over{t^n}}\sum_{p\in
M^G}i^*_p(\alpha)=|v|t^{d-n}\in {\Bbb Z}_2[t]$$ so if $d<n$, then $|v|$ must be zero. This means that for each $i<n$, $V_i^M$ is a subspace of $\mathcal{V}_{2r}$ and $V_{n-1}^M=\mathcal{V}_{2r}$ is obvious since $\dim V_{n-1}^M=2r-1$. In particular, when $\alpha=1$ is the identity of $H_G^*(M;{\Bbb Z}_2)$, $i^*(1)=\bigoplus\limits_{p\in M^G}i_p^*(1)=(1,...,1)^\top\in ({\Bbb
Z}_2)^{2r}$. Thus, $V^M_0\cong {\Bbb Z}_2$ is generated by $(1,...,1)^\top\in ({\Bbb Z}_2)^{2r}$ since $\dim V_0^M=b_0=1$. Since $H^*_G(M;{\Bbb Z}_2)=H^*(M;{\Bbb Z}_2)\otimes_{{\Bbb
Z}_2}H^*(BG;{\Bbb Z}_2)$, one has that $(1,...,1)^\top t\in
i^*(H^1_G(M;{\Bbb Z}_2))$. Thus, for any $v\in V_i^M$ with $i<n-1$, $(vt^i)\circ[(1,...,1)^\top t]=vt^{i+1}\in V_{i+1}^Mt^{i+1}$, so one has that $v\in V_{i+1}^M$. This completes the proof of Lemma \[p\](1).
.2cm
As for the proof of Lemma \[p\](2), for each $v^{(i)}_j\in V^M_i$, since $i^*: H_G^*(M;{\Bbb Z}_2)\longrightarrow ({\Bbb Z}_2)^{2r}[t]$ is injective, there is a class $\alpha_j^{(i)}$ of degree $i$ in $H^*_G(M;{\Bbb Z}_2)$ such that $i^*(\alpha_j^{(i)})=v^{(i)}_jt^i$. Since $i^*=\bigoplus\limits_{p\in M^G}i_p^*$ is also a ring homomorphism, one has that $$i^*(\prod_{i=0}^{n-1}\prod_{j=1}^{d_i}\alpha_j^{(i)})=\bigoplus_{p\in
M^G}i_p^*(\prod_{i=0}^{n-1}\prod_{j=1}^{d_i}\alpha_j^{(i)})
=\bigoplus_{p\in
M^G}\prod_{i=0}^{n-1}\prod_{j=1}^{d_i}i_p^*(\alpha_j^{(i)})=v_{\omega_{d_0}}\circ\cdots\circ
v_{\omega_{d_{n-1}}}t^d$$ so $v_{\omega_{d_0}}\circ\cdots\circ v_{\omega_{d_{n-1}}}\in V_d^M$.
\[ring\] An easy observation shows that the properties (1) and (2) of Lemma \[p\] exactly give a subring structure of $$\mathcal{R}_M=V^M_0+V_1^Mt+\cdots+V^M_{n-1}t^{n-1}+({\Bbb Z}_2)^{2r}(t^n+\cdots)$$ in $({\Bbb Z}_2)^{2r}[t]$.
.2cm Combining Lemmas \[dim1\], \[p\], and Remark \[ring\] one has
\[ring str\] Let $M\in \Lambda_n^{2r}$. Then there are subspaces $V^M_i$ with $\dim V^M_i=\sum\limits_{j=0}^ib_j (i=0,..., n-1)$ of $\mathcal{V}_{2r}$ such that $H^*_G(M;{\Bbb Z}_2)$ is isomorphic to the graded ring $$\mathcal{R}_M=V^M_0+ V^M_1t+\cdots+
V^M_{n-2}t^{n-2}+ V^M_{n-1}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^{n}+t^{n+1}+\cdots)$$ where the ring structure of $\mathcal{R}_M$ is given by
1. ${\Bbb Z}_2\cong V^M_0\subset V_1^M\subset\cdots\subset V_{n-2}^M\subset V_{n-1}^M=\mathcal{V}_{2r}$, where $V^M_0$ is generated by $(1,...,1)^\top\in ({\Bbb Z}_2)^{2r}$;
2. For $d=\sum\limits_{i=0}^{n-1}i d_i<n$ with each $d_i\geq0$, $v_{\omega_{d_0}}\circ\cdots\circ
v_{\omega_{d_{n-1}}}\in V^M_{d}$, where $v_{\omega_{d_i}}=v^{(i)}_1\circ\cdots\circ v^{(i)}_{d_i}$ with each $v^{(i)}_j\in V^M_i$.
\[module\] Since $H^*_G(M;{\Bbb Z}_2)$ is also a free $H^*(BG;{\Bbb
Z}_2)$-module, one has that $$\mathcal{R}_M=V^M_0+ V^M_1t+\cdots+
V^M_{n-2}t^{n-2}+ V^M_{n-1}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^{n}+t^{n+1}+\cdots)$$ is a free ${\Bbb Z}_2[t]$-module, too.
Next, let us determine the largest-dimensional space $V$ in $\{V_i^M| 0\leq i\leq n-1\}$ with the property that $u\circ v\in \mathcal{V}_{2r}$ for $u,v\in V$. .2cm
Given a vector $v\in \mathcal{V}_{2r}$, set $$\mathcal{V}(v):=\{x\in \mathcal{V}_{2r}|x\circ v\in
\mathcal{V}_{2r}\}.$$ Then it is easy to see that $\mathcal{V}(v)$ is a linear subspace of $\mathcal{V}_{2r}$, and $\mathcal{V}(\underline{1}+v)=\mathcal{V}(v)$, and $\mathcal{V}(v)=\mathcal{V}_{2r}\Longleftrightarrow v=\underline{0}
\text{ or } \underline{1}$ where $\underline{0}=(0,...,0)^\top$ and $\underline{1}=(1,...,1)^\top$.
\[d1\] Let $v\in \mathcal{V}_{2r}$ with $v\neq
\underline{0},\underline{1}$. Then $\dim\mathcal{V}(v) =2r-2$.
Let $\mathcal{V}_1(v)=\{x\in \mathcal{V}_{2r}| v\circ x=x\}$ and $\mathcal{V}_2(v)=\{x\in \mathcal{V}_{2r} | v\circ x=\underline{0}
\}$. Obviously, they are subspace of $\mathcal{V}_{2r}$ and $\mathcal{V}_1(v)\cap \mathcal{V}_2(v)=\{\underline{0}\}$. For any $x\in \mathcal{V}(x)$, $x=v\circ x+(v\circ x+x)$. Since $v\circ x\in
\mathcal{V}_1(v)$ and $v\circ x+x\in \mathcal{V}_2(v)$, one has that $\mathcal{V}(v)\subset \mathcal{V}_1(v)\oplus \mathcal{V}_2(v)$. However, obviously $\mathcal{V}_1(v)\oplus \mathcal{V}_2(v)\subset
\mathcal{V}(v)$. Thus, $\mathcal{V}(v)=\mathcal{V}_1(v)\oplus
\mathcal{V}_2(v)$.
.2cm
By $\sharp v$ one denotes the number of nonzero elements in $v$. Let $\sharp v=m$. Then $0<m<2r$ since $v\neq
\underline{0},\underline{1}$, and $m$ is even since $v\in
\mathcal{V}_{2r}$. An easy observation shows that $\dim\mathcal{V}_1(v)=m-1$ and $\dim\mathcal{V}_2(v)=(2r-m)-1$. Thus, $$\dim\mathcal{V}(v)=\dim\mathcal{V}_1(v) +
\dim\mathcal{V}_2(v)=m-1+(2r-m)-1=2r-2.$$
\[d2\] Let $u, v\in \mathcal{V}_{2r}$. Then $$\mathcal{V}(u+v)=[\mathcal{V}(u)\cap \mathcal{V}(v)]\cup[\mathcal{V}_{2r}\backslash
(\mathcal{V}(u)\cup \mathcal{V}(v))].$$
For any $x\in\mathcal{V}(u+v)$, one has that $x\circ(u+v)=x\circ u+x\circ v\in \mathcal{V}_{2r}$ so $\sharp
(x\circ u+x\circ v)$ is even. Since $\sharp (a+b)= \sharp a +\sharp
b -2\sharp (a\circ b)$ for any $a, b\in ({\Bbb Z}_2)^{2r}$, one has that both $\sharp (x\circ u)$ and $\sharp(x\circ v)$ are even or odd. Thus $$\begin{aligned}
&\quad \mathcal{V}(u+v)\\&=\{x\in \mathcal{V}_{2r}\vert x\circ u+x\circ v\in \mathcal{V}_{2r}\}\\
&=\{x\in \mathcal{V}_{2r}\vert \text{ $\sharp (x\circ u)$ and
$\sharp(x\circ v)$ are even}\}\cup\{x\in \mathcal{V}_{2r}\vert \text{ $\sharp (x\circ u)$ and $\sharp(x\circ v)$ are odd}\}\\
&=[\mathcal{V}(u)\cap \mathcal{V}(v)]\cup[\mathcal{V}_{2r}\backslash
(\mathcal{V}(u)\cup \mathcal{V}(v))].\end{aligned}$$
Let $v_1,...,v_k\in \mathcal{V}_{2r}$, and let $\mathcal{V}(v_1,...,v_k)$ denote $\mathcal{V}(v_1)\cap \cdots\cap
\mathcal{V}(v_k)$.
\[dim\] Suppose that $\underline{1},v_1,...,v_k\in \mathcal{V}_{2r}$ are linearly independent. Then $$\dim\mathcal{V}(v_1,...,v_k)=2r-1-k.$$
One uses induction on $k$. From Lemma \[d1\] one knows that the case $k=1$ holds. If $k\leq m$, suppose inductively that Proposition \[dim\] holds. Consider the case $k=m+1$. Since $\mathcal{V}(v_1,...,v_{m+1})\subset \mathcal{V}(v_1,...,v_m)$, one has $\dim\mathcal{V}(v_1,...,v_{m+1})\leq\dim
\mathcal{V}(v_1,...,v_m)$.
.2cm If $\dim\mathcal{V}(v_1,...,v_{m+1})=2r-1-m$, then $\mathcal{V}(v_1,...,v_{m+1})= \mathcal{V}(v_1,...,v_m)$. One claims that this is impossible. If so, by induction, $\dim\mathcal{V}(v_2,...,v_{m+1})=2r-1-m$ so $$\label{eq1}
\mathcal{V}(v_2,...,v_{m+1})=\mathcal{V}(v_1,...,v_m)$$ since $\mathcal{V}(v_1,...,v_m)=\mathcal{V}(v_1,...,v_{m+1})\subset\mathcal{V}(v_2,...,v_{m+1})$. In a similar way, one has also that $$\label{eq2}
\mathcal{V}(v_1, v_3,...,v_{m+1})=\mathcal{V}(v_1,...,v_m).$$ By Lemma \[d2\], $$\begin{aligned}
\mathcal{V}(v_1+v_2)\cap
\mathcal{V}(v_1)&=\{[\mathcal{V}(v_1)\cap
\mathcal{V}(v_2)]\cup[\mathcal{V}_{2r}\backslash
(\mathcal{V}(v_1)\cup \mathcal{V}(v_2))]\}\cap\mathcal{V}(v_1)\\
&=[\mathcal{V}(v_1)\cap \mathcal{V}(v_2)]\cap \emptyset\\
&=\mathcal{V}(v_1)\cap \mathcal{V}(v_2).\end{aligned}$$ Similarly, one also has that $\mathcal{V}(v_1+v_2)\cap
\mathcal{V}(v_2)=\mathcal{V}(v_1)\cap \mathcal{V}(v_2)$. Thus $$\begin{aligned}
&\quad \mathcal{V}(v_1+v_2,
v_3,...,v_{m+1})\cap\mathcal{V}(v_1,...,v_m)\\
&=\{\mathcal{V}(v_1+v_2)\cap\mathcal{V}( v_3,...,v_{m+1})\}\cap
\{\mathcal{V}(v_1)\cap
\mathcal{V}(v_2)\cap\mathcal{V}(v_3,...,v_m)\}\\
&= \mathcal{V}(v_1,...,v_m)\end{aligned}$$ so $$\mathcal{V}(v_1+v_2, v_3,...,v_{m+1})=\mathcal{V}(v_1,...,v_m).$$ On the other hand, $$\begin{aligned}
&\quad\mathcal{V}(v_1+v_2, v_3,...,v_{m+1})(=\mathcal{V}(v_1,...,v_m)=\mathcal{V}(v_1,...,v_{m+1})) \\
&=\mathcal{V}(v_1+v_2)\cap \mathcal{V}(v_3,...,v_{m+1})\\
&=\{[\mathcal{V}(v_1)\cap
\mathcal{V}(v_2)]\cup[\mathcal{V}_{2r}\backslash
(\mathcal{V}(v_1)\cup \mathcal{V}(v_2))]\}\cap
\mathcal{V}(v_3,...,v_{m+1}) \text{ by Lemma~\ref{d2}}\\
&=\mathcal{V}(v_1,...,v_{m+1})\cup \{[\mathcal{V}_{2r}\backslash
(\mathcal{V}(v_1)\cup \mathcal{V}(v_2))]\cap
\mathcal{V}(v_3,...,v_{m+1})\}.\\\end{aligned}$$ This implies that $$\label{eq3}
[\mathcal{V}_{2r}\backslash (\mathcal{V}(v_1)\cup
\mathcal{V}(v_2))]\cap \mathcal{V}(v_3,...,v_{m+1})=\emptyset.$$ Combining the formulae (\[eq1\]), (\[eq2\]) and (\[eq3\]), one obtains that $$\begin{aligned}
\mathcal{V}(v_1,...,v_m)&= [\mathcal{V}(v_1)\cup
\mathcal{V}(v_2)]\cap \mathcal{V}(v_3,...,v_{m+1}) \text{ by
(\ref{eq1}) and (\ref{eq2})}\\
&= \{\mathcal{V}(v_1)\cup
\mathcal{V}(v_2)\cup[\mathcal{V}_{2r}\backslash
(\mathcal{V}(v_1)\cup \mathcal{V}(v_2))]\}\cap
\mathcal{V}(v_3,...,v_{m+1})\text{ by
(\ref{eq3})}\\
&= \mathcal{V}_{2r}\cap
\mathcal{V}(v_3,...,v_{m+1})\\
&= \mathcal{V}(v_3,...,v_{m+1})\end{aligned}$$ so $\mathcal{V}(v_1,...,v_m)=\mathcal{V}(v_3,...,v_{m+1})$. However, by induction, $\dim\mathcal{V}(v_1,...,v_m)=2r-1-m$ but $\dim\mathcal{V}(v_3,...,v_{m+1})=2r-1-(m-1)$. This is a contradiction.
.2cm
Therefore, $\dim\mathcal{V}(v_1,...,v_{m+1})<\dim\mathcal{V}(v_1,...,v_m)$.
.2cm Now let $\{x_1,..., x_{2r-1-m}\}$ be a basis of $\mathcal{V}(v_1,...,v_m)$. Since $$\mathcal{V}(v_1,...,v_{m+1})\subsetneqq \mathcal{V}(v_1,...,v_m)$$ there must exist at least one element $x$ in $\{x_1,...,
x_{2r-1-m}\}$ such that $x\not\in\mathcal{V}(v_1,...,v_{m+1})$. With no loss, one may assume that $x_1\not\in\mathcal{V}(v_1,...,v_{m+1})$. Then $x_1\circ
v_{m+1}\not\in \mathcal{V}_{2r}$. If there is also another $x_i
(i\not=1)$ in $\{x_1,x_2,..., x_{2r-1-m}\}$ such that $x_i\not\in\mathcal{V}(v_1,...,v_{m+1})$, then $x_i\circ
v_{m+1}\not\in \mathcal{V}_{2r}$. However, one knows from the proof of Lemma \[d2\] that $x_1\circ v_{m+1}+x_i\circ v_{m+1}\in
\mathcal{V}_{2r}$ so $x_1+x_i\in \mathcal{V}(v_{m+1})$ and $x_1+x_i\in \mathcal{V}(v_1,...,v_{m+1})$. In this case, $\{x_1,...,
x_{i-1},x_1+x_i, x_{i+1},..., x_{2r-1-m}\}$ is still a basis of $\mathcal{V}(v_1,...,v_m)$ but $x_1+x_i\in
\mathcal{V}(v_1,...,v_{m+1})$. One can use this way to further modify the basis $\{x_1,..., x_{i-1},x_1+x_i, x_{i+1}, ..., x_{2r-1-m}\}$ into a basis $\{x_1, {x'}_2,..., {x'}_{2r-1-m}\}$, such that ${x'}_2,..., {x'}_{2r-1-m}\in
\mathcal{V}(v_1,...,v_{m+1})$ except for $x_1\not\in
\mathcal{V}(v_1,...,v_{m+1})$. Thus, $\dim\mathcal{V}(v_1,...,v_{m+1})=2r-1-(m+1)$. This completes the induction and the proof of Proposition \[dim\].
\[d3\] Suppose that $\underline{1},v_1,...,v_k\in \mathcal{V}_{2r}$ are linearly independent with $v_i\circ v_j\in \mathcal{V}_{2r}$ for any $i,j\in\{1,...,k\}$. Then $k\leq r-1$.
Since $v_i\circ v_j\in \mathcal{V}_{2r}$ for any $i,j\in\{1,...,k\}$, $$\text{Span}\{\underline{1},v_1,...,v_k\}\subset \mathcal{V}(v_1,...,v_k).$$ So $k+1\leq 2r-1-k$, i.e., $k\leq r-1$.
\[largest-dim\] The largest-dimensional space of preserving the operation $\circ$ closed in $\mathcal{V}_{2r}$ of $\{V_i^M| 0\leq i\leq n-1\}$ is $V^M_{[{n\over 2}]}$ with dimension $r$ if $n$ is odd, and $V^M_{{n\over 2}-1}$ with dimension $<r$ if $n$ is even but $b_{{n\over 2}}\not=0$, and $V^M_{{n\over 2}-1}=V^M_{{n\over 2}}$ with dimension $r$ if $n$ is even and $b_{{n\over 2}}=0$.
Since $b_0+b_1+\cdots+b_{n-1}+b_n=2r$ and $b_i=b_{n-i}$ (note $b_0=b_n=1$), if $n$ is odd, then $b_0+b_1+\cdots+b_{[{n\over
2}]}=r$ so $\dim V^M_{[{n\over 2}]}=r$. Since $\underline{1}\in
V_i^M$ for each $i$, by Corollary \[d3\], the largest-dimensional space of preserving the operation $\circ$ closed in $\mathcal{V}_{2r}$ of $\{V_i^M| 0\leq i\leq n-1\}$ must be $V^M_{[{n\over 2}]}$. If $n$ is even and $b_{{n\over 2}}\not=0$, then $b_0+b_1+\cdots+b_{{n\over 2}-1}<r$ but $b_0+b_1+\cdots+b_{{n\over 2}}>r$ so the required largest-dimensional subspace is $V^M_{{n\over 2}-1}$ with dimension $r-{{b_{{n\over 2}}}\over 2}<r$. If $n$ is even and $b_{{n\over
2}}=0$, then $$b_0+b_1+\cdots+b_{{n\over
2}-1}=b_0+b_1+\cdots+b_{{n\over 2}}=r$$ so the desired result holds.
2-dimensional case
==================
Let $M\in \Lambda_2^{2r}$. By Theorem \[ring str\], we know that $$H_G^*(M;{\Bbb Z}_2)\cong V^M_0+\mathcal{V}_{2r}t+({\Bbb
Z}_2)^{2r}(t^2+t^3+\cdots).$$ This means that the ring structure of $H_G^*(M;{\Bbb Z}_2)$ only depends upon the number $2r=|M^G|$. Thus we have
\[2-dim\] Let $M_1, M_2\in \Lambda_2$. Then $H_G^*(M_1;{\Bbb Z}_2)$ and $H_G^*(M_2;{\Bbb Z}_2)$ are isomorphic if and only if $|M_1^G|=|M_2^G|$.
An easy observation shows that for an orientable connected closed surface $\Sigma_g$ with genus $g\geq 0$, $\Sigma_g$ must admit a $G$-action such that $|\Sigma_g^G|=2(g+1)$. Thus, for each $r\geq
1$, $\Lambda_2^{2r}$ is non-empty.
However, one knows from [@b] that ${\Bbb R}P^2$ never admits a $G$-action such that the fixed point set is a finite set so ${\Bbb
R}P^2$ doesn’t belong to $\Lambda_2$. Actually, generally each non-orientable connected closed surface $S_g$ with genus $g$ odd must not belong to $\Lambda_2$. This is because the sum of all mod 2 Betti numbers of $S_g$ is $2+g$, which is odd.
As a consequence of Proposition \[2-dim\], one has
For each positive integer $r$, all $G$-manifolds in $\Lambda_2^{2r}$ determine a unique equivariant cohomology up to isomorphism.
For each $M\in \Lambda_2^{2r}$, its equivariant cohomology $H^*_G(M;{\Bbb Z}_2)$ may be expressed in a simpler way. Since $(1,...,1)^\top\in V_0^M\cong{\Bbb Z}_2$ and $|v|=0$ for $v\in
\mathcal{V}_{2r}$, one has that $$H_G^*(M;{\Bbb Z}_2)\cong \Big\{\alpha=(\alpha_1,...,\alpha_{2r})\in({\Bbb
Z}_2)^{2r}[t]\Big| \begin{cases}\alpha_1=\cdots=\alpha_{2r}
&\text{ if $\deg\alpha=0$}\\
\sum\limits_{i=1}^{2r}\alpha_i=0 &\text{ if } \deg\alpha=1
\end{cases}\Big\}.$$ Compare with [@gh Proposition 3.1], Goldin and Holm gave a description for the equivariant cohomolgy of a compact connected symplectic 4-dimensional manifold with an effective Hamiltonian $S^1$-action with a finite fixed set, so that they computed the equivariant cohomology of certain manifolds with a Hamiltonian action of a torus $T$ (see [@gh Theorem 2]). Similarly to the argument in [@gh], using this description of $H_G^*(M;{\Bbb Z}_2)$ and the main result of Chang and Skjelbred [@cs], one may give an explicit description in ${\Bbb
Z}_2[t_1,...,t_k]$ of the equivariant cohomology of a closed $({\Bbb
Z}_2)^k$-manifold $N$ with the following conditions:
1. The fixed point set is finite;
2. The equivariant cohomology of $N$ is a free $H^*(B({\Bbb
Z}_2)^k;{\Bbb Z}_2)$-module;
3. For $K< ({\Bbb Z}_2)^k$ a corank-1 2-torus, each component of $N^K$ has dimension at most 2.
We would like to leave it to readers as an exercise. For this description in ${\Bbb Z}_2[t_1,...,t_k]$ of $H^*_{({\Bbb
Z}_2)^k}(N;{\Bbb Z}_2)$, actually the above restriction condition (3) for each component of $N^K$ is the best possible since generally there can be different equivariant cohomology structures for $G$-manifolds in $\Lambda_n^{2r}$ when $n\geq 3$ (see, e.g., Section 6 of this paper).
An analytic description of ring isomorphisms and a necessary and sufficient condition
=====================================================================================
The purpose of this section is to give an analytic description for the isomorphism between equivariant cohomology rings of two $G$-manifolds in $\Lambda_n$ and to show a necessary and sufficient condition that the equivariant cohomology rings of two $G$-manifolds in $\Lambda_n$ are isomorphic.
.2cm
Let $$\mathcal{W}_{2r}=\{\sigma\in {\a} \mathcal{V}_{2r}\subset \G (2r,{\Bbb Z}_2)|
\text{ $\sigma (x\circ y)=\sigma (x)\circ\sigma (y)$ for any $x,
y\in \mathcal{V}_{2r}$}\}.$$ Given two $\sigma,\tau$ in $\mathcal{W}_{2r}$, it is easy to check that $\sigma\tau\in
\mathcal{W}_{2r}$. Thus one has that $\mathcal{W}_{2r}$ is a subgroup of $\a \mathcal{V}_{2r}$.
\[weyl\] $\mathcal{W}_{2r}$ is the Weyl subgroup of $\G(2r,{\Bbb Z}_2)$.
By [@ab1], it suffices to prove that $\mathcal{W}_{2r}$ is isomorphic to the symmetric group $\mathcal{S}_{2r}$ of rank $2r$. Let $\sigma=(a_{ij})_{2r\times 2r}$ be an element of $\mathcal{W}_{2r}\subset \G(2r,{\Bbb Z}_2)$. Since $\sigma$ is an automorphism of $\mathcal{V}_{2r}$, there exists a vector $\underline{x}\in\mathcal{V}_{2r}$ such that $\sigma(\underline{x})=\underline{1}$ where $\underline{1}=(1\underbrace{,...,}_{2r}1)^\top$. Since $\underline{1}\circ x=x$ for any $x\in\mathcal{V}_{2r}$, one has that $\underline{1}=\sigma(\underline{x})=\sigma(\underline{1}\circ\underline{x})=\sigma(\underline{1})\circ
\sigma(\underline{x})=\sigma(\underline{1})$ so $$\label{e1}
\sum_{j=1}^{2r}a_{ij}=1, \text{ for } i=1,2,...,2r.$$ On the other hand, for any $x=(x_1,...,x_{2r})^\top$ and $y=(y_1,...,y_{2r})^\top$ in $\mathcal{V}_{2r}$, one has that $\sigma(x\circ y)=\sigma(x)\circ\sigma(y)$ so $$\label{e2}
\sum_{l=1}^{2r}a_{il}x_ly_l=(\sum_{j=1}^{2r}a_{ij}x_j)(\sum_{k=1}^{2r}a_{ik}y_k),
\text{ for } i=1,2,...,2r.$$ From (\[e1\]) one knows that for each $i$, the number $q(i)$ of nonzero elements in $a_{i1},...,a_{i2r}$ is odd.
.2cm
Now let us show that for each $i$, $q(i)$ actually must be 1. Taking an $i$, without loss of generality one may assume that $a_{i1}=\cdots=a_{iq(i)}=1$, and $a_{i(q(i)+1)}=\cdots=a_{i2r}=0$. Then from (\[e2\]) one has $$\label{e3}
(\sum_{j=1}^{q(i)}x_j)(\sum_{k=1}^{q(i)}y_k)+\sum_{l=1}^{q(i)}x_ly_l=0.$$ If $q(i)>1$, taking $x$ with $x_1=x_{q(i)}=1$ and $x_j=0$ for $j\not=1, q(i)$ and $y$ with $y_2=y_{q(i)}=1$ and $y_k=0$ for $k\not=2,q(i)$, the left side of (\[e3\]) then becomes 1, but this is impossible. Thus $q(i)=1$.
.2cm
Since $q(i)=1$ for each $i$, this means that $\sigma$ is actually obtained by doing a permutation on all rows (or all columns) of the identity matrix. The lemma then follows from this.
\[analyze\] Let $M_1$ and $M_2$ in $\Lambda_n^{2r}$. Suppose that $f$ is an isomorphism between graded rings $$\mathcal{R}_{M_1}=V^{M_1}_0+\cdots+ V^{M_1}_{n-2}t^{n-2}+\mathcal{V}_{2r}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^n+\cdots)$$ and $$\mathcal{R}_{M_2}=V^{M_2}_0+\cdots+ V^{M_2}_{n-2}t^{n-2}+\mathcal{V}_{2r}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^n+\cdots).$$ Then there is an element $\sigma\in
\mathcal{W}_{2r}$ such that $f=\sum\limits_{i=0}^\infty \sigma t^i$ is analytic, where $f(\beta)=\sum\limits_{i=0}^\infty \sigma(v_i)
t^i$ for $\beta=\sum\limits_{i=0}^\infty v_i t^i\in
\mathcal{R}_{M_1}$.
Since the restriction $f|_{\mathcal{V}_{2r}t^{n-1}}:\mathcal{V}_{2r}t^{n-1}\longrightarrow\mathcal{V}_{2r}t^{n-1}$ is a linear isomorphism, there exists an automorphism $\sigma$ of $
\mathcal{V}_{2r}$ such that $f|_{\mathcal{V}_{2r}t^{n-1}}=\sigma
t^{n-1}$. .2cm
First, let us show that $\sigma\in \mathcal{W}_{2r}$. Since $x\circ
x=x$ for any $x\in ({\Bbb Z}_2)^{2r}$ and $f$ is a ring isomorphism, one has that for any $v\in \mathcal{V}_{2r}$, $$f(vt^{2n-2})=f(v\circ
vt^{2n-2})=f(vt^{n-1})f(vt^{n-1})=\sigma(v)\circ\sigma(v)t^{2n-2}=\sigma(v)t^{2n-2}$$ so $f|_{\mathcal{V}_{2r}t^{2n-2}}=\sigma t^{2n-2}$. Furthermore, for any $v_1, v_2\in \mathcal{V}_{2r}$, one has that $$\sigma(v_1\circ v_2)t^{2n-2}=f(v_1\circ
v_2t^{2n-2})=f(v_1t^{n-1})f(v_2t^{n-1})=\sigma(v_1)\circ\sigma(v_2)t^{2n-2}$$ so $\sigma(v_1\circ v_2)=\sigma(v_1)\circ\sigma(v_2)$. Thus, $\sigma\in \mathcal{W}_{2r}$.
.2cm
By Theorem \[ring str\], for each $i<n-1$, $\underline{1}\in
V_i^M\subset \mathcal{V}_{2r}$, and one knows from the proof of Lemma \[weyl\] that $\sigma(\underline{1})=\underline{1}$ so $f|_{V_0^M}=\sigma$. By Remark \[module\] one knows that $\mathcal{R}_{M_i}, i=1,2,$ are free ${\Bbb Z}_2[t]$-modules, and it is easy to see that $f$ is also an isomorphism between free ${\Bbb
Z}_2[t]$-modules $\mathcal{R}_{M_1}$ and $\mathcal{R}_{M_2}$. For $0\leq i<n-1$, let $v\in V_i^M\subset \mathcal{V}_{2r}$, since $f|_{\mathcal{V}_{2r}t^{n-1}}=\sigma t^{n-1}$, one has that $$f(vt^i)t^{n-1-i}=f(vt^{n-1})=
\sigma(v)t^{n-1}$$ so $f(vt^i)=\sigma(v)t^i$. Thus, $f|_{V_i^Mt^i}=\sigma t^i$ for $0\leq i<n-1$.
.2cm
Let $\ell=a(n-1)+b\geq n$ with $b<n-1$. For any $v\in
\mathcal{V}_{2r}$, one has that $$f(vt^\ell)=f(v\underbrace{\circ \cdots\circ}_avt^{a(n-1)})t^b
=[f(vt^{n-1})]^at^b=\sigma(v)\underbrace{\circ
\cdots\circ}_a\sigma(v)t^\ell=\sigma(v)t^\ell.$$ Thus, for $i\geq
n$, $f|_{\mathcal{V}_{2r}^Mt^i}=\sigma t^i$.
.2cm Since $\mathcal{V}_{2r}$ is not closed with respect to the operation $\circ$ by Corollary \[d3\], there must be $u,
w\in\mathcal{V}_{2r}$ such that $u\circ w\not\in \mathcal{V}_{2r}$. Since $\mathcal{V}_{2r}$ has dimension $2r-1$, one then has that $({\Bbb Z}_2)^{2r}=\mathcal{V}_{2r}+\text{Span}\{u\circ w\}$. Actually, this is a direct sum decomposition of $({\Bbb Z}_2)^{2r}$, i.e., $({\Bbb Z}_2)^{2r}=\mathcal{V}_{2r}\oplus\text{Span}\{u\circ
w\}$. Furthermore, let $x\in({\Bbb Z}_2)^{2r}$, then there is a vector $v\in \mathcal{V}_{2r}$ such that $x$ can be written as $v+\varepsilon u\circ w$ where $\varepsilon=0$ or 1. For $i=a(n-1)+b\geq n$ with $b<n-1$, since $f|_{\mathcal{V}_{2r}^Mt^j}=\sigma t^j$ for any $j\geq n-1$, one has that $$\begin{aligned}
f(xt^i)t^{n-1-b}&=f( vt^i+\varepsilon u\circ wt^i)t^{n-1-b}\\&=
f(vt^i)t^{n-1-b}
+\varepsilon f(u\circ wt^i)t^{n-1-b}\\
&= f(vt^{(a+1)(n-1)})+\varepsilon f([ut^{n-1}]\circ [wt^{a(n-1)}])\\
&= \sigma(v)t^{(a+1)(n-1)}+\varepsilon f(ut^{n-1})f(wt^{a(n-1)})\\
&=\sigma(v)t^{(a+1)(n-1)}+\varepsilon \sigma(u)\circ
\sigma(w)t^{(a+1)(n-1)}\\
&= [\sigma(v)+\varepsilon \sigma(u)\circ
\sigma(w)]t^{(a+1)(n-1)}\\
&= \sigma(v+\varepsilon u\circ
w)t^{(a+1)(n-1)}\text{ since $\sigma\in\mathcal{W}_{2r}$}\\
&= \sigma(x)t^{i+(n-1-b)}\\\end{aligned}$$ so $f(xt^i)=\sigma(x)t^i$. Thus $f|_{({\Bbb Z}_2)^{2r}t^i}=\sigma t^i$ for $i\geq n$.
.2cm Combining the above argument, we complete the proof.
\[ns\] Let $M_1$ and $M_2$ in $\Lambda_n^{2r}$. Then $H^*_G(M_1;{\Bbb
Z}_2)$ and $H^*_G(M_2;{\Bbb Z}_2)$ are isomorphic if and only if there exists an element $\sigma\in \mathcal{W}_{2r}$ such that $\sigma$ isomorphically maps $V^{M_1}_i$ onto $V^{M_2}_i$ for $i<
n-1$.
Suppose that $H^*_G(M_1;{\Bbb Z}_2)$ and $H^*_G(M_2;{\Bbb Z}_2)$ are isomorphic. Then by Theorem \[ring str\] there is an isomorphism $f$ between graded rings $$\mathcal{R}_{M_1}=V^{M_1}_0+\cdots+ V^{M_1}_{n-2}t^{n-2}+\mathcal{V}_{2r}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^n+\cdots)$$ and $$\mathcal{R}_{M_2}=V^{M_2}_0+\cdots+ V^{M_2}_{n-2}t^{n-2}+\mathcal{V}_{2r}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^n+\cdots).$$ One knows from Theorem \[analyze\] that there is an element $\sigma\in \mathcal{W}_{2r}$ such that $f=\sum\limits_{i=0}^\infty \sigma t^i$. Then the restriction $f|_{V_i^{M_1}t^i}=\sigma t^i$, which is an isomorphism from $V^{M_1}_it^i$ to $V^{M_2}_it^i$ for $i< n-1$. Thus $\sigma$ isomorphically maps $V^{M_1}_i$ onto $V^{M_2}_i$ for $i< n-1$.
.2cm
Conversely, if there exists an element $\sigma\in
\mathcal{W}_{2r}$ such that $\sigma$ isomorphically maps $V^{M_1}_i$ onto $V^{M_2}_i$ for $i< n-1$, then by Theorem \[analyze\], $\sum\limits_{i=0}^\infty \sigma t^i$ gives an isomorphism between graded rings $\mathcal{R}_{M_1}=V^{M_1}_0+\cdots+
V^{M_1}_{n-2}t^{n-2}+\mathcal{V}_{2r}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^n+\cdots)$ and $\mathcal{R}_{M_2}=V^{M_2}_0+\cdots+
V^{M_2}_{n-2}t^{n-2}+\mathcal{V}_{2r}t^{n-1}+ ({\Bbb
Z}_2)^{2r}(t^n+\cdots)$. Then $H^*_G(M_1;{\Bbb Z}_2)$ and $H^*_G(M_2;{\Bbb Z}_2)$ are isomorphic by Theorem \[ring str\].
Let $M_1, M_2\in \Lambda_n$. One sees from Theorem \[ns\] that if $|M_1^G|\not=|M_2^G|$, then $H_G^*(M_1;{\Bbb Z}_2)$ and $H_G^*(M_2;{\Bbb Z}_2)$ must not be isomorphic.
In the case $n=3$, for $M\in \Lambda_3^{2r}$, since $$H_G^*(M;{\Bbb Z}_2)\cong V_0^M+V_1^Mt+\mathcal{V}_{2r}t^2+({\Bbb
Z}_2)^{2r}(t^3+\cdots),$$ one sees that the structure of $H_G^*(M;{\Bbb Z}_2)$ actually depends upon that of $V_1^M$. Thus, Theorem \[ns\] has a simpler expression in this case.
\[3ns\] Let $M_1$ and $M_2$ in $\Lambda_3^{2r}$. Then $H^*_G(M_1;{\Bbb
Z}_2)$ and $H^*_G(M_2;{\Bbb Z}_2)$ are isomorphic if and only if there exists an element $\sigma\in \mathcal{W}_{2r}$ such that $\sigma$ isomorphically maps $V^{M_1}_1$ onto $V^{M_2}_1$.
The number of equivariant cohomology structures
===============================================
In this section we shall consider the number of equivariant cohomology rings up to isomorphism of all 3-dimensional $G$-manifolds in $\Lambda_3^{2r}$. For $M\in \Lambda_3^{2r}$ one knows that $ V_1^M$ has dimension $r$ and is the largest-dimensional subspace of $\mathcal{V}_{2r}$ with the property that $u\circ v\in \mathcal{V}_{2r}$ for $u, v\in V_1^M$ by Corollary \[largest-dim\].
.2cm
Let $\mathcal{M}_r$ denote the set of those $2r\times r$ matrices $(v_1,...,v_r)$ with rank $r$ over ${\Bbb Z}_2$ such that $v_i\circ
v_j\in \mathcal{V}_{2r}$ for any $1\leq i,j\leq r$. $\mathcal{M}_r$ admits the following two actions.
.2cm
One action is the right action of $\text{GL}(r,{\Bbb Z}_2)$ on $\mathcal{M}_r$ defined by $(v_1,...,v_r)\lambda$ for $\lambda\in\text{GL}(r,{\Bbb Z}_2)$. It is easy to see that such action is free. Obviously, all column vectors of each matrix $(v_1,...,v_r)$ span the same linear space as all column vectors of $(v_1,...,v_r)\lambda$ for $\lambda\in\text{GL}(r,{\Bbb Z}_2)$.
.2cm
The other action is the left action of the Weyl group $\mathcal{W}_{2r}=\mathcal{S}_{2r}$ on $\mathcal{M}_r$ defined by $\sigma(v_1,...,v_r)=(\sigma v_1,...,\sigma v_r)$ for $(v_1,...,v_r)\in \mathcal{M}_r$ and $\sigma\in \mathcal{S}_{2r}$. In general, this action is not free.
.2cm
For $M\in \Lambda_3^{2r}$, since the space $V_1^M\subset\mathcal{V}_{2r}$ may be spanned by all column vectors of some matrix $(v_1,...,v_r)$ in $\mathcal{M}_r$, the number of all possible spaces $V_1^M\subset\mathcal{V}_{2r}$ is at most $\big|\mathcal{M}_r/\G(r,{\Bbb
Z}_2)\big|={{|\mathcal{M}_r|}\over{|\G(r,{\Bbb Z}_2)|}}$.
.2cm
Together with the above understood and Corollary \[3ns\], one has
\[number\] The number of equivariant cohomology rings up to isomorphism of all ${\Bbb Z}_2$-manifolds in $\Lambda_3^{2r}$ is at most $$|\mathcal{S}_{2r}\backslash\mathcal{M}_r/\G(r,{\Bbb Z}_2)|.$$
\[vp\] There is a natural map $g$ from $\Lambda_3^{2r}$ to $\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)$. If this map is surjective, then the number of equivariant cohomology rings up to isomorphism of all ${\Bbb Z}_2$-manifolds in $\Lambda_3^{2r}$ is exactly $|\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)|.$ To determine whether $g$ is surjective or not is an interesting thing, but it seems to be quite difficult.
.2cm
Generally, the computation of the number $|\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)|$ is not an easy thing. Next, we shall analyze this number.
.2cm
Taking an element $A\in\mathcal{M}_r$, there must be $\sigma\in \mathcal{S}_{2r}$ and $\lambda\in\text{GL}(r,{\Bbb Z}_2)$ such that $$\sigma A\lambda=
\begin{pmatrix}
I_r \\
P
\end{pmatrix}$$ so each orbit of the orbit set $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb
Z}_2)$ contains the representative of the form $
\begin{pmatrix}
I_r \\
P
\end{pmatrix}$, where $I_r$ is the $r\times r$ identity matrix. It is easy to check that $P\in O(r,{\Bbb Z}_2)$, where $O(r,{\Bbb Z}_2)$ is the orthogonal matrix group over ${\Bbb Z}_2$. Obviously, $O(r,{\Bbb Z}_2)$ always admits the left and right actions of the Weyl group $\mathcal{S}_r$ in $\text{GL}(r,{\Bbb
Z}_2)$.
\[bound\] $|\mathcal{S}_{2r}\backslash\mathcal{M}_r/\G(r,{\Bbb Z}_2)|\leq
|\mathcal{S}_r\backslash O(r,{\Bbb Z}_2)/ \mathcal{S}_r|.$
Suppose that $P_1,P_2\in O(r,{\Bbb Z}_2)$ belong to the same orbit in $\mathcal{S}_r\backslash O(r,{\Bbb Z}_2)/ \mathcal{S}_r$. Then there exist $\tau, \rho$ in $\mathcal{S}_r$ such that $$\tau P_1\rho=P_2.$$ Let $\sigma=\begin{pmatrix}
\rho^{-1} & 0 \\
0 & \tau
\end{pmatrix}$. Then $\sigma\in \mathcal{S}_{2r}$. Obviously, $$\sigma\begin{pmatrix}
I_r \\
P_1
\end{pmatrix}\rho=\begin{pmatrix}
\rho^{-1}\rho \\
\tau P_1\rho
\end{pmatrix}=\begin{pmatrix}
I_r \\
P_2
\end{pmatrix}$$ so $
\begin{pmatrix}
I_r \\
P_1
\end{pmatrix}$ and $
\begin{pmatrix}
I_r \\
P_2
\end{pmatrix}$ belong to the same orbit in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb
Z}_2)$. This means that $$|\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)|\leq |\mathcal{S}_r\backslash
O(r,{\Bbb Z}_2)/ \mathcal{S}_r|.$$ This completes the proof.
Generally, $|\mathcal{S}_{2r}\backslash\mathcal{M}_r
/\text{GL}(r,{\Bbb Z}_2)|$ is not equal to $|\mathcal{S}_r\backslash
O(r,{\Bbb Z}_2)/ \mathcal{S}_r|.$ For example, take $$P_1=\begin{pmatrix}
0&1&1&1&1&1\\1&0&1&1&1&1\\1&1&0&1&1&1\\1&1&1&0&1&1\\1&1&1&1&0&1\\1&1&1&1&1&0
\end{pmatrix} \text{ and } P_2=\begin{pmatrix}
0&1&1&1&1&1\\1&1&0&0&0&1\\1&0&1&0&0&1\\1&0&0&1&0&1\\1&0&0&0&1&1\\1&1&1&1&1&0
\end{pmatrix}$$ it is easy to check that $P_1$ and $P_2$ don’t belong to the same orbit in $\mathcal{S}_6\backslash O(6,{\Bbb Z}_2)/ \mathcal{S}_6$, but $\begin{pmatrix}
I_6 \\
P_1
\end{pmatrix}$ and $\begin{pmatrix}
I_6 \\
P_2
\end{pmatrix}$ belong to the same orbit in $\mathcal{S}_{12}\backslash\mathcal{M}_6
/\text{GL}(6,{\Bbb Z}_2)$.
Let $A=(v_1,...,v_r)\in \mathcal{M}_r$. One says that $A$ is [*irreducible*]{} if the space $\text{Span}\{v_1,...,v_r\}$ cannot be decomposed as a direct sum of some nonzero subspaces $V_1,...,V_l,
l>1,$ with the property that $V_i\circ V_j=\{\underline{0}\}$ for $i\not=j$, where $V_i\circ V_j=\{\underline{0}\}$ means that for any $x\in V_i$ and $y\in V_j$, $x\circ y=\underline{0}$.
We would like to point out that if one can find out all possible irreducible matrices of $\mathcal{M}_r$ for any $r$, then one can construct a representative of each class in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)$, with the form $$\begin{pmatrix}
A_1&0&...&0\\0&A_2&...&0\\
& &...
\\0&0&...&A_s
\end{pmatrix}$$ where the blocks $A_i$’s are irreducible matrices.
.2cm
The following result shows that there exists at least one irreducible matrix in $\mathcal{M}_r$ for almost any positive integer $r$. Let $$A(l)=\left(
\begin{matrix}
1&0&0&0&......&0&0\\
1&0&0&0&......&0&1\\
1&0&0&0&......&1&0\\
&&&&......
\\1&0&0&1&......&0&0\\
1&0&1&0&......&0&0\\
1&1&0&0&......&0&0
\\1&1&1&1&......&1&1\\
0&1&1&1&......&1&1
\\0&1&0&0&......&0&0\\
0&0&1&0&......&0&0\\
0&0&0&1&......&0&0\\
&&&&......
\\0&0&0&0&......&1&0\\
0&0&0&0&......&0&1
\end{matrix} \right)_{(2l-1)\times (l-1)}$$ be a $(2l-1)\times (l-1)$ matrix with $l\geq 4$ and $\underline{1}_{l\times 1}=(1\underbrace{,...,}_l1)^\top$. Note that only when $l\geq 4$ is even, the first column vector $v$ of $A(l)$ exactly has the property $|v|=0$ in ${\Bbb Z}_2$.
\[ir\] $(a)$ For even $r\geq 4$, there exists an irreducible $2r\times r$ matrix $$\begin{pmatrix} A(r)&\underline{1}_{(2r-1)\times 1}\\0&1
\end{pmatrix}.$$ $(b)$ For odd $r\geq 7$, there exist irreducible $2r\times r$ matrices of the following form $$\begin{pmatrix}A(s)&0&\underline{1}_{(2s-1)\times 1}\\0&A(t)&\underline{1}_{(2t-1)\times 1}
\end{pmatrix}$$ with only two blocks $A(l)$’s, where $s+t=r+1$ and $s,t\geq4$ are even. In particular, both $
\begin{pmatrix}A(s_1)&0&\underline{1}_{(2s_1-1)\times1}\\0&A(t_1)&\underline{1}_{(2t_1-1)\times 1}
\end{pmatrix}$ and $
\begin{pmatrix}A(s_2)&0&\underline{1}_{(2s_2-1)\times1}\\0&A(t_2)&\underline{1}_{(2t_2-1)\times1}
\end{pmatrix}$ belong to the same orbit in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\G(r,{\Bbb Z}_2)$ if and only if $\{s_1,t_1 \} = \{s_2,t_2 \}$, where $s_j+t_j=r+1$ and $s_j,t_j\geq 4$ are even for $ j=1,2$.
\(i) A direct observation shows that when $r=1$, $\mathcal{M}_1$ contains a unique matrix $\begin{pmatrix} 1\\1
\end{pmatrix}$, which is irreducible, and when $r=2,3,5$, there is no any irreducible matrix. However, we don’t know whether those irreducible matrices stated in Lemma \[ir\], with $\begin{pmatrix} 1\\1
\end{pmatrix}$ together, give all possible irreducible matrices.
.2cm (ii) For odd $r\geq7$, let $\lambda(r)$ denote the number of the orbit classes of irreducible matrices in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)$. By Lemma \[ir\], $\lambda(r)$ is equal to or more than the number of the solutions $(x, y)$ of the following equation $$\begin{cases}
\ x+y={{r+1}\over 2}\\
\ x\geq y\geq 2.
\end{cases}$$ Then, an easy argument shows that the number of solutions is $[{{{(r+1)/2}-1-1}\over2}]=[{{r-3}\over4}]$, so $\lambda(r)\geq
[{{r-3}\over4}]$.
Let $A=(v_1,...,v_r)\in \mathcal{M}_r$. Set $$\mathcal{V}(A):=\text{Span}\{v_1,...,v_r\}$$ and $$\mathcal{X}(A):=\text{Span}\{v_i\circ v_j| i,j=1,...,r\}.$$ It is easy to see the following properties:
1. $\mathcal{V}(A)\subset\mathcal{X}(A)\subset
\mathcal{V}_{2r}$ so $r\leq \dim\mathcal{X}(A)\leq 2r-1$;
2. If $A, B\in \mathcal{M}_r$ belong to the same orbit in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)$, then $\mathcal{X}(A)$ is linearly isomorphic to $\mathcal{X}(B)$.
.2cm
[**Fact 1.**]{} [*Let $A\in \mathcal{M}_r$. If $A$ is not irreducible, then there exists a $2k\times k$ matrix $A_{2k\times k}$ and a $2l\times l$ matrix $A_{2l\times l}$ with $k+l=r$ such that $A$ and $\begin{pmatrix} A_{2k\times k}&0\\0&
A_{2l\times l}
\end{pmatrix}$ belong to the same orbit in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\G(r,{\Bbb Z}_2)$.*]{}
If $A$ is not irreducible, by definition, there exists a $k_1\times
k_2$ matrix $A_{k_1\times k_2}$ and a $l_1\times l_2$ matrix $A_{l_1\times l_2}$ with $k_1+l_1=2r$ and $k_2+l_2=r$ such that $A$ and $\begin{pmatrix} A_{k_1\times k_2}&0\\0& A_{l_1\times l_2}
\end{pmatrix}$ belong to the same orbit in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)$. By Proposition \[dim\] and the proof method of Corollary \[d3\], one has that $k_2\leq k_1/2$ and $l_2\leq l_1/2$. Then the relations $k_1+l_1=2r$ and $k_2+l_2=r$ force $k_2= k_1/2$ and $l_2=l_1/2$.
[**Fact 2.**]{} [*Let $A\in \mathcal{M}_r$. If $\dim\mathcal{X}(A)= 2r-1$, then $A$ is irreducible.*]{}
Suppose that $A$ is not irreducible. By Fact 1, there is a matrix $\begin{pmatrix} A_{2k\times k}&0\\0& A_{2l\times l}
\end{pmatrix}$ that is in the same orbit as $A$. Thus $$\dim \mathcal{X}(A)=\dim \mathcal{X}(A_{2k\times k})+\dim
\mathcal{X}(A_{2l\times l})
\leq (2k-1)+(2l-1)=2r-2$$ by (P1). This is a contradiction.
Let $A\in \mathcal{M}_r$. Then there is a matrix $P\in O(r, {\Bbb Z}_2)$ such that $A$ and $\begin{pmatrix}I_r\\P
\end{pmatrix}$ belong to the same orbit in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\G(r,{\Bbb Z}_2)$. Furthermore, an easy argument shows by (P2) that if $\dim\mathcal{X}(A)= r$, then $P$ can be chosen as being $I_r$, so $A$ is not irreducible when $r\not=1$.
Now let us give the proof of Lemma \[ir\].
.2cm
[*Proof of Lemma \[ir\]*]{}. Let $A=\begin{pmatrix}
A(r)&\underline{1}_{(2r-1)\times 1}\\0&1
\end{pmatrix}=(v_1,...,v_{r-1}, \underline{1}_{2r\times1})$. By Fact 2, it suffices to check that $\dim\mathcal{X}(A)=2r-1$. A direct observation shows that $$\{v_1\circ v_2, v_1\circ v_3,...,v_1\circ v_{r-1},v_2\circ v_3,
v_1,v_2,v_3,...,v_{r-1},\underline{1}_{2r\times1} \}$$ are linearly independent, so $\dim\mathcal{X}(A)\geq 2r-1$. Since $\dim\mathcal{X}(A)\leq 2r-1$ by (P1), $\dim\mathcal{X}(A)=2r-1$. This completes the proof of Lemma \[ir\](a).
.2cm
In a similar way to the above argument, one may show that $
\begin{pmatrix}A(s)&0&\underline{1}_{(2s-1)\times 1}\\0&A(t)&\underline{1}_{(2t-1)\times 1}
\end{pmatrix}$ is irreducible, too.
.2cm
Suppose that $
\begin{pmatrix}A(s_1)&0&0&...&0&\underline{1}_{(2s_1-1)\times 1}\\0&A(s_2)&0&...&0&\underline{1}_{(2s_2-1)\times 1}\\
0&0&A(s_3)&...&0&\underline{1}_{(2s_3-1)\times 1}\\
&&&...
\\0&0&0&...&A(s_k)&\underline{1}_{(2s_k-1)\times 1}
\end{pmatrix}$ is a $2r\times r$ matrix and is irreducible. Then one must have that $\sum\limits_{i=1}^k(s_i-1)+1=r$ and $\sum\limits_{i=1}^k(2s_i-1)=2r$. From these two equations, it is easy to check that $k$ must be 2.
.2cm The proof of the last part of Lemma \[ir\](b) is immediate. $\hfill\Box$\
By a direct computation, one may give the first 6 numbers for $N(r)= |\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb
Z}_2)|$ as follows: $$
$r$ 1 2 3 4 5 6
-------- --- --- --- --- --- ---
$N(r)$ 1 1 1 2 2 3
$$ The representatives of all orbits in $\mathcal{S}_{2r}\backslash\mathcal{M}_r/\text{GL}(r,{\Bbb Z}_2)$ for $r\leq 6$ are stated as follows:
$$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$r$ 1 2 3 4 5 6
------ ---------------------- -------------------------- -------------------------- -------------------------- ------------------------------------------ ---------------------------------------------
Rep. $\begin{pmatrix}1\\1 $\begin{pmatrix}I_2\\I_2 $\begin{pmatrix}I_3\\I_3 $\begin{pmatrix}I_4\\I_4 $\begin{pmatrix}I_5\\I_5 $\begin{pmatrix}I_6\\I_6
\end{pmatrix}$ \end{pmatrix}$ \end{pmatrix}$ \end{pmatrix}$, $B(4)$ \end{pmatrix}$, $\begin{pmatrix}B(4)&0\\ \end{pmatrix}$, $\begin{pmatrix} B(4)&0&0\\
0&\underline{1}_{2\times 1} 0& \underline{1}_{2\times 1}& 0\\
\end{pmatrix}$ \\0&0&\underline{1}_{2\times 1}
\end{pmatrix}$, $B(6)$
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$$
where $B(l)=\begin{pmatrix}
A(l)&\underline{1}_{(2l-1)\times 1}\\0&1
\end{pmatrix}$. Note that for $ r\geq 7$, one also can find the lower bound of $N(r)$ as follows: $$
$r$ 7 8 9 10 11 12 13 14 $\cdots$
------------ --- --- --- ---- ---- ---- ---- ---- ----------
$N(r)\geq$ 4 6 7 9 12 16 20 25 $\cdots$
$$
.2cm
[99]{} J.L. Alperin and R.B. Bell, [*Groups and representations*]{}, Graduate Texts in Mathematics [**162**]{} (1995), Springer-Verlag. M. Atiyah and R. Bott, [*The moment map and equivariant cohomology*]{}, Topology [**23**]{} (1984), 1-28. C. Allday and V. Puppe, [*Cohomological Methods in Transformation Groups*]{}, Cambridge Studies in Advanced Mathematics, [**32**]{}, Cambridge University Press, 1993. G. E. Bredon, [*Intrduction to compact transformation groups*]{}, Pure and Applied Mathematics, Vol. [**46**]{} Academic Press, New York-London, 1972. D. Biss, V. Guillemin and T. S. Holm, [*The mod 2 cohomology of fixed point sets of anti-symplectic involutions*]{}, Adv. Math. [**185**]{} (2004), 370–399. N. Berline and M. Vergne, [*Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante*]{}, C. R. Acad. Sci. Paris Sr. I Math. [**295**]{} (1982), 539-541. P.E. Conner and E.E. Floyd, [*Differentiable periodic maps*]{}, Ergebnisse Math. Grenzgebiete, N. F., Bd. [**33**]{}, Springer-Verlag, Berlin, 1964. T. Chang and T. Skjelbred, [*The topological Schur lemma and related results*]{}, Ann. of Math. [**100**]{} (1974), 307-321. T. tom Dieck, [*Characteristic numbers of $G$-manifolds. I*]{}, Invent. Math. [**13**]{} (1971), 213-224. R. Goldin and T. S. Holm, [*The equivariant cohomology of Hamiltonian $G$-spaces from residual $S^1$ actions*]{}, Math. Res. Lett. [**8**]{} (2001), 67-77. M. Goresky, R. Kottwitz, and R. MacPherson [*Equivariant cohomology, Koszul duality, and the localization theorem*]{}, Invent. Math. [**131**]{} (1998), 25-83. V. Guillemin and C. Zara, [*1-Skeleta, Betti numbers, and equivariant cohomology*]{}, Duke Math. J. [**107**]{} (2001), 283-349. W. Y. Hsiang, [*Cohomology theory of topological transformation groups*]{}, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band [**85**]{}. Springer-Verlag, New York-Heidelberg, 1975. C. Kosniowski and R.E. Stong, [*$({\Bbb Z}_2)^k$-actions and characteristic numbers*]{}, Indiana Univ. Math. J. [**28**]{} (1979), 723-743. M. Kreck and V. Puppe, [*Involutions on 3-manifolds and self-dual, binary codes*]{}, arXiv:0707.1599. Z. Lü, [*Graphs and $({\Bbb Z}_2)^k$-actions*]{}, arXiv: math.AT/0508643. V. Puppe, [*Group actions and codes*]{}, Canad. J. math. [**l53**]{} (2001), 212-224.
|
---
author:
- 'H. Kaneda, D. Ishihara, T.Suzuki, N. Ikeda, T. Onaka, M. Yamagishi, Y. Ohyama, T. Wada,'
- 'A. Yasuda'
date: 'Received; accepted'
title: 'Large-scale distributions of mid- and far-infrared emission from the center to the halo of M 82 revealed with AKARI'
---
Introduction
============
M 82 is a nearby starburst galaxy in a group of galaxies, where an appreciable amount of material can be pushed out of a galaxy into the intergalactic medium by both internal (e.g. starburst activities) and external forces (e.g. tidal interactions between member galaxies). In fact, M 82 shows prominent galactic superwinds in H$\alpha$ [e.g. @Bla88; @Dev99] and X-rays [e.g. @Bre95; @Str97] accelerated out of the galactic plane, which are attributed to violent nuclear starbursts. The X-ray emission spatially correlates well with the H$\alpha$ emission [@Wat84; @Str04]. The ionized galactic superwinds seem to entrain various phases of neutral gas (e.g. CO: Walter et al. 2002; H$_2$: Veilleux et al. 2009) and dust [@Alt99; @Ohy02; @Hoo05; @Lee09]. In addition, M 82 shows large-scale molecular and atomic streamers anchoring around the edges of the galactic disk [@Wal02; @Yun93]. The large-scale streamers extend mostly in parallel to the galactic plane and thus in entirely different directions from the superwinds. In particular, neutral hydrogen gas is largely extended around the intergalactic space of the M 81–M 82 group, including the halo regions of M 82 [@Yun94]. The presence of dust residing between the group members is also revealed through systematic reddening of photometric color of background galaxies viewed through the intergalactic medium [@Xil06]. The streamers are likely caused by the close encounter with M 81 that M 82 experienced about 100 Myr ago [@Yun93]. Then the hydrogen gas and dust in the intergalactic medium can be regarded as leftover from the interaction with M 81.
As a result of its proximity ($\sim$3.5 Mpc) and nearly edge-on orientation with an inclination angle of about 80$^{\circ}$, M 82 is a valuable target for the study of extraplanar dust grains and their properties in its superwinds and galactic halos. To answer questions about how far, how much, and what kind of dust grains are carried out of the galaxy is of great importance for the understanding of material circulation and evolution in the galactic halo. The enrichment of the intergalactic medium with dust could affect observations of high-redshift objects [e.g. @Hei88; @Dav98]. The dust expulsion from a galaxy would also play an important role in galactic chemical evolution acting as a sink for heavy elements [e.g. @Eal96].
Nevertheless, information on the dust components in the halo of M 82 is relatively scarce in contrast to abundant information on the gaseous components. The dust in the halo was observed in reflection in the UV [@Hoo05] and optical [@Ohy02], extinction [@Hec00], and submillimeter dust continuum emission [@Lee09]. However, these are rather indirect or inefficient ways to detect largely-extended dust. High-sensitivity FIR observations from space are undoubtedly most effective to study the properties of faint extended emission from extraplanar dust, since dust emission typically peaks in the FIR and low photon backgrounds in space enable us to detect faint diffuse emission. However one serious problem is that the central starburst core is dazzlingly bright for space observations in the MIR and FIR, due to tremendous star-forming activity in the central region of M 82 [@Tel80]; M 82 is the brightest galaxy in the MIR and FIR after the Magellanic Clouds on the sky [@Col99]. Then, instrumental effects caused by saturation in observing very bright sources severely hamper reliable detection of low-level FIR emission outside the disk. The flux density of the central region of M 82 increases very rapidly towards wavelengths shorter than 300 $\mu$m; there is about two-orders-of-magnitude difference between FIR and submillimeter fluxes [@Thu00], making FIR detection of the extraplanar dust in M 82 extremely difficult.
The situation in the MIR is similar for the very bright core, however, spatial resolution is much better in the MIR than in the FIR; the AKARI telescope of 700 mm in diameter has diffraction-limited imaging performance at a wavelength of 7 $\mu$m [@Kan07]. In addition to MIR dust continuum emission, star-forming galaxies show a series of strong MIR spectral features emitted by polycyclic aromatic hydrocarbons (PAHs) or PAH clusters [e.g. @Smi07], which can be regarded as the smallest forms of carbonaceous dust particles. The PAH emission features are unexceptionally bright in the central regions of M 82 [@Stu00; @For03b]. With the Spitzer/IRAC and IRS, Engelbracht et al. (2006) and Beirão et al. (2008) showed that the PAH emission is largely extended throughout the halo up to 6 kpc from the galactic plane. Their origins, again, can be either outflows entrained by the superwind or leftover clouds from the past interaction with M 81. Engelbracht et al. (2006) favored the latter origin because significant emission is detected outside the superwinds, and thus some process to expel PAHs from all parts of the disk is needed prior to the starburst.
Far beyond the disk of the galaxy, extended emission named ’Cap’ in the H$\alpha$ and X-ray was discovered at $\sim 11$ kpc to the north of the center of M 82 [@Dev99; @Leh99]. The cap seen in the H$\alpha$ and X-ray may be the result of a collision between the hot superwind and a preexisting neutral cloud [@Leh99]. Hoopes et al. (2005) detected UV emission in the Cap suggesting that the emission is likely to be stellar UV light scattered by dust in the Cap. Tsuru et al. (2007) determined metal abundances of the X-ray plasma in the Cap region, which support the idea that the origin of the metal in the Cap is type-II supernova explosions that occurred in the central region of M 82. Hence the halo regions of M 82 are highly complicated; neither relationships among different phases of the outflowing material nor those between the superwinds and preexisting clouds is well understood.
In this paper, we report MIR and FIR imaging observations of M 82 performed with the Infrared Camera (IRC; Onaka et al. 2007) and the Far-Infrared Surveyor (FIS; Kawada et al. 2007), respectively, on board AKARI [@Mur07]. Thoughout this paper, we assume a distance of 3.53 Mpc for M 82 [@Kar02]. Our IRC data consist of 4 narrow-band images ($S7$, $S11$, $L15$, and $L24$) at reference wavelengths of 7 $\mu$m (effective band width: 1.75 $\mu$m), 11 $\mu$m (4.12 $\mu$m), 15 $\mu$m (5.98 $\mu$m), and 24 $\mu$m (5.34 $\mu$m), the allocation of which is ideal to discriminate between the PAH emission features ($S7$, $S11$) and the MIR dust continuum emission ($L15$, $L24$). The FIS has 4 photometric bands; 2 wide bands ($WIDE$-$S$ and $WIDE$-$L$) at central wavelengths of 90 $\mu$m (effective band width: 37.9 $\mu$m) and 140 $\mu$m (52.4 $\mu$m) and 2 narrow bands ($N60$ and $N160$) at 65 $\mu$m (21.7 $\mu$m) and 160 $\mu$m (34.1 $\mu$m). The wide bands provide high sensitivities, while the 2 narrow bands combined with the 2 wide bands are useful to accurately determine the temperatures of the FIR dust. Besides the fine allocation of the photometric bands, the special fast reset mode of the FIS as well as the combination of short and long exposure data of the IRC provide high signal saturation levels; we can safely observe very bright sources without serious saturation effects. Hence, the uniqueness of the IRC and FIS as compared to any other previous or currently existing instruments is a combination of their high saturation limits and high sensitivities (i.e. large dynamic range) with relatively high spatial resolution, which is essential to detect faint PAH and dust emission extending to the halo of an edge-on galaxy that is very bright in the center.
Observations and data analyses
==============================
We observed M 82 with AKARI seven times from April 2006 to April 2007. The observation log is listed in Table 1. The two observations of the IDs starting with ’50’ were performed during the AKARI performance verification phase. The observations with IDs starting with ’51’ were performed during AKARI Director’s Time. The others were carried out in part of the AKARI mission program “ISM in our Galaxy and Nearby galaxies” (ISMGN; Kaneda et al. 2009a). In the two IRC observations with IDs starting with ’51’, we observed the Cap region, while the others were targeted at the galaxy body.
With the IRC, we obtained the $S7$, $S11$, $L15$, and $L24$ band images of M 82 using a standard staring observation mode, where each field-of-view has a size of about $10'\times 10'$. We also obtained the near-IR (NIR) $N3$ (reference wavelength of 3.2 $\mu$m) and $N4$ (4.1 $\mu$m) images simultaneously with the $S7$ and $S11$ images, but in this paper, we do not discuss the NIR images because there are no new findings different from the Spitzer/IRAC NIR images presented in Engelbracht et al. (2006). The MIR images were created by using the IRC imaging pipeline software version 20071017 (see IRC DATA User Manual; Lorente et al. 2007 for details). The background levels were estimated by averaging values from multiple apertures placed around the galaxy, while avoiding overlap with faint extended emission from the galaxy as much as possible, and were subtracted from the images.
The FIS observations were performed in the special fast reset mode called CDS (Correlated Double Sampling) mode in order to avoid signal saturation near the nuclear region of the galaxy. Fluxes were later cross-calibrated with the other ordinary integration modes by using the internal calibration lamp of the FIS. The FIR observations were performed three times with slightly shifted positions (table 1). A $10'\times 30'$ region was covered with every observation using a standard 2-round-trip slow scan mode. As a result, we covered an area of about $15'\times 15'$ for $N60$ and $WIDE$-$S$ and $20'\times 20'$ for $WIDE$-$L$ and $N160$ around M 82. The FIR images were processed from the FIS Time Series Data (TSD) using the AKARI official pipeline being developed by the AKARI data reduction team (Verdugo et al. 2007). The background levels were estimated from data taken near the beginning and the end of the slow scan observations and subtracted from the images.
To minimize detector artifacts due to the high surface brightness of the central starburst, we applied custom reduction procedures in addition to the above normal procedures. For the IRC, we combined the long exposure image with the short exposure image; the former has 28 times longer exposure than the latter, both obtained in one pointed observations. We replaced pixels significantly affected by the high surface brightness (i.e. saturated or even deviated from linearity) in the long exposure image by those unaffected in the short exposure image. In addition, very low-level ghost signals of the peak appeared at about $1'$ and $2'$ to the southwest direction of the center of M 82 for the $S7$ and $S11$ bands. We removed the ghosts by subtracting scaled images of the central $3'\times 3'$ area at the ghost positions for each band.
For the FIS, low-level ghost signals appeared at about $5'$ to the north or south direction of the center, depending on whether the observation was performed in April or October. The reason for the ghost is electrical cross talk in the multiplexer of the cryogenic readout electronics [@Kaw07]; the narrow band produces a ghost signal just at the timing when the wide band detects a strong signal, and vice versa. We removed the ghost signals by masking the TSD where the ghosts are predicted to appear before creating the images. We also removed residual artifacts due to cosmic-ray hits by masking the affected TSD. The lost data were replaced by utilizing data redundancy gained from the three observations. Note that we did not apply any high-pass filtering because the high-pass filtering not only removes streaking due to residual slow response variations of the detector but also filters out faint emission extended around the galaxy. Instead, only for $N160$ where the streaking was relatively strong and larger blank areas were obtained, the residual variations were corrected by using blank sky for flat-fielding in both upstream and downstream regions of the slow scan.
Instrument R.A. (J2000) Dec. (J2000) Observation ID Date
------------ -------------- --------------- ---------------- ------------- --
FIS 09 55 33.0 $+$69 41 55.3 5011011 2006 Apr 18
FIS 09 56 11.3 $+$69 39 42.1 5011012 2006 Apr 18
IRC 09 55 52.2 $+$69 40 46.9 1400586 2006 Oct 21
FIS 09 55 52.2 $+$69 40 46.9 5110031 2006 Oct 22
IRC 09 55 26.0 $+$69 48 48.6 5124075 2007 Apr 19
IRC 09 55 26.0 $+$69 48 48.6 5124076 2007 Apr 19
IRC 09 55 52.2 $+$69 40 46.9 1401057 2007 Apr 20
Results
=======
MIR images
----------
The MIR 4-band contour images of the central $8'\times 8'$ area of M 82 obtained with the IRC are shown in Fig.1, where the bin size for all the maps is set to be 23. The FWHM of the point spread function (PSF) is about $5''$ for each band [@Ona07]. These maps exhibit distributions of surface brightness over a range as large as 4 orders of magnitude, demonstrating the large dynamic range of the IRC. The peak surface brightness is 5130, 4341, 18344, and 31842 MJy sr$^{-1}$ for $S7$, $S11$, $L15$, and $L24$, respectively. The 1-sigma background fluctuation levels are approximately 0.09, 0.06, 0.1, and 0.9 MJy sr$^{-1}$, which correspond to $3-14$ % of the lowest contour levels in the MIR maps.
Based on the Spitzer/IRS spectroscopy of the central region of M 82 [@Bei08], the $S7$ and $S11$ band images are most likely dominated by the PAH emission features, while the $L15$ and $L24$ images are dominated by hot dust continuum emission by very small grains (VSGs). On the basis of the observed spectrum combined with the IRC system response curve [@Ona07], we estimate that PAH emission contributes approximately 85 % and 45 % of the band intensities of $S7$ and $S11$, respectively near the central region, and even higher in the halo because hot continuum emission is much less likely in regions far from the central starburst. This is supported by the fact that the $L24$ band image shows the most compact distribution, while the $L15$ band image shows less extended structures but may contain a small contribution from the PAH 17 $\mu$m broad emission feature [@Wer04; @Kan08b].
The bright emission in the central part extends more or less along the east-west direction for each map, corresponding to the major axis of the M 82 optical disk. The very central region exhibits a double peak structure, where the stronger peak does not coincide with the optical center of M 82, slightly shifting to the west. A similar double-lobed distribution of the inner disk was also observed in CO emission [@She95] and submillimeter dust continuum emission [@Lee09]. At larger scales, all the maps show extended emission structure in the northwest and southeast directions, although the $L24$ map apparently suffers from the diffraction spike pattern of the telescope truss. There is a striking similarity between the $S7$ and $S11$ band images; there are two filamentary structures in the northern halo and one in the southern halo. The similarity is well consistent with dominance of the PAH emission in both bands. The northern two filaments are of similar brightness in the $S7$ and $S11$ images, while only the western structure is clearly seen in the $L15$ and $L24$ band images. This implies a difference in spatial distribution between the PAHs and VSG emission in the two extended structures, presumably a difference not in material distribution but in radiation field intensity (see section 4.1). The southern filament is seen also in the $L15$ image, but not clearly in the $L24$ image due to the diffraction pattern. The central 25 maps of the 850 $\mu$m continuum emission [@Lee09] and integrated CO(2$-$1) line intensity [@Thu00] exhibit only the southern extended structure.
{width="50.00000%"} {width="50.00000%"}\
{width="50.00000%"} {width="50.00000%"}
FIR images
----------
The FIR 4-band contour images of the central $40'\times 40'$ area of M 82 obtained with the FIS are shown in Fig.2, where the bin size for all the maps is set to be $15''$. The FWHM of the PSF is $37''-39''$ for $N60$ and $WIDE$-$S$ and $58''-61''$ for $WIDE$-$L$ and $N160$ [@Kaw07]. The grey-scale image from the same data is also shown in order to reveal spatial coverage for each band. Note that the observed area is slightly different from band to band; $N160$ covers the northernmost and westernmost area, while $WIDE$-$L$ covers the largest area among the bands. The peak surface brightness is 9881, 7569, 5974, and 4195 MJy sr$^{-1}$ for $N60$, $WIDE$-$S$, $WIDE$-$L$, and $N160$, respectively. The 1-sigma background fluctuation levels are 2.0, 0.5, 1.0, and 1.5 MJy sr$^{-1}$ for $N60$, $WIDE$-$S$, $WIDE$-$L$, and $N160$, respectively. They correspond to $7-18$ % levels of the lowest contours that are as low as $0.1-0.2$ % levels of the peak brightness. The elongation in the east-west direction seen in the $N60$ and $WIDE$-$S$ images is a result of optical cross talk among the pixels of the monolithic arrays used for these bands [@Kaw07]. Telescope diffraction patterns similar to that in the $L24$ image (Fig.1d) are also visible in the $N60$ and $WIDE$-$S$ images. The elongation along the scan direction seen in the $WIDE$-$L$ image is probably due to the slow transient response of the detector used for this band. Other than these, we believe there are no additional substantial artifacts in the FIR images. Note that the spatial scale in Fig.2 is much larger than that in Fig.1; the MIR image size in Fig.1 is indicated by the dashed square in Fig.2d. Alton et al. (1999) stressed that the central FIR emission should be very compact juding from the compactness of the submillimeter emission, which is consistent with our results. It should be noted that the intensity scales in Fig.2 are stretched in order to emphasize the low surface brightness emission in the halo.
The emission component extending to the northwest direction from the center can be recognized in the $WIDE$-$S$, $WIDE$-$L$, and $N160$ images, and is the most prominent in the $N160$ band because the extended emission is cooler than the central emission. In addition, there are two emission regions away from the galaxy body clearly seen in both $WIDE$-$S$ and $WIDE$-$L$ images, one located at $12'$ to the west and the other at $8'$ to the southeast from the center. There are no counterparts found in the SIMBAD database. For the southeast emission region that is located inside the borders of the observed area in both bands, the ratio of the $WIDE$-$S$ to the $WIDE$-$L$ brightness corresponds to a color temperature of 23 K, which is significantly higher than typical temperatures of $16-18$ K for the Galactic cirrus [@Sod94] and thus unlikely of Galactic foreground origin. As shown later, these emission structures seem to be spatially correlated with the neutral gas streamers. There is another emission region seen only in the $N60$ band, at about $8'$ to the northwest from the center, which is located approximately on the line extending toward the direction of the above elongated structure seen in the $WIDE$-$S$, $WIDE$-$L$, and $N160$ images. The good spatial alignment might indicate that the source is associated with the elongated structure. The contribution of gas emission lines such as the \[\] 63 $\mu$m line could be very important because this line can dominate the flux of the IRAS 60 $\mu$m band, similar to the $N60$ band, in shocked regions [@Bur90]. However the previous mapping of M 82 by shock tracers such as SiO [@Gar01], \[FeII\] 1.644 $\mu$m [@Alo03], and molecular hydrogen [@Vei09] does not cover the above bright region that is located too far ($\sim 8'$) from the nucleus.
{width="50.00000%"} {width="50.00000%"}\
{width="50.00000%"} {width="50.00000%"}
Cap region
----------
With the IRC, we performed dedicated observations of the halo region including the X-ray Cap located at $\sim 11'$ to the north from the center of M 82. Figures 3a$-$d show the obtained images, where we combine the two $10'\times 10'$ fields-of-view of the IRC, one centered at the galaxy body and the other at the Cap region. The color scales are stretched to very low surface brightness levels to bring out faint diffuse emission in the halo, and therefore the disk of the galaxy that is presented in Fig.1 is saturated in Fig.3. For comparison, in Fig.3e, we show the FIR $N160$ band image of the same area as in Fig.3a$-$d; among the 4 FIR bands, only the $N160$ band covers the X-ray Cap. In creating the MIR images in Fig.3a$-$d, we slightly shift the dark levels of the images of the Cap region so that it can be connected smoothly to the images of the galaxy body. We have applied smoothing to the MIR images by a gaussian kernel of $24''$ in FWHM to increase S/N ratios for detecting faint diffuse emission.
There seems to be a faint emission component extended largely in the Cap region, especially in the $S9$ and $S11$ band images. The color scales of these images are logarithmically scaled down to 0.0007 % of the peak surface brightness of the central starburst, which are still approximately 6, 5, 7, and 6 times higher than the 1-sigma background fluctuation levels in the corresponding bands. Here, since there might be some offset in dark levels between the two fields-of-view, we re-estimate the background level and its fluctuation within a single aperture located at the darkest nearby sky of the smoothed image as shown in Fig.3d. Hence the presence of the largely-extended faint emission component is statistically significant. The FIR image shows the aforementioned emission extended toward the northwest direction, but no significant signal at the position of the Cap.
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}\
{width="33.00000%"} {width="33.00000%"} {width="33.00000%"}
Spectral energy distributions
-----------------------------
We derive the flux densities of M 82 by integrating the surface brightness within circular apertures of diameters of $2'$ ($\sim$2 kpc), $4'$, and $8'$ around the center of the galaxy in each photometric band image of Figs.1 and 2. For the IRC, no aperture corrections are performed since above aperture sizes are sufficiently large as compared to the PSF [@Ona07]. For the FIS, point-source aperture corrections are performed by using the correction table given in Shirahata et al. (2009), where the correction factors include the effects of the optical cross talk in the $N60$ and $WIDE$-$S$ images. We also derive the flux densities of the central regions within a diameter of $1'$ for the IRC and 125 for the FIS; the latter is the largest aperture covered by all the 4 FIR bands. For the former, the aperture size is as small as the FWHMs ($37''-61''$) of the PSFs for the FIS (Kawada et al. 2007) and thus we cannot obtain reasonable fluxes in the FIR since the central emission is not a point source.
We obtain the flux densities of the halo region within a diameter of $4'$, which is located at R.A. (J2000) $=$ 9 55 21.0 and Dec. (J2000) $=$ $+$69 46 46.0. The two 4-kpc apertures in the center and the halo are shown on the FIR image in Fig.2a and the same aperture in the halo is also indicated on the MIR image in Fig.3d. The flux densities in all the MIR and FIR bands thus obtained for the center ($d\leq 2'$, $d\leq 4'$), the halo ($d\leq 4'$), and the total region ($d\leq 8'$) are listed in Table 2.
The spectral energy distributions (SEDs) constructed from the flux densities in Table 2 are presented in Fig.4. The SEDs of the regions centered at the nucleus with diameters of $1'$, $2'$, $4'$, $8'$, and 125 are shown in the ascending order in Fig.4a, while the SED of the center ($d\leq 2'$), the differential SEDs of $4'-2'$, $8'-4'$, and 125$-8'$, and the SED of the halo region ($d\leq 4'$) are given in the descending order in Fig.4b, in units of Jy arcmin$^{-2}$ (flux densities divided by the corresponding area). The figures clearly show that the SED is getting softer toward regions farther away from the galactic center. In particular, the lowest two FIR SEDs in Fig.4b, for which the integrated areas are partially overlapped, indicate the presence of cold dust in the halo.
--------------------------- ----------------------------- --------------------- ------------------------------------ -------------------- --------------------- ----------------------------------
Band Center ($d\leq 1'$) Center ($d\leq 2'$) Center ($d\leq 4'$)$^{\mathrm{a}}$ Total ($d\leq 8'$) Total ($d\leq$ 125) Halo ($d\leq 4'$)$^{\mathrm{a}}$
(Jy) (Jy) (Jy) (Jy) (Jy ) (Jy)
IRC $S7$ 7 $\mu$m 54.1$\pm$1.2$^{\mathrm{b}}$ 64.2$\pm$1.5 72.7$\pm$1.7 75.3$\pm$1.7 … 0.045$\pm$0.001
IRC $S11$ 11 $\mu$m 43.7$\pm$1.0 53.8$\pm$1.3 62.8$\pm$1.5 65.5$\pm$1.5 … 0.048$\pm$0.001
IRC $L15$ 15 $\mu$m 112.0$\pm$3.2 128.0$\pm$3.6 137.0$\pm$3.9 140.8$\pm$4.0 … 0.091$\pm$0.002
IRC $L24$ 24 $\mu$m 307$\pm$14 342$\pm$16 361$\pm$17 365$\pm$17 … 0.21$\pm$0.01
FIS $N60$ 65 $\mu$m … 630$\pm$130 1530$\pm$310 1700$\pm$340 1920$\pm$380 16.4$\pm$3.3
FIS $WIDE$-$S$ 90 $\mu$m … 529$\pm$110 1550$\pm$310 1840$\pm$370 2040$\pm$410 16.0$\pm$3.2
FIS $WIDE$-$L$ 140 $\mu$m … 520$\pm$160 1580$\pm$480 2000$\pm$600 2250$\pm$670 26.2$\pm$7.9
FIS $N160$ 160 $\mu$m … 360$\pm$110 1270$\pm$380 1580$\pm$470 1940$\pm$580 47$\pm$14
--------------------------- ----------------------------- --------------------- ------------------------------------ -------------------- --------------------- ----------------------------------
The sizes and positions of the aperture regions are indicated by the dashed circles in Figs.2a and 3d.
The flux density errors include both systematic effects associated with the detectors and absolute calibration uncertainties (Lorente et al. 2007; Verdugo et al. 2007).
{width="50.00000%"} {width="50.00000%"}
Comparison with other wavelength images
---------------------------------------
Figure 5 shows comparison between the $S7$ contour map and the continuum-subtracted H$\alpha$ image; the former is the same as presented in Fig.1a, while the latter is taken from the NASA/IPAC Extragalactic Database. We find that they are remarkably similar to each other. The H$\alpha$ image is dominated by a biconical structure that defines the superwinds. The excellent spatial correspondence indicates that the PAHs responsible for the $S7$ band flux are well mixed with the ionized gas and entrained by the galactic superwinds. Engelbracht et al. (2006) pointed out that the morphology of the Spitzer 8 $\mu$m image of M 82 is similar to that of the H$\alpha$ emission, but also that the 8 $\mu$m image differs in that the emission is bright all around the galaxy rather than being dominated by a cone perpendicular to the disk. As far as the central $7'$ ($\simeq$ 7 kpc) area is concerned, however, our result reveals a marked similarity and no significant deviation in morphology between the PAH and the H$\alpha$ emission.
We calculate the linear-correlation coefficient, $R$, between the low-level $S7$ and H$\alpha$ images for a bin size of 23 and the $S7$ brightness ranging from 1 % to 0.01 % of the peak. We obtain $R=+0.80$ (for a total of 21014 data points). In general, as observed in our Galaxy and nearby galaxies, most PAHs are associated with neutral gas and their emission is very weak in ionized regions with strong radiation field probably due to destruction [e.g. @Bou88; @Des90; @Ben08]. Therefore the strong positive correlation observed between the PAH and H$\alpha$ emission is rather unusual. This may be due to a difference in the length of time over which PAHs have been exposed to a harsh environment. The tight correlation would then suggest that the PAHs have been traveling fast enough to reach their present locations, $\sim 3$ kpc above the disk, in less than a destruction timescale.
Figure 6 shows comparison between the $WIDE$-$L$ and the contour map; the former is the same but enlarged $25'\times 25'$ image as presented in Fig.2c, while the latter is taken from Yun et al. (1994). The map shows the large-scale streamers expanding almost in parallel to the disk. In addition, there are at least three prominent extending structures toward the north, west, and southeast directions in spatial scales of $5'-10'$, which might be related to the three filamentary structures in the PAH emission. Although the overall morphology of the FIR image is somewhat different from the map, the FIR-bright regions at $12'$ to the west and $8'$ to the southeast as well as the extended structures connecting to them from the center show some spatial resemblance to the distribution. The difference in overall distribution between the dust and the gas might be explained by inhomogeneity of the intergalactic UV radiation heating the dust, which is likely attenuated more along the major axis than the minor axis. The large-scale neutral streamers are probably caused by a past tidal interaction of M 82 with M 81 (Yun et al. 1993). The intergalactic FIR dust can thus be attributed to leftover clouds ejected out of the galaxy by the tidal interaction and residing in the intergalactic medium.
In Fig.3, we superpose an X-ray contour map of M 82 from the XMM/Newton archival data on the MIR and FIR images. The conspicuous X-ray superwind extends along the northwest direction, correlating very well with the FIR emission. We obtain $R=+0.60$ ($N=2417$) between the low-level FIR ($N160$) and X-ray images for a bin size of $15''$, specifically the $N160$ brightness higher than 0.1 % (the lowest color level in Fig.3e), and the X-ray brightness lower than 0.1 % of the peak (an intermediate contour level in Fig.3e). The spatial correlation between X-ray and FIR suggests that the FIR dust is entrained by the superwind and outflowing from the galactic plane.
In contrast, there is no clear correlation between the X-ray superwind and the PAH/VSG emission. For example, we obtain $R=+0.21$ ($N=13568$) between the low-level $S7$ and X-ray images for a bin size of $4''$, the $S7$ brightness higher than 0.0007 % (the lowest color level in Fig.3a), and the X-ray brightness lower than 0.1 % of the peak (the same as the above). There might be even some anti-correlation between them especially in the $S7$ image; the X-ray plasma seems to be situated in between the two filamentary structures in the PAH emission. Moreover in the Cap region, where there is significant diffuse MIR emission, the PAH emission is somewhat reduced locally at the position of the X-ray Cap (see section 4.3). This might reflect that the PAHs in preexisting diffuse neutral clouds are destroyed by collision of the energetic superwind. The difference between the PAHs and the FIR dust indicates that the PAHs are more easily destroyed in hot plasma.
{width="50.00000%"}
{width="50.00000%"}
Discussion
==========
PAH properties
--------------
The striking similarity between the $S7$ and $S11$ band images in Fig.1 indicates that spatial variations in properties of PAHs are quite small; the PAH interband strength ratio of the $6-9$ $\mu$m to the $11-13$ $\mu$m emission is nearly constant throughout the observed regions of M 82. In Fig.7, we calculate the ratios of surface brightness at 11 $\mu$m to that at 7 $\mu$m. Prior to dividing the images, we have applied smoothing to both images by a gaussian kernel of $24''$ in FWHM. The area in the $S7$ image with brightness levels lower than 1.0 MJy/str that corresponds to the lowest contour in Fig.1 is masked in calculating the ratios. As a result, the ratio map in Fig.7 reveals that the values are nearly constant at $\sim$ 0.8, which is consistent with the spectroscopic results on M 82 by ISO [@For03b] and Spitzer [@Bei08].
The relative strength of the different PAH bands is expected to vary with the size and the ionization state of the PAHs [e.g. @Dra07]. The C-C stretching modes at 6.2 and 7.7 $\mu$m are predominantly emitted by PAH cations, while the C-H out-of-plane mode at 11.3 $\mu$m arises mainly from neutral PAHs [@All89; @Dra07]. Neutral PAHs emit significantly less in the 6–8 $\mu$m emission [@Job94; @Kan08a]. The size effect on the PAH 11.3 $\mu$m/7.7 $\mu$m ratio is relatively small [@Dra07], although, in general, smaller PAHs can emit features at shorter wavelengths. Hence, the variations of the $S11$/$S7$ ratio in the halo regions where there is no extinction effects show support for variation mostly in PAH ionization, i.e. a larger fraction of neutral PAHs for higher $S11$/$S7$ ratios, with small contributions from changes in PAH size distribution. As seen in Fig.7, regions with higher ratios have a reasonable tendency to be located in the outskirts of the PAH emission distribution, where the UV radiation is weaker. Apart from the overall tendency, there is a region showing systematically higher $S11$/$S7$ ratios toward the east direction from the center, where the UV radiation field is expected to be relatively weak in such a large scale. This relative weakness might reflect the spatial distribution of dense gas near the central region shielding UV light from the nucleus; the central 3 kpc CO map of M 82 in Walter et al. (2002) exhibits a distribution of molecular gas with larger viewing angles toward the east direction from the nucleus.
Beirão et al. (2008) observed an enhancement of the 11.3 $\mu$m PAH feature relative to the underlying continuum emission outward from the galactic plane, and suggested that the UV radiation field excites PAHs and VSGs differently. In the halo regions, the radiation is still intense enough to excite PAHs but no longer intense enough to excite VSGs to the same temperatures as in the galactic plane. Thus it is reasonable that the $S7$ and $S11$ images show more extended emission than the $L15$ and $L24$ images in Fig.1. Moreover the ratio map in Fig.7 suggests that the radiation field is a little weaker in the northeast filament than in the northwest filament, which may explain the difference in the PAH and VSG emission intensity between these filaments (Fig.1).
{width="50.00000%"}
Dust mass and luminosity
------------------------
We fit the SEDs in Fig.4 by a three-temperature dust grey-body plus PAH component model (Fig.8). The PAH parameters are taken from Draine & Li (2007) by adopting the PAH size distribution and fractional ionization typical of diffuse ISM and assuming the interstellar radiation field in the solar neighborhood. The latter assumption does not influence the spectral shape much unless the radiation field is as much as $10^4$ times higher [@Dra07], while Colbert et al. (1999) estimated $2.8$ times higher than the solar neighborhood value from photodissociation regions in the central $\sim 1$ kpc area of M 82.
For the dust grey-body model, we adopt an emissivity power-law index of $\beta=1$ for every component. We started with the initial conditions of 140 K, 60 K, and 28 K for the dust temperatures of the three components. These components are just representatives to reproduce the shape of the dust continuum spectra as precisely as possible and we below consider only the total luminosity and dust mass by summing up the corresponding values from the three components. We calculate dust mass by using the equation [e.g. @Hil83]: $$M_{\rm dust}=\frac{4a\rho D^2}{3}\frac{F_{\nu}}{Q_{\nu}B_{\nu}(T)},$$ where $M_{\rm dust}$, $D$, $a$, and $\rho$ are the dust mass, the galaxy distance, the average grain radius, and the specific dust mass density, respectively. $F_{\nu}$, $Q_{\nu}$, and $B_{\nu}(T)$ are the observed flux density, the grain emissivity, and the value of the Planck function at the frequency of $\nu$ and the dust temperature of $T$. We adopt the grain emissivity factor given by Hildebrand (1983), the average grain radius of 0.1 $\mu$m, and the specific dust mass density of 3 g cm$^{-3}$. We use the 90 $\mu$m flux density and the dust temperature, both given by each dust component. The dust mass thus derived as well as the total luminosity of the PAH and dust components, $L_{\rm PAH}$ and $L_{\rm dust}$, are listed in Table 3. We should note that the spectral fitting to the SED of the total ($d\leq 8$ kpc) region does not require additional component, but the differential SED in Fig.4b shows the presence of colder dust emission peaking at a wavelength longer than 160 $\mu$m. Hence the dust mass in the total region is considered to be lower limits to the real dust mass contained in this region.
The ISO/LWS spectroscopy of the central $\sim 1$ kpc region of M 82 showed that the SED over the wavelength range of 43-197 $\mu$m is well fitted with a 48 K dust temperature and $\beta=1$, giving a total infrared flux of $3.8\times 10^{10}$ $L_{\odot}$ [@Col99]. Our result shows overall consistency with the ISO result. Thuma et al. (2000) estimated from their 240 GHz measurement that the total dust mass in the inner 3 kpc of the galaxy is $7.5\times 10^6$ $M_{\odot}$, which is in an excellent agreement with our result. The total mass of atomic hydrogen gas in M 82 is estimated to be $8\times 10^8$ $M_{\odot}$ [@Yun94], while the mass of molecular gas in an area of $2.8\times 3.9$ kpc$^2$ is $1.3\times 10^9$ $M_{\odot}$ [@Wal02]. Hence the gas-to-dust mass ratio for the total 8 kpc region is $\sim$ 200, similar to the accepted value of $100-200$ for our Galaxy [@Sod97]. In the inner $2$ kpc, the molecular gas mass is estimated to be $1.0\times 10^9$ $M_{\odot}$ from Fig.1 and Table 1 in Walter et al. (2002), and thus the gas-to-dust mass ratio is $\sim$400, where contribution of atomic hydrogen gas is not included. Hence the gas-to-dust ratio is considerably higher in the central region than in the halo; similar results were also obtained for NGC 253 [@Rad01; @Kan09b]. The possible presence of colder dust in the halo would further increase the difference. In much larger spatial scales, Xilouris et al. (2006) obtained the -to-dust mass ratio of $\sim$20 from a systematic shift in the color of background galaxies viewed through the intergalactic medium of the M 81-M 82 group.
The variations of $L_{\rm PAH}$, $L_{\rm dust}$, and $M_{\rm dust}$ in Table 3 reveal a rapid decrease in the radiation field intensity but not in the dust mass away from the galactic plane. The ratio $L_{\rm PAH}$/$L_{\rm dust}$ is fairly constant ($\sim 0.1$) for all the regions except the central 2 kpc region where $L_{\rm PAH}$/$L_{\rm dust}$ is about 0.2; the PAH emission in the central region is relatively bright due to the nuclear starburts. The relatively small change in $M_{\rm dust}$ indicates that the total amount of the dust residing in the halo is so large that it can be comparable to that of the dust contained in the galaxy body. Hence the dust-enriched gas injected by the starburst significantly contributes to the total mass of wind material that is tranported out of the disk.
{width="50.00000%"} {width="50.00000%"}\
{width="50.00000%"} {width="50.00000%"}
Interplay of dust and PAHs with various gas components
------------------------------------------------------
There are at least three filamentary structures of the PAH emission (Fig.1). The tight correlation between the PAH and H$\alpha$ emission provides evidence that the PAHs are well mixed in the ionized superwind gas and outflowing from the disk. In contrast, the H$_{2}$ 2.12 $\mu$m emission [@Vei09] shows a relatively loose correlation with the PAH emission in the northern halo, while they are well correlated in the southern halo. Veilleux et al. (2009) suggested that UV radiation is important for the excitation of the warm H$_2$ in the southeast extension because this is a region where photoionization by OB stars dominates over shocks [@Sho98], whereas a dominant heating mechanism is not well determined among shock, UV pumping, and X-ray heating in the northern halo. Our result favors the scenario that shock or X-ray heating is more important for the warm H$_2$ in the northen halo rather than the UV radiation that heats the PAHs, which could explain the loose correlation between the H$_2$ and PAH emission in the northern halo.
The deprojected outflow velocity of the H$\alpha$ filaments is $525-655$ km s$^{-1}$ [@Sho98]. Therefore the PAHs seem to have survived in a harsh environment for about 5 Myr to reach the observed positions at $\sim 3$ kpc above the plane. The dominant excitation mechanism for the H$\alpha$ filaments is most likely photoionization by the nuclear starburst; UV radiation is escaping from the disk along a channel excavated by the hot superwind [@Sho98]. Shock ionization begins to contribute toward larger radii, and beyond 4 kpc above the disk in the northern halo, X-ray hot plasma is a dominant phase of the superwind. In the hot plasma phase, the PAH emission is significantly reduced, as can be seen in Fig.3, which might imply that the PAHs are destroyed in the hot plasma. This would happen in $\sim$6 Myr after being ejected with a hot plasma whose outflow velocity is $\sim$ 700 km s$^{-1}$ [@Leh99].
Consistently with the fact that the dust temperature is expected to decrease with attenuating radiation field away from the disk, the VSG emission cannot be seen in the halo beyond 4 kpc above the disk, either (Fig.3). Nevertheless, as seen in Fig.3e, the FIR dust emission is still observed in the hot plasma superwind beyond 4 kpc. As discussed in Tsuru et al. (2007), the lifetime of dust of a 0.1 $\mu$m size against sputtering destruction in the hot plasma superwinds of M 82 is 20$\times f^{0.5}$ Myr, where $f$ is the volume filling factor of the hot plasma that is expected to be as low as $\sim$ 0.01 [@Str00]. If we assume that the observed dust is homogeneously mixed in the hot plasma, the dust grains have to survive during a traveling time of $5-10$ Myr to reach their present locations, which is comparable to the sputtering destruction time scale. Indeed we have direct evidence that there is a close morphological correpondence between the PAHs/dust and the hotter phases of the galactic winds probed in the H$\alpha$ and X-ray emission. As seen in Fig.6, the extended FIR dust emission is also detected from part of the streamers, the surface of which is probably illuminated by UV light escaping from the disk along a channel excavated by the hot superwind. The streamers are thought to provide evidence that the gas within the optical disk of M 82 is disrupted by the interaction of M 82 with M 81 100 Myr ago, and likely triggers the starburst activitiy in the center of M 82 [@Wal02]. Furthermore, the PAH emission in the halo also exhibits significant enhancement in surface brightness in the eastern edge of the Cap region, which spatially corresponds to the edge of the clouds (Fig.3f). Hence the streamers should contain dust and PAHs, obviously not of a primordial origin, but rather as leftover clouds of the past interaction of M 82 with M 81. Detailed modeling of starburst activity in M 82 on the basis of NIR-MIR spectroscopy suggested the occurrence of starburst in two successive episodes, about 10 and 5 Myr ago, each lasting a few million years [@For03a]. From the Spitzer/IRS spectroscopy, Beirão et al. (2008) indicated that the star formation rate has decreased significantly in the last 5 Myr. The above dynamical time scales of the superwinds are consistent with these results. The inclusion of the dust and PAHs in the streamers implies that the streamers were already enriched by metals prior to the starburst episodes at the M 82 nucleus.
As for the Cap region, we detect significant signals at wavelengths of 7 and 11 $\mu$m, which are diffusely extended near the X-ray and H$\alpha$ Cap but appear to be reduced locally at the position of the Cap (Figs.3a and 3b). Lehnert et al. (1999) suggested that the Cap is the result of a collision between the hot superwind and a preexisting neutral cloud. We adopt a size of $3.7\times 0.9$ kpc$^2$ for the X-ray Cap according to Lehnert et al. (1999). By integrating signal within the $3.7\times 0.9$ kpc$^2$ aperture centered at R.A. (J2000) $=$ 9 55 17.0 and DEC (J2000) $=$ $+$69 51 13.0, the X-ray peak position of the Cap, we obtain the flux densities of 4.4 mJy and 9.1 mJy at 7 $\mu$m and 11 $\mu$m, respectively. From the same aperture but at the different position of R.A. (J2000) $=$ 9 54 48.0 and DEC (J2000) $=$ $+$69 48 26.0 that is located in the middle of the faint diffuse emission region, we obtain the flux densities of 8.8 mJy and 11 mJy at 7 $\mu$m and 11 $\mu$m, respectively. The 1-sigma statistical errors are estimated to be 0.3 mJy and 0.5 mJy at 7 $\mu$m and 11 $\mu$m, respectively, from the nearby darkest blank sky. Therefore the signal reduction at the Cap seems to have a statistical significance, which is higher at 7 $\mu$m possibly reflecting that PAHs of smaller sizes are easier to be destroyed there. By adopting the flux ratio between the $S7$ and the $N160$ band in the annular region of radii $2'$ to $4'$ from the galactic center, we estimate the $N160$ surface brightness to be $\sim$2.6 MJy sr$^{-1}$ at the position of the Cap, which is only 1.8 times higher than the 1-sigma background fluctuation level in the $N160$ band, and thus consistent with non-detection of FIR signals from the corresponding area.
An integration of the surface brightness of 2.6 MJy leads to the flux density of 0.7 Jy in the $N160$ band for the $3.7\times 0.9$ kpc$^2$ area of the X-ray Cap. By assuming the same FIR dust SED as observed in the 4 kpc halo region (Fig.8d), dust mass in the Cap is estimated to be $6\times 10^4$ $M_{\odot}$. From the $\sim$50 % signal reduction of the PAH emission at the Cap, a comparable amount of dust might have been already lost there by sputtering destruction. Thus the dust sputtering provides a potential impact on the metal abundances measured for the X-ray plasma in the Cap as pointed out by Tsuru et al. (2007); the masses of Si and Fe in the hot plasma phase are $1.4\times 10^3\times f^{0.5}$ $M_{\odot}$ and $1.8\times 10^3\times f^{0.5}$ $M_{\odot}$, respectively [@Tsu07]. The charge-exchange process could be important in X-ray emission from the Cap [@Lal04], where the ionized superwind from M 82 can be assumed to collide with cool ambient gas located at the Cap. From the destruction of the PAHs at the Cap, we expect that the hot plasma is somewhat enriched with carbon there, which might be related to the marginal detection of the emission line at 0.459 keV due to the charge-exchange process by Tsuru et al. (2007).
We find that observable amounts of dust and PAHs are included in the phases of both ionized (superwinds) and neutral (streamers) gas, which are spatially separated from each other in the northern halo (Fig.3f). A significant fraction of PAHs seem to have been destroyed in the hot plasma phase of the northern superwind beyond 4 kpc from the disk, but still partly remaining in the X-ray Cap. Moreover PAHs seem to be present widely around the Cap region far beyond the disk, which may have been strewn into the intergalactic space by a past tidal interaction with M 81 before the starburst began at the nucleus of M 82. Hoopes et al. (2005) concluded that the most likely mechanism for the UV emission in the halo of M 82 is scattering of stellar continuum from the starburst by dust in the halo because the brightness of the UV wind is too high to be explained by photoionized or shock-heated gas. Our result supports their conclusion as far as the galactic superwind regions are concerned. But for the Cap at 11 kpc north of M 82, where UV light is also seen in spatial scales similar to the X-ray [@Hoo05], light scattered by dust may not be a substantial component of the UV emission. The diffuse distribution of the intergalactic material represented by PAHs would scatter UV light from the starburst in much wider area rather than the observed region limited to the X-ray Cap.
--------------------- ---------------------- ----------------------- ----------------------
Region $L_{\rm PAH}$ $L_{\rm dust}$ $M_{\rm dust}$
$10^{9}$ $L_{\odot}$ $10^{10}$ $L_{\odot}$ $10^{6}$ $M_{\odot}$
Center ($d\leq 2'$) 6.0 3.1 2.3
Center ($d\leq 4'$) 6.4 5.6 7.6
Total ($d\leq 8'$) 6.9 6.1 10.3
Halo ($d\leq 4'$) 0.0041 0.0057 3.9
--------------------- ---------------------- ----------------------- ----------------------
: Luminosities and dust masses derived from the spectral fitting to the observed SEDs[]{data-label="log"}
Summary
=======
We have presented new MIR and FIR images of M 82 obtained by AKARI, which reveal both faint extended emission in the halo and very bright emission in the center with signal dynamic ranges as large as five and three orders of magnitude for the MIR and FIR, respectively. Our observations cover wider areas than previous IR observations up to the X-ray/H$\alpha$ Cap at 11 kpc above the disk, which complements previous studies. We detect MIR and FIR emission in the regions far away from the disk of the galaxy, reflecting the presence of dust and PAHs in the halo of M 82. We show that the ionization state of the PAHs is fairly constant throughout the halo of M 82 with small variations in some areas probably due to reduction in the UV radiation escaping from the disk. We find that the dust and PAHs are contained in both ionized and neutral gas components, implying that they have been expelled into the halo of M 82 by both starbursts and galaxy interaction. In particular, we obtain an tight correlation between the PAH and the H$\alpha$ filamentary structures, which provides evidence that the PAHs are well mixed in the ionized superwind gas and outflowing from the disk in a short timescale. We also find that the dust is contained even in the X-ray hot plasma while PAHs are widely spread over the Cap region. Both suggest that the gas in the halo of M 82 is highly enriched with dust, connecting to the results of Xilouris et al. (2006) that dust exists in the intergalactic medium on much larger scales.
We thank all the members of the AKARI projects, particularly those belonging to the working group for the ISMGN mission program. We would also express many thanks to the anonymous referee for giving us useful comments. AKARI is a JAXA project with the participation of ESA. This research is supported by the Grants-in-Aid for the scientific research No. 19740114 and the Nagoya University Global COE Program, “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos”, both from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
Allamandola, L. J., Tielens, A. G. G. M., & Barker, J. R. 1989, , 71, 733 Alonso-Herrero, A., Rieke, G. H., Rieke, M. J., & Kelly, D. M. 2003, , 125, 1210 Alton, P. B., Davies, J. I., & Bianchi, S. 1999, , 343, 51 Beirão, P., Brandl, B. R., Appleton, P. N., et al. 2008, , 676, 304 Bendo, G. J., Draine, B. T., Engelbracht, C. W., et al. 2008, MNRAS, 389, 629 Bland, J. & Tully, B. 1988, , 334, 43B Boulanger, F., Beichman, C., Désert, F. X., Helou, G., Pérault, M., & Ryter, C. 1988, , 332, 328 Bregman, J. N., Schulman, E., & Tomisaka, K. 1995, , 439, 155 Burton, M. G., Hollenbach, D. J., Haas, M. R. & Erickson, E. F. 1990, , 355, 197 Désert, F. X. Boulanger, F., & Puget J. L. 1990, , 237, 215 Colbert, J. W., Malkan, M. A., Clegg, P. E., et al. 1999, , 511, 721 Davies, J. I., Alton, P., Bianchi, S., & Trewhella, M. 1998, MNRAS, 300, 1006 Devine, D. & Bally, J. 1999, , 510, 197 Draine, B. T. & Li, A. 2007, , 657, 810 Eales, S. & Edmunds, M. 1996, MNRAS, 280, 1167 Engelbracht, C. W., Kundurthy, P., Gordon, K. D., et al. 2006, , 642, L12 Förster Schreiber, N. M., Genzel, R., Lutz, D., & Sternberg, A. 2003a, , 599, 193 Förster Schreiber, N. M., Sauvage, M., Charmandaris, V., Laurent, O., Gallais, P., Mirabel, I. F., & Vigroux, L. 2003b, , 399, 833 García-Burillo, S., Martín-Pintado, J., & Fuente, A. 2001, , 563, L27 Heckman, T. M., Lehnert, M. D., Strickland, D. K., & Armus, L. 2000, , 129, 493 Heisler, J. & Ostriker, J. P. 1988, , 332, 543 Hildebrand, R. H. 1983, QJRAS, 24, 267 Hoopes, C. G., Heckman, T. M., Strickland, D. K., et al. 2005, , 619, L99 Joblin, C., D’Hendecourt, L., Leger, A., & Defourneau, D. 1994, , 281, 923 Kaneda, H., Kim, W.-J., Onaka, T., Wada, T., Ita, Y., Sakon, I., & Takagi, T. 2007, , 59, S423 Kaneda, H., Onaka, T., Sakon, I., Kitayama, T., Okada, Y., & Suzuki, T. 2008a, 684, 270 Kaneda, H., Suzuki, T., Onaka, T., Okada, Y., & Sakon, I. 2008b, , 60, S467 Kaneda, H., Koo, B.-C., Onaka, T., & Takahashi, H. 2009a, Adv.Sp.Res., 44, 1038 Kaneda, H., Yamagishi, M., Suzuki, T., & Onaka, T. 2009b, , 698, L125 Karachentsev, I. D., Dolphin, A. E., Geisler, D., et al. 2002, , 383, 125 Kawada, M., Baba, H., Barthel, P. D., et al. 2007, , 59, 389 Lallement, R. 2004, , 422, 391 Leeuw, L. L. & Robson, E. I. 2009, , 137, 517 Lehnert, M. D., Heckman, T. M., & Weaver, K. A. 1999, , 523, 575 Lorente, R., Onaka, T., Ita, Y., Ohyama, Y., & Pearson, C. P. 2007, AKARI IRC Data User Manual Version 1.2 Murakami, H., Baba, H., Barthel, P., et al. 2007, , 59, 369 Ohyama, Y., Taniguchi, Y., Iye, M., et al. 2002, , 54, 891 Onaka, T., Matsuhara, H., Wada, T., et al. 2007, , 59, 4010 Radovich, M., Kahanpää, J., & Lemke, D. 2001, A&A, 377, 73 Shen, J. & Lo, K. Y. 1995, , 445, L99 Shirahata, M., Matsuura, S., Hasegawa, S., et al. 2009, , 61, 737 Shopbell, P. L. & Bland-Hawthorn, J. 1998, 493,129 Smith, J. D., Draine, B. T., Dale, D. A., et al. 2007, , 656, 770 Sodroski, T. J., Bennett, C., Boggess, N., et al. 1994, , 428, 638 Sodroski, T. J., Odegard, N., Arendt, R. G., Dwek, E., Weiland, J. L., Hauser, M. G., & Kelsall, T. 1997, , 480, 173 Strickland, D. K., Ponman, T. J., & Stevens, I. R. 1997, , 320, 378 Strickland, D. K. & Stevens, I. R. 2000, MNRAS, 314, 511 Strickland, D. K., Heckman, T. M., Colbert, E. J. M., Hoopes, C. G., & Weaver, K. A. 2004, , 151, 193 Sturm, E., Lutz, D., Tran, D., et al. 2000, , 358, 481 Telesco, C. M. & Harper, D. A. 1980, 235, 392 Thuma, G., Neininger, N., Klein, U., & Wielebinski, R. 2000, , 358, 65 Tsuru, T. G., Ozawa, M., Hyodo, Y. et al. 2007, , 59, S269 Veilleux, S., Rupke, D. S. N., & Swaters, R. 2009, , 700, L149 Verdugo, E., Yamamura, I., & Pearson, C. P. 2007, AKARI FIS Data User Manual Version 1.2 Walter, F., Weiss, A., & Scoville, N. 2002, , 580, L21 Watson, M. G., Stanger, V., & Griffiths, R. E. 1984, , 286, 144 Werner, M. W., Uchida, K. I., Sellgren, K., et al. 2004, , 154, 309 Xilouris, E., Alton, P., Alikakos, J., Xilouris, K., Boumis, P., & Goudis, C. 2006, , 651, L107 Yun, M. S., Ho, P. T. P., & Lo, K. Y. 1994, , 372, 530 Yun, M. S., Ho, P. T. P., & Lo, K. Y. 1993, , 411, L17
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.